Sui procedimenti iterativi composti

In this paper some results of composite iterations are enlarged upon. In particular we discuss the efficiency of the resulting iteration and we show how in some cases the efficiency index may be greater than that of each one of the constituent iterations. In the case of multipoints iterations without memory, in sufficiently wide conditions, we show that the order can not be more than 2^m−1 according to the conjecture of Kung and Traub, and we give a necessary and sufficient condition so that such a limit can be reached. Ricevuto 29/4/75.      

Convergence properties of discrete analogs of orthogonal polynomials

Summary If (,) is an inner product on [a, b], and if [,] _N is a discrete inner product analogous to (,), and such that [1, 1] _N =(1, 1), then, a sufficient condition that the discrete orthogonal polynomials converge to the corresponding continuous orthogonal polynomials likeN ^–p , is that [1,t ^k ] _N =(1,t ^k )+O(N ^–p ),k=1, 2, ... A similar result holds for correspondingFourier segments.

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Konvergenzeigenschaften von Polynomen, die orthogonal sind in bezug auf ein diskretes Skalarprodukt

Zusammenfassung Falls (,) ein Skalarprodukt auf [a, b] darstellt und [,] _N das entsprechende diskrete Skalarprodukt bedeutet, wobei [1, 1] _N =(1, 1) gilt, dann ist die Bedingung [1,t ^k ] _N =(1,t ^k )+0(N ^–p ),k=1, 2, ..., hinreichend dafür, daß die diskreten orthogonalen Polynome zu den entsprechenden kontinuierlichen Polynomen wieN ^–p konvergieren. Ein ähnliches Ergebnis gilt für entsprechendeFouriersegmente.

     

Multipoint iterative parallel methods for solving equations

In this paper we show the results of some research carried out on parallel iterative methods to solve equations. In particular we study general classes of one point parallel methods and multipoint ones without memory, and we point out the convergence order of these methods and the conditions which are both necessary and sufficient for them to be optimal. In addition we prove that the convergence order for multipoint parallel procedures without memory cannot be more thenr(r+) ^m−1 , wherer indicates the number of the parallel processor used andm the number of the functions and eventual derivatives, calculated not simultaneously in every iteration.

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Sommario In questo articolo presentiamo alcuni risultati concernenti i metodi iterativi paralleli per risolvere equazioni. In particolare analizziamo alcune classi generali di procedimenti ad un punto ed a più punti senza memoria, il loro ordine di convergenza e le condizioni necessarie e sufficienti per ottenere l'ottimalità. Inoltre dimostriamo che l'ordine di convergenza di un procedimento iterativo senza memoria non può eccedere:r(r+1) ^m−1 , dover indica il numero di processor in parallelo usati edm indica il numero di funzioni ed eventuali derivate calcolate non simultaneamente in ogni iterazione.

     

Qualche contributo alla risoluzione delle disequazioni variazionali inR n

Riassunto Viene indicato un metodo di risoluzione di disequazioni variazionali in un convesso diR ⁿ definito da funzioni convesse non necessariamente affini: il metodo è la naturale estensione di quello indicato in [1] per il caso di convessi diR ⁿ definiti da funzioni affini.

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Here we show a method of computing solutions of variational inequalities in a convex subset ofR ⁿ defined by convex, not necessarily affine, functions: this method is the natural extension of that related in [1] for convex subset ofR ⁿ defined by affine functions.

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Lavoro estratto dalla Tesi di Laurea [5] svolta presso l'Istituto di Elaborazione della Informazione, Pisa, (Rel. A. Laratta).     

Sequentielle Analyse kontextfreier Sprachen

Summary Sequential analysis of context-free languages. Since 1965 several methods to solve the word problem for context-free languages in no more than cn ³steps have been published where c is a constant and n the length of the word [2, 5, 7]. The procedure to be described is a sequential procedure that means that the word which is to be analyzed has to be read monotonic from left to right and that the word problem is decided for each partial word. The method consists in constructing a growing automaton whose wiring (in a physical way of speaking) grows only quadratically. From a monotonicity of property of signal propagation in our automata we can conclude that the procedure can be implemented on a RAM (random access machine) in such a way that the word problem is decided in at most cn ³steps.     

