DEFINABLE SUBCATEGORIES OVER PURE SEMISIMPLE RINGS

2012 ◽  
Vol 11 (05) ◽  
pp. 1250099 ◽  
Author(s):  
NGUYEN VIET DUNG ◽  
JOSÉ LUIS GARCÍA
Keyword(s):  
Module Category ◽  
Indecomposable Modules ◽  
Functor Category ◽  
The Family ◽  
Finite Set ◽  
Semisimple Ring

Let R be a right pure semisimple ring, and [Formula: see text] be a family of sources of left almost split morphisms in mod-R. We study the definable subcategory [Formula: see text] in Mod-R determined by the family [Formula: see text], and show that [Formula: see text] has several nice properties similar to those of the category Mod-R. For example, its functor category [Formula: see text] is a module category, and preinjective objects of [Formula: see text] are sources of left almost split morphisms in [Formula: see text] and in mod-R. As an application, it is shown that if R is a right pure semisimple ring with no nonzero homomorphisms from preinjective modules to non-preinjective indecomposable modules in mod-R (in particular, if R is right pure semisimple hereditary), then any definable subcategory of Mod-R determined by a finite set of indecomposable right R-modules contains only finitely many non-isomorphic indecomposable modules.

scholarly journals FAIR FACILITY ALLOCATION IN EMERGENCY SERVICE SYSTEM

2020 ◽  
Vol 21 (4) ◽  
pp. 1058-1071
Author(s):  
Jaroslav Janáček ◽  
Lýdia Gábrišová ◽  
Miroslav Plevný
Keyword(s):  
Emergency Service ◽  
Service System ◽  
Allocation Problem ◽  
Dynamic Programming Principle ◽  
Objective Functions ◽  
The Family ◽  
Finite Set ◽  
Problem Instances ◽  
Original Approach ◽  
Average User

The request of equal accessibility must be respected to some extent when dealing with problems of designing or rebuilding of emergency service systems. Not only the disutility of the average user but also the disutility of the worst situated user must be taken into consideration. Respecting this principle is called fairness of system design. Unfairness can be mitigated to a certain extent by an appropriate fair allocation of additional facilities among the centres. In this article, two criteria of collective fairness are defined in the connection with the facility allocation problem. To solve the problems, we suggest a series of fast algorithms for solving of the allocation problem. This article extends the family of the original solving techniques based on branch-and-bound principle by newly suggested techniques, which exploit either dynamic programming principle or convexity and monotony of decreasing nonlinearities in objective functions. The resulting algorithms were tested and compared performing numerical experiments with real-sized problem instances. The new proposed algorithms outperform the original approach. The suggested methods are able to solve general min-sum and min-max problems, in which a limited number of facilities should be assigned to individual members from a finite set of providers.

A new type of coding problem

2001 ◽  
Vol 38 (1-4) ◽  
pp. 139-147 ◽  
Author(s):  
G. Brightwell ◽  
Gyula Katona
Keyword(s):  
The Family ◽  
Finite Set ◽  
New Type

Let X be an n-element finite set, and 0 Let X be an n-element finite set, and are pairs of disjoint k-element subsets of X (that is, {A1  =  B1} = {A2  =  B2} = k, A1 \ B1 = A2 \ B2 = Define the distance between these pairs by d(f A1;B1 g; f A2; B2 g)=min fj A1 - A2 j +  B1 - B2 j; j A1 - B2 j + j B1 - A2 jg . Itisknown ([2]) that the family of all k-element subsets of X can be paired (with one exception if their number is odd) in such a way that the distance between any two pairs is at least k. Here we answer questions arising for distances larger than k.

On cardinality questions concerning automaton mappings

2003 ◽  
Vol 40 (1-2) ◽  
pp. 71-82
Author(s):  
A. Ádám ◽  
M. Laczkovich
Keyword(s):  
The Family ◽  
Finite Set ◽  
Positive Length ◽  
Automaton Mapping

Let F+(X) be the set of words of positive length over a finite set X. By an automaton mapping (over (X,Y)) we understand a mapping of F+(X) into a finite set Y where |Y|?1). The family of all mappings over (X,Y) may be considered as an infinite automaton U having 2? states. U has at most 2^{2?} subautomata and at most 2? countable subautomata. We show that these bounds are actually attained.

