11 results
INTERCHANGE RINGS
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- CHARLES C. EDMUNDS
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- Journal:
- Journal of the Australian Mathematical Society / Volume 101 / Issue 3 / December 2016
- Published online by Cambridge University Press:
- 12 May 2016, pp. 310-334
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- December 2016
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An interchange ring,$(R,+,\bullet )$, is an abelian group with a second binary operation defined so that the interchange law$(w+x)\bullet (y+z)=(w\bullet y)+(x\bullet z)$ holds. An interchange near ring is the same structure based on a group which may not be abelian. It is shown that each interchange (near) ring based on a group $G$ is formed from a pair of endomorphisms of $G$ whose images commute, and that all interchange (near) rings based on $G$ can be characterized in this manner. To obtain an associative interchange ring, the endomorphisms must be commuting idempotents in the endomorphism semigroup of $G$. For $G$ a finite abelian group, we develop a group-theoretic analogue of the simultaneous diagonalization of idempotent linear operators and show that pairs of endomorphisms which yield associative interchange rings can be diagonalized and then put into a canonical form. A best possible upper bound of $4^{r}$ can be given for the number of distinct isomorphism classes of associative interchange rings based on a finite abelian group $A$ which is a direct sum of $r$ cyclic groups of prime power order. If $A$ is a direct sum of $r$ copies of the same cyclic group of prime power order, we show that there are exactly ${\textstyle \frac{1}{6}}(r+1)(r+2)(r+3)$ distinct isomorphism classes of associative interchange rings based on $A$. Several examples are given and further comments are made about the general theory of interchange rings.
Constructing Double Magma on Groups Using Commutation Operations
- Charles C. Edmunds
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- Journal:
- Canadian Mathematical Bulletin / Volume 58 / Issue 3 / 01 September 2015
- Published online by Cambridge University Press:
- 20 November 2018, pp. 497-506
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- 01 September 2015
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A magma$\left( M,\star \right)$ is a nonempty set with a binary operation. A double magma$\left( M,\star ,\bullet \right)$ is a nonempty set with two binary operations satisfying the interchange law$\left( w\star x \right)\bullet \left( y\star z \right)=\left( w\bullet y \right)\star \left( x\bullet z \right)$. We call a double magma proper if the two operations are distinct, and commutative if the operations are commutative. A double semigroup, first introduced by Kock, is a double magma for which both operations are associative. Given a non-trivial group $G$ we define a system of two magma $\left( G,\star ,\bullet \right)$ using the commutator operations $x\star y=\left[ x,y \right]\left( ={{x}^{-1}}{{y}^{-1}}xy \right)$ and $x\bullet y=\left[ y,x \right]$. We show that $\left( G,\star ,\bullet \right)$ is a double magma if and only if $G$ satisfies the commutator laws $\left[ x,y;x,z \right]=1$ and ${{\left[ w,x;y,z \right]}^{2}}=1$. We note that the first law defines the class of 3-metabelian groups. If both these laws hold in $G$, the double magma is proper if and only if there exist ${{x}_{0}},{{y}_{0}}\in G$ for which ${{\left[ {{x}_{0}},{{y}_{0}} \right]}^{2}}\ne 1$. This double magma is a double semigroup if and only if $G$ is nilpotent of class two. We construct a specific example of a proper double semigroup based on the dihedral group of order 16. In addition, we comment on a similar construction for rings using Lie commutators.
