A catalog server provides a single point of access that allows users to centrally search for information across a distributed network. In other words, it indexes databases, files and information across large network and allows keywords, Boolean and other searches. If you need to provide a comprehensive searching service for your intranet, extranet or even the Internet, a catalog server is a standard solution.
Sanchar Saathi
Sanchar Saathi (lit. 'Communication Partner' or 'Communication Companion') is an Indian state-owned app and web portal, operated by the Department of Telecommunications, designed to assist Indian mobile users in tracking and blocking stolen or lost mobile devices. In late 2025, a government order requiring Sanchar Saathi to be pre-installed on all mobile devices sold nationwide, with explicit provisions on preventing users from deleting the app or disabling any of its broad functionalities, triggered widespread backlash. The order was subsequently withdrawn. == Background == The Telecommunications Act 2023 introduced an exceptionally broad definition of the term "telecommunications" and conferred wide-ranging powers on the government. Although the Department of Telecommunications (DoT) assured reporters that this definition would not be used to justify government overreach, a November 2024 amendment to the Telecom Cyber Security Rules expanded it further and introduced the concept of the Telecommunication Identifier User Entity (TIEU), enabling users to be personally identified through their phone numbers. Sanchar Saathi was launched amid a widespread rise in cybercrime and hacking, as part of the Indian government's effort to prevent stolen phones from being used for fraud and to promote a state-backed application. In an official statement, the DoT said, "India has big second-hand mobile device market. Cases have also been observed where stolen or blacklisted devices are being re-sold. It makes the purchaser abettor in crime and causes financial loss to them." == Launch == Sanchar Saathi was originally launched as a web portal in May 2023. It was later launched as a mobile app in January 2025. Describing itself as a "citizen-centric" safety tool, Sanchar Saathi allows users to check a device's IMEI, report and block lost or stolen phones, and flag suspected fraud communications. Under Sanchar Saathi's privacy policy, it can make and manage phone calls, view and send messages, read call logs, access photos and files, access the location and camera of the device in which the app is used, as well as read and write into the device's storage. According to official government data, by December 2025, the Sanchar Saathi app had helped recover more than 700,000 lost and stolen mobile devices across India. Users report around 2,000 fraud incidents through the app each day. == Pre-installation controversy == On 28 November 2025, the Bharatiya Janata Party government, led by prime minister Narendra Modi, privately ordered phone manufacturers, including Apple, Samsung, Xiaomi, Vivo, Oppo, among others, to pre-install the Sanchar Saathi app on new devices sold in the country, alongside mandating that old devices get issued a software update for the installation of the app. The order had a 90-day deadline and further included explicit provisions to ensure that the app is to be "readily visible and accessible to the end users at the time of first use or device setup" and that users should neither be able to delete the app nor disable or restrict any of its broad functionalities. The order caused widespread political backlash. K. C. Venugopal, a general secretary of the main opposition party, the Indian National Congress (or simply the Congress), called the order "beyond unconstitutional" and said, "A pre-loaded government app that cannot be uninstalled is a dystopian tool to monitor every Indian. It is a means to watch over every movement, interaction and decision of each citizen", adding, "Big Brother cannot watch us." Another Congress general secretary, Priyanka Gandhi, termed Sanchar Saathi a "snooping app", and attacked the government for "turning this country into a dictatorship". Uddhav Thackeray, former chief minister of Maharashtra, compared Sanchar Saathi to the Pegasus spyware. Sanjay Hegde, a senior advocate at the Supreme Court of India, said "Here in the garb of security, the intrusion is vast, unfettered, unguided and is totally disproportionate. The app ought to be struck down on that account". The Internet Freedom Foundation (IFF), an Indian digital rights advocacy organisation, said, "Forcing every smartphone to carry a permanent government app for a simple verification task is excessive and violates the Puttaswamy proportionality standard", referring to Puttaswamy v. Union of India, a 2017 landmark decision of the Supreme Court, which asserted that the right to privacy should be protected as a fundamental right. The IFF further said, "For this to work in practice, the app will almost certainly need system level or root level access, similar to carrier or OEM system apps, so that it cannot be disabled. That design choice erodes the protections that normally prevent one app from peering into the data of others, and turns Sanchar Saathi into a permanent, non-consensual point of access sitting inside the operating system of every Indian smartphone user." Moreover, the organisation said that while the app was being "framed as a benign IMEI checker", a server-side update could allow the app to engage in "client side scanning for 'banned' applications, flag VPN usage, correlate SIM activity, or trawl SMS logs in the name of fraud detection. Nothing in the order constrains these possibilities." In reaction to the controversy, Jyotiraditya Scindia, the union minister of communications, said, "There is no snooping or call monitoring", adding, "Obviously you can delete it. There is no problem. This is a matter of customer protection. It is