The Information Coding Classification (ICC) is a classification system covering almost all extant 6500 knowledge fields (knowledge domains). Its conceptualization goes beyond the scope of the well known library classification systems, such as Dewey Decimal Classification (DDC), Universal Decimal Classification (UDC), and Library of Congress Classification (LCC), by extending also to knowledge systems that so far have not afforded to classify literature. ICC actually presents a flexible universal ordering system for both literature and other kinds of information, set out as knowledge fields. From a methodological point of view, ICC differs from the above-mentioned systems along the following three lines: Its main classes are not based on disciplines but on nine live stages of development, so-called ontical levels. It breaks them roughly down into hierarchical steps by further nine categories which makes decimal number coding possible. The contents of a knowledge field is earmarked via a digital position scheme, which makes the first hierarchical step refer to the nine ontical levels (object areas as subject categories), and the second hierarchical step refer to nine functionally ordered form categories. Respective knowledge fields permit to step down by the same principle to a third and forth level, and even further to a fifth and sixth level. Finally, knowledge field subdivisions will have to conform to said digital position scheme. Hence, for a given knowledge field identical codes will mark identical categories under respective numbers of the coding system. This mnemotechnical aspect of the system helps memorizing and straightaway retrieving the whereabouts of respective interdisciplinary and transdisciplinary fields. The first two hierarchical levels may be regarded as a top- or upper ontology for ontologies and other applications. The terms of the first three hierarchical levels were set out in German and English in Wissensorganisation. Entwicklung, Aufgabe, Anwendung, Zukunft, on pp. 82 to 100. It was published in 2014 and available so far only in German. In the meantime, also the French terms of the knowledge fields have been collected. Competence for maintenance and further development rests with the German Chapter of the International Society for Knowledge Organization (ISKO) e.V. == Historical development == At the end of 1970, Prof. Alwin Diemer, Univ.of Düsseldorf proposed to Ingetraut Dahlberg to undertake a philosophical dissertation on The universal classification system of knowledge, its ontological, epistemological, and information theoretical foundations. Diemer had in mind an innovating ontological approach for such a system based on the whole spectrum of kinds of being and complying with epistemological requirements. The third requirement had already been taken up somehow in the Indian Colon Classification, yet it still called for explanations and additions. In 1974, the dissertation was published in German entitled Grundlagen universaler Wissensordnung. It started with conceptual clarifications, and why and how the term „universal“ was linked to knowledge, including knowledge fields, such as commodity science, artefacts, statistics, patents, standardization, communication, utility services et al. In chapter 3, six universal classification systems (DDC, UDC, LCC, BC, CC and BBK) were presented, analyzed and compared. While preparing the dissertation, Dahlberg started with elaborating the new universal system by first gleaning a lot of extant designations of knowledge fields from whatever available reference works. This was funded by the German Documentation Society (DGD) (1971-2) under the title of Order system of knowledge fields. In addition, the syllabuses of German universities and polytechniques were explored for relevant terms and documented (1975). Thereafter, it seemed necessary to add definitions from special dictionaries and encyclopediae; it soon appeared that the 12.500 terms included numerous synonyms, so that the whole collection boiled down to about 6.500 concept designations (Project Logstruktur, supported by the German Science Foundation (DFG) 1976-78). The outcome of this work was the formulation of 30 theses which ended up in 12 principles for the new system, published 40 years later under. These principles refer not only to theoretical foundations but also to structure and other organizational aspects of the whole array of knowledge fields. In 1974, the digital position scheme for field subdivision had already been developed to allow for classifying classification literature in the bibliographical section of the first issue of the Journal International Classification. In 1977, the entire ICC was ready for presentation at a seminar in Bangalore, India. A publication of the first three hierarchical levels appeared however only in 1982. It was applied to the bibliography of classification systems and