Shift in critical temperature for random spatial permutations with cycle weights

We examine a phase transition in a model of random spatial permutations which originates in a study of the interacting Bose gas. Permutations are weighted according to point positions; the low-temperature onset of the appearance of arbitrarily long cycles is connected to the phase transition of Bose-Einstein condensates. In our simplified model, point positions are held fixed on the fully occupied cubic lattice and interactions are expressed as Ewens-type weights on cycle lengths of permutations. The critical temperature of the transition to long cycles depends on an interaction-strength parameter $\alpha$. For weak interactions, the shift in critical temperature is expected to be linear in $\alpha$ with constant of linearity $c$. Using Markov chain Monte Carlo methods and finite-size scaling, we find $c = 0.618 \pm 0.086$. This finding matches a similar analytical result of Ueltschi and Betz. We also examine the mean longest cycle length as a fraction of the number of sites in long cycles, recovering an earlier result of Shepp and Lloyd for non-spatial permutations.Comment: v2 incorporated reviewer comments. v3 removed two extraneous figures which appeared at the end of the PDF

Repeatability and reproducibility of cerebral 23Na imaging in healthy subjects

Abstract Background Initial reports of 23Na magnetic resonance imaging (MRI) date back to the 1970s. However, methodological challenges of the technique hampered its widespread adoption for many years. Recent technical developments have overcome some of these limitations and have led to more optimal conditions for 23Na-MR imaging. In order to serve as a reliable tool for the assessment of clinical stroke or brain tumor patients, we investigated the repeatability and reproducibility of cerebral sodium (23Na) imaging in healthy subjects. Methods In this prospective, IRB approved study 12 consecutive healthy volunteers (8 female, age 31 ± 8.3) underwent three cerebral 23Na-MRI examinations at 3.0 T (TimTrio, Siemens Healthineers) distributed between two separate visits with an 8 day interval. For each scan a T1w MP-RAGE sequence for anatomical referencing and a 3D-density-adapted, radial GRE-sequence for 23Na-imaging were acquired using a dual-tuned (23Na/1H) head-coil. On 1 day, these scans were repeated consecutively; on the other day, the scans were performed once. 23Na-sequences were reconstructed according to the MP-RAGE sequence, allowing direct cross-referencing of ROIs. Circular ROIs were placed in predetermined anatomic regions: gray and white matter (GM, WM), head of the caudate nucleus (HCN), pons, and cerebellum. External 23Na-reference phantoms were used to calculate the tissue sodium content. Results Excellent correlation was found between repeated measurements on the same day (r2 = 0.94), as well as on a different day (r2 = 0.86). No significant differences were found based on laterality other than in the HCN (63.1 vs. 58.7 mmol/kg WW on the right (p = 0.01)). Pronounced inter-individual differences were identified in all anatomic regions. Moderate to good correlation (0.310 to 0.701) was found between the readers. Conclusion Our study has shown that intra-individual 23Na-concentrations in healthy subjects do not significantly differ after repeated scans on the same day and a pre-set time interval. This confirms the repeatability and reproducibility of cerebral 23Na-imaging. However, with manual ROI placement in predetermined anatomic landmarks, fluctuations in 23Na-concentrations can be observed

Critical behavior for the model of random spatial permutations

We examine a phase transition in a model of random spatial permutations which originates in a study of the interacting Bose gas. Permutations are weighted according to point positions; the low-temperature onset of the appearance of arbitrarily long cycles is connected to the phase transition of Bose-Einstein condensates. In our simplified model, point positions are held fixed on the fully occupied cubic lattice and interactions are expressed as Ewens-type weights on cycle lengths of permutations. The critical temperature of the transition to long cycles depends on an interaction-strength parameter α. For weak interactions, the shift in critical temperature is expected to be linear in α with constant of linearity c. Using Markov chain Monte Carlo methods and finite-size scaling, we find c = 0.618 ± 0.086. This finding matches a similar analytical result of Ueltschi and Betz. We also examine the mean longest cycle length as a fraction of the number of sites in long cycles, recovering an earlier result of Shepp and Lloyd for non-spatial permutations. The plan of this paper is as follows. We begin with a non-technical discussion of the historical context of the project, along with a mention of alternative approaches. Relevant previous works are cited, thus annotating the bibliography. The random-cycle approach to the BEC problem requires a model of spatial permutations. This model it is of its own probabilistic interest; it is developed mathematically, without reference to the Bose gas. Our Markov-chain Monte Carlo algorithms for sampling from the random-cycle distribution - the swap-only, swap-and-reverse, band-update, and worm algorithms - are presented, compared, and contrasted. Finite-size scaling techniques are used to obtain information about infinite-volume quantities from finite-volume computational data

APPROVED: CURVES AND CODES

This paper gives an example-driven overview of algebraic-geometry codes, with attention confined to bivariate Goppa codes. J. Walker’s Codes and Curves uses notation that is standard for graduate mathematics, but unfortunately does not discuss decoding; O. Pretzel’s Codes and Algebraic Curves gives a full discussion, but using non-standard notation. The current work is a synthesis of the two, extending Walker’s work by including a discussion of the Skorobogatov-Vlǎdut¸ (SV) decoding algorithm. First, notation for finite fields is given; then, the engineering problem is defined. Selected concepts from algebraic geometry are introduced, illustrated by examples. The construction and encoding of Goppa codes is described, followed by an exposition of the SV decoding algorithm. Several worked examples are shown. Software implementations of the encoder and decoder are discussed, followed by performance analysis and comparison with other codes. Finally, directions for further research are sketched. iii This thesis is dedicated to Dr. Philip Leonard, Professor Emeritus of Mathematics at Arizona State University, who showed me — much to my astonishment — the connections between linear feedback shift registers and finite fields, revealing that engineering and abstract algebra need not be mutually exclusive. i

Critical behavior for the model of random spatial permutations

We examine a phase transition in a model of random spatial permutations which originates in a study of the interacting Bose gas. Permutations are weighted according to point positions; the low-temperature onset of the appearance of arbitrarily long cycles is connected to the phase transition of Bose-Einstein condensates. In our simplified model, point positions are held fixed on the fully occupied cubic lattice and interactions are expressed as Ewens-type weights on cycle lengths of permutations. The critical temperature of the transition to long cycles depends on an interaction-strength parameter α. For weak interactions, the shift in critical temperature is expected to be linear in α with constant of linearity c. Using Markov chain Monte Carlo methods and finite-size scaling, we find c = 0.618±0.086. This finding matches a similar analytical result of Ueltschi and Betz. We also examine the mean longest cycle length as a fraction of the number of sites in long cycles, recovering an earlier result of Shepp and Lloyd for non-spatial permutations. The plan of this paper is as follows. We begin with a non-technical discussion of the historical context of the project, along with a mention of alternative approaches. Relevant previous works are cited, thus annotating the bibliography. The random-cycle approach to the BEC problem requires a model of spatial permutations. This model it is of its own probabilistic interest; it is developed mathematically, without reference to the Bose gas. Our Markov-chain Monte Carlo algorithms for sampling from the random-cycle distribution—the swap-only, swap-and-reverse, band-update, and worm algorithms—are presented, compared, and contrasted. Finite-size scaling techniques are used to obtain information about infinite-volume quantities from finite-volume computational data

The Turbulent Terrain : An Exhibition of Works by Five Alberta Artists

Ginn appraises the significance of the landscape genre for Alberta artists in the face technology, industry and development. Statements by the artists. Biographical notes