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Journal articles on the topic 'Fractional Four Color Theorem'

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Published: 4 June 2021
Last updated: 4 February 2022

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1

Koch, John A., Rudolf Fritsch, Gerda Fritsch, and J. Peschke. "The Four-Color Theorem." American Mathematical Monthly 106, no. 8 (October 1999): 785. http://dx.doi.org/10.2307/2589042.

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2

Wang Weiqing, Wang Weihua, and Wang Weifu. "Study on the Four Color Theorem." Journal of Convergence Information Technology 8, no. 7 (April 15, 2013): 1220–28. http://dx.doi.org/10.4156/jcit.vol8.issue7.150.

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3

邹, 山中. "Mathematical Proof of Four-Color Theorem." Pure Mathematics 09, no. 03 (2019): 410–13. http://dx.doi.org/10.12677/pm.2019.93054.

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4

Eppelbaum, Lev V. "Four Color Theorem and Applied Geophysics." Applied Mathematics 05, no. 04 (2014): 658–66. http://dx.doi.org/10.4236/am.2014.54062.

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5

Huson, D. H. "A four-color theorem for periodic tilings." Geometriae Dedicata 51, no. 1 (May 1994): 47–61. http://dx.doi.org/10.1007/bf01264100.

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6

Eliahou, Shalom, and Cédric Lecouvey. "Signed permutations and the four color theorem." Expositiones Mathematicae 27, no. 4 (2009): 313–40. http://dx.doi.org/10.1016/j.exmath.2009.04.002.

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7

Bar-Natan, Dror. "Lie algebras and the Four Color Theorem." Combinatorica 17, no. 1 (March 1997): 43–52. http://dx.doi.org/10.1007/bf01196130.

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8

Das Sarma, Atish, Amita S. Gajewar, Richard L. Lipton, and Danupon Nanongkai. "An Approximate Restatement of the Four-Color Theorem." Journal of Graph Algorithms and Applications 17, no. 5 (2013): 567–73. http://dx.doi.org/10.7155/jgaa.00304.

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9

Matiyasevich, Yuri. "Four color theorem from three points of view." Illinois Journal of Mathematics 60, no. 1 (2016): 185–205. http://dx.doi.org/10.1215/ijm/1498032030.

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10

Eliahou, Shalom. "Signed Diagonal Flips and the Four Color Theorem." European Journal of Combinatorics 20, no. 7 (October 1999): 641–47. http://dx.doi.org/10.1006/eujc.1999.0312.

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11

Kauffman*, Louis, and Robin Thomas†. "Temperely-Lieb Algebras and the Four-Color Theorem." COMBINATORICA 23, no. 4 (December 2003): 653–67. http://dx.doi.org/10.1007/s00493-003-0039-7.

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12

Sipka, Timothy. "Alfred Bray Kempe's “Proof” of the Four-Color Theorem." Math Horizons 10, no. 2 (November 2002): 21–26. http://dx.doi.org/10.1080/10724117.2002.11974616.

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13

Ohnishi, Koji. "Towards a brief proof of the Four-Color Theorem without using a computer: theorems to be used for proving the Four-Color Theorem." Artificial Life and Robotics 14, no. 4 (December 2009): 551–56. http://dx.doi.org/10.1007/s10015-009-0745-3.

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14

Jr., John W. Oller. "The Four-Color Theorem of Map-Making Proved by Construction." OALib 03, no. 12 (2016): 1–12. http://dx.doi.org/10.4236/oalib.1103089.

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15

韩, 文镇. "Tait’s Conjecture Continue—The Proof of the Four-Color Theorem." Pure Mathematics 09, no. 08 (2019): 949–60. http://dx.doi.org/10.12677/pm.2019.98121.

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16

Cooper, Bobbe, Eric Rowland, and Doron Zeilberger. "Toward a language theoretic proof of the four color theorem." Advances in Applied Mathematics 48, no. 2 (February 2012): 414–31. http://dx.doi.org/10.1016/j.aam.2011.11.002.

