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ON THE SET OF BETTI ELEMENTS OF A PUISEUX MONOID
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- SCOTT T. CHAPMAN, JOSHUA JANG, JASON MAO, SKYLER MAO
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- Bulletin of the Australian Mathematical Society , First View
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- 20 May 2024, pp. 1-13
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Let M be a Puiseux monoid, that is, a monoid consisting of nonnegative rationals (under standard addition). In this paper, we study factorisations in atomic Puiseux monoids through the lens of their associated Betti graphs. The Betti graph of $b \in M$ is the graph whose vertices are the factorisations of b with edges between factorisations that share at least one atom. If the Betti graph associated to b is disconnected, then we call b a Betti element of M. We explicitly compute the set of Betti elements for a large class of Puiseux monoids (the atomisations of certain infinite sequences of rationals). The process of atomisation is quite useful in studying the arithmetic of Puiseux monoids, and it has been actively considered in recent literature. This leads to an argument that for every positive integer n, there exists an atomic Puiseux monoid with exactly n Betti elements.
A Novel Algorithm for Improving the Prehospital Diagnostic Accuracy of ST-Segment Elevation Myocardial Infarction
- Mat Goebel, Lauren M. Westafer, Stephanie A. Ayala, El Ragone, Scott J. Chapman, Masood R. Mohammed, Marc R. Cohen, James T. Niemann, Marc Eckstein, Stephen Sanko, Nichole Bosson
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- Journal:
- Prehospital and Disaster Medicine / Volume 39 / Issue 1 / February 2024
- Published online by Cambridge University Press:
- 04 December 2023, pp. 37-44
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- February 2024
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Introduction:
Early detection of ST-segment elevation myocardial infarction (STEMI) on the prehospital electrocardiogram (ECG) improves patient outcomes. Current software algorithms optimize sensitivity but have a high false-positive rate. The authors propose an algorithm to improve the specificity of STEMI diagnosis in the prehospital setting.
Methods:A dataset of prehospital ECGs with verified outcomes was used to validate an algorithm to identify true and false-positive software interpretations of STEMI. Four criteria implicated in prior research to differentiate STEMI true positives were applied: heart rate <130, QRS <100, verification of ST-segment elevation, and absence of artifact. The test characteristics were calculated and regression analysis was used to examine the association between the number of criteria included and test characteristics.
Results:There were 44,611 cases available. Of these, 1,193 were identified as STEMI by the software interpretation. Applying all four criteria had the highest positive likelihood ratio of 353 (95% CI, 201-595) and specificity of 99.96% (95% CI, 99.93-99.98), but the lowest sensitivity (14%; 95% CI, 11-17) and worst negative likelihood ratio (0.86; 95% CI, 0.84-0.89). There was a strong correlation between increased positive likelihood ratio (r2 = 0.90) and specificity (r2 = 0.85) with increasing number of criteria.
Conclusions:Prehospital ECGs with a high probability of true STEMI can be accurately identified using these four criteria: heart rate <130, QRS <100, verification of ST-segment elevation, and absence of artifact. Applying these criteria to prehospital ECGs with software interpretations of STEMI could decrease false-positive field activations, while also reducing the need to rely on transmission for physician over-read. This can have significant clinical and quality implications for Emergency Medical Services (EMS) systems.
