Social advertising is advertising that relies on social information or networks in generating, targeting, and delivering marketing communications. Many current examples of social advertising use a particular Internet service to collect social information, establish and maintain relationships with consumers, and for delivering communications. For example, the advertising platforms provided by Google, Twitter, and Facebook involve targeting and presenting ads based on relationships articulated on those same services. Social advertising can be part of a broader social media marketing strategy designed to connect with consumers. == Social targeting == Since a pair of consumers connected via a relationship are more likely to be similar than an unconnected pair, information about such relationships can be used to infer characteristics of consumers useful for targeting. For example, predictions of an individual's home location can be improved using geographic information about their peers. Existing advertising platforms can allow advertisers to explicitly target the peers (e.g., Facebook friends, Twitter followers) of consumers who have a known affiliation with their brand. Thus, one way social advertising is expected to be effective is because social networks encode information about unobserved characteristics of consumers, including their susceptibility to adopt a product and to influence their peers to adopt. Social advertisement targets audiences' demographics based on customers browsing histories. This helped companies understand users' interests and target a specific group of users. Whether it is location or personal interest, different categories of companies can make the consumers on social media rely heavily on their advertisements. This is one of the reasons why social advertising has grown over time. Targeting their audience to real life stakeholders generally increase the attention of the advertised deal which brings up more profits for companies. Subsequently, the psychological effects that social media gives off to its users play a huge role in advertisement companies keeping their customers online. One of the main reasons users rely on social media is because it's a source of entertainment that provides them with a feeling of inclusiveness. In making the customers feel the inclusiveness, social advertising targeting a specific group of users is presented as if these advertisements are customized for the users in their perspective making them feel the attention that they do not often feel in the real world. You can use Social signals checker tool to find more information about links. Social signals are metrics that measure how much people interact with your content on social media. From likes, to shares, to comments; each of these signals contributes to an overall number that tells search engines like Google how much people like your content. The more social signals your website gets, the more likely it is to rank higher in Google. The reason for this is two-fold. First, social media is used by millions of people every day, and if your content is being shared and interacted with on these sites, it shows that it’s worthy of being seen. And second, social media sites are highly trusted by Google. So if you can get your content seen and interacted with on these platforms, you’ll be off to a great start. == Social cues in advertisements == Social ads often include information about the affiliation of a peer with an advertised entity. For example, a social ad might indicate a friend has endorsed a product, highly rated a restaurant, or watched a particular film. In fact, some definitions make these personalized social signals a necessary condition for the advertising being social advertising. Inclusion of personalized social signals creates a channel for social influence. Experiments that remove peers' names or images from social advertisements provide evidence that their presence increases proximal outcomes (e.g., clicks on advertisements). This is technically how trends are started on social media. Since social media links a single profile to thousands of other accounts some being real-life friends or even acquaintances, the opinions and the bias a user has for other users who are also a customer of an advertisement on the feed can heavily affect whether to click on the advertisement or not. Once this pattern continues, the brand benefits from increased customers, profit, and attention. Social networking can spread rapidly because 71 percent of the world's population contributes and uses social media which means social advertising gives companies a better marketing technique than a physical poster advertisement. == Word of mouth == Advertisers often attempt to use word of mouth to affect consumers and their decisions to adopt products and services. Ads and other inducements targeted at a seed set of individuals can be designed to produce a larger cascade of adoption through influence. Businesses are also using social media to attempt to identify and persuade influential consumers to spread positive messages about their products or services. Consequently, not only on social platforms but also in physical settings, users start talking to each other. When individuals develop an intimate relationship with each other, it is quite heavily based on shared characteristics, interests, and personalities. If one social media user becomes a regular customer to a well-known company that advertises often, there is a higher chance that all the other people who have intimate relationships with that one customer will be exposed to the online advertisement more than another user who might be completely new to a brand that is being advertised on screen. In reality, this happens to not only one user but to most of the users which mean a single brand advertisement online can have to potential of being talked about between billions and trillions of people all around the globe. == Relationship marketing == To accurately conduct relationship marketing, businesses must develop and manage six marketplaces: internal, customer, referral, supplier, influencer and employee. To maintain relationship marketing, customers often see social media influencers getting free sponsorships or PR boxes just to advertise their products. At times, users who become customers through these social influencers will get a better deal than regular customers which stands as a very commonly used marketing technique. By doing this, users think they are receiving special treatment when in reality it very much benefits social influencers and brands. Especially for brands that are just starting, they use this marketing technique so that their names can be out there, and people will start talking, which is their initial goal.
