Note on a theorem of R. Baer

1949 ◽  
Vol 45 (3) ◽  
pp. 321-327 ◽  
Author(s):  
G. Higman
Keyword(s):  
Finite Number ◽  
Present Note ◽  
The Other ◽  
Fixed Integer ◽  
Other Hand

It is trivial that a group all of whose elements except the identity have order two is Abelian; and F. Levi and B. L. van der Waerden(1) have shown that a group all of whose elements except the identity have order three has class less than or equal to three. On the other hand, R. Baer(3) has shown that if the fact that all the elements of a group have orders dividing n implies a limitation on the class of the group, then n is a prime. The object of the present note is to extend this result by showing that if M is a fixed integer there are at most a finite number of prime powers n other than primes, such that the fact that all the elements of a group have orders dividing n implies a limitation on the class of its Mth derived group.

Minimal Size of Counters for (Real-Time) Multicounter Automata

2021 ◽  
Vol 181 (2-3) ◽  
pp. 99-127
Author(s):  
Viliam Geffert ◽  
Zuzana Bednárová
Keyword(s):  
Finite Number ◽  
Real Time ◽  
The Other ◽  
Binary Representation ◽  
Minimal Size ◽  
Nondeterministic Automaton ◽  
Other Hand ◽  
Weak Space ◽  
Fixed Constant

We show that, for automata using a finite number of counters, the minimal space that is required for accepting a nonregular language is (log n)ɛ. This is required for weak space bounds on the size of their counters, for real-time and one-way, and for nondeterministic and alternating versions of these automata. The same holds for two-way automata, independent of whether they work with strong or weak space bounds, and of whether they are deterministic, nondeterministic, or alternating. (Here ɛ denotes an arbitrarily small—but fixed—constant; the “space” refers to the values stored in the counters, rather than to the lengths of their binary representation.) On the other hand, we show that the minimal space required for accepting a nonregular language is nɛ for multicounter automata with strong space bounds, both for real-time and one-way versions, independent of whether they are deterministic, nondeterministic, or alternating, and also for real-time and one-way deterministic multicounter automata with weak space bounds. All these bounds are optimal both for unary and general nonregular languages. However, for automata equipped with only one counter, it was known that one-way nondeterministic automata cannot recognize any unary nonregular languages at all, even if the size of the counter is not restricted, while, with weak space bound log n, we present a real-time nondeterministic automaton recognizing a binary nonregular language here.

A System of Linear Equations with an Infinity of Unknowns

1924 ◽  
Vol 22 (3) ◽  
pp. 282-286
Author(s):  
E. C. Titchmarsh
Keyword(s):  
Present Note ◽  
Linear Equations ◽  
The Other ◽  
System Of Linear Equations ◽  
Other Hand ◽  
Image Position ◽  
Monotonic Character

I have collected in the present note some theorems regarding the solution of a certain system of linear equations with an infinity of unknowns. The general form of the equations isthe numbers a1, a2, … c1, c2, … being given. Equations of this type are of course well known; but in studying them it is generally assumed that the series depend for convergence on the convergence-exponent of the sequences involved, e.g. that and are convergent. No assumptions of this kind are made here, and in fact the series need not be absolutely convergent. On the other hand rather special assumptions are made with regard to the monotonic character of the sequences an and cn.

The No-Three-In-Line Problem

1968 ◽  
Vol 11 (4) ◽  
pp. 527-531 ◽  
Author(s):  
Richard K. Guy ◽  
Patrick A. Kelly
Keyword(s):  
Finite Number ◽  
The Other ◽  
Probabilistic Argument ◽  
Number Of Solutions ◽  
Other Hand ◽  
Image Position

