The GeoNetwork opensource (GNOS) project is a free and open source (FOSS) cataloging application for spatially referenced resources. It is a catalog of location-oriented information. == Outline == It is a standardized and decentralized spatial information management environment designed to enable access to geo-referenced databases, cartographic products and related metadata from a variety of sources, enhancing the spatial information exchange and sharing between organizations and their audience, using the capacities of the internet. Using the Z39.50 protocol it both accesses remote catalogs and makes its data available to other catalog services. As of 2007, OGC Web Catalog Service are being implemented. Maps, including those derived from satellite imagery, are effective communicational tools and play an important role in the work of decision makers (e.g., sustainable development planners and humanitarian and emergency managers) in need of quick, reliable and up-to-date user-friendly cartographic products as a basis for action and to better plan and monitor their activities; GIS experts in need of exchanging consistent and updated geographical data; and spatial analysts in need of multidisciplinary data to perform preliminary geographical analysis and make reliable forecasts. == Deployment == The software has been deployed to various organizations, the first being FAO GeoNetwork and WFP VAM-SIE-GeoNetwork, both at their headquarters in Rome, Italy. Furthermore, the WHO, CGIAR, BRGM, ESA, FGDC and the Global Change Information and Research Centre (GCIRC) of China are working on GeoNetwork opensource implementations as their spatial information management capacity. It is used for several risk information systems, in particular in the Gambia. Several related tools are packaged with GeoNetwork, including GeoServer. GeoServer stores geographical data, while GeoNetwork catalogs collections of such data.
Sycophancy (artificial intelligence)
In the field of artificial intelligence, sycophancy is a tendency of large language models (LLMs) and other AI assistants to tailor their responses to what they predict the user wants to hear rather than to what is accurate or warranted. The behavior takes several forms: an assistant may agree with a user's stated opinion even when the user is mistaken; it may abandon a correct answer after a challenge such as "are you sure?"; it may validate beliefs, decisions or self-presentation regardless of merit; or it may praise the user, their work or their ideas in unwarranted terms. The word is borrowed from the ordinary English term for fawning flattery, and is used in AI alignment and AI safety research to describe a class of misalignment failures associated with training on human feedback. Researchers at Anthropic first documented the behavior systematically in 2022. They found that models fine-tuned with reinforcement learning from human feedback (RLHF) were more likely than untuned models to repeat back a user's preferred answer. A 2023 follow-up paper, "Towards Understanding Sycophancy in Language Models", showed that five frontier assistants from OpenAI, Anthropic and Meta all exhibited the behavior, and traced its origin to biases in the human preference data used during training. Later work documented sycophancy in mathematics, medicine, academic peer review and other domains, and identified a broader category called "social sycophancy" affecting an assistant's emotional and interpersonal responses. The issue drew widespread public attention in April 2025 after OpenAI rolled back an update to its GPT-4o model. Users had reported that the assistant praised dangerous decisions, endorsed delusional thinking and offered exaggerated compliments for trivial prompts. OpenAI's post-mortem attributed the change in behavior to an additional training signal based on user thumbs-up and thumbs-down feedback. That episode, together with reporting in The New York Times, Rolling Stone and elsewhere on users drawn into delusional thinking through prolonged chatbot interaction, has been cited in litigation and in academic studies as evidence that sycophancy poses risks to user well-being. Proposed mitigations include fine-tuning on synthetic data that rewards disagreement with incorrect user statements, editing the small subset of model parameters causally responsible for the behavior, changes to the dialogue or system prompt, and benchmarks designed to surface sycophantic behavior before models are released. == Causes == The dominant explanation points to RLHF, the standard technique for aligning chat assistants with user expectations. Human annotators rank candidate model responses; a reward model is trained to predict those rankings; and the language model is then optimized against the reward model. Because human raters tend to prefer outputs that confirm their existing beliefs or flatter their work, the pipeline systematically rewards responses that agree with the annotator. Perez and colleagues at Anthropic published the first large-scale empirical evidence of the effect in 2022. They reported that RLHF training increased the probability that a model would repeat back a dialog user's preferred answer, and that larger models exhibited the behavior more strongly. Sharma and colleagues, the following year, went further and examined Anthropic's own preference data directly. Both the human raters and the reward models trained on their judgments preferred convincingly written sycophantic responses to truthful ones at a non-negligible rate. Wei and co-authors at Google DeepMind