scholarly journals Nondeterministic Automatic Complexity of Overlap-Free and Almost Square-Free Words

10.37236/4851 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Kayleigh K. Hyde ◽  
Bjørn Kjos-Hanssen
Keyword(s):  
Hausdorff Dimension ◽  
Upper Bound ◽  
Decision Problem ◽  
The Other ◽  
Binary Word ◽  
Other Hand

Shallit and Wang studied deterministic automatic complexity of words.  They showed that the automatic Hausdorff dimension $I(\mathbf t)$ of the infinite Thue word satisfies $1/3\le I(\mathbf t)\le 1/2$.   We improve that result by showing that $I(\mathbf t)= 1/2$.  We prove that the nondeterministic automatic complexity $A_N(x)$ of a word $x$ of length $n$ is bounded by $b(n):=\lfloor n/2\rfloor + 1$.  This enables us to define the complexity deficiency $D(x)=b(n)-A_N(x)$.  If $x$ is square-free then $D(x)=0$. If $x$ is almost square-free in the sense of Fraenkel and Simpson, or if $x$ is a overlap-free binary word such as the infinite Thue--Morse word, then $D(x)\le 1$.  On the other hand, there is no constant upper bound on $D$ for overlap-free words over a ternary alphabet, nor for cube-free words over a binary alphabet.The decision problem whether $D(x)\ge d$ for given $x$, $d$ belongs to $\mathrm{NP}\cap \mathrm{E}$.

scholarly journals The -transformation with a hole at 0

2019 ◽  
Vol 40 (9) ◽  
pp. 2482-2514
Author(s):  
CHARLENE KALLE ◽  
DERONG KONG ◽  
NIELS LANGEVELD ◽  
WENXIA LI
Keyword(s):  
Hausdorff Dimension ◽  
Upper Bound ◽  
The Other ◽  
Bifurcation Set ◽  
Other Hand ◽  
Accumulation Points ◽  
Isolated Points ◽  
Image Position ◽  
Doubling Map ◽  
Null Set

For $\unicode[STIX]{x1D6FD}\in (1,2]$ the $\unicode[STIX]{x1D6FD}$-transformation $T_{\unicode[STIX]{x1D6FD}}:[0,1)\rightarrow [0,1)$ is defined by $T_{\unicode[STIX]{x1D6FD}}(x)=\unicode[STIX]{x1D6FD}x\hspace{0.6em}({\rm mod}\hspace{0.2em}1)$. For $t\in [0,1)$ let $K_{\unicode[STIX]{x1D6FD}}(t)$ be the survivor set of $T_{\unicode[STIX]{x1D6FD}}$ with hole $(0,t)$ given by $$\begin{eqnarray}K_{\unicode[STIX]{x1D6FD}}(t):=\{x\in [0,1):T_{\unicode[STIX]{x1D6FD}}^{n}(x)\not \in (0,t)\text{ for all }n\geq 0\}.\end{eqnarray}$$ In this paper we characterize the bifurcation set $E_{\unicode[STIX]{x1D6FD}}$ of all parameters $t\in [0,1)$ for which the set-valued function $t\mapsto K_{\unicode[STIX]{x1D6FD}}(t)$ is not locally constant. We show that $E_{\unicode[STIX]{x1D6FD}}$ is a Lebesgue null set of full Hausdorff dimension for all $\unicode[STIX]{x1D6FD}\in (1,2)$. We prove that for Lebesgue almost every $\unicode[STIX]{x1D6FD}\in (1,2)$ the bifurcation set $E_{\unicode[STIX]{x1D6FD}}$ contains infinitely many isolated points and infinitely many accumulation points arbitrarily close to zero. On the other hand, we show that the set of $\unicode[STIX]{x1D6FD}\in (1,2)$ for which $E_{\unicode[STIX]{x1D6FD}}$ contains no isolated points has zero Hausdorff dimension. These results contrast with the situation for $E_{2}$, the bifurcation set of the doubling map. Finally, we give for each $\unicode[STIX]{x1D6FD}\in (1,2)$ a lower and an upper bound for the value $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FD}}$ such that the Hausdorff dimension of $K_{\unicode[STIX]{x1D6FD}}(t)$ is positive if and only if $t<\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FD}}$. We show that $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FD}}\leq 1-(1/\unicode[STIX]{x1D6FD})$ for all $\unicode[STIX]{x1D6FD}\in (1,2)$.

scholarly journals A BOUND FOR THE INDEX OF A QUADRATIC FORM AFTER SCALAR EXTENSION TO THE FUNCTION FIELD OF A QUADRIC

2018 ◽  
Vol 19 (2) ◽  
pp. 421-450 ◽  
Author(s):  
Stephen Scully
Keyword(s):  
Quadratic Form ◽  
Upper Bound ◽  
Function Field ◽  
The Other ◽  
Direct Generalization ◽  
Other Hand ◽  
General Field ◽  
The One ◽  
Anisotropic Quadratic Form

