On some spaces which can be partitioned by the rational line

1984 ◽  
Vol 49 (1) ◽  
pp. 1-8
Author(s):  
Wang Shutang
Keyword(s):  
Rational Line

QUANTIFIED MODAL LOGIC ON THE RATIONAL LINE

2014 ◽  
Vol 7 (3) ◽  
pp. 439-454 ◽  
Author(s):  
PHILIP KREMER
Keyword(s):  
Modal Logic ◽  
Real Line ◽  
Topological Spaces ◽  
Topological Semantics ◽  
Cantor Space ◽  
Quantified Modal Logic ◽  
Propositional Modal Logic ◽  
The Family ◽  
Modal Logic S4 ◽  
Rational Line

AbstractIn the topological semantics for propositional modal logic, S4 is known to be complete for the class of all topological spaces, for the rational line, for Cantor space, and for the real line. In the topological semantics for quantified modal logic, QS4 is known to be complete for the class of all topological spaces, and for the family of subspaces of the irrational line. The main result of the current paper is that QS4 is complete, indeed strongly complete, for the rational line.

Representations of free groups on rational line Q

2000 ◽  
Vol 4 (2) ◽  
pp. 101-105
Author(s):  
Bing-xin Lu ◽  
Zuo-tong Zhu
Keyword(s):  
Free Groups ◽  
Rational Line

scholarly journals Soft rational line integral

2021 ◽  
Vol 31 (4) ◽  
pp. 578-596
Author(s):  
S. Acharjee ◽  
D.A. Molodtsov
Keyword(s):  
Computer Science ◽  
Set Theory ◽  
Soft Set ◽  
Theoretical Development ◽  
Classical Analysis ◽  
Computing Systems ◽  
Line Integral ◽  
Fundamental Properties ◽  
Necessary And Sufficient ◽  
Rational Line

Soft set theory is a new area of mathematics that deals with uncertainties. Applications of soft set theory are widely spread in various areas of science and social science viz. decision making, computer science, pattern recognition, artificial intelligence, etc. The importance of soft set-theoretical versions of mathematical analysis has been felt in several areas of computer science. This paper suggests some concepts of a soft gradient of a function and a soft integral, an analogue of a line integral in classical analysis. The fundamental properties of soft gradients are established. A necessary and sufficient condition is found so that a set can be a subset of the soft gradient of some function. The inclusion of a soft gradient in a soft integral is proved. Semi-additivity and positive uniformity of a soft integral are established. Estimates are obtained for a soft integral and the size of its segment. Semi-additivity with respect to the upper limit of integration is proved. Moreover, this paper enriches the theoretical development of a soft rational line integral and associated areas for better functionality in terms of computing systems.

scholarly journals STRONG COMPLETENESS OF S4 FOR ANY DENSE-IN-ITSELF METRIC SPACE

2013 ◽  
Vol 6 (3) ◽  
pp. 545-570 ◽  
Author(s):  
PHILIP KREMER
Keyword(s):  
Metric Space ◽  
Real Line ◽  
General Question ◽  
Topological Semantics ◽  
Strong Completeness ◽  
Cantor Space ◽  
The Real ◽  
Special Cases ◽  
Modal Logic S4 ◽  
Rational Line

AbstractIn the topological semantics for modal logic, S4 is well-known to be complete for the rational line, for the real line, and for Cantor space: these are special cases of S4’s completeness for any dense-in-itself metric space. The construction used to prove completeness can be slightly amended to show that S4 is not only complete, but also strongly complete, for the rational line. But no similarly easy amendment is available for the real line or for Cantor space and the question of strong completeness for these spaces has remained open, together with the more general question of strong completeness for any dense-in-itself metric space. In this paper, we prove that S4 is strongly complete for any dense-in-itself metric space.

scholarly journals Rational lines on cubic hypersurfaces

2020 ◽  
pp. 1-14
Author(s):  
JULIA BRANDES ◽  
RAINER DIETMANN
Keyword(s):  
Recent Result ◽  
Single Point ◽  
Rational Points ◽  
Cubic Hypersurfaces ◽  
Rational Line

Abstract We show that any smooth projective cubic hypersurface of dimension at least 29 over the rationals contains a rational line. A variation of our methods provides a similar result over p-adic fields. In both cases, we improve on previous results due to the second author and Wooley. We include an appendix in which we highlight some slight modifications to a recent result of Papanikolopoulos and Siksek. It follows that the set of rational points on smooth projective cubic hypersurfaces of dimension at least 29 is generated via secant and tangent constructions from just a single point.

scholarly journals On the continued fraction algorithm

1970 ◽  
Vol 3 (3) ◽  
pp. 413-422 ◽  
Author(s):  
J. M. Mack
Keyword(s):  
Finite Number ◽  
Real Line ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Extreme Case ◽  
Rational Approximations ◽  
Real Numbers ◽  
Best Approximations ◽  
Rational Line ◽  
Fraction Algorithm

The fact that continued fractions can be described in terms of Farey sections is used to obtain a generalised continued fraction algorithm. Geometrically, the algorithm transfers the continued fraction process from the real line R to an arbitrary rational line l in Rn. Arithmetically, the algorithm provides a sequence of simultaneous rational approximations to a set of n real numbers θ1, …, θn in the extreme case where all of the numbers are rationally dependent on 1 and (say) θ1. All but a finite number of best approximations are given by the algorithm.

scholarly journals On cubic surfaces with a rational line

2012 ◽  
Vol 98 (3) ◽  
pp. 229-234 ◽  
Author(s):  
Andreas-Stephan Elsenhans ◽  
Jörg Jahnel
Keyword(s):  
Cubic Surfaces ◽  
Rational Line

QUANTIFIED INTUITIONISTIC LOGIC OVER METRIZABLE SPACES

2019 ◽  
Vol 12 (3) ◽  
pp. 405-425
Author(s):  
PHILIP KREMER
Keyword(s):  
Intuitionistic Logic ◽  
Metrizable Space ◽  
Topological Spaces ◽  
Topological Semantics ◽  
Constant Domain ◽  
Current Article ◽  
Image Position ◽  
Metrizable Spaces ◽  
Rational Line

AbstractIn the topological semantics, quantified intuitionistic logic, QH, is known to be strongly complete not only for the class of all topological spaces but also for some particular topological spaces — for example, for the irrational line, ${\Bbb P}$, and for the rational line, ${\Bbb Q}$, in each case with a constant countable domain for the quantifiers. Each of ${\Bbb P}$ and ${\Bbb Q}$ is a separable zero-dimensional dense-in-itself metrizable space. The main result of the current article generalizes these known results: QH is strongly complete for any zero-dimensional dense-in-itself metrizable space with a constant domain of cardinality ≤ the space’s weight; consequently, QH is strongly complete for any separable zero-dimensional dense-in-itself metrizable space with a constant countable domain. We also prove a result that follows from earlier work of Moerdijk: if we allow varying domains for the quantifiers, then QH is strongly complete for any dense-in-itself metrizable space with countable domains.

Maximal Chains of Copies of the Rational Line

Order ◽  
2012 ◽  
Vol 30 (3) ◽  
pp. 737-748 ◽  
Author(s):  
Miloš S. Kurilić
Keyword(s):  
Maximal Chains ◽  
Rational Line
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