Let Sn be the set of n2 points with integer coordinates n (x, y), 1 ≤ x, y <n. Let fn be the maximum cardinal of a subset T of Sn such that no three points of T are collinear. Clearly fn < 2n.For 2 ≤ n ≤ 10 it is known ([2], [3] for n = 8, [ 1] for n = 10, also [4], [6]) that fn = 2n, and that this bound is attained in 1, 1, 4, 5, 11, 22, 57, 51 and 156 distinct configurations for these nine values of n. On the other hand, P. Erdös [7] has pointed out that if n is prime, fn ≥ n, since the n points (x, x2) reduced modulo n have no three collinear. We give a probabilistic argument to support the conjecture that there is only a finite number of solutions to the no-three-in-line problem. More specifically, we conjecture that

scholarly journals The Core of Voting Games: A Partition Approach

2015 ◽  
Vol 17 (03) ◽  
pp. 1550001
Author(s):  
Aymeric Lardon
Keyword(s):  
Finite Number ◽  
Coalition Formation ◽  
The Other ◽  
Veto Power ◽  
The Core ◽  
Voting Games ◽  
Other Hand ◽  
The One

The purpose of this paper is to analyze a class of voting games in a partition approach. We consider a society in which coalitions can be formed and where a finite number of voters have to choose among a set of alternatives. A coalition is winning if it can veto any proposed alternative. In our model, the veto power of a coalition is dependent on the coalition formation of the outsiders. We show that whether or not the core is non-empty depends crucially on the expectations of each coalition regarding outsiders' behavior when it wishes to veto an alternative. On the one hand, if each coalition has pessimistic expectations, then the core is non-empty if and only if the dimension of the set of alternatives is equal to one. On the other hand, if each coalition has optimistic expectations, the non-emptiness of the core is not ensured.

Automata with cyclic move operations for picture languages

2018 ◽  
Vol 52 (2-3-4) ◽  
pp. 235-251
Author(s):  
Friedrich Otto ◽  
František Mráz
Keyword(s):  
Finite Number ◽  
Expressive Power ◽  
Single Step ◽  
The Other ◽  
Cyclic Extension ◽  
Two Dimensional ◽  
Other Hand ◽  
Picture Languages ◽  
Restarting Automaton

Here, we study the cyclic extensions of Sgraffito automata and of deterministic two-dimensional two-way ordered restarting automata for picture languages. Such a cyclically extended automaton can move in a single step from the last column (or row) of a picture to the first column (or row). For Sgraffito automata, we show that this cyclic extension does not increase the expressive power of the model, while for deterministic two-dimensional two-way restarting automata, the expressive power is strictly increased by allowing cyclic moves. In fact, for the latter automata, we take the number of allowed cyclic moves in any column or row as a parameter, and we show that already with a single cyclic move per column (or row) the deterministic two-dimensional extended two-way restarting automaton can be simulated. On the other hand, we show that two cyclic moves per column or row already give the same expressive power as any finite number of cyclic moves.

On the idea of frequency

1925 ◽  
Vol 22 (5) ◽  
pp. 726-727 ◽  
Author(s):  
W. Burnside
Keyword(s):  
Finite Number ◽  
The Other ◽  
Finite Collection ◽  
Other Hand ◽  
Statistical Table

A statistical table is in effect a classification of a finite number, N, of objects in respect of a finite number of different classes, A, B, C, … It is assumed that unambiguous rules have been laid down by which it is possible to determine whether any particular one of the objects does or does not belong to any particular one of the classes. The application of these rules to a particular object does not depend on the fact that that object is one of a finite number of N objects, so that the process of classification may be started on a non-finite collection of objects. When the collection of objects is non-finite the process can never end. On the other hand when the collection is finite the process must end; and when completed it will determine how many of the objects belong to any particular class. If in this way it is found that N1 (≤ N) of the objects belong to class A, the proper fraction N1/N is spoken of as the frequency of class A in the collection. In particular cases it may be zero or unity. In general it is a rational proper fraction. For its determination in complicated cases it may be convenient to suppose the collection to be arranged in some special way, but its value is absolutely independent of any such particular arrangement.

scholarly journals On Some Divisibility Properties of

1964 ◽  
Vol 7 (4) ◽  
pp. 513-518 ◽  
Author(s):  
P. Erdös
Keyword(s):  
Simple Proof ◽  
Present Note ◽  
The Other ◽  
Other Hand ◽  
Image Position ◽  
Divisibility Properties

L. Moser [3] recently gave a very simple proof that1.has no solutions. In the present note we shall first of all prove that for , which by the fact that there is a prime p satisfying n < p ≤ 2n immediately implies that2.has no solutions. It is easy to see on the other hand that3.has infinitely many non-trivial solutions.