found similar results in the PaLM family, observing that both model scale and instruction tuning increased sycophancy on opinion questions. The behavior is often classified as a form of reward hacking, in which an optimization process exploits a flaw in its reward signal rather than achieving the intended objective. OpenAI's post-mortem of the April 2025 GPT-4o incident identified a more specific mechanism. An additional reward signal based on aggregated thumbs-up and thumbs-down feedback from ChatGPT users had, in OpenAI's words, "weakened the influence of our primary reward signal, which had been holding sycophancy in check." Separately, an Anthropic interpretability paper from 2025 located a linear direction in a model's internal activations corresponding to sycophantic behavior, and showed that such "persona vectors" could be used to flag sycophancy-inducing training data and to steer models away from the trait at inference time. == Measurement == The Anthropic team released SycophancyEval with its 2023 paper, supplying test sets for each of the four canonical behaviors. Two further benchmarks from Stanford followed in 2025. SycEval, applied to mathematical and medical reasoning tasks, reported an overall sycophancy rate of 58 per cent across the GPT-4o, Claude and Gemini models tested. ELEPHANT, aimed at social sycophancy, found that the eleven LLMs evaluated affirmed posts that the Reddit community r/AmITheAsshole had judged inappropriate in 42 per cent of cases, and preserved a user's face 45 percentage points more often than human respondents did. Domain-specific benchmarks have followed. BrokenMath tests robustness to plausible-looking but false mathematical claims drawn from competition problems, and reports that the best evaluated model was sycophantic in 29 per cent of cases. SYCON-Bench measures how many dialogue turns are required before a model abandons a correct position. Visual sycophancy in multimodal models has been examined with MM-SY and PENDULUM. A 2026 study by researchers at the Massachusetts Institute of Technology reported that personalization features, which adapt assistants to individual users over repeated sessions, can intensify social sycophancy. == Notable incidents == === GPT-4o rollback (April 2025) === On 25 April 2025, OpenAI completed the rollout of an update to GPT-4o, the default model used in ChatGPT at the time. Within days, users reported that the assistant had begun praising trivial messages in extravagant terms, endorsing impulsive or dangerous decisions, and reinforcing strong emotional statements without pushback. Widely shared examples included the model congratulating a user who reported stopping prescribed psychiatric medication, and praising a business plan to sell "shit on a stick" as venture-capital ready. OpenAI's chief executive, Sam Altman, wrote on 27 April that recent updates had made the model "too sycophant-y and annoying" and said fixes were in progress. The company began reverting the update on 28 April and completed the rollback for free users by 30 April. Two post-mortems followed: a short note on 29 April and a longer technical follow-up, "Expanding on what we missed with sycophancy", on 2 May. Both attributed the regression to a new training signal based on user thumbs-up and thumbs-down feedback, to inadequate pre-launch evaluation for sycophantic drift, and to the dismissal of qualitative concerns raised by internal testers before release. Reporting in CNN, Fortune and Bloomberg News treated the incident as a turning point in public awareness of the problem. === Chatbot-related psychological harm === From mid-2025 onward, news reports began to link sycophantic chatbot behavior to acute psychological harm. In June 2025, The New York Times technology reporter Kashmir Hill published an investigation centered on Eugene Torres, a Manhattan accountant with no history of mental illness, who developed a sustained delusional episode after a series of conversations with ChatGPT about simulation theory. According to the article, the assistant encouraged Torres to stop taking prescribed medication, to cut off friends and family, and at one point told him that he could fly from a nineteen-story building if he "truly believed". Futurism and Rolling Stone ran parallel investigations documenting other cases in which heavy use of ChatGPT had been associated with delusional thinking, involuntary commitment or, in at least one case, the death of a user with a pre-existing psychiatric diagnosis. A 2026 paper by researchers at the Massachusetts Institute of Technology and the University of Washington put forward a formal Bayesian model. It showed that even an ideally rational user could be drawn into what the authors call "delusional spiraling" when interacting with a sufficiently sycophantic assistant, and that the effect was not eliminated by suppressing hallucinations or by warning users in advance. The lawsuit Raine v. OpenAI, filed in San Francisco Superior Court in August 2025 by the parents of a sixteen-year-old who had died by suicide, alleges that "heightened sycophancy" was a design feature of ChatGPT that contributed to their son's death; it is the first wrongful-death suit against a large language-model provider. === Wider commentary === Mainstream coverage in outlets including The New York Times, The Washington Pos
Interstellar communication