not mandatory. If you don't want to register, and don't want to use the app, don't use it; don't register, and it will lay dormant." Scindia compared the app to other pre-installed mobile apps such as Google Maps, which he said could be deleted if users wished so. However, contrary to Scindia's statement, on many phone brands, such pre-installed apps cannot be deleted, although users can disable them. Furthermore, upon enquiry, Scindia did not clarify whether his remarks applied to the app after the order took effect, making no comment on the provision in the order that would prevent users from deleting the app. When Congress member Renuka Chowdhury submitted an adjournment motion notice in the Rajya Sabha seeking the suspension of all other matters to discuss the Sanchar Saathi issue, Kiren Rijiju, the union minister of parliamentary affairs, accused the opposition of "manufacturing issues" to stall session proceedings. By 2 December, it had been reported that Apple did not plan to comply with the order, citing privacy and security concerns for the iOS ecosystem and the fact that the order would violate its internal policy against the pre-installation of third-party software in iPhones. Although it was clarified that Apple did not intend to take the matter to court or publicly oppose the government, it was said that Apple "can't do this. Period." The order would have also required Google to create a custom version of Android solely for India which would include the Sanchar Saathi app, a requirement described to "not be acceptable to the company". Following the backlash, the order was revoked on 3 December 2025. In a press release, the government said, "Given Sanchar Saathi's increasing acceptance, Government has decided not to make the pre-installation mandatory for mobile manufacturers".
Accumulated local effects
Accumulated local effects (ALE) is a machine learning interpretability method. == Concepts == ALE uses a conditional feature distribution as an input and generates augmented data, creating more realistic data than a marginal distribution. It ignores far out-of-distribution (outlier) values. Unlike partial dependence plots and marginal plots, ALE is not defeated in the presence of correlated predictors. It analyzes differences in predictions instead of averaging them by calculating the average of the differences in model predictions over the augmented data, instead of the average of the predictions themselves. == Example == Given a model that predicts house prices based on its distance from city center and size of the building area, ALE compares the differences of predictions of houses of different sizes. The result separates the impact of the size from otherwise correlated features. == Limitations == Defining evaluation windows is subjective. High correlations between features can defeat the technique. ALE requires more and more uniformly distributed observations than PDP so that the conditional distribution can be reliably determined. The technique may produce inadequate results if the data is highly sparse, which is more common with high-dimensional data (curse of dimensionality).
Genetic operator
A genetic operator is an operator used in evolutionary algorithms (EA) to guide the algorithm towards a solution to a given problem. There are three main types of operators (mutation, crossover and selection), which must work in conjunction with one another in order for the algorithm to be successful. Genetic operators are used to create and maintain genetic diversity (mutation operator), combine existing solutions (also known as chromosomes) into new solutions (crossover) and select between solutions (selection). The classic representatives of evolutionary algorithms include genetic algorithms, evolution strategies, genetic programming and evolutionary programming. In his book discussing the use of genetic programming for the optimization of complex problems, computer scientist John Koza has also identified an 'inversion' or 'permutation' operator; however, the effectiveness of this operator has never been conclusively demonstrated and this operator is rarely discussed in the field of genetic programming. For combinatorial problems, however, these and other operators tailored to permutations are frequently used by other EAs. Mutation (or mutation-like) operators are said to be unary operators, as they only operate on one chromosome at a time. In contrast, crossover operators are said to be binary operators, as they operate on two chromosomes at a time, combining two existing chromosomes into one new chromosome. == Operators == Genetic variation is a necessity for the process of evolution. Genetic operators used in evolutionary algorithms are analogous to those in the natural world: survival of the fittest, or selection; reproduction (crossover, also called recombination); and mutation. === Selection === Selection operators give preference to better candidate solutions (chromosomes), allowing them to pass on their 'genes' to the next generation (iteration) of the algorithm. The best solutions are determined using some form of objective function (also known as a 'fitness function' in evolutionary algorithms), before being passed to the crossover operator. Different methods for choosing the best solutions exist, for example, fitness proportionate selection and tournament selection. A further or the same selection operator is used to determine the individuals for being selected to form the next parental generation. The selection operator may also ensure that the best solution(s) from the current generation always become(s) a member of the next generation without being altered; this is known as elitism or elitist selection. === Crossover === Crossover is the process of taking more than one parent solutions (chromosomes) and producing a child solution from them. By recombining portions of good solutions, the evolutionary algorithm is more likely to create