Contributors
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- By Mitchell Aboulafia, Frederick Adams, Marilyn McCord Adams, Robert M. Adams, Laird Addis, James W. Allard, David Allison, William P. Alston, Karl Ameriks, C. Anthony Anderson, David Leech Anderson, Lanier Anderson, Roger Ariew, David Armstrong, Denis G. Arnold, E. J. Ashworth, Margaret Atherton, Robin Attfield, Bruce Aune, Edward Wilson Averill, Jody Azzouni, Kent Bach, Andrew Bailey, Lynne Rudder Baker, Thomas R. Baldwin, Jon Barwise, George Bealer, William Bechtel, Lawrence C. Becker, Mark A. Bedau, Ernst Behler, José A. Benardete, Ermanno Bencivenga, Jan Berg, Michael Bergmann, Robert L. Bernasconi, Sven Bernecker, Bernard Berofsky, Rod Bertolet, Charles J. Beyer, Christian Beyer, Joseph Bien, Joseph Bien, Peg Birmingham, Ivan Boh, James Bohman, Daniel Bonevac, Laurence BonJour, William J. Bouwsma, Raymond D. Bradley, Myles Brand, Richard B. Brandt, Michael E. Bratman, Stephen E. Braude, Daniel Breazeale, Angela Breitenbach, Jason Bridges, David O. Brink, Gordon G. 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- 05 August 2015
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- By Ashok Agarwal, Linda D. Applegarth, Nelson E. Bennett, Nancy L. Brackett, Melissa B. Brisman, Mark F. H. Brougham, Cara B. Cimmino, Owen K. Davis, Rian J. Dickstein, Michael L. Eisenberg, Mikkel Fode, Gretchen A. Gignac, Bruce R. Gilbert, Ellen R. Goldmark, Marc Goldstein, Wayne J. G. Hellstrom, Wayland Hsiao, Jack Huang, Kathleen Hwang, Ann A. Jakubowski, Keith Jarvi, Loren Jones, Hey-Joo Kang, Joanne Frankel Kelvin, Mohit Khera, Thomas F. Kolon, Kate H. Kraft, Andrew C. Kramer, Dolores J. Lamb, Andrew B. Lassman, Helen R. Levey, Larry I. Lipshultz, Charles M. Lynne, Akanksha Mehta, Marvin L. Meistrich, Gregory C. Mitchell, Mark A. Moyad, John P. Mulhall, Lauren Murray, Craig Niederberger, Ariella Noy, Robert D. Oates, Dana A. Ohl, Kutluk Oktay, Ndidiamaka Onwubalili, Fabio Firmbach Pasqualatto, Elena Pentsova, Susanne A. Quallich, Gwendolyn P. Quinn, Alex Ridgeway, Matthew T. Roberts, Kenny A. Rodriguez-Wallberg, Allison B. Rosen, Lisa Rosenzweig, Edmund S. Sabanegh, Hossein Sadeghi-Nejad, Mary K. Samplaski, Jay I. Sandlow, Peter N. Schlegel, Gunapala Shetty, Mark Sigman, Jens Sønksen, Peter J. Stahl, Eytan Stein, Doron S. Stember, Raanan Tal, Susan T. Vadaparampil, W. Hamish, B. Wallace, Leonard H. Wexler, Daniel H. Williams
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- Fertility Preservation in Male Cancer Patients
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- Infertility in the Male
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- 19 May 2010
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141 - Meningococcus and Miscellaneous Neisseriae
- from Part XVIII - Specific Organisms – Bacteria
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- By Edmund C. Tramont, National Institute of Allergy and Infectious Diseases, Charles Davis, University of Maryland School of Medicine
- Edited by David Schlossberg
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- Clinical Infectious Disease
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- 05 March 2013
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- 12 May 2008, pp 997-1004
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Summary
MENINGOCOCCAL INFECTION
Meningococcal infection, first recognized nearly 2 centuries ago as epidemic cerebrospinal fever, occurs worldwide as sporadic, endemic, and epidemic cases. Worldwide, most cases are caused by serogroups A and C, whereas in the United States, serogroups B and Y predominate (see “Culture and Laboratory Findings”). Humans are the only natural host for the bacteria. Transmission of the organism occurs from person to person by direct contact with contaminated respiratory secretions or airborne droplets with subsequent colonization of the nasopharynx. Nasopharyngeal carriage approximates 5% to 15% in nonepidemic periods but may approach 50% to 95% during epidemics, especially serogroup A epidemics. The carriage rate is also increased when there is crowding, such as in military barracks, dormitories, prisons, convocations, and sporting events. The oropharyngeal and nasopharyngeal carriage may persist for several weeks to several months. Sexual transmission of meningococci in women and homo sexual men may result in anogenital carriage. Most cases of disease (eg, bacteremia, meningitis) occur in children between 6 months and 5 years of age. With few exceptions, invasive meningococci have a polysaccharide capsule that forms the basis for serogrouping of strains. Invasive disease occurs almost exclusively in persons who lack specific antimeningococcal antibody to the colonizing meningococcal strain. Individuals with complement component deficiencies are at an increased risk for developing invasive meningococcal infections because their serum loses the ability of complement-antibody mediated lysis of the bacteria.
