Chapter 1: "Permanent Maxillary and Mandibular Incisors"\n
Chapter 2: "The Permanent Maxillary and Mandibular Premolar Teeth"\n
Chapter 3: "Dental Anatomical Features and Caries: A Relationship to be Investigated"\n
Chapter 4: "Anatomy Applied to Block Anaesthesia"\n
Chapter 5: "Treatment Considerations for Missing Teeth"\n
Chapter 6: "Anatomical and Functional Restoration of the Compromised Occlusion: From Theory to Materials"\n
Chapter 7: "Evaluation of the Anatomy of the Lower First Premolar"\n
Chapter 8: "A Comparative Study of the Validity and Reproducibility of Mesiodistal Tooth Size and Dental Arch with the iTero Intraoral Scanner and the Traditional Method"\n
Chapter 9: "Identification of Lower Central Incisors"\n
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She was born in Zonguldak, Turkey, on February 14, 1974; she is married and has two children. She graduated in 1997 from the Ankara University, Faculty of Dentistry, Ankara, Turkey. She aquired her PhD in 2004 from the Gazi University, Faculty of Dentistry, Department of Endodontics, Ankara, Turkey, and she is still an associate professor at the same department. She has published numerous articles and a book chapter in the areas of Operative Dentistry, Esthetic Dentistry and Endodontics. 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Vaz-Leal",slug:"francisco-j.-vaz-leal",email:"fjvazleal@gmail.com",position:null,institution:null},{id:"188719",title:"Dr.",name:"María Cristina",middleName:null,surname:"Álvarez Mateos",fullName:"María Cristina Álvarez Mateos",slug:"maria-cristina-alvarez-mateos",email:"cristinaalvarezmateos@gmail.com",position:null,institution:null},{id:"195142",title:"Dr.",name:"Laura",middleName:null,surname:"Rodríguez Santos",fullName:"Laura Rodríguez Santos",slug:"laura-rodriguez-santos",email:"laura@unex.es",position:null,institution:null},{id:"195143",title:"Dr.",name:"María I",middleName:null,surname:"Ramos Fuentes",fullName:"María I Ramos Fuentes",slug:"maria-i-ramos-fuentes",email:"miramos@unex.es",position:null,institution:null}]},book:{id:"5372",title:"Eating Disorders",subtitle:"A Paradigm of the Biopsychosocial Model of Illness",fullTitle:"Eating Disorders - A Paradigm of the Biopsychosocial Model of Illness",slug:"eating-disorders-a-paradigm-of-the-biopsychosocial-model-of-illness",publishedDate:"February 1st 2017",bookSignature:"Ignacio Jauregui-Lobera",coverURL:"https://cdn.intechopen.com/books/images_new/5372.jpg",licenceType:"CC BY 3.0",editedByType:"Edited by",editors:[{id:"55769",title:"Prof.",name:"Ignacio",middleName:null,surname:"Jáuregui Lobera",slug:"ignacio-jauregui-lobera",fullName:"Ignacio Jáuregui Lobera"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}}},ofsBook:{item:{type:"book",id:"10407",leadTitle:null,title:"Sediment Transport - Recent Advances",subtitle:null,reviewType:"peer-reviewed",abstract:"
\r\n\tThe effective management of many aquatic and terrestrial environments requires a detailed understanding of sediment dynamics. This has both environmental and economic implications, especially where there is any anthropogenic involvement. Numerical models are often the tool used for predicting the transport and fate of sediment movement in these situations, as they can estimate the various spatial and temporal fluxes. However, the physical sedimentary processes can vary quite considerably depending upon whether the local sediments are fully cohesive, non-cohesive, or a mixture of both types. For this reason for more than half a century, scientists, engineers, hydrologists, and mathematicians have all been continuing to conduct research into the many aspects which influence sediment transport.
\r\n\r\n\tThis book will draw on scientific research to examine the following river delta related issues: classification, circulation hydrodynamics, ecosystems, sedimentation, anthropogenic stresses, and response to extreme events. These key topics will each be supported by case study examples of river deltas from around the world.