Interstellar communication is the transmission of signals between planetary systems. Sending interstellar messages is potentially much easier than interstellar travel, being possible with technologies and equipment which are currently available. However, the distances from Earth to other potentially inhabited systems introduce prohibitive delays, assuming the limitations of the speed of light. Even an immediate reply to radio communications sent to stars tens of thousands of light-years away would take many human generations to arrive. == Radio == The SETI project has for the past several decades been conducting a search for signals being transmitted by extraterrestrial life located outside the Solar System, primarily in the radio frequencies of the electromagnetic spectrum. Special attention has been given to the Water Hole, the frequency of one of neutral hydrogen's absorption lines, due to the low background noise at this frequency and its symbolic association with the basis for what is likely to be the most common system of biochemistry (but see alternative biochemistry). The regular radio pulses emitted by pulsars were briefly thought to be potential intelligent signals; the first pulsar to be discovered was originally designated "LGM-1", for "Little Green Men." They were quickly determined to be of natural origin, however. Several attempts have been made to transmit signals to other stars as well. (See "Realized projects" at Active SETI.) One of the earliest and most famous was the 1974 radio message sent from the largest radio telescope in the world, the Arecibo Observatory in Puerto Rico. An extremely simple message was aimed at a globular cluster of stars known as M13 in the Milky Way Galaxy and at a distance of 30,000 light years from the Solar System. These efforts have been more symbolic than anything else, however. Further, a possible answer needs double the travel time, i.e. tens of years (near stars) or 60,000 years (M13). == Other methods == It has also been proposed that higher frequency signals, such as lasers operating at visible light frequencies, may prove to be a fruitful method of interstellar communication; at a given frequency it takes surprisingly small energy output for a laser emitter to outshine its local star from the perspective of its target. Other more exotic methods of communication have been proposed, such as modulated neutrino or gravitational wave emissions. These would have the advantage of being essentially immune to interference by intervening matter. Sending physical mail packets between stars may prove to be optimal for many applications. While mail packets would likely be limited to speeds far below that of electromagnetic or other light-speed signals (resulting in very high latency), the amount of information that could be encoded in only a few tons of physical matter could more than make up for it in terms of average bandwidth. The possibility of using interstellar messenger probes for interstellar communication — known as Bracewell probes — was first suggested by Ronald N. Bracewell in 1960, and the technical feasibility of this approach was demonstrated by the British Interplanetary Society's starship study Project Daedalus in 1978. Starting in 1979, Robert Freitas advanced arguments for the proposition that physical space-probes provide a superior mode of interstellar communication to radio signals, then undertook telescopic searches for such probes in 1979 and 1982.
Digital media service
A digital media service (DMS) is an online service provider that sells access to digital library of content such as films, software, games, images, literature, etc. While no transfer of property is made, a nearly perfect duplicate of the data (song movie, etc.) is made on a customer's computer. Content is either primarily hosted on a dedicated server, which is owned by the service provider, or it is hosted primarily on the hard drives of its customers using a P2P protocol with, perhaps, a dedicated server to supplement. == History == One example of the older business model is the iTunes Store, which still markets and prices data as individual retail products. There are no examples of the latter business model in operation yet, but one is currently in development by Global Gaming Factory X and expected to begin operation some time after they acquire The Pirate Bay domain on August 27, 2009. A key difference between the two models is that the model which relies on its customer base for offering their bandwidth for other customers to access customer hosted data can operate at significantly lower costs than a company that seeks to limit data access to a per-download fee in order to supplement the cost of using its own hosting and bandwidth. The P2P model holds the potential for companies to offer unlimited access to the largest data library in the history of the internet to its customers for a reasonably low membership rate that is relevant to the cost of operation. While the market is virtually untouched, the P2P supplemented model will need entrepreneurs who are able to overcome a series of challenges in order to compete with the older business model as well as that which is offered for free (and often against the wishes of copyright holders) by hundreds of P2P communities on the internet. These challenges include, but are not limited to: Offering better data quality, speed, convenience and ease of use, protocol, sense of security, indexing and search organization, site up time, data library size, customer support, advertising, artist/copyright holder incentives and compensation, incentives and compensation for customers hosting data and providing bandwidth, guaranteed seeding (available access to indexed data at all times), than competitors.
