Rejoyn is a prescription-only digital therapeutic smartphone app approved by the US FDA for the treatment of major depressive disorder (MDD) in adults ages 22 and up. It is prescribed in conjunction with standard antidepressant medication and professional guidance and support. Rejoyn was developed by Click Therapeutics and Otsuka America Pharmaceutical Inc., and gained FDA clearance as a "medical device" on March 30th, 2024. The smartphone app helps patients with depression using exercises based on cognitive behavioral therapy (CBT) along with timed notifications to keep the patient engaged and in treatment. Randomized controlled trials showed that the Rejoyn app was more effective at relieving depression symptoms compared to a "sham app", a placebo app that required similar effort but was not intended to be helpful. Dr. John Torous, MD, MBI,[a] a psychiatrist at the Beth Israel Deaconess Medical Center in Boston, said that the app seems to pose minimal risks, and is an important step forward in unlocking the power of smartphones in treating psychiatric disorders. Some experts have signaled that the claims should be taken with caution, since the app was "tested only in a narrow subset of patients." and its benefits are "not statistically significant," according to the study’s primary outcome."
Convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions f {\displaystyle f} and g {\displaystyle g} that produces a third function f ∗ g {\displaystyle fg} , as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The term convolution refers to both the resulting function and to the process of computing it. The integral is evaluated for all values of shift, producing the convolution function. The choice of which function is reflected and shifted before the integral does not change the integral result (see commutativity). Graphically, it expresses how the 'shape' of one function is modified by the other. Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, convolution f ∗ g {\displaystyle fg} differs from cross-correlation f ⋆ g {\displaystyle f\star g} only in that either f ( x ) {\displaystyle f(x)} or g ( x ) {\displaystyle g(x)} is reflected about the y-axis in convolution; thus it is a cross-correlation of g ( − x ) {\displaystyle g(-x)} and f ( x ) {\displaystyle f(x)} , or f ( − x ) {\displaystyle f(-x)} and g ( x ) {\displaystyle g(x)} . For complex-valued functions, the cross-correlation operator is the adjoint of the convolution operator. Convolution has applications that include probability, statistics, acoustics, spectroscopy, signal processing and image processing, computer vision and human vision, geophysics, engineering, physics, and differential equations. The convolution can be defined for functions on Euclidean space and other groups (as algebraic structures). For example, periodic functions, such as the discrete-time Fourier transform, can be defined on a circle and convolved by periodic convolution. (See row 18 at DTFT § Properties.) A discrete convolution can be defined for functions on the set of integers. Generalizations of convolution have applications in the field of numerical analysis and numerical linear algebra, and in the design and implementation of finite impulse response filters in signal processing. Computing the inverse of the convolution operation is known as deconvolution. == Definition == The convolution of f {\displaystyle f} and g {\displaystyle g} is written f ∗ g {\displaystyle fg} , denoting the operator with the symbol ∗ {\displaystyle } . It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. As such, it is a particular kind of integral transform: ( f ∗ g ) ( t ) := ∫ − ∞ ∞ f ( τ ) g ( t − τ ) d τ . {\displaystyle (fg)(t):=\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau .} An equivalent definition is (see commutativity): ( f ∗ g ) ( t ) := ∫ − ∞ ∞ f ( t − τ ) g ( τ ) d τ . {\displaystyle (fg)(t):=\int _{-\infty }^{\infty }f(t-\tau )g(\tau )\,d\tau .