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Journal articles on the topic 'Distribution of fractional parts'

Author: Grafiati

Published: 4 June 2021

Last updated: 14 February 2022

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1

Banks, William D., and Igor E. Shparlinski. "Fractional parts of Dedekind sums." International Journal of Number Theory 12, no. 05 (May 10, 2016): 1137–47. http://dx.doi.org/10.1142/s179304211650069x.

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Using a recent improvement by Bettin and Chandee to a bound of Duke, Friedlander and Iwaniec [Bilinear forms with Kloosterman fractions, Invent. Math. 128 (1997) 23–43] on double exponential sums with Kloosterman fractions, we establish a uniformity of distribution result for the fractional parts of Dedekind sums [Formula: see text] with [Formula: see text] and [Formula: see text] running over rather general sets. Our result extends earlier work of Myerson [Dedekind sums and uniform distribution, J. Number Theory 28 (1988) 233–239] and Vardi [A relation between Dedekind sums and Kloosterman sums, Duke Math. J. 55 (1987) 189–197]. Using different techniques, we also study the least denominator of the collection of Dedekind sums [Formula: see text] on average for [Formula: see text].

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2

Banks, William D., Moubariz Z. Garaev, Florian Luca, and Igor E. Shparlinski. "Uniform Distribution of Fractional Parts Related to Pseudoprimes." Canadian Journal of Mathematics 61, no. 3 (June 1, 2009): 481–502. http://dx.doi.org/10.4153/cjm-2009-025-2.

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Abstract.We estimate exponential sums with the Fermat-like quotientswhere g and n are positive integers, n is composite, and P (n) is the largest prime factor of n. Clearly, both fg (n) and hg (n) are integers if n is a Fermat pseudoprime to base g, and if n is a Carmichael number, this is true for all g coprime to n. Nevertheless, our bounds imply that the fractional parts ﹛ fg (n)﹜ and ﹛hg (n)﹜ are uniformly distributed, on average over g for fg (n), and individually for hg (n). We also obtain similar results with the functions and .

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3

Rochev, I. P. "On distribution of fractional parts of linear forms." Journal of Mathematical Sciences 182, no. 4 (March 29, 2012): 527–38. http://dx.doi.org/10.1007/s10958-012-0756-9.

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4

Shutov, A. V. "Inhomogeneous diophantine approximations and distribution of fractional parts." Journal of Mathematical Sciences 182, no. 4 (March 29, 2012): 576–85. http://dx.doi.org/10.1007/s10958-012-0762-y.

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5

Fomenko, O. M. "On the distribution of fractional parts of polynomials." Journal of Mathematical Sciences 184, no. 6 (July 11, 2012): 770–75. http://dx.doi.org/10.1007/s10958-012-0898-9.

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6

Zhuravlev, V. G. "Multidimensional Hecke theorem on the distribution of fractional parts." St. Petersburg Mathematical Journal 24, no. 1 (November 15, 2012): 71–97. http://dx.doi.org/10.1090/s1061-0022-2012-01232-x.

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7

HEATH–BROWN, D. R. "Pair correlation for fractional parts of αn2." Mathematical Proceedings of the Cambridge Philosophical Society 148, no. 3 (January 15, 2010): 385–407. http://dx.doi.org/10.1017/s0305004109990466.

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It was proved by Weyl [8] in 1916 that the sequence of values of αn2 is uniformly distributed modulo 1, for any fixed real irrational α. Indeed this result covered sequences αnd for any fixed positive integer exponent d. However Weyl's work leaves open a number of questions concerning the finer distribution of these sequences. It has been conjectured by Rudnick, Sarnak and Zaharescu [6] that the fractional parts of αn2 will have a Poisson distribution provided firstly that α is “Diophantine”, and secondly that if a/q is any convergent to α then the square-free part of q is q1+o(1). Here one says that α is Diophantine if one has (1.1) for every rational number a/q and any fixed ϵ > 0. In particular every real irrational algebraic number is Diophantine. One would predict that there are Diophantine numbers α for which the sequence of convergents pn/qn contains infinitely many squares amongst the qn. If true, this would show that the second condition is independent of the first. Indeed one would expect to find such α with bounded partial quotients.