thesauri in vol.1 of the International Classification and Indexing Bibliography; it has been updated. == Governing principles == These were published in full length in the book Wissensorganisation. Entwicklung, Aufgabe, Anwendung, Zukunft and the article Information Coding Classification. Geschichtliches, Prinzipien, Inhaltliches, hence it suffices to just mention their topics with some necessary additions. Principle 1: Concept theoretical approaches. Concepts are the contents of ICC, they are understood as being units of knowledge. The „birth“ of a concept. Where do the characteristics, the knowledge elements come from? How do conceptual relations arise? Principle 2: The four kinds of concept relations and their applications. Principle 3: Decimal numbers form the ICC codes as its universal language. Principle 4: The nine ontical levels of ICC. They were grouped under three captions: Prolegomena (1-3), life sciences (4-6) and human output (7-9): Structure and form Matter and energy Cosmos and earth Biosphere Anthroposphere Sociosphere Material products (economics and technology) Intellectual products (knowledge and information) Spiritual products (products of mind and culture) Principle 5: Knowledge fields are structured by categories, based on the Aristotelian form-categories, under a digital position scheme, a kind of scaling rule for subdividing a given field as follows: General area: problems, theories, principles (axiom and structure) Object area: objects, kinds, parts, properties of objects Activity area: methods, processes, activities Field properties or first characterization Persons or secondary characterization Societies or tertiary characterization Influences from outside Applications of the field to other fields Field information and synthesizing tasks The digital position scheme, called Systematifier, has also been used for structuring the entire system via the categories figuring on the upper zero level. An example of its application is the structure of the classification system for knowledge organization literature Gliederung der Klassifikationsliteratur. (A simplified version with an additional introduction is given in, p. 71) Principle 6: The ontical levels outlined under principle 4 conform to the „integrative level theory“ which means that every level is integrated in the following one. In addition, each knowledge area presumes the following one. Principle 7: The combination potential of knowledge fields (interdisciplinarity and transdisciplinarity)is determined by the digital position scheme. (Examples are given in, p. 103-4) Principle 8: The categories of the zero-level are general concepts, their possible subdivisions could once be used for classificatory statements. (These subdivisions still need elaboration) Principle 9 and 10: These relate to the combination potential of classificatory statements with space and time concepts. (Still to be elaborated) Principle 11: The system's mnemotechnical aspect relies on the fixed system position codes and on the 3x3 form- and subject-categories. Principle 12: The combination potential of system position 1, 8 and 9 make ICC to a self-networking system which complies with the present scientific development. == In matrix form == The first two levels of ICC can be represented by following matrix. The first hierarchical level of the 9 subject categories results from the first vertical array under codes 1-9. The second hierarchical level of subject categories is structured by the 9 functionally ordered form categories, listed in the first horizontal line under codes 01-09. Some exceptions are mentioned in principle 7. == Research == === Exploration of automatic classification === For classifying web documents as conceived by Jens Hartmann, University of Karlsruhe, Prof.Walter Koch, University of Graz, has explored in his Institute for Applied Information Technology Research Society (AIT) the application of ICC to automatically classifying metadata of some 350.000 documents. This was facilitated by data generated within the framework of an E
Structured sparsity regularization
Structured sparsity regularization is a class of methods, and an area of research in statistical learning theory, that extend and generalize sparsity regularization learning methods. Both sparsity and structured sparsity regularization methods seek to exploit the assumption that the output variable Y {\displaystyle Y} (i.e., response, or dependent variable) to be learned can be described by a reduced number of variables in the input space X {\displaystyle X} (i.e., the domain, space of features or explanatory variables). Sparsity regularization methods focus on selecting the input variables that best describe the output. Structured sparsity regularization methods generalize and extend sparsity regularization methods, by allowing for optimal selection over structures like groups or networks of input variables