} While the symbol t {\displaystyle t} is used above, it need not represent the time domain. At each t {\displaystyle t} , the convolution formula can be described as the area under the function f ( τ ) {\displaystyle f(\tau )} weighted by the function g ( − τ ) {\displaystyle g(-\tau )} shifted by the amount t {\displaystyle t} . As t {\displaystyle t} changes, the weighting function g ( t − τ ) {\displaystyle g(t-\tau )} emphasizes different parts of the input function f ( τ ) {\displaystyle f(\tau )} ; If t {\displaystyle t} is a positive value, then g ( t − τ ) {\displaystyle g(t-\tau )} is equal to g ( − τ ) {\displaystyle g(-\tau )} that slides or is shifted along the τ {\displaystyle \tau } -axis toward the right (toward + ∞ {\displaystyle +\infty } ) by the amount of t {\displaystyle t} , while if t {\displaystyle t} is a negative value, then g ( t − τ ) {\displaystyle g(t-\tau )} is equal to g ( − τ ) {\displaystyle g(-\tau )} that slides or is shifted toward the left (toward − ∞ {\displaystyle -\infty } ) by the amount of | t | {\displaystyle |t|} . For functions f {\displaystyle f} , g {\displaystyle g} supported on only [ 0 , ∞ ) {\displaystyle [0,\infty )} (i.e., zero for negative arguments), the integration limits can be truncated, resulting in: ( f ∗ g ) ( t ) = ∫ 0 t f ( τ ) g ( t − τ ) d τ for f , g : [ 0 , ∞ ) → R . {\displaystyle (fg)(t)=\int _{0}^{t}f(\tau )g(t-\tau )\,d\tau \quad \ {\text{for }}f,g:[0,\infty )\to \mathbb {R} .} For the multi-dimensional formulation of convolution, see domain of definition (below). === Notation === A common engineering notational convention is: f ( t ) ∗ g ( t ) := ∫ − ∞ ∞ f ( τ ) g ( t − τ ) d τ ⏟ ( f ∗ g ) ( t ) , {\displaystyle f(t)g(t)\mathrel {:=} \underbrace {\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau } _{(fg)(t)},} which has to be interpreted carefully to avoid confusion. For instance, f ( t ) ∗ g ( t − t 0 ) {\displaystyle f(t)g(t-t_{0})} is equivalent to ( f ∗ g ) ( t − t 0 ) {\displaystyle (fg)(t-t_{0})} , but f ( t − t 0 ) ∗ g ( t − t 0 ) {\displaystyle f(t-t_{0})g(t-t_{0})} is in fact equivalent to ( f ∗ g ) ( t − 2 t 0 ) {\displaystyle (fg)(t-2t_{0})} . === Relations with other transforms === Given two functions f ( t ) {\displaystyle f(t)} and g ( t ) {\displaystyle g(t)} with bilateral Laplace transforms (two-sided Laplace transform) F ( s ) = ∫ − ∞ ∞ e − s u f ( u ) d u {\displaystyle F(s)=\int _{-\infty }^{\infty }e^{-su}\ f(u)\ {\text{d}}u} and G ( s ) = ∫ − ∞ ∞ e − s v g ( v ) d v {\displaystyle G(s)=\int _{-\infty }^{\infty }e^{-sv}\ g(v)\ {\text{d}}v} respectively, the convolution operation ( f ∗ g ) ( t ) {\displaystyle (fg)(t)} can be defined as the inverse Laplace transform of the product of F ( s ) {\displaystyle F(s)} and G ( s ) {\displaystyle G(s)} . More precisely, F ( s ) ⋅ G ( s ) = ∫ − ∞ ∞ e − s u f ( u ) d u ⋅ ∫ − ∞ ∞ e − s v g ( v ) d v = ∫ − ∞ ∞ ∫ − ∞ ∞ e − s ( u + v ) f ( u ) g ( v ) d u d v {\displaystyle {\begin{aligned}F(s)\cdot G(s)&=\int _{-\infty }^{\infty }e^{-su}\ f(u)\ {\text{d}}u\cdot \int _{-\infty }^{\infty }e^{-sv}\ g(v)\ {\text{d}}v\\&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{-s(u+v)}\ f(u)\ g(v)\ {\text{d}}u\ {\text{d}}v\end{aligned}}} Let t = u + v {\displaystyle t=u+v} , then F ( s ) ⋅ G ( s ) = ∫ − ∞ ∞ ∫ − ∞ ∞ e − s t f ( u ) g ( t − u ) d u d t = ∫ − ∞ ∞ e − s t ∫ − ∞ ∞ f ( u ) g ( t − u ) d u ⏟ ( f ∗ g ) ( t ) d t = ∫ − ∞ ∞ e − s t ( f ∗ g ) ( t ) d t . {\displaystyle {\begin{aligned}F(s)\cdot G(s)&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{-st}\ f(u)\ g(t-u)\ {\text{d}}u\ {\text{d}}t\\&=\int _{-\infty }^{\infty }e^{-st}\underbrace {\int _{-\infty }^{\infty }f(u)\ g(t-u)\ {\text{d}}u} _{(fg)(t)}\ {\text{d}}t\\&=\int _{-\infty }^{\infty }e^{-st}(fg)(t)\ {\text{d}}t.\end{aligned}}} Note that F ( s ) ⋅ G ( s ) {\displaystyle F(s)\cdot G(s)} is the bilateral Laplace transform of ( f ∗ g ) ( t ) {\displaystyle (fg)(t)} . A similar derivation can be done using the unilateral Laplace transform (one-sided Laplace transform). The convolution operation also describes the output (in terms of the input) of an important class of operations known as linear time-invariant (LTI). See LTI system theory for a derivation of convolution as the result of LTI constraints. In terms of the Fourier transforms of the input and output of an LTI operation, no new frequency components are created. The existing ones are only modified (amplitude and/or phase). In other words, the output transform is the pointwise product of the input transform with a third transform (known as a transfer function). See Convolution theorem for a derivation of that property of convolution. Conversely, convolution can be derived as