\r\n\t
\nInterpolation is a method of constructing new data points within the range of a discrete set of known data points or the process of estimating the value of a function at a point from its values at nearby points. Although a large number of interpolation schemes are available in the mathematical field of numerical analysis, the majority of these conventional interpolation methods produce interpolants, i.e., functions used to generate interpolation, that are differentiable a number of times except possibly at a finite set of points. Taking into account that the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, the aforementioned interpolants are considered smooth.
\nOn the other hand, many real-world and experimental signals are intricate and rarely show a sensation of smoothness in their traces. Consequently, to model these signals, we require interpolants that are nondifferentiable in dense sets of points in the domain. To address this issue, interpolation by fractal (graph of) functions is introduced in [1, 2], which is based on the theory of iterated function system. A fractal interpolation function can be considered as a continuous function whose graph is the attractor, a fractal set, of an appropriately chosen iterated function system. If this graph has a Hausdorff-Besicovitch dimension between 1 and 2, the resulting attractor is called fractal interpolation curved line or fractal interpolation curve. If this graph has a Hausdorff-Besicovitch dimension between 2 and 3, the resulting attractor is called fractal interpolation surface. Various types of fractal interpolation functions have been constructed, and some significant properties of them, including calculus, dimension, smoothness, stability, perturbation error, etc., have been widely studied [3, 4, 5].
\n\nFractal interpolation is an advanced technique for analysis and synthesis of scientific and engineering data, whereas the approximation of natural curves and surfaces in these areas has emerged as an important research field. Fractal functions are currently being given considerable attention due to their applications in areas such as Metallurgy, Earth Sciences, Surface Physics, Chemistry and Medical Sciences. In the development of fractal interpolation theory, many researchers have generalised the notion in different ways [6, 7, 8, 9]. Two key issues should be addressed in constructing fractal interpolation functions. They regard to ensuring continuity and the existence of the contractivity, or vertical scaling, factors; see [10, 11]. In [12], nonlinear fractal interpolation surfaces resulting from Rakotch or Geraghty contractions together with some continuity conditions were introduced as well as explicit illustrative examples were given.
\nThe concept of iterated function system was originally introduced as a generalisation of the well-known Banach contraction principle. Since it has become a powerful tool for constructing and analysing fractal interpolation functions, one can use the well-known fixed point results obtained in the fixed point theory in order to construct them in a more general sense. A comparison of various definitions of contractive mappings as well as fixed point theorems that can be used to construct iterated function systems can be found in [13, 14, 15]. In [14], the authors proposed some iterated function systems by using various fixed point theorems, but unfortunately, one does not know whether fractal interpolation functions correspond to those may exist or not. As far as we know, the first significant generalisation of Banach’s principle was obtained by Rakotch [16] in 1962. Recently, a method to generate nonlinear fractal interpolation functions by using the Rakotch or Geraghty fixed point theorem instead of Banach fixed point theorem was presented in [12, 17, 18].
\nThe aim of our article is to provide the connections between several fractal interpolation functions and the contractions used to generate them; it is organised as follows. In Section 2, we recall the results obtained in construction of fractal interpolation curved lines and fractal interpolation surfaces by using Rakotch contractions (or Geraghty contractions) instead of Banach contractions. In Section 3, we only present the connection between fractal interpolation functions by using the Banach contractions and fractal interpolation functions by using the Rakotch contractions because in the case of Geraghty contractions, the existence of fractal interpolation curved lines and fractal interpolation surfaces is similar to the case of Rakotch contractions.
\nLet \n
for \n
An iterated function system, or IFS for short, is a collection of a complete metric space \n
where \n
If \n
Definition 2.1.Let\n\n
\nA self-map\n\n
\nWe say that\n\n
\nIf\n\n
From [14], we have the following.