a better solution. As with selection, there are a number of different methods for combining the parent solutions, including the edge recombination operator (ERO) and the 'cut and splice crossover' and 'uniform crossover' methods. The crossover method is often chosen to closely match the chromosome's representation of the solution; this may become particularly important when variables are grouped together as building blocks, which might be disrupted by a non-respectful crossover operator. Similarly, crossover methods may be particularly suited to certain problems; the ERO is considered a good option for solving the travelling salesman problem. === Mutation === The mutation operator encourages genetic diversity amongst solutions and attempts to prevent the evolutionary algorithm converging to a local minimum by stopping the solutions becoming too close to one another. In mutating the current pool of solutions, a given solution may change between slightly and entirely from the previous solution. By mutating the solutions, an evolutionary algorithm can reach an improved solution solely through the mutation operator. Again, different methods of mutation may be used; these range from a simple bit mutation (flipping random bits in a binary string chromosome with some low probability) to more complex mutation methods in which genes in the solution are changed, for example by adding a random value from the Gaussian distribution to the current gene value. As with the crossover operator, the mutation method is usually chosen to match the representation of the solution within the chromosome. == Combining operators == While each operator acts to improve the solutions produced by the evolutionary algorithm working individually, the operators must work in conjunction with each other for the algorithm to be successful in finding a good solution. Using the selection operator on its own will tend to fill the solution population with copies of the best solution from the population. If the selection and crossover operators are used without the mutation operator, the algorithm will tend to converge to a local minimum, that is, a good but sub-optimal solution to the problem. Using the mutation operator on its own leads to a random walk through the search space. Only by using all three operators together can the evolutionary algorithm become a noise-tolerant global search algorithm, yielding good solutions to the problem at hand.
Error-driven learning
In reinforcement learning, error-driven learning is a method for adjusting a model's (intelligent agent's) parameters based on the difference between its output results and the ground truth. These models stand out as they depend on environmental feedback, rather than explicit labels or categories. They are based on the idea that language acquisition involves the minimization of the prediction error (MPSE). By leveraging these prediction errors, the models consistently refine expectations and decrease computational complexity. Typically, these algorithms are operated by the GeneRec algorithm. Error-driven learning has widespread applications in cognitive sciences and computer vision. These methods have also found successful application in natural language processing (NLP), including areas like part-of-speech tagging, parsing, named entity recognition (NER), machine translation (MT), speech recognition (SR), and dialogue systems. == Formal Definition == Error-driven learning models are ones that rely on the feedback of prediction errors to adjust the expectations or parameters of a model. The key components of error-driven learning include the following: A set S {\displaystyle S} of states representing the different situations that the learner can encounter. A set A {\displaystyle A} of actions that the learner can take in each state. A prediction function P ( s , a ) {\displaystyle P(s,a)} that gives the learner's current prediction of the outcome of taking action a {\displaystyle a} in state s {\displaystyle s} . An error function E ( o , p ) {\displaystyle E(o,p)} that compares the actual outcome o {\displaystyle o} with the prediction p {\displaystyle p} and produces an error value. An update rule U ( p , e ) {\displaystyle U(p,e)} that adjusts the prediction p {\displaystyle p} in light of the error e {\displaystyle e} . == Algorithms == Error-driven learning algorithms refer to a category of reinforcement learning algorithms that leverage the disparity between the real output and the expected output of a system to regulate the system's parameters. Typically applied in supervised learning, these algorithms are provided with a collection of input-output pairs to facilitate the process of generalization. The widely utilized error backpropagation learning algorithm is known as GeneRec, a generalized recirculation algorithm primarily employed for gene prediction in DNA sequences. Many other error-driven learning algorithms are derived from alternative versions of GeneRec. == Applications == === Cognitive science === Simpler error-driven learning models effectively capture complex human cognitive phenomena and anticipate elusive behaviors. They provide a flexible mechanism for modeling the brain's learning process, encompassing perception, attention, memory, and decision-making. By using errors as guiding signals, these algorithms adeptly adapt to changing environmental demands and objectives, capturing statistical regularities and structure. Furthermore, cognitive science has led to the creation of new error-driven learning algorithms that are both biologically acceptable and computationally efficient. These algorithms, including deep belief networks, spiking neural networks, and reservoir computing, follow the principles and constraints of the brain and nervous system. Their primary aim is to capture the emergent properties and dynamics of neural circuits and systems. === Computer vision === Computer vision is a complex task