Content-based image retrieval, also known as query by image content (QBIC) and content-based visual information retrieval (CBVIR), is the application of computer vision techniques to the image retrieval problem, that is, the problem of searching for digital images in large databases (see this survey for a scientific overview of the CBIR field). Content-based image retrieval is opposed to traditional concept-based approaches (see Concept-based image indexing). "Content-based" means that the search analyzes the contents of the image rather than the metadata such as keywords, tags, or descriptions associated with the image. The term "content" in this context might refer to colors, shapes, textures, or any other information that can be derived from the image itself. CBIR is desirable because searches that rely purely on metadata are dependent on annotation quality and completeness. == Comparison with metadata searching == An image meta search requires humans to have manually annotated images by entering keywords or metadata in a large database, which can be time-consuming and may not capture the keywords desired to describe the image. The evaluation of the effectiveness of keyword image search is subjective and has not been well-defined. In the same regard, CBIR systems have similar challenges in defining success. "Keywords also limit the scope of queries to the set of predetermined criteria." and, "having been set up" are less reliable than using the content itself. == History == The term "content-based image retrieval" seems to have originated in 1992 when it was used by Japanese Electrotechnical Laboratory engineer Toshikazu Kato to describe experiments into automatic retrieval of images from a database, based on the colors and shapes present. Since then, the term has been used to describe the process of retrieving desired images from a large collection on the basis of syntactical image features. The techniques, tools, and algorithms that are used originate from fields such as statistics, pattern recognition, signal processing, and computer vision. === QBIC - Query By Image Content === The earliest commercial CBIR system was developed by IBM and was called QBIC (Query By Image Content). Recent network- and graph-based approaches have presented a simple and attractive alternative to existing methods. While the storing of multiple images as part of a single entity preceded the term BLOB (Binary Large OBject), the ability to fully search by content, rather than by description, had to await IBM's QBIC. === VisualRank === == Technical progress == The interest in CBIR has grown because of the limitations inherent in metadata-based systems, as well as the large range of possible uses for efficient image retrieval. Textual information about images can be easily searched using existing technology, but this requires humans to manually describe each image in the database. This can be impractical for very large databases or for images that are generated automatically, e.g. those from surveillance cameras. It is also possible to miss images that use different synonyms in their descriptions. Systems based on categorizing images in semantic classes like "cat" as a subclass of "animal" can avoid the miscategorization problem, but will require more effort by a user to find images that might be "cats", but are only classified as an "animal". Many standards have been developed to categorize images, but all still face scaling and miscategorization issues. Initial CBIR systems were developed to search databases based on image color, texture, and shape properties. After these systems were developed, the need for user-friendly interfaces became apparent. Therefore, efforts in the CBIR field started to include human-centered design that tried to meet the needs of the user performing the search. This typically means inclusion of: query methods that may allow descriptive semantics, queries that may involve user feedback, systems that may include machine learning, and systems that may understand user satisfaction levels. == Techniques == Many CBIR systems have been developed, but as of 2006, the problem of retrieving images on the basis of their pixel content remains largely unsolved. Different query techniques and implementations of CBIR make use of different types of user queries. === Query By Example === QBE (Query By Example) is a query technique that involves providing the CBIR system with an example image that it will then base its search upon. The underlying search algorithms may vary depending on the application, but result images should all share common elements with the provided example. Options for providing example images to the system include: A preexisting image may be supplied by the user or chosen from a random set. The user draws a rough approximation of the image they are looking for, for example with blobs of color or general shapes. This query technique removes the difficulties that can arise when trying to describe images with words. === Semantic retrieval === Semantic retrieval starts with a user making a request like "find pictures of Abraham Lincoln". This type of open-ended task is very difficult for computers to perform - Lincoln may not always be facing the camera or in the same pose. Many CBIR systems therefore generally