title: Asynchronous Transfer Mode

text: ATM uses the PT field to designate various special kinds of cells for operations, administration and management (OAM) purposes, and to delineate packet boundaries in some ATM adaptation layers (AAL). If the most significant bit (MSB) of the PT field is 0, this is a user data cell, and the other two bits are used to indicate network congestion and as a general purpose header bit available for ATM adaptation layers. If the MSB is 1, this is a management cell, and the other two bits indicate the type. (Network management segment, network management end-to-end, resource management, and reserved for future use.)


title: Asynchronous Transfer Mode

text: Several ATM link protocols use the HEC field to drive a CRC-based framing algorithm, which allows locating the ATM cells with no overhead beyond what is otherwise needed for header protection. The 8-bit CRC is used to correct single-bit header errors and detect multi-bit header errors. When multi-bit header errors are detected, the current and subsequent cells are dropped until a cell with no header errors is found.


title: Asynchronous Transfer Mode

text: A UNI cell reserves the GFC field for a local flow control/submultiplexing system between users. This was intended to allow several terminals to share a single network connection, in the same way that two Integrated Services Digital Network (ISDN) phones can share a single basic rate ISDN connection. All four GFC bits must be zero by default.


title: Asynchronous Transfer Mode

text: The NNI cell format replicates the UNI format almost exactly, except that the 4-bit GFC field is re-allocated to the VPI field, extending the VPI to 12 bits. Thus, a single NNI ATM interconnection is capable of addressing almost 2 VPs of up to almost 2 VCs each (in practice some of the VP and VC numbers are reserved).


title: Asynchronous Transfer Mode

text: ATM supports different types of services via AALs. Standardized AALs include AAL1, AAL2, and AAL5, and the rarely used AAL3 and AAL4. AAL1 is used for constant bit rate (CBR) services and circuit emulation. Synchronization is also maintained at AAL1. AAL2 through AAL4 are used for variable bitrate (VBR) services, and AAL5 for data. Which AAL is in use for a given cell is not encoded in the cell. Instead, it is negotiated by or configured at the endpoints on a per-virtual-connection basis.


title: Asynchronous Transfer Mode

text: Following the initial design of ATM, networks have become much faster. A 1500 byte (12000-bit) full-size Ethernet frame takes only 1.2 µs to transmit on a 10 Gbit/s network, reducing the need for small cells to reduce jitter due to contention. The increased link speeds by themselves do not alleviate jitter due to queuing. Additionally, the hardware for implementing the service adaptation for IP packets is expensive at very high speeds.


title: Asynchronous Transfer Mode

text: ATM provides a useful ability to carry multiple logical circuits on a single physical or virtual medium, although other techniques exist, such as Multi-link PPP, Ethernet VLANs, and multi-protocol support over SONET.


title: Asynchronous Transfer Mode

text: A network must establish a connection before two parties can send cells to each other. In ATM this is called a virtual circuit (VC). It can be a permanent virtual circuit (PVC), which is created administratively on the end points, or a switched virtual circuit (SVC), which is created as needed by the communicating parties. SVC creation is managed by signaling, in which the requesting party indicates the address of the receiving party, the type of service requested, and whatever traffic parameters may be applicable to the selected service. "Call admission" is then performed by the network to confirm that the requested resources are available and that a route exists for the connection.


title: Asynchronous Transfer Mode

text: ATM operates as a channel-based transport layer, using VCs. This is encompassed in the concept of the virtual paths (VP) and virtual channels. Every ATM cell has an 8- or 12-bit virtual path identifier (VPI) and 16-bit virtual channel identifier (VCI) pair defined in its header. The VCI, together with the VPI, is used to identify the next destination of a cell as it passes through a series of ATM switches on its way to its destination. The length of the VPI varies according to whether the cell is sent on the user-network interface (on the edge of the network), or if it is sent on the network-network interface (inside the network).


title: Asynchronous Transfer Mode

text: As these cells traverse an ATM network, switching takes place by changing the VPI/VCI values (label swapping). Although the VPI/VCI values are not necessarily consistent from one end of the connection to the other, the concept of a circuit "is" consistent (unlike IP, where any given packet could get to its destination by a different route than the others). ATM switches use the VPI/VCI fields to identify the virtual channel link (VCL) of the next network that a cell needs to transit on its way to its final destination. The function of the VCI is similar to that of the data link connection identifier (DLCI) in Frame Relay and the logical channel number and logical channel group number in X.25.


title: Asynchronous Transfer Mode

text: Another advantage of the use of virtual circuits comes with the ability to use them as a multiplexing layer, allowing different services (such as voice, Frame Relay, n* 64 channels, IP). The VPI is useful for reducing the switching table of some virtual circuits which have common paths.


title: Asynchronous Transfer Mode

text: ATM can build virtual circuits and virtual paths either statically or dynamically. Static circuits (permanent virtual circuits or PVCs) or paths (permanent virtual paths or PVPs) require that the circuit is composed of a series of segments, one for each pair of interfaces through which it passes.


title: Asynchronous Transfer Mode

text: PVPs and PVCs, though conceptually simple, require significant effort in large networks. They also do not support the re-routing of service in the event of a failure. Dynamically built PVPs (soft PVPs or SPVPs) and PVCs (soft PVCs or SPVCs), in contrast, are built by specifying the characteristics of the circuit (the service "contract") and the two endpoints.


title: Asynchronous Transfer Mode

text: ATM networks create and remove switched virtual circuits (SVCs) on demand when requested by an end piece of equipment. One application for SVCs is to carry individual telephone calls when a network of telephone switches are inter-connected using ATM. SVCs were also used in attempts to replace local area networks with ATM.


title: Asynchronous Transfer Mode

text: Most ATM networks supporting SPVPs, SPVCs, and SVCs use the Private Network Node Interface or the Private Network-to-Network Interface (PNNI) protocol to share topology information between switches and select a route through a network. PNNI is a link-state routing protocol like OSPF and IS-IS. PNNI also includes a very powerful route summarization mechanism to allow construction of very large networks, as well as a call admission control (CAC) algorithm which determines the availability of sufficient bandwidth on a proposed route through a network in order to satisfy the service requirements of a VC or VP.


title: Asynchronous Transfer Mode

text: Another key ATM concept involves the traffic contract. When an ATM circuit is set up each switch on the circuit is informed of the traffic class of the connection.


title: Asynchronous Transfer Mode

text: ATM traffic contracts form part of the mechanism by which "quality of service" (QoS) is ensured. There are four basic types (and several variants) which each have a set of parameters describing the connection.


title: Asynchronous Transfer Mode

text: VBR has real-time and non-real-time variants, and serves for "bursty" traffic. Non-real-time is sometimes abbreviated to vbr-nrt.


title: Asynchronous Transfer Mode

text: Most traffic classes also introduce the concept of Cell-delay variation tolerance (CDVT), which defines the "clumping" of cells in time.


title: Asynchronous Transfer Mode

text: To maintain network performance, networks may apply traffic policing to virtual circuits to limit them to their traffic contracts at the entry points to the network, i.e. the user–network interfaces (UNIs) and network-to-network interfaces (NNIs): usage/network parameter control (UPC and NPC). The reference model given by the ITU-T and ATM Forum for UPC and NPC is the generic cell rate algorithm (GCRA), which is a version of the leaky bucket algorithm. CBR traffic will normally be policed to a PCR and CDVt alone, whereas VBR traffic will normally be policed using a dual leaky bucket controller to a PCR and CDVt and an SCR and Maximum Burst Size (MBS). The MBS will normally be the packet (SAR-SDU) size for the VBR VC in cells.


title: Asynchronous Transfer Mode

text: If the traffic on a virtual circuit is exceeding its traffic contract, as determined by the GCRA, the network can either drop the cells or mark the Cell Loss Priority (CLP) bit (to identify a cell as potentially redundant). Basic policing works on a cell by cell basis, but this is sub-optimal for encapsulated packet traffic (as discarding a single cell will invalidate the whole packet). As a result, schemes such as partial packet discard (PPD) and early packet discard (EPD) have been created that will discard a whole series of cells until the next packet starts. This reduces the number of useless cells in the network, saving bandwidth for full packets. EPD and PPD work with AAL5 connections as they use the end of packet marker: the ATM user-to-ATM user (AUU) indication bit in the payload-type field of the header, which is set in the last cell of a SAR-SDU.


title: Asynchronous Transfer Mode

text: Traffic shaping usually takes place in the network interface card (NIC) in user equipment, and attempts to ensure that the cell flow on a VC will meet its traffic contract, i.e. cells will not be dropped or reduced in priority at the UNI. Since the reference model given for traffic policing in the network is the GCRA, this algorithm is normally used for shaping as well, and single and dual leaky bucket implementations may be used as appropriate.


title: Asynchronous Transfer Mode

text: The ATM network reference model approximately maps to the three lowest layers of the OSI reference model. It specifies the following layers:


title: Asynchronous Transfer Mode

text: ATM became popular with telephone companies and many computer makers in the 1990s. However, even by the end of the decade, the better price/performance of Internet Protocol-based products was competing with ATM technology for integrating real-time and bursty network traffic. Companies such as FORE Systems focused on ATM products, while other large vendors such as Cisco Systems provided ATM as an option. After the burst of the dot-com bubble, some still predicted that "ATM is going to dominate". However, in 2005 the ATM Forum, which had been the trade organization promoting the technology, merged with groups promoting other technologies, and eventually became the Broadband Forum.


title: Asynchronous Transfer Mode

text: Wireless ATM, or mobile ATM, consists of an ATM core network with a wireless access network. ATM cells are transmitted from base stations to mobile terminals. Mobility functions are performed at an ATM switch in the core network, known as "crossover switch", which is similar to the MSC (mobile switching center) of GSM networks. The advantage of wireless ATM is its high bandwidth and high speed handoffs done at layer 2. In the early 1990s, Bell Labs and NEC research labs worked actively in this field. Andy Hopper from the University of Cambridge Computer Laboratory also worked in this area. There was a wireless ATM forum formed to standardize the technology behind wireless ATM networks. The forum was supported by several telecommunication companies, including NEC, Fujitsu and AT&T. Mobile ATM aimed to provide high speed multimedia communications technology, capable of delivering broadband mobile communications beyond that of GSM and WLANs.