in X {\displaystyle X} . Common motivation for the use of structured sparsity methods are model interpretability, high-dimensional learning (where dimensionality of X {\displaystyle X} may be higher than the number of observations n {\displaystyle n} ), and reduction of computational complexity. Moreover, structured sparsity methods allow to incorporate prior assumptions on the structure of the input variables, such as overlapping groups, non-overlapping groups, and acyclic graphs. Examples of uses of structured sparsity methods include face recognition, magnetic resonance image (MRI) processing, socio-linguistic analysis in natural language processing, and analysis of genetic expression in breast cancer. == Definition and related concepts == === Sparsity regularization === Consider the linear kernel regularized empirical risk minimization problem with a loss function V ( y i , f ( x ) ) {\displaystyle V(y_{i},f(x))} and the ℓ 0 {\displaystyle \ell _{0}} "norm" as the regularization penalty: min w ∈ R d 1 n ∑ i = 1 n V ( y i , ⟨ w , x i ⟩ ) + λ ‖ w ‖ 0 , {\displaystyle \min _{w\in \mathbb {R} ^{d}}{\frac {1}{n}}\sum _{i=1}^{n}V(y_{i},\langle w,x_{i}\rangle )+\lambda \|w\|_{0},} where x , w ∈ R d {\displaystyle x,w\in \mathbb {R^{d}} } , and ‖ w ‖ 0 {\displaystyle \|w\|_{0}} denotes the ℓ 0 {\displaystyle \ell _{0}} "norm", defined as the number of nonzero entries of the vector w {\displaystyle w} . f ( x ) = ⟨ w , x i ⟩ {\displaystyle f(x)=\langle w,x_{i}\rangle } is said to be sparse if ‖ w ‖ 0 = s < d {\displaystyle \|w\|_{0}=s
ISO/IEC JTC 1/SC 24
ISO/IEC JTC 1/SC 24 Computer graphics, image processing and environmental data representation is a standardization subcommittee of the joint subcommittee ISO/IEC JTC 1 of the International Organization for Standardization (ISO) and the International Electrotechnical Commission (IEC), which develops and facilitates standards within the field of computer graphics, image processing, and environmental data representation. The international secretariat of ISO/IEC JTC 1/SC 24 is the British Standards Institute (BSI) located in the United Kingdom. == History == ISO/IEC JTC 1/SC 24 was formed in 1987 from ISO/TC 97 as a result of Resolution 21 at the ISO/IEC JTC 1 plenary. The group's origins began in computer graphics, the standardization of which was originally under ISO/IEC JTC 1/SC 21/WG 2. However, when ISO/IEC JTC 1/SC 24 was created, the standardization activity of ISO/IEC JTC 1/SC 21/WG 2 was carried over to the new subcommittee. The initial five working groups of ISO/IEC JTC 1/SC 24 were titled, “Architecture,” “Application programming interfaces,” “Metafiles and interfaces,” “Language bindings,” and “Validation, testing and registration.” The work of ISO/IEC JTC 1/SC 24 began with the Graphical Kernel System (GKS), which was adopted from ISO/IEC JTC 1/SC 21/WG 2. However, since GKS only addressed 2D functionality, attention turned to the standardization of 3D functionality. This resulted in two standards being published: GKS-3D in 1988 and PHIGS in 1989, both of which addressed 3D functionality. Since 1991, ISO/IEC JTC 1/SC 24 has held plenaries in a number of countries, including the Netherlands, Germany, United States, France, Canada, Japan, Sweden, Korea, United Kingdom, Australia, and Czech Republic. == Scope == The scope of ISO/IEC JTC 1/SC 24 is the “Standardization of interfaces for information technology based applications relating to”: Computer graphics Image processing Environmental data representation Support for the Mixed and Augmented Reality (MAR) Interaction with, and visual representation of, information Included are the following related areas: Modeling and simulation and related reference models Virtual reality with accompanying augmented reality/augmented virtuality aspects and related reference models Application program interfaces Functional specifications Representation models Interchange formats, encodings and their specifications, including metafiles Device interfaces Testing methods Registration procedures Presentation and support for creation of multimedia, hypermedia, and mixed reality documents Excluded are the following areas: Character and image coding Coding of multimedia, hypermedia, and mixed reality document interchange formats JTC 1 work in user system interfaces and document presentation ISO/TC 207 work on ISO 14000 environment management, ISO/TC 211 work on geographic information and geomatics Software environments as described by ISO/IEC JTC 1/SC 22 == Structure == ISO/IEC JTC 1/SC 24 is made up of four active working groups, each of which carries out specific tasks in standards development within the field of computer graphics, image processing and environmental data representation, together with ITU-T Study Group 16. As a response to changing standardization needs, working groups of ISO/IEC JTC 1/SC 24 can be disbanded if their area of work is no longer