759 research outputs found
Piazza Campitelli after Pope Alexander VII's Project
No full texthistorical view, G. B. Falda, print after Alexander VII's project for Piazza Campitelli, 166
Pisteis in Comparison.Examples and Enthymemes in the Rhetoric to Alexander and in Aristotle's Rhetoric.
No full textToday it is generally accepted that the main differences between the "Rhetoric to Alexander" and Aristotle’s "Rhetoric" depend on the different aims of each work’s author: essentially practical in the first case, theoretical in the second one. The difference is clearly visible in the classification of pisteis (proofs). Our basic assumption is that this difference depends on the two majors elements of novelty in Aristotle's Rhetoric: 1. the introduction in the rhetorical field of the logico-dialectical apparatus (above all, the concept of syllogismos); 2. the consideration of ethos and pathos as entechnoi pisteis. In order to specify this general consideration, the paper focuses on enthymema and paradeigma: thanks to the position these pisteis have in Aristotle's "Rhetoric", they represent a suitable starting point for the comparison with the Rh. ad Al
MR2741185 Talvila, Erik The regulated primitive integral. Illinois J. Math. 53 (2009), no. 4, 1187–1219. (Reviewer: Luisa Di Piazza) 46G12 (26A39 46E15 46F10)
No full textTalvila Erik, The regulated primitive integral. Illinois J. Math. 53 (2009), no. 4, 1187–1219, 46Gxx (26A39 46Exx) MR 2 741 185
A descriptive definition of an integral is a definition which provides a ``description'' of the space of primitives. The derivatives in some sense of the primitives are the integrands.
In this paper the author introduces a descriptive method of integrating distributions: the regulated primitive integral.
The set \textbf{B}_R= \{F: [-\infty,\infty]
\rightarrow {\bf R}
\ \ | \mbox{ F {\it is regulated and left continuous on }}\\
\ \ {\bf R},
\ \ F(-\infty)=0, \ \ F(\infty)\in {\bf R}\} is the family of primitives.
The derivative here is in the sense of the distributions (i.e. a distributional or weak derivative). Then the integrable distributions are those distributions (in the Schwartz's sense) that are the distributional derivative of a function in .
The regulated primitive integral is a proper extension of the integral of distribution defined by L. Schwartz [Théorie des distributions. (French) Publications de l'Institut de Mathématique de l'Université de Strasbourg, No. IX-X. Hermann, Paris 1966 xiii+420, 46.40 (44.00), MR0209834 (35730)]. Moreover it is proved that the space of regulated integrable distributions is the completion of the space of signed Radon measures in the Alexiewicz norm, but it is not the completion in this norm of the Henstock-Kurzweil integrable functions. The functions of bounded variation constitute its dual space and also the space of multipliers. In the introduction a wide
panorama of descriptive and constructive integration methods is given.
Reviewed by (L. Di Piazza
MR2569913: Rodríguez, José. Some examples in vector integration. Bull. Aust. Math. Soc. 80 (2009), no. 3, 384–392. (Reviewer: Luisa Di Piazza),
No full textThe paper deals with some classical examples in vector integration due to Phillips, Hagler and Talagrand, revisited from the point of view of the Birkhoff and McShane integrals. More precisely, the author considers:
- Phillips' example of a Pettis integrable function f which is not Birkhoff integrable [R. S. Phillips, Trans. Amer. Math. Soc. 47 (1940), 114--145; MR0002707 (2,103c)]. It is proved here that f is universally McShane integrable.
- Hagler's example of a scalarly measurable l∞-valued function g which is not strongly measurable. The function g is proved to be universally Birkhoff integrable.
- Talagrand's example of a bounded Pettis integrable function φ having no conditional expectation [M. Talagrand, Mem. Amer. Math. Soc. 51 (1984), no. 307, ix+224 pp.; MR0756174 (86j:46042)]. Here the author shows that φ is also Birkhoff integrable, giving a negative answer to the question whether conditional expectations exist within the Birkhoff theory.
Some interesting open problems are also stated.
Reviewed by Luisa Di Piazz
MR2553995 (2010h:26008): Mihail, Alexandru The Arzela-Ascoli theorem for partial defined functions. An. Univ. Bucureşti Mat. 57 (2008), no. 2, 259–268. (Reviewer: Luisa Di Piazza),
No full textIn this paper the author gives a generalization of the Arzela-Ascoli theorem for partial defined functions, i.e., for functions defined in a nonempty subset of a metric space X and taking values in a metric space Y. To this end suitable definitions of local and uniform convergence for partial defined functions are introduced. As an application a different proof of a known result concerning the existence of Lipschitz selections for Lipschitz multifunctions is given.