title: Iapetus

text: In Greek mythology, Iapetus (; ; ), also Japetus, is a Titan, the son of Uranus and Gaia and father of Atlas, Prometheus, Epimetheus, and Menoetius. He was also called the father of Buphagus and Anchiale in other sources.


title: Iapetus

text: Iapetus was linked to Japheth (יֶפֶת) one of the sons of Noah and a progenitor of mankind in biblical accounts. The practice by early historians and biblical scholars of identifying various historical nations and ethnic groups as descendants of Japheth, together with the similarity of their names, led to a fusion of their identities, from the early modern period to the present.


title: Iapetus

text: Iapetus ("the Piercer") is the one Titan mentioned by Homer in the "Iliad" as being in Tartarus with Cronus. He is a brother of Cronus, who ruled the world during the Golden Age but is now locked up in Tartarus along with Iapetus, where neither breeze nor light of the sun reaches them.


title: Iapetus

text: Iapetus' wife is usually described as a daughter of Oceanus and Tethys named either Clymene (according to Hesiod and Hyginus) or Asia (according to Apollodorus).


title: Iapetus

text: In Hesiod's "Works and Days" Prometheus is addressed as "son of Iapetus", and no mother is named. However, in Hesiod's "Theogony", Clymene is listed as Iapetus' wife and the mother of Prometheus. In Aeschylus's play "Prometheus Bound", Prometheus is son of the goddess Themis with no father named (but still with at least Atlas as a brother). However, in Horace's Odes, in Ode 1.3 Horace writes "audax Iapeti genus ... Ignem fraude mala gentibus intulit" ("The bold offspring of Iapetus [i.e. Prometheus] ... brought fire to peoples by wicked deceit").


title: Iapetus

text: Hesiod and other Greek scholars regarded the sons of Iapetus as mankind's ancestors and as such, some of humanity's worst qualities were said to have been inherited from these four gods, each of whom were described with a particular moral fault that often led to their own downfall. For instance, sly and clever Prometheus could perhaps represent crafty scheming; the inept and guileless Epimetheus, foolish stupidity; the enduring, strongest and powerful Atlas, excessive daring; and the arrogant Menoetius, rash violence.


title: Iapetus

text: Iapetus as the progenitor of mankind has been equated with Japheth (), the son of Noah, based on the similarity of their names and the tradition, reported by Josephus ("Antiquities of the Jews"), which made Japheth the ancestor of the "Japhetites", i.e. the Indo-European speaking peoples. Iapetus was linked to Japheth by 17th-century theologian Matthew Poole (and more recently by Robert Graves) and by John Pairman Brown.


title: Erythema multiforme

text: Erythema multiforme (EM) is a skin condition that appears with red patches evolving into target lesions, typically on both hands.


title: Erythema multiforme

text: It is a type of erythema possibly mediated by deposition of immune complexes (mostly IgM-bound complexes) in the superficial microvasculature of the skin and oral mucous membrane that usually follows an infection or drug exposure. It is an uncommon disorder, with peak incidence in the second and third decades of life. The disorder has various forms or presentations, which its name reflects ("multiforme", "multiform", from "multi-" + "formis"). Target lesions are a typical manifestation. Two types, one mild to moderate and one severe, are recognized (erythema multiforme minor and erythema multiforme major).


title: Erythema multiforme

text: The condition varies from a mild, self-limited rash (E. multiforme minor) to a severe, life-threatening form known as erythema multiforme major (or erythema multiforme majus) that also involves mucous membranes.


title: Erythema multiforme

text: Stevens–Johnson syndrome and toxic epidermal necrolysis used to be considered part of the erythema multiforme spectrum, but that is no longer the case.


title: Erythema multiforme

text: The mild form usually presents with mildly itchy (but itching can be very severe), pink-red blotches, symmetrically arranged and starting on the extremities. It often takes on the classical "target lesion" appearance, with a pink-red ring around a pale center. Resolution within 7–10 days is the norm.


title: Erythema multiforme

text: Individuals with persistent (chronic) erythema multiforme will often have a lesion form at an injury site, e.g. a minor scratch or abrasion, within a week. Irritation or even pressure from clothing will cause the erythema sore to continue to expand along its margins for weeks or months, long after the original sore at the center heals.


title: Erythema multiforme

text: EM minor is regarded as being triggered by HSV in almost all cases. A herpetic etiology also accounts for 55% of cases of EM major. Among the other infections, "Mycoplasma" infection appears to be a common cause.


title: Erythema multiforme

text: Herpes simplex virus suppression and even prophylaxis (with acyclovir) has been shown to prevent recurrent erythema multiforme eruption.


title: Erythema multiforme

text: Erythema multiforme is frequently self-limiting and requires no treatment. The appropriateness of glucocorticoid therapy can be uncertain, because it is difficult to determine if the course will be a resolving one.


title: Social identity theory

text: Social identity is the portion of an individual's self-concept derived from perceived membership in a relevant social group.


title: Social identity theory

text: As originally formulated by social psychologists Henri Tajfel and John Turner in the 1970s and the 1980s, social identity theory introduced the concept of a social identity as a way in which to explain intergroup behaviour. "Social identity theory explores the phenomenon of the 'ingroup' and 'outgroup', and is based on the view that identities are constituted through a process of difference defined in a relative or flexible way depends on the activities in which one engages" This theory is described as a theory that predicts certain intergroup behaviours on the basis of perceived group status differences, the perceived legitimacy and stability of those status differences, and the perceived ability to move from one group to another. This contrasts with occasions where the term "social identity theory" is used to refer to general theorizing about human social selves. Moreover, and although some researchers have treated it as such, social identity theory was never intended to be a general theory of social categorization. It was awareness of the limited scope of social identity theory that led John Turner and colleagues to develop a cousin theory in the form of self-categorization theory, which built on the insights of social identity theory to produce a more general account of self and group processes.


title: Social identity theory

text: The term social identity "approach", or social identity "perspective", is suggested for describing the joint contributions of both social identity theory and self-categorization theory. Social identity theory suggests that an organization can change individual behaviours if it can modify their self-identity or part of their self-concept that derives from the knowledge of, and emotional attachment to the group.


title: Social identity theory

text: The term 'social identity theory' achieved academic currency only in the late 1970s, but the basic underlying concepts associated with it had emerged by the early twentieth century. William G. Sumner, writing in 1906, captures the primary dynamics in this excerpt from his influential work "Folkways: A Study of the Sociological Importance of Usages, Manners, Customs, Mores, and Morals":


title: Social identity theory

text: By the late 1920s the collectivist perspective had all but disappeared from mainstream social psychology. Over fifty years later, around the time of the first formal use of the term 'social identity theory', Tajfel wrote this on the state of social psychology:


title: Social identity theory

text: Thus, social identity theory in part reflects a desire to reestablish a more collectivist approach to social psychology of the self and social groups.


title: Social identity theory

text: Social identity theory states that social behaviour will want a person to change their behaviour while in a group. It varies along a continuum between interpersonal behaviour and intergroup behaviour. Completely interpersonal behaviour would be behaviour determined solely by the individual characteristics and interpersonal relationships that exists between only two people. Completely intergroup behaviour would be behaviour determined solely by the social category memberships that apply to more than two people. The authors of social identity theory state that purely interpersonal or purely intergroup behaviour is unlikely to be found in realistic social situations. Rather, behaviour is expected to be driven by a compromise between the two extremes. The cognitive nature of personal vs. social identities, and the relationship between them, is more fully developed in self-categorization theory. Social identity theory instead focuses on the social structural factors that will predict which end of the spectrum will most influence an individual's behaviour, along with the forms that the behaviour may take.


title: Social identity theory

text: A key assumption in social identity theory is that individuals are intrinsically motivated to achieve positive distinctiveness. That is, individuals "strive for a positive self-concept". As individuals to varying degrees may be defined and informed by their respective social identities (as per the interpersonal-intergroup continuum) it is further derived in social identity theory that "individuals strive to achieve or to maintain positive social identity". The precise nature of this striving for positive self-concept is a matter of debate (see the self-esteem hypothesis). Both the interpersonal-intergroup continuum and the assumption of positive distinctiveness motivation arose as outcomes of the findings of minimal group studies. In particular, it was found that under certain conditions individuals would endorse resource distributions that would maximize the positive distinctiveness of an in-group in contrast top an out-group at the expense of personal self-interest.


title: Social identity theory

text: Building on the above components, social identity theory details a variety of strategies that may be invoked in order to achieve positive distinctiveness. The individual's choice of behaviour is posited to be dictated largely by the perceived intergroup relationship. In particular the choice of strategy is an outcome of the perceived permeability of group boundaries (e.g., whether a group member may pass from a low status group into a high status group), as well as the perceived stability and legitimacy of the intergroup status hierarchy. The self-enhancing strategies detailed in social identity theory are detailed below. Importantly, although these are viewed from the perspective of a low status group member, comparable behaviours may also be adopted by high status group members.


title: Social identity theory

text: It is predicted that under conditions where the group boundaries are considered permeable individuals are more likely to engage in individual mobility strategies. That is, individuals "disassociate from the group and pursue individual goals designed to improve their personal lot rather than that of their ingroup".


title: Social identity theory

text: Where group boundaries are considered impermeable, and where status relations are considered reasonably stable, individuals are predicted to engage in social creativity behaviours. Here, low-status ingroup members are still able to increase their positive distinctiveness without necessarily changing the objective resources of the ingroup or the outgroup. This may be achieved by comparing the ingroup to the outgroup on some new dimension, changing the values assigned to the attributes of the group, and choosing an alternative outgroup by which to compare the ingroup.