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8

Garifullina, R. L. "Distribution of the fractional parts of the matrix exponential function." Journal of Soviet Mathematics 41, no. 6 (June 1988): 1396–400. http://dx.doi.org/10.1007/bf01097066.

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9

Ford, Kevin, Xianchang Meng, and Alexandru Zaharescu. "Simultaneous distribution of the fractional parts of Riemann zeta zeros." Bulletin of the London Mathematical Society 49, no. 1 (December 29, 2016): 1–9. http://dx.doi.org/10.1112/blms.12001.

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10

Zhuravlev, V. G. "Embeddings of circular orbits and the distribution of fractional parts." St. Petersburg Mathematical Journal 26, no. 6 (September 21, 2015): 881–909. http://dx.doi.org/10.1090/spmj/1365.

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11

Levin, M. B. "Completely uniform distribution of fractional parts of the exponential function." Journal of Soviet Mathematics 31, no. 5 (December 1985): 3247–56. http://dx.doi.org/10.1007/bf02105146.

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12

SHAHVERDIAN, ASHOT, and ADEM KILICMAN. "EXACT FORMULA FOR DISTRIBUTION OF SEQUENCES {ωn}." International Journal of Number Theory 09, no. 01 (November 13, 2012): 179–87. http://dx.doi.org/10.1142/s1793042112501321.

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An exact formula for distribution of sequences of fractional parts is proved. This provides us with a possibility for computational study of their discrepancies by the means of integer-number programming.

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13

Fomenko, O. M. "On the distribution of fractional parts of polynomials in two variables." Journal of Mathematical Sciences 193, no. 1 (July 24, 2013): 129–35. http://dx.doi.org/10.1007/s10958-013-1441-3.

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14

MARKLOF, JENS. "The $\bm{n}$-point correlations between values of a linear form." Ergodic Theory and Dynamical Systems 20, no. 4 (August 2000): 1127–72. http://dx.doi.org/10.1017/s0143385700000626.

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We show that the $n$-point correlation function for the fractional parts of a random linear form in $m$ variables has a limit distribution with power-like tail. The existence of the limit distribution follows from the mixing property of flows on ${\rm SL}(m+1,{\Bbb R})/{\rm SL}(m+1,{\Bbb Z})$. Moreover, we prove similar limit theorems (i) for the probability to find the fractional part of a random linear form close to zero and (ii) also for related trigonometric sums. For large $m$, all of the above limit distributions approach the classical distributions for independent uniformly distributed random variables.

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15

Tolev, D. I. "On the joint distribution of the fractional parts of different prime powers." Russian Mathematical Surveys 44, no. 6 (December 31, 1989): 192–93. http://dx.doi.org/10.1070/rm1989v044n06abeh002296.

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16

Kulikova, A. A., and Yu V. Prokhorov. "Distribution of the Fractional Parts of Random Vectors: The Gaussian Case. I." Theory of Probability & Its Applications 48, no. 2 (January 2004): 355–59. http://dx.doi.org/10.1137/s0040585x97980452.

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17

Kulikova, A. A., Yu V. Prokhorov, and V. I. Khokhlov. "Distribution of the Fractional Parts of Random Vectors: The Gaussian Case. II." Theory of Probability & Its Applications 50, no. 4 (January 2006): 685–87. http://dx.doi.org/10.1137/s0040585x97982062.

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18

Rudnick, Zeév, Peter Sarnak, and Alexandru Zaharescu. "The distribution of spacings between the fractional parts of n 2 α." Inventiones Mathematicae 145, no. 1 (July 1, 2001): 37–57. http://dx.doi.org/10.1007/s002220100141.

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19

Bugeaud, Yann, Lingmin Liao, and Michał Rams. "Metrical Results on the Distribution of Fractional Parts of Powers of Real Numbers." Proceedings of the Edinburgh Mathematical Society 62, no. 2 (November 29, 2018): 505–21. http://dx.doi.org/10.1017/s0013091518000585.