make use of lower-level features like texture, color, and shape. These features are either used in combination with interfaces that allow easier input of the criteria or with databases that have already been trained to match features (such as faces, fingerprints, or shape matching). However, in general, image retrieval requires human feedback in order to identify higher-level concepts. === Relevance feedback (human interaction) === Combining CBIR search techniques available with the wide range of potential users and their intent can be a difficult task. An aspect of making CBIR successful relies entirely on the ability to understand the user intent. CBIR systems can make use of relevance feedback, where the user progressively refines the search results by marking images in the results as "relevant", "not relevant", or "neutral" to the search query, then repeating the search with the new information. Examples of this type of interface have been developed. === Iterative/machine learning === Machine learning and application of iterative techniques are becoming more common in CBIR. === Other query methods === Other query methods include browsing for example images, navigating customized/hierarchical categories, querying by image region (rather than the entire image), querying by multiple example images, querying by visual sketch, querying by direct specification of image features, and multimodal queries (e.g. combining touch, voice, etc.) == Content comparison using image distance measures == The most common method for comparing two images in content-based image retrieval (typically an example image and an image from the database) is using an image distance measure. An image distance measure compares the similarity of two images in various dimensions such as color, texture, shape, and others. For example, a distance of 0 signifies an exact match with the query, with respect to the dimensions that were considered. As one may intuitively gather, a value greater than 0 indicates various degrees of similarities between the images. Search results then can be sorted based on their distance to the queried image. Many measures of image distance (Similarity Models) have been developed. === Color === Computing distance measures based on color similarity is achieved by computing a color histogram for each image that identifies the proportion of pixels within an image holding specific values. Examining images based on the colors they contain is one of the most widely used techniques because it can be completed without regard to image size or orientation. However, research has also attempted to segment color proportion by region and by spatial relationship among several color regions. === Texture === Texture measures look for visual patterns in images and how they are spatially defined. Textures are represented by texels which are then placed into a number of sets, depending on how many textures are detected in the image. These sets not only define the texture, but also where in the image the texture is located. Texture is a difficult concept to represent. The identification of specific textures in an image is achieved primarily by modeling texture as a two-dimensional gray level variation. The relative brightness of pairs of pixels is computed such that degree of contrast, regularity, coarseness and directionality may be estimated. The problem is in identifying patterns of co-pixel variation and associating them with particular classes of textures such as silky, or rough. Other methods of classifying textures include: Co-occurrence matrix Laws texture energy Wavelet transform Orthogonal transforms (discrete Chebyshev moments) =
Document mosaicing
Document mosaicing is a process that stitches multiple, overlapping snapshot images of a document together to produce one large, high resolution composite. The document is slid under a stationary, over-the-desk camera by hand until all parts of the document are snapshotted by the camera's field of view. As the document slid under the camera, all motion of the document is coarsely tracked by the vision system. The document is periodically snapshotted such that the successive snapshots are overlap by about 50%. The system then finds the overlapped pairs and stitches them together repeatedly until all pairs are stitched together as one piece of document. The document mosaicing can be divided into four main processes. Tracking Feature detecting Correspondences establishing Images mosaicing. == Tracking (simple correlation process) == In this process, the motion of the document slid under the camera is coarsely tracked by the system. Tracking is performed by a process called simple correlation process. In the first frame of snapshots, a small patch is extracted from the center of the image as a correlation template. The correlation process is performed in the four times size of the patch area of the next frame. The motion of the paper is indicated by the peak in the correlation function. The peak in the correlation function indicates the motion of the paper. The template is resampled from this frame and the tracking continues until the template reaches the edge of the document. After the template reaches the edge of the document, another snapshot is taken and the tracking process performs repeatedly