Base Size Sets and Determining Sets

Bridging the work of Cameron, Harary, and others, we examine the base size set B(G) and determining set D(G) of several families of groups. The base size set is the set of base sizes of all faithful actions of the group G on finite sets. The determining set is the subset of B(G) obtained by restricting the actions of G to automorphism groups of finite graphs. We show that for finite abelian groups, B(G)=D(G)={1,2,...,k} where k is the number of elementary divisors of G. We then characterize B(G) and D(G) for dihedral groups of the form D_{p^k} and D_{2p^k}. Finally, we prove B(G) is not equal to D(G) for dihedral groups of the form D_{pq} where p and q are distinct odd primes.Comment: 10 pages, 1 figur

Minimal sets determining universal and phase-covariant quantum cloning

We study the minimal input sets which can determine completely the universal and the phase-covariant quantum cloning machines. We find that the universal quantum cloning machine, which can copy arbitrary input qubit equally well, however can be determined completely by only four input states located at the four vertices of a tetrahedron. The phase-covariant quantum cloning machine, which can copy all qubits located on the equator of the Bloch sphere, can be determined by three equatorial qubits with equal angular distance. These results sharpen further the well-known results that BB84 states and six-states used in quantum cryptography can determine completely the phase-covariant and universal quantum cloning machines. This concludes the study of the power of universal and phase-covariant quantum cloning, i.e., from minimal input sets necessarily to full input sets by definition. This can simplify dramatically the testing of whether the quantum clone machines are successful or not, we only need to check that the minimal input sets can be cloned optimally.Comment: 7 pages, 4 figure

Determining efficient temperature sets for the simulated tempering method

In statistical physics, the efficiency of tempering approaches strongly depends on ingredients such as the number of replicas $R$, reliable determination of weight factors and the set of used temperatures, ${\mathcal T}_R = \{T_1, T_2, \ldots, T_R\}$. For the simulated tempering (SP) in particular -- useful due to its generality and conceptual simplicity -- the latter aspect (closely related to the actual $R$) may be a key issue in problems displaying metastability and trapping in certain regions of the phase space. To determine ${\mathcal T}_R$'s leading to accurate thermodynamics estimates and still trying to minimize the simulation computational time, here it is considered a fixed exchange frequency scheme for the ST. From the temperature of interest $T_1$, successive $T$'s are chosen so that the exchange frequency between any adjacent pair $T_r$ and $T_{r+1}$ has a same value $f$. By varying the $f$'s and analyzing the ${\mathcal T}_R$'s through relatively inexpensive tests (e.g., time decay toward the steady regime), an optimal situation in which the simulations visit much faster and more uniformly the relevant portions of the phase space is determined. As illustrations, the proposal is applied to three lattice models, BEG, Bell-Lavis, and Potts, in the hard case of extreme first-order phase transitions, always giving very good results, even for $R=3$. Also, comparisons with other protocols (constant entropy and arithmetic progression) to choose the set ${\mathcal T}_R$ are undertaken. The fixed exchange frequency method is found to be consistently superior, specially for small $R$'s. Finally, distinct instances where the prescription could be helpful (in second-order transitions and for the parallel tempering approach) are briefly discussed.Comment: 10 pages, 14 figure

Resolving sets for breaking symmetries of graphs

This paper deals with the maximum value of the difference between the determining number and the metric dimension of a graph as a function of its order. Our technique requires to use locating-dominating sets, and perform an independent study on other functions related to these sets. Thus, we obtain lower and upper bounds on all these functions by means of very diverse tools. Among them are some adequate constructions of graphs, a variant of a classical result in graph domination and a polynomial time algorithm that produces both distinguishing sets and determining sets. Further, we consider specific families of graphs where the restrictions of these functions can be computed. To this end, we utilize two well-known objects in graph theory: $k$-dominating sets and matchings.Comment: 24 pages, 12 figure

Determining the quality of mathematical software using reference data sets

This paper describes a methodology for evaluating the numerical accuracy of software that performs mathematical calculations. The authors explain how this methodology extends the concept of metrological traceability, which is fundamental to measurement, to include software quality. Overviews of two European Union-funded projects are also presented. The first project developed an infrastructure to allow software to be verified by testing, via the internet, using reference data sets. The primary focus of the project was software used within systems that make physical measurements. The second project, currently underway, explores using this infrastructure to verify mathematical software used within general scientific and engineering disciplines. Publications on using reference data sets for the verification of mathematical software are usually intended for a readership specialising in measurement science or mathematics. This paper is aimed at a more general readership, in particular software quality specialists and computer scientists. Further engagement with experts in these disciplines will be helpful to the continued development of this application of software quality