The Total Weak Discrepancy of a Partially Ordered Set

We define the total weak discrepancy of a poset P as the minimum nonnegative integer k for which there exists a function f : V → Z satisfying (i) if a \prec b then f(a) + 1 ≤ f(b) and (ii) Σ|f(a) − f(b)| ≤ k, where the sum is taken over all unordered pairs {a, b} of incomparable elements. If we allow k and f to take real values, we call the minimum k the fractional total weak discrepancy of P. These concepts are related to the notions of weak and fractional weak discrepancy, where (ii) must hold not for the sum but for each individual pair of incomparable elements of P. We prove that, unlike the latter, the total weak and fractional total weak discrepancy of P are always the same, and we give a polynomial-time algorithm to find their common value. We use linear programming duality and complementary slackness to obtain this result

Implanted muon spin spectroscopy on 2-O-adamantane: a model system that mimics the liquid

The transition taking place between two metastable phases in 2-O-adamantane, namely the [Formula: see text] cubic, rotator phase and the lower temperature P21/c, Z  =  4 substitutionally disordered crystal is studied by means of muon spin rotation and relaxation techniques. Measurements carried out under zero, weak transverse and longitudinal fields reveal a temperature dependence of the relaxation parameters strikingly similar to those exhibited by structural glass[Formula: see text]liquid transitions (Bermejo et al 2004 Phys. Rev. B 70 214202; Cabrillo et al 2003 Phys. Rev. B 67 184201). The observed behaviour manifests itself as a square root singularity in the relaxation rates pointing towards some critical temperature which for amorphous systems is located some tens of degrees above that shown as the characteristic transition temperature if studied by thermodynamic means. The implications of such findings in the context of current theoretical approaches concerning the canonical liquid-glass transition are discussed.Postprint (author's final draft

Range of the Fractional Weak Discrepancy Function

In this paper we describe the range of values that can be taken by the fractional weak discrepancy of a poset and characterize semiorders in terms of these values. In [6], we defined the fractional weak discrepancy wdF (P) of a poset P=(V,≺) to be the minimum nonnegative k for which there exists a function f:V→R satisfying (1) if a≺b then f(a)+1≤f(b) and (2) if a∥b then |f(a)−f(b)|≤k. This notion builds on previous work on weak discrepancy in [3, 7, 8]. We prove here that the range of values of the function wdF is the set of rational numbers that are either at least one or equal to r [over] r+1 for some nonnegative integer r. Moreover, P is a semiorder if and only if wdF (P) \u3c 1, and the range taken over all semiorders is the set of such fractions r [over] r+1

Integral and fractional Quantum Hall Ising ferromagnets

We compare quantum Hall systems at filling factor 2 to those at filling factors 2/3 and 2/5, corresponding to the exact filling of two lowest electron or composite fermion (CF) Landau levels. The two fractional states are examples of CF liquids with spin dynamics. There is a close analogy between the ferromagnetic (spin polarization P=1) and paramagnetic (P=0) incompressible ground states that occur in all three systems in the limits of large and small Zeeman spin splitting. However, the excitation spectra are different. At filling factor 2, we find spin domains at half-polarization (P=1/2), while antiferromagnetic order seems most favorable in the CF systems. The transition between P=0 and 1, as seen when e.g. the magnetic field is tilted, is also studied by exact diagonalization in toroidal and spherical geometries. The essential role of an effective CF-CF interaction is discussed, and the experimentally observed incompresible half-polarized state is found in some models