title: Social identity theory

text: Here an ingroup seeks positive distinctiveness and requires positive differentiation via direct competition with the outgroup in the form of ingroup favoritism. It is considered competitive in that in this case favoritism for the ingroup occurs on a value dimension that is shared by all relevant social groups (in contrast to social creativity scenarios). Social competition is predicted to occur when group boundaries are considered impermeable, and when status relations are considered to be reasonably unstable. Although not privileged in the theory, it is this positive distinctiveness strategy that has received the greatest amount of attention.


title: Social identity theory

text: In political science, social identity theory has been incorporated as the subconsitituency politics theory of representation. This theory holds that political elites are individually rational, and they use identity instrumentally to cultivate minimum winning constituencies (e.g., via the "microtargeting" of ads). An example of microtargeting is Russian use of social media advertising alleged to have influenced the 2016 presidential election. Separately, a recent "Science Advances" article validates a computational model of in-group favoritism and political economy developed by Princeton political scientist Nolan McCarty using public opinion polling data.


title: Social identity theory

text: In-group favoritism (also known as "ingroup bias", despite Turner's objections to the term) is an effect where people give preferential treatment to others when they are perceived to be in the same ingroup. Social identity attributes the cause of ingroup favoritism to a psychological need for positive distinctiveness and describes the situations where ingroup favoritism is likely to occur (as a function of perceived group status, legitimacy, stability, and permeability). It has been shown via the minimal group studies that ingroup favoritism may occur for both arbitrary ingroups (e.g. a coin toss may split participants into a 'heads' group and a 'tails' group) as well as non-arbitrary ingroups (e.g. ingroups based on cultures, genders, sexual orientation, and first languages).


title: Social identity theory

text: Continued study into the relationship between social categorization and ingroup favoritism has explored the relative prevalences of the ingroup favoritism vs. outgroup discrimination, explored different manifestations of ingroup favoritism, and has explored the relationship between ingroup favoritism and other psychological constraints (e.g., existential threat).


title: Social identity theory

text: System justification theory was originally proposed by John Jost and Mahzarin Banaji in 1994 to build on social identity theory and to understand important deviations from ingroup favoritism, such as outgroup favoritism on the part of members of disadvantaged groups (Jost & Banaji, 1994; Jost, 2020).


title: Social identity theory

text: Social identification can lead individuals to engage in prosocial behaviours towards others. Examples include contexts such as food drives or even shared purchasing patterns, as might occur for motorcycle riders. Interestingly, consumers may have sub-identities that are nested into a larger identity. As a result, "[w]hen consumers identify with the overall community, they assist other consumers. However, consumers are less likely to help consumers in the overall community when identifying with a subgroup".


title: Social identity theory

text: Social identities are a valued aspect of the self, and people will sacrifice their pecuniary self-interest to maintain the self-perception that they belong to a given social group. Political partisans and fans of sports teams (e.g., Republicans and Democrats, or MLB, NFL, NCAA fans) are reluctant to bet against the success of their party or team because of the diagnostic cost such a bet would incur to their identification with it. As a result, partisans and fans will reject even very favorable bets against identity-relevant desired outcomes. More than 45% of N.C.A.A. basketball and hockey fans, for example, turned down a free, real chance to earn $5 if their team lost its upcoming game.


title: Social identity theory

text: Social identity theory proposes that people are motivated to achieve and maintain positive concepts of themselves. Some researchers, including Michael Hogg and Dominic Abrams, thus propose a fairly direct relationship between positive social identity and self-esteem. In what has become known as the "self-esteem hypothesis", self-esteem is predicted to relate to in-group bias in two ways. Firstly, successful intergroup discrimination elevates self-esteem. Secondly, depressed or threatened self-esteem promotes intergroup discrimination. Empirical support for these predictions has been mixed.


title: Social identity theory

text: Some social identity theorists, including John Turner, consider the self-esteem hypothesis as not canonical to social identity theory. In fact, the self-esteem hypothesis is argued to be conflictual with the tenets of the theory. It is argued that the self-esteem hypothesis misunderstands the distinction between a social identity and a personal identity. Along those lines, John Turner and Penny Oakes argue against an interpretation of positive distinctiveness as a straightforward need for self-esteem or "quasi-biological drive toward prejudice". They instead favour a somewhat more complex conception of positive self-concept as a reflection of the ideologies and social values of the perceiver. Additionally, it is argued that the self-esteem hypothesis neglects the alternative strategies to maintaining a positive self-concept that are articulated in social identity theory (i.e., individual mobility and social creativity).


title: Social identity theory

text: In what has been dubbed the Positive-Negative Asymmetry Phenomenon, researchers have shown that punishing the out-group benefits self-esteem less than rewarding the in-group. From this finding it has been extrapolated that social identity theory is therefore unable to deal with bias on negative dimensions. Social identity theorists, however, point out that for ingroup favouritism to occur a social identity "must be psychologically salient", and that negative dimensions may be experienced as a "less fitting basis for self-definition". This important qualification is subtly present in social identity theory, but is further developed in self-categorization theory. Empirical support for this perspective exist. It has been shown that when experiment participants can self-select negative dimensions that define the ingroup no positive–negative asymmetry is found.


title: Social identity theory

text: It has been posited that social identity theory suggests that similar groups should have an increased motivation to differentiate themselves from each other. Subsequently, empirical findings where similar groups are shown to possess increased levels of intergroup attraction and decreased levels of in-group bias have been interpreted as problematic for the theory. Elsewhere it has been suggested that this apparent inconsistency may be resolved by attending to social identity theory's emphasis on the importance of the perceived stability and legitimacy of the intergroup status hierarchy.


title: Social identity theory

text: Social identity theory has been criticised for having far greater explanatory power than predictive power. That is, while the relationship between independent variables and the resulting intergroup behaviour may be consistent with the theory in retrospect, that particular outcome is often not that which was predicted at the outset. A rebuttal to this charge is that the theory was never advertised as the definitive answer to understanding intergroup relationships. Instead it is stated that social identity theory must go hand in hand with sufficient understanding of the specific social context under consideration. The latter argument is consistent with the explicit importance that the authors of social identity theory placed on the role of "objective" factors, stating that in any particular situation "the effects of [social identity theory] variables are powerfully determined by the previous social, economic, and political processes".


title: Social identity theory

text: Some researchers interpret social identity theory as drawing a direct link between identification with a social group and ingroup favoritism. This is because social identity theory was proposed as a way of explaining the ubiquity of ingroup favoritism in the minimal group paradigm. For example, Charles Stangor and John Jost state that "a main premise of social identity theory is that ingroup members will favour their own group over other groups". This interpretation is rejected by other researchers. For example, Alex Haslam states that "although vulgarized versions of social identity theory argue that 'social identification leads automatically to discrimination and bias', in fact…discrimination and conflict are anticipated only in a limited set of circumstances". The likening of social identity theory with social competition and ingroup favouritism is partly attributable to the fact that early statements of the theory included empirical examples of ingroup favouritism, while alternative positive distinctiveness strategies (e.g., social creativity) were at that stage theoretical assertions. Regardless, in some circles the prediction of a straightforward identification-bias correlation has earned the pejorative title "social identity theory-lite". This raises the problem of whether social identity theory really does explain the ubiquity of ingroup favoritism in the minimal group paradigm without making recourse to "the generic norm hypothesis" originally proposed by Tajfel but later abandoned.


title: Black people in Japan

text: Yasuke, an African man, possibly from Mozambique, arrived in Japan in the late-16th century alongside Jesuit missionary Alessandro Valignano. He found favor with Oda Nobunaga, the daimyō and warlord, and ultimately became his retainer.


title: Black people in Japan

text: After WW2, with the Japanese economic miracle, many students from Africa began coming to Japan often to pursue relevant postgraduate education through MEXT and JICA. African Americans also joined the JET Programme to work as English teachers. Some African Americans arrive to serve in the United States Forces Japan.


title: Black people in Japan

text: In 2015, Ariana Miyamoto, who was born in Japan to a Japanese mother and an African-American father, became the first (meaning “mixed”) contestant to win the title of Miss Universe Japan. The decision to allow Miyamoto to win the title, due to her not being fully Japanese ethnically, was controversial.


title: Life After Life (novel)

text: Life After Life is a 2013 novel by Kate Atkinson. It is the first of two novels about the Todd family. The second, "A God in Ruins", was published in 2015. "Life After Life" garnered acclaim from critics.


title: Life After Life (novel)

text: The novel has an unusual structure, repeatedly looping back in time to describe alternative possible lives for its central character, Ursula Todd, who is born on 11 February 1910 to an upper-middle-class family near Chalfont St Peter in Buckinghamshire. In the first version, she is strangled by her umbilical cord and stillborn. In later iterations of her life she dies as a child - drowning in the sea, or when saved from that, by falling to her death from the roof when trying to retrieve a fallen doll. Then there are several sequences when she falls victim to the Spanish flu epidemic of 1918 - which repeats itself again and again, though she already has a foreknowledge of it, and only her fourth attempt to avert catching the flu succeeds.


title: Life After Life (novel)

text: Then there is an unhappy life where she is traumatised by being raped, getting pregnant and undergoing an illegal abortion, and finally becoming trapped in a highly oppressive marriage, and being killed by her abusive husband when trying to escape. In later lives she averts all this by being pre-emptively aggressive to the would-be rapist. In between, she also uses her half-memory of earlier lives to avert the young neighbour Nancy being raped and murdered by a child molester. The saved Nancy would play an important role in Ursula's later life(s), forming a deep love relationship with Ursula's brother Teddy, and would become a main character in the sequel, "A God in Ruins".


title: Life After Life (novel)

text: Still later iterations of Ursula's life take her into World War II, where she works in London for the War Office and repeatedly witnesses the results of the Blitz, including a direct hit on a bomb shelter in Argyll Road in November 1940 - with herself being among the victims in some lives and among the rescuers in others. There is also a life in which she marries a German in 1934, is unable to return to England and experiences the war in Berlin under the allied bombings.


