Giovanni Battista Rizza

Giovanni Battista Rizza
Giovanni Battista Rizza
	Università degli Studi di Genova
Known for	Integral representation of pluriharmonic functions; Rizza manifolds; Theory of functions on Algebras;
Spouse	Lucilla Bassotti
Awards	Unione Matematica Italiana (1954),; President of the Italian Republic) (1973);
	Scientific career
Fields	Hypercomplex analysis; Several complex variables; Differential geometry;
Institutions	Università degli Studi di Genova; Istituto Nazionale di Alta Matematica; Sapienza University of Rome; Università degli Studi di Parma;
Doctoral advisor	Enzo Martinelli

Giovanni Battista Rizza (7 February 1924 – 15 October 2018), officially known as Giambattista Rizza,

complex analysis of several variables and in differential geometry: he is known for his contribution to hypercomplex analysis, notably for extending Cauchy's integral theorem and Cauchy's integral formula to complex functions of a hypercomplex variable,^[4] the theory of pluriharmonic functions and for the introduction of the now called Rizza manifolds

.

Biography

Life and academic career

Born in

ordinary professor to the same chair.^[16] In 1979 he became ordinary professor of "Geometria superiore",^[17] holding that chair uninterruptedly until 1994:^[18] from 1994 up to his retirement in 1997, he was "professore fuori ruolo" in the same department of mathematics where he worked for more than 35 years.^[19]

Apart from his research and teaching work, he was actively involved as a member of the editorial board of the "Rivista di Matematica della Università di Parma", and served also as the journal director from 1992 to 1997.^[20]

Rizza died in Parma on 15 October 2018, at the age of 94.^[21]^[22]

Honors

In 1954 he was awarded the

Unione Matematica Italiana, jointly with Gabriele Darbo: the judging commission was composed by Giovanni Sansone (as the president), Alessandro Terracini, Beniamino Segre, Giuseppe Scorza-Dragoni, Carlo Miranda, Mario Villa and Enzo Martinelli (as the secretary).^[1]

In 1973 he was awarded the

President of the Italian Republic,^[2] as an acknowledgement his research and teaching and achievements as civil servant at the University of Parma.^[23]

In 1995, to celebrate his 70th birthday, an international conference on differential geometry was organized in Parma: the

proceedings were later published as a special issue of the "Rivista di Matematica della Università di Parma".^[24]

In 1999 the University of Parma, where he worked for more than 35 years, awarded him the title of

professor emeritus.^[25]

Rizza was an honorary member of the Balkan Society of Geometers and life member of the Tensor Society.^[26]

Personality traits

Enzo Martinelli described Giovanni Battista Rizza as a passionate researcher with a "strong intellectual force",^[27] and his scientific work as rich of geometrical ideas, denoting his strong algorithmic ability.^[28] According to Martinelli, Rizza is also a skilled organizer:^[29] his ability in organizational tasks is also acknowledged and praised by Schreiber (1973, p. 1), who also alludes the positive opinions of colleagues and students alike about his involvement in research, teaching and administrative duties at the mathematics department of the University of Parma.

Work

Research activity

Giovanni Battista Rizza authored 53 research papers and 30 other scientific works, including research announcements, short notes, surveys and reports: he also wrote didactic notes and papers on historical topics, including commemorations of other scientists.

theory of functions of several complex variables, and differential geometry

.

Theory of functions on algebras

The theory of functions on algebras, also referred to as hypercomplex analysis, is the study of functions whose domain is a subset of an algebra.^[31] The first works of Giovanni Battista Rizza belong to this field of research, and he was awarded the Premio Ottorino Pomini for his contributions.^[4]

His first main result is the extension of Cauchy's integral theorem to every monogenic function $F$ on a general complex algebra $A$ ,^[32]

\int _{\Gamma _{1}}\mathrm {F} (\mathrm {X} )\mathrm {d} \mathrm {X} =0

where $Γ 1$ is a

1-dimensional cycle

homologous to zero, and also satisfying other technical conditions.