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17

Hodneland, Erlend, Xue-Cheng Tai, and Hans-Hermann Gerdes. "Four-Color Theorem and Level Set Methods for Watershed Segmentation." International Journal of Computer Vision 82, no. 3 (December 17, 2008): 264–83. http://dx.doi.org/10.1007/s11263-008-0199-4.

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18

Doi, H. "The "four-color issue" in ecology for considering ecosystem boundaries." Web Ecology 13, no. 1 (October 28, 2013): 91–93. http://dx.doi.org/10.5194/we-13-91-2013.

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Abstract. Ecosystem boundaries are important structures in defining ecosystems. To date, ecologists have not extensively considered which boundaries are important in explaining ecological phenomena in order to simplify ecological theories. The four-color theorem in mathematics maintains that only four colors are required to color a set of regions so that no two adjacent regions have the same color. Before being proven in 1976, the theorem was considered the "four-color issue", which proposed that a small number of colors were required to separate regional boundaries. Applying the principle of "four-color issue" to the ecological field, we can also examine reducing the number of ecosystem boundaries considered. That is, we can ask ourselves the following question: "how many boundaries of an ecosystem should be considered for ecology"? Here, I suggest a principle of ecosystem boundaries as the "four-color issue of ecology", and propose that this will be an important step toward advancing knowledge in ecology and conservation biology. In addition, I introduce graph theory, developed from the four-color theorem, which can be useful for estimating ecosystem boundaries.
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19

Xue, Geng, Hui Liu, Shi Xian Li, and Wei Chen. "A New Method Based on Undirected Graph and Polygon Triangulation to Prove the Four-Color Theorem." Applied Mechanics and Materials 347-350 (August 2013): 2832–35. http://dx.doi.org/10.4028/www.scientific.net/amm.347-350.2832.

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The development of computer science and plane geometry brings a new favorable opportunity for the research about the Four-color Theorem. After converting original map to undirected graph, this paper proposes a new method based on undirected graph and polygon triangulation to prove the Four-color Theorem.
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20

Liu, Yuji. "A new method for converting boundary value problems for impulsive fractional differential equations to integral equations and its applications." Advances in Nonlinear Analysis 8, no. 1 (May 5, 2017): 386–454. http://dx.doi.org/10.1515/anona-2016-0064.

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Abstract In this paper, we present a new method for converting boundary value problems of impulsive fractional differential equations to integral equations. Applications of this method are given to solve some types of anti-periodic boundary value problems for impulsive fractional differential equations. Firstly by using iterative method, we prove existence and uniqueness of solutions of Cauchy problems of differential equations involving Caputo fractional derivative, Riemann–Liouville and Hadamard fractional derivatives with order {q\in(0,1)} , see Theorem 2, Theorem 4, Theorem 6 and Theorem 8. Then we obtain exact expression of piecewise continuous solutions of these fractional differential equations see Theorem 1, Theorem 2, Theorem 3 and Theorem 4. Finally, four classes of integral type anti-periodic boundary value problems of singular fractional differential equations with impulse effects are proposed. Sufficient conditions are given for the existence of solutions of these problems. See Theorems 4.1–4.4. We allow the nonlinearity {p(t)f(t,x)} in fractional differential equations to be singular at {t=0,1} and be involved a super-linear and sub-linear term. The analysis relies on Schaefer’s fixed point theorem. In order to avoid misleading readers, we correct the results in [28] and [65]. We establish sufficient conditions for the existence of solutions of an anti-periodic boundary value problem for impulsive fractional differential equation. The results in [68] are complemented. The results in [81] are corrected. See Lemma 5.1, Lemma 5.7, Lemma 5.10 and Lemma 5.13.
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21

Memory, J. D. "Dialog with Computer in the Proof of the Four-Color Theorem." Mathematics Magazine 74, no. 4 (October 1, 2001): 313. http://dx.doi.org/10.2307/2691102.

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22

Memory, J. D. "Dialog with Computer in the Proof of the Four-Color Theorem." Mathematics Magazine 74, no. 4 (October 2001): 313. http://dx.doi.org/10.1080/0025570x.2001.11953082.