\nTheorem 2.1.Let\n\n
Let \n
where the real numbers \n
Let \n
Now, consider an IFS of the form \n
where the transformations are constrained by the data according to
\nfor \n
Let us denote by \n
Let \n
Define a metric \n
for all \n
for \n
Theorem 2.2.Let\n\n
\nthere is a unique continuous function\n\n
\n\n
\nif\n\n
An extremely explicit simple example is the following; cf. [12].
\nExample 1.Let\n\n
where
\n\nThen, there exists a continuous function\n\n
Let \n
Set \n
Thus, for \n
Furthermore, for \n
where the transformations are constrained by the data according to
\nfor \n
Let \n
Let \n
Define an operator \n
for \n
Theorem 2.3.Let\n\n
\nthere is a unique bounded function\n\n
\n\n
\nif\n\n
Let for all \n
where \n
and
\nLet \n
for all \n
Let \n
for \n
Corollary 2.1(see [18]) Let\n\n
\nthere is a unique continuous function\n\n
\n\n
\nif\n\n
The most simple example is the following; cf. [12].
\nExample 2.Let\n\n
\nLet for\n\n
\nThen,\n\n
Let \n
and for some \n
Let
\nwhere \n
for \n
For the next, see [12] for details.
\nCorollary 2.2.Let\n\n
\nthere is a unique continuous function\n\n
\n\n
\nif\n\n
An especially simple example is the following; see [12].
\nExample 3.Let\n\n
\nLet for\n\n
\nThen,\n\n
In this section, we only present the interconnections between FIFs resulting from Banach contractions and FIFs resulting from Rakotch contractions because in the case of Geraghty contractions, the existence of FICs and FISs is derived similarly to the case of Rakotch contractions.
\n\n
Each Banach contraction is a Rakotch contraction, since a self-map is a Banach contraction if and only if it is a \n
The Rakotch’s functional condition for convergence of a contractive iteration in a complete metric space can be replaced by an equivalent (or another) functional condition; for instance, a map is a Rakotch contraction if and only if it is a \n
\n
\n\n
for all \n
\n\n
for all \n
\n\n
Let \n
Then, by the Differential Mean Value Theorem and the extreme value theorem, we can see that for some \n
where \n
\n
The function \n
Let \n
The functional condition \n
where \n
On the other hand, if \n
\n
In Theorem 2.2, for all \n
That is, each \n
In the case where the vertical scaling factors are constants, in [1], the existence of affine FIFs by using the Banach fixed point theorem was investigated, whereas in [20], a generalisation of affine FIFs by using vertical scaling factors as (continuous) ‘contraction functions’ and Banach’s fixed point theorem was introduced. Theorem 2.2 gives the existence of fractal interpolation curves by using the Rakotch fixed point theorem and vertical scaling factors as (continuous) ‘contraction functions’.
\nThe boundedness of \n
So, for all \n
In view of a \n
In the affine FIF (cf. [1], p. 308, Example 1), for all \n
where \n
In the FIF with vertical scaling factors as (continuous) ‘contraction functions’ (cf. [20], p. 3), for all \n
where \n
We refer to \n
where \n
we can see that
\nwhere \n
\n
The well-known FIS in theory and applications is generated by an IFS of the form \n
where \n
That is, each \n
and
\nwe can see that \n
In Theorem 2.3, for all \n
That is, each \n
In view of a \n
In the affine FIS (cf. [22]), for all \n
where \n
In the FIS with vertical scaling factors as function (cf. [21]), for all \n
where \n
The continuity of bivariable FIFs differ from the continuity of univariable FIFs.
The graphs of linear univariable FIFs are always continuous curves.
There are bivariable discontinuous functions that interpolate the given data; (see for instance [23], p. 630, 631).
Theorem 2.3 ensures that attractors of constructed IFSs are graphs of some bounded functions which interpolate the given data, but these graphs (i.e., the graphs of bivariable FIFs) are not always continuous surfaces. Some continuity conditions of bivariable FIFs are given explicitly by Corollary 2.1 and Corollary 2.2.