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- By Janice Capel Anderson, S. Wesley Ariarajah, Constantine Belezos, Ian Boxall, Marc Zvi Brettler, Edward Breuer, Daniel Bruno, Mark Chapman, W. T. Dickens, Mark W. Elliott, Eldon Epp, Tassilo Erhardt, Timothy Gorringe, Harriet Harris, Peter C. Hodgson, Leslie Howsam, Werner G. Jeanrond, Scott McLaren, Wayne A. Meeks, Néstor Míguez, Stephen D. Moore, Robert Morgan, Halvor Moxnes, Peter Neuner, Mark Noll, Jorunn Økland, Gaye Ortiz, John Riches, Christopher Rowland, Nicolaas A. Rupke, Edmund J. Rybarczyk, Lamin Sanneh, Constantine Scouteris, R. S. Sugirtharajah, Willard M. Swartley, William R. Telford, David Thompson, Elena Volkova, J. R. Watson, Gerald West, Michael Wheeler, Keith Whitelam
- Edited by John Riches, University of Glasgow
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- The New Cambridge History of the Bible
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- 09 June 2015
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- 13 April 2015, pp xi-xii
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THE CATENARY AND TAME DEGREES ON A NUMERICAL MONOID ARE EVENTUALLY PERIODIC
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- SCOTT T. CHAPMAN, MARLY CORRALES, ANDREW MILLER, CHRIS MILLER, DHIR PATEL
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- Journal:
- Journal of the Australian Mathematical Society / Volume 97 / Issue 3 / December 2014
- Published online by Cambridge University Press:
- 09 September 2014, pp. 289-300
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- December 2014
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Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}M$ be a commutative cancellative monoid. For $m$ a nonunit in $M$, the catenary degree of $m$, denoted $c(m)$, and the tame degree of $m$, denoted $t(m)$, are combinatorial constants that describe the relationships between differing irreducible factorizations of $m$. These constants have been studied carefully in the literature for various kinds of monoids, including Krull monoids and numerical monoids. In this paper, we show for a given numerical monoid $S$ that the sequences $\{c(s)\}_{s\in S}$ and $\{t(s)\}_{s\in S}$ are both eventually periodic. We show similar behavior for several functions related to the catenary degree which have recently appeared in the literature. These results nicely complement the known result that the sequence $\{\Delta (s)\}_{s\in S}$ of delta sets of $S$ also satisfies a similar periodicity condition.
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- By Jane E. Adcock, Yahya Aghakhani, A. Anand, Eva Andermann, Frederick Andermann, Alexis Arzimanoglou, Sandrine Aubert, Nadia Bahi-Buisson, Carman Barba, Agatino Battaglia, Geneviève Bernard, Nadir E. Bharucha, Laurence A. Bindoff, William Bingaman, Francesca Bisulli, Thomas P. Bleck, Stewart G. Boyd, Andreas Brunklaus, Harry Bulstrode, Jorge G. Burneo, Laura Canafoglia, Laura Cantonetti, Roberto H. Caraballo, Fernando Cendes, Kevin E. Chapman, Patrick Chauvel, Richard F. M. Chin, H. T. Chong, Fahmida A. Chowdhury, Catherine J. Chu-Shore, Rolando Cimaz, Andrew J. Cole, Bernard Dan, Geoffrey Dean, Alessio De Ciantis, Fernando De Paolis, Rolando F. Del Maestro, Irissa M. Devine, Carlo Di Bonaventura, Concezio Di Rocco, Henry B. Dinsdale, Maria Alice Donati, François Dubeau, Michael Duchowny, Olivier Dulac, Monika Eisermann, Brent Elliott, Bernt A. Engelsen, Kevin Farrell, Natalio Fejerman, Rosalie E. Ferner, Silvana Franceschetti, Robert Friedlander, Antonio Gambardella, Hector H. Garcia, Serena Gasperini, Lorenzo