until the whole document is imaged. The snapshots are stored in an ordered list to facilitate pairing the overlapped images in later processes. == Feature detecting for efficient matching == Feature detection is the process of finding the transformation that aligns one image with another. There are two main approaches for feature detection. Feature-based approach : Motion parameters are estimated from point correspondences. This approach is suitable for the case that there is plenty supply of stable and detectable features. Featureless approach : When the motion between the two images is small, the motion parameters are estimated using optical flow. On the other hand, when the motion between the two images is large, the motion parameters are estimated using generalised cross-correlation. However, this approach requires a computationally expensive resources. Each image is segmented into a hierarchy of columns, lines, and words to match the organised sets of features across images. Skew angle estimation and columns, lines and words finding are the examples of feature detection operations. === Skew angle estimation === Firstly, the angle that the rows of text make with the image raster lines (skew angle) is estimated. It is assumed to lie in the range of ±20°. A small patch of text in the image is selected randomly and then rotated in the range of ±20° until the variance of the pixel intensities of the patch summed along the raster lines is maximised. To ensure that the found skew angle is accurate, the document mosaic system performs calculation at many image patches and derive the final estimation by finding the average of the individual angles weighted by the variance of the pixel intensities of each patch. === Columns, lines and words finding === In this operation, the de-skewed document is intuitively segmented into a hierarchy of columns, lines and words. The sensitivity to illumination and page coloration of the de-skewed document can be removed by applying a Sobel operator to the de-skewed image and thresholding the output to obtain the binary gradient, de-skewed image. The operation can be roughly separated into 3 steps: column segmentation, line segmentation and word segmentation. Columns are easily segmented from the binary gradient, de-skewed images by summing pixels vertically. Baselines of each row are segmented in the same way as the column segmentation process but horizontally. Finally, individual words are segmented by applying the vertical process at each segmented row. These segmentations are important because the document mosaic is created by matching the lower right corners of words in overlapping images pair. Moreover, the segmentation operation can organize the list of images in the context of a hierarchy of rows and column reliably. The segmentation operation involves a considerable amount of summing in the binary gradient, de-skewed images, which done by construct a matrix of partial sums whose elements are given by p i y = ∑ u = 1 i ∑ v = 1 j b u v {\displaystyle p_{iy}=\sum _{u=1}^{i}\sum _{v=1}^{j}b_{uv}} The matrix of partial sums is calculated in one pass through the binary gradient, de-skewed image. ∑ u = u 1 u 2 ∑ v = v 1 v 2 b u v = p u 2 v 2 + p u 1 v 1 − p u 1 v 2 − p u 2 v 1 {\displaystyle \sum _{u=u_{1}}^{u_{2}}\sum _{v=v_{1}}^{v_{2}}b_{uv}=p_{u_{2}v_{2}}+p_{u_{1}v_{1}}-p_{u_{1}v_{2}}-p_{u_{2}v_{1}}} == Correspondences establishing == The two images are now organized in hierarchy of linked lists in following structure : image=list of columns row=list of words column=list of row word=length (in pixels) At the bottom of the structure, the length of each word is recorded for establishing correspondence between two images to reduce to search only the corresponding structures for the groups of words with the matching lengths. === Seed match finding === A seed match finding is done by comparing each row in image1 with each row in image2. The two rows are then compared to each other by every word. If the length (in pixel) of the two words (one from image1 and one from image2) and their immediate neighbours agree with each other within a predefined tolerance threshold (5 pixels, for example), then they are assumed to match. The row of each image is assumed a match if there are three or more word matches between the two rows. The seed match finding operation is terminated when two pairs of consecutive row match are found. === Match list building === After finishing a seed match finding operation, the next process is to build the match list to generate the correspondences points of the two images. The process is done by searching the matching pairs of rows away from the seed row. == Images mosaicing == Given the list of corresponding points of the two images, finding the transformation of the overlapping portion of the images is the next process. Assuming a pinhole camera model, the transformation between pixels (u,v) of image 1 and pixels (u0, v0) of image 2 is demonstrated by a plane-to-plane projectivity. [ s u ′ s v ′ s ] = [ p 11 p 12 p 13 p 21 p 22 p 23 p 31 p 32 1 ] [ u v 1 ] E q .1 {\displaystyle \left[{\begin{array}{c}su'\\sv'\\s\end{array}}\right]=\left[{\begin{array}{ccc}p_{11}&p_{12}&p_{13}\\p_{21}&p_{22}&p_{23}\\p_{31}&p_{32}&1\end{array}}\right]\left[{\begin{array}{c}u\\v\\1\end{array}}\right]\qquad