ON THE DENSITY OF HAUSDORFF DIMENSIONS OF BOUNDED TYPE CONTINUED FRACTION SETS: THE TEXAN CONJECTURE

2004 ◽  
Vol 04 (01) ◽  
pp. 63-76 ◽  
Author(s):  
OLIVER JENKINSON
Keyword(s):  
Hausdorff Dimension ◽  
Finite Subset ◽  
Continued Fraction ◽  
Cantor Set ◽  
Computational Method ◽  
Accurate Approximation ◽  
Natural Numbers ◽  
Important Special Case ◽  
The One ◽  
Special Case

Given a non-empty finite subset A of the natural numbers, let EA denote the set of irrationals x∈[0,1] whose continued fraction digits lie in A. In general, EA is a Cantor set whose Hausdorff dimension dim (EA) is between 0 and 1. It is shown that the set [Formula: see text] intersects [0,1/2] densely. We then describe a method for accurately computing dimensions dim (EA), and employ it to investigate numerically the way in which [Formula: see text] intersects [1/2,1]. These computations tend to support the conjecture, first formulated independently by Hensley, and by Mauldin & Urbański, that [Formula: see text] is dense in [0,1]. In the important special case A={1,2}, we use our computational method to give an accurate approximation of dim (E{1,2}), improving on the one given in [18].

scholarly journals On the metric theory of the optimal continued fraction expansion

1997 ◽  
Vol 56 (1) ◽  
pp. 69-79
Author(s):  
R. Nair
Keyword(s):  
Lebesgue Measure ◽  
Continued Fraction ◽  
Natural Numbers ◽  
Real Numbers ◽  
Continued Fraction Expansion ◽  
Image Position ◽  
Special Case

Suppose kn denotes either φ(n) or φ(rn) (n = 1, 2, …) where the polynomial φ maps the natural numbers to themselves and rk denotes the kth rational prime. Let denote the sequence of convergents to a real numbers x for the optimal continued fraction expansion. Define the sequence of approximation constants byIn this paper we study the behaviour of the sequence for all most all x with respect to Lebesgue measure. In the special case where kn = n (n = 1, 2, …) these results are due to Bosma and Kraaikamp.

scholarly journals Rotational-monad metaphysics

2021 ◽  
Author(s):  
Roman Kremen
Keyword(s):  
Three Dimensional ◽  
Physical Space ◽  
Discrete Element ◽  
Physical Reality ◽  
Foundations Of Mathematics ◽  
Important Special Case ◽  
Infinite Dimensional ◽  
The One ◽  
Special Case ◽  
Mathematics And Physics

The article presents a metaphysical concept, in the main tesa of which the simplest discrete element of physical reality is constituted — designated as a protomonad — which forms the basis of spatial forms of materiality, including space itself as such. The genesis of the protomonad is clarified by a certain way interpreted rotation of the spiritual essence, which in itself does not have an extension, and both the indicated essence and its rotation have a metaphysical order, and the dimension of physical space finds a rational interpretation through the characteristics of metaphysical rotation. The semantic aspects of complex mathematical constructs are considered that convey the semantics of rotations, and quite reasonably proposed by some mathematicians as the unified foundations of mathematics and physics, where the properties of constructs act as a mathematically strict co-proof of the validity of the concept. The meaning of number is explained as a method of restriction on infinite pre-physical multiplicity, and finite natural multiplicities are the result of such restrictions; the most important special case is the three-dimensional spatial metric given in the experiment, which appears as a restriction of an infinite-dimensional metaphysical space. The so-called principle of genetic inheritance is formulated, which makes it possible to remove the dialectical opposition between the one and the multiple and illustrates the categories of time and space as dialectical oppositions.

Stably continuous frames

1988 ◽  
Vol 104 (1) ◽  
pp. 7-19 ◽  
Author(s):  
B. Banaschewski ◽  
G. C. L. Brummer
Keyword(s):  
Lattice Theory ◽  
The Other ◽  
Hausdorff Spaces ◽  
Important Special Case ◽  
Compact Spaces ◽  
The One ◽  
Continuous Frames ◽  
Special Case

In the lattice theory that underlies topology, that is, in the study of frames, a class of frames arising naturally is that of the stably continuous frames (see §0 for definitions). On the one hand, they correspond to the most reasonable not necessarily Hausdorff compact spaces, and on the other, they are precisely the retracts of coherent frames. Moreover, an important special case of stably continuous frames are the compact regular frames which correspond to compact Hausdorff spaces.

