The Group of Measure Preserving Transformations of the Unit Interval is an Absolute Retract

An invertible measure preserving transformation T on the unit interval I generates a unitary operator U on the space L2(I) of Lebesque square integrable functions given by (Uf)(x) = f(Tx) for all f in L2(I) and x in I. By definitionfor all f , g in L2(I), the bar denoting complex conjugation.

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A Lochs-Type Approach via Entropy in Comparing the Efficiency of Different Continued Fraction Algorithms

Mathematics ◽

10.3390/math9030255 ◽

2021 ◽

Vol 9 (3) ◽

pp. 255

Author(s):

Dan Lascu ◽

Gabriela Ileana Sebe

Keyword(s):

Dynamical Systems ◽

Continued Fractions ◽

Continued Fraction ◽

Unit Interval ◽

Probability Measures ◽

Absolutely Continuous ◽

Continued Fraction Expansions ◽

Absolutely Continuous Invariant

We investigate the efficiency of several types of continued fraction expansions of a number in the unit interval using a generalization of Lochs theorem from 1964. Thus, we aim to compare the efficiency by describing the rate at which the digits of one number-theoretic expansion determine those of another. We study Chan’s continued fractions, θ-expansions, N-continued fractions, and Rényi-type continued fractions. A central role in fulfilling our goal is played by the entropy of the absolutely continuous invariant probability measures of the associated dynamical systems.

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Complex q-rung orthopair fuzzy Schweizer–Sklar Muirhead mean aggregation operators and their application in multi-criteria decision-making

Journal of Intelligent & Fuzzy Systems ◽

10.3233/jifs-202440 ◽

2021 ◽

pp. 1-23

Author(s):

Peide Liu ◽

Tahir Mahmood ◽

Zeeshan Ali

Keyword(s):

Decision Making ◽

Comparative Analysis ◽

Imaginary Part ◽

Fuzzy Set ◽

Score Function ◽

Aggregation Operators ◽

Unit Interval ◽

Multi Criteria Decision Making ◽

Accuracy Function ◽

Special Cases

Complex q-rung orthopair fuzzy set (CQROFS) is a proficient technique to describe awkward and complicated information by the truth and falsity grades with a condition that the sum of the q-powers of the real part and imaginary part is in unit interval. Further, Schweizer–Sklar (SS) operations are more flexible to aggregate the information, and the Muirhead mean (MM) operator can examine the interrelationships among the attributes, and it is more proficient and more generalized than many aggregation operators to cope with awkward and inconsistence information in realistic decision issues. The objectives of this manuscript are to explore the SS operators based on CQROFS and to study their score function, accuracy function, and their relationships. Further, based on these operators, some MM operators based on PFS, called complex q-rung orthopair fuzzy MM (CQROFMM) operator, complex q-rung orthopair fuzzy weighted MM (CQROFWMM) operator, and their special cases are presented. Additionally, the multi-criteria decision making (MCDM) approach is developed by using the explored operators based on CQROFS. Finally, the advantages and comparative analysis are also discussed.

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GENETICS OF ARGININE BIOSYNTHESIS IN NEUROSPORA CRASSA

Genetics ◽

10.1093/genetics/93.3.557 ◽

1979 ◽

Vol 93 (3) ◽

pp. 557-575

Author(s):

Rowland H Davis

Keyword(s):

Neurospora Crassa ◽

Gene Product ◽

Unit Interval ◽

Starting Material ◽

Limited Extent ◽

Arginine Biosynthesis ◽

Regulatory Mutants ◽

Intragenic Complementation ◽

Translocation Breakpoints ◽

Bifunctional Gene

ABSTRACT A large number of arginine-requiring mutants of Neurospora was isolated, using a strain already partially impaired in an enzyme of the pathway. Among the mutants, all previously described loci, except one, were represented, and several new loci were defined and mapped. Four groups of mutants were of particular interest. First, thc large group of arg-6 mutants, when tested for intragenic complementation, suggested a bifunctional gene, possibly controlling two steps in ornithine synthesis. This is consistent with the limited enzymic information about this locus. Second, the arg-13 locus was represented by 14 new mutants. All five tested were quite leaky. suggesting that the function controlled by this gene can be rarried out to a limited extent spontaneously or by another gene product. Third, a new locus, arg-14, was defined. It controls a step in ornithine synthesis. It lies in a 1 to 2 map-unit interval between arg-2 and pyr-3 on LG IVR, as shown by mapping in relation tG translocation breakpoints. Fourth, a second new locus whose mutants render the partial mutation in starting material auxotrophic was defined and mapped near the centromere of LG VIL. These new mutants are unable to derepress enzymes of the pathway and may qualify as regulatory mutants.

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