Genitori, Gioia Gioi, Flavio Giordano, Leif Gjerstad, Daniel G. Glaze, Howard P. Goodkin, Sidney M. Gospe, Andrea Grassi, William P. Gray, Renzo Guerrini, Marie-Christine Guiot, William Harkness, Andrew G. Herzog, Linda Huh, Margaret J. Jackson, Thomas S. Jacques, Anna C. Jansen, Sigmund Jenssen, Michael R. Johnson, Dorothy Jones-Davis, Reetta Kälviäinen, Peter W. Kaplan, John F. Kerrigan, Autumn Marie Klein, Matthias Koepp, Edwin H. Kolodny, Kandan Kulandaivel, Ruben I. Kuzniecky, Ahmed Lary, Yolanda Lau, Anna-Elina Lehesjoki, Maria K. Lehtinen, Holger Lerche, Michael P. T. Lunn, Snezana Maljevic, Mark R. Manford, Carla Marini, Bindu Menon, Giulia Milioli, Eli M. Mizrahi, Manish Modi, Márcia Elisabete Morita, Manuel Murie-Fernandez, Vivek Nambiar, Lina Nashef, Vincent Navarro, Aidan Neligan, Ruth E. Nemire, Charles R. J. C. Newton, John O'Donavan, Hirokazu Oguni, Teiichi Onuma, Andre Palmini, Eleni Panagiotakaki, Pasquale Parisi, Elena Parrini, Liborio Parrino, Ignacio Pascual-Castroviejo, M. Scott Perry, Perrine Plouin, Charles E. Polkey, Suresh S. Pujar, Karthik Rajasekaran, R. Eugene Ramsey, Rahul Rathakrishnan, Roberta H. Raven, Guy M. Rémillard, David Rosenblatt, M. Elizabeth Ross, Abdulrahman Sabbagh, P. Satishchandra, Swati Sathe, Ingrid E. Scheffer, Philip A. Schwartzkroin, Rod C. Scott, Frédéric Sedel, Michelle J. Shapiro, Elliott H. Sherr, Michael Shevell, Simon D. Shorvon, Adrian M. Siegel, Gagandeep Singh, S. Sinha, Barbara Spacca, Waney Squier, Carl E. Stafstrom, Bernhard J. Steinhoff, Andrea Taddio, Gianpiero Tamburrini, C. T. Tan, Raymond Y. L. Tan, Erik Taubøll, Robert W. Teasell, Mario Giovanni Terzano, Federica Teutonico, Suzanne A. Tharin, Elizabeth A. Thiele, Pierre Thomas, Paolo Tinuper, Dorothée Kasteleijn-Nolst Trenité, Sumeet Vadera, Pierangelo Veggiotti, Jean-Pierre Vignal, J. M. Walshe, Elizabeth J. Waterhouse, David Watkins, Ruth E. Williams, Yue-Hua Zhang, Benjamin Zifkin, Sameer M. Zuberi
- Edited by Simon D. Shorvon, Frederick Andermann, Renzo Guerrini
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- The Causes of Epilepsy
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- 05 March 2012
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- 14 April 2011, pp ix-xvi
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- By Rose Teteki Abbey, K. C. Abraham, David Tuesday Adamo, LeRoy H. Aden, Efrain Agosto, Victor Aguilan, Gillian T. W. Ahlgren, Charanjit Kaur AjitSingh, Dorothy B E A Akoto, Giuseppe Alberigo, Daniel E. Albrecht, Ruth Albrecht, Daniel O. Aleshire, Urs Altermatt, Anand Amaladass, Michael Amaladoss, James N. Amanze, Lesley G. Anderson, Thomas C. Anderson, Victor Anderson, Hope S. Antone, María Pilar Aquino, Paula Arai, Victorio Araya Guillén, S. Wesley Ariarajah, Ellen T. Armour, Brett Gregory Armstrong, Atsuhiro Asano, Naim Stifan Ateek, Mahmoud Ayoub, John Alembillah Azumah, Mercedes L. García Bachmann, Irena Backus, J. Wayne Baker, Mieke Bal, Lewis V. Baldwin, William Barbieri, António Barbosa da Silva, David Basinger, Bolaji Olukemi Bateye, Oswald Bayer, Daniel H. Bays, Rosalie Beck, Nancy Elizabeth Bedford, Guy-Thomas Bedouelle, Chorbishop Seely Beggiani, Wolfgang Behringer, Christopher M. Bellitto, Byard Bennett, Harold V. Bennett, Teresa Berger, Miguel A. Bernad, Henley Bernard, Alan E. Bernstein, Jon L. Berquist, Johannes Beutler, Ana María Bidegain, Matthew P. Binkewicz, Jennifer Bird, Joseph Blenkinsopp, Dmytro Bondarenko, Paulo Bonfatti, Riet en Pim Bons-Storm, Jessica A. Boon, Marcus J. Borg, Mark Bosco, Peter C. Bouteneff, François Bovon, William D. Bowman, Paul S. Boyer, David Brakke, Richard E. Brantley, Marcus