Few years later, he extended

isomorphic to a given complex algebra

A

:^[34]

precisely, he proves the formula

\int _{\Gamma _{1}}{\frac {\mathrm {F} (\mathrm {X} )}{\mathrm {X} -\Xi }}\mathrm {d} \mathrm {X} =2\pi i\sum _{s=1}^{k}\mathrm {N} ^{(s)}u^{(s)}\mathrm {F} (\Xi )

where

X \equiv x * \equiv x

identifies indifferently a

point

in the complex algebra

A

or in its isomorphic real algebra

A *

,

Γ 1

is again a

1-dimensional cycle

homologous to zero, and satisfying other technical conditions,

$N (s)$ is the winding number of the cycle $Γ 1$ respect to the zero divisor locus for the considered algebra.

Theory of analytic functions of several complex variables

All'estensione, tutt'altro che banale, allo spazio $R 2 n$ dei metodi di Martinelli per dimostrare la (3), è dedicata una Memoria [8] di Giovanni Battista Rizza, il quale, sempre nell'ipotesi $ρ (x 1, y 1,..., x n, y n) \in C ω$ , perviene a stabilire la (3) per $n$ qualsiasi. Anche questo lavoro, per quanto redatto in lingua inglese e pubblicato su una delle principali riviste matematiche, non ha nella letteratura attuale, la notorietà che meriterebbe.^[35]
— Gaetano Fichera, (Fichera 1982a, p. 135).

Rizza published only three work in this field:^[36] in the first one, the highly remarkable memoir (Rizza 1955),^[37] he extends to pluriharmonic functions of $2 n$ real variables, $n > 2$ , the methods introduced by Enzo Martinelli in order to give new proof of a result of Luigi Amoroso for pluriharmonic functions of four real variables.^[38] Precisely, he proves the following formula

{\frac {\partial u}{\partial \nu }}={\frac {\vert \nabla \rho \vert ^{3}}{Q(\rho )}}Eu

(1)

where

$u$ is a polyharmonic function defined on a bounded domain $Ω$ ,
$ρ$ is a real analytic function defining the boundary of $Ω$ by the equation

\partial \Omega =\{x\in \mathbb {R} ^{2n}|\rho (x)=0\},

$Q (ρ)$ is a linear combination of the
complex variables
,
$E$ is a linear tangential operator defined on $\partialΩ$ .

Formula

Laplacian,^[40] and also to establish an integro-differential equation boundary values of pluriharmonic functions must satisfy.^[41] Rizza's result motivated other works on the same topic by Gaetano Fichera, Paolo de Bartolomeis and Giuseppe Tomassini.^[42]

Selected publications

Research works

Rizza, Giovanni Battista (1950), "Sulle funzioni analitiche nelle algebre ipercomplesse" [On analytic functions on hypercomplex algebras], MR 0057350
. In this work Rizza extends the classical Cauchy's integral theorem to monogenic functions on a general complex algebra.

Rizza, Giovanni Battista (1952), "Contributi al problema della determinazione di una formula integrale per le funzioni monogene nelle algebre complesse dotate di modulo e commutative" [Contributions to the problem of determining an integral formula for monogenic functions on complex commutative algebras with modulus], Zbl 0047.32204
.

Rizza, Giovanni Battista (1952a), "Estensione della formula integrale di Cauchy alle algebre complesse dotate di modulo e commutative" [Extension of Cauchy's integral formula to commutative complex algebras with modulus], Rendiconti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Serie VIII (in Italian), XII (6): 667–669,
Zbl 0048.06101
.

Rizza, Giovanni Battista (1953), "Teoria delle funzioni nelle algebre complesse dotate di modulo e commutative" [Function theory on commutative complex algebras with modulus], Zbl 0123.15203
.

Rizza, Giovanni Battista (1954), "On Dirichlet's problem for components of analytic functions of several complex variables" (PDF), Proceedings of the International Congress of Mathematicians, 1954. Volume II,
North-Holland Publishing Company, pp. 161–162. A short research announcement describing briefly the results proved in (Rizza 1955
).

Rizza, G. B. (1955), "Dirichlet problem for $n$ -harmonic functions and related geometrical problems", Zbl 0067.33004, available at DigiZeitschirften
.

