Young’s modulus, atom volumes for boron and transition metals, and Goldschmidt criterion for SSS formation.
\r\n\tThe most common results are dental cavities which eventually lead to pulp infection and sometimes tooth loss if left untreated. The successful endodontic treatment cane save the tooth and alveolar bone as well. Nowadays, there are many innovative methods in order to preserve the vitality of tooth without invading the pulp, and to enhance the success rate of endodontic therapy when needed, reduce the chair time and avoid post-operative symptoms. Starting from diagnosis using the most advanced X-rays, to a professional access to the pulp chamber, and root canal shaping with the very effective instruments and irrigation in order to create the ideal space for an optimal canal filling; this academic book will provide a comprehensive approach to a successful endodontic therapy of the most challenging cases, and ways of managing many severe cases in order to upgrade the life quality of the patient.
",isbn:null,printIsbn:"979-953-307-X-X",pdfIsbn:null,doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"cb2c892352c013fcad77cef2f16a1da6",bookSignature:"Ph.D. Alaa Eddin Omar Al Ostwani",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/9071.jpg",keywords:"Root Canal Shaping, Ultrasonic, Reciprocating System, Rotary Files, Sodium Hypochlorite, Chlorhexidine Gel, Laser Activation, Apical Negative Pressure, Bioceramic, Gutta-percha, Vertical Compaction, MTA",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:0,numberOfDimensionsCitations:0,numberOfTotalCitations:0,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"May 7th 2019",dateEndSecondStepPublish:"September 4th 2019",dateEndThirdStepPublish:"November 3rd 2019",dateEndFourthStepPublish:"January 22nd 2020",dateEndFifthStepPublish:"March 22nd 2020",remainingDaysToSecondStep:"2 years",secondStepPassed:!0,currentStepOfPublishingProcess:5,editedByType:null,kuFlag:!1,biosketch:null,coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"240870",title:"Ph.D.",name:"Alaa Eddin Omar",middleName:null,surname:"Al Ostwani",slug:"alaa-eddin-omar-al-ostwani",fullName:"Alaa Eddin Omar Al Ostwani",profilePictureURL:"https://mts.intechopen.com/storage/users/240870/images/system/240870.jpeg",biography:"Dr. Al Ostwani Alaa Eddin Omar received his Master in dentistry from Damascus University in 2010, and his Ph.D. in Pediatric Dentistry from Damascus University in 2014. Dr. Al Ostwani is an assistant professor and faculty member at IUST University since 2014. \nDuring his academic experience, he has received several awards including the scientific research award from the Union of Arab Universities, the Syrian gold medal and the international gold medal for invention and creativity. Dr. Al Ostwani is a Member of the International Association of Dental Traumatology and the Syrian Society for Research and Preventive Dentistry since 2017. He is also a Member of the Reviewer Board of International Journal of Dental Medicine (IJDM), and the Indian Journal of Conservative and Endodontics since 2016.",institutionString:"International University for Science and Technology.",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"1",totalChapterViews:"0",totalEditedBooks:"1",institution:{name:"Islamic University of Science and Technology",institutionURL:null,country:{name:"India"}}}],coeditorOne:null,coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"16",title:"Medicine",slug:"medicine"}],chapters:null,productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"247865",firstName:"Jasna",lastName:"Bozic",middleName:null,title:"Ms.",imageUrl:"https://mts.intechopen.com/storage/users/247865/images/7225_n.jpg",email:"jasna.b@intechopen.com",biography:"As an Author Service Manager, my responsibilities include monitoring and facilitating all publishing activities for authors and editors. From chapter submission and review to approval and revision, copyediting and design, until final publication, I work closely with authors and editors to ensure a simple and easy publishing process. I maintain constant and effective communication with authors, editors and reviewers, which allows for a level of personal support that enables contributors to fully commit and concentrate on the chapters they are writing, editing, or reviewing. I assist authors in the preparation of their full chapter submissions and track important deadlines and ensure they are met. I help to coordinate internal processes such as linguistic review, and monitor the technical aspects of the process. As an ASM I am also involved in the acquisition of editors. 