Braybrooke, Ian Breward, Ênio José da Costa Brito, Jewel Spears Brooker, Johannes Brosseder, Nicholas Canfield Read Brown, Robert F. Brown, Pamela K. Brubaker, Walter Brueggemann, Bishop Colin O. Buchanan, Stanley M. Burgess, Amy Nelson Burnett, J. Patout Burns, David B. Burrell, David Buttrick, James P. Byrd, Lavinia Byrne, Gerado Caetano, Marcos Caldas, Alkiviadis Calivas, William J. Callahan, Salvatore Calomino, Euan K. Cameron, William S. Campbell, Marcelo Ayres Camurça, Daniel F. Caner, Paul E. Capetz, Carlos F. Cardoza-Orlandi, Patrick W. Carey, Barbara Carvill, Hal Cauthron, Subhadra Mitra Channa, Mark D. Chapman, James H. Charlesworth, Kenneth R. Chase, Chen Zemin, Luciano Chianeque, Philip Chia Phin Yin, Francisca H. Chimhanda, Daniel Chiquete, John T. Chirban, Soobin Choi, Robert Choquette, Mita Choudhury, Gerald Christianson, John Chryssavgis, Sejong Chun, Esther Chung-Kim, Charles M. A. Clark, Elizabeth A. Clark, Sathianathan Clarke, Fred Cloud, John B. Cobb, W. Owen Cole, John A Coleman, John J. Collins, Sylvia Collins-Mayo, Paul K. Conkin, Beth A. Conklin, Sean Connolly, Demetrios J. Constantelos, Michael A. Conway, Paula M. Cooey, Austin Cooper, Michael L. Cooper-White, Pamela Cooper-White, L. William Countryman, Sérgio Coutinho, Pamela Couture, Shannon Craigo-Snell, James L. Crenshaw, David Crowner, Humberto Horacio Cucchetti, Lawrence S. Cunningham, Elizabeth Mason Currier, Emmanuel Cutrone, Mary L. Daniel, David D. Daniels, Robert Darden, Rolf Darge, Isaiah Dau, Jeffry C. Davis, Jane Dawson, Valentin Dedji, John W. de Gruchy, Paul DeHart, Wendy J. Deichmann Edwards, Miguel A. De La Torre, George E. Demacopoulos, Thomas de Mayo, Leah DeVun, Beatriz de Vasconcellos Dias, Dennis C. Dickerson, John M. Dillon, Luis Miguel Donatello, Igor Dorfmann-Lazarev, Susanna Drake, Jonathan A. Draper, N. Dreher Martin, Otto Dreydoppel, Angelyn Dries, A. J. Droge, Francis X. D'Sa, Marilyn Dunn, Nicole Wilkinson Duran, Rifaat Ebied, Mark J. Edwards, William H. Edwards, Leonard H. Ehrlich, Nancy L. Eiesland, Martin Elbel, J. Harold Ellens, Stephen Ellingson, Marvin M. Ellison, Robert Ellsberg, Jean Bethke Elshtain, Eldon Jay Epp, Peter C. Erb, Tassilo Erhardt, Maria Erling, Noel Leo Erskine, Gillian R. Evans, Virginia Fabella, Michael A. Fahey, Edward Farley, Margaret A. Farley, Wendy Farley, Robert Fastiggi, Seena Fazel, Duncan S. Ferguson, Helwar Figueroa, Paul Corby Finney, Kyriaki Karidoyanes FitzGerald, Thomas E. FitzGerald, John R. Fitzmier, Marie Therese Flanagan, Sabina Flanagan, Claude Flipo, Ronald B. 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Phan, Isabel Apawo Phiri, William S. F. Pickering, Derrick G. Pitard, William Elvis Plata, Zlatko Plese, John Plummer, James Newton Poling, Ronald Popivchak, Andrew Porter, Ute Possekel, James M. Powell, Enos Das Pradhan, Devadasan Premnath, Jaime Adrían Prieto Valladares, Anne Primavesi, Randall Prior, María Alicia Puente Lutteroth, Eduardo Guzmão Quadros, Albert Rabil, Laurent William Ramambason, Apolonio M. Ranche, Vololona Randriamanantena Andriamitandrina, Lawrence R. Rast, Paul L. Redditt, Adele Reinhartz, Rolf Rendtorff, Pål Repstad, James N. Rhodes, John K. Riches, Joerg Rieger, Sharon H. Ringe, Sandra Rios, Tyler Roberts, David M. Robinson, James M. Robinson, Joanne Maguire Robinson, Richard A. H. Robinson, Roy R. Robson, Jack B. Rogers, Maria Roginska, Sidney Rooy, Rev. Garnett Roper, Maria José Fontelas Rosado-Nunes, Andrew C. Ross, Stefan Rossbach, François Rossier, John D. Roth, John K. Roth, Phillip Rothwell, Richard E. 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Yee, Viktor Yelensky, Yeo Khiok-Khng, Gustav K. K. Yeung, Angela Yiu, Amos Yong, Yong Ting Jin, You Bin, Youhanna Nessim Youssef, Eliana Yunes, Robert Michael Zaller, Valarie H. Ziegler, Barbara Brown Zikmund, Joyce Ann Zimmerman, Aurora Zlotnik, Zhuo Xinping
