Applications of Markov spectra for the weighted partition network by the substitution rule

Skeleton is a new notion designed for constructing space-filling curves of self-similar sets. In a previous paper by Dai and the authors [Space-filling curves of self-similar sets (II): Edge-to-trail substitution rule, Nonlinearity 32(5) (2019) 1772–1809] it was shown that for all the connected self-similar sets with a skeleton satisfying the open set condition, space-filling curves can be constructed. In this paper, we give a criterion of existence of skeletons by using the so-called neighbor graph of a self-similar set. In particular, we show that a connected self-similar set satisfying the finite-type condition always possesses skeletons: an algorithm is obtained here.

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FRACTAL TILINGS FROM SUBSTITUTION TILINGS

Fractals ◽

10.1142/s0218348x19500099 ◽

2019 ◽

Vol 27 (02) ◽

pp. 1950009

Author(s):

XINCHANG WANG ◽

PEICHANG OUYANG ◽

KWOKWAI CHUNG ◽

XIAOGEN ZHAN ◽

HUA YI ◽

...

Keyword(s):

Short Paper ◽

Self Similarity ◽

Substitution Rule ◽

Substitution Tiling ◽

Simple Strategy ◽

Substitution Tilings

A fractal tiling or [Formula: see text]-tiling is a tiling which possesses self-similarity and the boundary of which is a fractal. By substitution rule of tilings, this short paper presents a very simple strategy to create a great number of [Formula: see text]-tilings. The substitution tiling Equithirds is demonstrated to show how to achieve it in detail. The method can be generalized to every tiling that can be constructed by substitution rule.

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Inverse Problems for Partition Functions

Canadian Journal of Mathematics ◽

10.4153/cjm-2001-035-9 ◽

2001 ◽

Vol 53 (4) ◽

pp. 866-896

Author(s):

Yifan Yang

Keyword(s):

Asymptotic Behavior ◽

Partition Function ◽

Inverse Problems ◽

Large Class ◽

Generating Function ◽

Riemann Hypothesis ◽

Arithmetic Function ◽

Partition Functions ◽

Class Of Functions ◽

Weighted Partition

AbstractLet pw(n) be the weighted partition function defined by the generating function , where w(m) is a non-negative arithmetic function. Let be the summatory functions for pw(n) and w(n), respectively. Generalizing results of G. A. Freiman and E. E. Kohlbecker, we show that, for a large class of functions Φ(u) and λ(u), an estimate for Pw(u) of the formlog Pw(u) = Φ(u){1 + Ou(1/λ(u))} (u→∞) implies an estimate forNw(u) of the formNw(u) = Φ*(u){1+O(1/ log ƛ(u))} (u→∞) with a suitable function Φ*(u) defined in terms of Φ(u). We apply this result and related results to obtain characterizations of the Riemann Hypothesis and the Generalized Riemann Hypothesis in terms of the asymptotic behavior of certain weighted partition functions.

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Variation on a theme of Nathan Fine. New weighted partition identities

Journal of Number Theory ◽

10.1016/j.jnt.2016.12.011 ◽

2017 ◽

Vol 176 ◽

pp. 226-248 ◽

Cited By ~ 2

Author(s):

Alexander Berkovich ◽

Ali K. Uncu

Keyword(s):

Partition Identities ◽

Weighted Partition

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Analysis of substitution times in soccer

Journal of Quantitative Analysis in Sports ◽

10.1515/jqas-2015-0114 ◽

2016 ◽

Vol 12 (3) ◽

Author(s):

Rajitha M. Silva ◽

Tim B. Swartz

Keyword(s):

Decision Rule ◽

Substitution Rule

AbstractThis paper considers the problem of determining optimal substitution times in soccer. We review the substitution rule proposed by Myers (Myers, B. R. 2012. “A Proposed Decision Rule for the Timing of Soccer Substitutions.”

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Strong completeness of fragments of the propositional calculus

Journal of Symbolic Logic ◽

10.2307/2266391 ◽

1951 ◽

Vol 16 (3) ◽

pp. 204-204 ◽

Cited By ~ 1

Author(s):

Alan Rose

Keyword(s):

Propositional Calculus ◽

Modus Ponens ◽

Primitive Function ◽

Strong Completeness ◽

Substitution Rule ◽

Material Implication ◽

Rules Of Procedure

There has recently been developed a method of formalising any fragment of the propositional calculus, subject only to the condition that material implication is a primitive function of the fragmentary system considered. Tarski has stated, without proof, that when implication is the only primitive function a formulation which is weakly complete (i.e., has as theorems all expressible tautologies) is also strongly complete (i.e., provides for the deduction of any expressible formula from any which is not a tautology). The methods used by Henkin suggest the following proof of theTheorem. If in a fragment of the propositional calculus material implication can be defined in terms of the primitive functions, then any weakly complete formalisation of the fragmentary system which has for rules of procedure the substitution rule and modus ponens is also strongly complete.

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