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Venkateswarlu",coverURL:"https://cdn.intechopen.com/books/images_new/371.jpg",editedByType:"Edited by",editors:[{id:"58592",title:"Dr.",name:"Arun",surname:"Shanker",slug:"arun-shanker",fullName:"Arun Shanker"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"72",title:"Ionic Liquids",subtitle:"Theory, Properties, New Approaches",isOpenForSubmission:!1,hash:"d94ffa3cfa10505e3b1d676d46fcd3f5",slug:"ionic-liquids-theory-properties-new-approaches",bookSignature:"Alexander Kokorin",coverURL:"https://cdn.intechopen.com/books/images_new/72.jpg",editedByType:"Edited by",editors:[{id:"19816",title:"Prof.",name:"Alexander",surname:"Kokorin",slug:"alexander-kokorin",fullName:"Alexander Kokorin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"314",title:"Regenerative Medicine and Tissue Engineering",subtitle:"Cells and Biomaterials",isOpenForSubmission:!1,hash:"bb67e80e480c86bb8315458012d65686",slug:"regenerative-medicine-and-tissue-engineering-cells-and-biomaterials",bookSignature:"Daniel Eberli",coverURL:"https://cdn.intechopen.com/books/images_new/314.jpg",editedByType:"Edited by",editors:[{id:"6495",title:"Dr.",name:"Daniel",surname:"Eberli",slug:"daniel-eberli",fullName:"Daniel Eberli"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"57",title:"Physics and Applications of Graphene",subtitle:"Experiments",isOpenForSubmission:!1,hash:"0e6622a71cf4f02f45bfdd5691e1189a",slug:"physics-and-applications-of-graphene-experiments",bookSignature:"Sergey Mikhailov",coverURL:"https://cdn.intechopen.com/books/images_new/57.jpg",editedByType:"Edited by",editors:[{id:"16042",title:"Dr.",name:"Sergey",surname:"Mikhailov",slug:"sergey-mikhailov",fullName:"Sergey Mikhailov"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"1373",title:"Ionic Liquids",subtitle:"Applications and Perspectives",isOpenForSubmission:!1,hash:"5e9ae5ae9167cde4b344e499a792c41c",slug:"ionic-liquids-applications-and-perspectives",bookSignature:"Alexander Kokorin",coverURL:"https://cdn.intechopen.com/books/images_new/1373.jpg",editedByType:"Edited by",editors:[{id:"19816",title:"Prof.",name:"Alexander",surname:"Kokorin",slug:"alexander-kokorin",fullName:"Alexander Kokorin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"70913",title:"Peculiarities of Refractory Borides Formation during Mechanical Alloying IV-V Group Transition Metals with Boron in Planetary Mill",doi:"10.5772/intechopen.89401",slug:"peculiarities-of-refractory-borides-formation-during-mechanical-alloying-iv-v-group-transition-metal",body:'\nRefractory metal borides have a high melting point, high thermal and electrical conductivity, low linear coefficient of thermal expansion, excellent corrosion resistance, and very high microhardness. Therefore, they attract much attention as promising materials for application. The potential application of such materials could be jet engine parts, armor plates, cutting tools, dies, etc. [1]. A large number of studies are devoted to these materials and in particular to methods for their preparation. As a rule, borides are obtained at high temperature and, therefore, the grain size of the formed boride is big. However, high enthalpy of borides points that these compounds can be obtained at reduced temperature (for example, formation enthalpy Δ
We selected the Ta-B system with different tantalum-to-boron ratios as a model to study the peculiarities of structure transformation under milling transition group V metals (
The aim of the work was to study what phases can form during mechanical alloying of transition metals with different crystalline structures with boron in a planetary mill and how much the phase formation depends on the metal structure, the metal/boron atom size relationship, and the peculiarities of mechanical alloying process and to develop a model of solid solutions formation during milling using the Ta-B system as an example.