Rizza, G. B. (1957), "Su diverse estensioni dell'invariante di E. E. Levi nella teoria delle funzioni di più variabili complesse" [On different extensions of E. E. Levi invariant in the theory of functions of several complex variables], Zbl 0091.25903. In this work Rizza epitomizes all known extensions of the Levi invariant to hypersurfaces
in $\mathbb {C} ^{n}$ for $n > 2$ in a single tensor of hybrid type. This paper is also interesting since it traces the story of such extensions back to the pioneering work of Eugenio Elia Levi.

Rizza, G. B. (1958), "Appendice I. Rappresentazione esplicita di tipo integrale per le funzioni $r$ –armoniche. Estensione al caso di $r$ variabili complesse dell'invariante di E. E. Levi", in
Istituto Nazionale di Alta Matematica: the full course notes, published as a monograph, include also a chapter by Enzo Martinelli and an appendix by Mario Benedicty
). The topics he exposes are summarized by the two parts of the title, whose free English translations are "Explicit integral representation for $r$ –harmonic functions" and "Extension of the E. E. Levi invariant to the case of $r$ complex variables".

Rizza, Giovanni Battista (1962a), "Finsler structures on almost complex manifolds", Proceedings of the International Congress of Mathematicians, Stockholm. (PDF), ICM Proceedings, vol. P, Stockholm, p. 73{{citation}}: CS1 maint: location missing publisher (link). A short research announcement describing briefly the results proved in (Rizza 1963).

Rizza, Giovanni Battista (1962b), "Strutture di Finsler sulle varietà quasi complesse" [Finsler structures on almost complex manifolds], Rendiconti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Serie VIII (in Italian), 33 (5): 271–275,
Zbl 0113.37202. Another short presentation of the results proved in (Rizza 1963
).

Rizza, Giovanni Battista (1963), "Strutture di Finsler di tipo quasi Hermitiano" [Finsler structures of almost Hermitian type] (PDF), Zbl 0129.14101. The article gives the proofs of the results previously announced in references (Rizza 1962a) and Rizza (1962b)
.

Rizza, Giovanni Battista (1964), " $F$ -forme quadratiche ed hermitiane" [Hermitian and quadratic $F$ -forms], Zbl 0123.15203. Shoshichi Kobayashi
cites this article as the first one in the theory of Rizza manifolds.

Rizza, Giovanni Battista (1969), "Teoremi di rappresentazione per alcune classi di connessioni su di una varietà quasi complessa" [Representation theorems for some classes of connections on a quasi complessa] (PDF), Zbl 0183.50701
.

Rizza, Giovanni Battista (1969), "Connessioni metriche sulle varietà quasi hermitiane" [Metric connections quasi hermitian manifolds] (PDF), Zbl 0192.58903
.

Dentoni, Paolo; Rizza, Giovanni Battista (1972), "Una nuova classe di funzioni in un'algebra reale" [A new class of functions on a real algebra] (PDF), Zbl 0251.30050
. In this work the authors introduce a new class of functions on a real algebra in the attempt of unifying the research trends on functions on real algebras in the seventies.

Historical, commemorative and survey papers

Rizza, Giovanni Battista (December 12, 1973), "Contributi recenti alla teoria delle funzioni nelle algebre" [Recent contributions to the theory of functions on algebras], Zbl 0325.30040
. A short but comprehensive survey paper detailing the works on the field done by Italian mathematicians during the years from 1961 to 1973: however, it also includes several biographical references to other earlier works by non Italian mathematicians and to historical bibliographies on hypercomplex analysis.

Rizza, Giovanni Battista (1986), "Indirizzo di adesione", in Montalenti, G.;
Accademia Nazionale dei Lincei, pp. 29–30, archived from the original on February 23, 2011, retrieved February 16, 2014. The brief "participating address" presented to the International congress on the occasion of the celebration of the centenary of birth of Mauro Picone and Leonida Tonelli (held in Rome on May 6–9, 1985), by Giovanni Battista Rizza on behalf of the University of Parma: the scientific relations between Leonida Tonelli
and the Department of Mathematics in Parma are described.

Rizza, Giovanni Battista (1984), "Enzo Martinelli: Scienziato e Maestro" [Enzo Martinelli: Scientist and Master] (PDF), Zbl 0557.01011
. A celebrative paper written by Giovanni Battista Rizza to honor his former master.

Rizza, Giovanni Battista (1998), "Commemorazione del Prof. Francesco Speranza" [Commemoration of Prof. Francesco Speranza] (PDF), MR 1680985
.

