Best AI Photo Editor

Best AI Photo Editor — hands-on reviews, top picks, pricing, pros and cons and a practical how-to guide on Aizhi.

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  ImageMixer

  ImageMixer is a brand name of video editing software that edits digital video and still image in camcorders and authors to VCD and DVD. It is a second-party Japanese product, distributed by Pixela Corporation, a Japanese manufacturer of PC peripheral hardware and multimedia software. == Bundling == ImageMixer is widely used for several camcorder brands, such as JVC, Hitachi and Canon. Also, Sony has chosen to package ImageMixer with its DVD and HDD Handycam. == ImageMixer series == ImageMixer has other series of software for digital camera, such as ImageMixer Label Maker and ImageMixer DVD dubbing. ImageMixer also has movie editing solution for Macintosh. == Windows Vista version of ImageMixer == A Windows Vista version of ImageMixer has been developed (ImageMixer3).

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  Query rewriting

  Query rewriting is a typically automatic transformation that takes a set of database tables, views, and/or queries, usually indices, often gathered data and query statistics, and other metadata, and yields a set of different queries, which produce the same results but execute with better performance (for example, faster, or with lower memory use). Query rewriting can be based on relational algebra or an extension thereof (e.g. multiset relational algebra with sorting, aggregation and three-valued predicates i.e. NULLs as in the case of SQL). The equivalence rules of relational algebra are exploited, in other words, different query structures and orderings can be mathematically proven to yield the same result. For example, filtering on fields A and B, or cross joining R and S can be done in any order, but there can be a performance difference. Multiple operations may be combined, and operation orders may be altered. The result of query rewriting may not be at the same abstraction level or application programming interface (API) as the original set of queries (though often is). For example, the input queries may be in relational algebra or SQL, and the rewritten queries may be closer to the physical representation of the data, e.g. array operations. Query rewriting can also involve materialization of views and other subqueries; operations that may or may not be available to the API user. The query rewriting transformation can be aided by creating indices from which the optimizer can choose (some database systems create their own indexes if deemed useful), mandating the use of specific indices, creating materialized and/or denormalized views, or helping a database system gather statistics on the data and query use, as the optimality depends on patterns in data and typical query usage. Query rewriting may be rule based or optimizer based. Some sources discuss query rewriting as a distinct step prior to optimization, operating at the level of the user accessible algebra API (e.g. SQL). There are other, largely unrelated concepts also named similarly, for example, query rewriting by search engines.