The key difficulty in constructing fractal interpolation surfaces (or volumes) involves ensuring continuity. Another important element necessary in modelling complicated surfaces of this type is the existence of the contractivity, or vertical scaling, factors.
In order to ensure continuity of a fractal interpolation surface, in [22], the interpolation points on the boundary was assumed collinear, whereas in [21], vertical scaling factors as (continuous) ‘contraction functions’ were used.
A new bivariable fractal interpolation function by using the Matkowski fixed point theorem and the Rakotch contraction is presented in [18]. In order to ensure the continuity of nonlinear FIS, the coplanarity of all the interpolation points on the boundaries instead of collinearity of interpolation points on the boundary was assumed in [18], whereas in [12], vertical scaling factors as (continuous) ‘contraction functions’ were used.
\n
In Theorem 2.2, we can see that
\n
In Corollary 2.1, we can see that
\n
In Corollary 2.2, we can see that (compare with above coefficients)
\nFigures 1(a) and 2(a) are associated with Banach contractions, whereas Figures 1(b) and 2(b) are not necessarily associated with Banach contractions.
\nThe graph of a fractal interpolation function (a) that is associated with Banach contractions, (b) that is not necessarily associated with Banach contractions.
A fractal interpolation surface (a) that is associated with Banach contractions, (b) that is not necessarily associated with Banach contractions.
We reviewed nonlinear fractal interpolation functions by using the Geraghty fixed point theorem instead of the Banach fixed point theorem (or the Rakotch fixed point theorem) since Banach contraction (or Rakotch contraction) is a special case of Geraghty contraction. Theorems 2.1, 2.2 and 2.3 ensure that attractors of constructed nonlinear iterated function systems are graphs of some continuous functions which interpolate the given data. In particular, Examples 1, 2 and 3 show that our results remain still true under essentially weaker conditions on the maps of iterated function systems. The methods presented here can be directly extended to piecewise fractal interpolation functions that are based on recurrent IFS. A premise for future work is to extend these methods to hidden-variable fractal interpolation surfaces as well as to identify the parameters of such surfaces.
\nWe believe financial barriers should not prevent researchers from publishing their findings. With the need to make scientific research more publicly available and support the benefits of Open Access, more and more institutions and funders are dedicating resources to assist faculty members and researchers cover Open Access Publishing Fees (OAPFs). In addition, IntechOpen provides several further options presented below, all of which are available to researchers, and could secure the financing of your Open Access publication.
",metaTitle:"Waiver Policy",metaDescription:"We feel that financial barriers should never prevent researchers from publishing their research. With the need to make scientific research more publically available and support the benefits of Open Access, more institutions and funders have dedicated funds to assist their faculty members and researchers cover the APCs associated with publishing in Open Access. Below we have outlined several options available to secure financing for your Open Access publication.",metaKeywords:null,canonicalURL:"/page/waiver-policy",contentRaw:'[{"type":"htmlEditorComponent","content":"At IntechOpen, the majority of OAPFs are paid by an Author’s institution or funding agency - Institutions (73%) vs. Authors (23%).
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\\n"}]'},components:[{type:"htmlEditorComponent",content:'At IntechOpen, the majority of OAPFs are paid by an Author’s institution or funding agency - Institutions (73%) vs. Authors (23%).
\n\nThe first step in obtaining funds for your Open Access publication begins with your institution or library. IntechOpen’s publishing standards align with most institutional funding programs. Our advice is to petition your institution for help in financing your Open Access publication.
\n\nHowever, as Open Access becomes a more commonly used publishing option for the dissemination of scientific and scholarly content, in addition to institutions, there are a growing number of funders who allow the use of grants for covering OA publication costs, or have established separate funds for the same purpose.
\n\nPlease consult our Open Access Funding page to explore some of these funding opportunities and learn more about how you could finance your IntechOpen publication. Keep in mind that this list is not definitive, and while we are constantly updating and informing our Authors of new funding opportunities, we recommend that you always check with your institution first.