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23

Bhapkar, H. R., and J. N. Salunke. "Proof of Four Color Map Theorem by Using PRN of Graph." Bulletin of Society for Mathematical Services and Standards 11 (September 2014): 26–30. http://dx.doi.org/10.18052/www.scipress.com/bsmass.11.26.

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This paper intends to study the relation between PRN and chromatic number of planar graphs. In this regard we investigate that isomorphic or 1 isomorphic graph may or may not have equal PRN and few other related results. Precisely, we give simple proof of Four Color Map Theorem.
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24

Calude, Cristian S., and Elena Calude. "The complexity of the four colour theorem." LMS Journal of Computation and Mathematics 13 (August 27, 2010): 414–25. http://dx.doi.org/10.1112/s1461157009000461.

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AbstractThe four colour theorem states that the vertices of every planar graph can be coloured with at most four colours so that no two adjacent vertices receive the same colour. This theorem is famous for many reasons, including the fact that its original 1977 proof includes a non-trivial computer verification. Recently, a formal proof of the theorem was obtained with the equational logic program Coq [G. Gonthier, ‘Formal proof–the four color theorem’,Notices of Amer. Math. Soc.55 (2008) no. 11, 1382–1393]. In this paper we describe an implementation of the computational method introduced by C. S. Calude and co-workers [Evaluating the complexity of mathematical problems. Part 1’,Complex Systems18 (2009) 267–285; A new measure of the difficulty of problems’,J. Mult. Valued Logic Soft Comput.12 (2006) 285–307] to evaluate the complexity of the four colour theorem. Our method uses a Diophantine equational representation of the theorem. We show that the four colour theorem is in the complexity class ℭU,4. For comparison, the Riemann hypothesis is in class ℭU,3while Fermat’s last theorem is in class ℭU,1.
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25

MASHOOD, BAHMAN. "On Geometrical Methods that Provide a Short Proof of Four Color Theorem." American Journal of Mathematical Analysis 3, no. 3 (September 12, 2015): 65–71. http://dx.doi.org/10.12691/ajma-3-3-2.

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26

Ellen, Gethner, Bopanna Kallichanda, Alexander Mentis, Sarah Braudrick, Sumeet Chawla, Andrew Clune, Rachel Drummond, Panagiota Evans, William Roche, and Nao Takano. "How false is Kempe’s proof of the Four Color Theorem? Part II." Involve, a Journal of Mathematics 2, no. 3 (October 3, 2009): 249–65. http://dx.doi.org/10.2140/involve.2009.2.249.

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27

Bouridah, Mohammed Salah, Toufik Bouden, and Müştak Erhan Yalçin. "Chaos Synchronization of Fractional-Order Lur’e Systems." International Journal of Bifurcation and Chaos 30, no. 14 (November 2020): 2050206. http://dx.doi.org/10.1142/s0218127420502065.

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Based on some essential concepts of fractional calculus and the theorem related to the fractional extension of Lyapunov direct method, we present in this paper a synchronization scheme of fractional-order Lur’e systems. A quadratic Lyapunov function is chosen to derive the synchronization criterion. The derived criterion is a suffcient condition for the asymptotic stability of the error system, formulated in the form of linear matrix inequality (LMI). The controller gain can be achieved by solving the LMI. The proposed scheme is illustrated for fractional-order Chua’s circuits and fractional-order four-cell CNN. Numerical results, which agree well with the proposed theorem, are given to show the effectiveness of this scheme.
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28

Chen, Lei, and Hai-Lin Liu. "An Evolutionary Algorithm Based on the Four-Color Theorem for Location Area Planning." Mathematical Problems in Engineering 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/271935.