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  Artificial empathy

  Artificial empathy or computational empathy is the development of AI systems—such as companion robots or virtual agents—that can detect emotions and respond to them in an empathic way. Although such technology can be perceived as scary or threatening, it could also have a significant advantage over humans for roles in which emotional expression can be important, such as in the health care sector. An October 2025 review and meta-analysis in the British Medical Bulletin found that AI chatbots were rated as showing more empathy than human healthcare professionals in 13 of 15 studies that compared them. Care-givers who perform emotional labor above and beyond the requirements of paid labor can experience chronic stress or burnout, and can become desensitized to patients. Artificial empathy could also help the socialization of care-givers, or serve as role model for emotional detachment. A broader definition of artificial empathy is "the ability of nonhuman models to predict a person's internal state (e.g., cognitive, affective, physical) given the signals (s)he emits (e.g., facial expression, voice, gesture) or to predict a person's reaction (including, but not limited to internal states) when he or she is exposed to a given set of stimuli (e.g., facial expression, voice, gesture, graphics, music, etc.)". A 2025 study reported that some multimodal large language models can recognize basic facial expressions with human-level accuracy on a commonly used research dataset of posed facial expressions. == Areas of research == There are a variety of philosophical, theoretical, and applicative questions related to artificial empathy. For example: Which conditions would have to be met for a robot to respond competently to a human emotion? What models of empathy can or should be applied to Social and Assistive Robotics? Must the interaction of humans with robots imitate affective interaction between humans? Can a robot help science learn about affective development of humans? Would robots create unforeseen categories of inauthentic relations? What relations with robots can be considered authentic? How can we assess artificial empathy in AI systems? == Examples of artificial empathy research and practice == People often communicate and make decisions based on inferences about each other's internal states (e.g., emotional, cognitive, and physical states) that are in turn based on signals emitted by the person such as facial expression, body gesture, voice, and words. Broadly speaking, artificial empathy focuses on developing non-human models that achieve similar objectives using similar data. === Streams of artificial empathy research === Artificial empathy has been applied in various research disciplines, including artificial intelligence and business. Two main streams of research in this domain are: the use of nonhuman models to predict a person's internal state (e.g., cognitive, affective, physical) given the signals he or she emits (e.g., facial expression, voice, gesture) the use of nonhuman models to predict a person's reaction when he or she is exposed to a given set of stimuli (e.g., facial expression, voice, gesture, graphics, music, etc.). Research on affective computing, such as emotional speech recognition and facial expression detection, falls within the first stream of artificial empathy. Contexts that have been studied include oral interviews, call centers, human-computer interaction, sales pitches, and financial reporting. The second stream of artificial empathy has been researched more in marketing contexts, such as advertising, branding, customer reviews, in-store recommendation systems, movies, and online dating. === Artificial empathy applications in practice === With the increasing volume of visual, audio, and text data in commerce, many business applications for artificial empathy have followed. For example, Affectiva analyses viewers' facial expressions from video recordings while they are watching video advertisements in order to optimize the content design of video ads. Software like HireVue, BarRaiser, a hiring intelligence firm, helps firms make recruitment decisions by analyzing audio and video information from candidates' video interviews. Lapetus Solutions develops a model to estimate an individual's longevity, health status, and disease susceptibility from a face photo. Their technology has been applied in the insurance industry. == Artificial empathy and human services == Although artificial intelligence cannot yet replace social workers themselves, the technology has been deployed in that field. Florida State University published a study about Artificial Intelligence being used in the human services field. The research used computer algorithms to analyze health records for combinations of risk factors that could predict a future suicide attempt. The article reports, "machine learning—a future frontier for artificial intelligence—can predict with 80% to 90% accuracy whether someone will attempt suicide as far off as two years into the future. The algorithms become even more accurate as a person's suicide attempt gets closer. For example, the accuracy climbs to 92% one week before a suicide attempt when artificial intelligence focuses on general hospital patients". Such algorithmic machines can help social workers. Social work operates on a cycle of engagement, assessment, intervention, and evaluation with clients. Earlier assessment for risk of suicide can lead to earlier interventions and prevention, therefore saving lives. The system would learn, analyze, and detect risk factors, alerting the clinician of a patient's suicide risk score (analogous to a patient's cardiovascular risk score). Then, social workers could step in for further assessment and preventive intervention.

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  Informedia Digital Library

  The Informedia Digital Library is an ongoing research program at Carnegie Mellon University to build search engines and information visualization technology for many types of media. The program has carried out research on spoken document retrieval, video information retrieval, video segmentation, face recognition, and cross-language information retrieval. The Lycos search engine was an early product of the Informedia Digital Library Project. The project is led by Howard Wactlar. Researchers on the project have included: Michael Mauldin, Alex Hauptmann, Michael Christel, Michael Witbrock, Raj Reddy, Takeo Kanade and Scott Stevens.