title: Life After Life (novel)

text: Ursula eventually comes to realize, through a particularly strong sense of "deja vu," from her previous lives, and decides to try to prevent the war by killing Adolf Hitler in late 1930. Memory of her earlier lives also provides the means of doing that: the knowledge that by befriending Eva Braun - in 1930 an obscure shop girl in Munich - Ursula would be able to get close to Hitler with a loaded gun in her bag; the inevitable price, however, is to be herself shot dead by Hitler's Nazi followers immediately after killing him.


title: Life After Life (novel)

text: What is left unclear - since each of the time sequences end with "darkness" and Ursula's death and does not show what followed - is whether in fact all these lives actually occurred in an objective world, or were only subjectively experienced by her. Specifically it is not clear whether or not her killing Hitler in 1930 actually produced an altered timeline where the Nazis did not take power in Germany, or possibly took power under a different leader with a different course of the Second World War. Although in her 1967 incarnation Ursula speculates with her nephew on this "might have been", the book avoids giving a clear answer.


title: Life After Life (novel)

text: Alex Clark of "The Guardian" gave "Life After Life" a positive review, saying that domestic details of daily life are conveyed beautifully, and that traumatic shifts in British society are also captured well "precisely because she cuts directly from one war to the next, only later going back to fill in, partially, what happened in between." Clark argued that the novel "[co-opts] the family [...] and [uses] it to show how fiction works and what it might mean to us [...] with an emotional delicacy and understanding that transcend experiment or playfulness. "Life After Life" gives us a heroine whose fictional underpinning is permanently exposed, whose artificial status is never in doubt; and yet one who feels painfully, horribly real to us." "The Daily Telegraph"'s Helen Brown likewise praised it, calling it Atkinson's best book to date. "The Independent" found the central character to be sympathetic, and argued that the book's central message was that World War II was preventable and should not have been allowed to happen.


title: Life After Life (novel)

text: Janet Maslin of "The New York Times Book Review" praised "Life After Life" as Atkinson's "very best" book and "full of mind games, but they are purposeful rather than emptily playful. [...] this one connects its loose ends with facile but welcome clarity." She described it as having an "engaging cast of characters" and called the depiction of the British experience of World War II "gutsy and deeply disturbing, just as the author intends it to be." Francine Prose of "The New York Times" wrote that Atkinson "nimbly succeeds in keeping the novel from becoming confusing" and argued that the work "makes the reader acutely conscious of an author’s power: how much the novelist can do."


title: Life After Life (novel)

text: "The Wall Street Journal"'s Sam Sacks dubbed "Life After Life" a "formidable bid" for the Man Booker Prize (though the novel was ultimately not longlisted). He said the high-concept premise of "Ursula [contriving] to avoid the accident that previously killed her [...] blends uneasily with what is otherwise a deft and convincing portrayal of an English family's evolution across two world wars [...] all the other characters seem complexly armed with free will." He found the resolution related to the prologue as "rushed and anti-climactic". But Sacks also said that "she [brings] characters to life with enviable ease", referring to the erosion of Sylvie and Hugh's marriage as "poignantly charted". Also, like Maslin, he lauded the novella-length Blitz chapter as "gorgeous and nerve-racking".


title: Life After Life (novel)

text: In NPR, novelist Meg Wolitzer suggested that the book proves that "a fully-realised world" is more important to the success of a fiction work than the progression of its story, and dubbed it a "major, serious yet playfully experimental novel". She argued that by not choosing one path for Ursula, Atkinson "opened her novel outward, letting it breathe unrestricted".


title: Life After Life (novel)

text: "The Guardian"'s Sam Jordison expressed mixed feelings. He commended the depiction of Ursula and her family, and Atkinson's "fine storytelling and sharp eye for domestic detail". He argued, "There is real playfulness in these revisited moments and repetition never breeds dullness. Instead, we try to spot the differences and look for refractions of the same scene, considering the permutations of what is said and done. It can provide an enjoyable and interactive experience." He criticised the portions outside Britain, however, and said overall that the book has "an abundance of human warmth, but it just isn't convincing. There is much to enjoy – but not quite enough to admire."


title: Life After Life (novel)

text: In 2019, "Life After Life" was ranked by "The Guardian" as the 20th best book since 2000. It was written that the "dizzying fictional construction is grounded by such emotional intelligence that her heroine’s struggles always feel painfully, joyously real." The novel was 20th in "Paste"'s list of the 40 best novels of the 2010s, with Alexis Gunderson arguing, "No one gets to live as many lives and have as many second chances to get the next step right as protagonist Ursula Todd. But in a decade where the real world swung between wars and elections, there are few more clarifying literary escapes than "Life After Life". [...] Atkinson’s sage weaves a heartbreaking, frightening and beautiful journey that’s written with tenacity and grace."


title: Life After Life (novel)

text: It was listed as one of the decade's top 10 fiction works by "Time", where it was billed as "a defining account of wartime London, as Ursula experiences the devastation of the Blitz from various perspectives, highlighting the senselessness of bombing raids. The story of her multiple lives is both moving and lighthearted, filled with comic asides and evocative language about life’s many joys and sorrows." "Entertainment Weekly" ranked it second, with David Canfield arguing that "Life After Life" "seamlessly executes an idiosyncratic premise [...] and contains a seemingly endless capacity to surprise", but that it "will stand the test of time for its in-between moments — its portraits of wartime, its glimpses into small domestic worlds, its understanding of one woman’s life as filled with infinite possibilities." The novel was among the honourable mentions on the Literary Hub list of the 20 best novels of the decade.


title: Life After Life (novel)

text: It won the 2013 Costa Book Awards (Novel). It was shortlisted for the 2013 Baileys Women's Prize for Fiction, Waterstones Book of the Year (2013), and the Walter Scott Prize (2014). It was selected as one of the 10 Best Books of 2013 by the editors of the "New York Times Book Review", an ALA Notable Books for Adults (2014), The Morning News Tournament of Books (Zombie Selection and Finalist 2014), Goodreads Choice Awards (Historical Fiction 2013), Andrew Carnegie Medal longlist (2014), "The South Bank Show" Annual Award for Literature (2014).


title: Life After Life (novel)

text: The BBC commissioned a four-part adaptation of Atkinson's work in December 2020. In April 2021, it was announced that Thomasin McKenzie and Sian Clifford would star in the series alongside James McArdle, Jessica Brown Findlay, and Jessica Hynes with Patsy Ferran, Harry Michell, Laurie Kynaston, Joshua Hill, and Maria Laird completing the cast. The series began broadcast on 19 April 2022.


title: Fokker–Planck equation

text: In statistical mechanics, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as in Brownian motion. The equation can be generalized to other observables as well.


title: Fokker–Planck equation

text: It is named after Adriaan Fokker and Max Planck, who described it in 1914 and 1917. It is also known as the Kolmogorov forward equation, after Andrey Kolmogorov, who independently discovered it in 1931. When applied to particle position distributions, it is better known as the Smoluchowski equation (after Marian Smoluchowski), and in this context it is equivalent to the convection–diffusion equation. The case with zero diffusion is the continuity equation. The Fokker–Planck equation is obtained from the master equation through Kramers–Moyal expansion.


title: Fokker–Planck equation

text: The first consistent microscopic derivation of the Fokker–Planck equation in the single scheme of classical and quantum mechanics was performed by Nikolay Bogoliubov and Nikolay Krylov.


title: Fokker–Planck equation

text: In one spatial dimension "x", for an Itô process driven by the standard Wiener process formula_1 and described by the stochastic differential equation (SDE)


title: Fokker–Planck equation

text: with drift formula_3 and diffusion coefficient formula_4, the Fokker–Planck equation for the probability density formula_5 of the random variable formula_6 is


title: Fokker–Planck equation

text: The "transition probability" formula_11, the probability of going from formula_12 to formula_13, is introduced here; the expectation can be written as


title: Fokker–Planck equation

text: Now we replace in the definition of formula_9, multiply by formula_11 and integrate over formula_17. The limit is taken on


title: Fokker–Planck equation

text: which is the Chapman–Kolmogorov theorem. Changing the dummy variable formula_20 to formula_21, one gets


title: Fokker–Planck equation

text: From here, the Kolmogorov backward equation can be deduced. If we instead use the adjoint operator of formula_9, formula_25, defined such that


title: Fokker–Planck equation

text: then we arrive to the Kolmogorov forward equation, or Fokker–Planck equation, which, simplifying the notation formula_27, in its differential form reads


title: Fokker–Planck equation

text: Remains the issue of defining explicitly formula_9. This can be done taking the expectation from the integral form of the Itô's lemma:


title: Fokker–Planck equation

text: While the Fokker–Planck equation is used with problems where the initial distribution is known, if the problem is to know the distribution at previous times, the Feynman–Kac formula can be used, which is a consequence of the Kolmogorov backward equation.


title: Fokker–Planck equation

text: The stochastic process defined above in the Itô sense can be rewritten within the Stratonovich convention as a Stratonovich SDE:


title: Fokker–Planck equation

text: It includes an added noise-induced drift term due to diffusion gradient effects if the noise is state-dependent. This convention is more often used in physical applications. Indeed, it is well known that any solution to the Stratonovich SDE is a solution to the Itô SDE.


title: Fokker–Planck equation

text: The zero-drift equation with constant diffusion can be considered as a model of classical Brownian motion:


title: Fokker–Planck equation

text: This model has discrete spectrum of solutions if the condition of fixed boundaries is added for formula_37:


title: Fokker–Planck equation

text: It has been shown that in this case an analytical spectrum of solutions allows deriving a local uncertainty relation for the coordinate-velocity phase volume:


title: Fokker–Planck equation

text: Here formula_41 is a minimal value of a corresponding diffusion spectrum formula_42, while formula_43 and formula_44 represent the uncertainty of coordinate–velocity definition.