the inverse Fourier transform of the pointwise product of two Fourier transforms. == Visual explanation == == Historical developments == One of the earliest uses of the convolution integral appeared in D'Alembert's derivation of Taylor's theorem in Recherches sur différents points importants du système du monde, published in 1754. Also, an expression of the type: ∫ f ( u ) ⋅ g ( x − u ) d u {\displaystyle \int f(u)\cdot g(x-u)\,du} is used by Sylvestre François Lacroix on page 505 of his book entitled Treatise on differences and series, which is the last of 3 volumes of the encyclopedic series: Traité du calcul différentiel et du calcul intégral, Chez Courcier, Paris, 1797–1800. Soon thereafter, convolution operations appear in the works of Pierre Simon Laplace, Jean-Baptiste Joseph Fourier, Siméon Denis Poisson, and others. The term itself did not come into wide use until the 1950s or 1960s. Prior to that it was sometimes known as Faltung (which means folding in German), composition product, superposition integral, and Carson's integral. Yet it appears as early as 1903, though the definition is rather unfamiliar in older uses. The operation: ∫ 0 t φ ( s ) ψ ( t − s ) d s , 0 ≤ t < ∞ , {\displaystyle \int _{0}^{t}\varphi (s)\psi (t-s)\,ds,\quad 0\leq t<\infty ,} is a particular case of composition products considered by the Italian mathematician Vito Volterra in 1913. == Circular c
List of color palettes
The following is a list that contains color palettes for notable computer graphics, terminals and video game consoles. Only a simulated image using a palette and its name are given. Main articles are linked from the name of each palette, test charts, sample colours, simulated images, and further technical details (including references). During older eras of computing, manufacturers developed many different display systems often in a competitive, non-collaborative basis (with a few exceptions in the VESA consortium), creating many proprietary, non-standard different instances of display hardware. Often, as with early personal and home computers, a given machine employed its unique display subsystem, also with its unique color palette. Furthermore, software developers had made use of the color abilities of distinct display systems in many different ways. The result is that there is no single common standard nomenclature or classification taxonomy which can encompass every computer color palette. In order to organize the material, color palettes have been grouped following certain criteria. First, generic monochrome and full RGB repertories common to various computer display systems are listed. Then, usual color repertories used for display systems that employ indexed color techniques. And finally, specific manufacturers' color palettes implemented in many representative early personal computers and video game consoles of various brands. The list for personal computer palettes is split into two categories: 8-bit and 16-bit machines. This is not intended as a true strict categorization of such machines, because mixed architectures also exist (16-bit processors with an 8-bit data bus or 32-bit processors with a 16-bit data bus, among others). The distinction is based more on broad 8-bit and 16-bit computer ages or generations (around 1975–1985 and 1985–1995, respectively) and their associated state of the art in color display capabilities. The following is the common color test chart and sample image used to render each palette in this list: See further details in the summary paragraph of the corresponding article. == List of monochrome and RGB palettes == In this article, the term monochrome palette means a set of intensities for a monochrome display, and the term RGB palette is defined as the complete set of combinations a given RGB display can offer by mixing all the possible intensities of the red, green, and blue primaries available in its hardware. These are generic complete repertories of colors to produce black and white and RGB color pictures by the display hardware, not necessarily the total number of such colors that can be simultaneously displayed in a given text or graphic mode of any machine. RGB is the most common method to produce colors for displays; so these complete RGB color repertories have every possible combination of R-G-B triplets within any