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As an important constituent of wireless network planning, location area planning (LAP) directly affects the stability, security, and performance of wireless network. This work proposes a novel evolutionary algorithm (EA) to solve the LAP problem. The difference between the proposed algorithm and the previous EA is mainly how to encode. The new coding method is inspired by the famous four-color theorem in graph theory. Only four numbers are needed to encode all chromosomes by this method. The encoding and decoding process is fast and easy to implement. What is more, illegal solutions can be processed easily in the process of decoding. The design of effective and efficient genetic operators can also benefit from this coding method. The modified evolutionary algorithm with this coding method is especially effective for LAP problem. The use of the principle of fuzzy clustering in initialization can effectively compress the search space in this new algorithm. The computer simulation has been conducted, and the quality of proposed algorithm is confirmed by comparing the results of proposed algorithm with EA and simulated annealing (SA).
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29

Luckett Shuler Jr., Robert. "Entropy-Like State Counting Leads to Human Readable Four Color Map Theorem Proof." Pure and Applied Mathematics Journal 7, no. 3 (2018): 37. http://dx.doi.org/10.11648/j.pamj.20180703.12.

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30

Tian, Yuansheng, Sujing Sun, and Zhanbing Bai. "Positive Solutions of Fractional Differential Equations with p-Laplacian." Journal of Function Spaces 2017 (2017): 1–9. http://dx.doi.org/10.1155/2017/3187492.

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The multiplicity of positive solution for a new class of four-point boundary value problem of fractional differential equations with p-Laplacian operator is investigated. By the use of the Leggett-Williams fixed-point theorem, the multiplicity results of positive solution are obtained. An example is given to illustrate the main results.
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31

Gaber, M., and M. G. Brikaa. "Existence Results for a Coupled System of Nonlinear Fractional Differential Equation with Four-Point Boundary Conditions." ISRN Mathematical Analysis 2011 (December 15, 2011): 1–14. http://dx.doi.org/10.5402/2011/468346.

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This paper studies a coupled system of nonlinear fractional differential equation with four-point boundary conditions. Applying the Schauder fixed-point theorem, an existence result is proved for the following system: , , , , , , , , where satisfy certain conditions.
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32

Song, Chuan-Jing, and Yi Zhang. "Noether symmetry and conserved quantity for fractional Birkhoffian mechanics and its applications." Fractional Calculus and Applied Analysis 21, no. 2 (April 25, 2018): 509–26. http://dx.doi.org/10.1515/fca-2018-0028.

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AbstractNoether theorem is an important aspect to study in dynamical systems. Noether symmetry and conserved quantity for the fractional Birkhoffian system are investigated. Firstly, fractional Pfaff actions and fractional Birkhoff equations in terms of combined Riemann-Liouville derivative, Riesz-Riemann-Liouville derivative, combined Caputo derivative and Riesz-Caputo derivative are reviewed. Secondly, the criteria of Noether symmetry within combined Riemann-Liouville derivative, Riesz-Riemann-Liouville derivative, combined Caputo derivative and Riesz-Caputo derivative are presented for the fractional Birkhoffian system, respectively. Thirdly, four corresponding conserved quantities are obtained. The classical Noether identity and conserved quantity are special cases of this paper. Finally, four fractional models, such as the fractional Whittaker model, the fractional Lotka biochemical oscillator model, the fractional Hénon-Heiles model and the fractional Hojman-Urrutia model are discussed as examples to illustrate the results.
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33

Burger, Edward B., and Frank Morgan. "Fermat's Last Theorem, the Four Color Conjecture, and Bill Clinton for April Fools' Day." American Mathematical Monthly 104, no. 3 (March 1997): 246–55. http://dx.doi.org/10.1080/00029890.1997.11990629.

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34

Zhang, Ruiliang, Xavier Bresson, Tony F. Chan, and Xue-Cheng Tai. "Four color theorem and convex relaxation for image segmentation with any number of regions." Inverse Problems & Imaging 7, no. 3 (2013): 1099–113. http://dx.doi.org/10.3934/ipi.2013.7.1099.

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35

Donnelly, John, Ryan Hicks, and Kurt Virgin. "Generators and normal forms of Richard Thompson’s group F and the four-color theorem." Journal of Algebraic Combinatorics 43, no. 3 (October 30, 2015): 485–93. http://dx.doi.org/10.1007/s10801-015-0643-9.

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36

Burger, Edward B., and Frank Morgan. "Fermat's Last Theorem, the Four Color Conjecture, and Bill Clinton for April Fools' Day." American Mathematical Monthly 104, no. 3 (March 1997): 246. http://dx.doi.org/10.2307/2974790.