title: Fokker–Planck equation

text: where formula_46 and formula_47 are -dimensional random vectors, formula_48 is an formula_49 matrix and formula_50 is an "M"-dimensional standard Wiener process, the probability density formula_51 for formula_46 satisfies the Fokker–Planck equation


title: Fokker–Planck equation

text: Here the drift term is zero and the diffusion coefficient is 1/2. Thus the corresponding Fokker–Planck equation is


title: Fokker–Planck equation

text: which is the simplest form of a diffusion equation. If the initial condition is formula_61, the solution is


title: Fokker–Planck equation

text: with formula_64. Physically, this equation can be motivated as follows: a particle of mass formula_65 with velocity formula_66 moving in a medium, e.g., a fluid, will experience a friction force which resists motion whose magnitude can be approximated as being proportional to particle's velocity formula_67 with formula_68. Other particles in the medium will randomly kick the particle as they collide with it and this effect can be approximated by a white noise term; formula_69. Newton's second law is written as


title: Fokker–Planck equation

text: Taking formula_71 for simplicity and changing the notation as formula_72 leads to the familiar form formula_73.


title: Fokker–Planck equation

text: In plasma physics, the distribution function for a particle species formula_77, formula_78, takes the place of the probability density function. The corresponding Boltzmann equation is given by


title: Fokker–Planck equation

text: where the third term includes the particle acceleration due to the Lorentz force and the Fokker–Planck term at the right-hand side represents the effects of particle collisions. The quantities formula_80 and formula_81 are the average change in velocity a particle of type formula_77 experiences due to collisions with all other particle species in unit time. Expressions for these quantities are given elsewhere. If collisions are ignored, the Boltzmann equation reduces to the Vlasov equation.


title: Fokker–Planck equation

text: The Smoluchowski Diffusion equation is the Fokker–Planck equation restricted to Brownian particles affected by an external force formula_83.


title: Fokker–Planck equation

text: Where formula_85 is the diffusion constant and formula_86. The importance of this equation is it allows for both the inclusion of the effect of temperature on the system of particles and a spatially dependent diffusion constant.


title: Fokker–Planck equation

text: Starting with the Langevin Equation of a Brownian particle in external field formula_83, where formula_88 is the friction term, formula_89 is a fluctuating force on the particle, and formula_90 is the amplitude of the fluctuation.


title: Fokker–Planck equation

text: At equilibrium the frictional force is much greater than the inertial force, formula_92. Therefore, the Langevin equation becomes,


title: Fokker–Planck equation

text: Where formula_96. Note, the diffusion coefficient may not necessarily be spatially independent if formula_90 or formula_88 are spatially dependent.


title: Fokker–Planck equation

text: Therefore, the flux of particles can be determined by taking the time derivative of the number of particles in a given volume, plugging in the Fokker–Planck equation, and then applying Gauss's Theorem.


title: Fokker–Planck equation

text: In equilibrium, it is assumed that the flux goes to zero. Therefore, Boltzmann statistics can be applied for the probability of a particles location at equilibrium, where formula_102 is a conservative force and the probability of a particle being in a state formula_103 is given as formula_104.


title: Fokker–Planck equation

text: This relation is a realization of the fluctuation–dissipation theorem. Now applying formula_107 to formula_108 and using the Fluctuation-dissipation theorem,


title: Fokker–Planck equation

text: Brownian motion follows the Langevin equation, which can be solved for many different stochastic forcings with results being averaged (canonical ensemble in molecular dynamics). However, instead of this computationally intensive approach, one can use the Fokker–Planck equation and consider the probability formula_113 of the particle having a velocity in the interval formula_114 when it starts its motion with formula_115 at time 0.


title: Fokker–Planck equation

text: Starting with a linear potential of the form formula_116 the corresponding Smoluchowski equation becomes,


title: Fokker–Planck equation

text: Where the diffusion constant, formula_85, is constant over space and time. The boundary conditions are such that the probability vanishes at formula_119 with an initial condition of the ensemble of particles starting in the same place, formula_120.


title: Fokker–Planck equation

text: The simulation on the right was completed using a Brownian dynamics simulation. Starting with a Langevin equation for the system,


title: Fokker–Planck equation

text: where formula_88 is the friction term, formula_89 is a fluctuating force on the particle, and formula_90 is the amplitude of the fluctuation. At equilibrium the frictional force is much greater than the inertial force, formula_132. Therefore, the Langevin equation becomes,


title: Fokker–Planck equation

text: For the Brownian dynamic simulation the fluctuation force formula_134 is assumed to be Gaussian with the amplitude being dependent of the temperature of the system formula_135. Rewriting the Langevin equation,


title: Fokker–Planck equation

text: where formula_137 is the Einstein relation. The integration of this equation was done using the Euler–Maruyama method to numerically approximate the path of this Brownian particle.


title: Fokker–Planck equation

text: Being a partial differential equation, the Fokker–Planck equation can be solved analytically only in special cases. A formal analogy of the Fokker–Planck equation with the Schrödinger equation allows the use of advanced operator techniques known from quantum mechanics for its solution in a number of cases. Furthermore, in the case of overdamped dynamics when the Fokker–Planck equation contains second partial derivatives with respect to all spatial variables, the equation can be written in the form of a master equation that can easily be solved numerically.


title: Fokker–Planck equation

text: In many applications, one is only interested in the steady-state probability distribution formula_138, which can be found from formula_139.


title: Fokker–Planck equation

text: The computation of mean first passage times and splitting probabilities can be reduced to the solution of an ordinary differential equation which is intimately related to the Fokker–Planck equation.


title: Fokker–Planck equation

text: In mathematical finance for volatility smile modeling of options via local volatility, one has the problem of deriving a diffusion coefficient formula_140 consistent with a probability density obtained from market option quotes. The problem is therefore an inversion of the Fokker–Planck equation: Given the density f(x,t) of the option underlying "X" deduced from the option market, one aims at finding the local volatility formula_140 consistent with "f". This is an inverse problem that has been solved in general by Dupire (1994, 1997) with a non-parametric solution. Brigo and Mercurio (2002, 2003) propose a solution in parametric form via a particular local volatility formula_140 consistent with a solution of the Fokker–Planck equation given by a mixture model. More information is available also in Fengler (2008), Gatheral (2008), and Musiela and Rutkowski (2008).


title: Fokker–Planck equation

text: Every Fokker–Planck equation is equivalent to a path integral. The path integral formulation is an excellent starting point for the application of field theory methods. This is used, for instance, in critical dynamics.


title: Fokker–Planck equation

text: A derivation of the path integral is possible in a similar way as in quantum mechanics. The derivation for a Fokker–Planck equation with one variable formula_21 is as follows. Start by inserting a delta function and then integrating by parts:


title: Fokker–Planck equation

text: The formula_21-derivatives here only act on the formula_146-function, not on formula_147. Integrate over a time interval formula_148,


title: Fokker–Planck equation

text: This equation expresses formula_153 as functional of formula_147. Iterating formula_155 times and performing the limit formula_156 gives a path integral with action


title: Fokker–Planck equation

text: Although formally equivalent, different problems may be solved more easily in the Fokker–Planck equation or the path integral formulation. The equilibrium distribution for instance may be obtained more directly from the Fokker–Planck equation.


title: The Secret Saturdays

text: The Secret Saturdays is an American animated television series created by Canadian cartoonist Jay Stephens for Cartoon Network. It debuted on October 3, 2008 in the United States. The series follows the adventures of the Saturdays, a family of cryptozoologists that work to keep the truth about cryptids from getting out, to protect both the human race and the creatures themselves. The Saturdays travel the Earth searching for cryptids to study and battling twisted villains like the megalomaniac V.V. Argost. The series is influenced by 1960's-era Hanna-Barbera action series and is combined with Jay Stephens' own personal interest in cryptozoology. The show finished its run on January 30, 2010, but aired reruns on Boomerang from December 5, 2011 to June 1, 2014.


title: The Secret Saturdays

text: "The Secret Saturdays" was the first Cartoon Network original series to be broadcast in high-definition.


title: The Secret Saturdays

text: The Saturday family later appear in the "" episode "TGIS" where they team up with Ben Tennyson and Rook Blonko to stop Ben's enemy Dr. Animo who has revived the Saturdays' archenemy V.V. Argost in a plot to collaborate with him to cause Cryptid-based havoc.


title: The Secret Saturdays

text: "The Secret Saturdays" revolves around the titular Saturdays, a family of cryptozoologists consisting of parents Doc and Drew Saturday and their 11-year-old son, Zak Saturday. The Saturdays are members of the Secret Scientists, a global organization with the goal of studying and safeguarding sciences considered too dangerous to be general public knowledge. As the foremost experts in cryptids, the Saturdays are responsible for studying and protecting cryptids around the world, while also keeping their existence a secret from the rest of humanity and averting cryptid-related threats. The Saturdays travel in their airship with their cryptid companions Fiskerton, Komodo, and Zon, while the young Zak Saturday strives to help his parents with their missions and to master his mysterious innate ability to influence and control the actions of cryptids.


title: The Secret Saturdays

text: At the beginning of the series, the Secret Scientists are attacked en masse by V.V. Argost, the masked host of the cryptid-centric television series "V.V. Argost's Weird World" and a longtime enemy of the Secret Scientists. Argost and his henchmen steal the pieces of the Kur Stone, a Sumerian artifact which can lead its wielder to the location of Kur, an allegedly all-powerful cryptid. In the first season of the show, the Saturdays are also pitted against bounty hunter Leonidas Van Rook and his apprentice, who is later revealed to be Drew's long-lost brother Doyle Blackwell, who was separated from her after their parents died in an accident in the Himalayas. The Saturdays also discover an ancient mirror which leads to a dimension made of antimatter, and encounter their evil counterparts, whom Zak dubs the "Mondays." At the end of the first season, the Saturdays and Argost race to Kur's supposed resting place in Antarctica, only to discover that Kur is actually Zak; when the Kur stone first shattered years ago, Kur's soul entered Zak's unborn body, which is the origin of Zak's cryptid-influencing powers.