Eq.1} The parameters of the projectivity is found from four pairs of matching points. RANSAC regression technique is used to reject outlying matches and estimate the projectivity from the remaining good matches. The projectivity is fine-tuned using correlation at the corners of the overlapping portion to obtain four correspondences to sub-pixel accuracy. Therefore, image1 is then transformed into image2's coordinate system using Eq.1. The typical result of the process is shown in Figure 5. === Many images coping === Finally, the whole page composition is built up by mapping all the images into the coordinate system of an "anchor" image, which is normally the one nearest the page center. The transformations to the anchor frame are calculated by concatenating the pair-wise transformations found earlier. The raw document mosaic is shown in Figure 6. However, there might be a problem of non-consecutive images that are overlap. This problem can be solved by performing Hierarchical sub-mosaics. As shown in Figure 7, image1 and image2 are registered, as are image3 and image4, creating two sub-mosaics. These two sub-mosaics are later stitched together in another mosaicing process. == Applied areas == There are various areas that the technique of document mosaicing can be applied to such as : Text segmentation of images of documents Document Recognition Interaction with paper on the digital desk Video mosaics for virtual environments Image registration techniques == Relevant research papers == Huang, T.S.; Netravali, A.N. (1994). "Motion and structure from feature correspondences: A review". Proceedings of the IEEE. 82 (2): 252–268. doi:10.1109/5.265351. D.G. Lowe. [1] Perceptual Organization and Visual Recognition. Kluwer Academic Publishers, Boston, 1985. Irani, M.; Peleg, S. (1991). "Improving resolution by image registration". CVGIP: Graphical Models and Image Processing. 53 (3): 231–239. doi:10.1016/1049-9652(91)90045-L. S2CID 4834546. Shivakumara, P.; Kumar, G. Hemantha; Guru, D. S.; Nagabhushan, P. (2006). "
Jarosław Królewski
Jarosław Królewski ([jaˈrɔswaf kruˈlɛfskʲi]; born September 26, 1986) is a Polish entrepreneur, programmer, sociologist, investor, and philanthropist from Hańczowa, Poland. He is a researcher and lecturer at the AGH University of Krakow. He was selected as a Young Global Leader by the World Economic Forum in 2025. Królewski is a cofounder and chief executive of the software development company Synerise that develops its namesake business intelligence software based on artificial intelligence and big data. He is also the president and a majority stakeholder of the Polish soccer club Wisła Kraków. == Biography == === Scientific activities === Królewski graduated from the AGH University of Kraków and the University of Banking and Management in Kraków. He completed two fields of study: a master's degree in sociology, and an engineer's degree in computer science. He co-created innovative study programs, including social informatics and electronic business, recognized as the most innovative field of study in Poland in 2012 by the Ministry of Science and Higher Education, which led to the AGH receiving a PLN 1 million award for the development of the program. Królewski is a research and teaching employee at AGH, where since 2010 he has been conducting classes and lectures on the Internet, mobile technologies, and UX/UI. He has been preparing a PhD thesis. He is the brand ambassador of the Academy. He is also a mentor of the Polish Development Fund network. In 2019, on the occasion of the AGH University's 100th anniversary, Królewski was honored the title of "AGH Graduate Junior 2018." Królewski is the co-originator of the "Data Science in Business and Administration" doctoral studies organized by the Faculty of Computer Science and Electronic Economy of the Poznań University of Economics. He is a co-author of a textbook E-marketing. Contemporary trends. Starter package (2013), and an Book on algorithmic governance Algocracy. How and why artificial intelligence changes everything (with Krzysztof Rybiński, 2023). === Business career === Throughout the 2000s, Królewski was responsible for issues of usability and user experience at the advertising agency Eskadra in Kraków. In 2012, along with programmer Miłosz Baluś and graphic designer Krzysztof Kochmański, he founded the software house Humanoit Group. The company created a project management software using machine learning and artificial intelligence. In 2013, HG Intelligence was established to create a platform for analytics and automation of business processes called "Synerise" that combined big data with artificial intelligence mechanisms. Królewski became the president of the company's management board. In 2016, the company rebranded itself after its own platform. It is one of the fastest growing enterprises in Poland – in 2019 it was valued at USD 85 million (PLN 323.5 million), and its value is still growing, in 2022 it announced an investment of USD 23 million. Królewski is a supporter of releasing some software in open-source form, an example of which is the open library Cleora.ai. Królewski has been described "one of the most promising young Polish businessmen in the technology industry." According to Forbes, he is a "visionary