Explicit solutions to constrained linear MPC problems can be obtained by solving multi-parametric quadratic programs (mp-QP) where the parameters are the components of the state vector. We study the properties of the polyhedral partition of the state-space induced by the multiparametric piecewise linear solution and propose a new mp-QP solver. Compared to existing algorithms, our approach adopt...

We establish pointwise characterizations of functions in the HardySobolev spaces H within the range p ∈ (n/(n + 1), 1]. In particular, a locally integrable function u belongs to H(R) if and only if u ∈ L(R) and it satisfies the Hajlasz type condition |u(x)− u(y)| ≤ |x − y|(h(x) + h(y)), x, y ∈ R \ E, where E is a set of measure zero and h ∈ L(R). We also investigate HardySobolev spaces on subdo...

Given two Banach function spaces we study the pointwise product space E · F , especially for the case that the pointwise product of their unit balls is again convex. We then give conditions on when the pointwise product E ·M(E,F ) = F , where M(E,F ) denotes the space of multiplication operators from E into F .

Fix an odd prime p, an algebraic closure Qp, and an isomorphism C ∼ → Qp. Fix an integer N ≥ 1 prime to p, and let T be the polynomial algebra over Z generated by the operators T!, ! ! Np, Up and 〈d〉 , d ∈ (Z/NZ). Set W = Spf(Zp[[Zp ]]), and let W = W rig be the rigid analytic space of characters of Zp together with its universal character χW : Z × p → O(W ) ; we embed Z in W (Qp) by mapping k ...

The ideals of the ring Zp are {0} and pZp, n ≥ 0. From this it follows that Zp is a discrete valuation ring, a principal ideal domain with exactly one maximal ideal, namely pZp; Zp is the valuation ring of Qp with the valuation vp. For n ≥ 1, Zp/pZp is isomorphic as a ring with Z/pZ. |x|p = p−vp(x), dp(x, y) = |x− y|p. With the topology induced by the metric dp, Qp is a locally compact abelian ...

Let G be a locally Qp-analytic group and K a finite extension of Qp with residue field k. Adapting a strategy of B. Mazur (cf. [Maz89]) we use deformation theory to study the possible liftings of a given smooth G-representation ρ over k to unitary G-Banach space representations over K. The main result proves the existence of a universal deformation space in case ρ admits only scalar endomorphis...

We present preliminary numerical findings concerning measure synchronization in a pair of coupled Nonlinear Hamiltonian Systems (NLHS) derived from a Nonlinear Schrödinger Equation (NLSE). The dynamics of the two coupled NLHS were found to exhibit a transition to coherent invariant measure; their orbits sharing the same phase space as the coupling strength is increased. Transitions from quasipe...

Exercise 1 (Maximal compact subgroups of G). A lattice in Qp is a finitelygenerated Zp-submodule of Qp that generates Qp as vector space. In particular, it’s free of rank n. Note that G acts transitively on the set of lattices in Qp . (i) Show that K = StabG(Zp ). (ii) Suppose that K ′ is a compact subgroup of G. Show that K ′ stabilises a lattice. (Hint: show that the K ′-orbit of Zp is finite...

For a fairly general reductive group G/Qp , we explicitly compute the space of locally algebraic vectors in the Breuil-Herzig construction Π(ρ)ord, for a potentially semistable Borel-valued representation ρ of Gal(Q̄p/Qp). The point being we deal with the whole representation, not just its socle – and we go beyond GLn(Qp). In the case of GL2(Qp), this relation is one of the key properties of the...