Sul calcolo approssimato di particolari funzioni ipergeometriche confluenti con la tecnica del «τ-method»

For linear homogeneous differential equations of orderm, with polynomial coefficients, the τ-method is described and a relative error valnation is obtained. The method is than used for the numerical approximation of particular confluent hypergeometric-functions.

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Lavoro eseguito nell’ambito del Gruppo di Ricerca n⁰ 18 del C. N. R. (1969).     

Proportions in the Architecture Curriculum

Roger Herz-Fischler presents a revised version of a chapter entitled “Proportions” that appeared in the problems part of his book, Space, Shape and Form /An Algorithmic Approach, developed for a mathematics course he taught in the School of Architecture at Carleton University from 1973-1984.

Example of reckoning of a pyramid, Height 250, base 360 cubits. What is its seked?......(Answer:5+1/25 hands). From the Rhind Papyrus¹

¹The Rhind Papyrus is from the Fifteenth Dynasty (c. - 1800), but is known to be based on a Twelfth Dynasty text. For details of the pyramid problems, as well as a discussion of the archaeological, historical and philosophical aspects, related to the problem of determining how the Great Pyramid was designed, see [Herz-Fischler 2000].     

Un metodo alle direzioni alternate esplicito per la risoluzione numerica delle equazioni della diffusione

Sommario Si presenta un metodo esplicito alle direzioni alternate per la risoluzione numerica delle equazioni a multigruppi della diffusione neutronica, dipendenti dal tempo, in domini bidimensionali. Questo metodo ha le proprietà di avere un errore locale dell'ordineO (h ³) (doveh è il passo temporale) edi essere incondizionatamente stabile. L'efficacia di questo metodo è stata messa in evidenza attraverso alcune esperienze pratiche.

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An explicit alternating direction method is presented for the numerical solution of the time dependent multigroup neutron diffusion equations in bidimensional domains. This method has the properties to have a truncation error that behaves likeO (h ³) (whereh is the time step size) and to be unconditionally stable. The effectiveness of this method has been verified in many pratical problems.

     

Studio del resto di formule di quadratura mediante l'esame del segno della funzione d'influenza

Sommario Si sa che le proprietà del resto nelle formule di quadatura dipendono essenzialmente dalla struttura della funzione d'influenza. Pertanto, in questo lavoro, si è introdotta una nuova tecnica che permette di riconoscere casi, parzialmente noti, in cui la funzione d'influenza è a segno costante e, di conseguenza, il resto assume la formaE=k.f ^(N+1) (ξ).

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Summary We know that the properties of the remainder in the quadrature formulae depend essentially from the structure of the influence's function. Therefore, in this work we have introduced a new technique that let us study partially well-known cases, in which the influence's function is of constant sign and, consequently, the remainder assumes the form:E=k.f ^(N+1) (ξ).

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Lavoro svolto nell'ambito delle attività del G.N.A.F.A. e dell'Istituto per le Applicazioni del Calcolo.     

Verallgemeinerung des Begriffes einer gewöhnlichen Differentialgleichung n-ter Ordnung im Komplexen sowie Existenz- und Eindeutigkeitsaussagen

Zusammenfassung In der vorliegenden Arbeit wird der Begriff einer gew?hnlichen Differentialgleichung n-ter Ordnung mit komplexen Variablen verallgemeinert. Weiter werden Fragen über Existenz und Eindeutigkeit der L?sungen einer derart verallgemeinerten Differentialgleichung untersucht.

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Summary The present paper contains a generalization of the concept of an ordinary differential nquation of n^th order with complex variables and deals with existence and uniqueness problems of such gereralized differential equations.