SD-WAN
A Software-Defined Wide Area Network (SD-WAN) is a wide area network that uses software-defined networking technology, such as communicating over the Internet using overlay tunnels which are encrypted when destined for internal organization locations. If standard tunnel setup and configuration messages are supported by all of the network hardware vendors, SD-WAN simplifies the management and operation of a WAN by decoupling the networking hardware from its control mechanism. This concept is similar to how software-defined networking implements virtualization technology to improve data center management and operation. In practice, proprietary protocols are used to set up and manage an SD-WAN, meaning there is no decoupling of the hardware and its control mechanism. A key application of SD-WAN is to allow companies to build higher-performance WANs using lower-cost and commercially available Internet access, enabling businesses to partially or wholly replace more expensive private WANs connection technologies such as MPLS. When SD-WAN traffic is carried over the Internet, there are no end-to-end performance guarantees. Carrier MPLS VPN WAN services are not carried as Internet traffic, but rather over carefully controlled carrier capacity, and do come with an end-to-end performance guarantee. == History == WANs were very important for the development of networking in general and for a long time one of the most important applications of networks both for military and enterprise applications. The ability to communicate data over long distances was one of the main driving factors for the development of data communications, as it made it possible to overcome the distance limitations, as well as shortening the time necessary to exchange messages with other parties. Legacy WANs allowed communication over circuits connecting two or more endpoints. Earlier networking supported point-to-point communication over a slow speed circuit, usually between two fixed locations. As networking progressed, WAN circuits became faster and more flexible. Innovations like circuit and packet switching (in the form of X.25, ATM and later Internet Protocol or Multiprotocol Label Switching) allowed communication to become more dynamic, supporting ever-growing networks. The need for strict control, security and quality of service (QOS) meant that multinational corporations were very conservative in leasing and operating their WANs. National regulations restricted the companies that could provide local service in each country, and complex arrangements were necessary to establish truly global networks. All that changed with the growth of the Internet, which permitted entities around the world to connect to each other. However, over the first years, the uncontrolled nature of the Internet was not considered adequate or safe for private corporate use. Independent of safety concerns, connectivity to the Internet became a necessity to the point where every branch required Internet access. At first, due to safety concerns, private communications were still done via WAN, and communication with other entities (including customers and partners) moved to the Internet. As the Internet grew in reach and maturity, companies started to evaluate how to leverage it for private corporate communications. During the early 2000s, application delivery over the WAN became an important topic of research and commercial innovation. Over the next decade, increasing computing power made it possible to create software-based appliances that were able to analyze traffic and make informed decisions without delays, making it possible to create large-scale overlay networks over the public Internet that could replicate all the functionality of legacy WANs, at a fraction of the cost. SD-WAN combines several networking aspects to create full-fledged private networks, with the ability to dynamically share network bandwidth across the connection points. Additional enhancements include central controllers, zero-touch provisioning, integrated analytics and on-demand circuit provisioning, with some network intelligence based in the cloud, allowing centralized policy management and security. Networking publications started using the term SD-WAN to describe this new networking trend as early as 2014. With the rapid shift to remote work as a result of lockdowns and stay at home orders during the COVID-19 pandemic, SD-WAN grew in popularity as a way of connecting remote workers. == Overview == WANs allow companies to extend their computer networks over large distances, connecting remote branch offices to data centers and to each other, and delivering applications and services required to perform business functions. Due to the physical constraints imposed by the propagation time over large distances, and the need to integrate multiple service providers to cover global geographies (often crossing nation boundaries), WANs face important operational challenges, including network congestion, packet delay variation, packet loss, and even service outages. Modern applications such as VoIP calling, videoconferencing, streaming media, and virtualized applications and desktops require low latency. Bandwidth requirements are also increasing, especially for applications featuring high-definition video. It can be expensive and difficult to expand WAN capability, with corresponding difficulties related to network management and troubleshooting. SD-WAN products are designed to address these network problems. By enhancing or even replacing traditional branch routers with virtualization appliances that can control application-level policies and offer a network overlay, less expensive consumer-grade Internet links can act more like a dedicated circuit. This simplifies the setup process for branch personnel. SD-WAN products can be physical appliances or software based only. === Components === The MEF Forum has defined an SD-WAN architecture consisting of an SD-WAN edge, SD-WAN gateway, SD-WAN controller and SD-WAN orchestrator. ==== SD-WAN edge ==== The SD-WAN edge is a physical or virtual network function that is placed at an organization's branch/regional/central office site, data center, and in public or private cloud platforms. MEF Forum has published the first SD-WAN service standard, MEF 70 which defines the fundamental