computer scientist who in many respects resembles the young Bill Gates." Królewski considers himself a “technological determinist and optimist.” He never wants to be a millionaire or billionaire, he spends 80 percent of his private income on education, sports and charities. === Sports === In his youth (2002–2006) he was a football player of the (then 4th-league) club Glinik Gorlice, and represented it at the then-highest level of junior competitions in Poland. He played there with Rafał Wisłocki, later president of Wisła Kraków and vice-president of Bruk-Bet Termalica Nieciecza. In early 2019, Królewski was the initiator of a rescue operation that saved Wisła Kraków from bankruptcy, as well as the originator of the crowdfunding issue of shares of Wisła Kraków, pioneering in Polish sports, during restructuring and searching for a strategic investor. The offered shares constituted 5.1 percent. all the company's shares, which meant that the club was valued at PLN 74.4 million. 40,000 shares were put up for sale, each worth PLN 100. Within 24 hours, they were purchased by 9,124 investors through an equity crowdfunding platform Beesfund, earning the club PLN 4 million. In March 2019, Królewski became vice-chairman of Wisła's supervisory board, a position he held until 2021. In April 2020, he became Wisła's co-owner, along with the footballer Jakub Błaszczykowski, and Tomasz Jażdżyński, president of Gremi Media (publisher of the news outlets Rzeczpospolita and Parkiet). The three granted a bridging loan to the club of PLN 4 million, each supporting PLN 1.33 million. The funds were used to repay the club's debts to players. In November 2022, the supervisory board of Wisła Kraków appointed Królewski as the president of the club's management board. In December 2022, Królewski took over a majority stake in the club. In January 2024, based on match statistics, he used AI tools to select Wisła's new coach, Albert Rudé. === Social activities === Królewski is the creator and originator of the nationwide educational project "AI Schools & Academy", the first artificial intelligence teaching program in Polish kindergartens, primary and secondary schools in Polish history. Launched in 2018, the project was financed by Synerise business partners: Carrefour, CCC, Ernst & Young, IDC, Media Expert, Microsoft, Orange Foundation, Oriflame, Bank Pekao, Photon, PZU, and Żabka. Physicists, mathematicians, and computer scientists conduct special classes in 1,500 kindergartens, primary and secondary schools. Outstanding students and teachers are awarded scholarships. The project was appreciated by experts. In the years 2018–2020, Królewski was the main sponsor of Glinik Gorlice. He also supported the women's football team Staszkówka Jelna (of Staszkówka). After taking over the shares of Wisła Kraków in 2020, he launched socially conscience initiatives along with other shareholders, including a women's football team, the amp football section, and the blind football section. He has privately sponsored social charities. == Accolades and awards == In 2017, Królewski along with the Synerise co-founders Baluś and Kochmański was included in the “New Europe 100” list of eastern Europe's brightest and best citizens changing the region's societies, politics, or business environments, according to Res Publica, along with the International Visegrad Fund, Google and the Financial Times. Królewski was included on Ernst & Young's list of the 30 most promising technology entrepreneurs in the world. In 2018, he was honored with the Special Jury Award in the Polish edition of the Ernst & Young Entrepreneur of the Year Award competition, for combining scientific activities with entrepreneurship. The same year, Królewski won an award in the competition Digital Shapers, distinguishing outstanding tech personalities by the Digital Poland Foundation. He was also selected to Ernst & Young startup program EY Accelerating Entrepreneurs for businesses that focus on disruptive fields. In 2019, as part of the AI Awards competition, Królewski received the title of AI Person of the Year. == Private life == Królewski comes from a Lemko family from Hańczowa in the Low Beskids. He is married to Aleksandra Królewska.
Project Mariner
Project Mariner was a research prototype developed by Google DeepMind that explored human-agent interactions, particularly within web browsers. It automated tasks such as online shopping, information retrieval, and form-filling, aiming to enhance user productivity by delegating routine web-based tasks to an AI agent. Project Mariner operated as an experimental Chrome extension that understands the contents of your screen, including images, code, forms, and more. It could interpret complex goals, plan actionable steps, and navigate websites to carry out tasks, while keeping the user informed and allowing them to intervene at any time. As of May 2025, Project Mariner was available to Google AI Ultra subscribers in the US and was being integrated into the Gemini API and Vertex AI, allowing developers to build applications powered by the agent Google plans to bring Project Mariner’s capabilities to more countries and integrate it into Google Search's AI Mode, which was currently in the Search Labs testing phase. Project Mariner was discontinued on May 4, 2026.
