scholarly journals Retract Rationality and Algebraic Tori

2019 ◽  
Vol 63 (1) ◽  
pp. 173-186 ◽  
Author(s):  
Federico Scavia
Keyword(s):  
Prime Number ◽  
Affirmative Answer ◽  
Positive Answer ◽  
Algebraic Tori ◽  
Finite Set ◽  
Multiplicative Type

AbstractFor any prime number $p$ and field $k$, we characterize the $p$-retract rationality of an algebraic $k$-torus in terms of its character lattice. We show that a $k$-torus is retract rational if and only if it is $p$-retract rational for every prime $p$, and that the Noether problem for retract rationality for a group of multiplicative type $G$ has an affirmative answer for $G$ if and only if the Noether problem for $p$-retract rationality for $G$ has a positive answer for all $p$. For every finite set of primes $S$ we give examples of tori that are $p$-retract rational if and only if $p\notin S$.

Nonrecursive tilings of the plane. I

1974 ◽  
Vol 39 (2) ◽  
pp. 283-285 ◽  
Author(s):  
William Hanf
Keyword(s):  
Turing Machine ◽  
Recursive Function ◽  
Positive Answer ◽  
Additional Restriction ◽  
Detailed History ◽  
Finite Set

A finite set of tiles (unit squares with colored edges) is said to tile the plane if there exists an arrangement of translated (but not rotated or reflected) copies of the squares which fill the plane in such a way that abutting edges of the squares have the same color. The problem of whether there exists a finite set of tiles which can be used to tile the plane but not in any periodic fashion was proposed by Hao Wang [9] and solved by Robert Berger [1]. Raphael Robinson [7] gives a more detailed history and a very economical solution to this and related problems; we will assume that the reader is familiar with §4 of [7]. In 1971, Dale Myers asked whether there exists a finite set of tiles which can tile the plane but not in any recursive fashion. If we make an additional restriction (called the origin constraint) that a given tile must be used at least once, then the positive answer is given by the main theorem of this paper. Using the Turing machine constructed here and a more complicated version of Berger and Robinson's construction, Myers [5] has recently solved the problem without the origin constraint.Given a finite set of tiles T1, …, Tn, we can describe a tiling of the plane by a function f of two variables ranging over the integers. f(i, j) = k specifies that the tile Tk is to be placed at the position in the plane with coordinates (i, j). The tiling will be said to be recursive if f is a recursive function.

Perturbations of AF-Algebras

1979 ◽  
Vol 31 (5) ◽  
pp. 1012-1016 ◽  
Author(s):  
John Phillips ◽  
Iain Raeburn
Keyword(s):  
Hilbert Space ◽  
Unit Ball ◽  
Von Neumann Algebra ◽  
Unitary Operator ◽  
Affirmative Answer ◽  
Positive Answer ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Image Position ◽  
Af Algebras

Let A and B be C*-algebras acting on a Hilbert space H, and letwhere A1 is the unit ball in A and d(a, B1) denotes the distance of a from B1. We shall consider the following problem: if ‖A – B‖ is sufficiently small, does it follow that there is a unitary operator u such that uAu* = B?Such questions were first considered by Kadison and Kastler in [9], and have received considerable attention. In particular in the case where A is an approximately finite-dimensional (or hyperfinite) von Neumann algebra, the question has an affirmative answer (cf [3], [8], [12]). We shall show that in the case where A and B are approximately finite-dimensional C*-algebras (AF-algebras) the problem also has a positive answer.

On tamely ramified pro-p-extensions over -extensions of

2013 ◽  
Vol 156 (2) ◽  
pp. 281-294
Author(s):  
TSUYOSHI ITOH ◽  
YASUSHI MIZUSAWA
Keyword(s):  
Galois Group ◽  
Prime Number ◽  
Number Field ◽  
Rational Number ◽  
Prime Numbers ◽  
Finite Set

AbstractFor an odd prime number p and a finite set S of prime numbers congruent to 1 modulo p, we consider the Galois group of the maximal pro-p-extension unramified outside S over the ${\mathbb Z}_p$-extension of the rational number field. In this paper, we classify all S such that the Galois group is a metacyclic pro-p group.

The Lattice of all Topologies is Complemented

1968 ◽  
Vol 20 ◽  
pp. 805-807 ◽  
Author(s):  
A. C. M. Van Rooij
Keyword(s):  
Affirmative Answer ◽  
Alternative Proof ◽  
Finite Set

In (3), J. Hartmanis raised the question whether the lattice of all topologies in a given set is complemented and gave the affirmative answer for the case of a finite set. H. Gaifman (2), has extended this result to denumerable sets. Using Gaifman's paper, Anne K. Steiner (4) has proved that the lattice is always complemented. Our aim in this article is to give an alternative proof, independent of Gaifman's results. So far, Steiner's proof has not been available to the author.