given maximum number of levels per component. For specific hardware and different methods to produce colors than RGB, see the List of computer hardware palettes and the List of video game consoles sections. For various software arrangements and sorts of colors, including other possible full RGB arrangements within 8-bit depth displays, see the List of software palettes section. === Monochrome palettes === These palettes only have shades of gray. === Dichrome palettes === Each permuted pair of red, green, and blue (16-bit color palette, with 65,536 colors). For example, "additive red green" has zero blue and "subtractive red green" has full blue. === Regular RGB palettes === These full RGB palettes employ the same number of bits to store the relative intensity for the red, green and blue components of every image's pixel color. Thus, they have the same number of levels per channel and the total number of possible colors is always the cube of a power of two. It should be understood that 'when developed' many of these formats were directly related to the size of some host computers 'natural word length' in bytes—the amount of memory in bits held by a single memory address such that the CPU can grab or put it in one operation. === Non-regular RGB palettes === These are also RGB palettes, in the sense defined above (except for 4-bit RGBI, which has an intensity bit that affects all channels at once), but either they do not have the same number of levels for each primary channel, or the numbers are not powers of two, so are not represented as separate bit fields. All of these have been used in popular personal computers. == List of software palettes == Systems that use a 4-bit or 8-bit pixel depth can display up to 16 or 256 colors simultaneously. Many personal computers in the later 1980s and early 1990s displayed at most 256 different colors, freely selected by software (either by the user or by a program) from their wider hardware's color palette. Usual selections of colors in limited subsets (generally 16 or 256) of the full palette includes some RGB level arrangements commonly used with the 8 bpp palettes as master palettes or universal palettes (i.e., palettes for multipurpose uses). These are some representative software palettes, but any selection can be made in such types of systems. === System specific palettes === These are selections of colors officially employed as system palettes in some popular operating systems for personal computers that feature 8-bit displays. === RGB arrangements === These are selections of colors based on evenly ordered RGB levels, mainly used as master palettes to display any kind of image within the limitations of the 8-bit pixel depth. === Other common uses of software palettes === == List of computer hardware palettes == In old personal computers and terminals that offered color displays, some color palettes were chosen algorithmically to provide the most diverse set of colors for a given palette size, and others were chosen to assure the availability of certain colors. In many early home computers, especially when the palette choices were determined at the hardware level by resistor combinations, the palette was determined by the manufacturer. Many early models output composite video colors. When seen on TV devices, the perception of the colors may not correspond with the value levels for the color values employed (most noticeable with NTSC TV color system). For current RGB display systems for PCs (Super VGA, etc.), see the 16-bit RGB and 24-bit RGB for High Color (thousands) and True Color (millions of colors) modes. For video game consoles, see the List of video game consoles section. For every model, their main different graphical color modes are listed based exclusively in the way they handle colors on screen, not all their different screen modes. The list is organized roughly historically by video hardware, not by branch. They are listed according to the original model of each system, which means that extended versions, clones, and compatibles also support the original palette. === Terminals and 8-bit machines === === 16-bit machines === === Video game console palettes === Color palettes of some of the most popular video game consoles. The criteria are the same as those of the List of computer hardware palettes section.