Rizza, Giovanni Battista (1999), "Commemorazione della professoressa Bianca Manfredi" [Commemoration of professor Bianca Manfredi] (PDF), Zbl 1073.01521
.

Rizza, Giovanni Battista (April 2002), "Commemorazione di Enzo Martinelli" [Commemoration of Enzo Martinelli], Zbl 1194.01133
.

See also

Almost complex manifold

Complex manifold

Kähler manifold

Pluriharmonic function

Pseudoconvexity

Rizza manifold

Several complex variables

References

^ ^a ^b The detailed motivation for the award is reported in the Bollettino UMI 1954, pp. 477–478. The high scientific value of the works of the two young mathematicians induced the commission to ask the benefactors supporting the prize for a double award: their request was accepted.

^ ^a ^b See the list of the recipients of the medal.

Professor Emeritus
".

^
Clifford algebras
").

doctoral student
.

^ He, Giuseppe Arcidiacono and Dario Del Pasqua, were awarded the scholarship without sustaining the "colloquio" ("colloquium" in English translation), an oral exam where the candidate was asked to answer questions posed by a scientific jury, according to Roghi (2005, p. 46) who reports also an excerpt of the motivation given by the commission for the awarding of the scholarship to Rizza: "... perché trattasi di giovani di cui è nota l'attività scientifica...", i.e. (English translation): "...because they are young researchers whose scientific activity is known, ...").

Luigi Fantappie, Giulio Krall, Enrico Bompiani and Mauro Picone
.

INdAM. See (Roghi 2005
) for further details.

^ See (Roghi 2005, p. 50).

^ See (Roghi 2005, p. 50) and Severi (1958, p. III)

^ See (Rizza 1958).

INdAM
to fund this course.

^ "Analytic geometry with elements of projective geometry and descriptive geometry with drawing" (English translation).

^ See the announce on the Bollettino UMI (1962, p. 454).

^ See (Venturini 1963, p. 15).

^ See the 1965 Yearbook of the University of Parma, p. 207: the exact date of this career advancement is 16th January 1965.

^ Literally "higher geometry": it is an Italian university course on advanced geometry topics.

^ See the 1980 Yearbook of the University of Parma, p. 209.

^ See the 1995 Yearbook of the University of Parma, pp. 887 and 1036: the locution, literally meaning "out of role professor", identifies a nearly retired professor which is not in charge of any particular university course.

^ According to the timeline of Editors in Chief of the "Rivista", as reported in the historical section of the journal web site.

^ "Giambattista Rizza". Necrologi Italia. Retrieved February 18, 2023.

^ "È Scomparsa Lucilla Bassotti Rizza". Università degli Studi di Roma "La Sapienza". Retrieved February 18, 2023.

^ See (Schreiber 1973, p. 1).

plane crash in China few weeks before the proceedings of the conference were published (p. iii
).

^ According to Decreto ministeriale 17 febbraio 1999.

^ See the list of members of the Balkan Society of Geometers (2011) and of the Tensor Society (2010).

^ Martinelli (1994, p. 1) precisely characterizes Rizza's scientific work as developed with "...molta passione e forza intellettuale...", i.e. with (English translation) "...much passion and intellectual force...".

^ Again according to Martinelli (1994, p. 2): "Queste poche righe mi auguro siano servite a dimostrare che Rizza è un matematico ricco di idee geometriche e dotato di forti capacità algoritmiche.", i.e. (free English translation) "I hope those few lines have been of some help in demonstrating that Rizza is a mathematician rich of geometrical ideas and gifted with a strong algorithmic ability."

^ See (Martinelli 1994, p. 2).

^ See, for example, (Rizza 1984), (Rizza 1986) and (Rizza 2002).

^ For more information see the survey article by Rizza (1973) and the references cited therein.

^ See (Rizza 1950).

^ See (Rizza 1952), (Rizza 1952a) and the survey (Rizza 1973).

^ In the terminology of Rizza (1952, 1952a), the algebra $A *$ is said to be the real image of (precisely, l'immagine reale di) $A$ .

