scholarly journals The generalized Auslander–Reiten duality on an exact category

2018 ◽  
Vol 17 (12) ◽  
pp. 1850227 ◽  
Author(s):  
Pengjie Jiao
Keyword(s):  
Finite Interval ◽  
Indecomposable Object ◽  
Locally Finite ◽  
Exact Category ◽  
The Third ◽  
Stable Categories ◽  
Finitely Presented

We introduce a notion of generalized Auslander–Reiten duality on a Hom-finite Krull–Schmidt exact category [Formula: see text]. This duality induces the generalized Auslander–Reiten translation functors [Formula: see text] and [Formula: see text]. They are mutually quasi-inverse equivalences between the stable categories of two full subcategories [Formula: see text] and [Formula: see text] of [Formula: see text]. A non-projective indecomposable object lies in the domain of [Formula: see text] if and only if it appears as the third term of an almost split conflation; dually, a non-injective indecomposable object lies in the domain of [Formula: see text] if and only if it appears as the first term of an almost split conflation. We study the generalized Auslander–Reiten duality on the category of finitely presented representations of locally finite interval-finite quivers.

Noncommutative G-semihereditary rings

2018 ◽  
Vol 17 (01) ◽  
pp. 1850014 ◽  
Author(s):  
Jian Wang ◽  
Yunxia Li ◽  
Jiangsheng Hu
Keyword(s):  
Associative Ring ◽  
Sufficient Condition ◽  
Projective Modules ◽  
The Third ◽  
Flat Modules ◽  
Gorenstein Projective Modules ◽  
Direct Limits ◽  
Gorenstein Flat ◽  
Finitely Presented ◽  
Semihereditary Rings

In this paper, we introduce and study left (right) [Formula: see text]-semihereditary rings over any associative ring, and these rings are exactly [Formula: see text]-semihereditary rings defined by Mahdou and Tamekkante provided that [Formula: see text] is a commutative ring. Some new characterizations of left [Formula: see text]-semihereditary rings are given. Applications go in three directions. The first is to give a sufficient condition when a finitely presented right [Formula: see text]-module is Gorenstein flat if and only if it is Gorenstein projective provided that [Formula: see text] is left coherent. The second is to investigate the relationships between Gorenstein flat modules and direct limits of finitely presented Gorenstein projective modules. The third is to obtain some new characterizations of semihereditary rings, [Formula: see text]-[Formula: see text] rings and [Formula: see text] rings.

scholarly journals On the Computation of Certain Homotopical Functors

1998 ◽  
Vol 1 ◽  
pp. 25-41 ◽  
Author(s):  
Graham Ellis
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Integral Homology ◽  
Finitely Presented Groups ◽  
The Third ◽  
Finite Module ◽  
Nonabelian Tensor ◽  
A Finite Group ◽  
Finitely Presented

AbstractThis paper provides details of a Magma computer program for calculating various homotopy-theoretic functors, defined on finitely presented groups. A copy of the program is included as an Add-On. The program can be used to compute: the nonabelian tensor product of two finite groups, the first homology of a finite group with coefficients in the arbirary finite module, the second integral homology of a finite group relative to its normal subgroup, the third homology of the finite p-group with coefficients in Zp, Baer invariants of a finite group, and the capability and terminality of a finite group. Various other related constructions can also be computed.

scholarly journals Hamiltonicity in Locally Finite Graphs: Two Extensions and a Counterexample

10.37236/6773 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Karl Heuer
Keyword(s):  
Finite Graph ◽  
Infinite Graphs ◽  
Alternative Proof ◽  
Sufficient Condition ◽  
Locally Finite ◽  
Finite Graphs ◽  
The Third ◽  
Locally Finite Graph ◽  
Locally Finite Graphs ◽  
A Minor

We state a sufficient condition for the square of a locally finite graph to contain a Hamilton circle, extending a result of Harary and Schwenk about finite graphs. We also give an alternative proof of an extension to locally finite graphs of the result of Chartrand and Harary that a finite graph not containing $K^4$ or $K_{2,3}$ as a minor is Hamiltonian if and only if it is $2$-connected. We show furthermore that, if a Hamilton circle exists in such a graph, then it is unique and spanned by the $2$-contractible edges. The third result of this paper is a construction of a graph which answers positively the question of Mohar whether regular infinite graphs with a unique Hamilton circle exist.