\nHigh purity tantalum and niobium powders with 10–60 μm particles were used. They were produced at the “Silmet” metallurgical plant by electrolytic reduction with hydrogenation and dehydrogenation for further purification. Titanium powder was produced at the Zaporizhzhya titanium-magnesium plant with a specific surface area of 0.1 m2/g brand TG-TB (−2 + 1) mm, zirconium (Dneprodzerzhinsky PA “PCP”) was obtained by the calcium thermal method, vanadium by thermal dissociation of vanadium iodide. Black amorphous boron powder (B-99 grade, as per 1-92-15490) with a specific surface area of 11.2 m2/g was used. The brand of such black amorphous boron powder was obtained by gaseous boron halogenide reduction with hydrogen. The metal-to-boron atoms ratio depended on a specific metal-boron compound to be produced: MeB (1Me:1B), and MeB2 (1Me:2B). Milling was performed in an argon medium in a planetary mill AIR 015 M, which provides an acceleration of 45 g at an rotation speeds of the disk and vials of 735 and 1840 rev/min, respectively. The balls-to-powder mass ratio was 20:1. XRD analysis was carried out on an installation DRON3 under copper K
Using titanium as an example, let us consider how borides are formed during milling
XRD patterns of Ti-B mixture in the initial state (a), upon milling for (b) 5, (c) 6, and (d) 7 min (the indices a, b, c denote Ti, and d TiB2).
Change in the crystal lattice volume of titanium after milling with boron calculated by the full-profile Rietveld method [
TEM microphotograph of TiB2 obtained upon milling Ti and B powders: (b) increased area accented in (a); (c) microelectron diffraction pattern.
Zirconium like titanium has the
XRD patterns of ZrB2 formed upon 6 min milling of Zr-B powders.
A peculiarity of the group V transitions metals is that they have a
Figure 5 shows changes in the XRD patterns of V-B mixtures depending on the milling time at V:B ratios of 1:1 and 1:2. The diffraction lines shift toward smaller angles within 5 min milling, which indicates an increase in the lattice parameters at the both atomic ratios of the components. The vanadium lattice parameter (
XRD patterns of V-B powders upon milling for (а) (1) 0, (2) 5, (3) 20 min, (b) (1) 0, (2) 5, (3) 15 min [the indices on lines 2 denote V, on lines 3: (a) VB, (b) VB2].
XRD patterns for the transformation of niobium into the borides NbB and NbB2 under milling Nb-B mixtures are shown in Figure 6. Unlike vanadium characterized by increasing the lattice parameter within 5 min milling, in this case, at the ratio Nb:B = 1:1, the lattice parameter does not change, whereas at the ratio Nb:B = 1:2, it reduces and is equal to 0.3293 nm (a = 0.3300 nm for Nb), which indicates the formation of a substitutional solid solution. After niobium transformation into borides, in the XRD patterns, a strong niobium line (110) appears, shifted toward smaller angles, that is, the lattice parameter increases. This remaining niobium with an increased lattice parameter is evidence to the existence of parallel mechanisms of formation of boron-in-niobium solid solutions: on the one hand, a SSS is formed, which leads to a decrease in the lattice parameter; on the other hand, an ISS is formed, which leads to the lattice parameter increasing.
\nXRD patterns of Nb-B mixture upon milling for (а) (1) 0, (2) 5, (3) 20 min, (b) (1) 0, (2) 5 and (3) 15 min [the indices on lines 1 denote Nb, on lines 3: (a) NbB, (b) NbB2].
In the case of 5 min milling of the Ta:B = 1:1 mixture, the resulted structure is defective, which is confirmed by weakening and broadening X-ray lines with remaining their positions (Figure 7). After 8 min milling, the tantalum lines positions are still the same (Figure 8). The 10 min milling transforms the tantalum powder into TaB with residual tantalum, whose line (110) is shifted toward smaller angles (Figure 9), which indicates that the lattice parameter increases owing to the formation of an interstitial boron-in-tantalum solid solution under milling. The facts that the tantalum lattice parameter does not change after milling for 5 and 10 min and that there is residual tantalum with increased lattice parameter after 10 min milling may be indicative of running two processes in parallel which differently influence the lattice parameter. Therefore, it can be assumed that both interstitial and substitutional solid solutions of boron in tantalum can be formed under milling. The lattice parameter in the Ta:B = 1:1 mixture increases for ISS and decreases for SSS. The powder mixtures containing 50 at% B show no change in the lattice parameter.