^ (English translation): "To the far from trivial extension to the $R 2n$ space of Martinelli's methods in order to prove (3) a Memoir [8] of Giovanni Battista Rizza is devoted, who, again under the hypothesis that $ρ (x 1, y 1,..., x n, y n) \in C ω$ , succeeds in proving (3) for every $n$ . Even this work, despite being written in English and published in a major mathematical journal, has not, in the current literature, the notoriety it deserves".

^ The work (Rizza 1954) is only a research announcement related to the (Rizza 1955), while (Rizza 1958) is set of course notes based on the same paper and on (Rizza 1957).

^ According to Fichera (1982b, p. 24), who praises this work as "molto considerevole": see also his comments in (Fichera 1982a, p. 135).

^ See (Fichera 1982a, p. 135), (Fichera 1982b, pp. 24–25) and (Martinelli 1941).

^ See (Fichera 1982a, p. 135), (Fichera 1982b, pp. 24–25) and (Fuks 1963, p. 277, footnote 1).

^ See (Fichera 1982a, p. 134), (Fichera 1982b, p. 33) and (Martinelli 1941, p. 162).

^ It is the Amoroso integro-differential equation: see (Fichera 1982a, p. 134) and (Fichera 1982b, pp. 33).

^ See the hystorical survey sections in (Fichera 1982b, p. 25) and the work (de Bartolomeis & Tomassini 1981, p. 33).

Sources

Biographical

Balkan Society of Geometers (July 24, 2011), The list of members of the Balkan Society of Geometers (PDF), retrieved April 19, 2011.

Bollettino UMI (1954), "Notizie" [Notices], Bollettino dell'Unione Matematica Italiana, Serie III (in Italian), 9 (4): 467–490. The official relation of the judging commission for the awarding of the Ottorino Pomini Prize in 1954, jointly won by Gabriele Darbo
and Giovanni Battista Rizza.

Bollettino UMI (1962), "Notizie" [Notices], Bollettino dell'Unione Matematica Italiana, Serie III (in Italian), 17 (1): 120–157. The official announcement of the winning by Giovanni Battista Rizza of the chair of "Geometria analitica con elementi di Geometria Proiettiva e Geometria Descrittiva con Disegno" awarded by the University of Parma
.

The Editorial Board, ed. (1965), "Professori ordinari", Annuario dell'Università di Parma [Yearbook of the University of Parma], vol. A.A. 1964/1965,
Università degli Studi di Parma
.

The Editorial Board, ed. (1980), "Professori ordinari", Annuario dell'Università di Parma [Yearbook of the University of Parma], vol. A.A. 1979/80,
Università degli Studi di Parma
.

The Editorial Board, ed. (1995), "Professori ordinari", Annuario dell'Università di Parma [Yearbook of the University of Parma], vol. A.A. 1994/95,
Università degli Studi di Parma
.

Martinelli, E. (1994), "Omaggio a Giovanni Battista Rizza in occasione del suo 70° compleanno" (PDF), in Donnini, S.; Gigante, G.; Mangione, V. (eds.), Geometria differenziale – Analisi complessa. Convegno internazionale – Parma, 19–20 maggio 1994 in occasione del 70° compleanno di G. B. Rizza, 5^a Serie (in Italian), vol. 3, Rivista di Matematica della Università di Parma, pp. 1–2. "Homage to Giovanni Battista Rizza on his 70th birthday" (English translation of the title) a tribute to Giovanni Battista Rizza by his former master Enzo Martinelli.

Il Ministro dell'Università e della Ricerca Scientifica e Tecnologica
(February 19, 1999), Decreto Ministeriale 17 Febbraio 1999 [Ministerial Decree 17 February 1999] (in Italian). The "Ministerial Decree" awarding the title of "Professor Emeritus" to Giovanni Battista Rizza.

Archivio Necrologi (October 15, 2018), Funerale di Giambattista Rizza, retrieved February 18, 2023.

Presidenza della Repubblica Italiana (July 31, 1973), Medaglia d'oro ai benemeriti della scuola della cultura e dell'arte: Giovanni Battista Rizza, retrieved May 31, 2011.

Rivista di Matematica della Università di Parma, (the Editorial Board of) (December 12, 2013), History, retrieved January 12, 2013.