applicable, or established if new working areas arise. The focus of each working group is described in the group's terms of reference. Active working groups of ISO/IEC JTC 1/SC 24 are: == Collaborations == ISO/IEC JTC 1/SC 24 works in close collaboration with a number of other organizations or subcommittees, both internal and external to ISO or IEC, in order to avoid conflicting or duplicative work. Organizations internal to ISO or IEC that collaborate with or are in liaison to ISO/IEC JTC 1/SC 24 include: ISO/IEC JTC 1/WG 7, Sensor Networks ISO/IEC JTC 1/SC 29, Coding of audio, picture, multimedia and hypermedia information ISO/IEC JTC 1/SC 32, Data management and interchange ISO/TAG 14, Imagery and technology ISO/TC 130, Graphic Technology ISO/TC 184/SC 4, Industrial data ISO/TC 211, Geographic information/Geomatics ISO/TC 215, Health informatics IEC TC 100, Audio, video and multimedia system and equipment Some organizations external to ISO or IEC that collaborate with or are in liaison to ISO/IEC JTC 1/SC 24 include: Defence Geospatial Information Working Group (DGIWG) Digital Imaging and Communications in Medicine (DICOM) International Hydrographic Organization (IHO) The Khronos Group NATO - Joint Intelligence Surveillance and Reconnaissance Capability Group (JISRCG) OMG Robotics DTF Open CGM Open Geospatial Consortium (OGC) SEDRIS Organization Simulation Interoperability Standards Organization (SISO) US National Imagery Transmission Format Standard (NITFS) Technical Board (US NTB) Web3D Consortium World Intellectual Property Organization (WIPO) World Wide Web Consortium (W3C) == Member countries == Countries pay a fee to ISO to be members of subcommittees. The 11 "P" (participating) members of ISO/IEC JTC 1/SC 24 are: Australia, China, Egypt, France, India, Japan, Republic of Korea, Portugal, Russian Federation, United Kingdom, and United States. The 22 "O" (observer) members of ISO/IEC JTC 1/SC 24 are: Argentina, Austria, Belgium, Bosnia and Herzegovina, Bulgaria, Canada, Cuba, Czech Republic, Finland, Ghana, Hungary, Iceland, Indonesia, Islamic Republic of Iran, Italy, Kazakhstan, Malaysia, Poland, Romania, Serbia, Slovakia, Switzerland, and Thailand. == Published standards == ISO/IEC JTC 1/SC 24 currently has 80 published standards under their direct responsibility within the field of computer graphics, image processing, and environmental data representation, including:
MinID
MinID is an electronic login system used to secure a range of internet services in the Norwegian public sector. The communication done with MinID is encrypted to secure information from unauthorized usage. Everyone registered in the Norwegian Population Register over the age of 13 years can create a public ID with MinID. As of April 2010, more than 2 million people living in Norway had created user accounts with MinID. To create a public ID, PIN-codes from the Norwegian Tax Administration are needed. == Purpose == The purpose of MinID is to communicate an electronic identity, so that users are authorized to use electronic services, in a secure way. MinID has a user database where social security numbers and PIN-codes are saved. MinID can be used to access more than 50 online services from various Norwegian public agencies, including the Norwegian Labour and Welfare Administration, the Directorate of Taxes and the State Educational Loan. == Controller == The Norwegian Digitalisation Agency (Digdir) is the controller of the personal data handled by MinID. The Norwegian Digitalisation Agency (Norwegian: Digitaliseringsdirektoratet) or Digdir is a government agency subordinate to the Ministry of Digitalisation and Public Governance. It is responsible for help the public sector achieve quality, efficiency, user friendliness, openness and participation, as well as helping the public sector be organized and led in a good way with good intersectoral cooperation. == User profile == Users of MinID have a user profile that contains their mobile phone number and/or e-mail address. This data is used to administrate MinID use. The e-mail address is needed in order to send the user a temporary password if he or she forgets the password. The phone number is needed in order to send an SMS-code at log in or a temporary password if the user forgets the password. == Transparency, correction and deletion == According to the law users can claim full access of the handling of their own personal data. Users also have the right to information about how this data are handled and saved, and how they can correct or delete inaccurate data. Users can at any time choose to delete themselves as a user of MinID. The user profile will then be deleted from the MinID user database. == Extradition to others == MinID passes on the user's social security number and chosen language to the public services he or she logs on to, so that the user can go to other public services without a new login.