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  Time-compressed speech

  Time-compressed speech refers to an audio recording of verbal text in which the text is presented in a much shorter time interval than it would through normally-paced real time speech. The basic purpose is to make recorded speech contain more words in a given time, yet still be understandable. For example: a paragraph that might normally be expected to take 20 seconds to read, might instead be presented in 15 seconds, which would represent a time-compression of 25% (5 seconds out of 20). The term "time-compressed speech" should not be confused with "speech compression", which controls the volume range of a sound, but does not alter its time envelope. == Methods == While some voice talents are capable of speaking at rates significantly in excess of general norms, the term "time-compressed speech" most usually refers to examples in which the time-reduction has been accomplished through some form of electronic processing of the recorded speech. In general, recorded speech can be electronically time-compressed by: increasing its speed (linear compression); removing silences (selective editing); a combination of the two (non-linear compression). The speed of a recording can be increased, which will cause the material to be presented at a faster rate (and hence in a shorter amount of time), but this has the undesirable side-effect of increasing the frequency of the whole passage, raising the pitch of the voices, which can reduce intelligibility. There are normally silences between words and sentences, and even small silences within certain words, both of which can be reduced or removed ("edited-out") which will also reduce the amount of time occupied by the full speech recording. However, this can also have the effect of removing verbal "punctuation" from the speech, causing words and sentences to run together unnaturally, again reducing intelligibility. Vowels are typically held a minimum of 20 milliseconds, over many cycles of the fundamental pitch. DSP systems can detect the beginning and end of each cycle and then skip over some fraction of those cycles, causing the material to be presented at a faster rate, without changing the pitch, maintaining a "normal" tone of voice. The current preferred method of time-compression is called "non-linear compression", which employs a combination of selectively removing silences; speeding up the speech to make the reduced silences sound normally-proportioned to the text; and finally applying various data algorithms to bring the speech back down to the proper pitch. This produces a more acceptable result than either of the two earlier techniques; however, if unrestrained, removing the silences and increasing the speed can make a selection of speech sound more insistent, possibly to the point of unpleasantness. == Applications == === Advertising === Time-compressed speech is frequently used in television and radio advertising. The advantage of time-compressed speech is that the same number of words can be compressed into a smaller amount of time, reducing advertising costs, and/or allowing more information to be included in a given radio or TV advertisement. It is usually most noticeable in the information-dense caveats and disclaimers presented (usually by legal requirement) at the end of commercials—the aural equivalent of the "fine print" in a printed contract. This practice, however, is not new: before electronic methods were developed, spokespeople who could talk extremely quickly and still be understood were widely used as voice talents for radio and TV advertisements, and especially for recording such disclaimers. === Education === Time-compressed speech has educational applications such as increasing the information density of trainings, and as a study aid. A number of studies have demonstrated that the average person is capable of relatively easily comprehending speech delivered at higher-than-normal rates, with the peak occurring at around 25% compression (that is, 25% faster than normal); this facility has been demonstrated in several languages. Conversational speech (in English) takes place at a rate of around 150 wpm (words per minute), but the average person is able to comprehend speech presented at rates of up to 200-250 wpm without undue difficulty. Blind and severely visually impaired subjects scored similar comprehension levels at even higher rates, up to 300-350 wpm. Blind people have been found to use time-compressed speech extensively, for example, when reviewing recorded lectures from high school and college classes, or professional trainings. Comprehension rates in older blind subjects have been found to be as good, or in some cases better than those found in younger sighted subjects. Other studies have determined that the ability to comprehend highly time-compressed speech tends to fall off with increased age, and is also reduced when the language of the time-compressed speech is not the listener's native language. Non-native speakers can, however, improve their comprehension level of time-compressed speech with multiday training. === Voice Mail === Voice mail systems have employed time-compressed speech since as far back as the 1970s. In this application, the technology enables the rapid review of messages in high-traffic systems, by a relatively small number of people. === Streaming Multimedia === Time-compressed speech has been explored as one of a variety of interrelated factors which may be manipulated to increase the efficiency of streaming multimedia presentations, by significantly reducing the latency times involved in the transfer of large digitally encoded media files.

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  Ballin' (Mustard and Roddy Ricch song)