that involves understanding and interpreting visual data, such as images or videos. In the context of error-driven learning, the computer vision model learns from the mistakes it makes during the interpretation process. When an error is encountered, the model updates its internal parameters to avoid making the same mistake in the future. This repeated process of learning from errors helps improve the model's performance over time. For NLP to do well at computer vision, it employs deep learning techniques. This form of computer vision is sometimes called neural computer vision (NCV), since it makes use of neural networks. NCV therefore interprets visual data based on a statistical, trial and error approach and can deal with context and other subtleties of visual data. === Natural Language Processing === ==== Part-of-speech tagging ==== Part-of-speech (POS) tagging is a crucial component in Natural Language Processing (NLP). It helps resolve human language ambiguity at different analysis levels. In addition, its output (tagged data) can be used in various applications of NLP such as information extraction, information retrieval, question Answering, speech eecognition, text-to-speech conversion, partial parsing, and grammar correction. ==== Parsing ==== Parsing in NLP involves breaking down a text into smaller pieces (phrases) based on grammar rules. If a sentence cannot be parsed, it may contain grammatical errors. In the context of error-driven learning, the parser learns from the mistakes it makes during the parsing process. When an error is encountered, the parser updates its internal model to avoid making the same mistake in the future. This iterative process of learning from errors helps improve the parser's performance over time. In conclusion, error-driven learning plays a crucial role in improving the accuracy and efficiency of NLP parsers by allowing them to learn from their mistakes and adapt their internal models accordingly. ==== Named entity recognition (NER) ==== NER is the task of identifying and classifying entities (such as persons, locations, organizations, etc.) in a text. Error-driven learning can help the model learn from its false positives and false negatives and improve its recall and precision on (NER). In the context of error-driven learning, the significance of NER is quite profound. Traditional sequence labeling methods identify nested entities layer by layer. If an error occurs in the recognition of an inner entity, it can lead to incorrect identification of the outer entity, leading to a problem known as error propagation of nested entities. This is where the role of NER becomes crucial in error-driven learning. By accurately recognizing and classifying entities, it can help minimize these errors and improve the overall accuracy of the learning process. Furthermore, deep learning-based NER methods have shown to be more accurate as they are capable of assembling words, enabling them to understand the semantic and syntactic relationship between various words better. ==== Machine translation ==== Machine translation is a complex task that involves converting text from one language to another. In the context of error-driven learning, the machine translation model learns from the mistakes it makes during the translation process. When an error is encountered, the model updates its internal parameters to avoid making the same mistake in the future. This iterative process of learning from errors helps improve the model's performance over time. ==== Speech recognition ==== Speech recognition is a complex task that involves converting spoken language into written text. In the context of error-driven learning, the speech recognition model learns from the mistakes it makes during the recognition process. When an error is encountered, the model updates its internal parameters to avoid making the same mistake in the future. This iterative process of learning from errors helps improve the model's performance over time. ==== Dialogue systems ==== Dialogue systems are a popular NLP task as they have promising real-life applications. They are also complicated tasks since many NLP tasks deserving study are involved. In the context of error-driven learning, the dialogue system learns from the mistakes it makes during the dialogue process. When an error is encountered, the model updates its internal parameters to avoid making the same mistake in the future. This iterative process of learning from errors helps improve the model's performance over time. == Advantages == Error-driven learning has several advantages over other types of machine learning algorithms: They can learn from feedback and correct their mistakes, which makes them adaptive and robust to noise and changes in the data. They can handle large and high-dimensional data sets, as they do not require explicit feature engineering or prior knowledge of the data distribution. They can achieve high accuracy and performance, as they can learn complex and nonlinear relationships between the input and the output. == Limitations == Although error driven learning has its advantages, their algorithms also have the following limitations: They can suffer from overfitting, which means that they memorize the training data and fail to generalize to new and unseen data. This can be mitigated by using regularization techniques, such as adding a penalty term to the loss function, or reducing the complexity of the model. They can be sensitive to the choice of
Termcap