title: The Secret Saturdays

text: In the second season, Argost, taking an interest in Zak as the true Kur, bargains with Zak to teach him how to use his powers. Meanwhile, the Secret Scientists have turned against the Saturdays, wishing to capture Zak and place him into cryogenic sleep to prevent him from becoming a serious threat. As Doyle and a reformed Van Rook investigate the origins of V.V. Argost, they discover that Argost is actually a cryptid, the yeti, and was responsible for the murder of Drew and Doyle's parents decades ago. Argost captures Zak and reveals himself as the yeti, then uses the mirror artifact to summon Zak Monday, whose powers he steals, giving him the same powers as Zak. Argost and Zak wage a global cryptid war against each other using their Kur abilities, but Zak ultimately overpowers Argost, forcing him to retreat. After killing Van Rook, Argost captures Zak and tries to absorb his powers in addition to Zak Monday's, but the combination of matter and antimatter destroys Argost and the power of Kur permanently.


title: The Secret Saturdays

text: When Stephens first pitched the show, it was called "Cryptids", after the undiscovered scientific monsters. It involved a team of globe-trotting cryptozoologists that were uniquely qualified for the job because they were once cryptids themselves. Okapi, Komodo, and Megamouth, along with their 9-year-old human mascot, Francis, served as a sort of anti-detective team. They scooted across land, sea, and air in what Stephens called a Campercopter and their goal was to preserve the mystery of the Earth's strangest creatures before they were exposed by humans or destroyed by the cryptids' nemesis, Monsieur Dodo. This version got him nowhere with many networks, though they all did show interest for some time.


title: The Secret Saturdays

text: Months later, the show was picked up by Cartoon Network, due to the network wanting more action shows at the time due to the success of "Ben 10" and "". 9-year-old Francis became 11-year-old Zak Saturday and many of the original characters were deleted. For a time, the network wanted the show's name to be "The Secret Adventures Of Zak Saturday", but it was later changed. Unlike most mystery shows, the goal of the heroes of "The Secret Saturdays" is not to reveal the existence of cryptids, but to hide it. The art style of the show was influenced by the artwork of Alex Toth and 1960s Hanna-Barbera action cartoons, such as "Jonny Quest" and "The Herculoids". Michael Tavera created the music score for the show after finishing his work on "Time Squad", "¡Mucha Lucha!", "", and "Yin Yang Yo!".


title: The Secret Saturdays

text: In July 2008, Cartoon Network started a viral campaign for the show. Several commercials for a website called "CryptidsAreReal.com" aired on Cartoon Network. Each commercial featured a pet of the Saturday family: one showed Komodo, another Fiskerton, and a final one showed Zon. One area of the program's website also held a "secret" document: while most of the words on it were blackedout, the names of Drew and Doc (two of the main characters) were revealed. The document also talked about a creature discovered outside of Nottinghamshire, England. Though the first word was blacked-out, the paper said a "phantom" had been the newly discovered creature. It was implied that the creature was the Fiskerton Phantom, a cryptid reported to have lived outside of Nottinghamshire and commonly done by this website, would be "Fiskerton", a main character in the show. To back this conclusion up, Zak stated in one episode that Fiskerton was lucky because Doc and Drew saved him, which was also stated in the document. Typed at the bottom of the document was the name "Agent Epsilon", a character in the show. The website also showed a blueprint of the Saturdays family's airship. Another site was linked to CryptidsAreReal.com: the page for an organization called "Ten Hero Tusk" (an anagram of "the Kur stone"), which states that its explorers are "D. Blackwell, B. Finster, and L. Van Rook", all of which are characters in the show. During San Diego Comic-Con 2008, booklets about a show called "Weirdworld" (the show-within-a-show) were given out, with the website for it on the back of them. Unlike CryptidsAreReal.com, this one could not be traced back to Cartoon Network. Later, commercials for "Weirdworld" aired on Cartoon Network for a few weeks before a commercial for "The Secret Saturdays" itself was aired on October 3, 2008.


title: The Secret Saturdays

text: A DVD featuring the first five episodes of the series (along with bonus features such as animatics) was released on July 21, 2009. A second DVD with similar features was released on October 27, 2009. An encyclopedia entitled "The Secret Saturdays: The Official Cryptid Field Guide" was released on August 11, 2009. Graphic novels and chapter books were released by Del Ray Manga in Summer 2009. A video game called "" was released by D3 Publisher for the PlayStation 2, PlayStation Portable, Nintendo DS and Wii on October 20, 2009. Comic adventures of "The Secret Saturdays" appeared semi-monthly in 2010 as part of "Cartoon Network Action Pack" comic books published by DC Comics. A line of action figures and playsets was released by Mattel. One of the toys made by Mattel, the Secret Saturdays Fire Sword, received some controversy for the loudness of the toy, which could reach up to 121 dBs.


title: The Eternal Daughter

text: The Eternal Daughter is a 2022 Gothic mystery drama film written, produced and directed by Joanna Hogg, and starring Tilda Swinton. It also features Joseph Mydell and Carly-Sophia Davies in supporting roles.


title: The Eternal Daughter

text: It had its world premiere at the 79th Venice International Film Festival on 6 September 2022, and was released in the United States on 2 December 2022 by A24.


title: The Eternal Daughter

text: Returning to a hotel now haunted by its mysterious past, an artist and her elderly mother confront long-buried secrets in their former family home.


title: The Eternal Daughter

text: On January 18, 2021, it was announced that a secret film written and directed by Joanna Hogg and starring Tilda Swinton has wrapped filming. On January 27, 2021, it was confirmed that A24 acquired the worldwide distribution rights.


title: The Eternal Daughter

text: The film had its world premiere at the 79th Venice International Film Festival on 6 September 2022. It also screened at the 2022 Toronto International Film Festival. and the 60th New York Film Festival. It was released in the United States on 2 December 2022.


title: Julian Robertson

text: Julian Hart Robertson Jr. (June 25, 1932 – August 23, 2022) was an American billionaire hedge fund manager, and philanthropist.


title: Julian Robertson

text: Robertson founded Tiger Management, one of the first hedge funds, in 1980. From its inception in 1980 to its 1998 asset peak, his fund returned 31.7% per year after fees, compared to a 12.7% annual return from the S&P 500 over the same period. However, a sharp decline thereafter led to the fund closing in March 2000. Tiger showed losses in only four of its 21 years. Robertson later mentored and provided seed funding to many notable hedge fund managers, known as the "Tiger cubs", including Ole Andreas Halvorsen, Stephen Mandel of Lone Pine Capital, Lee Ainslie of Maverick Capital, Bill Hwang, and Chase Coleman III.


title: Julian Robertson

text: During his lifetime, Robertson contributed more than US$2 billion to charity. He was also a signatory to The Giving Pledge. At the time of his death, his net worth was estimated at $4.8 billion.


title: Julian Robertson

text: Robertson was the son of Julian Hart Robertson Sr., a textile company executive, and Blanche Spencer, a local activist. He claimed that his father was a descendant of Pocahontas. Robertson first became interested in stocks at age 6.


title: Julian Robertson

text: He graduated from Episcopal High School in 1951 and the University of North Carolina at Chapel Hill in 1955. While at Chapel Hill, he was admitted to Zeta Psi fraternity and was a member of the Reserve Officers' Training Corps. He then served as an officer in the U.S. Navy, traveling the world aboard a munitions ship until 1957.


title: Julian Robertson

text: After leaving the Navy, Robertson moved to New York City and worked for a time as a stockbroker for Kidder, Peabody & Company. At Kidder, he eventually headed the firm's asset management division, Webster Securities. In 1978, he took a sabbatical and moved with his family to New Zealand for a year to write a novel.


title: Julian Robertson

text: On his return to the United States, in 1980, with $8 million of funding from family, friends, and his own wealth, he founded Tiger Management. The Tiger funds reached a peak of $22 billion in assets in 1998. Robertson's Tiger Fund accurately predicted the dot-com bubble, purposely underweighting the technology sector. Tiger's largest holding was US Airways; it controlled 25% of the company. Its troubles led to significant losses for the fund. Tiger also realized significant losses in the Japanese Yen. Such missteps ultimately led him to close his investment company in late March 2000, at the peak of the dot-com bubble, and return all outside capital to investors.


title: Julian Robertson

text: Robertson said in 2008 that he shorted subprime securities and used credit default swaps to make a 76.7% return on investment in 2007. From the closure of his fund in 2000 until January 2008, his return on his personal fortune was 403%. After closing his fund in 2000, Robertson supported and financed upcoming hedge fund managers in return for a stake in their fund management companies. Robertson later mentored and provided seed funding to many notable hedge fund managers, known as the "Tiger cubs", including Ole Andreas Halvorsen, Stephen Mandel of Lone Pine Capital, Lee Ainslie of Maverick Capital, Bill Hwang, and Chase Coleman III.


title: Julian Robertson

text: Robertson was an investor and developer in New Zealand and owned three lodges: Kauri Cliffs Lodge near Matauri Bay in Northland; Matakauri Lodge Queenstown; and The Farm at Cape Kidnappers, Hawkes Bay, as well as several wineries.


title: Julian Robertson

text: Julian married Josephine Tucker Robertson in 1972. She died in June 2010 from breast cancer. They had three children.


title: Julian Robertson

text: In August 2010, Robertson signed The Giving Pledge, an initiative by software mogul Bill Gates and investor Warren Buffett in which the wealthy pledge at least half their assets to charity. He was involved in donating to several educational causes. Robertson was the founder and benefactor of the Robertson Scholars Program which awards a merit scholarship that provides four-year full-tuition, room, board, and travel funding for 36 Duke University and University of North Carolina at Chapel Hill students each year. In April 2016, Robertson gave $25 million to Success Academy Charter Schools in New York. In 2019, he donated $2.7 million to the Liggins Institute at the University of Auckland.


title: Julian Robertson

text: In 2009, it was announced that Robertson would donate art valued at $115 million to the Auckland Art Gallery Toi o Tāmaki. The donation included works by Paul Cézanne, Paul Gauguin, Pablo Picasso, Henri Matisse, Piet Mondrian, Salvador Dalí, Georges Braque, André Derain, Fernand Léger, Pierre Bonnard, and Henri Fantin-Latour and was the largest of its kind in Australasia. In May 2010, the New York Stem Cell Foundation, a private research institution, received a $27 million gift from Robertson to fund its research.


