Conjecture on the Physical Implications of the Scale Anomaly

Murray Gell-Mann, after co-inventing QCD, recognized the interplay of the scale anomaly, the renormalization group, and the origin of the strong scale, Lambda_{QCD}. I tell a story, then elaborate this concept, and for the sake of discussion, propose a conjecture that the physical world is scale invariant in the classical, \hbar -> 0, limit. This principle has implications for the dimensionality of space-time, the cosmological constant, the weak scale, and Planck scale.Comment: Invited talk delivered at the Santa Fe Institute on the Occasion of the Celebration of the 75th Birthday of Murray Gell-Mann. July 23, 200

Covering Dimension of C*-Algebras and 2-Coloured Classification

Research partially supported by EPSRC (grant no. I019227/1-2), by NSF (grant no. DMS-1201385), by JSPS (the Grant-in-Aid for Research Activity Start-up 25887031), by NSERC (PDF, held by AT), by an Alexander von Humboldt foundation fellowship (held by SW) and by the DFG (SFB 878).Postprin

The State of Pakistan’s Dairy Sector: An Assessment

While there is a plethora of research documenting a multitude of dimensions of the crop sector of Pakistan, the virtual absence of meaningful economic analysis of the dairy economy is surprising. No serious attempt has been made in the past to clarify the microlevel potential of this sector to impact rural economy. This paper is a pioneering attempt to provide an objective assessment of the state of Pakistan’s dairy and to point out areas of further research. The paper analyses some core issues, highlights the potential of this sector, and recommends the measures to be adopted towards such a goal.Dairy Industry, Pakistan

The State of Pakistan’s Dairy Sector : An Assessment

While there is a plethora of research documenting a multitude of dimensions of the crop sector of Pakistan, there is virtual absence of meaningful economic analysis of the dairy economy that is surprising. No serious attempt has been made in the past to clarify the micro-level potential of this sector in creating an impact on rural economy. This paper is a pioneering attempt to provide an objective assessment of the state of Pakistans dairy and to point out areas of further research. The paper analyzes some core issues and highlights the potentials, and recommends measures that could be adopted.dairy, rural economy, research areas

Singular inflation

We prove that a homogeneous and isotropic universe containing a scalar field with a power-law potential, V(ϕ)=Aϕ^n, with 00 always develops a finite-time singularity at which the Hubble rate and its first derivative are finite, but its second derivative diverges. These are the first examples of cosmological models with realistic matter sources that possess weak singularities of “sudden” type. We also show that a large class of models with even weaker singularities exists for noninteger n>1. More precisely, if k<n<k+1 where k is a positive integer then the first divergence of the Hubble rate occurs with its (k+2)th derivative. At early times these models behave like standard large-field inflation models but they encounter a singular end state when inflation ends. We term this singular inflation.A.A.H.G. and J.D.B. are supported by the STFC.This is the author accepted manuscript. The final version is available from APS via http://dx.doi.org/10.1103/PhysRevD.91.08351

Arbeitsgemeinschaft: Ergodic Theory and Combinatorial Number Theory

The aim of this Arbeitsgemeinschaft was to introduce young researchers with various backgrounds to the multifaceted and mutually perpetuating connections between ergodic theory, topological dynamics, combinatorics, and number theory

The complexity of energy eigenstates as a mechanism for equilibration

Understanding the mechanisms responsible for the equilibration of isolated quantum many-body systems is a long-standing open problem. In this work we obtain a statistical relationship between the equilibration properties of Hamiltonians and the complexity of their eigenvectors, provided that a conjecture about the incompressibility of quantum circuits holds. We quantify the complexity by the size of the smallest quantum circuit mapping the local basis onto the energy eigenbasis. Specifically, we consider the set of all Hamiltonians having complexity C, and show that almost all such Hamiltonians equilibrate if C is super-quadratic with the system size, which includes the fully random Hamiltonian case in the limit C to infinity, and do not equilibrate if C is sub-linear. We also provide a simple formula for the equilibration time-scale in terms of the Fourier transform of the level density. Our results are statistical and, therefore, do not apply to specific Hamiltonians. Yet, they establish a fundamental link between equilibration and complexity theory.Comment: improved version (6 pages + appendix