A study of discrepancy results in partially ordered sets

In 2001, Fishburn, Tanenbaum, and Trenk published a pair of papers that introduced the notions of linear and weak discrepancy of a partially ordered set or poset. Linear discrepancy for a poset is the least k such that for any ordering of the points in the poset there is a pair of incomparable points at least distance k away in the ordering. Weak discrepancy is similar to linear discrepancy except that the distance is observed over weak labelings (i.e. two points can have the same label if they are incomparable, but order is still preserved). My thesis gives a variety of results pertaining to these properties and other forms of discrepancy in posets. The first chapter of my thesis partially answers a question of Fishburn, Tanenbaum, and Trenk that was to characterize those posets with linear discrepancy two. It makes the characterization for those posets with width two and references the paper where the full characterization is given. The second chapter introduces the notion of t-discrepancy which is similar to weak discrepancy except only the weak labelings with at most t copies of any label are considered. This chapter shows that determining a poset's t-discrepancy is NP-Complete. It also gives the t-discrepancy for the disjoint sum of chains and provides a polynomial time algorithm for determining t-discrepancy of semiorders. The third chapter presents another notion of discrepancy namely total discrepancy which minimizes the average distance between incomparable elements. This chapter proves that finding this value can be done in polynomial time unlike linear discrepancy and t-discrepancy. The final chapter answers another question of Fishburn, Tanenbaum, and Trenk that asked to characterize those posets that have equal linear and weak discrepancies. Though determining the answer of whether the weak discrepancy and linear discrepancy of a poset are equal is an NP-Complete problem, the set of minimal posets that have this property are given. At the end of the thesis I discuss two other open problems not mentioned in the previous chapters that relate to linear discrepancy. The first asks if there is a link between a poset's dimension and its linear discrepancy. The second refers to approximating linear discrepancy and possible ways to do it.Ph.D.Committee Chair: Trotter, William T.; Committee Member: Dieci, Luca; Committee Member: Duke, Richard; Committee Member: Randall, Dana; Committee Member: Tetali, Prasa

On Generalizations of Network Design Problems with Degree Bounds

Iterative rounding and relaxation have arguably become the method of choice in dealing with unconstrained and constrained network design problems. In this paper we extend the scope of the iterative relaxation method in two directions: (1) by handling more complex degree constraints in the minimum spanning tree problem (namely, laminar crossing spanning tree), and (2) by incorporating `degree bounds' in other combinatorial optimization problems such as matroid intersection and lattice polyhedra. We give new or improved approximation algorithms, hardness results, and integrality gaps for these problems.Comment: v2, 24 pages, 4 figure

Quantitative spectroscopic analysis of and distance to SN1999em

This work presents a detailed quantitative spectroscopic analysis of, and the determination of the distance to, the type II supernovae (SN) SN1999em with CMFGEN (Dessart & Hillier 2005a), based on spectrophotometric observations at eight dates up to 40 days after discovery. We use the same iron-group metal content for the ejecta, the same power-law density distribution (with exponent n~10), and a Hubble-velocity law at all times. We adopt a H/He/C/N/O abundance pattern compatible with CNO-cycle equilibrium values for a RSG/BSG progenitor, with C/O enhanced and N depleted at later times. Based on our synthetic fits to spectrophotometric observations of SN1999em, we obtain a distance of 11.5Mpc, similar to that of Baron et al. (2004) and the Cepheid distance to the galaxy host of 11.7Mpc (Leonard et al. 2003). Similarly, based on such models, the Expanding Photosphere Method (EPM) delivers a distance of 11.6Mpc, with negligible scatter between photometric bandpass sets; there is thus nothing wrong with the EPM as such. Previous determinations using the tabulated correction factors of Eastman et al. (1996) all led to 30-50% underestimates: we find that this is caused by 1) an underestimate of the correction factors compared to the only other study of the kind by Dessart & Hillier (2005b), 2) a neglect of the intrinsic >20% scatter of correction factors, and 3) the use of the EPM at late times when severe line blanketing makes the method inaccurate. The need of detailed model computations for reliable EPM distance estimates thus defeats the appeal and simplicity of the method. However, detailed fits to SN optical spectra, based on tailored models for individual SN observations, offers a promising approach to obtaining distances with 10-20% accuracy, either through the EPM or a la Baron et al. (2004).Comment: 20 pages, 13 figures, accepted for publication in A&

Doping effects on the electronic and structural properties of CoO2: An LSDA+U study

A systematic LSDA+U study of doping effects on the electronic and structural properties of single layer CoO2 is presented. Undoped CoO2 is a charge transfer insulator within LSDA+U and a metal with a high density of states (DOS) at the Fermi level within LSDA. (CoO2)$^{1.0-}$, on the other hand, is a band insulator with a gap of 2.2 eV. Systems with fractional doping are metals if no charge orderings are present. Due to the strong interaction between the doped electron and other correlated Co d electrons, the calculated electronic structure of (CoO2)$^{x-}$ depends sensitively on the doping level x. Zone center optical phonon energies are calculated under the frozen phonon approximation and are in good agreement with measured values. Softening of the $E_g$ phonon at doping x ~0.25 seems to indicate a strong electron-phonon coupling in this system. Possible intemediate spin states of Co ions, Na ordering, as well as magnetic and charge orderings in this system are also discussed.Comment: 11 pages, 12 figure