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AbstractWe establish several new metrical results on the distribution properties of the sequence ({xn})n≥1, where {·} denotes the fractional part. Many of them are presented in a more general framework, in which the sequence of functions (x ↦ xn)n≥1 is replaced by a sequence (fn)n≥1, under some growth and regularity conditions on the functions fn.

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20

Aistleitner, Christoph, and Gerhard Larcher. "Additive Energy and Irregularities of Distribution." Uniform distribution theory 12, no. 1 (June 27, 2017): 99–107. http://dx.doi.org/10.1515/udt-2017-0006.

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Abstract We consider strictly increasing sequences (an)n≥1 of integers and sequences of fractional parts ({anα})n≥1 where α ∈ R. We show that a small additive energy of (an)n≥1 implies that for almost all α the sequence ({anα})n≥1 has large discrepancy. We prove a general result, provide various examples, and show that the converse assertion is not necessarily true.

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21

Shutov, A. V. "Local discrepancies in the problem of fractional parts distribution of a linear function." Russian Mathematics 61, no. 2 (February 2017): 74–82. http://dx.doi.org/10.3103/s1066369x17020098.

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22

Filaseta, Michael, and Ognian Trifonov. "The Distribution of Fractional Parts with Applications to Gap Results in Number Theory." Proceedings of the London Mathematical Society s3-73, no. 2 (September 1996): 241–78. http://dx.doi.org/10.1112/plms/s3-73.2.241.

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23

Pustyl'nikov, L. D. "Distribution of the fractional parts of a polynomial, Weyl sums, and ergodic theory." Russian Mathematical Surveys 48, no. 4 (August 31, 1993): 143–79. http://dx.doi.org/10.1070/rm1993v048n04abeh001053.

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24

Timan, A. F. "Distribution of fractional parts and approximation of functions with singularities by Bernstein polynomials." Journal of Approximation Theory 50, no. 2 (June 1987): 167–74. http://dx.doi.org/10.1016/0021-9045(87)90007-4.

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25

Feng, Xiaobing, and Mitchell Sutton. "A new theory of fractional differential calculus." Analysis and Applications 19, no. 04 (February 20, 2021): 715–50. http://dx.doi.org/10.1142/s0219530521500019.

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This paper presents a self-contained new theory of weak fractional differential calculus in one-dimension. The crux of this new theory is the introduction of a weak fractional derivative notion which is a natural generalization of integer order weak derivatives; it also helps to unify multiple existing fractional derivative definitions and characterize what functions are fractionally differentiable. Various calculus rules including a fundamental theorem calculus, product and chain rules, and integration by parts formulas are established for weak fractional derivatives. Additionally, relationships with classical fractional derivatives and detailed characterizations of weakly fractional differentiable functions are also established. Furthermore, the notion of weak fractional derivatives is also systematically extended to general distributions instead of only to some special distributions. This new theory lays down a solid theoretical foundation for systematically and rigorously developing new theories of fractional Sobolev spaces, fractional calculus of variations, and fractional PDEs as well as their numerical solutions in subsequent works.

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26

Pustyl'Nikov, L. D. "New estimates of Weyl sums and the remainder term in the law of distribution of the fractional part of a polynomial." Ergodic Theory and Dynamical Systems 11, no. 3 (September 1991): 515–34. http://dx.doi.org/10.1017/s0143385700006313.

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27

Bugeaud, Yann. "Linear mod one transformations and the distribution of fractional parts {ξ(p/q)n}." Acta Arithmetica 114, no. 4 (2004): 301–11. http://dx.doi.org/10.4064/aa114-4-1.

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28

Tolev, D. I. "On the simultaneous distribution of the fractional parts of different powers of prime numbers." Journal of Number Theory 37, no. 3 (March 1991): 298–306. http://dx.doi.org/10.1016/s0022-314x(05)80045-9.

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29

Zhai, Wenguang. "On the Simultaneous Distribution of the Fractional Parts of Different Powers of Prime Numbers." Journal of Number Theory 86, no. 1 (January 2001): 133–55. http://dx.doi.org/10.1006/jnth.2000.2563.