characteristics of an SD-WAN service plus service requirements and attributes. ==== SD-WAN gateway ==== SD-WAN gateways provide access to the SD-WAN service in order to shorten the distance to cloud-based services or the user, and reduce service interruptions. A distributed network of gateways may be included in an SD-WAN service by the vendor or setup and maintained by the organization using the service. By sitting outside the headquarters in the cloud, the gateway also reduces headquarters traffic. ==== SD-WAN orchestrator ==== The SD-WAN orchestrator is a cloud hosted or on-premises web management tool that allows configuration, provisioning and other functions when operating an SD-WAN. It simplifies application traffic management by allowing central implementation of an organization's business policies. ==== SD-WAN controller ==== The SD-WAN controller functionality, which can be placed in the orchestrator or in an SD-WAN gateway, is used to make forwarding decisions for application flows. Application flows are IP packets that have been classified to determine their user application or grouping of applications to which they are associated. The grouping of application flows based on a common type, e.g., conferencing applications, is referred to as an Application Flow Group in MEF 70. Per MEF 70, the SD-WAN Edge classifies incoming IP packets at the SD-WAN UNI (SD-WAN user network interface), determines, via OSI Layer 2 through Layer 7 classification, which application flow the IP packets belong to, and then applies the policies to block the application flow or allow the application flows to be forwarded based on the availability of a route to the destination SD-WAN UNI on a remote SD-WAN Edge. This helps ensure that application performance meets service level agreements (SLAs). == Required characteristics == The Gartner research firm has defined an SD-WAN as having four required characteristics: The ability to support multiple connection types, such as MPLS, last mile fiber optic network or through high speed cellular networks e.g. 4G LTE and 5G wireless technologies The ability to do dynamic path selection, for load sharing and resiliency purposes A simple interface that is easy to configure and manage The ability to support VPNs, and third party services such as WAN optimization controllers, firewalls and web gateways == Features == Features of SD-WANs include resilience, quality of service (QoS), security, and performance, with flexible deployment options; simplified administration and troubleshooting; and online traffic engineering. === Resilience === A resilient SD-WAN reduces network downtime. To
Semantic compression
In natural language processing, semantic compression is a process of compacting a lexicon used to build a textual document (or a set of documents) by reducing language heterogeneity, while maintaining text semantics. As a result, the same ideas can be represented using a smaller set of words. In most applications, semantic compression is a lossy compression. Increased prolixity does not compensate for the lexical compression and an original document cannot be reconstructed in a reverse process. == By generalization == Semantic compression is basically achieved in two steps, using frequency dictionaries and semantic network: determining cumulated term frequencies to identify target lexicon, replacing less frequent terms with their hypernyms (generalization) from target lexicon. Step 1 requires assembling word frequencies and information on semantic relationships, specifically hyponymy. Moving upwards in word hierarchy, a cumulative concept frequency is calculating by adding a sum of hyponyms' frequencies to frequency of their hypernym: c u m f ( k i ) = f ( k i ) + ∑ j c u m f ( k j ) {\displaystyle cumf(k_{i})=f(k_{i})+\sum _{j}cumf(k_{j})} where k i {\displaystyle k_{i}} is a hypernym of k j {\displaystyle k_{j}} . Then a desired number of words with top cumulated frequencies are chosen to build a target lexicon. In the second step, compression mapping rules are defined for the remaining words in order to handle every occurrence of a less frequent hyponym as its hypernym in output text. Example The below fragment of text has been processed by the semantic compression. Words in bold have been replaced by their hypernyms. They are both nest building social insects, but paper wasps and honey bees organize their colonies in very different ways. In a new study, researchers report that despite their differences, these insects rely on the same network of genes to guide their social behavior.The study appears in the Proceedings of the Royal Society B: Biological Sciences. Honey bees and paper wasps are separated by more than 100 million years of evolution, and there are striking differences in how they divvy up the work of maintaining a colony. The procedure outputs the following text: They are both facility building insect, but insects and honey insects arrange their biological groups in very different structure. In a new study, researchers report that despite their difference of opinions, these insects act the same network of genes to steer their party demeanor. The study appears in the proceeding of the institution bacteria Biological Sciences. Honey insects and insect are separated by more than hundred million years of organic processes, and there are impinging differences of opinions in how they divvy up the work of affirming a biological group. == Implicit semantic compression == A natural tendency to keep natural language expressions concise can be perceived as a form of implicit semantic compression, by omitting unmeaningful words or redundant meaningful words (especially to avoid pleonasms). == Applications and advantages == In the vector space model, compacting a lexicon leads to a reduction of dimensionality, which results in less computational complexity and a positive influence on efficiency. Semantic compression is advantageous in information retrieval tasks, improving their effectiveness (in terms of both precision and recall). This is due to more precise descriptors (reduced effect of language diversity – limited language redundancy, a step towards a controlled dictionary). As in the example above, it is possible to display the output as natural text (re-applying inflexion, adding stop words).