\n\nFor Authors who are unable to obtain funding from their institution or research funding bodies and still need help in covering publication costs, IntechOpen offers the possibility of applying for a Waiver.
\n\nOur mission is to support Authors in publishing their research and making an impact within the scientific community. Currently, 14% of Authors receive full waivers and 6% receive partial waivers.
\n\nWhile providing support and advice to all our international Authors, waiver priority will be given to those Authors who reside in countries that are classified by the World Bank as low-income economies. In this way, we can help ensure that the scientific work being carried out can make an impact within the worldwide scientific community, no matter where an Author might live.
\n\nThe application process is open after your submitted manuscript has been accepted for publication. To apply, please fill out a Waiver Request Form and send it to your Author Service Manager. If you have an official letter from your university or institution showing that funds for your OA publication are unavailable, please attach that as well. The Waiver Request will normally be addressed within one week from the application date. All chapters that receive waivers or partial waivers will be designated as such online.
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\n\nFeel free to contact us at oapf@intechopen.com if you have any questions about Funding options or our Waiver program. If you have already begun the process and require further assistance, please contact your Author Service Manager, who is there to assist you!
\n\nNote: All data represented above was collected by IntechOpen from 2013 to 2017.
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I am also a member of the team in charge for the supervision of Ph.D. students in the fields of development of silicon based planar waveguide sensor devices, study of inelastic electron tunnelling in planar tunnelling nanostructures for sensing applications and development of organotellurium(IV) compounds for semiconductor applications. I am a specialist in data analysis techniques and nanosurface structure. I have served as the editor for many books, been a member of the editorial board in science journals, have published many papers and hold many patents.",institutionString:null,institution:{name:"Sheffield Hallam University",country:{name:"United Kingdom"}}},{id:"54525",title:"Prof.",name:"Abdul Latif",middleName:null,surname:"Ahmad",slug:"abdul-latif-ahmad",fullName:"Abdul Latif Ahmad",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"20567",title:"Prof.",name:"Ado",middleName:null,surname:"Jorio",slug:"ado-jorio",fullName:"Ado Jorio",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:{name:"Universidade Federal de Minas Gerais",country:{name:"Brazil"}}},{id:"47940",title:"Dr.",name:"Alberto",middleName:null,surname:"Mantovani",slug:"alberto-mantovani",fullName:"Alberto Mantovani",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"12392",title:"Mr.",name:"Alex",middleName:null,surname:"Lazinica",slug:"alex-lazinica",fullName:"Alex Lazinica",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/12392/images/7282_n.png",biography:"Alex Lazinica is the founder and CEO of IntechOpen. After obtaining a Master's degree in Mechanical Engineering, he continued his PhD studies in Robotics at the Vienna University of Technology. Here he worked as a robotic researcher with the university's Intelligent Manufacturing Systems Group as well as a guest researcher at various European universities, including the Swiss Federal Institute of Technology Lausanne (EPFL). During this time he published more than 20 scientific papers, gave presentations, served as a reviewer for major robotic journals and conferences and most importantly he co-founded and built the International Journal of Advanced Robotic Systems- world's first Open Access journal in the field of robotics. Starting this journal was a pivotal point in his career, since it was a pathway to founding IntechOpen - Open Access publisher focused on addressing academic researchers needs. Alex is a personification of IntechOpen key values being trusted, open and entrepreneurial. Today his focus is on defining the growth and development strategy for the company.",institutionString:null,institution:{name:"TU Wien",country:{name:"Austria"}}},{id:"19816",title:"Prof.",name:"Alexander",middleName:null,surname:"Kokorin",slug:"alexander-kokorin",fullName:"Alexander Kokorin",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/19816/images/1607_n.jpg",biography:"Alexander I. Kokorin: born: 1947, Moscow; DSc., PhD; Principal Research Fellow (Research Professor) of Department of Kinetics and Catalysis, N. Semenov Institute of Chemical Physics, Russian Academy of Sciences, Moscow.