Instance selection
Instance selection (or dataset reduction, or dataset condensation) is an important data pre-processing step that can be applied in many machine learning (or data mining) tasks. Approaches for instance selection can be applied for reducing the original dataset to a manageable volume, leading to a reduction of the computational resources that are necessary for performing the learning process. Algorithms of instance selection can also be applied for removing noisy instances, before applying learning algorithms. This step can improve the accuracy in classification problems. Algorithm for instance selection should identify a subset of the total available data to achieve the original purpose of the data mining (or machine learning) application as if the whole data had been used. Considering this, the optimal outcome of IS would be the minimum data subset that can accomplish the same task with no performance loss, in comparison with the performance achieved when the task is performed using the whole available data. Therefore, every instance selection strategy should deal with a trade-off between the reduction rate of the dataset and the classification quality. == Instance selection algorithms == The literature provides several different algorithms for instance selection. They can be distinguished from each other according to several different criteria. Considering this, instance selection algorithms can be grouped in two main classes, according to what instances they select: algorithms that preserve the instances at the boundaries of classes and algorithms that preserve the internal instances of the classes. Within the category of algorithms that select instances at the boundaries it is possible to cite DROP3, ICF and LSBo. On the other hand, within the category of algorithms that select internal instances, it is possible to mention ENN and LSSm. In general, algorithm such as ENN and LSSm are used for removing harmful (noisy) instances from the dataset. They do not reduce the data as the algorithms that select border instances, but they remove instances at the boundaries that have a negative impact on the data mining task. They can be used by other instance selection algorithms, as a filtering step. For example, the ENN algorithm is used by DROP3 as the first step, and the LSSm algorithm is used by LSBo. There is also another group of algorithms that adopt different selection criteria. For example, the algorithms LDIS, CDIS and XLDIS select the densest instances in a given arbitrary neighborhood. The selected instances can include both, border and internal instances. The LDIS and CDIS algorithms are very simple and select subsets that are very representative of the original dataset. Besides that, since they search by the representative instances in each class separately, they are faster (in terms of time complexity and effective running time) than other algorithms, such as DROP3 and ICF. Besides that, there is a third category of algorithms that, instead of selecting actual instances of the dataset, select prototypes (that can be synthetic instances). In this category it is possible to include PSSA, PSDSP and PSSP. The three algorithms adopt the notion of spatial partition (a hyperrectangle) for identifying similar instances and extract prototypes for each set of similar instances. In general, these approaches can also be modified for selecting actual instances of the datasets. The algorithm ISDSP adopts a similar approach for selecting actual instances (instead of prototypes).
1tik
1tik, pronounced Antik (Arabic: أنتيك; lit. "Everything is going well") is a fully Algerian instant messaging, social media and mobile payment app. designed, developed and built locally by the Algerian start-up, INTAJ Digital, with backing from the state-owned company ATM Mobilis (who's the company's main sponsor). It is described as Algeria's first super-app that is entirely designed and built by local developers. == Etymology == The name "1tik" (Arabic: أنتيك) is drawn from the popular Algerian vernacular (Antik), the neologism, which appeared several years ago, means "everything is going well" or "it's all good". == History == 1tik was officially launched and announced the 20th December 2025 by INTAJ Digital's founder Youcef Toulaib and a team of 50 employees, making it the first ever Algerian instant messaging, social media and mobile payment app, rivaling with the growing influence of Yassir in Algeria. it grew in popularity after the presidency of Algeria and several other state-owned companies, medias, and ministries opened official accounts on the app.