Corrado De Concini. It is almost exclusively based on sources from the institute archives: the wealth and variety of materials included, jointly with its appendices and indexes, make this monograph a useful reference not only for the history of the institute itself, but also for the history of many mathematicians
who taught, followed the institute courses or simply worked there.

Schreiber, Bruno (1973), Curriculum Vitæ di Giambattista Rizza [Curriculum Vitæ of Giambattista Rizza] (in Italian), Istituto di Matematica dell'Università di Parma, p. 4. The official 1973 CV of Giovanni Battista Rizza, available from the Institute of Mathematics of the University of Parma.

Tensor Society (2010), List of life members of the Tensor Society (PDF), retrieved July 14, 2013.

Venturini, Giancarlo (1963), "Prolusione all'apertura dell'A.A. 1962/63", Annuario dell'Università di Parma [Yearbook of the University of Parma], vol. A.A. 1962/63,
Magnifico Rettore
prof. G. Venturini.

Scientific

Aikou, Tadashi (2004), "Finsler Geometry on Complex Vector Bundles" (PDF), in Bao, David;
Zbl 1073.53093
.

de Bartolomeis, Paolo;
Zbl 0484.32007
.

Donnini, S.; Gigante, G.; Mangione, V., eds. (1994), "Geometria differenziale – Analisi complessa. Convegno internazionale – Parma, 19–20 maggio 1994 in occasione del 70° compleanno di G. B. Rizza" [Differential Geometry – Complex analysis" held in Parma on May 19–20, 1994 to celebrate Giovanni Battista Rizza's 70th birthday], proceedings of an international meeting celebrating Giovanni Battista Rizza, published by the Rivista di Matematica della Università di Parma. The first speaker was his former master Enzo Martinelli
.

Henri Poincare and analyzing several earlier results of Enzo Martinelli, Giovanni Battista Rizza and Francesco Severi, as well as works of Aldo Andreotti
among the others.

Fichera, Gaetano (1982b), "Valori al contorno delle funzioni pluriarmoniche: estensione allo spazio $R 2 n$ di un teorema di L. Amoroso" [Boundary values of pluriharmonic functions: extension to the space $R 2 n$ of a theorem of L. Amoroso], Zbl 0569.31006. In this work Gaetano Fichera proves another trace condition for pluriharmonic functions and surveys other recent works in the fields, notably the one of de Bartolomeis & Tomassini (1981)
.

Zbl 0138.30902
.

Ichijyō, Yoshihiro (1988), "Finsler metrics on almost complex manifolds" (PDF), Zbl 0885.53031, archived from the original
(PDF) on June 18, 2022.

Zbl 0326.32016. In this paper, Shoshichi Kobayashi acknowledges Giovanni Battista Rizza as the first one to study complex manifolds with Finsler structure, now called Rizza manifolds
.

Zbl 0025.40503. In this work Martinelli proves an earlier result of Luigi Amoroso on the boundary values of pluriharmonic function by using tensor calculus
.

Scharnhorst, K. (2001), "Angles in Complex Vector Spaces", Zbl 0993.51010
.

Istituto Nazionale di Alta Matematica, including appendices of Enzo Martinelli, Giovanni Battista Rizza and Mario Benedicty
.

Authority control databases
International

ISNI

VIAF

National

France

BnF data

Germany

United States

Vatican

Academics

zbMATH

Other

IdRef

Retrieved from "https://en.wikipedia.org/w/index.php?title=Giovanni_Battista_Rizza&oldid=1152170271"

[Pomini-1] The detailed motivation for the award is reported in the Bollettino UMI 1954, pp. 477–478. The high scientific value of the works of the two young mathematicians induced the commission to ask the benefactors supporting the prize for a double award: their request was accepted.

[Ben-2] See the list of the recipients of the medal.

[3] Professor Emeritus
".

[Pomini_motivation-4] 
Clifford algebras
").

[mar1-5] toral student
.

[6] He, Giuseppe Arcidiacono and Dario Del Pasqua, were awarded the scholarship without sustaining the "colloquio" ("colloquium" in English translation), an oral exam where the candidate was asked to answer questions posed by a scientific jury, according to Roghi (2005, p. 46) who reports also an excerpt of the motivation given by the commission for the awarding of the scholarship to Rizza: "... perché trattasi di giovani di cui è nota l'attività scientifica...", i.e. (English translation): "...because they are young researchers whose scientific activity is known, ...").