Let $q$ be an anisotropic quadratic form defined over a general field $F$. In this article, we formulate a new upper bound for the isotropy index of $q$ after scalar extension to the function field of an arbitrary quadric. On the one hand, this bound offers a refinement of an important bound established in earlier work of Karpenko–Merkurjev and Totaro; on the other hand, it is a direct generalization of Karpenko’s theorem on the possible values of the first higher isotropy index. We prove its validity in two key cases: (i) the case where $\text{char}(F)\neq 2$, and (ii) the case where $\text{char}(F)=2$ and $q$ is quasilinear (i.e., diagonalizable). The two cases are treated separately using completely different approaches, the first being algebraic–geometric, and the second being purely algebraic.

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.

Postnikov “Invariants” in 2004

2005 ◽  
Vol 12 (1) ◽  
pp. 139-155
Author(s):  
Julio Rubio ◽  
Francis Sergeraert
Keyword(s):  
Decision Problem ◽  
The Other ◽  
Topological Spaces ◽  
Exact Nature ◽  
Other Hand

Abstract The very nature of the so-called Postnikov invariants is carefully studied. Two functors, precisely defined, explain the exact nature of the connection between the category of topological spaces and the category of Postnikov towers. On one hand, these functors are in particular effective and lead to concrete machine computations through the general machine program Kenzo. On the other hand, the Postnikov “invariants” will be actual invariants only when an arithmetical decision problem – currently open – will be solved; it is even possible this problem is undecidable.

On the reduction of the decision problem. First paper. Ackermann prefix, a single binary predicate

1939 ◽  
Vol 4 (1) ◽  
pp. 1-9 ◽  
Author(s):  
László Kalmár
Keyword(s):  
Decision Problem ◽  
Reduction Process ◽  
The Other ◽  
First Order ◽  
Other Hand ◽  
Special Cases ◽  
Binary Predicate ◽  
Ternary Variables ◽  
Order Predicate Calculus ◽  
Predicate Variable

1. Although the decision problem of the first order predicate calculus has been proved by Church to be unsolvable by any (general) recursive process, perhaps it is not superfluous to investigate the possible reductions of the general problem to simple special cases of it. Indeed, the situation after Church's discovery seems to be analogous to that in algebra after the Ruffini-Abel theorem; and investigations on the reduction of the decision problem might prepare the way for a theory in logic, analogous to that of Galois.It has been proved by Ackermann that any first order formula is equivalent to another having a prefix of the form(1) (Ex1)(x2)(Ex3)(x4)…(xm).On the other hand, I have proved that any first order formula is equivalent to some first order formula containing a single, binary, predicate variable. In the present paper, I shall show that both results can be combined; more explicitly, I shall prove theTheorem. To any given first order formula it is possible to construct an equivalent one with a prefix of the form (1) and a matrix containing no other predicate variable than a single binary one.2. Of course, this theorem cannot be proved by a mere application of the Ackermann reduction method and mine, one after the other. Indeed, Ackermann's method requires the introduction of three auxiliary predicate variables, two of them being ternary variables; on the other hand, my reduction process leads to a more complicated prefix, viz.,(2) (Ex1)…(Exm)(xm+1)(xm+2)(Exm+3)(Exm+4).

scholarly journals The structure and the list 3-dynamic coloring of outer-1-planar graphs

2021 ◽  
Vol vol. 23, no. 3 (Graph Theory) ◽  
Author(s):  
Yan Li ◽  
Xin Zhang
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Upper Bound ◽  
Local Structure ◽  
Planar Graphs ◽  
Minimum Degree ◽  
Outer Region ◽  
The Other ◽  
Structural Theorem ◽  
Other Hand

An outer-1-planar graph is a graph admitting a drawing in the plane so that all vertices appear in the outer region of the drawing and every edge crosses at most one other edge. This paper establishes the local structure of outer-1-planar graphs by proving that each outer-1-planar graph contains one of the seventeen fixed configurations, and the list of those configurations is minimal in the sense that for each fixed configuration there exist outer-1-planar graphs containing this configuration that do not contain any of another sixteen configurations. There are two interesting applications of this structural theorem. First of all, we conclude that every (resp. maximal) outer-1-planar graph of minimum degree at least 2 has an edge with the sum of the degrees of its two end-vertices being at most 9 (resp. 7), and this upper bound is sharp. On the other hand, we show that the list 3-dynamic chromatic number of every outer-1-planar graph is at most 6, and this upper bound is best possible.