Transaction logic
Transaction Logic is an extension of predicate logic that accounts in a clean and declarative way for the phenomenon of state changes in logic programs and databases. This extension adds connectives specifically designed for combining simple actions into complex transactions and for providing control over their execution. The logic has a natural model theory and a sound and complete proof theory. Transaction Logic has a Horn clause subset, which has a procedural as well as a declarative semantics. The important features of the logic include hypothetical and committed updates, dynamic constraints on transaction execution, non-determinism, and bulk updates. In this way, Transaction Logic is able to declaratively capture a number of non-logical phenomena, including procedural knowledge in artificial intelligence, active databases, and methods with side effects in object databases. Transaction Logic was originally proposed in 1993 by Anthony Bonner and Michael Kifer and later described in more detail in An Overview of Transaction Logic and Logic Programming for Database Transactions. The most comprehensive description appears in Bonner & Kifer's technical report from 1995. In later years, Transaction Logic was extended in various ways, including concurrency, defeasible reasoning, partially defined actions, and other features. In 2013, the original paper on Transaction Logic has won the 20-year Test of Time Award of the Association for Logic Programming as the most influential paper from the proceedings of ICLP 1993 conference in the preceding 20 years. == Examples == === Graph coloring === Here tinsert denotes the elementary update operation of transactional insert. The connective ⊗ is called serial conjunction. === Pyramid stacking === The elementary update tdelete represents the transactional delete operation. === Hypothetical execution === Here <> is the modal operator of possibility: If both action1 and action2 are possible, execute action1. Otherwise, if only action2 is possible, then execute it. === Dining philosophers === Here | is the logical connective of parallel conjunction of Concurrent Transaction Logic. == Implementations == A number of implementations of Transaction Logic exist: The original implementation. An implementation of Concurrent Transaction Logic. Transaction Logic enhanced with tabling. An implementation of Transaction Logic has also been incorporated as part of the Flora-2 knowledge representation and reasoning system. All these implementations are open source.
Vujak
VuJak is an early video sampler, a VJ remix and mashup tool created in 1992 by Brian Kane, Lisa Eisenpresser, and Jay Haynes. The original name of the project was Mideo, but it was later changed to VuJak. VuJak was based on MIDI control of video in real-time. It was created with MAX from Opcode Systems, and utilized the newly released QuickTime 1.0 movie object. The first working version of the program was built on a Mac IIfx with 8 megs of ram, and could jump in real-time across a 160 x 120 pixel QuickTime movie via a midi keyboard. Later versions could manipulate full screen video, included the first real-time video scratch feature, had looping, vari-speed, and random play features, and allowed for recording and editing of video sequences within the application. VuJak also had networking capabilities which allowed artists to "jam" in real time across standard phone lines. The first public exhibition of VuJak was at the Digital Hollywood conference in Beverly Hills in 1993, where it was promoted by Timothy Leary. VuJak was featured in Mondo 2000, CBS Evening News, Wired Magazine, Electronic Musician, Billboard Magazine, The Hollywood Reporter, and it was used to create promotional videos for MTV. In 1994, VuJak was a featured interactive exhibition at the Exploratorium in San Francisco. Development of VuJak ceased in 1995.
Mathematical model
A mathematical model is an abstract description of a concrete system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in many fields, including applied mathematics, natural sciences, social sciences and engineering. In particular, the field of operations research studies the use of mathematical modelling and related tools to solve problems in business or military operations. A model may help to characterize a system by studying the effects of different components, which may be used to make predictions about behavior or solve specific problems. == Elements of a mathematical model == Mathematical models can take many forms, including dynamical systems, statistical models, differential equations, or game theoretic models. These and other types of models can overlap, with a given model involving a variety of abstract structures. In many cases, the quality of a scientific field depends on how well the mathematical models developed on the theoretical side agree with results of repeatable experiments. Lack of agreement between theoretical mathematical models and experimental measurements often leads to important advances as better theories are developed. In the physical sciences, a traditional mathematical model contains most of the following elements: Governing equations Supplementary sub-models Defining equations Constitutive equations Assumptions and constraints Initial and boundary conditions Classical constraints and kinematic equations == Classifications == Mathematical models are of different types: === Linear vs. nonlinear === If all the operators in a mathematical model exhibit linearity, the resulting mathematical model is defined as linear. All other models are considered nonlinear. The definition of linearity and nonlinearity is dependent on context, and linear models may have nonlinear expressions in them. For example, in a statistical linear model, it is assumed that a relationship is linear in the parameters, but it may be nonlinear in the predictor variables. Similarly, a differential equation is said to be linear if it can be written with linear differential operators, but it can still have nonlinear expressions in it. In a mathematical programming model, if the objective functions and constraints are represented entirely by linear equations, then the model is regarded as a linear model. If one or more of the objective functions or constraints are represented with a nonlinear equation, then the model is known as a nonlinear model. Linear