\nXRD patterns of Ta:B = 1:1 powder mixture: before milling (1) upon milling for 5 min (2) (the indices in
XRD patterns of Ta:B = 1:1 powder mixture upon 8 min milling.
XRD patterns of Ta:B = 1:1 powder mixture upon milling for 10 min (the indices denote TaB).
Figure 10 compares XRD patterns recorded from powders milled during different periods of time. As low-intensity peaks for the mixture milled for 15 min are not seen in pattern 3 (Figure 10a) because of the scale factor, this pattern is enlarged in Figure 10b, where one can see the peaks corresponding to the TaB2 phase. After milling for 5 min, the lines shift toward smaller angles and reveal that the lattice parameter changes from a = 0.3302 nm to a = 0.3315 nm, which indicates the fact of formation of an ISS. The intensive milling of the Ta:B = 1:2 powders for 5 min leads, like for Ta:B = 1:1 mixtures, to tantalum lattice distortion, which is confirmed by the broadening the X-ray lines and a lower intensity of reflection peaks under the impact of shock loads and shear deformation in the milling process. For the Ta:B = 1:2 mixture milled for 5 min, coherent scattering domain (CSD) size D is 28.91 nm and distortion ε is 0.004578. Figure 11 shows XRD patterns from the Ta:B = 1:2 mixture milled for 15, 30, and 50 min. Milling for 15 min leads to an increase in
XRD patterns of Ta:B = 1:2 powders in the initial state (1), upon milling for 5 (2), and 15 (3) min (the indices in a denote Ta). (b) Enlarged scale of above pattern 3 in (a).
XRD patterns of Ta:B = 1:2 powders mixture upon milling for 15 (1), 30 (2), 50 (3) min.
As established in [12], the change in the lattice parameter Δ
In order to produce stable solid solutions, the authors first estimated stability of boron-in-tantalum ISS and SSS through determination of their Gibbs energy with taking into account that the latter is formed by the elastic energy (owing to the distortions in solid solutions) and the enthalpy of the system milling. Then, the solid solutions were studied in the frame of the regular solution model [13], using the following formula for the enthalpy of mixing n component alloys [14]:
\nwhere Ω
According to the Boltzmann hypothesis, the entropy of mixing n elements in a regular solution can be expressed as follows:
\nwhere
The solid solutions are crystals with a distorted lattice because their atoms have different sizes. Elastic distortions, arising from size discrepancy, can affect the free energy of the alloy as well. It is important to take into consideration different factors that contribute to the total enthalpy, such as the elastic energy Δ
We consider that the atomic volumes and local bulk moduli for solid solutions correspond to those for single-component systems. Since the components have different sizes, the lattice becomes distorted. Taking
Here α is the parameter that characterizes the pore volume fraction corresponding to one tantalum atom. We take into account that
where
And for SSS and ISS with boron in tetrahedral pores:
\nFunction INT[
Eq. (4) can be used to obtain the expression for average atomic volume
In this case, the equation allowing for change in the free Gibbs energy in transition to the solid solution state is as follows:
\nTo calculate the composition dependences of Δ
The calculated Gibbs free energy of solid solutions (ΔG) for combined Ta-B solid solution with boron in tetrahedral (a) and octahedral (b) pores in the bcc lattice of Ta (cBs and cBi are the concentrations of substitutional and interstitial boron atoms, respectively).
The concentration dependence for χ is presented in Figure 13. As shown in Figure 12, both mechanisms decrease the solution energy. The SSS reaches minimum Δ
Change in the parameter χ of the combined (substitutional and interstitial) Ta-B solid solution with boron in tetrahedral (a) and octahedral (b) pores in the bcc lattice of Ta (cBs and cBi are the concentrations of substitutional and interstitial boron atoms, respectively).
