Seno, coseno & Co. Spunti e idee per una didattica della trigonometria

Alcuni capitoli della matematica godono di una brutta reputazione e sono spesso considerati dagli studenti (e non solo) particolarmente aridi, noiosi, astrusi. Il modulo di 10 ore, inserito nell’offerta didattica disciplinare “Didattica dell’analisi e laboratorio” del Tirocinio Formativo Attivo (TFA), classe A049 (Matematica e Fisica), dell’Università degli Studi di Trieste, proponeva di prendere in considerazione, come esempio tra i tanti, quello relativo alla trigonometria.Some chapters of mathematics have a bad reputation and are often considered by students (and not only) especially barren, boring, abstruse. The 10 hour module, included in the course “Didattica dell’analisi e laboratorio” of the Tirocinio Formativo Attivo (a Teacher Training Class offered to obtain the qualification of high school teacher), Class 049 (Mathematics and Physics), at the University of Trieste, proposed taking into account, as an example among many, the one regarding trigonometry

Optimal profiles in a phase-transition model with a saturating flux

It is well known that for the Allen\u2013Cahn equation, the minimizing transition in an infinite cylinder R 7\u3c9R 7\u3c9 is one-dimensional and unique up to a translation in the first variable. We analyze in this paper the existence and symmetry of optimal profiles for transitions in a similar phase-separation model with a saturating flux. This amounts to consider transitions in the space of BV functions as we consider the area integral instead of the Dirichlet energy to penalize the creation of wild interfaces

Corrigendum to “Some notes on weakly Whyburn spaces” Topology Appl. 128 (2003) 257–262

In our paper [O] the proof of Theorem 2.7 is not correct. In that proof we constructed a space X 7 X and said that it had a subspace Z that was not weakly Whyburn. In fact X 7 X is regular and scattered, hence by Corollary 2.9 [TY] hereditarily weakly Whyburn. Thus the following problem raised in [TY] remains open: are sequential spaces hereditarily weakly Whyburn? We will now describe a Hausdorff counterexample to this problem, hence what remains open is the questions does there exist a sequential Tychonoff (or even regular) space that is not hereditarily weakly Whyburn

The Dirichlet problem for gradient dependent prescribed mean curvature equations in the Lorentz-Minkowski space

We discuss existence, multiplicity, localisation and stability properties of solutions of the Dirichlet problem associated with the gradient dependent prescribed mean curvature equation in the Lorentz-Minkowski space { 12div( 07u/ 1a1 12| 07u|\ub2)=f(x,u, 07u) in \u2126, u=0 on 02\u2126 . The obtained results display various peculiarities, which are due to the special features of the involved differential operator and have no counterpart for elliptic problems driven by other quasilinear differential operators. This research is also motivated by some recent achievements in the study of prescribed mean curvature graphs in certain Friedmann\u2013Lema\ueetre\u2013Robertson\u2013Walker, as well as Schwarzschild\u2013Reissner\u2013Nordstr\uf6m, spacetimes

Positive Solutions of the Dirichlet Problem for the One-dimensional Minkowski-Curvature Equation

We discuss existence and multiplicity of positive solutions of the Dirichlet problem for the quasilinear ordinary differential equation-(u' / root 1 - u'(2))' = f(t, u). Depending on the behaviour of f = f(t, s) near s = 0, we prove the existence of either one, or two, or three, or infinitely many positive solutions. In general, the positivity of f is not required. All results are obtained by reduction to an equivalent non-singular problem to which variational or topological methods apply in a classical fashion

On the Dirichlet problem associated with bounded perturbations of positively-(p, q)- homogeneous Hamiltonian systems

The existence of solutions for the Dirichlet problem associated to bounded perturbations of positively-(p; q)-homogeneous Hamiltonian systems is considered both in nonresonant and resonant situations. In order to deal with the resonant case, the existence of a couple of lower and upper solutions is assumed. Both the well-ordered and the non-well-ordered cases are analysed. The proof is based on phase-plane analysis and topological degree theory

Qualitative analysis of a curvature equation modelling MEMS with vertical loads

We investigate existence, multiplicity and qualitative properties of the solutions of the Dirichlet problem for a singularly perturbed prescribed mean curvature equation, which appears in the theory of micro-electro-mechanical systems (MEMS) when the effects of capillarity and vertical forces are taken into account

A prescribed anisotropic mean curvature equation modeling the corneal shape: a paradigm of nonlinear analysis

In this paper we survey, complete and refine some recent results concerning the Dirichlet problem for the prescribed anisotropic mean curvature equation egin{equation*} { m -div}left({ abla u}/{sqrt{1 + | abla u|^2}} ight) = -au + {b}/{sqrt{1 + | abla u|^2}}, end{equation*} in a bounded Lipschitz domain $Omega subset RR^N$, with $a,b>0$ parameters. This equation appears in the description of the geometry of the human cornea, as well as in the modeling theory of capillarity phenomena for compressible fluids. Here we show how various techniques of nonlinear functional analysis can successfully be applied to derive a complete picture of the solvability patterns of the problem

Existence, regularity and boundary behaviour of bounded variation solutions of a one-dimensional capillarity equation

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