On Polynomial Representations of the DP Color Function: Theta Graphs and Their Generalizations

DP-coloring (also called correspondence coloring) is a generalization of list coloring that has been widely studied in recent years after its introduction by Dvo\v{r}\'{a}k and Postle in 2015. As the analogue of the chromatic polynomial $P(G,m)$, the DP color function of a graph $G$, denoted $P_{DP}(G,m)$, counts the minimum number of DP-colorings over all possible $m$-fold covers. It is known that, unlike the list color function $P_{\ell}(G,m)$, for any $g \geq 3$ there exists a graph $G$ with girth $g$ such that $P_{DP}(G,m) < P(G,m)$ when $m$ is sufficiently large. Thus, two fundamental open questions regarding the DP color function are: (i) for which $G$ does there exist an $N \in \mathbb{N}$ such that $P_{DP}(G,m) = P(G,m)$ whenever $m \geq N$, (ii) Given a graph $G$ does there always exist an $N \in \mathbb{N}$ and a polynomial $p(m)$ such that $P_{DP}(G,m) = p(m)$ whenever $m \geq N$? In this paper we give exact formulas for the DP color function of a Theta graph based on the parity of its path lengths. This gives an explicit answer, including the formulas for the polynomials that are not the chromatic polynomial, to both the questions above for Theta graphs. We extend this result to Generalized Theta graphs by characterizing the exact parity condition that ensures the DP color function eventually equals the chromatic polynomial. To answer the second question for Generalized Theta graphs, we confirm it for the larger class of graphs with a feedback vertex set of size one.Comment: 21 pages. arXiv admin note: text overlap with arXiv:2009.08242, arXiv:1904.0769

Development and validation of an objective swine feet and leg conformation procedure using digital imagery

The objectives of this thesis were i) to develop and assess repeatability of an objective method for evaluating feet and leg conformation in high parity sows and ii) to validate and examine measurements using the developed objective method with a group of pedigreed gilts from age at selection through their first parity and into their second gestation. For the first objective, 45 multiparous sows (average parity 6.7 ± 2.5; parity range 5 to 14) from two commercial farms (n = 21 farm one and n = 24 farm two) were used. Sows were moved to a pen where digital images of the profile and rear stance were captured. On average, 2.8 and 8.1 final profile images were used per sow at farm one and farm two respectively. Farm two had over twice the number of profile images as farm one, as farm one was taken from the right side only whereas farm two had both left (average 4.2) and right (average 3.9) profile images. Additionally, 2.6 rear stance images were used for measurement per sow. A joint angle measuring system was devised to collect angle measurements on the knee, front and rear pastern, hock and rear stance. Joint measurements were analyzed using mixed model methods, including farm, side of measurement and parity as fixed effects and sow as a random effect. Intraclass correlation coefficients were calculated to evaluate process repeatability. No significant farm or parity differences were observed for joint angles measured except for the knee angle between farms (P \u3c 0.05) and the hock angle between sows parity six and ≥ seven (P \u3c 0.05). Side was significantly different in all joints measured (P \u3c 0.05), except for in the rear stance measurement where side is not applicable. Joint angle measurement repeatability ranged from 0.58 to 0.87. Lowest and highest repeatabilities were observed for the knee and hock angle measurements, respectively. For the second objective, gilts were selected from a single population and moved to three different farms. Profile and rear stance images were obtained from gilts at selection (319; average age 21.6 ± 1.8 weeks; range 19 to 25) and during their second gestation (277; average gestation 26.7 ± 17.2 days; range 0 to 87). Knee, front and rear pastern, hock, and rear stance joint angles were measured using image analysis software. To evaluate symmetry and joint angle differences due to age between the same individual, only females with repeated measures at selection and post first parity, when second gestation days were between 0 to 21 (126 females), were used. Mixed model equations were used including parity (zero or one) and profile side (left or right) as fixed effects. Parity was included as a repeated variable with the animal as the subject. Knee and rear pastern angles decreased (weakened) and hock angles increased (straightened) as age progressed (P \u3c 0.05). All joint measurements were symmetric between left and right legs (P \u3e 0.05) except for the hock where a difference (P \u3c 0.05) was observed. To evaluate gestation age effects on joint angles, only the measurements taken during the second gestation were used. Data was analyzed using mixed model equations including farm and side as fixed effects and gestation age as continuous covariate and animal was included as random effect. Farm was a significant source of variation for knee, front and rear pasterns, and rear stance angle measurements (P \u3c 0.05). Additionally, asymmetry was detected in knee, and front and rear pasterns (P \u3c 0.05). Front pastern and hock angles increased (straightened) as gestation age increased, while knee angle decreased (weakened) (P \u3c 0.05). Heritability estimates were low to moderate for profile angles and was not estimable for the rear stance position. Results suggest that as age increases leg structure changes, with the rear leg joints showing greater variation from selection to first parity. Results also suggest that environmental factors such as farm where animals are housed could contribute to angle differences. Small angle changes in the front leg could indicate structure may change over the life of the animal; however, rear leg structure and its impact on longevity still require further investigation. Results from this body of work have set the ground work for an objective feet and leg joint conformation method using digital imagery. It is still necessary to look further into the life of the animal and understand the full genetic control over the change in structure until complete physical maturity and its association with lifetime productivity in the sow