Lattice Points of Cut Cones

1994 ◽  
Vol 3 (2) ◽  
pp. 191-214 ◽  
Author(s):  
Michel Deza ◽  
Viatcheslav Grishukhin
Keyword(s):  
Finite Number ◽  
Complete Graph ◽  
Lattice Points ◽  
The Other ◽  
Other Hand

Let ℝ+(ℋn),ℤ(ℋn),ℤ+(ℋn) be, respectively, the cone over ℝ, the lattice and the cone over ℤ, generated by all cuts of the complete graph on n nodes. For i ≥ 0, let has exactly i realizations in ℤ+(ℋn)}. We show that is infinite, except for the undecided case and empty and for i = 0, n ≤ 5 and for i ≥ 2, n ≤ 3. The set contains 0,1,∞ nonsimplicial points for n ≤ 4, n = 5, n ≥ 6, respectively. On the other hand, there exists a finite number t(n) such that t(n)d ∈ ℤ+(ℋn) for any ; we also estimate such scales for classes of points. We construct families of points of and ℤ+(ℋn), especially on a 0-lifting of a simplicial facet, and points d ∈ ℝ+(ℋn) with di, n = t for 1 ≤ i ≤ n − 1.

scholarly journals Note on Legendre's and Bertrand's Proofs of the Parallel-Postulate by Infinite Areas

1911 ◽  
Vol 30 ◽  
pp. 31-36
Author(s):  
D. M. Y. Sommerville
Keyword(s):  
Finite Number ◽  
Plane Surface ◽  
Second Order ◽  
The Other ◽  
Linear Segment ◽  
First Order ◽  
Other Hand ◽  
Straight Lines

One of the most plausible of the host of “proofs” that have ever been offered for Euclid's parallel-postulate is that known as Bertrand's, which is based upon a consideration of infinite areas. The area of the whole plane being regarded as an infinity of the second order, the area of a strip of plane surface bounded by a linear segment AB and the rays AA′, BB perpendicular to AB is an infinity of the first order, since a single infinity of such strips is required to cover the plane. On the other hand, the area contained between two intersecting straight lines is an infinity of the same order as the plane, since the plane can be covered by a finite number of such sectors. Hence if AP is drawn making any angle, however small, with AA′, the area A′AP, an infinity of the second order, cannot be contained within the area A′ABB′, an infinity of the first order, and therefore AP must cut BB′. And this is just Euclid's postulate.

scholarly journals Parameterized Synthesis for Fragments of First-Order Logic Over Data Words

2020 ◽  
pp. 97-118
Author(s):  
Béatrice Bérard ◽  
Benedikt Bollig ◽  
Mathieu Lehaut ◽  
Nathalie Sznajder
Keyword(s):  
Finite Number ◽  
Classical Case ◽  
Order Logic ◽  
The Other ◽  
Synthesis Problem ◽  
First Order Logic ◽  
First Order ◽  
Bounded Number ◽  
Other Hand

AbstractWe study the synthesis problem for systems with a parameterized number of processes. As in the classical case due to Church, the system selects actions depending on the program run so far, with the aim of fulfilling a given specification. The difficulty is that, at the same time, the environment executes actions that the system cannot control. In contrast to the case of fixed, finite alphabets, here we consider the case of parameterized alphabets. An alphabet reflects the number of processes, which is static but unknown. The synthesis problem then asks whether there is a finite number of processes for which the system can satisfy the specification. This variant is already undecidable for very limited logics. Therefore, we consider a first-order logic without the order on word positions. We show that even in this restricted case synthesis is undecidable if both the system and the environment have access to all processes. On the other hand, we prove that the problem is decidable if the environment only has access to a bounded number of processes. In that case, there is even a cutoff meaning that it is enough to examine a bounded number of process architectures to solve the synthesis problem.

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