The parameter χ decreases for the substitutional mechanism and increases for the interstitial mechanism. A greater its increase is observed when boron atoms occupy tetrahedral pores.
\nModeling for process of formation of boron-in-tantalum solid solutions allows a supposition that the formation of SSS under intense milling of Me-B mixtures in a planetary mill takes place owing to replacement of a metal atom by two (or three for zirconium and hafnium) boron atoms. The possibility of SSS formation through replacement of a metal atom by two boron atoms is due to their close sizes [so-called Goldschmidt criterion (G.c.)]. G.c. was calculated if suppose that two or three boron atoms take place a metal atom knocked out of the crystal lattice under milling according to the formula.
\nwhere (
The G.c., atom volumes, and Young moduli for transition metals of IV–VI groups are presented in Table 1. G.c. is valid for all of the transition metals except Gr. As for the Young modulus, it is very high for tungsten and molybdenum. That is why those atoms cannot be knocked out of the crystal lattice under milling. Also, a marked difference in the niobium and tantalum Young moduli explains the domination of SSS over ISS in the Nb + 2B mixture in our case. However, under the conditions of less intense milling, ISS prevails [10].
\nGroup | В | IV | V | VI | ||||||
---|---|---|---|---|---|---|---|---|---|---|
Ме | Ti | Zr | Hf | V | Nb | Ta | Cr | Mo | W | |
Young modulus, GPa | 120 | 98 | 141 | 131 | 105 | 186 | 279 | 325 | 415 | |
Atomic volume, cm3/g-atоm | 4.6 | 10.6 | 14.1 | 13.4 | 8.3 | 10.3 | 10.9 | 7.3 | 9.4 | 9.5 |
G.c. | 13% | 2% | +3% | +11% | 11% | 15% | 21% | 2% | 3% |
Young’s modulus, atom volumes for boron and transition metals, and Goldschmidt criterion for SSS formation.
Taking into account the atom volumes for transition metals and boron, one can reveal that replacement of a vanadium atom by two boron atoms results in some increase in the lattice parameter, whereas replacement of a tantalum (as well as niobium and titanium) atom by two boron atoms results in decreasing lattice parameter (Table 1). Hagg’s rule (
In addition, the modeling has showed that at 50 at% B in Ta, the Gibbs energy is minimal. Perhaps at the same boron concentration, an abrupt formation of the TiB2, VB, VB2, NbB, NbB2, and TaB phases occurs due to the minimal SSS stability. The presence of lattice sites replaced by two or three boron atoms in the
The formation of a particular phase under milling in a planetary mill of
This chapter deals with the nonoscillatory solutions of 3D nonlinear dynamical systems on time scales. In addition, it is very critical to discuss whether or not there exist such solutions. Therefore, the existence along with limit behaviors are also studied in this chapter by using double/triple integrals and fixed point theorems. Stefan Hilger, a German mathematician, introduced a theory in his PhD thesis in 1988 [1] that unifies continuous and discrete analysis and extend it in one comprehensive theory, which is called the
Now we explain what we mean by continuous and discrete analysis in details. Assuming readers are all familiar with differential and difference equations; the results are valid for differential equations when
3D nonlinear dynamical systems on time scales have recently gotten a valuable attention because of its potential in applications of control theory, population dynamics and mathematical biology and Physics. For example, Akn, Güzey and Öztürk [3] considered a 3D dynamical system to control a wheeled mobile robots on time scales
where
to show the existence and uniqueness and properties of solutions for flows of thin viscous films over solid surfaces, where
We assume that readers may not be familiar with the time scale basics, so we give an introductory section to the time scale calculus. We refer the books [6, 7] for more details and information about time scales. Structure of the rest of this chapter is as follows: In Section 3.1 and 3.2 we consider a system with different values, 1 and − 1, respectively, and show the qualitative behavior of solutions. In Section 4, we give some examples for readers to comprehend our theoretical results. Finally, we give a short conclusion about the summary of our results and open problems in the last section.