MacDowell symmetry and fermion regge trajectories

Phase shift analyses of pion-nucleon scattering have led to the discovery of a large number of excited baryonic states having positive and negative parity. A fascinating challenge is presented by the classification of these states, and the search for the fundamental laws of Nature which determine their spectrum.Previous study of this problem has taken place along two different lines. In one, the use of symmetry groups is made, and the pion-nucleon resonances are allocated to different representations of these groups. The other has been the study of the underlying forces involved, including dynamical models such as bootstrap theory. Both these approaches have been adequately discussed in the report of the Trieste Conference (19651, and are not treated further in this work.Recently, the importance of complex angular momentum and Regge theory in this problem was demonstrated by Barger and Cline, who showed that the known pion-nucleon resonances could be fitted on families of Regge trajectories.An important theoretical concept in baryonic systems is MacDowell symmetry which is a relationship between parity conserving partial wave amplitudes for one parity at positive energy, to the wave having opposite parity and negative energy. The application of MacDowell symmetry and Regge theory to the pion nucleon system shows that two Regge trajectories α(±w) may be defined. The physical Regge recurrences on α( +w) have one parity, and the trajectory α( -w) corresponds to a trajectory in which the Regge recurrences have opposite parity.The work of Barger and Cline is discussed in detail in Chapter II, and from the experimental fits it is shown that the two trajectories α(w), α( -w) are approximately the same, so parity degeneracy occurs. There are some notable exceptions to this result. Several states predicted by this parity degeneracy are missing, such as the lowest member of the highest ranking N₈ trajectory (the S₁₁), the lowest member of the Nδ (P₁₃), and the lowest members of the Δγ (D₃₃, G₃₇) . The usual spectroscopic notation L₂₁,2₂ⱼ is used in the classification of the baryons.This thesis is concerned with a study of the pion-nucleon resonances in the framework of Regge theory and MacDowell symmetry. Attempts .are made to explain the form of the Regge trajectories for the system, and special attention is paid to the missing mass states. The scope has been restricted to the nucleon Nα and Nß trajectories, but the theory may be generalised to other trajectories using SU symmetry.In Chapter I the concept of MacDowell symmetry is stated and proved for the parity conserving partial wave amplitudes of Gell-Mann, Goldberger, Low and Zachariasen. A discussion of generalised MacDowell symmetry which depends on field theoretic arguments has been given by Hara. The approach to MacDowell symmetry used in this thesis depends on crossing symmetry, and to the author's knowledge this has not been done before.Chapter II is an introduction to pion-nucleon scattering and the application of MacDowell symmetry and Regge poles. The original work in this thesis starts at section 2.5, in which a discussion of Riemann sheets and their application to missing mass states, is given.In Chapter III a potential scattering model is described, and its possible applications to the pion-nucleon system and missing mass states is discussed.Chapter IV is concerned with parametrisations of Regge trajectories, and a critical discussion is given of models which produce distortions of the Regge trajectory near the missing mass states.Finally, in Chapter V possible dynamical models for fernion Regge trajectories are discussed, and a review is given of their applications to the higher pion-nucleon resonances

The correlation of voiding variables between non-instrumented uroflowmetery and pressure-flow studies in women with pelvic organ prolapse

Aims To (1) correlate peak and maximum flow rates from non-instrumented flow (NIF) and pressure-flow studies (PFS) in women with pelvic organ prolapse (POP); (2) measure the impact of voided volume and degree of prolapse on correlations. Methods We compared four groups of women with stages II–IV POP. Groups 1 and 2 were symptomatically stress continent women participating in the colpopexy and urinary reduction efforts (CARE) trial; during prolapse reduction before sacrocolpopexy, Group 1 (n = 67) did not have and Group 2 (n = 84) had urodynamic stress incontinence (USI). Group 3 (n = 74) and Group 4 participants (n = 73), recruited specifically for this study, had stress urinary incontinence (SUI) symptoms. Group 3 planned sacrocolpopexy. Group 4 planned a different treatment option. Participants completed standardized uroflowmetry and pressure voiding studies. Results Subjects' median age was 61 years; median parity 3% and 80% had stage III or IV POP. Based on the Blaivas–Groutz nomogram, 49% of all women were obstructed. NIF and PFS peak and average flow rates had low correlations with one another (0.31, P  < 0.001 and 0.35, P  < 0.001, respectively). When NIF and PFS voided volumes were within 25% of each other, the peak and average flow rate correlations improved (0.52, P  < 0.001 and 0.57, P  < 0.001, respectively). As vaginal prolapse increased, correlations between NIF and PFS peak and average flow rates decreased. Conclusion Peak and average flow rates are highly dependent on voided volume in women with prolapse. As the prolapse stage increases, correlations between NIF and PFS variables decrease. Neurourol. Urodynam. 27:515–521, 2008. © 2008 Wiley-Liss, Inc.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/60897/1/20568_ftp.pd