\r\nArea of research interests: physical chemistry of complex-organized molecular and nanosized systems, including polymer-metal complexes; the surface of doped oxide semiconductors. He is an expert in structural, absorptive, catalytic and photocatalytic properties, in structural organization and dynamic features of ionic liquids, in magnetic interactions between paramagnetic centers. The author or co-author of 3 books, over 200 articles and reviews in scientific journals and books. He is an actual member of the International EPR/ESR Society, European Society on Quantum Solar Energy Conversion, Moscow House of Scientists, of the Board of Moscow Physical Society.",institutionString:null,institution:{name:"Semenov Institute of Chemical Physics",country:{name:"Russia"}}},{id:"62389",title:"PhD.",name:"Ali Demir",middleName:null,surname:"Sezer",slug:"ali-demir-sezer",fullName:"Ali Demir Sezer",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/62389/images/3413_n.jpg",biography:"Dr. Ali Demir Sezer has a Ph.D. from Pharmaceutical Biotechnology at the Faculty of Pharmacy, University of Marmara (Turkey). He is the member of many Pharmaceutical Associations and acts as a reviewer of scientific journals and European projects under different research areas such as: drug delivery systems, nanotechnology and pharmaceutical biotechnology. Dr. Sezer is the author of many scientific publications in peer-reviewed journals and poster communications. Focus of his research activity is drug delivery, physico-chemical characterization and biological evaluation of biopolymers micro and nanoparticles as modified drug delivery system, and colloidal drug carriers (liposomes, nanoparticles etc.).",institutionString:null,institution:{name:"Marmara University",country:{name:"Turkey"}}},{id:"61051",title:"Prof.",name:"Andrea",middleName:null,surname:"Natale",slug:"andrea-natale",fullName:"Andrea Natale",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"100762",title:"Prof.",name:"Andrea",middleName:null,surname:"Natale",slug:"andrea-natale",fullName:"Andrea Natale",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:{name:"St David's Medical Center",country:{name:"United States of America"}}},{id:"107416",title:"Dr.",name:"Andrea",middleName:null,surname:"Natale",slug:"andrea-natale",fullName:"Andrea Natale",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:{name:"Texas Cardiac Arrhythmia",country:{name:"United States of America"}}},{id:"64434",title:"Dr.",name:"Angkoon",middleName:null,surname:"Phinyomark",slug:"angkoon-phinyomark",fullName:"Angkoon Phinyomark",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/64434/images/2619_n.jpg",biography:"My name is Angkoon Phinyomark. I received a B.Eng. degree in Computer Engineering with First Class Honors in 2008 from Prince of Songkla University, Songkhla, Thailand, where I received a Ph.D. degree in Electrical Engineering. My research interests are primarily in the area of biomedical signal processing and classification notably EMG (electromyography signal), EOG (electrooculography signal), and EEG (electroencephalography signal), image analysis notably breast cancer analysis and optical coherence tomography, and rehabilitation engineering. I became a student member of IEEE in 2008. During October 2011-March 2012, I had worked at School of Computer Science and Electronic Engineering, University of Essex, Colchester, Essex, United Kingdom. In addition, during a B.Eng. I had been a visiting research student at Faculty of Computer Science, University of Murcia, Murcia, Spain for three months.\n\nI have published over 40 papers during 5 years in refereed journals, books, and conference proceedings in the areas of electro-physiological signals processing and classification, notably EMG and EOG signals, fractal analysis, wavelet analysis, texture analysis, feature extraction and machine learning algorithms, and assistive and rehabilitative devices. I have several computer programming language certificates, i.e. Sun Certified Programmer for the Java 2 Platform 1.4 (SCJP), Microsoft Certified Professional Developer, Web Developer (MCPD), Microsoft Certified Technology Specialist, .NET Framework 2.0 Web (MCTS). 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