scholarly journals Rings with a finite set of nonnilpotents

1979 ◽  
Vol 2 (1) ◽  
pp. 121-126 ◽  
Author(s):  
Mohan S. Putcha ◽  
Adil Yaqub
Keyword(s):  
Division Ring ◽  
Nonnegative Integer ◽  
Finite Ring ◽  
Structure Theory ◽  
Nilpotent Elements ◽  
Finite Set ◽  
Semisimple Ring ◽  
Nil Ring ◽  
Primitive Ring

LetRbe a ring and letNdenote the set of nilpotent elements ofR. Letnbe a nonnegative integer. The ringRis called aθn-ring if the number of elements inRwhich are not inNis at mostn. The following theorem is proved: IfRis aθn-ring, thenRis nil orRis finite. Conversely, ifRis a nil ring or a finite ring, thenRis aθn-ring for somen. The proof of this theorem uses the structure theory of rings, beginning with the division ring case, followed by the primitive ring case, and then the semisimple ring case. Finally, the general case is considered.

Comment on ‘A multiple process solution …’ (B. Macwhinney)

2004 ◽  
Vol 31 (4) ◽  
pp. 941-943
Author(s):  
PARTHA NIYOGI
Keyword(s):  
Learning Algorithm ◽  
Computational Learning Theory ◽  
Vc Dimension ◽  
Empirical Process Theory ◽  
Generalization Problem ◽  
The Family ◽  
Finite Set ◽  
Multiple Process ◽  
Probably Approximately Correct ◽  
Class Of Functions

The central question posed by the so called ‘logical problem of language acquisition’ is how it comes to be that children are able to GENERALIZE from a finite set of linguistic data to acquire (learn, develop, grow) a computational system (grammar) that applies to novel examples not encountered before. The difficulty of this generalization problem was first posed cogently by Gold and while Macwhinney discusses the Gold framework and the linguistic literature on this matter, it is worth noting that the Gold framework is not the only one. There are at least two important new sources of insight from computational learning theory in the decades following Gold that need to be kept in mind. First, there is the development of empirical process theory that forms the basis of any analysis of statistical learning (see summary in Vapnik, 1998). Applying this approach to language (see Niyogi, 1998 for a treatment), one concludes that the family of learnable grammars must have a finite Vapnik Chervonenkis (VC) dimension. The VC dimension is a combinatorial measure of the complexity of a class of functions. Grammars may be viewed as functions mapping sentences to their grammaticality value. In this more sophisticated sense of the VC dimension, the class of grammars must be constrained. Second, there is the development of the theory of computational complexity suggesting that while a learning algorithm might exist, it may not be efficient, i.e. run in polynomial time. These two developments come together in the influential Probably Approximately Correct (PAC) model (Valiant, 1984).

scholarly journals Pairs of Disjoint $q$-element Subsets Far from Each Other

10.37236/1606 ◽  
2001 ◽  
Vol 8 (2) ◽  
Author(s):  
Hikoe Enomoto ◽  
Gyula O. H. Katona
Keyword(s):  
Type Problem ◽  
Similar Nature ◽  
The Family ◽  
Finite Set

Let $n$ and $q$ be given integers and $X$ a finite set with $n$ elements. The following theorem is proved for $n>n_0(q)$. The family of all $q$-element subsets of $X$ can be partitioned into disjoint pairs (except possibly one if $n\choose q$ is odd), so that $|A_1\cap A_2|+|B_1\cap B_2|\leq q$, $|A_1\cap B_2|+|B_1\cap A_2| \leq q$ holds for any two such pairs $\{ A_1,B_1\} $ and $\{ A_2,B_2\} $. This is a sharpening of a theorem in [2]. It is also shown that this is a coding type problem, and several problems of similar nature are posed.

scholarly journals Intersecting Systems of Signed Sets

10.37236/959 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Peter Borg
Keyword(s):  
Property A ◽  
The Family ◽  
Finite Set