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- By Ralph Adolphs, Bernard J. Baars, John A. Bargh, Jesse M. Bering, David F. Bjorklund, Joseph E. Bogen, Rebekah Bradley, Wallace Chafe, Michael C. Corballis, Diego Cosmelli, Jean-Marie Danion, Richard J. Davidson, Steven W. Day, Georges Dreyfus, John D. Dunne, Stan Franklin, Helena Hong Gao, Lisa Geraci, Deborah E. Hannula, J. Allan Hobson, Caroline Huron, John F. Kihlstrom, Asher Koriat, Uriah Kriegel, Jean-Philippe Lachaux, Charles D. Laughlin, Antoine Lutz, Drew McDermott, Katharine McGovern, Keith Oately, Suparna Rajaram, Henry L. Roediger III, Edmund T. Rolls, Daniel L. Schachter, William Seager, Daniel J. Simons, Scott D. Slotnick, Henry Stapp, Petra Stoerig, Ron Sun, Evan Thompson, C. Jason Throop, Rebecca Todd, Carlo Umiltà, David E. Warren, Joel Weinberger, Drew Westen, Dan Zahavi, Philip David Zelazo
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Powers of Genus two in Free Groups
- Charles C. Edmunds, Gerhard Rosenberger
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- Canadian Mathematical Bulletin / Volume 33 / Issue 3 / 01 September 1990
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- 20 November 2018, pp. 342-344
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- 01 September 1990
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The general problem is to express an element of the derived group of a free group as a product of a minimal number of commutators. An old conjecture is settled in the negative, and a new conjecture and a number of related questions are posed.
On words of minimal length under endomorphisms of a free group
- Charles C. Edmunds
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- Journal:
- Mathematical Proceedings of the Cambridge Philosophical Society / Volume 83 / Issue 2 / March 1978
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- 24 October 2008, pp. 191-194
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- March 1978
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In (1) it is remarked that theorem 5·10 is an analogue, for endomorphisms of a free group, of a result of A. Shenitzer ((3), theorem 2) about automorphisms. With this in mind, D. Solitar asked the author if there might also be an analogue for the corollary on page 276 of(3). There is such an analogue and, after a preliminary discussion of definitions and notation, we will describe it.
Products of Commutators as Products of Squares
- Charles C. Edmunds
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- Journal:
- Canadian Journal of Mathematics / Volume 27 / Issue 6 / 01 December 1975
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- 20 November 2018, pp. 1329-1335
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- 01 December 1975
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In any group G, the commutator subgroup G' is contained in G2, the subgroup of G generated by the squares in G. Thus any product of commutators can be written as a product of squares in G. For instance, the commutator [x, y] ( = xyx-1y-1) can be expressed as the product of three squares: [x, y] = x2(x-1y)2(y-1)2. Roger Lyndon and Morris Newman have made the interesting observation [4, Theorem 1] that, in this case, the number three is minimal in the sense that there are groups which contain commutators not expressible as the product of fewer than three squares.
A Short Combinatorial Proof of the Vaught Conjecture
- Charles C. Edmunds
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- Journal:
- Canadian Mathematical Bulletin / Volume 18 / Issue 4 / October 1975
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- 20 November 2018, pp. 607-608
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- October 1975
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In [5] R. C. Lyndon gave the first proof of the Vaught conjecture: that if a, b9 and c are elements of a free group F such that a2b2=c2, then ab=ba. Lyndon's proof has been followed by many alternative proofs and generalizations [1, 2, 3, 4, 6, 8, 9, 10, 11, 13, 14] all of which involve rather long combinatorial arguments or group theoretical arguments of a noncombinatorial nature.This note provides a short, purely combinatorial proof of the Vaught conjecture.