Termcap (terminal capability) is a legacy software library and database used on Unix-like computers that enables programs to use display computer terminals in a terminal-independent manner, which greatly simplifies the process of writing portable text mode applications. It was superseded by the terminfo database used by ncurses, tput, and other programs. A termcap database can describe the capabilities of hundreds of different display terminals. This allows programs to have character-based display output, independent of the type of terminal. On-screen text editors such as vi and Emacs are examples of programs that may use termcap. Other programs are listed in the Termcap category. Access to the termcap database was usually provided by separate libraries, e.g. GNU Termcap. Examples of what the database describes: how many columns wide the display is what string to send to move the cursor to an arbitrary position (including how to encode the row and column numbers) how to scroll the screen up one or several lines how much padding is needed for such a scrolling operation. == History == Bill Joy wrote the first termcap library in 1978 for the Berkeley Unix operating system; it has since been ported to most Unix and Unix-like environments, even OS-9. Joy's design was reportedly influenced by the design of the terminal data store in the earlier Incompatible Timesharing System. == Data model == Termcap databases consist of one or more descriptions of terminals. === Indices === Each description must contain the canonical name of the terminal. It may also contain one or more aliases for the name of the terminal. The canonical name or aliases are the keys by which the library searches the termcap database. === Data values === The description contains one or more capabilities, which have conventional names. The capabilities are typed: boolean, numeric and string. The termcap library has no predetermined type for each capability name. It determines the types of each capability by the syntax: string capabilities have an "=" between the capability name and its value, numeric capabilities have a "#" between the capability name and its value, and boolean capabilities have no associated value (they are always true if specified). Applications which use termcap do expect specific types for the commonly used capabilities, and obtain the values of capabilities from the termcap database using library calls that return successfully only when the database contents matches the assumed type. === Hierarchy === Termcap descriptions can be constructed by including the contents of one description in another, suppressing capabilities from the included description or overriding or adding capabilities. No matter what storage model is used, the termcap library constructs the terminal description from the requested description, including, suppressing or overriding at the time of the request. == Storage model == Termcap data is stored as text, making it simple to modify. The text can be retrieved by the termcap library from files or environment variables. === Environment variables === The TERM environment variable contains the terminal type name. The TERMCAP environment variable may contain a termcap database. It is most often used to store a single termcap description, set by a terminal emulator to provide the terminal's characteristics to the shell and dependent programs. The TERMPATH environment variable is supported by newer termcap implementations and defines a search path for termcap files. === Flat file === The original (and most common) implementation of the termcap library retrieves data from a flat text file. Searching a large termcap file, e.g., 500 kB, can be slow. To aid performance, a utility such as reorder is used to put the most frequently used entries near the beginning of the file. === Hashed database === 4.4BSD based implementations of termcap store the terminal description in a hashed database (e.g., something like Berkeley DB version 1.85). These store two types of records: aliases which point to the canonical entry, and the canonical entry itself. The text of the termcap entry is stored literally. == Limitations and extensions == The original termcap implementation was designed to use little memory: the first name is two characters, to fit in 16 bits capability names are two characters descriptions are limited to 1023 characters. only one termcap entry with its definitions can be included, and must be at the end. Newer implementations of the termcap interface generally do not require the two-character name at the beginning of the entry. Capability names are still two characters in all implementations. The tgetent function used to read the terminal description uses a buffer whose size must be large enough for the data, and is assumed to be 1024 characters. Newer implementations of the termcap interface may relax this constraint by allowing a null pointer in place of the fixed buffer, or by hiding the data which would not fit, e.g., via the ZZ capability in NetBSD termcap. The terminfo library interface also emulates the termcap interface, and does not actually use the fixed-size buffer. The terminfo library's emulation of termcap allows multiple other entries to be included without restricting the position. A few other newer implementations of the termcap library may also provide this ability, though it is not well documented. == Obsolete features == A special capability, the "hz" capability, was defined specifically to support the Hazeltine 1500 terminal, which had the unfortunate characteristic of using the ASCII tilde character ('~') as a control sequence introducer. In order to support that terminal, not only did code that used the database have to know about using the tilde to introduce certain control sequences, but it also had to know to substitute another printable character for any tildes in the displayed text, since a tilde in the text would be interpreted by the terminal as the start of a control sequence, resulting in missing text and screen garbling. Additionally, attribute markers (such as start and end of underlining) themselves took up space on the screen. Comments in the database source code often referred to this as "Hazeltine braindamage". Since the Hazeltine 1500 was a widely used terminal in the late 1970s, it was important for applications to be able to deal with its limitations.