A family ${\cal F}$ of sets is said to be (strictly) EKR if no non-trivial intersecting sub-family of ${\cal F}$ is (as large as) larger than some trivial intersecting sub-family of ${\cal F}$. For a finite set $X := \{x_1, ..., x_{|X|}\}$ and an integer $k \geq 2$, we define ${\cal S}_{X,k}$ to be the family of signed sets given by $${\cal S}_{X,k} := \Big\{\big\{(x_1,a_1), ..., (x_{|X|},a_{|X|})\big\} \colon a_i \in [k], i = 1, ..., |X|\Big\}.$$ For a family ${\cal F}$, we define ${\cal S}_{{\cal F},k} := \bigcup_{F \in {\cal F}}{\cal S}_{F,k}$. We conjecture that for any ${\cal F}$ and $k \geq 2$, ${\cal S}_{{\cal F},k}$ is EKR, and strictly so unless $k=2$ and ${\cal F}$ has a particular property. A well-known result (stated by Meyer and proved in different ways by Deza and Frankl, Engel, Erdős et al., and Bollobás and Leader) supports this conjecture for ${\cal F} = {[n] \choose r}$. The main theorem in this paper generalises this result by establishing the truth of the conjecture for families ${\cal F}$ that are compressed with respect to some $f^* \in \bigcup_{F \in {\cal F}}F$ (i.e. $f \in F \in {\cal F}, f^* \notin F \Rightarrow (F \backslash \{f\}) \cup \{f^*\} \in {\cal F}$). We also confirm the conjecture for families ${\cal F}$ that are uniform and EKR.

scholarly journals GLOBAL PERIODICITY CONDITIONS FOR MAPS AND RECURRENCES VIA NORMAL FORMS

2013 ◽  
Vol 23 (11) ◽  
pp. 1350182 ◽  
Author(s):  
ANNA CIMA ◽  
ARMENGOL GASULL ◽  
VÍCTOR MAÑOSA
Keyword(s):  
Normal Forms ◽  
Necessary Conditions ◽  
Three Dimensional ◽  
Rational Maps ◽  
Normal Form Theory ◽  
The Family ◽  
Finite Set ◽  
Form Theory ◽  
Parametric Families ◽  
Formal Linearization

We face the problem of characterizing the periodic cases in parametric families of rational diffeomorphisms of 𝕂k, where 𝕂 is ℝ or ℂ, having a fixed point. Our approach relies on the Normal Form Theory, to obtain necessary conditions for the existence of a formal linearization of the map, and on the introduction of a suitable rational parametrization of the parameters of the family. Using these tools we can find a finite set of values p for which the map can be p-periodic, reducing the problem of finding the parameters for which the periodic cases appear to simple computations. We apply our results to several two- and three-dimensional classes of polynomial or rational maps. In particular, we find the global periodic cases for several Lyness-type recurrences.

scholarly journals Characterization of Infinite LSP Words and Endomorphisms Preserving the LSP Property

2019 ◽  
Vol 30 (01) ◽  
pp. 171-196 ◽  
Author(s):  
Gwenaël Richomme
Keyword(s):  
Infinite Words ◽  
Infinite Word ◽  
The Family ◽  
Finite Set ◽  
Special Factors

Answering a question of G. Fici, we give an [Formula: see text]-adic characterization of the family of infinite LSP words, that is, the family of infinite words having all their left special factors as prefixes. More precisely we provide a finite set of morphisms [Formula: see text] and an automaton [Formula: see text] such that an infinite word is LSP if and only if it is [Formula: see text]-adic and one of its directive words is recognizable by [Formula: see text]. Then we characterize the endomorphisms that preserve the property of being LSP for infinite words. This allows us to prove that there exists no set [Formula: see text] of endomorphisms for which the set of infinite LSP words corresponds to the set of [Formula: see text]-adic words. This implies that an automaton is required no matter which set of morphisms is used.

scholarly journals Equivalence classes of functions on finite sets

1982 ◽  
Vol 5 (4) ◽  
pp. 745-762
Author(s):  
Chong-Yun Chao ◽  
Caroline I. Deisher
Keyword(s):  
Equivalence Class ◽  
Equivalence Classes ◽  
Permutation Groups ◽  
Weak Equivalence ◽  
Finite Sets ◽  
The Family ◽  
Polya's Theorem ◽  
Finite Set ◽  
Pólya’S Theorem

By using Pólya's theorem of enumeration and de Bruijn's generalization of Pólya's theorem, we obtain the numbers of various weak equivalence classes of functions inRDrelative to permutation groupsGandHwhereRDis the set of all functions from a finite setDto a finite setR,Gacts onDandHacts onR. We present an algorithm for obtaining the equivalence classes of functions counted in de Bruijn's theorem, i.e., to determine which functions belong to the same equivalence class. We also use our algorithm to construct the family of non-isomorphicfm-graphs relative to a given group.

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