In the introduction section, we have only mentioned the time scales
For the sake of the rest of the chapter, Table 1 summarizes how
0 | |||
Some time scales with
As we know, the set of real numbers are dense and set of integers are scattered. Now we show how we classify the points on general time scales. For any
Classification of points.
Now, let us introduce the derivative for general time scales. Note that
for any
provided
Suppose
exists as a finite number.
If
A function
After derivative and its properties, we also introduce integrals for any time scale
Every rd-continuous function has an antiderivative. Moreover,
is an antiderivative of
The following theorem leads us to the properties of integrals on time scales, which are similar to continuous case.
If
Table 2 shows how the derivative and integral are defined for some time scales for
Derivative and integral for some time scales.
This chapter assumes that
Motivated by [16, 17], we deal with the nonlinear system
where
The other continuous and discrete cases of system (3) were studied in [18, 19, 20]. We first give the following definitions to help readers understand the terminology.
for
Finally, let us define nonoscillatory solutions of system (3).
For the sake of simplicity, let us set
where
Suppose that
It was shown in [21] that any nonoscillatory solution of system (3) for
Next, we consider system (3) for
for some nonnegative
In this section, we consider system (3) with
Nonoscillatory solutions in
Nonoscillatory solutions in
Therefore, for a nonoscillatory solution
Let
(i)
where
Now, we start with our first main result which shows that the existence of a nonoscillatory solution in
where
and define an operator
for
by (2). Also, it is trivial to show that
Then taking the derivative of (4) gives us
By setting
and taking the derivative of (5), we have
Finally letting
and taking the derivative yield
that leads us to
Showing existence of a nonoscillatory solution in
Similarly, for any nonoscillatory solution of system (3) in
(i)
where
where
Next, we show the existence of nonoscillatory solutions of (3) in those subclasses by using fixed point theorems. Observe that we have some additional assumption in theorems such that
where
Set an operator
One can show that
Second, we need to show
So the Lebesgue dominated convergence theorem, continuity of
we have that
and
and by a similar discussion as in Theorem 3.5, we get
Next, we focus on the existence of nonoscillatory solutions in
There does exist a nonoscillatory solution in
There does exist a nonoscillatory solution in
Finally, the last theorem in this section leads us to the fact that there must be a solution such that
and
where
and an operator
One can easily show that
and
one can have
This section deals with system (3) for
In the next section, we examine the solutions in each class
For any nonoscillatory solution
where
and
where
Let
and an operator
for
where
i.e.,
Then the continuity of
we have
Setting
for
Finally, differentiating
gives
Consequently
The following theorems can be proven very similarly to Theorem 3.11 with appropriate operators. Therefore, the proof is left to the reader, see [17].
Suppose
If both
If
If
We continue with the case when
and
where
and set an operator
The rest of the proof can be done as in proofs of the previous theorems by using the fact
Assuming
where
The first result of this section considers the case when each of the component solutions converges.
where
and an operator
One can prove that
Now for
and
Then, since
Consequently
In this section, we provide some examples to highlight our theoretical claims. The following theorem help us evaluate the integrals on a specific time scale, see ([6] Theorem 1.79 (ii)).
where
First we show
Similarly one can obtain
Now we consider
since
The sum formula for a finite geometric series,
So the claim indeed holds, and consequently we have
Also, we obtain
by (11). Therefore, as
where
by (12). Since the above integral converges as
We show that
So as
by the ratio test. We can also easily show
Hence, by the geometric series, and taking the limit of the latter inequality as
Therefore, we have
in
In this chapter, we consider a 3D time scale system and show the asymptotic properties of the nonoscillatory solutions along with the existence of such solutions. We are able to show the existence of solutions in most subclasses. On the other hand, it is still an open problem to show the existence in
Another significance of our system that we consider in this chapter is the following system
which is known as the third order Emden-Fowler system. Here,
I would like to dedicate this chapter to my beloved friend Dr. Serdar Çağlak, who always will be remembered as a fighter for his life. Also, I would like to thank to my wife for her tremendous support for writing this chapter.
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