Distributed concurrency control
Distributed concurrency control is the concurrency control of a system distributed over a computer network (Bernstein et al. 1987, Weikum and Vossen 2001). In database systems and transaction processing (transaction management) distributed concurrency control refers primarily to the concurrency control of a distributed database. It also refers to the concurrency control in a multidatabase (and other multi-transactional object) environment (e.g., federated database, grid computing, and cloud computing environments. A major goal for distributed concurrency control is distributed serializability (or global serializability for multidatabase systems). Distributed concurrency control poses special challenges beyond centralized one, primarily due to communication and computer latency. It often requires special techniques, like distributed lock manager over fast computer networks with low latency, like switched fabric (e.g., InfiniBand). The most common distributed concurrency control technique is strong strict two-phase locking (SS2PL, also named rigorousness), which is also a common centralized concurrency control technique. SS2PL provides both the serializability and strictness. Strictness, a special case of recoverability, is utilized for effective recovery from failure. For large-scale distribution and complex transactions, distributed locking's typical heavy performance penalty (due to delays, latency) can be saved by using the atomic commitment protocol, which is needed in a distributed database for (distributed) transactions' atomicity.
Concurrency control
In information technology and computer science, especially in the fields of computer programming, operating systems, multiprocessors, and databases, concurrency control ensures that correct results for concurrent operations are generated, while getting those results as quickly as possible. Computer systems, both software and hardware, consist of modules, or components. Each component is designed to operate correctly, i.e., to obey or to meet certain consistency rules. When components that operate concurrently interact by messaging or by sharing accessed data (in memory or storage), a certain component's consistency may be violated by another component. The general area of concurrency control provides rules, methods, design methodologies, and theories to maintain the consistency of components operating concurrently while interacting, and thus the consistency and correctness of the whole system. Introducing concurrency control into a system means applying operation constraints which typically result in some performance reduction. Operation consistency and correctness should be achieved with as good as possible efficiency, without reducing performance below reasonable levels. Concurrency control can require significant additional complexity and overhead in a concurrent algorithm compared to the simpler sequential algorithm. For example, a failure in concurrency control can result in data corruption from torn read or write operations. == Concurrency control in databases == Comments: This section is applicable to all transactional systems, i.e., to all systems that use database transactions (atomic transactions; e.g., transactional objects in Systems management and in networks of smartphones which typically implement private, dedicated database systems), not only general-purpose database management systems (DBMSs). DBMSs need to deal also with concurrency control issues not typical just to database transactions but rather to operating systems in general. These issues (e.g., see Concurrency control in operating systems below) are out of the scope of this section. Concurrency control in Database management systems (DBMS; e.g., Bernstein et al. 1987, Weikum and Vossen 2001), other transactional objects, and related distributed applications (e.g., Grid computing and Cloud computing) ensures that database transactions are performed concurrently without violating the data integrity of the respective databases. Thus concurrency control is an essential element for correctness in any system where two database transactions or more, executed with time overlap, can access the same data, e.g., virtually in any general-purpose database system. Consequently, a vast body of related research has been accumulated since database systems emerged in the early 1970s. A well established concurrency control theory for database systems is outlined in the references mentioned above: serializability theory, which allows to effectively design and analyze concurrency control methods and mechanisms. An alternative theory for concurrency control of atomic transactions over