170,044 research outputs found
A useful subdifferential in the Calculus of Variations
No full textConsider the basic problem in the Calculus of Variations of minimizing an energy functional depending on absolutely continuous functions Under suitable assumptions on the Lagrangian, a well-known result establishes that the minimizers satisfy the Du Bois-Reymond equation. Recent work (cf. Bettiol and Mariconda, 2020 [1], 2023; Mariconda, 2023 [2], 2021, 2024) highlights not only that a Du Bois-Reymond condition for minimizers can be broadened to cover the case of nonsmooth extended valued Lagrangians, but also that a particular subdifferential (associated with the generalized Du Bois-Reymond condition) plays an important role in the approximation of the energy via its values along Lispchitz functions, no matter minimizers exist. A crucial point is establishing boundedness properties of this subdifferential, based on weak local boundedness properties of the Lagrangian. This is the main objective of this paper. Our approach is based on a refined analysis of the metric that can be employed to evaluate the distance from the complementary of the effective domain of the reference Lagrangian. As an application of our findings we show how it is possible to deduce the non-occurrence of the Lavrentiev phenomenon, providing a new general result
G. Capaldo, Le pubblicità, p. 1-448 con G. Mariconda, N. Atlante, C. Verde
No full textIl lavoro ricostruisce il regime di pubblicità degli atti aventi ad oggetto diritti su beni, riconducibili, in via di prima approssimazione, alla locuzione «ricchezza mobiliare
Alkyne Hydroamination Promoted by NHC-Gold(I) Complexes: Activity and Mechanistic Insight
No full textThe pharmaceutical and chemical industries heavily rely on the production of N-containing
compounds, as these molecules serve as scaffolds for the synthesis of both fine chemicals and
polymers of interest. The hydroamination reaction can be hailed as a paradigm of a
contemporary, sustainable, catalytically promoted chemical process, since it can form C-N
bonds by adding amines to multiple carbon bonds with a 100% atom economy.1 By reducing
the high energy barriers connected to alkyne activation, use of metal complexes can facilitate
this process; lately, gold(I) N-heterocyclic (NHC) complexes were demonstrated to be
promising in this regard.2 Herein, we present a library of gold(I)-NHC complexes which were
identified as promising catalysts for regioselective Markovnikov addition of aromatic and
aliphatic alkynes to anilines, with high yields.3 Accurate catalyst’ design, experimental reaction
scope and computational mechanistic insight will be presented, by means of which the nonmonotonic
impact of substrate substituents on the reaction course was elucidated.
References
(1) Müller, T. E.; Hultzsch, K. C.; Yus, M.; Foubelo, F.; Tada, M. Chem. Rev. 2008, 108, 3795–3892.
(2) Mariconda, A.; Sirignano, M.; Troiano, R.; Russo, S.; Longo, P. Catalysts 2022, 12(8), 836.
(3) Sirignano, M.; D’Amato, A.; Costabile, C.; Mariconda, A.; Crispini, A.; Scarpelli, F.; Longo, P. Front. Chem.