structure implies that a problem can be decomposed into simpler parts that can be treated independently or analyzed at a different scale, and therefore that the results will remain valid if the initial is recomposed or rescaled. Nonlinearity, even in fairly simple systems, is often associated with phenomena such as chaos and irreversibility. Although there are exceptions, nonlinear systems and models tend to be more difficult to study than linear ones. A common approach to nonlinear problems is linearization, but this can be problematic if one is trying to study aspects such as irreversibility, which are strongly tied to nonlinearity. === Static vs. dynamic === A dynamic model accounts for time-dependent changes in the state of the system, while a static (or steady-state) model calculates the system in equilibrium, and thus is time-invariant. Dynamic models are typically represented by differential equations or difference equations. === Explicit vs. implicit === If all of the input parameters of the overall model are known, and the output parameters can be calculated by a finite series of computations, the model is said to be explicit. But sometimes it is the output parameters which are known, and the corresponding inputs must be solved for by an iterative procedure, such as Newton's method or Broyden's method. In such a case the model is said to be implicit. For example, a jet engine's physical properties such as turbine and nozzle throat areas can be explicitly calculated given a design thermodynamic cycle (air and fuel flow rates, pressures, and temperatures) at a specific flight condition and power setting, but the engine's operating cycles at other flight conditions and power settings cannot be explicitly calculated from the constant physical properties. === Discrete vs. continuous === A discrete model treats objects as discrete, such as the particles in a molecular model or the states in a statistical model; while a continuous model represents the objects in a continuous manner, such as the velocity field of fluid in pipe flows, temperatures and stresses in a solid, and electric field that applies continuously over the entire model due to a point charge. === Deterministic vs. probabilistic (stochastic) === A deterministic model is one in which every set of variable states is uniquely determined by parameters in the model and by sets of previous states of these variables; therefore, a deterministic model always performs the same way for a given set of initial conditions. Conversely, in a stochastic model—usually called a "statistical model"—randomness is present, and variable states are not described by unique values, but rather by probability distributions. === Deductive, inductive, or floating === A deductive model is a logical structure based on a theory. An inductive model arises from empirical findings and generalization from them. If a model rests on neither theory nor observation, it may be described as a 'floating' model. Application of mathematics in social sciences outside of economics has been criticized for unfounded models. Application of catastrophe theory in science has been characterized as a floating model. === Strategic vs. non-strategic === Models used in game theory are distinct in the sense that they model agents with incompatible incentives, such as competing species or bidders in an auction. Strategic models assume that players are autonomous decision makers who rationally choose actions that maximize their objective function. A key challenge of using strategic models is defining and computing solution concepts such as the Nash equilibrium. An interesting property of strategic models is that they separate reasoning about rules of the game from reasoning about behavior of the players. == Construction == In business and engineering, mathematical models may be used to maximize a certain output. The system under consideration will require certain inputs. The system relating inputs to outputs depends on other variables too: decision variables, state variables, exogenous variables, and random variables. Decision variables are sometimes known as independent variables. Exogenous variables are sometimes known as parameters or constants. The variables are not independent of each other as the state variables are dependent on the decision, input, random, and exogenous variables. Furthermore, the output variables are dependent on the state of the system (represented by the state variables). Objectives and constraints of the system and its users can be represented as functions of the output variables or state variables. The objective functions will depend on the perspective of the model's user. Depending on the context, an objective function is also known as an index of performance, as it is some measure of interest to the user. Although there is no limit to the number of objective functions and constraints a model can have, using or optimizing the model becomes more involved (computationally) as the number increases. For example, economists often apply linear algebra when using input–output models. Complicated mathematical models that have many variables may be consolidated by use of vectors where one symbol represents several variables. === A priori information === Mathematical modeling problems are often classified into black box or white box models, according to how much a priori information on the system is available. A black-box model is a system of which there is no a priori information available. A white-box model (also called glass box or clear box) is a system where all necessary information is available. Practically all systems are somewhere between the black-box and white-box models, so this concept is useful only as an intuitive guide for deciding which approach to take. Usually, it is preferable to use as much a priori information as possible to make the model more accurate. Therefore, the white-box models are usually considered easier, because if you have used the information correctly, then the model will behave correctly. Often the a priori information comes in forms of knowing the type of functions relating different variables. For example, if we make a model of how a medicine works in a human system, we know that usually the amount of medicine in the blood is an exponentially decaying function, but we are still left with several unknown parameters; how