Algorithmic radicalization
Algorithmic radicalization is the concept that recommender algorithms on popular social media sites, such as YouTube and Facebook, drive users toward progressively more extreme content over time, leading to the development of radicalized extremist political views. Algorithms meticulously record user interactions, encompassing likes, dislikes and the duration of time watching content, with the objective of generating an endless stream of media designed to sustain user engagement. The phenomenon of echo chamber channels has been demonstrated to exacerbate the polarization of consumers, primarily through the reinforcement of media preferences and the validation of one's existing beliefs. Algorithmic radicalization remains a controversial phenomenon as it is often not in the best interest of social media companies to remove echo chamber channels. To what extent recommender algorithms are actually responsible for radicalization remains disputed. Studies have found contradictory results regarding the promotion of extremist content by algorithms. == Social media echo chambers and filter bubbles == Social media platforms learn the interests and likes of the user to modify their experiences in their feed to keep them engaged and scrolling, known as a filter bubble. An echo chamber is formed when users come across beliefs that magnify or reinforce their thoughts and form a group of like-minded users in a closed system. Echo chambers spread information without any opposing beliefs and can possibly lead to confirmation bias. According to group polarization theory, an echo chamber can potentially lead users and groups towards more extreme radicalized positions. According to the National Library of Medicine, "Users online tend to prefer information adhering to their worldviews, ignore dissenting information, and form polarized groups around shared narratives. Furthermore, when polarization is high, misinformation quickly proliferates." == By site == === Facebook === Facebook's algorithm focuses on recommending content that makes the user want to interact. They rank content by prioritizing popular posts by friends, viral content, and sometimes divisive content. Each feed is personalized to the user's specific interests which can sometimes lead users towards an echo chamber of troublesome content. Users can find their list of interests the algorithm uses by going to the "Your ad Preferences" page. According to a Pew Research study, 74% of Facebook users did not know that list existed until they were directed towards that page in the study. It is also relatively common for Facebook to assign political labels to their users. In recent years, Facebook has started using artificial intelligence to change the content users see in their feed and what is recommended to them. A document known as The Facebook Files has revealed that their AI system prioritizes user engagement over everything else. The Facebook Files has also demonstrated that controlling the AI systems has proven difficult to handle. In an August 2019 internal memo leaked in 2021, Facebook has admitted that "the mechanics of our platforms are not neutral", concluding that in order to reach maximum profits, optimization for engagement is necessary. In order to increase engagement, algorithms have found that hate, misinformation, and politics are instrumental for app activity. As referenced in the memo, "The more incendiary the material, the more it keeps users engaged, the more it is boosted by the algorithm." According to a 2018 study, "false rumors spread faster and wider than true information... They found falsehoods are 70% more likely to be retweeted on Twitter than the truth, and reach their first 1,500 people six times faster. This effect is more pronounced with political news than other categories." === YouTube === YouTube has been around since 2005 and has more than 2.5 billion monthly users. YouTube discovery content systems focus on the user's personal activity (watched, favorites, likes) to direct them to recommended content. YouTube's algorithm is accountable for roughly 70% of users' recommended videos and what drives people to watch certain content. According to a 2022 study by the Mozilla Foundation, users have little power to keep unsolicited videos out of their suggested recommended content. This includes videos about hate speech, livestreams, etc. YouTube has been identified as an influential platform for spreading radicalized content. Al-Qaeda and similar extremist groups have been linked to using YouTube for recruitment videos and engaging with international media outlets. In a research study published by the American Behavioral Scientist Journal, they researched "whether it is possible to identify a set of attributes that may help explain part of the YouTube algorithm's decision-making process". The results of the study showed that YouTube's algorithm recommendations for extremism content factor into the presence of radical keywords in a video's title. In February 2023, in the case of Gonzalez v. Google, the question at hand is whether or not Google, the parent company of YouTube, is protected from lawsuits claiming that the site's algorithms aided terrorists in recommending ISIS videos to users. Section 230 is known to generally protect online platforms from civil liability for the content posted by its users. Multiple studies have found little to no evidence to suggest that YouTube's algorithms direct attention towards far-right content to those not already engaged with it. === TikTok === TikTok is a platform that recommends videos to a user's 'For You Page' (FYP), making every users' page different. With the nature of the algorithm behind the app, TikTok's FYP has been linked to showing more explicit and radical videos over time based on users' previous interactions on the app. Since TikTok's inception, the app has been scrutinized for misinformation and hate speech as those forms of media usually generate more interactions to the algorithm. Various extremist groups, including jihadist organizations, have utilized TikTok to disseminate propaganda, recruit followers, and incite violence. The platform's algorithm, which recommends content based on user engagement, can expose users to extremist content that aligns with their interests or interactions. As