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30

Usol’tsev, L. P. "On the distribution of the sequence of fractional parts of a slowly increasing exponential function." Mathematical Notes 65, no. 1 (January 1999): 124–27. http://dx.doi.org/10.1007/bf02675017.

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31

Polanco, Geremías, Daniel Schultz, and Alexandru Zaharescu. "Continuous distributions arising from the Three Gap Theorem." International Journal of Number Theory 12, no. 07 (September 6, 2016): 1743–64. http://dx.doi.org/10.1142/s1793042116501074.

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The well-known Three Gap Theorem states that there are at most three gap sizes in the sequence of fractional parts [Formula: see text]. It is known that if one averages over [Formula: see text], the distribution becomes continuous. We present an alternative approach, which establishes this averaged result and also provides good bounds for the error terms.

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32

Malik, Amita, and Arindam Roy. "On the distribution of zeros of derivatives of the Riemann ξ-function." Forum Mathematicum 32, no. 1 (January 1, 2020): 1–22. http://dx.doi.org/10.1515/forum-2018-0081.

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AbstractFor the completed Riemann zeta function {\xi(s)}, it is known that the Riemann hypothesis for {\xi(s)} implies the Riemann hypothesis for {\xi^{(m)}(s)}, where m is any positive integer. In this paper, we investigate the distribution of the fractional parts of the sequence {(\alpha\gamma_{m})}, where α is any fixed non-zero real number and {\gamma_{m}} runs over the imaginary parts of the zeros of {\xi^{(m)}(s)}. We also obtain a zero density estimate and an explicit formula for the zeros of {\xi^{(m)}(s)}. In particular, all our results hold uniformly for {0\leq m\leq g(T)}, where the function {g(T)} tends to infinity with T and {g(T)=o(\log\log T)}.

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33

Levins, Jess W. "Converting Remote Distribution Centers to a Frame Relay Based Wide Area Network." Project Management Journal 29, no. 1 (March 1998): 44–52. http://dx.doi.org/10.1177/875697289802900109.

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This case study reviews a project that involved the selection and installation of a new information system for unifying four auto parts distribution centers within an overall corporate system. The system consists of local area networks joined together in a wide area network utilizing Fractional T1s and frame relay technology. The project included the implementation of a warehouse management system with radio frequency bar code scanners integrated with inventory, operating and other software for enhancing overall warehouse management.

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34

Deng, Lih–Yuan, and Raj S. Chhikara. "On the characterization of the exponential distribution by the independence of its integer and fractional parts." Statistica Neerlandica 44, no. 2 (June 1990): 83–85. http://dx.doi.org/10.1111/j.1467-9574.1990.tb01529.x.

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35

Pustyl'nikov, L. D. "Probability and ergodic laws in the distribution of the fractional parts of the values of polynomials." Sbornik: Mathematics 192, no. 2 (February 28, 2001): 261–76. http://dx.doi.org/10.1070/sm2001v192n02abeh000545.

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36

Wang, Shaowei, Moli Zhao, Xicheng Li, Xi Chen, and Yanhui Ge. "Exact Solutions of Electro-Osmotic Flow of Generalized Second-Grade Fluid with Fractional Derivative in a Straight Pipe of Circular Cross Section." Zeitschrift für Naturforschung A 69, no. 12 (December 1, 2014): 697–704. http://dx.doi.org/10.5560/zna.2014-0066.

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AbstractThe transient electro-osmotic flow of generalized second-grade fluid with fractional derivative in a narrow capillary tube is examined. With the help of the integral transform method, analytical expressions are derived for the electric potential and transient velocity profile by solving the linearized Poisson-Boltzmann equation and the Navier-Stokes equation. It was shown that the distribution and establishment of the velocity consists of two parts, the steady part and the unsteady one. The effects of retardation time, fractional derivative parameter, and the Debye-Hückel parameter on the generation of flow are shown graphically.