- Edited by Daniel Patte, Vanderbilt University, Tennessee
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- The Cambridge Dictionary of Christianity
- Published online:
- 05 August 2012
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- 20 September 2010, pp xi-xliv
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ON FACTORIZATION IN BLOCK MONOIDS FORMED BY $\{\bar{1},\bar{a}\}$ IN $\mathbb{Z}_{n}$
- Scott T. Chapman, William W. Smith
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 46 / Issue 2 / June 2003
- Published online by Cambridge University Press:
- 04 July 2003, pp. 257-267
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We consider the factorization properties of block monoids on $\mathbb{Z}_n$ determined by subsets of the form $S_a=\{\bar{1},\bar{a}\}$. We denote such a block monoid by $\mathcal{B}_a(n)$. In §2, we provide a method based on the division algorithm for determining the irreducible elements of $\mathcal{B}_a(n)$. Section 3 offers a method to determine the elasticity of $\mathcal{B}_a(n)$ based solely on the cross number. Section 4 applies the results of §3 to investigate the complete set of elasticities of Krull monoids with divisor class group $\mathbb{Z}_n$.
AMS 2000 Mathematics subject classification: Primary 20M14; 20D60; 13F05
Overrings of Half-Factorial Domains
- David F. Anderson, Scott T. Chapman, William W. Smith
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- Journal:
- Canadian Mathematical Bulletin / Volume 37 / Issue 4 / 01 December 1994
- Published online by Cambridge University Press:
- 20 November 2018, pp. 437-442
- Print publication:
- 01 December 1994
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An atomic integral domain D is a half-factorial domain (HFD) if for any irreducible elements α1,..., αn, β1,..., βm of D with α1... αn = β1 ...βm, then n = m. In [3], Anderson, Anderson, and Zafrullah explore factorization problems in overrings of HFDs and ask whether a localization of a HFD is again a HFD. We construct an example of a Dedekind domain which is a HFD, but with a localization which is not a HFD. We also give an example of a Dedekind domain where each localization is a HFD, but with an overring which is not a HFD.
On the k-HFD property in Dedekind domains with small class group
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- Scott T. Chapman, William W. Smith
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- Journal:
- Mathematika / Volume 39 / Issue 2 / December 1992
- Published online by Cambridge University Press:
- 26 February 2010, pp. 330-340
- Print publication:
- December 1992
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Let D be an atomic integral domain (i.e., a domain in which each nonzero nonunit of D can be written as a product of irreducible elements) and k any positive integer. D is known as a half factorial domain (HFD) if for any irreducible elements α1, …, αn, β1, …, βm of D the equality α1… αn = β1… βm implies that n = m. In [5] the present authors define D to be a k-half factorial domain (k-HFD) if the equality above along with the fact that n or m ≤ k implies that n = m. In this paper we consider the k-HFD property in Dedekind domains with small class group and prove the following Theorem: if D is a Dedekind domain with class group of order less than 16 then D is k-HFD for some integer k > 1, if, and only if, D is HFD.