     

Schnelle Berechnung von Kettenbruchentwicklungen

Zusammenfassung Ein von D. E. Knuth angegebenes Verfahren, für ganze Zahlen u, v den größten gemeinsamen Teiler und den Kettenbruch für u/v zu berechnen, wird so modifiziert, daß für n-stellige Zahlen nur O(n(lg n)² (lglg n)) elementare Schritte gebraucht werden.

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Summary A method, given by D. E. Knuth for the computation of the greatest common divisor of two integers u, v and of the continued fraction for u/v is modified in such a way that only O(n(lg n)²(lglg n)) elementary steps are used for u,v<.2 ⁿ.

     

Condizioni necessarie e sufficienti per un problema di programmazione non lineare

Sommario In questa nota si considera un problema di programmazione non lineare, dove la funzione obiettivo è differenziabile e non lineare, ed i vincoli sono generati da funzioni differenziabili e quasi convesse. Sotto alcune condizioni di regolarità, si dimostra che se il gradiente della funzione obiettivo (f. o.) in un puntox ⁰ è diverso da zero, allora le condizioni di Kuhn-Tucker, verificate in tale punto sono sufficienti affinchéx ⁰ sia un minimo della f. o. Si dimostra poi che sex ⁰ è un minimo locale per la f. o. e soddisfa i vincoli, sono necessariamente verificate le condizioni di Kuhn-Tucker.

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In this paper a problem of non linear programming is considered. It is assumed the differentiability of object function (o. f.), and the differentiability and quasi-convexity of constraints. Under some regularity conditions, it is proved that the Kuhn-Tucker conditions are sufficient iff(x ⁰ )=0, thatx ⁰ should be the point of local minimum of the objective function. If a pointx ⁰ is a local minimum of the o. f., then the Kuhn-Tucker conditions are satisfied in the pointx ⁰.

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Lavoro svolto nell'ambito del gruppo G.N.I.M. del C.N.R.     

Mehrdimensionale Zuordnungsprobleme und Netzwerkflüsse mit vorgegebenen Wertemengen

Zusammenfassung Es wird eine Klasse von mehrdimensionalen Zuordnungsproblemen beschrieben, die in der Praxis häufig auftreten und wesentlich allgemeiner sind als das klassische Zuordnungsproblem. Die Probleme lassen sich auf die Bestimmung zulässiger Netzwerkflüsse mit vorgebenenen Wertemengen zurückführen. Der beschriebene Lösungsalgorithmus beruht auf einem Dekompositionsverfahren.

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Optimal assignment problems and network flows with given sets of values

Summary A class of optimal assignment problems is defined. These problems often occur in practice and are essentially more general than the classical assignment problem. The problems may be reduced to the task of finding feasible network flows with given sets of values. The described algorithm is based on a decomposition procedure.

     

Il metodo di Treffiz-Fichera per i semigruppi. potenziale di tipo potenza

A variation of the Trefftz-Fichera method is presented to compute lower bounds for the eigenvalues of a positive self-adjoint operator with discrete spectrum with grow at least in a logarithmic way as the index diverges. As suggested by Barnes et al. [2] to compute ground state, the semigroupe ^−βH, β>0, is used rather than the iterated resolvent(H+β) ⁻ⁿ,n=1,2,... As an example, the method is applied to the operatorH=−Δ+|x|^γ. inL ²(R), 1≤γ≤4.      

Sul calcolo delle radici complesse di particolari equazioni algebriche con l'ausilio dei polinomi di Tchebyshev

In this paper a process is suggested for the numerical approximation of the complex roots equations.

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Lavoro eseguito nell'ambito del gruppo di ricerca n^o 22 del C. N. R. (1963–64).     

Qualche contributo alla teoria delle formule di Runge-Kutta

Riassunto Utilizzando i coefficienti delle formule elementari di quadratura si ritrovauo in modo sistematico tutte le classiche formule di Runge-Kutta di ordine ≤4 e, sfruttando una opportuna transformazione suggerita da Fehlberg, si determinano anche formule di ordine superiore numericamente semplici.

