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.

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.

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.

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 $p\notin S$.

Mars: was There an Ancient Eden?

1997 ◽  
Vol 161 ◽  
pp. 203-218 ◽  
Author(s):  
Tobias C. Owen
Keyword(s):  
Positive Answer ◽  
Biogenic Elements ◽  
Habitable Zones ◽  
The Face ◽  
The Origin Of Life ◽  
Early Mars ◽  
Favorable Conditions ◽  
Open Question ◽  
Rocky Planets ◽  
Impact Erosion

AbstractThe clear evidence of water erosion on the surface of Mars suggests an early climate much more clement than the present one. Using a model for the origin of inner planet atmospheres by icy planetesimal impact, it is possible to reconstruct the original volatile inventory on Mars, starting from the thin atmosphere we observe today. Evidence for cometary impact can be found in the present abundances and isotope ratios of gases in the atmosphere and in SNC meteorites. If we invoke impact erosion to account for the present excess of129Xe, we predict an early inventory equivalent to at least 7.5 bars of CO2. This reservoir of volatiles is adequate to produce a substantial greenhouse effect, provided there is some small addition of SO2(volcanoes) or reduced gases (cometary impact). Thus it seems likely that conditions on early Mars were suitable for the origin of life – biogenic elements and liquid water were present at favorable conditions of pressure and temperature. Whether life began on Mars remains an open question, receiving hints of a positive answer from recent work on one of the Martian meteorites. The implications for habitable zones around other stars include the need to have rocky planets with sufficient mass to preserve atmospheres in the face of intensive early bombardment.

Normative Concepts

2017 ◽  
Author(s):  
Matti Eklund
Keyword(s):  
Positive Answer ◽  
Thick Concepts ◽  
Normative Concepts ◽  
The Right ◽  
The Way

What is it for a concept to be normative? Some possible answers are explored and rejected, among them that a concept is normative if it ascribes a normative property. The positive answer defended is that a concept is normative if it is in the right way associated with a normative use. Among issues discussed along the way are the nature of analyticity, and there being a notion of analyticity—what I call semantic analyticity—such that a statement can be analytic in this sense while failing to be true. Considerations regarding thick concepts and slurs are brought to bear on the issues that come up.

Epilogue

2017 ◽  
Author(s):  
Robert Schütze
Keyword(s):  
Internal Market ◽  
Positive Answer ◽  
The Past ◽  
Output Legitimacy ◽  
The Eu

Can the judicial creation of the EU internal market be justified? A famous—positive—answer has, in the past, been suggested by Miguel Maduro’s We the Court; and the first section explores the credentials of his ‘majoritarian activism’ thesis. The second section surveys alternative forms of legitimacy, such as ‘output legitimacy’ and ‘messianic legitimacy’, but it also offers a new Kantian approach to the legitimacy question.

History, the Hindu Right, and Subversion of Brahmanical-Hindu Political Thought

2019 ◽  
pp. 239-260
Author(s):  
Stuart Gray
Keyword(s):  
Political Theory ◽  
Political Thought ◽  
Positive Answer ◽  
Source Material ◽  
Caste System ◽  
Hindu Tradition ◽  
Early Twenty ◽  
Indian Nationalism ◽  
Late Twentieth ◽  
Locus Classicus

How can scholars critically engage premodern Indic traditions without falling prey to Hindu conservatism or Brahmanical-Hindu apologism? This question is pressing for Indic political theory and contemporary Indian democracy because of ethnically exclusivist, Hindu nationalist movements that have emerged in the late twentieth and early twenty-first centuries. This chapter argues that a positive answer to the question must begin by taking seriously the tremendous pluralism in India’s political and philosophical history, which requires systematically engaging with premodern source material and uncovering the internal pluralism within a longer and larger Brahmanical-Hindu tradition of political thought. The author explains how it is both possible and politically necessary to internally subvert Brahmanical-Hindu political thought, which can help diffuse essentialist and exclusivist arguments coming from the Hindu right. Locating such plurality and engaging in internal subversion can help challenge historical justifications for Indian nationalism and contribute to decolonization, thus contesting the Hindu right on its own conceptual and genealogical turf. To advance this argument, the author provides a critical reinterpretation of the infamous “Puruṣa Sūkta,” which is often viewed as the locus classicus of the modern caste system, providing a novel interpretation that challenges caste hierarchy and supplies new resources for democratic thought and practice in India.

Non-repetitive sequences

1970 ◽  
Vol 68 (2) ◽  
pp. 267-274 ◽  
Author(s):  
P. A. B. Pleasants
Keyword(s):  
Repetitive Sequences ◽  
Finite Set ◽  
Infinite Sequences

This note is concerned with infinite sequences whose terms are chosen from a finite set of symbols. A segment of such a sequence is a set of one or more consecutive terms, and a repetition is a pair of finite segments that are adjacent and identical. A non-repetitive sequence is one that contains no repetitions.

The inverse problem of recovering the coefficients of a differential equations on a graph

2020 ◽  
Vol 28 (5) ◽  
pp. 727-738
Author(s):  
Victor Sadovnichii ◽  
Yaudat Talgatovich Sultanaev ◽  
Azamat Akhtyamov
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Finite Number ◽  
Characteristic Equation ◽  
Arbitrary Number ◽  
New Class ◽  
Finite Set

AbstractWe consider a new class of inverse problems on the recovery of the coefficients of differential equations from a finite set of eigenvalues of a boundary value problem with unseparated boundary conditions. A finite number of eigenvalues is possible only for problems in which the roots of the characteristic equation are multiple. The article describes solutions to such a problem for equations of the second, third, and fourth orders on a graph with three, four, and five edges. The inverse problem with an arbitrary number of edges is solved similarly.

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