Randonautica
Randonautica (a portmanteau of "random" + "nautica") is an app launched on February 22, 2020 founded by Auburn Salcedo and Joshua Lengfelder. It randomly generates coordinates that encourages the user to explore their local area and report what is found. According to its creators, the app is "an attractor of strange things," letting one choose specific coordinates based on a specific theme. It gained controversy after a report of two teenagers coincidentally finding a corpse while using the application. == Overview == The app, which creators claim to be inspired by chaos theory and Guy Debord's Theory of the Dérive, offers its users three types of coordinates to choose from: an attractor, a void, or an anomaly. The app has a cult following on YouTube and TikTok and there is a subreddit made by the creators for users of the app. == History == 29-year-old circus performer Joshua Lengfelder discovered a bot called Fatum Project in a fringe science chat group on Telegram in January 2019. According to The New York Times, "He absorbed the project’s theories about how random exploration could break people out of their predetermined realities, and how people could influence random outcomes with their minds." Lengfelder then created a Telegram bot using Fatum Project's code, generating coordinates. He then created the subreddit r/randonauts in March. In October, developer Simon Nishi McCorkindale made the bot's webpage. With the help of Auburn Salcedo, chief executive of a TV agency, both created Randonauts LLC. Salcedo became the chief operating officer while Lengfelder was the CEO. The app, called Randonautica, was launched on February 22, 2020. Later the same year the app and back-end got completely overhauled by a new team of developers and got a more visual and friendlier design and logo. In April 2022 Lengfelder exited Randonauts LLC and Auburn Salcedo became CEO. == Reception == The app has as many as 10.8 million users as of July 2020, gaining popularity amid the COVID-19 pandemic in the United States as restrictions have been lightened. Emma Chamberlain made a YouTube video about the app that helped increase its following. i-D reported that the hashtag #randonautica has gained 176.5 million views on TikTok, although it has not marketed itself yet. === Controversy === With the app's popularity, users started reporting coincidences which many find unsettling. The majority of reports were from TikTok and Reddit, as well as Telegram. The most notable controversy involved a group of people heading to a beach in Duwamish Head, Puget Sound, West Seattle per the app, where they found a bag with two dead bodies, a 27-year-old male and a 36-year-old female, as reported by the Seattle Police homicide detectives. In August 2020, police arrested and charged their landlord, Michael Lee Dudley, in connection with the murders. In March 2021, Dudley was denied bail while other people were under suspicion of aiding Dudley in the dismemberment and disposal of the bodies, but no one else had been charged. This has caused speculation that the app has an intended, puzzle-like theme. However, Lengfelder stated that it is "a shocking coincidence." Salcedo called the videos fake, and that "It’s so hard to manage, because people are really taking creative liberties after seeing how much traction the app is getting in that fear factor." In 2022, Michael Dudley was convicted of second degree murder for killing both victims, who were identified as Jessica Lewis and Austin Wenner. He was sentenced to 46 years in prison the following year. In their questions page, Randonautica's creators have said that if the app generates coordinates inside a private property, it is a violation of their terms and conditions to trespass. In addition, Randonautica has also received allegations that the app is used for human trafficking, which its creators have denied, saying that data collected by the app are anonymous. It also ensured that the app is not designed to violate religious customs, saying that "the app is simply a tool. Just as a knife can be used either to prepare dinner or to cut somebody."
