Slow Continued Fractions and Permutative Representations of

Representations of the Cuntz algebra ${\mathcal{O}}_{N}$ are constructed from interval dynamical systems associated with slow continued fraction algorithms introduced by Giovanni Panti. Their irreducible decomposition formulas are characterized by using the modular group action on real numbers, as a generalization of results by Kawamura, Hayashi, and Lascu. Furthermore, a certain symmetry of such an interval dynamical system is interpreted as a covariant representation of the $C^{\ast }$-dynamical system of the “flip-flop” automorphism of ${\mathcal{O}}_{2}$.

C*-algebras generated by isometries and true representations

Let (G; P) be a quasi-lattice ordered group. In this paper we present a modied proof of Laca and Raeburn's theorem about the covariant isometric representations of amenable quasi-lattice ordered groups [7, Theorem 3.7], by following a two stage strategy. First, we construct a universal covariant representation for a given quasi-lattice ordered group (G; P) and show that it is unique. The construction of this object is new; we have not followed either Nica's approach in [10] or Laca and Raeburn's approach in [7], although all three objects are essentially the same. Our approach is a very natural one and avoids some of the intricacies of the other approaches. Then we show if (G; P) is amenable, true representations of (G; P) generate C-algebras which are canonically isomorphic to the universal object.

Projection-operator methods for classical transport in magnetized plasmas. Part 1. Linear response, the Braginskii equations and fluctuating hydrodynamics

An introduction to the use of projection-operator methods for the derivation of classical fluid transport equations for weakly coupled, magnetised, multispecies plasmas is given. In the present work, linear response (small perturbations from an absolute Maxwellian) is addressed. In the Schrödinger representation, projection onto the hydrodynamic subspace leads to the conventional linearized Braginskii fluid equations when one restricts attention to fluxes of first order in the gradients, while the orthogonal projection leads to an alternative derivation of the Braginskii correction equations for the non-hydrodynamic part of the one-particle distribution function. The projection-operator approach provides an appealingly intuitive way of discussing the derivation of transport equations and interpreting the significance of the various parts of the perturbed distribution function; it is also technically more concise. A special case of the Weinhold metric is used to provide a covariant representation of the formalism; this allows a succinct demonstration of the Onsager symmetries for classical transport. The Heisenberg representation is used to derive a generalized Langevin system whose mean recovers the linearized Braginskii equations but that also includes fluctuating forces. Transport coefficients are simply related to the two-time correlation functions of those forces, and physical pictures of the various transport processes are naturally couched in terms of them. A number of appendices review the traditional Chapman–Enskog procedure; record some properties of the linearized Landau collision operator; discuss the covariant representation of the hydrodynamic projection; provide an example of the calculation of some transport effects; describe the decomposition of the stress tensor for magnetised plasma; introduce the linear eigenmodes of the Braginskii equations; and, with the aid of several examples, mention some caveats for the use of projection operators.

An Isbell duality theorem for type refinement systems

Any refinement system (= functor) has a fully faithful representation in the refinement system of presheaves, by interpreting types as relative slice categories, and refinement types as presheaves over those categories. Motivated by an analogy between side effects in programming andcontext effectsin linear logic, we study logical aspects of this ‘positive’ (covariant) representation, as well as of an associated ‘negative’ (contravariant) representation. We establish several preservation properties for these representations, including a generalization of Day's embedding theorem for monoidal closed categories. Then, we establish that the positive and negative representations satisfy an Isbell-style duality. As corollaries, we derive two different formulas for the positive representation of a pushforward (inspired by the classical negative translations of proof theory), which express it either as the dual of a pullback of a dual or as the double dual of a pushforward. Besides explaining how these constructions on refinement systems generalize familiar category-theoretic ones (by viewing categories as special refinement systems), our main running examples involve representations of Hoare logic and linear sequent calculus.

Geometric constrained variational calculus. III: The second variation (Part II)

The problem of minimality for constrained variational calculus is analyzed within the class of piecewise differentiable extremaloids. A fully covariant representation of the second variation of the action functional based on a family of local gauge transformations of the original Lagrangian is proposed. The necessity of pursuing a local adaptation process, rather than the global one described in [1] is seen to depend on the value of certain scalar attributes of the extremaloid, here called the corners’ strengths. On this basis, both the necessary and the sufficient conditions for minimality are worked out. In the discussion, a crucial role is played by an analysis of the prolongability of the Jacobi fields across the corners. Eventually, in the appendix, an alternative approach to the concept of strength of a corner, more closely related to Pontryagin’s maximum principle, is presented.

Geometric constrained variational calculus. II: The second variation (Part I)

Within the geometrical framework developed in [Geometric constrained variational calculus. I: Piecewise smooth extremals, Int. J. Geom. Methods Mod. Phys. 12 (2015) 1550061], the problem of minimality for constrained calculus of variations is analyzed among the class of differentiable curves. A fully covariant representation of the second variation of the action functional, based on a suitable gauge transformation of the Lagrangian, is explicitly worked out. Both necessary and sufficient conditions for minimality are proved, and reinterpreted in terms of Jacobi fields.

Isometry generators in momentum representation of the Dirac theory on the de Sitter spacetime

In this paper, it is shown that the covariant representation (CR) transforming the Dirac field under de Sitter isometries is equivalent to a direct sum of two unitary irreducible representations (UIRs) of the Sp(2, 2) group transforming alike the particle and antiparticle field operators in momentum representation. Their basis generators and Casimir operators are written down for the first time finding that these representations are equivalent to an UIR from the principal series whose canonical labels are determined by the fermion mass and spin. The properties of the conserved observables (i.e. one-particle operators) associated to the de Sitter isometries via Noether theorem and of the corresponding Pauli–Lubanski type operator are also pointed out.