of 2022, TikTok's head of US Security has put out a statement that "81,518,334 videos were removed globally between April – June for violating our Community Guidelines or Terms of Service" to cut back on hate speech, harassment, and misinformation. Studies have noted instances where individuals were radicalized through content encountered on TikTok. For example, in early 2023, Austrian authorities thwarted a plot against an LGBTQ+ pride parade that involved two teenagers and a 20-year-old who were inspired by jihadist content on TikTok. The youngest suspect, 14 years old, had been exposed to videos created by Islamist influencers glorifying jihad. These videos led him to further engagement with similar content, eventually resulting in his involvement in planning an attack. Another case involved the arrest of several teenagers in Vienna, Austria, in 2024, who were planning to carry out a terrorist attack at a Taylor Swift concert. The investigation revealed that some of the suspects had been radicalized online, with TikTok being one of the platforms used to disseminate extremist content that influenced their beliefs and actions. == Self-radicalization == The U.S. Department of Justice defines 'Lone-wolf' (self) terrorism as "someone who acts alone in a terrorist attack without the help or encouragement of a government or a terrorist organization". Through social media outlets on the internet, 'Lone-wolf' terrorism has been on the rise, being linked to algorithmic radicalization. Through echo-chambers on the internet, viewpoints typically seen as radical were accepted and quickly adopted by other extremists. These viewpoints are encouraged by forums, group chats, and social media to reinforce their beliefs. == References in media == === The Social Dilemma === The Social Dilemma is a 2020 docudrama about how algorithms behind social media enables addiction, while possessing abilities to manipulate people's views, emotions, and behavior to spread conspiracy theories and disinformation. The film repeatedly uses buzz words such as 'echo chambers' and 'fake news' to prove psychological manipulation on social media, therefore leading to political manipulation. In the film, Ben falls deeper into a social media addiction as the algorithm found that his social media page has a 62.3% chance of long-term engagement. This leads into more videos on the recommended feed for Ben and he eventually becomes more immersed into propaganda and conspiracy theories, becoming more polarized with each video. == Proposed solutions == === United States: Weakening Section 230 protections === In the Communications Decency Act, Section 230 states t
Electronic game
An electronic game is a game that uses electronics to create an interactive system with which a player can play. Video games are the most common form today, and for this reason the two terms are often used interchangeably. There are other common forms of electronic games, including handheld electronic games, standalone arcade game systems (e.g. pinball, slot machines), and exclusively non-visual products (e.g. audio games). == Arcade games == === Arcade video games === Electronic video arcade games make extensive use of solid state electronics and integrated circuits. In the past coin-operated arcade video games generally used custom per-game hardware often with multiple CPUs, highly specialized sound and graphics chips and/or boards, and the latest in computer graphics display technology. Recent arcade game hardware is often based on modified video game console hardware or high end pc components. Arcade games may feature specialized ambiance or control accessories, including fully enclosed dynamic cabinets with force feedback controls, dedicated lightguns, rear-projection displays, reproductions of car or plane cockpits and even motorcycle or horse-shaped controllers, or even highly dedicated controllers such as dancing mats and fishing rods. These accessories are usually what set modern arcade games apart from PC or console games, and they provide an experience that some gamers consider more immersive and realistic. Examples of arcade video games include: Galaxy Game (1971) Pong (1972) Space Invaders (1978) Galaxian (1979) Pac-Man (1980) Battlezone (1980) Donkey Kong (1981) Street Fighter II (1991) Mortal Kombat (1992) Fatal Fury (1992) Killer Instinct (1994) King of Fighters (1994–2005) Time Crisis (1995) Dance Dance Revolution (1998) DrumMania (1999) House of the Dead (1998) === Pinball and pachinko machines === Since the introduction of electromechanics to the pinball machine in 1933's Contact, pinball has become increasingly dependent on electronics as a means to keep score on the backglass and to provide quick impulses on the playfield (as with bumpers and flippers) for exciting gameplay. Unlike games with electronic visual displays, pinball has retained a physical display that is viewed on a table through glass. Similar games such as pachinko have also become increasingly dependent on electronics in modern times. Examples of pinball games include: The Addams Family (1991) Indiana Jones: The Pinball Adventure (1993) Star Trek: The Next Generation (1993) List of pinball machines === Redemption games and merchandisers === Redemption games such as Skee-Ball have been around since the days of the carnival game - well earlier than the development of the electronic game, however with modern advances many of these games have been re-worked to employ electronic scoring and other game mechanics. The use of electronic scoring mechanisms has allowed carnival or arcade attendants to take a more passive role, simply exchanging prizes for electronically dispensed coupons and occasionally emptying out the coin boxes or banknote acceptors of the more popular games. Merchandisers such as the Claw Crane are more recent electronic games in which the player must accomplish a seemingly simple task (e.g. remotely controlling a mechanical arm) with sufficient ability to earn a reward. Examples of redemption games include: Whac-A-Mole (1976) Skee-Ball - modern electric versions Examples of merchandisers include: Claw crane (1980) === Slot machines === The slot machine is a casino gambling machine with three or more reels which spin when a button is pushed. Though slot machines were originally operated mechanically by a lever on the