Public computer
A public computer (or public access computer) is any of various computers available in public areas. Some places where public computers may be available are libraries, schools, or dedicated facilities run by government. Public computers share similar hardware and software components to personal computers, however, the role and function of a public access computer is entirely different. A public access computer is used by many different untrusted individuals throughout the course of the day. The computer must be locked down and secure against both intentional and unintentional abuse. Users typically do not have authority to install software or change settings. A personal computer, in contrast, is typically used by a single responsible user, who can customize the machine's behavior to their preferences. Public access computers are often provided with tools such as a PC reservation system to regulate access. The world's first public access computer center was the Marin Computer Center in California, co-founded by David and Annie Fox in 1977. == Kiosks == A kiosk is a special type of public computer using software and hardware modifications to provide services only about the place the kiosk is in. For example, a movie ticket kiosk can be found at a movie theater. These kiosks are usually in a secure browser with zero access to the desktop. Many of these kiosks may run Linux, however, ATMs, a kiosk designed for depositing money, often run Windows XP. == Public computers in the United States == === Library computers === In the United States and Canada, almost all public libraries have computers available for the use of patrons, though some libraries will impose a time limit on users to ensure others will get a turn and keep the library less busy. Users are often allowed to print documents that they have created using these computers, though sometimes for a small fee. ==== Privacy ==== Privacy is an important part of the public library institution, since the libraries entitle the public to intellectual freedom. Use of any computer or network may create records of users' activities that can jeopardize their privacy. It is possible for a patron to jeopardize their privacy if they do not delete cache, clear cookies, or documents from the public computer. In order for a member of the public to remain private on a computer, the American Library Association (ALA) has guidelines. These give patrons an idea of the right way to keep using public library computers. In their provision of services to library users, librarians have an ethical responsibility, expressed in the ALA Code of Ethics, to preserve users' right to privacy. A librarian is also responsible for giving users an understanding of private patron use and access. Libraries must ensure that users have the following rights when browsing on public computers: the computer automatically will clear a users history; libraries should display privacy screens so users do not see another patron's screen; updating software for effective safety measures; restoration data software to clear documents that users may have left on their computers and to combat possible malware; security practices; and making users aware of any possible monitoring of their browsing activities. Users can also view the Library Privacy Checklist for Public Access Computers and Networks to better understand what libraries strive for when protecting privacy. === School computers === The U.S. government has given money to many school boards to purchase computers for educational applications. Schools may have multiple computer labs, which contain these computers for students to use. There is usually Internet access on these machines, but some schools will put up a blocking service to limit the websites that students are able to access to only include educational resources, such as Google. In addition to controlling the content students are viewing, putting up these blocks can also help to keep the computers safe by preventing students from downloading malware and other threats. However, the effectiveness of such content filtering systems is questionable since it can easily be circumvented by using proxy websites, Virtual Private Networks, and for some weak security systems, merely knowing the IP address of the intended website is enough to bypass the filter. School computers often have advanced operating system security to prevent tech-savvy students from inflicting damage (i.e. the Windows Registry Editor and Task Manager, etc.) are disabled on Microsoft Windows machines. Schools with very advanced tech services may also install a locked down BIOS/firmware or make kernel-level changes to the operating system, precluding the possibility of unauthorized activity.
BevQ
BevQ is a queue management mobile application developed by Faircode Technologies of Kochi, Kerala. It is provided by the Kerala State Beverages Corporation under Government of Kerala. == History == This app was released together by the Government of Kerala and the Kerala State Beverages Corporation in order to implement social distancing in the liquor stores Kerala in the case of the COVID-19 pandemic in Kerala and to reduce the congestion of people. The BevQ App was released by Faircode Technologies on 27 May 2020 on the Google Play Store. In January 2021, the app was withdrawn as bars had opened. In June 2021, there was a commitment from the Kerala CM that the App will be relaunched again. It has been reported that over 132,000 new users downloaded the app in the 48 hours after the announcement. == Achievements == The BEVQ app, which works only in the state of Kerala, beat all other Indian food and drink apps in 2020 to see the highest growth in year-on-year sessions, according to the State of Mobile 2021 report by App Annie. The app even beat the likes of Domino’s, which is used all across India. Around 300 government Liquor shops and 900 private liquor shops were enlisted in the platform. More than 200 million unique users registered in the platform. About 250,000 tokens were given out a day.