scholarly journals Congruence pairs of principal MS-algebras and perfect extensions

2020 ◽  
Vol 70 (6) ◽  
pp. 1275-1288
Author(s):  
Abd El-Mohsen Badawy ◽  
Miroslav Haviar ◽  
Miroslav Ploščica
Keyword(s):  
Boolean Algebra ◽  
Congruence Lattices ◽  
Descriptive Solution ◽  
The One ◽  
Special Case ◽  
Congruence Pair

AbstractThe notion of a congruence pair for principal MS-algebras, simpler than the one given by Beazer for K2-algebras [6], is introduced. It is proved that the congruences of the principal MS-algebras L correspond to the MS-congruence pairs on simpler substructures L°° and D(L) of L that were associated to L in [4].An analogy of a well-known Grätzer’s problem [11: Problem 57] formulated for distributive p-algebras, which asks for a characterization of the congruence lattices in terms of the congruence pairs, is presented here for the principal MS-algebras (Problem 1). Unlike a recent solution to such a problem for the principal p-algebras in [2], it is demonstrated here on the class of principal MS-algebras, that a possible solution to the problem, though not very descriptive, can be simple and elegant.As a step to a more descriptive solution of Problem 1, a special case is then considered when a principal MS-algebra L is a perfect extension of its greatest Stone subalgebra LS. It is shown that this is exactly when de Morgan subalgebra L°° of L is a perfect extension of the Boolean algebra B(L). Two examples illustrating when this special case happens and when it does not are presented.

Shelah's work on non-semi-proper iterations, II

2001 ◽  
Vol 66 (4) ◽  
pp. 1865-1883 ◽  
Author(s):  
Chaz Schlindwein
Keyword(s):  
Closed Subset ◽  
Stationary Subset ◽  
Finite Support ◽  
Countable Support ◽  
Final Model ◽  
Axiom A ◽  
Mahlo Cardinal ◽  
The One ◽  
Image Position ◽  
Special Case

One of the main goals in the theory of forcing iteration is to formulate preservation theorems for not collapsing ω1 which are as general as possible. This line leads from c.c.c. forcings using finite support iterations to Axiom A forcings and proper forcings using countable support iterations to semi-proper forcings using revised countable support iterations, and more recently, in work of Shelah, to yet more general classes of posets. In this paper we concentrate on a special case of the very general iteration theorem of Shelah from [5, chapter XV]. The class of posets handled by this theorem includes all semi-proper posets and also includes, among others, Namba forcing.In [5, chapter XV] Shelah shows that, roughly, revised countable support forcing iterations in which the constituent posets are either semi-proper or Namba forcing or P[W] (the forcing for collapsing a stationary co-stationary subset ofwith countable conditions) do not collapse ℵ1. The iteration must contain sufficiently many cardinal collapses, for example, Levy collapses. The most easily quotable combinatorial application is the consistency (relative to a Mahlo cardinal) of ZFC + CH fails + whenever A ∪ B = ω2 then one of A or B contains an uncountable sequentially closed subset. The iteration Shelah uses to construct this model is built using P[W] to “attack” potential counterexamples, Levy collapses to ensure that the cardinals collapsed by the various P[W]'s are sufficiently well separated, and Cohen forcings to ensure the failure of CH in the final model.In this paper we give details of the iteration theorem, but we do not address the combinatorial applications such as the one quoted above.These theorems from [5, chapter XV] are closely related to earlier work of Shelah [5, chapter XI], which dealt with iterated Namba and P[W] without allowing arbitrary semi-proper forcings to be included in the iteration. By allowing the inclusion of semi-proper forcings, [5, chapter XV] generalizes the conjunction of [5, Theorem XI.3.6] with [5, Conclusion XI.6.7].

scholarly journals Free vs. locally free Kleinian groups

2019 ◽  
Vol 2019 (746) ◽  
pp. 149-170
Author(s):  
Pekka Pankka ◽  
Juan Souto
Keyword(s):  
Hausdorff Dimension ◽  
Kleinian Group ◽  
Cantor Set ◽  
Cantor Sets ◽  
Kleinian Groups ◽  
The Other ◽  
Limit Set ◽  
Limit Sets ◽  
Other Hand

Abstract We prove that Kleinian groups whose limit sets are Cantor sets of Hausdorff dimension < 1 are free. On the other hand we construct for any ε > 0 an example of a non-free purely hyperbolic Kleinian group whose limit set is a Cantor set of Hausdorff dimension < 1 + ε.