Reviewed by Luisa Di Piazz
Santa Maria della Pace
No full textClose view of the upper façade, detail left side; Cortona had more opportunity to work as an architect under Pope Alexander VII, and in his façade (1656-1659) for the 15th-century church of S Maria della Pace, Rome, and the replanning of its piazza, he realized one of the most impressive works of the Roman Baroque. This project ingeniously incorporates ideas analogous to his theatre-like designs. The patron, Alexander VII, had the piazza in front of the church enlarged to a regular trapezoidal area to permit him to arrive in his carriage in front of the church, which contained his redesigned family chapel, the Chigi Chapel. Source: Grove Art Online; http://www.groveart.com/ (accessed 12/27/2007
MR2886259 Naralenkov, Kirill Several comments on the Henstock-Kurzweil and McShane integrals of vector-valued functions. Czechoslovak Math. J. 61(136) (2011), no. 4, 1091–1106. (Reviewer: Luisa Di Piazza) 26A39 (28B05)
No full textIn this paper the author essentially discusses the difference between the Henstock-Kurzweil and McShane integrals of vector-valued functions from the descriptive point of view. He first considers three notions of absolute continuity for vector-valued functions AC, AC*, AC_{\delta}) and studies the relationships between the corresponding classes of functions. Then he uses such notions to give descriptive characterizations of the Henstock-Kurzweil and McShane integrable functions
MR3191427 Naralenkov, Kirill M., A Lusin type measurability property for vector- valued functions. J. Math. Anal. Appl. 417 (2014), no. 1, 293307. 28A20
No full textIn the paper under review the author introduces the notion of Riemann measurability for vector-valued
functions, generalizing the classical Lusin condition, which is equivalent to the Lebesgue measurability
for real valued functions. Let X be a Banach space, let f : [a; b] ! X and let E be a measurable subset of [a; b]. The function
f is said to be Riemann measurable on E if for each " > 0 there exist a closed set F E with
(E n F) < 0 (where is the Lebesgue measure) and a positive number such that
k XK
k=1
ff(tk) ?? f(t0
k)g (Ik)k < "
whenever fIkgKk
=1 is a nite collection of pairwise non-overlapping intervals with max1 k K (Ik) <
and tk; t0
k 2 Ik
T
F.
The Riemann measurability is more relevant to Riemann type integration theory, such as those of
McShane and Henstock, rather than the classical notion of Bochner or scalar measurability. In par-
ticular the author studies the relationship between the Riemann measurability and the M and the H
integrals that are obtained if we assume that the gauge in the de nitions of McShane and Henstock
integral can be chosen Lebesgue measurable.
The main results are the following
If f : [a; b] ! X is H-integrable on a measurable subset E of [a; b], then f is Riemann measurable
on E.
If f : [a; b] ! X is both bounded and Riemann measurable on a measurable subset E of [a; b], then
f is M-integrable on E.
If f : [a; b] ! X is both Riemann measurable and McShane (Henstock) integrable on a measurable
subset E of [a; b], then f is M-integrable (H-integrable) on E.
Suppose X separable. If f : [a; b] ! X is McShane (Henstock) integrable, then f is M-integrable
(H-integrable.)
The author concludes the paper with the following open problem: for which families of non-separable
Banach spaces does the McShane (or even the Pettis) integrability imply Riemann measurability?
Reviewed by (L. Di Piazza
MR2684422 Deville, Robert; Rodríguez, José Integration in Hilbert generated Banach spaces. Israel J. Math. 177 (2010), 285–306. (Reviewer: Luisa Di Piazza)
No full text2010), 285–306, 46Exx (46J10)
It is known that each McShane integrable function is also Pettis integrable, while the reverse implication in general is not true.
The equivalence of McShane and Pettis integrability depends on
the target Banach space X and has been proven: by R. A. Gordon [Illinois J. Math. 34 (1990), no. 3, 557–567, 26A42 (28B15 46G10 49Q15)], and by D. H. Fremlin and J. Mendoza [Illinois J. Math. 38 (1994), no. 1, 127–147, 46G10 (28B05)] if X is separable, by D. Preiss and the reviewer [Illinois J. Math. 47 (2003), no. 4, 1177–1187. 28B05 (26A39 26E25 46G10)] if X=c_0(\Gamma) (for any set
\Gamma) or X is super-reflexive, by the second author of the present paper [J. Math. Anal. Appl. 341 (2008), no. 1, 80–90, 46G10 (28B05 46B99 47B10)] if X=L^1(\nu) (for any probability measure
\nu).
Here the authors show that the McShane and Pettis integrability coincide for functions taking values in a subspace of a Hilbert generated Banach space. This result includes all previous known ones concerning the above mentioned equivalence.
The used approach relies heavily on some special properties of the Markushevich bases of those Banach spaces. They also give a ZFC example of a scalarly negligible function which is not McShane integrable.
Moreover they prove that, whenever the target Banach space is super-reflexive generated, the Birkhoff integrability lies strictly between Bochner and McShane integrability. Reviewed by L. Di Piazz
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