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37

Wang, Ruijie, Feng Yan, and Yanjiao Wang. "Vegetation Growth Status and Topographic Effects in the Pisha Sandstone Area of China." Remote Sensing 12, no. 17 (August 25, 2020): 2759. http://dx.doi.org/10.3390/rs12172759.

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Fractional vegetation coverage (FVC) plays an important role in monitoring vegetation growth status and evaluating restoration efforts in ecological environments. In this study, FVC was calculated using a binary pixel model and analyzed in the Pisha Sandstone area of China, using MODIS-EVI data from 2000 to 2019. Topographic effects were analyzed from elevation, slope and aspect using a terrain niche index model. The results were as follows. (1) From 2000 to 2019, FVC in the Pisha Sandstone area of China gradually increased at a mean rate of 0.0074/a, and the growth status of vegetation gradually improved. (2) The spatial distribution of FVC steadily decreased from southeast to northwest. FVC was the lowest in bare parts of the Pisha Sandstone area, whereas those in the sand- and soil-covered parts were the middle and highest, respectively. (3) With increasing elevation, the inferior coverage area and terrain niche index increased, and inferior coverage distribution changed from non-dominant to dominant. Meanwhile, the low, medium and high coverage areas decreased, and their distributions changed from dominance to non-dominance. (4) With increasing slope, distributions of the inferior, medium and high coverage areas changed from dominant to non-dominant, while the low coverage area had a dominant distribution. (5) Analyses of aspect effects revealed that the inferior coverage area was the dominant distribution in shady slopes but was non-dominant in semi-shady, semi-sunny and sunny slopes. The low, medium and high coverage areas were non-dominant in shady slopes, but dominant in semi-shady, semi-sunny and sunny slopes.

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38

Krasil’shchikov, V. V., and A. V. Shutov. "Description and exact maximum and minimum values of the remainder in the problem of the distribution of fractional parts." Mathematical Notes 89, no. 1-2 (February 2011): 59–67. http://dx.doi.org/10.1134/s0001434611010068.

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39

Wang, Xiaoping, Haitao Qi, and Huanying Xu. "Transient electro-osmotic flow of generalized second-grade fluids under slip boundary conditions." Canadian Journal of Physics 95, no. 12 (December 2017): 1313–20. http://dx.doi.org/10.1139/cjp-2017-0179.

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This work investigates the transient slip flow of viscoelastic fluids in a slit micro-channel under the combined influences of electro-osmotic and pressure gradient forcings. We adopt the generalized second-grade fluid model with fractional derivative as the constitutive equation and the Navier linear slip model as the boundary conditions. The analytical solution for velocity distribution of the electro-osmotic flow is determined by employing the Debye–Hückel approximation and the integral transform methods. The corresponding expressions of classical Newtonian and second-grade fluids are obtained as the limiting cases of our general results. These solutions are presented as a sum of steady-state and transient parts. The combined effects of slip boundary conditions, fluid rheology, electro-osmotic, and pressure gradient forcings on the fluid velocity distribution are also discussed graphically in terms of the pertinent dimensionless parameters. By comparison with the two cases corresponding to the Newtonian fluid and the classical second-grade fluid, it is found that the fractional derivative parameter β has a significant effect on the fluid velocity distribution and the time when the fluid flow reaches the steady state. Additionally, the slip velocity at the wall increases in a noticeable manner the flow rate in an electro-osmotic flow.

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40

Khuzhayorov, Bakhtiyor, Azizbek Usmonov, N. M. A. Nik Long, and Bekzodjon Fayziev. "Anomalous Solute Transport in a Cylindrical Two-Zone Medium with Fractal Structure." Applied Sciences 10, no. 15 (August 3, 2020): 5349. http://dx.doi.org/10.3390/app10155349.

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In this paper, a problem of anomalous solute transport in a coaxial cylindrical two-zone porous medium with fractal structure is posed and numerically solved. The porous medium is studied in the form of cylinder with two parts: macropore—with high permeability characteristics in the central part and micropore—with low permeability around it. Anomalous solute transport is modeled by differential equations with a fractional derivative. The solute concentration and pressure fields are determined. Based on numerical results, the influence of the fractional derivatives order on the solute transport process is analysed. It was shown that with a decrease in the order of the derivatives in the diffusion term of the transport equation in the macropore leads to a “fast diffusion” in both zones. Characteristics of the solute transport in both zones mainly depend on the concentration distribution and other hydrodynamic parameters in the macropore.