[7] Luigi Fantappie, Giulio Krall, Enrico Bompiani and Mauro Picone
.

[8] INdAM. See (Roghi 2005
) for further details.

[GRp50-9] See (Roghi 2005, p. 50).

[10] See (Roghi 2005, p. 50) and Severi (1958, p. III)

[11] See (Rizza 1958).

[12] INdAM
to fund this course.

[13] "Analytic geometry with elements of projective geometry and descriptive geometry with drawing" (English translation).

[14] See the announce on the Bollettino UMI (1962, p. 454).

[15] See (Venturini 1963, p. 15).

[16] See the 1965 Yearbook of the University of Parma, p. 207: the exact date of this career advancement is 16th January 1965.

[17] Literally "higher geometry": it is an Italian university course on advanced geometry topics.

[18] See the 1980 Yearbook of the University of Parma, p. 209.

[19] See the 1995 Yearbook of the University of Parma, pp. 887 and 1036: the locution, literally meaning "out of role professor", identifies a nearly retired professor which is not in charge of any particular university course.

[20] According to the timeline of Editors in Chief of the "Rivista", as reported in the historical section of the journal web site.

[21] "Giambattista Rizza". Necrologi Italia. Retrieved February 18, 2023.

[22] "È Scomparsa Lucilla Bassotti Rizza". Università degli Studi di Roma "La Sapienza". Retrieved February 18, 2023.

[23] See (Schreiber 1973, p. 1).

[24] rash in China few weeks before the proceedings of the conference were published (p. iii
).

[25] According to Decreto ministeriale 17 febbraio 1999.

[26] See the list of members of the Balkan Society of Geometers (2011) and of the Tensor Society (2010).

[Marp1-27] Martinelli (1994, p. 1) precisely characterizes Rizza's scientific work as developed with "...molta passione e forza intellettuale...", i.e. with (English translation) "...much passion and intellectual force...".

[28] Again according to Martinelli (1994, p. 2): "Queste poche righe mi auguro siano servite a dimostrare che Rizza è un matematico ricco di idee geometriche e dotato di forti capacità algoritmiche.", i.e. (free English translation) "I hope those few lines have been of some help in demonstrating that Rizza is a mathematician rich of geometrical ideas and gifted with a strong algorithmic ability."

[Marp2-29] See (Martinelli 1994, p. 2).

[30] See, for example, (Rizza 1984), (Rizza 1986) and (Rizza 2002).

[31] For more information see the survey article by Rizza (1973) and the references cited therein.

[32] See (Rizza 1950).

[33] See (Rizza 1952), (Rizza 1952a) and the survey (Rizza 1973).

[34] In the terminology of Rizza (1952, 1952a), the algebra $A *$ is said to be the real image of (precisely, l'immagine reale di) $A$ .

[35] (English translation): "To the far from trivial extension to the $R 2n$ space of Martinelli's methods in order to prove (3) a Memoir [8] of Giovanni Battista Rizza is devoted, who, again under the hypothesis that $ρ (x 1, y 1,..., x n, y n) \in C ω$ , succeeds in proving (3) for every $n$ . Even this work, despite being written in English and published in a major mathematical journal, has not, in the current literature, the notoriety it deserves".

[36] The work (Rizza 1954) is only a research announcement related to the (Rizza 1955), while (Rizza 1958) is set of course notes based on the same paper and on (Rizza 1957).

[37] According to Fichera (1982b, p. 24), who praises this work as "molto considerevole": see also his comments in (Fichera 1982a, p. 135).

[38] See (Fichera 1982a, p. 135), (Fichera 1982b, pp. 24–25) and (Martinelli 1941).

[39] See (Fichera 1982a, p. 135), (Fichera 1982b, pp. 24–25) and (Fuks 1963, p. 277, footnote 1).

[40] See (Fichera 1982a, p. 134), (Fichera 1982b, p. 33) and (Martinelli 1941, p. 162).

[41] It is the Amoroso integro-differential equation: see (Fichera 1982a, p. 134) and (Fichera 1982b, pp. 33).

[42] See the hystorical survey sections in (Fichera 1982b, p. 25) and the work (de Bartolomeis & Tomassini 1981, p. 33).

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