title: Julian Robertson

text: While Robertson was a Republican, he urged the party to support clean energy policies and contributed $500,000 to the ClearPath Foundation in 2016. Robertson supported the Mitt Romney 2012 presidential campaign, and Romney attended Robertson's 80th birthday party in 2012. In January 2012, Robertson donated $1.25 million to Restore Our Future, a Super PAC supporting the Mitt Romney 2012 presidential campaign. In 2015, Robertson gave $1 million to a Super PAC supporting the Jeb Bush 2016 presidential campaign. Although Robertson did not support Donald Trump in the 2016 election, instead supporting libertarian Gary Johnson, Robertson was a supporter of the presidency of Donald Trump and the Tax Cuts and Jobs Act of 2017. Robertson supported the estate tax.


title: Julian Robertson

text: Robertson owned residences in Manhattan on Central Park South, Nassau County, Long Island, New Zealand, Sun Valley, and the Hamptons. In 2020, Robertson completed construction of three new homes, each approximately 4,000 square feet, for his family in New Zealand.


title: Julian Robertson

text: Robertson kept track of where he spent his time and won a legal case after he proved that he did not spend enough time in New York to be liable for income taxes in the state.


title: Julian Robertson

text: In 2008, he was inducted into Institutional Investors Alpha's Hedge Fund Manager Hall of Fame. Robertson was appointed an Honorary Knight Companion of the New Zealand Order of Merit, for services to business and philanthropy, in the 2010 New Year Honours, and in 2017, he was one of nine people awarded a Carnegie Medal of Philanthropy.


title: Julian Robertson

text: On April 1, 1996, "BusinessWeek" carried a cover story written by reporter Gary Weiss, called "Fall of the Wizard", that was critical of Robertson's performance and behavior as founder and manager of Tiger Management. Robertson subsequently sued Weiss and "BusinessWeek" for $1 billion for defamation. The suit was settled with no money changing hands and "BusinessWeek" standing by the substance of its reporting.


title: False discovery rate

text: In statistics, the false discovery rate (FDR) is a method of conceptualizing the rate of type I errors in null hypothesis testing when conducting multiple comparisons. FDR-controlling procedures are designed to control the FDR, which is the expected proportion of "discoveries" (rejected null hypotheses) that are false (incorrect rejections of the null). Equivalently, the FDR is the expected ratio of the number of false positive classifications (false discoveries) to the total number of positive classifications (rejections of the null). The total number of rejections of the null include both the number of false positives (FP) and true positives (TP). Simply put, FDR = FP / (FP + TP). FDR-controlling procedures provide less stringent control of Type I errors compared to family-wise error rate (FWER) controlling procedures (such as the Bonferroni correction), which control the probability of "at least one" Type I error. Thus, FDR-controlling procedures have greater power, at the cost of increased numbers of Type I errors.


title: False discovery rate

text: The modern widespread use of the FDR is believed to stem from, and be motivated by, the development in technologies that allowed the collection and analysis of a large number of distinct variables in several individuals (e.g., the expression level of each of 10,000 different genes in 100 different persons). By the late 1980s and 1990s, the development of "high-throughput" sciences, such as genomics, allowed for rapid data acquisition. This, coupled with the growth in computing power, made it possible to seamlessly perform a very high number of statistical tests on a given data set. The technology of microarrays was a prototypical example, as it enabled thousands of genes to be tested simultaneously for differential expression between two biological conditions.


title: False discovery rate

text: As high-throughput technologies became common, technological and/or financial constraints led researchers to collect datasets with relatively small sample sizes (e.g. few individuals being tested) and large numbers of variables being measured per sample (e.g. thousands of gene expression levels). In these datasets, too few of the measured variables showed statistical significance after classic correction for multiple tests with standard multiple comparison procedures. This created a need within many scientific communities to abandon FWER and unadjusted multiple hypothesis testing for other ways to highlight and rank in publications those variables showing marked effects across individuals or treatments that would otherwise be dismissed as non-significant after standard correction for multiple tests. In response to this, a variety of error rates have been proposed—and become commonly used in publications—that are less conservative than FWER in flagging possibly noteworthy observations. The FDR is useful when researchers are looking for "discoveries" that will give them followup work (E.g.: detecting promising genes for followup studies), and are interested in controlling the proportion of "false leads" they are willing to accept.


title: False discovery rate

text: The FDR concept was formally described by Yoav Benjamini and Yosef Hochberg in 1995 (BH procedure) as a less conservative and arguably more appropriate approach for identifying the important few from the trivial many effects tested. The FDR has been particularly influential, as it was the first alternative to the FWER to gain broad acceptance in many scientific fields (especially in the life sciences, from genetics to biochemistry, oncology and plant sciences). In 2005, the Benjamini and Hochberg paper from 1995 was identified as one of the 25 most-cited statistical papers.


title: False discovery rate

text: Prior to the 1995 introduction of the FDR concept, various precursor ideas had been considered in the statistics literature. In 1979, Holm proposed the Holm procedure, a stepwise algorithm for controlling the FWER that is at least as powerful as the well-known Bonferroni adjustment. This stepwise algorithm sorts the "p"-values and sequentially rejects the hypotheses starting from the smallest "p"-values.


title: False discovery rate

text: Benjamini (2010) said that the false discovery rate, and the paper Benjamini and Hochberg (1995), had its origins in two papers concerned with multiple testing:


title: False discovery rate

text: The BH procedure was proven to control the FDR for independent tests in 1995 by Benjamini and Hochberg. In 1986, R. J. Simes offered the same procedure as the "Simes procedure", in order to control the FWER in the weak sense (under the intersection null hypothesis) when the statistics are independent.


title: False discovery rate

text: Based on definitions below we can define as the proportion of false discoveries among the discoveries (rejections of the null hypothesis):


title: False discovery rate

text: where formula_8 is the expected value of formula_9. The goal is to keep FDR below a given threshold "q". To avoid division by zero, formula_9 is defined to be 0 when formula_11. Formally, formula_12.


title: False discovery rate

text: The settings for many procedures is such that we have formula_13 null hypotheses tested and formula_14 their corresponding "p"-values. We list these "p"-values in ascending order and denote them by formula_15. A procedure that goes from a small "p"-value to a large one will be called a step-up procedure. In a similar way, in a "step-down" procedure we move from a large corresponding test statistic to a smaller one.


title: False discovery rate

text: The "Benjamini–Hochberg procedure" (BH step-down procedure) controls the FDR at level formula_16. It works as follows:


title: False discovery rate

text: Geometrically, this corresponds to plotting formula_21 vs. (on the and axes respectively), drawing the line through the origin with slope formula_22 , and declaring discoveries for all points on the left up to and including the last point that is below the line.


title: False discovery rate

text: The BH procedure is valid when the tests are independent, and also in various scenarios of dependence, but is not universally valid. It also satisfies the inequality:


title: False discovery rate

text: If an estimator of formula_1 is inserted into the BH procedure, it is no longer guaranteed to achieve FDR control at the desired level. Adjustments may be needed in the estimator and several modifications have been proposed.


title: False discovery rate

text: Note that the mean formula_16 for these tests is formula_26, the Mean(FDR formula_16) or MFDR, formula_16 adjusted for independent or positively correlated tests (see AFDR below). The MFDR expression here is for a single recomputed value of formula_16 and is not part of the Benjamini and Hochberg method.


title: False discovery rate

text: The "Benjamini–Yekutieli" procedure controls the false discovery rate under arbitrary dependence assumptions. This refinement modifies the threshold and finds the largest such that:


title: False discovery rate

text: Using MFDR and formulas above, an adjusted MFDR, or AFDR, is the min(mean formula_16) for  dependent tests formula_37.


title: False discovery rate

text: Using a multiplicity procedure that controls the FDR criterion is adaptive and scalable. Meaning that controlling the FDR can be very permissive (if the data justify it), or conservative (acting close to control of FWER for sparse problem) - all depending on the number of hypotheses tested and the level of significance.


title: False discovery rate

text: The FDR criterion "adapts" so that the same number of false discoveries (V) will have different implications, depending on the total number of discoveries (R). This contrasts with the family-wise error rate criterion. For example, if inspecting 100 hypotheses (say, 100 genetic mutations or SNPs for association with some phenotype in some population):


title: False discovery rate

text: The FDR criterion is "scalable" in that the same proportion of false discoveries out of the total number of discoveries (Q), remains sensible for different number of total discoveries (R). For example:


title: False discovery rate

text: Controlling the FDR using the linear step-up BH procedure, at level q, has several properties related to the dependency structure between the test statistics of the null hypotheses that are being corrected for. If the test statistics are:


title: False discovery rate

text: If all of the null hypotheses are true (formula_45), then controlling the FDR at level guarantees control over the FWER (this is also called "weak control of the FWER"): formula_46, simply because the event of rejecting at least one true null hypothesis formula_47 is exactly the event formula_48, and the event formula_49 is exactly the event formula_50 (when formula_51, formula_52 by definition). But if there are some true discoveries to be made (formula_53) then . In that case there will be room for improving detection power. It also means that any procedure that controls the FWER will also control the FDR.


title: False discovery rate

text: The discovery of the FDR was preceded and followed by many other types of error rates. These include:


title: False discovery rate

text: The false coverage rate (FCR) is, in a sense, the FDR analog to the confidence interval. FCR indicates the average rate of false coverage, namely, not covering the true parameters, among the selected intervals. The FCR gives a simultaneous coverage at a formula_87 level for all of the parameters considered in the problem. Intervals with simultaneous coverage probability 1−q can control the FCR to be bounded by "q". There are many FCR procedures such as: Bonferroni-Selected–Bonferroni-Adjusted, Adjusted BH-Selected CIs (Benjamini and Yekutieli (2005)), Bayes FCR (Yekutieli (2008)), and other Bayes methods.