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41

Pustyl'nikov, L. D. "Probability laws in the distribution of the fractional parts of the values of polynomials and evidence for the quantum chaos conjecture." Russian Mathematical Surveys 54, no. 6 (December 31, 1999): 1259–60. http://dx.doi.org/10.1070/rm1999v054n06abeh000244.

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42

Davidson, James, and Nigar Hashimzade. "REPRESENTATION AND WEAK CONVERGENCE OF STOCHASTIC INTEGRALS WITH FRACTIONAL INTEGRATOR PROCESSES." Econometric Theory 25, no. 6 (December 2009): 1589–624. http://dx.doi.org/10.1017/s0266466609990260.

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This paper considers the asymptotic distribution of the sample covariance of a nonstationary fractionally integrated process with the stationary increments of another such process—possibly itself. Questions of interest include the relationship between the harmonic representation of these random variables, which we have analyzed in a previous paper (Davidson and Hashimzade, 2008), and the construction derived from moving average representations in the time domain. Depending on the values of the long memory parameters and choice of normalization, the limiting integral is shown to be expressible as the sum of a constant and two Itô-type integrals with respect to distinct Brownian motions. In certain cases the latter terms are of small order relative to the former. The mean is shown to match that of the harmonic representation, where the latter is defined, and satisfies the required integration by parts rule. The advantages of our approach over the harmonic analysis include the facts that our formulas are valid for the full range of the long memory parameters and that they extend to non-Gaussian processes.

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Marklof, Jens, and Nadav Yesha. "Pair correlation for quadratic polynomials mod 1." Compositio Mathematica 154, no. 5 (March 20, 2018): 960–83. http://dx.doi.org/10.1112/s0010437x17008028.

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It is an open question whether the fractional parts of non-linear polynomials at integers have the same fine-scale statistics as a Poisson point process. Most results towards an affirmative answer have so far been restricted to almost sure convergence in the space of polynomials of a given degree. We will here provide explicit Diophantine conditions on the coefficients of polynomials of degree two, under which the convergence of an averaged pair correlation density can be established. The limit is consistent with the Poisson distribution. Since quadratic polynomials at integers represent the energy levels of a class of integrable quantum systems, our findings provide further evidence for the Berry–Tabor conjecture in the theory of quantum chaos.

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Karls, Caleb, Kevin Shinners, and Dan Schaefer. "PSIX-15 Intake of corn stover plant parts by feedlot beef steers." Journal of Animal Science 98, Supplement_4 (November 3, 2020): 425. http://dx.doi.org/10.1093/jas/skaa278.740.

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Abstract Corn stover supports feedlot operations in intensive corn producing regions. A single-pass corn grain and stover harvest system was developed to increase efficiency of field operations and capture different anatomical fractions than are typically harvested with conventional corn stover. The objectives were to feed beef steers diets that included a roughage component consisting of harvested corn residue in chopped form from conventional corn stover bales (CST) or single-pass bales (SPB). Whole-plant corn silage (WPCS) served as a control. Steers (n = 90, 5 pens/treatment) were fed during Grow (84 d) and Finish (66 d) phases to assess consumption of corn plant botanical fractions and calculate net energy values of the stover feeds. Cattle consumed a larger proportion of stover as cob (P < 0.001) and less as stalk (P = 0.001) when stover was offered as SPB rather than CST. These differences are consistent with the fractional distribution of botanical components offered. During the Grow phase, cattle fed WPCS had greater (P = 0.018) daily gains (1.27 kg d-1) than cattle fed the SPB (1.14 kg d-1) and CST (1.08 kg d-1), and were more efficient than CST cattle. Steers sorted corn stover during both phases and consumed 52.5% of corn stover offered. SPB cob intake was 70% greater than CST cob intake (P < 0.01) indicating if more cob fraction is available, cattle will consume more. There was no treatment effect on final body weight (P = 0.37) or growth rate (P = 0.12) during the Finish phase. Stover NEm and NEg were calculated using Dairy NRC (2001) methods for SPB (1.04 and 0.49 Mcal kg-1) and CST (0.98 and 0.44 Mcal kg-1), respectively. In conclusion, there is evidence that CST and SPB can substitute for WPCS in beef feedlot diets without adverse effects on overall steer performance.

