By Andre Weil
From the reviews:
"вЂ¦All of WeilвЂ™s works aside from books and lecture notes are compiled right here, in strict chronological order for simple reference.
But the price вЂ¦ is going past the benefit of straightforward reference and accessibility. within the first position, those volumes comprise numerous essays, letters, and addresses which have been both released in vague areas (вЂ¦) or no longer released in any respect.
Even extra priceless are the long commentaries on a few of the articles, written by way of Weil himself. those comments function a consultant, assisting the reader position the papers of their right context. furthermore, now we have the infrequent chance of seeing an outstanding mathematician in his later existence reflecting at the improvement of his rules and people of his contemporaries at quite a few levels of his career.
The sheer variety of mathematical papers of basic importance could earn WeilвЂ™s accrued Papers a spot within the library of a mathematician with an curiosity in quantity thought, algebraic geometry, representations conception, or comparable parts. the extra import of the mathematical heritage and tradition in those volumes makes them much more essential."
Neal Koblitz in Mathematical Reviews
"вЂ¦AndrГ© WeilвЂ™s mathematical paintings has deeply prompted the maths of the 20 th century and the enormous (...) "Collected papers" emphasize this influence."
O. Fomenko in Zentralblatt der Mathematik
Volume I (1926-1951) ISBN 978-3-540-85888-1
Volume III (1964-1978) ISBN 978-3-540-87737-0
Read Online or Download Oeuvres scientifiques, Collected papers, - (1951-1964) PDF
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Extra info for Oeuvres scientifiques, Collected papers, - (1951-1964)
1 allows us to obtain the following characterization of a diﬀerentiable quasiconvex function in more than one variable. 2. Let S ⊆ n be a convex set and let f be a diﬀerentiable function on S. Then, f is quasiconvex on S if and only if the following implication holds: x1 , x2 ∈ S, f (x1 ) ≥ f (x2 ) ⇒ ∇f (x1 )T (x2 − x1 ) ≤ 0. 2) Proof. Assume f is quasiconvex and let x1 , x2 ∈ S with f (x1 ) ≥ f (x2 ). Consider the restriction ϕ(t) = f (x1 + t(x2 − x1 )), t ∈ [0, 1]. 2) holds. 2) holds. 1, there exist t1 , t2 ∈ [0, 1] such that ϕ(t1 ) ≥ ϕ(t2 ) and ϕ (t1 )(t2 − t1 ) > 0.
Case t0 ∈ (a, b). We will prove that ϕ is non-increasing in [a, t0 ) and non-decreasing in (t0 , b]. If not, there exist t1 , t2 , tˆ1 , tˆ2 ∈ [a, b] such that t1 < t2 < t0 < tˆ1 < tˆ2 with ϕ(t1 ) < ϕ(t2 ) and ϕ(tˆ1 ) > ϕ(tˆ2 ). Since is the inﬁmum value of ϕ, there exists n ¯ such that t2 < tn¯ < tˆ1 with ϕ(tn¯ ) < ϕ(t2 ), ϕ(tn¯ ) < ϕ(tˆ1 ) and this contradicts the quasiconvexity of ϕ in the intervals [t1 , tn¯ ], [tn¯ , tˆ2 ], respectively. It remains to be proven that at least one of the two intervals [a, t0 , t0 , b] is closed.
Let f (x) = Ax + b be an aﬃne function, where A is an m × n matrix, b ∈ m , and let g : m → be a convex function. Show that z(x) = g(Ax+ b) is a convex function. State and prove a similar result for concave functions. 29. Show that if z(x) = log f (x) is a convex function, then f (x) is convex. Give an example showing that the reverse is not true. 30. Let f be a negative convex function. Show that z(x) = 1 f (x) is concave. 31. Give an example showing that the reciprocal of a positive convex function is not concave.
Oeuvres scientifiques, Collected papers, - (1951-1964) by Andre Weil