Set-theoretical basis for real numbers

1950 ◽  
Vol 15 (4) ◽  
pp. 241-247
Author(s):  
Hao Wang
Keyword(s):  
Upper Bound ◽  
Theoretical Basis ◽  
The Other ◽  
Real Numbers ◽  
Full Generality ◽  
Ordered Field ◽  
Other Hand

In [1] we have considered a certain system L and shown that although its axioms are considerably weaker than those of [2], it suffices for purposes of the topics covered in [2]. The purpose of the present paper is to consider the system L more carefully and to show that with suitably chosen definitions for numbers, the ordinary theory of real numbers is also obtainable in it. For this purpose, we shall indicate that we can prove in L a certain set of twenty axioms used by Tarski which are sufficient for the arithmetic of real numbers and are to the effect that real numbers form a complete ordered field. Indeed, we cannot prove in L all Tarski's twenty axioms in their full generality. One of them, stating in effect that every bounded class of real numbers possesses a least upper bound, can only be proved as a metatheorem which states that every bounded nameable class of real numbers possesses a least upper bound. However, all the other nineteen axioms can be proved in L without any modification.This result may be of some interest because the axioms of L are considerably weaker than those commonly employed for the same purpose. In L variables need to take as values only classes each of whose members has no more than two members. In other words, only classes each with no more than two members are to be elements. On the other hand, it is usual to assume for the purpose of natural arithmetic that all finite classes are elements, and, for the purpose of real arithmetic, that all enumerable classes are elements.

scholarly journals Triebel-Lizorkin capacity and hausdorff measure in metric spaces

2020 ◽  
Vol 70 (3) ◽  
pp. 617-624
Author(s):  
Nijjwal Karak
Keyword(s):  
Hausdorff Measure ◽  
Upper Bound ◽  
Metric Spaces ◽  
Measure Zero ◽  
Gauge Function ◽  
The Other ◽  
Other Hand ◽  
Zero Capacity

AbstractWe provide a upper bound for Triebel-Lizorkin capacity in metric settings in terms of Hausdorff measure. On the other hand, we also prove that the sets with zero capacity have generalized Hausdorff h-measure zero for a suitable gauge function h.

scholarly journals Dimensions of Sets Which Uniformly Avoid Arithmetic Progressions

2017 ◽  
Vol 2019 (14) ◽  
pp. 4419-4430 ◽  
Author(s):  
Jonathan M Fraser ◽  
Kota Saito ◽  
Han Yu
Keyword(s):  
Hausdorff Dimension ◽  
Upper Bound ◽  
Arithmetic Progressions ◽  
The Other ◽  
Kakeya Problem ◽  
Higher Dimensional ◽  
Finite Arithmetic

AbstractWe provide estimates for the dimensions of sets in $\mathbb{R}$ which uniformly avoid finite arithmetic progressions (APs). More precisely, we say $F$ uniformly avoids APs of length $k \geq 3$ if there is an $\epsilon>0$ such that one cannot find an AP of length $k$ and gap length $\Delta>0$ inside the $\epsilon \Delta$ neighbourhood of $F$. Our main result is an explicit upper bound for the Assouad (and thus Hausdorff) dimension of such sets in terms of $k$ and $\epsilon$. In the other direction, we provide examples of sets which uniformly avoid APs of a given length but still have relatively large Hausdorff dimension. We also consider higher dimensional analogues of these problems, where APs are replaced with arithmetic patches lying in a hyperplane. As a consequence, we obtain a discretized version of a “reverse Kakeya problem:” we show that if the dimension of a set in $\mathbb{R}^d$ is sufficiently large, then it closely approximates APs in every direction.

Analysis of Central Bursting Defects in Plane Strain Drawing and Extrusion

1986 ◽  
Vol 108 (4) ◽  
pp. 317-321 ◽  
Author(s):  
B. Avitzur ◽  
J. C. Choi
Keyword(s):  
Plane Strain ◽  
Upper Bound ◽  
The Other ◽  
Major Conclusion ◽  
Upper Bound Theorem ◽  
Other Hand ◽  
Identical Manner ◽  
Bound Theorem ◽  
Inclined Angle ◽  
Proportional Flow

Based on the upper-bound theorem in limit analysis, the central bursting defect in plane strain drawing and extrusion is analyzed by comparing the proportional flow with the central bursting flow for the metal with voids at the center. A criterion for the unique conditions that promote this defect has been derived. The metal with voids may flow in the identical manner to that of solid strip with no voids to form a sound flow, deterring central bursting. A solid strip, on the other hand, or a material with voids, may flow in a manner so as to produce central bursting defects. A major conclusion of the study is that, for a range of combinations of inclined angle of the die, reduction, and friction, central bursting is expected whether or not the material originally had any voids. On the other hand, central bursting can be prevented even if the original rod contains small-size voids.

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