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45

Chen, Ming. "Fractional-Order Adaptive P -Laplace Equation-Based Art Image Edge Detection." Advances in Mathematical Physics 2021 (August 31, 2021): 1–10. http://dx.doi.org/10.1155/2021/2337712.

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In recent years, with the rapid development of image processing research, the study of nonstandard images has gradually become a research hotspot, for example, fabric images, remote sensing images, and gear images. Some of the remote sensing images have a complex background and low illumination compared with standard images and are easy to be mixed with noise during acquisition; some of the fabric images have rich texture information, which adds difficulty to the related processing, and are also easy to be mixed with noise during acquisition. In this paper, we propose a fractional-order adaptive P -Laplace equation image edge detection algorithm for the problem of image edge detection in which the edge and texture information of the image is lost. The algorithm can apply for the order adaptively to filter the noise according to the noise distribution of the image, and the adaptive diffusion factor is determined by both the fractional-order curvature and fractional-order gradient of the iso-illumination line and combined with the iterative approach to realize the fine-tuning of the noisy image. The experimental results demonstrate that the algorithm can remove the noise while preserving the texture and details of the image. A fractional-order partial differential equation image edge detection model with a fractional-order fidelity term is proposed for Gaussian noise. The model incorporates a fractional-order fidelity term because this fidelity term smoothes out the rougher parts of the image while preserving the texture in the original image in greater detail and eliminating the step effect produced by other models such as the Perona-Malik (PM) and Rudin-Osher-Fatemi (ROF) models. By comparing with other algorithms, the image edge detection effect is measured with the help of evaluation metrics such as peak signal-to-noise ratio and structural similarity, and the optimal value is selected iteratively so that the image with the best edge detection result is retained. A convolutional mask image edge detection model based on adaptive fractional-order calculus is proposed for the scattered noise in medical images. The adaption is mainly reflected in the model algorithm by constructing an exponential parameter relation that is closely related to the image, which can dynamically adjust the parameter values, thus making the model algorithm more practical. The model achieves the scattering noise removal in four steps.

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Bournez, Olivier, Oleksiy Kurganskyy, and Igor Potapov. "Reachability Problems for One-Dimensional Piecewise Affine Maps." International Journal of Foundations of Computer Science 29, no. 04 (June 2018): 529–49. http://dx.doi.org/10.1142/s0129054118410046.

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Piecewise affine maps (PAMs) are frequently used as a reference model to discuss the frontier between known and open questions about the decidability for reachability questions. In particular, the reachability problem for one-dimensional PAM is still an open problem, even if restricted to only two intervals. As the main contribution of this paper we introduce new techniques for solving reachability problems based on [Formula: see text]-adic norms and weights as well as showing decidability for two classes of maps. Then we show the connections between topological properties for PAM’s orbits, reachability problems and representation of numbers in a rational base system. Finally we construct an example where the distribution properties of well studied sequences can be significantly disrupted by taking fractional parts after regular shifts. The study of such sequences could help with understanding similar sequences generated in PAMs or in well known Mahler’s [Formula: see text] problem.

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Pustyl'nikov, L. D. "Generalized continuous fractions and estimates of weyl sums and the remainder term in the distribution law for the fractional parts of the values of a polynomial." Mathematical Notes 56, no. 6 (December 1994): 1315–17. http://dx.doi.org/10.1007/bf02266704.

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Blais, Sylvain, Bernard Auvray, Bor-ming Jahn, and Kalle Taipale. "Processus de fractionnement dans les coulées komatiitiques archéennes: cas des laves à spinifex de la ceinture de roches vertes de Tipasjärvi (Finlande orientale)." Canadian Journal of Earth Sciences 24, no. 5 (May 1, 1987): 953–66. http://dx.doi.org/10.1139/e87-093.