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From the elementary quadrature rules the classical Rnuge-Kutta formulae of order ≤4 are derived; simple formulae of higher order are also obtained by means of a transformation suggested by Fehlberg.

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Lavoro eseguito nell'ambito del gruppo di ricerca n⁰ 22 del C. N. R. (1963–64).     

<Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript>-interpolation on quadrilateral and hexahedral meshes with hanging nodes

We propose a Scott-Zhang type finite element interpolation operator of first order for the approximation of H ¹-functions by means of continuous piecewise mapped bilinear or trilinear polynomials. The novelty of the proposed interpolation operator is that it is defined for general non-affine equivalent quadrilateral and hexahedral elements and so-called 1-irregular meshes with hanging nodes. We prove optimal local approximation properties of this interpolation operator for functions in H ¹. As necessary ingredients we provide a definition of a hanging node and a rigorous analysis of the issue of constrained approximation which cover both the two- and three-dimensional case in a unified fashion.      

Die Zeitkomplexität des Normalisierungsproblems bei kontextsensitiven Grammatiken

Zusammenfassung Sei G eine kontextsensitive Grammatik. G^c bezeichne den kontextfreien Kern von G. In dieser Arbeit wird die Zeitkomplexität des folgenden Problems untersucht. Das NormalisierungsproblemSei ein Ableitungsbaum bezüglich G^c; ist auch ein Ableitungsbaum bezüglich G? Es wird gezeigt, daß im allgemeinen das Normalisierungs-problem NP-vollständig ist. Andererseits gibt es zu jeder kontextsensitiven Sprache L eine kontextsensitive Grammatik G, für welche das Normalisierungsproblem in Polynomzeit lösbar ist.

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The time complexity of the normalization problem of contextsensitive grammars

Summary Let G be a Contextsensitive grammar. G ^cf represents the context free core of G. In this paper the time complexity of the following problem will be discussed. The Normalization ProblemLet be a derivation tree with respect to G^c; is also a derivation tree with respect to G? It is shown that in general the normalization problem is NP-complete. On the other hand, for every context sensitive language L there is a corresponding context sensitive grammar G for which this normalization problem is solvable in polynomial time.

     

Study on the cooling effects of urban parks on surrounding environments using Landsat TM data: a case study in Guangzhou,southern China

The temperature cooling effects of ten urban parks on surrounding environments in Guangzhou, southern China, are analysed and quantified using Landsat Thematic Mapper data. The results show that there is a temperature rise (about 1.74°C) between green spaces of parks and bare-ground areas of the surroundings. For those parks whose green area percentage is more than 69% and length:width ratio is close to 1, the average temperature differences between boundaries and surrounding sites of parks have linear relationships with the green areas of parks (R ² > 0.82). Moreover, the nonlinear relationship between the average cooling distance of parks and green areas can be simulated very well using a logarithmic curve (R ² > 0.93). When the green areas of parks are smaller than 10 566 m², parks will have no temperature cooling effects on their surrounding environments. When the green areas of parks reach 740 000 m², the increase of temperature cooling distance is less than 1 m per 10 000 m² increase of the green area. The most appropriate size of green areas of urban parks should fall between 10 566 and 740 000 m². For those parks with water areas larger than 128 889 m², the temperature cooling effects are usually more remarkable. When the length:width ratios of the green areas of urban parks are more than or equal to 2, their temperature cooling distances are always larger than those with length:width ratios equal to 1 given similar green area. Parks with larger green areas (37 163 m²) or larger water areas (>128 889 m²) will have more significant temperature cooling effects in June than in October.     

Sulla convergenza del metodo delle approssimazioni successive inR 1

In this paper, by means of the concept of chaos, we characterise the class of iterating functions inR ¹ for which the method of successive approximation converges from any starting point with the exception of a contable set.

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Lavoro svolto nell'ambito del GNIM-CNR.