Subgroups of finitely presented groups

1961 ◽  
Vol 262 (1311) ◽  
pp. 455-475 ◽  
Keyword(s):  
Finite Group ◽  
Genetic Characterization ◽  
Recursively Enumerable Set ◽  
Locally Finite ◽  
Finitely Presented Groups ◽  
Defining Relations ◽  
Finitely Presented Group ◽  
Recursively Enumerable ◽  
Finitely Presented

The main theorem of this paper states that a finitely generated group can be embedded in a finitely presented group if and only if it has a recursively enumerable set of defining relations. It follows that every countable A belian group, and every countable locally finite group can be so embedded; and that there exists a finitely presented group which simultaneously embeds all finitely presented groups. A nother corollary of the theorem is the known fact that there exist finitely presented groups with recursively insoluble word problem . A by-product of the proof is a genetic characterization of the recursively enumerable subsets of a suitable effectively enumerable set.

scholarly journals Planar Transitive Graphs

10.37236/7888 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Matthias Hamann
Keyword(s):  
Fundamental Group ◽  
Homology Group ◽  
Cayley Graphs ◽  
Finitely Generated ◽  
Transitive Graph ◽  
Locally Finite ◽  
Transitive Graphs ◽  
Finitely Presented

We prove that the first homology group of every planar locally finite transitive graph $G$ is finitely generated as an $\Aut(G)$-module and we prove a similar result for the fundamental group of locally finite planar Cayley graphs. Corollaries of these results include Droms's theorem that planar groups are finitely presented and Dunwoody's theorem that planar locally finite transitive graphs are accessible. 

HOMOLOGY OF GROUPS AND THIRD BUSY BEAVER FUNCTION

2010 ◽  
Vol 20 (06) ◽  
pp. 769-791
Author(s):  
PASCAL MICHEL
Keyword(s):  
Finite Rank ◽  
Homology Group ◽  
Turing Degree ◽  
Finitely Presented Groups ◽  
The Third ◽  
Finitely Presented Group ◽  
Finitely Presented ◽  
Standing Problem

In 2007, Nabutovsky and Weinberger provided a solution to a long-standing problem: to find naturally defined functions that grow faster than any function with Turing degree of unsolvability 0'. They considered the functions bksuch that, for a natural integer N, bk(N) is the rank of the kth homology group Hk(G) of maximum finite rank, among the finitely presented groups G with presentation length ≤ N. They proved that, for k ≥ 3, function bkgrows as the third busy beaver function, and so grows faster than any function with degree of unsolvability 0″.Can more be said about these functions bk? We give some results on the function b2, we study the challenge of computing Hk(G) for a finitely presented group G, and we compute bk(N) for small values of N.

scholarly journals On triangle equivalences of stable categories

2019 ◽  
Vol 150 (2) ◽  
pp. 955-974
Author(s):  
Zhenxing Di ◽  
Zhongkui Liu ◽  
Jiaqun Wei
Keyword(s):  
Approximation Theory ◽  
Noetherian Rings ◽  
Triangulated Category ◽  
Gorenstein Rings ◽  
Gorenstein Algebras ◽  
The Third ◽  
Stable Categories

AbstractWe apply the Auslander–Buchweitz approximation theory to show that the Iyama and Yoshino's subfactor triangulated category can be realized as a triangulated quotient. Applications of this realization go in three directions. Firstly, we recover both a result of Iyama and Yang and a result of the third author. Secondly, we extend the classical Buchweitz's triangle equivalence from Iwanaga–Gorenstein rings to Noetherian rings. Finally, we obtain the converse of Buchweitz's triangle equivalence and a result of Beligiannis, and give characterizations for Iwanaga–Gorenstein rings and Gorenstein algebras.

Progress report on 21-cm observations at low latitude between longitudes (l 11) 0° and 66°

1967 ◽  
Vol 31 ◽  
pp. 177-179
Author(s):  
W. W. Shane
Keyword(s):  
Preliminary Report ◽  
Progress Report ◽  
Galactic Centre ◽  
Low Latitude ◽  
The Third ◽  
Introductory Lecture ◽  
Centre Region

In the course of several 21-cm observing programmes being carried out by the Leiden Observatory with the 25-meter telescope at Dwingeloo, a fairly complete, though inhomogeneous, survey of the regionl11= 0° to 66° at low galactic latitudes is becoming available. The essential data on this survey are presented in Table 1. Oort (1967) has given a preliminary report on the first and third investigations. The third is discussed briefly by Kerr in his introductory lecture on the galactic centre region (Paper 42). Burton (1966) has published provisional results of the fifth investigation, and I have discussed the sixth in Paper 19. All of the observations listed in the table have been completed, but we plan to extend investigation 3 to a much finer grid of positions.

The orbit of Pluto over a long interval of time

1966 ◽  
Vol 25 ◽  
pp. 227-229 ◽  
Author(s):  
D. Brouwer
Keyword(s):  
Periodic Orbit ◽  
Numerical Integration ◽  
Restricted Problem ◽  
Three Dimensional ◽  
The Third ◽  
Circular Restricted Problem ◽  
Resonance Relation

The paper presents a summary of the results obtained by C. J. Cohen and E. C. Hubbard, who established by numerical integration that a resonance relation exists between the orbits of Neptune and Pluto. The problem may be explored further by approximating the motion of Pluto by that of a particle with negligible mass in the three-dimensional (circular) restricted problem. The mass of Pluto and the eccentricity of Neptune's orbit are ignored in this approximation. Significant features of the problem appear to be the presence of two critical arguments and the possibility that the orbit may be related to a periodic orbit of the third kind.

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