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Komatiitic flows (ultramafic lavas with MgO > 18%) from the small Tipasjärvi Archean greenstone belt (eastern Finland) are of two types: (1) thin lava flows with microspinifex textures showing no chemical variation along a vertical cross section; (2) a thicker flow with important chemical variation in which macroscopic spinifex textures are found within small veins in the upper part of this flow. Two different trends of fractional crystallization can be demonstrated by major- and some trace-element variations in the upper and the lower parts of the flow: olivine crystallizes and settles to the base of the flow; in the upper part, the major phase that crystallizes seems to be clinopyroxene.Such double-trend fractionation has also been shown in some other komatiitic flows of similar ages from Australia and Canada. However, some komatiitic flows indicate a simpler fractionation process involving olivine only.Rare-earth-element (REE) distribution patterns in the thick Finnish flow show two quite different patterns. The upper part of the flow has strongly LREE-depleted patterns, whilst those in the lower part of the flow are flat. It is suggested that the flow is composite and that two source regions are involved.

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Mukhamatdinov, Irek I., Indad Sh S. Salih, Ilfat Z. Rakhmatullin, Nikita N. Sviridenko, Galina S. Pevneva, Rakesh K. Sharma, and Alexey V. Vakhin. "Transformation of Resinous Components of the Ashalcha Field Oil during Catalytic Aquathermolysis in the Presence of a Cobalt-Containing Catalyst Precursor." Catalysts 11, no. 6 (June 18, 2021): 745. http://dx.doi.org/10.3390/catal11060745.

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The aim of this work was to study the fractional composition of super-viscous oil resins from the Ashalcha field, as well as the catalytic aquathermolysis product in the presence of a cobalt-containing catalyst precursor and a hydrogen donor. The study was conducted at various durations of thermal steam exposure. In this regard, the work enabled the identification of the distribution of resin fractions. These fractions, obtained by liquid adsorption chromatography, were extracted with individual solvents and their binary mixtures in various ratios. The results of MALDI spectroscopy revealed a decrease in the molecular mass of all resin fractions after catalytic treatment, mainly with a hydrogen donor. However, the elemental analysis data indicated a decrease in the H/C ratio for resin fractions as a result of removing alkyl substituents in resins and asphaltenes. Moreover, the data of 1H NMR spectroscopy of resin fractions indicated an increase in the aliphatic hydrogen index during catalytic aquathermolysis at the high molecular parts of the resins R3 and R4. Finally, a structural group analysis was carried out in this study, and hypothetical structures of the initial oil resin molecules and aquathermolysis products were constructed as well.

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Lee, Joo-Heon, Hyun-Han Kwon, Ho-Won Jang, and Tae-Woong Kim. "Future Changes in Drought Characteristics under Extreme Climate Change over South Korea." Advances in Meteorology 2016 (2016): 1–19. http://dx.doi.org/10.1155/2016/9164265.

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This study attempts to analyze several drought features in South Korea from various perspectives using a three-month standard precipitation index. In particular, this study aims to evaluate changes in spatial distribution in terms of frequency and severity of droughts in the future due to climate change, using IPCC (intergovernmental panel on climate change) GCM (general circulation model) simulations. First, the Mann-Kendall method was adopted to identify drought trends at the five major watersheds. The simulated temporal evolution of SPI (standardized precipitation index) during the winter showed significant drying trends in most parts of the watersheds, while the simulated SPI during the spring showed a somewhat different feature in the GCMs. Second, this study explored the low-frequency patterns associated with drought by comparing global wavelet power, with significance test. Future spectra decreased in the fractional variance attributed to a reduction in the interannual band from 2 to 8 years. Finally, the changes in the frequency and the severity under climate change were evaluated through the drought spell analyses. Overall features of drought conditions in the future showed a tendency to increase (about 6%) in frequency and severity of droughts during the dry season (i.e., from October to May) under climate change.

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