Charge based boundary element fast multipole method
The charge-based formulation of the boundary element method (BEM) is a dimensionality reduction numerical technique that is used to model quasistatic electromagnetic phenomena in highly complex conducting media (targeting, e.g., the human brain) with a very large (up to approximately 1 billion) number of unknowns. The charge-based BEM solves an integral equation of the potential theory written in terms of the induced surface charge density. This formulation is naturally combined with fast multipole method (FMM) acceleration, and the entire method is known as charge-based BEM-FMM. The combination of BEM and FMM is a common technique in different areas of computational electromagnetics and, in the context of bioelectromagnetism, it provides improvements over the finite element method. == Historical development == Along with more common electric potential-based BEM, the quasistatic charge-based BEM, derived in terms of the single-layer (charge) density, for a single-compartment medium has been known in the potential theory since the beginning of the 20th century. For multi-compartment conducting media, the surface charge density formulation first appeared in discretized form (for faceted interfaces) in the 1964 paper by Gelernter and Swihart. A subsequent continuous form, including time-dependent and dielectric effects, appeared in the 1967 paper by Barnard, Duck, and Lynn. The charge-based BEM has also been formulated for conducting, dielectric, and magnetic media, and used in different applications. In 2009, Greengard et al. successfully applied the charge-based BEM with fast multipole acceleration to molecular electrostatics of dielectrics. A similar approach to realistic modeling of the human brain with multiple conducting compartments was first described by Makarov et al. in 2018. Along with this, the BEM-based multilevel fast multipole method has been widely used in radar and antenna studies at microwave frequencies as well as in acoustics. == Physical background - surface charges in biological media == The charge-based BEM is based on the concept of an impressed (or primary) electric field E i {\displaystyle \mathbf {E} ^{i}} and a secondary electric field E s {\displaystyle \mathbf {E} ^{s}} . The impressed field is usually known a priori or is trivial to find. For the human brain, the impressed electric field can be classified as one of the following: A conservative field E i {\displaystyle \mathbf {E} ^{i}} derived from an impressed density of EEG or MEG current sources in a homogeneous infinite medium with the conductivity σ {\displaystyle \sigma } at the source location; An instantaneous solenoidal field E i {\displaystyle \mathbf {E} ^{i}} of an induction coil obtained from Faraday's law of induction in a homogeneous infinite medium (air), when transcranial magnetic stimulation (TMS) problems are concerned; A surface field E i {\displaystyle \mathbf {E} ^{i}} derived from an impressed surface current density J i = σ E i {\displaystyle \mathbf {J} ^{i}=\sigma \mathbf {E} ^{i}} of current electrodes injecting electric current at a boundary of a compartment with conductivity σ {\displaystyle \sigma } when transcranial direct-current stimulation (tDCS) or deep brain stimulation (DBS) are concerned; A conservative field E i {\displaystyle \mathbf {E} ^{i}} of charges deposited on voltage electrodes for tDCS or DBS. This specific problem requires a coupled treatment since these charges will depend on the environment; In application to multiscale modeling, a field E i {\displaystyle \mathbf {E} ^{i}} obtained from any other macroscopic numerical solution in a small (mesoscale or microscale) spatial domain within the brain. For example, a constant field can be used. When the impressed field is "turned on", free charges located within a conducting volume D immediately begin to redistribute and accumulate at the boundaries (interfaces) of regions of different conductivity in D. A surface charge density ρ ( r ) {\displaystyle \rho (\mathbf {r} )} appears on the conductivity interfaces. This charge density induces a secondary conservative electric field E s {\displaystyle \mathbf {E} ^{s}} following Coulomb's law. One example is a human under a direct current powerline with the known field E i {\displaystyle \mathbf {E} ^{i}} directed down. The superior surface of the human's conducting body will be charged negatively while its inferior portion is charged positively. These surface charges create