side of the machine (the one arm) instead of an electronic button on the front panel as used on today's models, many modern machines still have a "legacy lever" in addition to the button on the front. Slot machines include a currency detector that validates the coin or money inserted to play. The machine pays off based on patterns of symbols visible on the front of the machine when it stops. Modern computer technology has resulted in many variations on the slot machine concept. == Audio games == An audio game is a game played on an electronic device such as—but not limited to—a personal computer. It is similar to a video game save that the only feedback device is audible rather than visual. Audio games originally started out as 'blind accessible'-games, but recent interest in audio games has come from sound artists, game accessibility researchers, mobile game developers, and mainstream video gamers. Most audio games run on a computer platform, although there are a few audio games for handhelds and video game consoles. Audio games feature the same variety of genres as video games, such as adventure games, racing games, etc. Examples of audio games include: Real Sound: Kaze no Regret (1997) Chillingham (2004) BBBeat (2005) === Tabletop games === A tabletop audio game is an audio game that is designed to be played on a table rather than a handheld game. Examples of tabletop audio games include: Brain Shift (1998) Who Wants to be a Millionaire? (2000) Electronic Battleship (1977) (Milton Bradley) Electronic battleship is a portable game with the objective of marking all enemy ships. When an enemy ship is marked, an electronic battleship makes an explosion sound. Milton Bradley created the Electronic battleship game in 1977 and was later acquired by Hasbro in 1984. Modern day electronic battleship features an interactive missile launching platform and advanced mode that features custom special attack pegs. Tabletop non-audio games include: Electronic Chess Boards (DGT) DGT is a line of electronic chess boards that are commonly used in FIDE chess tournaments and national tournaments such as USCF. Electronic Chess boards can be used to broadcast games live. == Electronic handhelds == The earliest form of dedicated console, handheld electronic games are characterized by their size and portability. Used to play interactive games, handheld electronic games are often miniaturized versions of video games. The controls, display and speakers are all part of a single unit, and rather than a general-purpose screen made up of a grid of small pixels, they usually have custom displays designed to play one game. This simplicity means they can be made as small as a digital watch, which they sometimes are. The visual output of these games can range from a few small light bulbs or LED lights to calculator-like alphanumerical screens; later these were mostly displaced by liquid crystal and Vacuum fluorescent display screens with detailed images and in the case of VFD games, color. Handhelds were at their most popular from the late 1970s into the early 1990s. They are both the predecessors to and inexpensive alternatives to the handheld game console. Examples of handheld electronic games include: Mattel Auto Race (1976) Simon (1978) Merlin (1978) Game & Watch (1980) MB Omni (1980) Bandai LCD Solarpower (1982) Entex Adventure Vision (1982) Lights Out (1995) == Home video games == A video game is a game that involves interaction with a user interface to generate visual feedback on a video device. The word video in video game traditionally referred to a raster display device. However, with the popular use of the term "video game", it now implies any type of display device. Term "digital game" has been offered by some in academia as an alternative term. === Computer games === A personal computer video game (also known as a computer game or simply PC game) is a video game played on a personal computer. This is opposed to video game consoles or arcade machines, which are not considered personal computers. Computer games became a form of video games, and since the earliest days of the medium, visual displays such as the cathode-ray tube have been used to relay game information. === Console games === A console game is a form of interactive multimedia used for entertainment. The game consists of manipulable images (and usually sounds) generated by a video game console, and displayed on a television or similar audio-video system. The game itself is usually controlled and manipulated using a handheld device connected to the console called a controller. The controller generally contains a number of buttons and directional controls (such as analog joysticks) each of which has been assigned a purpose for interacting with and controlling the images on the screen. The display, speakers, console, and controls of a console can also be incorporated into one small object known as a handheld game console. Console games are most frequently differentiated between by their compatibility with consoles belonging in the following categories: Traditional console, also called "home console" - A multi-game system that uses the screen of a television to produce graphics. Handheld game console - A multi-game system the screen and controls of which are compacted into a singl
Texture atlas
In computer graphics, a texture atlas (also called a spritesheet or an image sprite in 2D game development) is an image containing multiple smaller images, usually packed together to reduce overall dimensions. An atlas can consist of uniformly-sized images or images of varying dimensions. A sub-image is drawn using custom texture coordinates to pick it out of the atlas. == Benefits == In an application where many small textures are used frequently, it is often more efficient to store the textures in a texture atlas which is treated as a single unit by the graphics hardware. This reduces both the disk I/O overhead and the overhead of a context switch by increasing memory locality. Careful alignment may be needed to avoid bleeding between sub textures when used with mipmapping and texture compression. In web development, images are packed into a sprite sheet to reduce the number of image resources that need to be fetched in order to display a page. == Gallery ==
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}^{