General time- and transfer constant analysis
The general time- and transfer-constants (TTC) analysis is the generalized version of the Cochran-Grabel (CG) method, which itself is the generalized version of zero-value time-constants (ZVT), which in turn is the generalization of the open-circuit time constant method (OCT). While the other methods mentioned provide varying terms of only the denominator of an arbitrary transfer function, TTC can be used to determine every term both in the numerator and the denominator. Its denominator terms are the same as that of Cochran-Grabel method, when stated in terms of time constants (when expressed in Rosenstark notation). however, the numerator terms are determined using a combination of transfer constants and time constants, where the time constants are the same as those in CG method. Transfer constants are low-frequency ratios of the output variable to input variable under different open- and short-circuited active elements. In general, a transfer function (which can characterize gain, admittance, impedance, trans-impedance, etc., based on the choice of the input and output variables) can be written as: H ( s ) = a 0 + a 1 s + a 2 s 2 + … + a m s m 1 + b 1 s + b 2 s 2 + … + b n s n {\displaystyle H(s)={\frac {a_{0}+a_{1}s+a_{2}s^{2}+\ldots +a_{m}s^{m}}{1+b_{1}s+b_{2}s^{2}+\ldots +b_{n}s^{n}}}} == The denominator terms == The first denominator term b 1 {\textstyle b_{1}} can be expressed as the sum of zero value time constants (ZVTs): b 1 = ∑ i = 1 N τ i 0 {\displaystyle b_{1}=\sum _{i=1}^{N}\tau _{i}^{0}} where τ i 0 {\textstyle \tau _{i}^{0}} is the time constant associated with the reactive element i {\textstyle i} when all the other sources are zero-valued (hence the superscript '0'). Setting a capacitor value to zero corresponds to an open circuit, while a zero-valued inductor is a short circuit. So for calculation of the τ i 0 {\textstyle \tau _{i}^{0}} , all other capacitors are open-circuited and all other inductors are short-circuited. This is the essence of the ZVT method, which reduces to OCT when only capacitors are involved. All independent sources are also zero-valued during the time constant calculations (voltage sources short-circuited and current source open-circuited). In this case, if the element in question (element i {\textstyle i} ) is a capacitor, the time constant is given by τ i 0 = R i 0 C i {\displaystyle \tau _{i}^{0}=R_{i}^{0}C_{i}} and when element i {\textstyle i} is an inductor is it given by: τ i 0 = L i / R i 0 {\displaystyle \tau _{i}^{0}=L_{i}/R_{i}^{0}} . where in both cases, the resistance R i 0 {\textstyle R_{i}^{0}} , is the resistance seen by elements i {\textstyle i} (denoted by subscript), when all the other elements are zero-valued (denoted by the zero superscript). The second-order denominator term is equal to: b 2 = ∑ i = 1 N − 1 ∑ j = i + 1 N τ i 0 τ j i = ∑ i 1 ⩽ i ∑ j < j ⩽ N τ i 0 τ j i {\displaystyle b_{2}=\sum _{i=1}^{N-1}\sum _{j=i+1}^{N}\tau _{i}^{0}\tau _{j}^{i}=\sum _{i}^{1\leqslant i}\sum _{j}^{