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This study investigates the structure of affine systems through an exploration of time-scaling properties induced by control inputs. By leveraging the Frobenius theorem and the principles of involutive distributions, we uncover that system dynamics can be effectively decomposed into time-scaled flows of commuting vector fields. This decomposition enhances the geometric understanding of affine systems and provides a theoretical perspective that may inform future developments in control design and performance evaluation. The proposed framework is general and flexible, and is expected to provide a foundation for future theoretical advances in nonlinear control theory, which may potentially be extended to practical applications such as plant design or performance optimization.

For any given integer rr a closed manifold is constructed which has a smooth free action of the rr-torus, and hence has rank at least rr, but for which -1 is not a multiple root of the Poincaré polynomial.

To a torus action on a complex vector space, Gelfand, Kapranov and Zelevinsky introduce a system of differential equations, which are now called the GKZ hypergeometric system. Its solutions are GKZ hypergeometric functions. We study the p -adic counterpart of the GKZ hypergeometric system. The p -adic GKZ hypergeometric complex is a twisted relative de Rham complex of overconvergent differential forms with logarithmic poles. It is an over-holonomic object in the derived category of arithmetic D -modules with Frobenius structures. Traces of Frobenius on fibers at Techmüller points of the GKZ hypergeometric complex define the hypergeometric function over the finite field introduced by Gelfand and Graev. Over the non-degenerate locus, the GKZ hypergeometric complex defines an overconvergent F -isocrystal. It is the crystalline companion of the ℓ -adic GKZ hypergeometric sheaf that we constructed before. Our method is a combination of Dwork’s theory and the theory of arithmetic D -modules of Berthelot.

We show how entanglement shared between encoder and decoder can simplify the theory of quantum error correction. The entanglement-assisted quantum codes we describe do not require the dual-containing constraint necessary for standard quantum error--correcting codes, thus allowing us to \"quantize\" all of classical linear coding theory. In particular, efficient modern classical codes that attain the Shannon capacity can be made into entanglement-assisted quantum codes attaining the hashing bound (closely related to the quantum capacity). For systems without large amounts of shared entanglement, these codes can also be used as catalytic codes, in which a small amount of initial entanglement enables quantum communication.

We study quadratic integrability of systems with velocity dependent potentials in three-dimensional Euclidean space. Unlike in the case with only scalar potential, quadratic integrability with velocity dependent potentials does not imply separability in the configuration space. The leading order terms in the pairs of commuting integrals can either generalize or have no relation to the forms leading to separation in the absence of a vector potential. We call such pairs of integrals generalized, to distinguish them from the standard ones, which would correspond to separation. Here we focus on three cases of generalized non-subgroup type integrals, namely elliptic cylindrical, prolate/oblate spheroidal and circular parabolic integrals, together with one case not related to any coordinate system. We find two new integrable systems, non-separable in the configuration space, both with generalized elliptic cylindrical integrals. In the other cases, all systems found were already known and possess standard pairs of integrals. In the limit of vanishing vector potential, both systems reduce to free motion and therefore separate in every orthogonal coordinate system. Graphical abstract

The aim of this article is to demonstrate how the vector field method of Klainerman can be adapted to the study of transport equations. After an illustration of the method for the free transport operator, we apply the vector field method to the Vlasov-Poisson system in dimension 3 or greater. The main results are optimal decay estimates and the propagation of global bounds for commuted fields associated with the conservation laws of the free transport operators, under some smallness assumption. Similar decay estimates had been obtained previously by Hwang, Rendall and Velázquez using the method of characteristics, but the results presented here are the first to contain the global bounds for commuted fields and the optimal spatial decay estimates. In dimension 4 or greater, it suffices to use the standard vector fields commuting with the free transport operator while in dimension 3, the rate of decay is such that these vector fields would generate a logarithmic loss. Instead, we construct modified vector fields where the modification depends on the solution itself. The methods of this paper, being based on commutation vector fields and conservation laws, are applicable in principle to a wide range of systems, including the Einstein-Vlasov and the Vlasov-Nordström system.

We study a popular kinetic model introduced by Saintillan and Shelley for the dynamics of suspensions of active elongated particles where the particles are described by a distribution in space and orientation. The uniform distribution of particles is the stationary state of incoherence which is known to exhibit a phase transition. We perform an extensive study of the linearised evolution around the incoherent state. We show (i) in the non-diffusive regime corresponding to spectral (neutral) stability that the suspensions experience a mixing phenomenon similar to Landau damping and we provide optimal pointwise in time decay rates in weak topology. Further, we show (ii) in the case of small rotational diffusion ν that the mixing estimates persist up to time scale ν - 1 / 2 until the exponential decay at enhanced dissipation rate ν 1 / 2 takes over. The interesting feature is that the usual velocity variable of kinetic models is replaced by an orientation variable on the sphere. The associated orientation mixing leads to limited algebraic decay for macroscopic quantities. For the proof, we start with a general pointwise decay result for Volterra equations that may be of independent interest. While, in the non-diffusive case, explicit formulas on the sphere allow to conclude the desired decay, much more work is required in the diffusive case: here we prove mixing estimates for the advection-diffusion equation on the sphere by combining an optimized hypocoercive approach with the vector field method. One main point in this context is to identify good commuting vector fields for the advection-diffusion operator on the sphere. Our results in this direction may be useful to other models in collective dynamics, where an orientation variable is involved.

We obtain intertwining dilation theorems for non-commutative regular domains f and non-commutative varieties J in B( )n, which generalize Sarason and Szőkefalvi-Nagy and Foiaş's commutant lifting theorem for commuting contractions. We present several applications including a new proof for the commutant lifting theorem for pure elements in the domain f (respectively, variety J ) as well as a Schur-type representation for the unit ball of the Hardy algebra associated with the variety J. We provide Andô-type dilations and inequalities for bi-domains f ×c g consisting of all pairs (X,Y ) of tuples X := (X1,…, Xn1) ∊ f and Y := (Y1,…, Yn2) ∊ g that commute, i.e. each entry of X commutes with each entry of Y . The results are new, even when n1 = n2 = 1. In this particular case, we obtain extensions of Andô's results and Agler and McCarthy's inequality for commuting contractions to larger classes of commuting operators. All the results are extended to bi-varieties J1 ×c J2, where J1 and J2 are non-commutative varieties generated by weak-operator-topology-closed two-sided ideals in non-commutative Hardy algebras. The commutative case and the matrix case when n1 = n2 = 1 are also discussed.

We define a Lie bracket on a certain set of local vector fields along a null curve in a 4-dimensional semi-Riemannian space form. This Lie bracket will be employed to study integrability properties of evolution equations for null curves in a pseudo-Euclidean space. In particular, a geometric recursion operator generating infinitely many local symmetries for the null localized induction equation is provided.