Unbounded Recursion and Non-size-increasing Functions

We investigate the computing power of function algebras defined by means of unbounded recursion on notation. We introduce two function algebras which contain respectively the regressive logspace computable functions and the non-size-increasing logspace computable functions. However, such algebras are unlikely to be contained in the set of logspace computable functions because this is equivalent to L=P . Finally, we introduce a function algebra based on simultaneous recursion on notation for the non-size-increasing functions computable in polynomial time and linear space

Rothberger gaps in fragmented ideals

The~\emph{Rothberger number} $\mathfrak{b} (\mathcal{I})$ of a definable ideal $\mathcal{I}$ on $\omega$ is the least cardinal $\kappa$ such that there exists a Rothberger gap of type $(\omega,\kappa)$ in the quotient algebra $\mathcal{P} (\omega) / \mathcal{I}$. We investigate $\mathfrak{b} (\mathcal{I})$ for a subclass of the $F_\sigma$ ideals, the fragmented ideals, and prove that for some of these ideals, like the linear growth ideal, the Rothberger number is $\aleph_1$ while for others, like the polynomial growth ideal, it is above the additivity of measure. We also show that it is consistent that there are infinitely many (even continuum many) different Rothberger numbers associated with fragmented ideals.Comment: 28 page

Study of the one dimensional Holstein model using the augmented space approach

A new formalism using the ideas of the augmented space recursion (introduced by one of us) has been proposed to study the ground state properties of ordered and disordered one-dimensional Holstein model. For ordered case our method works equally well in all parametric regime and matches with the existing exact diagonalization and DMRG results. On the other hand the quenched substitutionally disordered model works in low and intermediate regime of electron phonon coupling. Effect of phononic and substitutional disorder are treated on equal footing.Comment: Accepted for publication in Physica

Template iterations with non-definable ccc forcing notions

We present a version with non-definable forcing notions of Shelah's theory of iterated forcing along a template. Our main result, as an application, is that, if $\kappa$ is a measurable cardinal and $\theta<\kappa<\mu<\lambda$ are uncountable regular cardinals, then there is a ccc poset forcing $\mathfrak{s}=\theta<\mathfrak{b}=\mu<\mathfrak{a}=\lambda$. Another application is to get models with large continuum where the groupwise-density number $\mathfrak{g}$ assumes an arbitrary regular value.Comment: To appear in the Annals of Pure and Applied Logic, 45 pages, 2 figure

Quantitative Approximation of the Probability Distribution of a Markov Process by Formal Abstractions

The goal of this work is to formally abstract a Markov process evolving in discrete time over a general state space as a finite-state Markov chain, with the objective of precisely approximating its state probability distribution in time, which allows for its approximate, faster computation by that of the Markov chain. The approach is based on formal abstractions and employs an arbitrary finite partition of the state space of the Markov process, and the computation of average transition probabilities between partition sets. The abstraction technique is formal, in that it comes with guarantees on the introduced approximation that depend on the diameters of the partitions: as such, they can be tuned at will. Further in the case of Markov processes with unbounded state spaces, a procedure for precisely truncating the state space within a compact set is provided, together with an error bound that depends on the asymptotic properties of the transition kernel of the original process. The overall abstraction algorithm, which practically hinges on piecewise constant approximations of the density functions of the Markov process, is extended to higher-order function approximations: these can lead to improved error bounds and associated lower computational requirements. The approach is practically tested to compute probabilistic invariance of the Markov process under study, and is compared to a known alternative approach from the literature.Comment: 29 pages, Journal of Logical Methods in Computer Scienc