Locally finite minimal non-FC-groups

1989 ◽  
Vol 105 (3) ◽  
pp. 417-420 ◽  
Author(s):  
Mahmut Kuzucuoglu ◽  
Richard E. Phillips
Keyword(s):  
Simple Group ◽  
Proper Subgroup ◽  
Positive Answer ◽  
Locally Finite ◽  
Finite Set

We recall that a group G is an FC-group if for every x∈G the set of conjugates {xg|g∈G} is a finite set. Our interest here is with those groups G which are not FC groups while every proper subgroup of G is an FC-group: such groups are called minimal non-FC-groups. Locally finite minimal non-FC-groups with (G ≠ G′ are studied in [1] and the structure of these groups is reasonably well understood. In [2] Belyaev has shown that a perfect, locally finite, minimal non-FC-group is either a simple group or a p-group for some prime p. Here we make use of the results of [5] to refine the result of Belyaev and provide a positive answer to problem 5·1 of [11]; in particular, we prove the followingTheorem. There exists no simple, locally finite, minimal non-FC-group.

ON A PARTIAL AFFIRMATIVE ANSWER FOR A PĂUN'S CONJECTURE

2011 ◽  
Vol 22 (01) ◽  
pp. 55-64 ◽  
Author(s):  
IGNACIO PÉREZ-HURTADO ◽  
MARIO J. PÉREZ-JIMÉNEZ ◽  
AGUSTÍN RISCOS-NÚÑEZ ◽  
MIGUEL A. GUTIÉRREZ-NARANJO ◽  
MIQUEL RIUS-FONT
Keyword(s):  
Affirmative Answer ◽  
Positive Answer ◽  
Dependency Graph ◽  
P Systems ◽  
Complete Proof ◽  
Active Membranes ◽  
Hard Problems

At the beginning of 2005, Gheorghe Păun formulated a conjecture stating that in the framework of recognizer P systems with active membranes (evolution rules, communication rules, dissolution rules and division rules for elementary membranes), polarizations cannot be avoided in order to solve computationally hard problems efficiently (assuming that P ≠ NP ). At the middle of 2005, a partial positive answer was given, proving that the conjecture holds if dissolution rules are forbidden. In this paper we give a detailed and complete proof of this result modifying slightly the notion of dependency graph associated with recognizer P systems.

scholarly journals Numerical Investigation of Aeroelastic Mode Distribution for Aircraft Wing Model in Subsonic Air Flow

2010 ◽  
Vol 2010 ◽  
pp. 1-23 ◽  
Author(s):  
Marianna A. Shubov ◽  
Stephen Wineberg ◽  
Robert Holt
Keyword(s):  
Boundary Value Problem ◽  
Numerical Investigation ◽  
Air Flow ◽  
Affirmative Answer ◽  
Positive Answer ◽  
Aircraft Wing ◽  
Aircraft Wings ◽  
Flutter Speed ◽  
Wing Model ◽  
Analytical Formulas

In this paper, the numerical results on two problems originated in aircraft wing modeling have been presented.The first problemis concerned with the approximation to the set of the aeroelastic modes, which are the eigenvalues of a certain boundary-value problem. The affirmative answer is given to the following question: can the leading asymptotical terms in the analytical formulas be used as reasonably accurate description of the aeroelastic modes? The positive answer means that these leading terms can be used by engineers for practical calculations.The second problemis concerned with the flutter phenomena in aircraft wings in a subsonic, incompressible, inviscid air flow. It has been shown numerically that there exists a pair of the aeroelastic modes whose behavior depends on a speed of an air flow. Namely, when the speed increases, the distance between the modes tends to zero, and at some speed that can be treated as the flutter speed these two modes merge into one double mode.

scholarly journals Wild Primes of a Higher Degree Self-Equivalence of a Number Field

2016 ◽  
Vol 30 (1) ◽  
pp. 17-38
Author(s):  
Alfred Czogała ◽  
Beata Rothkegel ◽  
Andrzej Sładek
Keyword(s):  
Prime Number ◽  
Number Field ◽  
Class Group ◽  
Sufficient Condition ◽  
Finite Set

AbstractLet ℓ > 2 be a prime number. Let K be a number field containing a unique ℓ-adic prime and assume that its class is an ℓth power in the class group CK. The main theorem of the paper gives a sufficient condition for a finite set of primes of K to be the wild set of some Hilbert self-equivalence of K of degree ℓ.

Remarks on Complementation in the Lattice of all Topologies

1966 ◽  
Vol 18 ◽  
pp. 83-88 ◽  
Author(s):  
Haim Gaifman
Keyword(s):  
Positive Answer ◽  
Finite Set

Our aim is to prove that certain topologies have complements in the lattice of all the topologies on a given set. Lattices of topologies were studied in (1-8). In (7) Hartmanis points out that the lattice of all the topologies on a finite set is complemented and poses the question whether this is so if the set is infinite. A positive answer is given here for denumerable sets. This result was announced in (6). The case of higher powers remains unsettled, although quite a few topologies turn out to have complements. As far as the author knows, no one has proved the existence of a topology that has no complement.

scholarly journals An affirmative answer to two questions concerning special case of Simsek numbers and open problems

2020 ◽  
Vol 14 (1) ◽  
pp. 94-105
Author(s):  
Mouloud Goubi
Keyword(s):  
Affirmative Answer ◽  
Positive Answer ◽  
Euler Numbers ◽  
Open Problems ◽  
Negative Order ◽  
Special Case

The purpose of this work is to give a positive answer to two questions asked by professor Yilmaz Simsek in a recent paper [6] concerning special numbers B(n,k) for computing negative order Euler numbers.

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