Space partitioning
In geometry, space partitioning is the process of dividing an entire space (usually a Euclidean space) into two or more disjoint subsets (see also partition of a set). In other words, space partitioning divides a space into non-overlapping regions. Any point in the space can then be identified to lie in exactly one of the regions. == Overview == Space-partitioning systems are often hierarchical, meaning that a space (or a region of space) is divided into several regions, and then the same space-partitioning system is recursively applied to each of the regions thus created. The regions can be organized into a tree, called a space-partitioning tree. Most space-partitioning systems use planes (or, in higher dimensions, hyperplanes) to divide space: points on one side of the plane form one region, and points on the other side form another. Points exactly on the plane are usually arbitrarily assigned to one or the other side. Recursively partitioning space using planes in this way produces a BSP tree, one of the most common forms of space partitioning. == Uses == === In computer graphics === Space partitioning is particularly important in computer graphics, especially heavily used in ray tracing, where it is frequently used to organize the objects in a virtual scene. A typical scene may contain millions of polygons. Performing a ray/polygon intersection test with each would be a very computationally expensive task. Storing objects in a space-partitioning data structure (k-d tree or BSP tree for example) makes it easy and fast to perform certain kinds of geometry queries—for example in determining whether a ray intersects an object, space partitioning can reduce the number of intersection test to just a few per primary ray, yielding a logarithmic time complexity with respect to the number of polygons. Space partitioning is also often used in scanline algorithms to eliminate the polygons out of the camera's viewing frustum, limiting the number of polygons processed by the pipeline. There is also a usage in collision detection: determining whether two objects are close to each other can be much faster using space partitioning. === In integrated circuit design === In integrated circuit design, an important step is design rule check. This step ensures that the completed design is manufacturable. The check involves rules that specify widths and spacings and other geometry patterns. A modern design can have billions of polygons that represent wires and transistors. Efficient checking relies heavily on geometry query. For example, a rule may specify that any polygon must be at least n nanometers from any other polygon. This is converted into a geometry query by enlarging a polygon by n/2 at all sides and query to find all intersecting polygons. === In probability and statistical learning theory === The number of components in a space partition plays a central role in some results in probability theory. See Growth function for more details. === In geography and GIS === There are many studies and applications where Geographical Spatial Reality is partitioned by hydrological criteria, administrative criteria, mathematical criteria or many others. In the context of cartography and GIS - Geographic Information System, is common to identify cells of the partition by standard codes. For example the for HUC code identifying hydrographical basins and sub-basins, ISO 3166-2 codes identifying countries and its subdivisions, or arbitrary DGGs - discrete global grids identifying quadrants or locations. == Data structures == Common space-partitioning systems include: BSP trees Quadtrees Octrees k-d trees Bins == Number of components == Suppose the n-dimensional Euclidean space is partitioned by r {\displaystyle r} hyperplanes that are ( n − 1 ) {\displaystyle (n-1)} -dimensional. What is the number of components in the partition? The largest number of components is attained when the hyperplanes are in general position, i.e, no two are parallel and no three have the same intersection. Denote this maximum number of components by C o m p ( n , r ) {\displaystyle Comp(n,r)} . Then, the following recurrence relation holds: C o m p ( n , r ) = C o m p ( n , r − 1 ) + C o m p ( n − 1 , r − 1 ) {\displaystyle Comp(n,r)=Comp(n,r-1)+Comp(n-1,r-1)} C o m p ( 0 , r ) = 1 {\displaystyle Comp(0,r)=1} - when there are no dimensions, there is a single point. C o m p ( n , 0 ) = 1 {\displaystyle Comp(n,0)=1} - when there are no hyperplanes, all the space is a single component. And its solution is: C o m p ( n , r ) = ∑ k = 0 n ( r k ) {\displaystyle Comp(n,r)=\sum _{k=0}^{n}{r \choose k}} if r ≥ n {\displaystyle r\geq n} C o m p ( n , r ) = 2 r {\displaystyle Comp(n,r)=2^{r}} if r ≤ n {\displaystyle r\leq n} (consider e.g. r {\displaystyle r} perpendicular hyperplanes; each additional hyperplane divides each existing component to 2). which is upper-bounded as: C o m p ( n , r ) ≤ r n + 1 {\displaystyle Comp(n,r)\leq r^{n}+1}