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37

Sanders, Daniel P., and Yue Zhao. "On the Entire Coloring Conjecture." Canadian Mathematical Bulletin 43, no. 1 (March 1, 2000): 108–14. http://dx.doi.org/10.4153/cmb-2000-017-7.

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AbstractThe Four Color Theorem says that the faces (or vertices) of a plane graph may be colored with four colors. Vizing’s Theorem says that the edges of a graph with maximum degree Δ may be colored with Δ + 1 colors. In 1972, Kronk and Mitchem conjectured that the vertices, edges, and faces of a plane graph may be simultaneously colored with Δ + 4 colors. In this article, we give a simple proof that the conjecture is true if Δ ≥ 6.
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38

Prasad, Kapula Rajendra, Boddu Muralee Bala Krushna, and L. T. Wesen. "Existence results for positive solutions to iterative systems of four-point fractional-order boundary value problems in a Banach space." Asian-European Journal of Mathematics 13, no. 04 (November 30, 2018): 2050070. http://dx.doi.org/10.1142/s1793557120500709.

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We investigate the eigenvalue intervals of [Formula: see text] for which the iterative system of four-point fractional-order boundary value problem has at least one positive solution by utilizing Guo–Krasnosel’skii fixed point theorem under suitable conditions. The obtained results in the paper are illustrated with an example for their feasibility.
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39

Verloren van Themaat, W. A. "The own character of mathematics discussed with consideration of the proof of the four-color theorem." Zeitschrift für Allgemeine Wissenschaftstheorie 20, no. 2 (September 1989): 340–50. http://dx.doi.org/10.1007/bf01801483.

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40

Xia, Zhile. "Design of Static Output Feedback Controller for Fractional-Order T-S Fuzzy System." Mathematical Problems in Engineering 2020 (June 15, 2020): 1–9. http://dx.doi.org/10.1155/2020/7898109.

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This paper studies the design of fuzzy static output feedback controllers for two kinds of fractional-order T-S fuzzy systems. The fractional order α satisfies 0<α<1 and 1≤α<2. Based on the fractional order theory, matrix decomposition technique, and projection theorem, four new sufficient conditions for the asymptotic stability of the system and the corresponding controller design methods are given. All the results can be expressed by linear matrix inequalities, and the relationship between fuzzy subsystems is also considered. These have great advantages in solving the results and reducing the conservatism. Finally, a simulation example is given to show the effectiveness of the proposed method.
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41

Zada, Akbar, Sartaj Ali, and Tongxing Li. "Analysis of a New Class of Impulsive Implicit Sequential Fractional Differential Equations." International Journal of Nonlinear Sciences and Numerical Simulation 21, no. 6 (October 25, 2020): 571–87. http://dx.doi.org/10.1515/ijnsns-2019-0030.

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AbstractIn this paper, we study an implicit sequential fractional order differential equation with non-instantaneous impulses and multi-point boundary conditions. The article comprehensively elaborate four different types of Ulam’s stability in the lights of generalized Diaz Margolis’s fixed point theorem. Moreover, some sufficient conditions are constructed to observe the existence and uniqueness of solutions for the proposed model. The proposed model contains both the integer order and fractional order derivatives. Thus, the exponential function appearers in the solution of the proposed model which will lead researchers to study fractional differential equations with well known methods of integer order differential equations. In the last, few examples are provided to show the applicability of our main results.
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42

Keeports, David. "A Map-coloring Algorithm." Mathematics Teacher 84, no. 9 (December 1991): 759–63. http://dx.doi.org/10.5951/mt.84.9.0759.

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Among the most tantalizing propositions of mathematics are those generalizations that are easily and concisely stated and are readily shown to be true for specific cases yet, despite their apparent simplicity, defy concise proof. Outstanding examples of such propositions include Fermat's last theorem (see Vanden Eynden [1989]), the Goldbach conjecture, and the four-color theorem. Because such propositions can be understood by students with almost no previous background in mathematics, they are easily introduced in mathematics courses intended for the liberal arts student.
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43

Yang, Wengui. "Existence results for nonlinear fractional q-difference equations with nonlocal Riemann-Liouville q-integral boundary conditions." Filomat 30, no. 9 (2016): 2521–33. http://dx.doi.org/10.2298/fil1609521y.

