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We say that a finite group G is DRR-detecting if, for every subset S of G, either the Cayley digraph Cay(G,S) is a digraphical regular representation (that is, its automorphism group acts regularly on its vertex set) or there is a nontrivial group automorphism phi of G such that phi(S) = S. We show that every nilpotent DRR-detecting group is a p-group, but that the wreath product of two cyclic groups of order p is not DRR-detecting, for every odd prime p. We also show that if G and H are nontrivial groups that admit a digraphical regular representation and either gcd(|G|,|H|) = 1, or H is not DRR-detecting, then the direct product G x H is not DRR-detecting. Some of these results also have analogues for graphical regular representations.
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In this paper we extend the classical notion of digraphical and graphical regular representation of a group and we classify, by means of an explicit description, the finite groups satisfying this generalization. A graph or digraph is called regular if each vertex has the same valency, or, the same out-valency and the same in-valency, respectively. An m-(di)graphical regular representation (respectively, m-GRR and m-DRR, for short) of a group G is a regular (di)graph whose automorphism group is isomorphic to G and acts semiregularly on the vertex set with m orbits. When m=1, this definition agrees with the classical notion of GRR and DRR. Finite groups admitting a 1-DRR were classified by Babai in 1980, and the analogue classification of finite groups admitting a 1-GRR was completed by Godsil in 1981. Pivoting on these two results in this paper we classify finite groups admitting an m-GRR or an m-DRR, for arbitrary positive integers m. For instance, we prove that every non-identity finite group admits an m-GRR, for every m>4.
The problem of photoemission from a quasi-1D material is studied. We identify two issues that play a key role in the detection of gapless Tomonaga-Luttinger liquid (TLL) phase. Firstly, we show how a disorder -- backward scattering as well as forward scattering component, is able to significantly obscure the TLL states, hence the initial state of ARPES. Secondly, we investigate the photo-electron propagation towards a samples surface. We focus on the scattering path operator contribution to the final state of ARPES. We show that, in the particular conditions set by the 1D states, one can derive exact analytic solution for this intermediate stage of ARPES. The solution shows that for particular energies of incoming photons the intensity of photo-current may be substantially reduced. Finally, we put together the two aspects (the disorder and the scattering path operator) to show the full, disruptive force of any inhomogeneities on the ARPES amplitude.
Theoretical predictions suggest that the distribution of planets in very young stars could be very different to that typically observed in Gyr old systems that are the current focus of radial velocity surveys. However, the detection of planets around young stars is hampered by the increased stellar activity associated with young stars, the signatures of which can bias the detection of planets. In this paper we place realistic limitations on the possibilities for detecting planets around young active G and K dwarfs. The models of stellar activity based on tomographic imaging of the G dwarf HD 141943 and the K1 dwarf AB Dor and also include contributions from plage and many small random starspots. Our results show that the increased stellar activity levels present on young Solar-type stars strongly impacts the detection of Earth-mass and Jupiter mass planets and that the degree of activity jitter is directly correlated with stellar vsinis. We also show that for G and K dwarfs, the distribution of activity in individual stars is more important than the differences in induced radial velocities as a function of spectral type. We conclude that Jupiter mass planets can be detected close-in around fast-rotating young active stars, Neptune-mass planets around moderate rotators and that Super-Earths are only detectable around very slowly rotating stars. The effects of an increase in stellar activity jitter by observing younger stars can be compensated for by extending the observational base-line to at least 100 epochs.
The axion, originated from the Peccei-Quinn mechanism proposed to solve the strong-CP problem, is a well motivated and popular dark matter candidate. Experimental searches for this hypothetical particle are starting to reach theoretically interesting sensitivity levels. However, only a small fraction of the allowed parameter space has been explored so far, mostly in the $mu$eV (GHz) region, relying on large volume solenoid magnetic fields and microwave resonators with signals read out by quantum noise limited amplifiers. There have been intensive experimental efforts to widen the search range by devising various techniques as well as to enhance sensitivities by implementing advanced technologies. The developments and improvements in these orthogonal approaches will enable us to explore most of the parameter space of the axion and axion-like particles within the next five to ten years. We review the experimental aspects of axion physics and discuss the past, present and future of the individual search programs.
In this paper, we classify regular polytopes with automorphism groups of order $2^n$ and Schlafli types ${4, 2^{n-3}}, {4, 2^{n-4}}$ and ${4, 2^{n-5}}$ for $n geq 10$, therefore giving a partial answer to a problem proposed by Schulte and Weiss in [Problems on polytopes, their groups, and realizations, Periodica Math. Hungarica 53(2006) 231-255].
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