abstract data types is presented in (Lynch et al. 1993), and not utilized below. This theory is more refined, complex, with a wider scope, and has been less utilized in the Database literature than the classical theory above. Each theory has its pros and cons, emphasis and insight. To some extent they are complementary, and their merging may be useful. To ensure correctness, a DBMS usually guarantees that only serializable transaction schedules are generated, unless serializability is intentionally relaxed to increase performance, but only in cases where application correctness is not harmed. For maintaining correctness in cases of failed (aborted) transactions (which can always happen for many reasons) schedules also need to have the recoverability (from abort) property. A DBMS also guarantees that no effect of committed transactions is lost, and no effect of aborted (rolled back) transactions remains in the related database. Overall transaction characterization is usually summarized by the ACID rules below. As databases have become distributed, or needed to cooperate in distributed environments (e.g., Federated databases in the early 1990, and Cloud computing currently), the effective distribution of concurrency control mechanisms has received special attention. === Database transaction and the ACID rules === The concept of a database transaction (or atomic transaction) has evolved in order to enable both a well understood database system behavior in a faulty environment where crashes can happen any time, and recovery from a crash to a well understood database state. A database transaction is a unit of work, typically encapsulating a number of operations over a database (e.g., reading a database object, writing, acquiring lock, etc.), an abstraction supported in database and also other systems. Each transaction has well defined boundaries in terms of which program/code executions are included in that transaction (determined by the transaction's programmer via special transaction commands). Every database transaction obeys the following rules (by support in the database system; i.e., a database system is designed to guarantee them for the transactions it runs): Atomicity - Either the effects of all or none of its operations remain ("all or nothing" semantics) when a transaction is completed (committed or aborted respectively). In other words, to the outside world a committed transaction appears (by its effects on the database) to be indivisible (atomic), and an aborted transaction does not affect the database at all. Either all the operations are done or none of them are. Consistency - Every transaction must leave the database in a consistent (correct) state, i.e., maintain the predetermined integrity rules of the database (constraints upon and among the database's objects). A transaction must transform a database from one consistent state to another consistent state (however, it is the responsibility of the transaction's programmer to make sure that the transaction itself is correct, i.e., performs correctly what it intends to perform (from the application's point of view) while the predefined integrity rules are enforced by the DBMS). Thus since a database can be normally changed only by transactions, all the database's states are consistent. Isolation - Transactions cannot interfere with each other (as an end result of their executions). Moreover, usually (depending on concurrency control method) the effects of an incomplete transaction are not even visible to another transaction. Providing isolation is the main goal of concurrency control. Durability - Effects of successful (committed) transactions must persist through crashes (typically by recording the transaction's effects and its commit event in a non-volatile memory). The concept of atomic transaction has been extended during the years to what has become Business transactions which actually implement types of Workflow and are not atomic. However also such enhanced transactions typically utilize atomic transactions as components. === Why is concurrency control needed? === If transactions are executed serially, i.e., sequentially with no overlap in time, no transaction concurrency exists. However, if concurrent transactions with interleaving operations are allowed in an uncontrolled manner, some unexpected, undesirable results may occur, such as: The lost update problem: A second transaction writes a second value of a data-item (datum) on top of a first value written by a first concurrent transaction, and the first value is lost to other transactions running concurrently which need, by their precedence, to read the first value. The transactions that have read the wrong