title: False discovery rate

text: Connections have been made between the FDR and Bayesian approaches (including empirical Bayes methods), thresholding wavelets coefficients and model selection, and generalizing the confidence interval into the false coverage statement rate (FCR).


title: Differentiable manifold

text: In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart.


title: Differentiable manifold

text: In formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a vector space. To induce a global differential structure on the local coordinate systems induced by the homeomorphisms, their compositions on chart intersections in the atlas must be differentiable functions on the corresponding vector space. In other words, where the domains of charts overlap, the coordinates defined by each chart are required to be differentiable with respect to the coordinates defined by every chart in the atlas. The maps that relate the coordinates defined by the various charts to one another are called "transition maps."


title: Differentiable manifold

text: The ability to define such a local differential structure on an abstract space allows one to extend the definition of differentiability to spaces without global coordinate systems. A locally differential structure allows one to define the globally differentiable tangent space, differentiable functions, and differentiable tensor and vector fields.


title: Differentiable manifold

text: Differentiable manifolds are very important in physics. Special kinds of differentiable manifolds form the basis for physical theories such as classical mechanics, general relativity, and Yang–Mills theory. It is possible to develop a calculus for differentiable manifolds. This leads to such mathematical machinery as the exterior calculus. The study of calculus on differentiable manifolds is known as differential geometry.


title: Differentiable manifold

text: "Differentiability" of a manifold has been given several meanings, including: continuously differentiable, "k"-times differentiable, smooth (which itself has many meanings), and analytic.


title: Differentiable manifold

text: The emergence of differential geometry as a distinct discipline is generally credited to Carl Friedrich Gauss and Bernhard Riemann. Riemann first described manifolds in his famous habilitation lecture before the faculty at Göttingen. He motivated the idea of a manifold by an intuitive process of varying a given object in a new direction, and presciently described the role of coordinate systems and charts in subsequent formal developments:


title: Differentiable manifold

text: The works of physicists such as James Clerk Maxwell, and mathematicians Gregorio Ricci-Curbastro and Tullio Levi-Civita led to the development of tensor analysis and the notion of covariance, which identifies an intrinsic geometric property as one that is invariant with respect to coordinate transformations. These ideas found a key application in Albert Einstein's theory of general relativity and its underlying equivalence principle. A modern definition of a 2-dimensional manifold was given by Hermann Weyl in his 1913 book on Riemann surfaces. The widely accepted general definition of a manifold in terms of an atlas is due to Hassler Whitney.


title: Differentiable manifold

text: Let be a topological space. A chart on consists of an open subset of , and a homeomorphism from to an open subset of some Euclidean space . Somewhat informally, one may refer to a chart , meaning that the image of is an open subset of , and that is a homeomorphism onto its image; in the usage of some authors, this may instead mean that is itself a homeomorphism.


title: Differentiable manifold

text: The presence of a chart suggests the possibility of doing differential calculus on ; for instance, if given a function and a chart on , one could consider the composition , which is a real-valued function whose domain is an open subset of a Euclidean space; as such, if it happens to be differentiable, one could consider its partial derivatives.


title: Differentiable manifold

text: This situation is not fully satisfactory for the following reason. Consider a second chart on , and suppose that and contain some points in common. The two corresponding functions and are linked in the sense that they can be reparametrized into one another:


title: Differentiable manifold

text: the natural domain of the right-hand side being . Since and are homeomorphisms, it follows that is a homeomorphism from to . Consequently, even if both functions and are differentiable, their differential properties will not necessarily be strongly linked to one another, as is not necessarily sufficiently differentiable for the chain rule to be applicable. The same problem is found if one considers instead functions ; one is led to the reparametrization formula


title: Differentiable manifold

text: This is resolved by the introduction of a "differentiable atlas" of charts, which specifies a collection of charts on for which the transition maps are all differentiable. This makes the situation quite clean: if is differentiable, then due to the reparametrization formula, the map is also differentiable on the region . Moreover, the derivatives of these two maps are linked to one another by the chain rule. Relative to the given atlas, this facilitates a notion of differentiable mappings whose domain or range is , as well as a notion of the derivative of such maps.


title: Differentiable manifold

text: Formally, the word "differentiable" is somewhat ambiguous, as it is taken to mean different things by different authors; sometimes it means the existence of first derivatives, sometimes the existence of continuous first derivatives, and sometimes the existence of infinitely many derivatives. The following gives a formal definition of various (nonambiguous) meanings of "differentiable atlas". Generally, "differentiable" will be used as a catch-all term including all of these possibilities, provided .


title: Differentiable manifold

text: Since every real-analytic map is smooth, and every smooth map is for any , one can see that any analytic atlas can also be viewed as a smooth atlas, and every smooth atlas can be viewed as a atlas. This chain can be extended to include holomorphic atlases, with the understanding that any holomorphic map between open subsets of can be viewed as a real-analytic map between open subsets of .


title: Differentiable manifold

text: Given a differentiable atlas on a topological space, one says that a chart is differentiably compatible with the atlas, or differentiable relative to the given atlas, if the inclusion of the chart into the collection of charts comprising the given differentiable atlas results in a differentiable atlas. A differentiable atlas determines a maximal differentiable atlas, consisting of all charts which are differentiably compatible with the given atlas. A maximal atlas is always very large. For instance, given any chart in a maximal atlas, its restriction to an arbitrary open subset of its domain will also be contained in the maximal atlas. A maximal smooth atlas is also known as a smooth structure; a maximal holomorphic atlas is also known as a complex structure.


title: Differentiable manifold

text: An alternative but equivalent definition, avoiding the direct use of maximal atlases, is to consider equivalence classes of differentiable atlases, in which two differentiable atlases are considered equivalent if every chart of one atlas is differentiably compatible with the other atlas. Informally, what this means is that in dealing with a smooth manifold, one can work with a single differentiable atlas, consisting of only a few charts, with the implicit understanding that many other charts and differentiable atlases are equally legitimate.


title: Differentiable manifold

text: According to the invariance of domain, each connected component of a topological space which has a differentiable atlas has a well-defined dimension . This causes a small ambiguity in the case of a holomorphic atlas, since the corresponding dimension will be one-half of the value of its dimension when considered as an analytic, smooth, or atlas. For this reason, one refers separately to the "real" and "complex" dimension of a topological space with a holomorphic atlas.


title: Differentiable manifold

text: A differentiable manifold is a Hausdorff and second countable topological space , together with a maximal differentiable atlas on . Much of the basic theory can be developed without the need for the Hausdorff and second countability conditions, although they are vital for much of the advanced theory. They are essentially equivalent to the general existence of bump functions and partitions of unity, both of which are used ubiquitously.


title: Differentiable manifold

text: The notion of a manifold is identical to that of a topological manifold. However, there is a notable distinction to be made. Given a topological space, it is meaningful to ask whether or not it is a topological manifold. By contrast, it is not meaningful to ask whether or not a given topological space is (for instance) a smooth manifold, since the notion of a smooth manifold requires the specification of a smooth atlas, which is an additional structure. It could, however, be meaningful to say that a certain topological space cannot be given the structure of a smooth manifold. It is possible to reformulate the definitions so that this sort of imbalance is not present; one can start with a set (rather than a topological space ), using the natural analogue of a smooth atlas in this setting to define the structure of a topological space on .


title: Differentiable manifold

text: One can reverse-engineer the above definitions to obtain one perspective on the construction of manifolds. The idea is to start with the images of the charts and the transition maps, and to construct the manifold purely from this data. As in the above discussion, we use the "smooth" context but everything works just as well in other settings.


title: Differentiable manifold

text: Given an indexing set formula_3 let formula_4 be a collection of open subsets of formula_5 and for each formula_6 let formula_7 be an open (possibly empty) subset of formula_8 and let formula_9 be a smooth map. Suppose that formula_10 is the identity map, that formula_11 is the identity map, and that formula_12 is the identity map. Then define an equivalence relation on the disjoint union formula_13 by declaring formula_14 to be equivalent to formula_15 With some technical work, one can show that the set of equivalence classes can naturally be given a topological structure, and that the charts used in doing so form a smooth atlas.


title: Differentiable manifold

text: A real valued function "f" on an "n"-dimensional differentiable manifold "M" is called differentiable at a point if it is differentiable in any coordinate chart defined around "p". In more precise terms, if formula_16 is a differentiable chart where formula_17 is an open set in formula_18 containing "p" and formula_19 is the map defining the chart, then "f" is differentiable at "p" if and only if


title: Differentiable manifold

text: is differentiable at formula_21, that is formula_22 is a differentiable function from the open set formula_23, considered as a subset of formula_24, to formula_25. In general, there will be many available charts; however, the definition of differentiability does not depend on the choice of chart at "p". It follows from the chain rule applied to the transition functions between one chart and another that if "f" is differentiable in any particular chart at "p", then it is differentiable in all charts at "p". Analogous considerations apply to defining "C" functions, smooth functions, and analytic functions.


title: Differentiable manifold

text: There are various ways to define the derivative of a function on a differentiable manifold, the most fundamental of which is the directional derivative. The definition of the directional derivative is complicated by the fact that a manifold will lack a suitable affine structure with which to define vectors. Therefore, the directional derivative looks at curves in the manifold instead of vectors.


title: Differentiable manifold

text: Given a real valued function "f" on an "n" dimensional differentiable manifold "M", the directional derivative of "f" at a point "p" in "M" is defined as follows. Suppose that γ("t") is a curve in "M" with , which is "differentiable" in the sense that its composition with any chart is a differentiable curve in R. Then the directional derivative of "f" at "p" along γ is


title: Differentiable manifold

text: then, by the chain rule, "f" has the same directional derivative at "p" along "γ" as along "γ". This means that the directional derivative depends only on the tangent vector of the curve at "p". Thus, the more abstract definition of directional differentiation adapted to the case of differentiable manifolds ultimately captures the intuitive features of directional differentiation in an affine space.


title: Differentiable manifold

text: A tangent vector at is an equivalence class of differentiable curves "γ" with , modulo the equivalence relation of first-order contact between the curves. Therefore,