Revisiting Generation and Meshing Properties of Beveloid Gears

2017 ◽  
Vol 139 (9) ◽  
Author(s):  
Alessio Artoni ◽  
Massimo Guiggiani
Keyword(s):  
Tooth Surface ◽  
Generation Process ◽  
Spur Gears ◽  
Helical Gears ◽  
Additional Degree ◽  
Beveloid Gears ◽  
Fixed Space ◽  
Fixed Axis ◽  
The One ◽  
Special Case

The teeth of ordinary spur and helical gears are generated by a (virtual) rack provided with planar generating surfaces. The resulting tooth surface shapes are a circle-involute cylinder in the case of spur gears, and a circle-involute helicoid for helical gears. Advantages associated with involute geometry are well known. Beveloid gears are often regarded as a generalization of involute cylindrical gears involving one additional degree-of-freedom, in that the midplane of their (virtual) generating rack is inclined with respect to the axis of the gear being generated. A peculiarity of their generation process is that the motion of the generating planar surface, seen from the fixed space, is a rectilinear translation (while the gear blank is rotated about a fixed axis); the component of such translation that is orthogonal to the generating plane is the one that ultimately dictates the shape of the generated, envelope surface. Starting from this basic fact, we set out to revisit this type of generation-by-envelope process and to profitably use it to explore peculiar design layouts, in particular for the case of motion transmission between skew axes (and intersecting axes as a special case). Analytical derivations demonstrate the possibility of involute helicoid profiles (beveloids) transmitting motion between skew axes through line contact and, perhaps more importantly, they lead to the derivation of designs featuring insensitivity of the transmission ratio to all misalignments within relatively large limits. The theoretical developments are confirmed by various numerical examples.

Mathematical Knowledge and the Origin of Phenomenology

2021 ◽  
Vol 21 ◽  
pp. 273-294
Author(s):  
Gabriele Baratelli ◽  
Keyword(s):  
Mathematical Knowledge ◽  
Cardinal Number ◽  
The Other ◽  
Mathematical Science ◽  
Natural Numbers ◽  
Symbolic Thinking ◽  
The One

The paper is divided into two parts. In the first one, I set forth a hypothesis to explain the failure of Husserl’s project presented in the Philosophie der Arithmetik based on the principle that the entire mathematical science is grounded in the concept of cardinal number. It is argued that Husserl’s analysis of the nature of the symbols used in the decadal system forces the rejection of this principle. In the second part, I take into account Husserl’s explanation of why, albeit independent of natural numbers, the system is nonetheless correct. It is shown that its justification involves, on the one hand, a new conception of symbols and symbolic thinking, and on the other, the recognition of the question of “the formal” and formalization as pivotal to understand “the mathematical” overall.

An Elementary Proof of Suslin Reciprocity

2005 ◽  
Vol 48 (2) ◽  
pp. 221-236 ◽  
Author(s):  
Matt Kerr
Keyword(s):  
Elementary Proof ◽  
Function Fields ◽  
Algebraic Cycles ◽  
Important Special Case ◽  
Special Case

AbstractWe state and prove an important special case of Suslin reciprocity that has found significant use in the study of algebraic cycles. An introductory account is provided of the regulator and norm maps on Milnor K2-groups (for function fields) employed in the proof.

Kruskal's theorem for matroids

1968 ◽  
Vol 64 (1) ◽  
pp. 3-4 ◽  
Author(s):  
D. J. A. Welsh
Keyword(s):  
Vector Space ◽  
Finite Subset ◽  
Spanning Tree ◽  
General Theorem ◽  
Independent Set ◽  
Easy Method ◽  
Maximal Weight ◽  
Linearly Independent ◽  
Kruskal’S Theorem ◽  
Special Case

AbstractKruskal's theorem for obtaining a minimal (maximal) spanning tree of a graph is shown to be a special case of a more general theorem for matroid spaces in which each element of the matroid has an associated weight. Since any finite subset of a vector space can be regarded as a matroid space this theorem gives an easy method of selecting a linearly independent set of vectors of minimal (maximal) weight.

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