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This paper deals with the existence and uniqueness of solutions for a class of nonlinear fractional q-difference equations boundary value problems involving four-point nonlocal Riemann-Liouville q-integral boundary conditions of different order. Our results are based on some well-known tools of fixed point theory such as Banach contraction principle, Krasnoselskii fixed point theorem, and the Leray-Schauder nonlinear alternative. As applications, some interesting examples are presented to illustrate the main results.
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44

Joshi, Madhusudan, Chandra Shakher, and Kehar Singh. "Fractional Fourier transform based image multiplexing and encryption technique for four-color images using input images as keys." Optics Communications 283, no. 12 (June 2010): 2496–505. http://dx.doi.org/10.1016/j.optcom.2010.02.024.

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45

Garrido, Santiago, Concepción A. Monje, Fernando Martín, and Luis Moreno. "Design of Fractional Order Controllers Using the PM Diagram." Mathematics 8, no. 11 (November 13, 2020): 2022. http://dx.doi.org/10.3390/math8112022.

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This work presents a modeling and controller tuning method for non-rational systems. First, a graphical tool is proposed where transfer functions are represented in a four-dimensional space. The magnitude is represented in decibels as the third dimension and a color code is applied to represent the phase in a fourth dimension. This tool, which is called Phase Magnitude (PM) diagram, allows the user to visually obtain the phase and the magnitude that have to be added to a system to meet some control design specifications. The application of the PM diagram to systems with non-rational transfer functions is discussed in this paper. A fractional order Proportional Integral Derivative (PID) controller is computed to control different non-rational systems. The tuning method, based on evolutionary computation concepts, relies on a cost function that defines the behavior in the frequency domain. The cost value is read in the PM diagram to estimate the optimum controller. To validate the contribution of this research, four different non-rational reference systems have been considered. The method proposed here contributes first to a simpler and graphical modeling of these complex systems, and second to provide an effective tool to face the unsolved control problem of these systems.
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46

Abd-Elhameed, W. M., and Y. H. Youssri. "Sixth-Kind Chebyshev Spectral Approach for Solving Fractional Differential Equations." International Journal of Nonlinear Sciences and Numerical Simulation 20, no. 2 (April 26, 2019): 191–203. http://dx.doi.org/10.1515/ijnsns-2018-0118.

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AbstractThe basic aim of this paper is to develop new numerical algorithms for solving some linear and nonlinear fractional-order differential equations. We have developed a new type of Chebyshev polynomials, namely, Chebyshev polynomials of sixth kind. This type of polynomials is a special class of symmetric orthogonal polynomials, involving four parameters that were constructed with the aid of the extended Sturm–Liouville theorem for symmetric functions. The proposed algorithms are basically built on reducing the fractional-order differential equations with their initial/boundary conditions to systems of algebraic equations which can be efficiently solved. The new proposed algorithms are supported by a detailed study of the convergence and error analysis of the sixth-kind Chebyshev expansion. New connection formulae between Chebyshev polynomials of the second and sixth kinds were established for this study. Some examples were displayed to illustrate the efficiency of the proposed algorithms compared to other methods in literature. The proposed algorithms have provided accurate results, even using few terms of the proposed expansion.
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47

Nageswara Rao, S., and M. Zico Meetei. "Positive Solutions for a Coupled System of Nonlinear Semipositone Fractional Boundary Value Problems." International Journal of Differential Equations 2019 (February 3, 2019): 1–9. http://dx.doi.org/10.1155/2019/2893857.