a secondary electric field that effectively cancels or blocks the primary field everywhere in the body so that no current will flow within the body under DC steady state conditions. Another example is a human head with electrodes attached. At any conductivity interface with a normal vector n {\displaystyle \mathbf {n} } pointing from an "inside" (-) compartment of conductivity σ − {\displaystyle \sigma ^{-}} to an "outside" (+) compartment of conductivity σ + {\displaystyle \sigma ^{+}} , Kirchhoff's current law requires continuity of the normal component of the electric current density. This leads to the interfacial boundary condition in the form for every facet at a triangulated interface. As long as σ ± {\displaystyle \sigma ^{\pm }} are different from each other, the two normal components of the electric field, E ± ⋅ n {\displaystyle \mathbf {E} ^{\pm }\cdot \mathbf {n} } , must also be different. Such a jump across the interface is only possible when a sheet of surface charge exists at that interface. Thus, if an electric current or voltage is applied, the surface charge density follows. The goal of the numerical analysis is to find the unknown surface charge distribution and thus the total electric field E = E i + E s {\displaystyle \mathbf {E} =\mathbf {E} ^{i}+\mathbf {E} ^{s}} (and the total electric potential if required) anywhere in space. == System of equations for surface charges == Below, a derivation is given based on Gauss's law and Coulomb's law. All conductivity interfaces, denoted by S, are discretized into planar triangular facets t m {\displaystyle t_{m}} with centers r m {\displaystyle \mathbf {r} _{m}} . Assume that an m-th facet with the normal vector n m {\displaystyle \mathbf {n} _{m}} and area A m {\displaystyle A_{m}} carries a uniform surface charge density ρ m {\displaystyle \rho _{m}} . If a volumetric tetrahedral mesh were present, the charged facets would belong to tetrahedra with different conductivity values. We first compute the electric field E m + {\displaystyle \mathbf {E} _{m}^{+}} at the point r m + δ n m {\displaystyle \mathbf {r} _{m}+\delta \mathbf {n} _{m}} , for δ → 0 + {\displaystyle \delta \rightarrow 0^{+}} i.e., just outside facet 𝑚 at its center. This field contains three contributions: The continuous impressed electric field E i {\displaystyle \mathbf {E} ^{i}} itself; An electric field of the m-th charged facet itself. Very close to the facet, it can be approximated as the electric field of an infinite sheet of uniform surface charge ρ m {\displaystyle \rho _{m}} . By Gauss's law, it is given by + ρ m / 2 ε 0 ⋅ n m {\displaystyle +\rho _{m}/2\varepsilon _{0}\cdot \mathbf {n} _{m}} where ε 0 {\displaystyle \varepsilon _{0}} is a background electrical permittivity; An electric field generated by all other facets t n {\displaystyle t_{n}} , which we approximate as point charges of charge A n ρ n {\displaystyle A_{n}\rho _{n}} at each center r n {\displaystyle \mathbf {r} _{n}} . A similar treatment holds for the electric field E m − {\displaystyle \mathbf {E} _{m}^{-}} just inside facet 𝑚, but the electric field of the flat sheet of charge changes its sign. Using Coulomb's law to calculate the contribution of facets different from t m {\displaystyle t_{m}} , we find From this equation, we see that the normal component of the electric field indeed undergoes a jump through the charged interface. This is equivalent to a jump relation of the potential theory. As a second step, the two expressions for E m ± {\displaystyle \mathbf {E} _{m}^{\pm }} are substituted into the interfacial boundary condition σ − E m − ⋅ n m = σ + E m + ⋅ n m {\displaystyle \sigma ^{-}\mathbf {E} _{m}^{-}\cdot \mathbf {n} _{m}=\sigma ^{+}\mathbf {E} _{m}^{+}\cdot \mathbf {n} _{m}} , applied to every facet 𝑚. This operation leads to a system of linear equations for unknown charge densities ρ m {\displaystyle \rho _{m}} which solves the problem: where K m = σ − − σ + σ − + σ + {\displaystyle K_{m}={\frac {\sigma ^{-}-\sigma ^{+}}{\sigma ^{-}+\sigma ^{+}}}} is the electric conductivity contrast at the m-th facet. The normalization constant ε 0 {\displaystyle \varepsilon _{0}} will cancel out after the solution is substituted in the expression for E s {\displaystyle \mathbf {E} ^{s}} and becomes redundant. == Application of fast multipole method == For modern characterizations of brain topologies with ever-increasing levels of complexity, the above system of equations for ρ m {\displaystyle \rho _{m}} is very large; it is t