Investigating the evolution of the nuclear magic numbers via single-neutron transfer populating 26Na

The changing shell structure close to the N = 20 shell gap has been explored by studying one-neutron transfer on 26Na in inverse kinematics. The experiment employed a 5MeV per nucleon beam of 25Na, incident on a 0.5mgcm-2 (CD2)n target, produced by the ISAC-II facility at TRIUMF, in Vancouver. A compact, highly-segmented silicon array, SHARC was positioned close to the target for the detection of the ejected protons. Coincident y rays, from the de-excitation of the 26Na recoil nucleus, were detected in the fully-suppressed γ- ray array, TIGRESS. Through the use of P γ: and p γ γ -gating, states in 26Na populated close in excitation energy were able to be resolved, and the proton angular distributions extracted. The proton angular distributions have been analysed for seven states directly populated by (d,p) in 26Na; and compared to theoretical cross-sections generated using TWOFNR. These states, amongst others, were compared to shell model calculations and to the N = 15 isotone 28 Al in order to tentatively assign spin and parity. Different interactions were em- ployed and compared to the experimental data, including USD, USD-A and USD-B calcu- . lations. The experimental data were also compared to SDPF-M Monte-Carlo Shell Model predictions, as well as WBP shell model calculations, in which the single-particle energies of the pf-shell were artificially lowered by 0.7 and 1.0MeV. A new level scheme with spin and parity assignments for states, including the negative parity states from f7/2 transfer, has been determined. It has been found that the gap observed between the f7/2 and SI/2 shells is approximately right, based on the shell model predictions with a 0.7MeV reduction in the shell gap. The gap between the sd- and pf-shells is found to be reduced by 1. 2 MeV, indicating the closure of the N = 20 shell gap for nuclei far from stability.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Does urethral competence affect urodynamic voiding parameters in women with prolapse?

Aims To (1) compare voiding parameters and (2) correlate symptoms and urodynamic findings in women with pelvic organ prolapse (POP) and varying degrees of urethral competence. Methods We compared three groups of women with stages II–IV POP. Groups 1 and 2 were symptomatically stress continent women participating in the Colpopexy and Urinary Reduction Efforts (CARE) trial; during prolapse reduction before sacrocolpopexy, Group 1 (n = 67) did not have and Group 2 (n = 84) had urodynamic stress incontinence (USI) during prolapse reduction. Group 3 participants (n = 74), recruited specifically for this study, had stress urinary incontinence (SUI) symptoms and planned sacrocolpopexy. Participants completed standardized uroflowmetry, pressure voiding studies, and validated symptom questionnaires. Results Subjects' median age was 61 years, median parity 3 and 87% had stage III or IV POP. Fourteen percent of women in Group 3 demonstrated USI without, and 70% with, prolapse reduction. Women in Groups 2 and 3 had more detrusor overactivity (DO) than Group 1 (17 and 24% vs. 6%, P  = 0.02) and detrusor overactivity incontinence (DOI) (15 and 8% vs. 0%, P  = 0.004). Based on the Blaivis–Groutz nomogram, 60% of all women were obstructed. Post-void residual volume (PVR), peak flow rate, detrusor pressure at peak flow, voiding mechanisms, voiding patterns, obstruction and urinary retention did not differ among groups. Women in Group 3 had higher irritative and obstructive symptom scores than Group 1 or 2; neither score differed by presence of DO nor obstruction, respectively. Conclusion Women with POP have significant rates of urodynamic obstruction and retention, independent of their continence status. Symptoms of obstruction and retention correlate poorly with urodynamic findings. Neurourol. Urodynam. 26:1030–1035, 2007. © 2007 Wiley-Liss, Inc.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/57365/1/20436_ftp.pd

On The Structure Of The Chan-Paton Factors For D-Branes In Type II Orientifolds

We determine the structure of the Chan-Paton factors of the open strings ending on space filling D-branes in Type II orientifolds. Through the analysis, we obtain a rule concerning possible distribution of O-plane types. The result is applied to classify the topology of D-branes in terms of Fredholm operators and K-theory, deriving a proposal made earlier and extending it to more general cases. It is also applied to compactifications with N=1 supersymmetry in four-dimensions. We adapt and develop the language of category in this context, and use it to describe some decay channels.Comment: 137 page