Symmetries in reversible programming: from symmetric rig groupoids to reversible programming languages

The Pi family of reversible programming languages for boolean circuits is presented as a syntax of combinators witnessing type isomorphisms of algebraic data types. In this paper, we give a denotational semantics for this language, using weak groupoids à la Homotopy Type Theory, and show how to derive an equational theory for it, presented by 2-combinators witnessing equivalences of type isomorphisms. We establish a correspondence between the syntactic groupoid of the language and a formally presented univalent subuniverse of finite types. The correspondence relates 1-combinators to 1-paths, and 2-combinators to 2-paths in the universe, which is shown to be sound and complete for both levels, forming an equivalence of groupoids. We use this to establish a Curry-Howard-Lambek correspondence between Reversible Logic, Reversible Programming Languages, and Symmetric Rig Groupoids, by showing that the syntax of Pi is presented by the free symmetric rig groupoid, given by finite sets and bijections. Using the formalisation of our results, we perform normalisation-by-evaluation, verification and synthesis of reversible logic gates, motivated by examples from quantum computing. We also show how to reason about and transfer theorems between different representations of reversible circuits.

Elements of ∞-Category Theory

The language of ∞-categories provides an insightful new way of expressing many results in higher-dimensional mathematics but can be challenging for the uninitiated. To explain what exactly an ∞-category is requires various technical models, raising the question of how they might be compared. To overcome this, a model-independent approach is desired, so that theorems proven with any model would apply to them all. This text develops the theory of ∞-categories from first principles in a model-independent fashion using the axiomatic framework of an ∞-cosmos, the universe in which ∞-categories live as objects. An ∞-cosmos is a fertile setting for the formal category theory of ∞-categories, and in this way the foundational proofs in ∞-category theory closely resemble the classical foundations of ordinary category theory. Equipped with exercises and appendices with background material, this first introduction is meant for students and researchers who have a strong foundation in classical 1-category theory.

Late time cosmology in f(R,G)- gravity with interacting fluids

Cosmological models are obtained in a $f(R)$ modified gravity with a coupled Gauss-Bonnet (GB) terms in the gravitational action. The dynamical role of the GB terms is explored with a coupled dilaton field in two different cases (I) $f(R)= R + \gamma R^2- \lambda \left( \frac{R}{3m_s^2} \right)^{\delta}$ where $\gamma$, $\lambda$ and $\delta$ are arbitrary constants and (II) $f(R)=R$ and estimate the constraints on the model parameters. In the first case we choose GB terms coupled with a free scalar field in the presence of interacting fluid and in the second case GB terms coupled with scalar field in a self interacting potential to compare the observed universe. The evolutionary scenario of the universe is obtained adopting a numerical technique as the field equations are highly non-linear. Defining a new density parameter $\Omega_{H}$, a ratio of the dark energy density to the present energy density of the non-relativistic matter, we look for a late accelerating universe. The state finder parameters $\Omega_{H}$, deceleration parameter ($q$), jerk parameter ($j$) are plotted. It is noted that a non-singular universe with oscillating cosmological parameters for a given strength of interactions is admitted in Model-I. The gravitational coupling constant $\lambda$ is playing an important role. The Lagrangian density of $f(R)$ is found to dominate over the GB terms when oscillating phase of dark energy arises. In Model-II, we do not find oscillation of the cosmological parameters as the universe evolves. In the presence of interaction the energy from radiation sector of matter cannot flow to the other two sectors of fluid. The range of values of the strengths of interaction of the fluids are estimated for a stable universe assuming the primordial gravitational wave speed equal to unity.

A thermodynamic origin for the Cohen-Kaplan-Nelson bound

The Cohen-Kaplan-Nelson bound is imposed on the grounds of logical consistency (with classical General Relativity) upon local quantum field theories. This paper puts the bound into the context of a thermodynamic principle applicable to a field with a particular equation of state in an expanding universe. This is achieved without overtly appealing to either a decreasing density of states or a minimum coupling requirement, though they might still be consistent with the results described. We do so by defining an appropriate Helmholtz free energy which when extremized relative to a key parameter (the Hubble radius L) provides a scaling formula for the entropy with the Hubble radius (an exponent r used in the text). We deduce that the CKN bound is one possible solution to this extremization problem (with r = 3/2 ), but there are others consistent with r = 2. The paper establishes that the holographic principle applied to cosmology is consistent with minimizing the free energy of the universe in the canonical ensemble, upon the assumption that the ultraviolet cutoff is a function of the causal horizon scale.

Hamiltonian formalism for nonlocal gravity models

Nonlocal gravity models are constructed to explain the current acceleration of the universe. These models are inspired by the infrared correction appearing in Einstein–Hilbert action. Here, we develop the Hamiltonian formalism of a nonlocal model by considering only terms to quadratic order in Riemann tensor, Ricci tensor and Ricci scalar. We show how to count degrees of freedom using Hamiltonian formalism including Ricci tensor and Ricci scalar terms. In this model, we have also worked out with a choice of a nonlocal action which has only two degrees of freedom equivalent to GR. Finally, we find the existence of additional constraints in Hamiltonian required to remove the ghosts in our full action. We also compare our results with that of obtained using Lagrangian formalism.

The Interstellar Medium of Dwarf Galaxies

Dwarf galaxies are by far the most numerous galaxies in the Universe, showing properties that are quite different from those of their larger and more luminous cousins. This review focuses on the physical and chemical properties of the interstellar medium of those dwarfs that are known to host significant amounts of gas and dust. The neutral and ionized gas components and the impact of the dust will be discussed, as well as first indications for the existence of active nuclei in these sources. Cosmological implications are also addressed, considering the primordial helium abundance and the similarity of local Green Pea galaxies with young, sometimes protogalactic sources in the early Universe.