Some algorithms solving the unilateral dirichlet problem with two constraints

We consider the Dirichlet problem with two obstacles in the theory of variational inequalities and a finite-dimensional discretization related with a matrix belonging to the class (P)∩(Z). We construct some algorithms which produce monotone sequences. These sequences converge to the solution of the discrete problem starting from lower or upper obstacle.

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Sommario Consideriamo il problema di Dirichlet dei due ostacoli, ben conosciuto nella teoria delle disequazioni variazionali ed una sua discretizzazione relative ad una matrice della classe (P)∩(Z). In questo conteto costruiamo alcuni algoritmi atti a produrre successioni approssimanti, monotone non decrescenti e non crescenti, convergenti alla soluzione del problema discreto a partire rispettivamente dall’ostacolo inferiore e superiore.

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This paper has been partially supported by GNAFA-CNR.     

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].     

Sulla convergenza delle formule di quadratura del tipo Gauss-Lobatto per il calcolo di integrali singolari V.P.

Sommario L’autore considera formule di quadratura del tipo Gauss-Lobatto per il calcolo di integrali a valor principale secondo Cauchy rispetto ad un peso generalizzato regolare di Jacobi. Vengono dimostrati teoremi di convergenza e stabilite alcune stime dell’errore. Si generalizzano e migliorano, inoltre, precedenti risultati.

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The author considers Gauss-Lobatto rules for evaluating of Cauchy singular integrals with respect to a generalized smooth Jacobi weight. Convergence theorems are given, and some estimates of the remainder are established. The previous results are generalized and improved.

     

Über die Nullstellen der ersten Ableitung von Besselfunktionen

Zusammenfassung Die Arbeit liefert einen Beitrag zur derLense'schen Vermutung über die Bewegung der Doppelnullstellen der AbleitungI′ _ν (z) der Besselfunktionen bei ver?nderlichem negativen ν.

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Summary This paper gives a contribution to the hypothesis ofLense regarding the movements of the second order zeros of the derivativesI′ _ν (z) of Bessel functions for every negativ variable ν.

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Mit 3 Textabbildungen

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HerrnJ. Lense zum 75. Geburtstag gewidmet.     

Is Grover’s Algorithm a Quantum Hidden Subgroup Algorithm?

The arguments given in this paper suggest that Grover’s and Shor’s algorithms are more closely related than one might at first expect. Specifically, we show that Grover’s algorithm can be viewed as a quantum algorithm which solves a non-abelian hidden subgroup problem (HSP). But we then go on to show that the standard non-abelian quantum hidden subgroup (QHS) algorithm can not find a solution to this particular HSP. This leaves open the question as to whether or not there is some modification of the standard non-abelian QHS algorithm which is equivalent to Grover’s algorithm.      

Palladio’s Villa Emo: The Golden Proportion Hypothesis Rebutted

In a most thoughtful and persuasive paper Rachel Fletcher comes close to convincing that Palladio may well have made use of the ‘golden section’, or extreme and mean ratio, in the design of the Villa Emo at Fanzolo. What is surprising is that a visually gratifying result is so very wrong when tested by the numbers. Lionel March provides an arithmetic analysis of the dimensions provided by Palladio in the Quattro libri to reach new conclusions about Palladio’s design process.

Not all that tempts your wand’ring eyes And heedless hearts, is lawful prize; Nor all that glisters, gold (Thomas Gray, Ode on the Death of a Favourite Cat)

     

Un metodo per l’ordinamento dei sottoprogrammi e l’ottimizzazione della assegnazione della memoria in un programma assemblatore

Sommario Si descrive il metodo nsato in un programma assemblatore per l’ordinamento dei sottoprogrammi componenti un lavoro e per l’ottimizzazione dell’assegnazione della memoria centrale ai sottoprogrammi medesimi.

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Summary A method is described which allows the ordering of subroutines in a job and the optimization of the memory allocation to the subroutines. The method was applied in writing the assembler for the computer CEP.

     

The Education of the Classical Architect from Plato to Vitruvius

Plato divided science (episteme) into ‘science of action’ (praktike) and ‘science of mere knowing’ (gnostike). His argument is the first known attempt to distinguish what is now recognised as technology, as distinct from more purely rational science. Aristotle coined the compound term technologia and thereby established this new department of science within the general system of knowledge. Plato did not develop his novel characterisation of the architect any further, for the ancient Greeks did not consider architecture a fine or estimable art. The best available source of Greek architectural pedagogy is the Roman Vitruvius. Graham Pont discusses Vitruvius’s distinction between the ‘practical’ side of architecture (fabrica) and the ‘theoretical’ (ratiocinatio), and examines the mathematical preparation of ancient Greek and Roman architects

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Diramazioni di soluzioni di equazioni differenziali ordinarie non lineari

Sommario In questa nota vengono studiate proprietà delle soluzioni non banaliw(x) (0≤x≤1) del problema non linearew″+λw=g(w) w′ ²,w(0)=w(1)=0, dove λ è reale eg(w) è intera; sono studiati in dettaglio alcuni casi corrispondenti a scelte particolari dig.

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Properties are obtained of the non trrivial solutionsw (x) (0≤x≤1) of the non linear boundary problemw″+λw=g(w)w′ ²,w(0)=w(1)=0, where λ is real andg(w) is entire. Some special cases (corresponding to special choices ofg) are studied in detail.

     

Un approssimante delle,funzioni di Bessel

Sommario Facendo ricorso al metodo τ, si determina un nuovo approssimante polinomiale diJ _λ(x), valido perx≥0 e λ reale, e di esso si dà una stima dell'errore. Si individua, poi, sperimentalmente una regioe del piano (x,λ) in cui i risultati numerici sono esatti fino alla quinta cifra significativa.