2023, 11, 1260726
Formal power series
No full textWe begin here the subject of formal power series, objects of the form ∑n=0∞anXn (an∈ R or C) which can be thought as a generalization of polynomials. We focus here on their algebraic properties and basic applications to combinatorics. The reader must not be confused by the many technical, though simple, details that are needed in a book to justify rigorously every step. In Chap. 3, we have learned how to count sequences and collections with occupancy constraints: the number of possible codes of 10 digits that use only four 1’s, five 2’s and one 3 is easily obtained: 10!4!5!. What about counting the possible codes of 10 digits that use an even number of 1’s, an odd number of 2’s and any number of 3’s? What about the validity of the “Latin teacher’s random choice” which selects the students to test by opening randomly a book and summing up the digits of the page? Many counting problems can be solved using the formal power series! These are extremely useful in studying recurrences, notable sequences, probability and many other arguments. Moreover, they constitute a rich and interesting environment in their own right. The chapter ends with a combinatorial proof of the celebrated Euler pentagonal theorem
Non-occurrence of gap for one-dimensional non-autonomous functionals
No full textLet F(y):=∫tTL(s,y(s),y′(s))ds be a positive functional defined on the space W1,p([t, T] ; Rn) (p≥ 1) of Sobolev functions with, possibly, one or both end point conditions. It is important, especially for the applications, to be able to approximate the infimum of F with the values of F along a sequence of Lipschitz functions satisfying the same boundary condition(s). Sometimes this is not possible, i.e., the so called Lavrentiev phenomenon occurs. This is the case of the seemingly innocent Manià’s Lagrangian L(s,y,y′)=(y3-s)2(y′)6 and boundary data y(0) = 0 , y(1) = 1 ; nevertheless in this situation the phenomenon does not occur with just the end point condition y(1) = 1. The paper focuses about the different set of conditions that are needed to avoid the Lavrentiev phenomenon for problems depending on the number of end point conditions that are considered. Under minimal assumptions on the (possibly) extended value, Lagrangian, we ensure the non-occurrence of the Lavrentiev phenomenon with just one end point condition, thus extending a milestone result by Alberti and Serra Cassano to non-autonomous case. We then introduce an additional hypothesis, satisfied when the Lagrangian is bounded on bounded sets, in order to ensure the non-occurrence of the phenomenon when dealing with both end point conditions; the result gives some new light even in the autonomous case
Avoidance of the Lavrentiev gap for one-dimensional non-autonomous functionals with constraints
No full textConsider a positive functional F(y):= ∫tT L (s, y (s), y ′ (s)) ds, defined on the space of Sobolev functions W1, p ([t, T]; Rn, where p ≥ 1. This functional is minimized among functions y that may satisfy one or both endpoint conditions. The Lagrangian L is allowed to assume the value + ∞. In numerous applications minimizers may not be explicit or even may not exist. In such circumstances, it is crucial to know that the infimum of F can be approximated using a sequence of Lipschitz functions that meet the given boundary conditions. However, there are instances where this approximation is not feasible, even with polynomial Lagrangians that meet Tonelli's existence conditions: this situation is referred to as the Lavrentiev phenomenon. Some results are present in the literature if one requires the Lipschitz approximations to preserve just one endpoint constraint or when the Lagrangian is finite valued. The present paper deals with problems with two endpoint constraints and Lagrangians that are allowed to take extended values. As a byproduct, our Lipschitz approximations may preserve given state constraints. The extended-valued case is challenging since the phenomenon may occur even when the Lagrangian is constant on its effective domain, whose topology becomes relevant. Once assumed that the Lagrangian is radial convex on the rays of the last variable, our findings offer new insights, even when the Lagrangian L is real-valued, autonomous, and there are no state constraints
The vector measures whose range is strictly convex
No full textLet mu be a measure on a measure space (X, Lambda) with values in R-n and f be the density of mu with respect to its total variation. We show that the range R(mu)= {mu(E) : E is an element of Lambda} of mu is strictly convex if and only if the determinant det[f(x(1)),..,f(x(n))] is nonzero a.e, on X-n. We apply the result to a class of measeres containing those that are generated by Chebyshev systems. (C) 1999 Academic Press
Linear recurrence relations
No full textIn the following chapter we address the techniques for the resolution of some celebrated recurrence relations. We will discuss in detail the linear recurrences with constant coefficients. Our emphasis goes to the application of the theory: the proofs, though elementary, are relegated to the end of the chapter. We proceed step by step in showing first how to solve just homogeneous recurrences, then how to find a particular solution in some special cases and only finally how to obtain all the solutions to the original problem. We also consider linear recurrences with variable coefficients and the divide and conquer recurrences: here we focus on the order of magnitude of the solutions, a fact which has an impact in the analysis of algorithms. There are about 40 examples and 50 classified problems
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