In this paper, we will report on the developments carried out in Isabelle/HOL, ACL2 and Coq/SSReflect on Computational Algebraic Topology, in the frame of the ForMath European project. The aim is to illustrate, trough concrete examples, the role of formalization technologies in Computational Mathematics in general.

In the past decade there has been a flurry of activity at the intersection of spectral theory and symplectic geometry. In this paper we review recent results on semiclassical spectral theory for commuting Berezin-Toeplitz and $\hbar$-pseudodifferential operators. The paper emphasizes the interplay between spectral theory of operators (quantum theory) and symplectic geometry of Hamiltonians (classical theory), with an eye towards recent developments on the geometry of finite dimensional integrable systems.

We study a generalized version of Yao's Millionaires problem. Namely, $n$ millionaires want to find out the $k$ richest persons among them without revealing their amounts of wealth. We propose an efficient solution that takes $2n$ secure comparison operations on average to find out the correct result. Our solution protects the privacy of every participant.

We give a description, in terms of ``pseudo-stable increasing process'', of the Lamperti process associated with the inverse local time of a radial Ornstein-Uhlenbeck process. Following Bertoin-Yor, we also express, in two particular cases, the law of the perpetuity associated with this inverse local time.

We prove the Haagerup property for an infinite discrete group constructed using surgery on a Euclidean Tits building of type $\tilde A_2$.

We study a mathematical model for a quasistatic process of contact with normal compliance and friction when the wear of the contact surface due to friction is taken into account. The material is electro-viscoelastic with long memory. We establish a variational formulation for the model and prove the existence and uniqueness of the weak solution. The proof is based on classical results for elliptic variational inequalities and fixed point arguments.

Attaching to a compact ball $\overline{\mathbb{B}_{r}}$ in the quaternion field $\mathbb{H}$ and to analytic functions in Weierstrass sense (slice regular functions on $\overline{\mathbb{B}_{r}}$) some convolution operators, the exact orders of approximation in $\overline{\mathbb{B}_{r}}$ for these operators are obtained. The results in this paper extend to quaternionic variables those in the case of approximation of analytic functions of a complex variable in disks by convolution operators of a complex variable, extensively studied in the Chapter 3 of the book [5]. More in general, the results extend also to the setting of analytic functions of paravector variable with coefficients in a Clifford algebra.

This paper considers the existence of solutions for two boundary value problems of fractional $p$-Laplacian equation at resonance. Under certain nonlinear growth condition of the nonlinearity, two new existence results are obtained by using the coincidence degree theory. As an application, an example to illustrate our results is given.

We consider the fourth-order problem $$ \left\{ \begin{array}{l} \epsilon^4\Delta^2u + V(x)u = f(u) + \gamma|u|^{2_{**}-2}u \mbox{in $\mathbb{R}^N$}\\ u\in H^2(\mathbb{R}^N), \end{array} \right. $$ where $\epsilon > 0$, $N\geq 5$, $V$ is a positive continuous potential, $f$ is a function with subcritical growth and $\gamma \in \{0,1\}$. We relate the number of solutions with the topology of the set where $V$ attain its minimum values. We consider the subcritical case $\gamma=0$ and the critical case $\gamma=1$. In the proofs we apply Ljusternik-Schnirelmann theory.

We consider the nonautonomous differential equation of second order $x''- a(t)x+b(t) x^2+c(t)x^{3}=0$, where $a(t),b(t),c(t)$ are $T$-periodic functions. This is a biomathematical model of an aneurysm in the circle of Willis. We prove the existence of at least two $T$-periodic solution for this equation, using coincidence degree theories.

In this paper, for the fourth-order parabolic equation modeling epitaxial thin film growth, the optimal control under boundary condition is given, and the existence of optimal solution to the equation is proved.

In this paper we consider a cyclic $\varphi_A$-contraction mapping defined on a partially ordered orbitally complete metric space and prove some fixed point and best proximity point theorems. We also discuss some relationship between points of coincidence and common best proximal points. It is shown that, under certain condition, a point of coincidence, a common best proximal point and a common fixed point coincide.