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In this paper, we consider a four-point coupled boundary value problem for system of the nonlinear semipositone fractional differential equation D0+αu(t)+λf(t,u(t),v(t))=0, 0<t<1, D0+αv(t)+μg(t,u(t),v(t))=0, 0<t<1, u(0)=v(0)=0, a1D0+βu(1)=b1D0+βv(ξ), a2D0+βv(1)=b2D0+βu(η), η,ξ∈(0,1), where the coefficients ai,bi,i=1,2 are real positive constants, α∈(1,2],β∈(0,1],D0+α, D0+β are the standard Riemann-Liouville derivatives. Values of the parameters λ and μ are determined for which boundary value problem has positive solution by utilizing a fixed point theorem on cone.
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48

Goel, Navdeep, and Kulbir Singh. "A modified convolution and product theorem for the linear canonical transform derived by representation transformation in quantum mechanics." International Journal of Applied Mathematics and Computer Science 23, no. 3 (September 1, 2013): 685–95. http://dx.doi.org/10.2478/amcs-2013-0051.

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Abstract The Linear Canonical Transform (LCT) is a four parameter class of integral transform which plays an important role in many fields of signal processing. Well-known transforms such as the Fourier Transform (FT), the FRactional Fourier Transform (FRFT), and the FreSnel Transform (FST) can be seen as special cases of the linear canonical transform. Many properties of the LCT are currently known but the extension of FRFTs and FTs still needs more attention. This paper presents a modified convolution and product theorem in the LCT domain derived by a representation transformation in quantum mechanics, which seems a convenient and concise method. It is compared with the existing convolution theorem for the LCT and is found to be a better and befitting proposition. Further, an application of filtering is presented by using the derived results.
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49

Singh, Ashutosh Kumar, and Rajiv Saxena. "DFRFT: A Classified Review of Recent Methods with Its Application." Journal of Engineering 2013 (2013): 1–13. http://dx.doi.org/10.1155/2013/214650.

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In the literature, there are various algorithms available for computing the discrete fractional Fourier transform (DFRFT). In this paper, all the existing methods are reviewed, classified into four categories, and subsequently compared to find out the best alternative from the view point of minimal computational error, computational complexity, transform features, and additional features like security. Subsequently, the correlation theorem of FRFT has been utilized to remove significantly the Doppler shift caused due to motion of receiver in the DSB-SC AM signal. Finally, the role of DFRFT has been investigated in the area of steganography.
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50

Moosavi, Horieh, and Fatemeh Darvishzadeh. "The Influence of Post Bleaching Treatments in Stain Absorption and Microhardness." Open Dentistry Journal 10, no. 1 (March 25, 2016): 69–78. http://dx.doi.org/10.2174/1874210616021000069.

Full text
Abstract:
Objectives: This study investigated the effects of post bleaching treatments to prevent restaining and the change of enamel surface microhardness after dental bleaching in vitro. Methods: Sixty intact human incisor teeth were stained in tea solution and randomly assigned into four groups (n=15). Then samples were bleached for two weeks (8 hours daily) by 15% carbamide peroxide. Tooth color was determined both with a spectrophotometer and visually before bleaching (T1) and immediately after bleaching (T2). Next, it was applied in group 1 fluoride (Naf 2%) gel for 2 minutes, and in group 2 a fractional CO2 laser (10 mJ, 200 Hz, 10 s), and in group 3, nanohydroxyapatite gel for 2 minutes. The bleached teeth in group 4 remained untreated (control group). Then teeth placed in tea solution again. Color examinations were repeated after various post bleaching treatments (T3) and restaining with tea (T4) and color change values recorded. The microhardness was measured at the enamel surface of samples. Data was analyzed using ANOVA, Tukey HSD test and Dunnett T3 (α = 0.05). Results: Directly after bleaching (ΔE T3-T2), the treatment with nanohydroxyapatite showed significantly the least color lapse in colorimetric evaluation. In experimental groups, the color change between T3 and T4 stages (ΔE T4-T3) was significantly lower than control group (P < 0.05). Different methods of enamel treatment caused a significant increase in surface microhardness compared to control group (P < 0.05). Significance: Application of fluoride, fractional CO2 laser and nanohydroxyapatite as post bleaching treatments are suggested for prevention of stain absorption and increasing the hardening of bleached enamel.
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