value end with incorrect results. The dirty read problem: Transactions read a value written by a transaction that has been later aborted. This value disappears from the database upon abort, and should not have been read by any transaction ("dirty read"). The reading transactions end with incorrect results. The incorrect summary problem: While one transaction takes a summary over the values of all the instances of a repeated data-item, a second transaction updates some instances of that data-item. The resulting summary does not reflect a correct result for any (usually needed for correctness) precedence order between the two transactions (if one is executed before the other), but rather some random result, depending on the timing of the updates, and whether certain update results have been included in the summary or not. Most high-performance transactional systems need to run transactions concurrently to meet their performance requirements. Thus, without concurrency control such systems can neither provide correct results nor maintain their databases consistently. === Concurrency control mechanisms === ==== Categories ==== The main categories of concurrency control mechanis
PDE surface
PDE surfaces are used in geometric modelling and computer graphics for creating smooth surfaces conforming to a given boundary configuration. PDE surfaces use partial differential equations to generate a surface which usually satisfy a mathematical boundary value problem. PDE surfaces were first introduced into the area of geometric modelling and computer graphics by two British mathematicians, Malcolm Bloor and Michael Wilson. == Technical details == The PDE method involves generating a surface for some boundary by means of solving an elliptic partial differential equation of the form ( ∂ 2 ∂ u 2 + a 2 ∂ 2 ∂ v 2 ) 2 X ( u , v ) = 0. {\displaystyle \left({\frac {\partial ^{2}}{\partial u^{2}}}+a^{2}{\frac {\partial ^{2}}{\partial v^{2}}}\right)^{2}X(u,v)=0.} Here X ( u , v ) {\displaystyle X(u,v)} is a function parameterised by the two parameters u {\displaystyle u} and v {\displaystyle v} such that X ( u , v ) = ( x ( u , v ) , y ( u , v ) , z ( u , v ) ) {\displaystyle X(u,v)=(x(u,v),y(u,v),z(u,v))} where x {\displaystyle x} , y {\displaystyle y} and z {\displaystyle z} are the usual cartesian coordinate space. The boundary conditions on the function X ( u , v ) {\displaystyle X(u,v)} and its normal derivatives ∂ X / ∂ n {\displaystyle \partial {X}/\partial {n}} are imposed at the edges of the surface patch. With the above formulation it is notable that the elliptic partial differential operator in the above PDE represents a smoothing process in which the value of the function at any point on the surface is, in some sense, a weighted average of the surrounding values. In this way, a surface is obtained as a smooth transition between the chosen set of boundary conditions. The parameter a {\displaystyle a} is a special design parameter which controls the relative smoothing of the surface in the u {\displaystyle u} and v {\displaystyle v} directions. When a = 1 {\displaystyle a=1} , the PDE is the biharmonic equation: X u u u u + 2 X u u v v + X v v v v = 0 {\displaystyle X_{uuuu}+2X_{uuvv}+X_{vvvv}=0} . The biharmonic equation is the equation produced by applying the Euler-Lagrange equation to the simplified thin plate energy functional X u u 2 + 2 X u v 2 + X v v 2 {\displaystyle X_{uu}^{2}+2X_{uv}^{2}+X_{vv}^{2}} . So solving the PDE with a = 1 {\displaystyle a=1} is equivalent to minimizing the thin plate energy functional subject to the same boundary conditions. == Applications == PDE surfaces can be used in many application areas. These include computer-aided design, interactive design, parametric design, computer animation, computer-aided physical analysis and design optimisation. == Related publications == M.I.G. Bloor and M.J. Wilson, Generating Blend Surfaces using Partial Differential Equations, Computer Aided Design, 21(3), 165–171, (1989). H. Ugail, M.I.G. Bloor, and M.J. Wilson, Techniques for Interactive Design Using the PDE Method, ACM Transactions on Graphics, 18(2), 195–212, (1999). J. Huband, W. Li and R. Smith, An Explicit Representation of Bloor-Wilson PDE Surface Model by using Canonical Basis for Hermite Interpolation, Mathematical Engineering in Industry, 7(4), 421-33 (1999). H. Du and H. Qin, Direct Manipulation and Interactive Sculpting of PDE surfaces, Computer Graphics Forum, 19(3), C261-C270, (2000). H. Ugail, Spine Based Shape Parameterisations for PDE surfaces, Computing, 72, 195–204, (2004). L. You, P. Comninos, J.J. Zhang, PDE Blending Surfaces with C2 Continuity, Computers and Graphics, 28(6), 895–906, (2004).