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A new polynomial approximation ofJ _λ(x) is found, forx≥0 and real λ, using the τ-method. The error is evaluated.,A region of the (x,λ)-plane is empirically determined, where the numerical results are correct to the fifth significant digit.

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Lavoro eseguito nell'ambito dell'attività dei Gruppi di Ricerca Matematica del C.N.R.     

Die Berechnung einiger elementarer transzendenter Funktionen mit Hilfe des Richardson-Algorithmus

Zusammenfassung In dieser Arbeit wird gezeigt, wie man mit Hilfe desRichardson-Algorithmus Funktionswerte der Inversen einiger elementarer transzendenter Funktionen in einfacher Weise berechnen kann. Hierzu wurden auf einer Digitalrechenanlage auch einige Vergleichsrechnungen durchgeführt.

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Summary In this paper we present a method for the computation of some inverse elementary transcendental functions usingRichardsons “Deferred Approach to the Limit”. A comparison of this method with other methods is also given.

     

Analysis numerica di modelli di simulazione per le acque basse

In this paper we study a bidimensional model which describes the rising of the sea water caused by wind and atmospheric pressure. In the first paragraph we present the physical conditions which justify the construction of the model based on the shallow water theory. In the second paragraph it is demonstrated why the problem related to the mathematical model, using natural boundary conditions, is well posed. Finally in the third paragraph we show some finite difference schemes for the numerical solution figuring out some stability aspects.

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Questo lavoro è stato svolto nell’ambito di una collaborazione esistente tra l’Istituto per le Applicazioni del Calcolo del C.N.R. e l’Istituto di Matematica del Politecnico diMilano, in un programma di ricerca la cui direzione scientifica è tenuta dal prof. G. Prouse.     

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).     

Su di un problema di A. M. Hobbs

Sommario Si dimostra in questa nota un Teorema di esistenza relativo alle classiF _n,12 (n≥12) di grafi planari con 12 vertici di valenza 5 edn−12 vertici di valenza 6, risolvendo così un problema posto da A. M. Hobbs.

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In this article we prove, in opposition to a conjecture stated by R. H. Fox, the existence of a class of maximal planar graphs with exactly 12 vertices of valence 5 and all other vertices of valence 6.

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Lavoro svolto nell'ambito del GNAFA, CNR.     

Sugli zeri delle funzioni di bessel

In this paper we give some inequalities for the zerosc _ν,n of the cilinder functionJ _ν (x)cosα−Y _ν (x)sindα, 0≤α<π, for the zerosc′_ν,n ofC′_ν(x) and we make an application to negative zerosa _n ,b _n of Airy functions Ai(x), Bi(x) of first and second kind respectively. We also make a comparison between inequalities related to the first zeroj _ν,1 of the Bessel functionJ _ν (x) of first kind with ν>0 and other known inequalities. Some results are new, other not, but in this case they are derived in a simpler way than usual ones.

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Lavoro eseguito nell'ambito del Gruppo Nazionale per l'Informatica Matematica del C.N.R.     

An a posteriori error estimate and a comparison theorem for the nonconforming <Emphasis Type="Italic">P</Emphasis><Subscript>1</Subscript> element

A posteriori error estimates for the nonconforming P ₁ element are easily determined by the hypercircle method via Marini’s observation on the relation to the mixed method of Raviart–Thomas. Another tool is Ainsworth’s application of the hypercircle method to mixed methods. The relation on the finite element solutions is also extended to an a priori relation of the errors, and the errors of four different finite element methods can be compared.      

The structure of the overhanging relations associated with some types of closure systems

Over many different kinds of cryptomorphisms equivalent with closure systems (and so with closure operators), we focus here on implication relations and the related overhanging relations, as introduced and axiomatized in a previous paper (Domenach and Leclerc, Math Soc Sci 47(3):349–366, 2004). In relation with data analysis motivations, we particularize the axioms on overhanging relations in order to account for some types of closure systems, such as nested or distributive ones. We also examine the lattice structure of overhanging relations, which is isomorphic to the lattice of closure systems, and derived structures for particular sets of overhangings.      

Autosoluzioni positive di un’equazione non lineare di diffusione

Sommario Vengono studiati lo spettro positivo e le proprietà degli autovettori positivi dell’analogo discreto di una equazione non lineare di diffusione. Si analizza poi l’applicazione del metodo di Newton-Kantarovich per il calcolo di questi autovettori.

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The positive spectrum and the properties of the positive eigenvectors of the discrete analog of a non-linear diffusion equation are studied. Applications of the Newton-Kantarovich method have been analyzed for calculating positive eigenvectors.

     

On the div-curl lemma in a Galerkin setting

Given a sequence of Galerkin spaces X _h of square-integrable vector fields, we state necessary and sufficient conditions on X _h under which it is true that for any two sequences of vector fields u _h ,u _h ′∈X _h converging weakly in L² and such that u _h is discrete divergence free and curl u _h ′ is precompact in H⁻¹, the scalar product u _h ⋅u′ _h converges in the sense of distributions to the right limit. The conditions are related to super-approximation and discrete compactness results for mixed finite elements, and are satisfied for Nédélec’s edge elements. We also provide examples of sequences of discrete divergence free edge element vector fields converging weakly to 0 in L² but whose divergence is not precompact in H _loc ⁻¹.      

Una discretizzazione stabile per una equazione parabolica autoaggiunta con condizioni al contorno implicanti sia la funzione che la derivata spaziale

A self-adjoint parabolic equation in one space variable is considered, under boundary conditions which involve the function and its space derivative. A type of numerical instability which can arise, traceable to the boundary conditions, is reduced through a new difference approximation.

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Lavoro eseguito nell’ambito dell’attività dei gruppi di ricerca matematica del C. N. R. per l’anno accademico 1968–69.