  "Ballin'" is a song by American record producer Mustard featuring American rapper Roddy Ricch. The track was released as the third single from Mustard's third studio album, Perfect Ten, on August 20, 2019, though it was available as early as the end of June 2019. The song and its accompanying video received acclaim from music critics, with Complex magazine naming it the Best Song of 2019. It peaked at number 11 on the Billboard Hot 100, marking Mustard's highest charting song in the US. The song received a nomination for Best Rap/Sung Performance at the 2020 Grammy Awards, making it the first time Ricch has been nominated for a Grammy and Mustard's first nomination as an artist. Later in 2019, the two released another collaboration, "High Fashion". == Background == Roddy Ricch revealed in an interview that the song was composed in late 2018, but Mustard wanted to keep it for his album, Perfect Ten, which he was still working on. The song was later included on the album, released in June 2019. Ricch said he knew the song was "hard enough" the first time he heard it, while Mustard proclaimed "this is going to be the one". == Composition and lyrics == "Ballin'" has a "rags to riches" theme. In its intro, the song samples girl group 702's 1997 top ten hit "Get It Together". The song features a "smooth, bouncy beat", with Roddy Ricch rapping about his come-up and ascent in the music industry. In the first verse, Ricch salutes fellow Los Angeles rapper, the late Nipsey Hussle and his girlfriend Lauren London: "I run these racks up with my queen like London and Nip". The line simultaneously references Ricch and Hussle's collaboration "Racks in the Middle", released earlier in 2019 as Hussle's last single before his death. Billboard's Heran Mamo noted that "in typical Hussle fashion", Roddy Ricch "narrates his life's hardships before delving into his newfound treasures". == Critical reception == The song was widely acclaimed by music critics. Charles Holmes of Rolling Stone magazine called it "a song of the year contender", while Complex and Billboard both named it as a "standout track" on the album. Pitchfork magazine included "Ballin'" in its list of The Best Rap Songs of 2019 and called it "the centerpiece of Mustard's underappreciated album Perfect Ten". Complex later named it the Best Song of 2019, calling it "a feel-good anthem so infectious you'll need antibiotics just to stop running it back". == Chart performance == "Ballin'" was at the time Mustard's highest charting song in the US, peaking at number 11 on the Billboard Hot 100. It was also Roddy Ricch's highest charting song, until he surpassed it a week later, with the release of his album track "The Box", which eventually reached number 1 on the chart. It reached number one on Billboard's Rhythmic Songs chart, becoming Mustard's second number one following "Pure Water" and Ricch's first number one. The song also topped the Rap Airplay chart. == Music video == The music video for the track was teased by Mustard on his Instagram page on September 29, 2019. The music video for the track was eventually released on October 2, 2019 to critical acclaim. The video features Mustard and Roddy Ricch driving a Lamborghini Aventador in Los Angeles, where they both are from, playing poker in a casino, and going to a strip club. This is contrasted with scenes in which Mustard and Roddy Ricch as children play cards with Monopoly money and playing with miniature toy Lamborghinis together, aspiring for wealth and luxury, representing how they went from "rags to riches". The video also pays tribute to rapper Nipsey Hussle, who had been killed a few months ago. == Live performances == On December 16, 2019, Roddy Ricch performed the song live, alongside an 8-piece orchestra, at Peppermint Club in Los Angeles for Audiomack's Trap Symphony series. Along with Mustard, he performed it at The Pop Out: Ken & Friends on June 19, 2024. == Other uses == The song can be heard on "Elyse's Skit", track 10 off Roddy Ricch's debut album Please Excuse Me for Being Antisocial. In the skit, which is an actual voicenote recording, the mother of a woman named Elyse sends her daughter a voicenote, with "Ballin'" playing in the background, while the mother proceeds to say "I can't get that damn song out my head", jokingly calling it "inappropriate music". Ricch called the skit "something natural". In 2023, AI covers of the song using models based on pop culture characters and real-world celebrities gained viral popularity. == Awards and nominations == 62nd Annual Grammy Awards == Charts == == Certifications ==

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  Bibliometrician

  A bibliometrician is a researcher or a specialist in bibliometrics. It is near-synonymous with an informetrican (who studies informetrics), a scientometrican (who study scientometrics) and a webometrician, who study webometrics. == Notable bibliometricians == Christine L. Borgman Samuel C. Bradford Blaise Cronin Margaret Elizabeth Egan Eugene Garfield (developer of the Science Citation Index and the Impact factor) Jorge E. Hirsch (developer of the h-index) Alfred J. Lotka Vasily Nalimov Derek J. de Solla Price Ronald Rousseau George Kingsley Zipf

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  In-place algorithm

  In computer science, an in-place algorithm is an algorithm that operates directly on the input data structure without requiring extra space proportional to the input size. In other words, it modifies the input in place, without creating a separate copy of the data structure. An algorithm which is not in-place is sometimes called not-in-place or out-of-place. In-place can have slightly different meanings. In its strictest form, the algorithm can only have a constant amount of extra space, counting everything including function calls and pointers. However, this form is very limited as simply having an index to a length n array requires O(log n) bits. More broadly, in-place means that the algorithm does not use extra space for manipulating the input but may require a small though non-constant extra space for its operation. Usually, this space is O(log n), though sometimes anything in o(n) is allowed. Note that space complexity also has varied choices in whether or not to count the index lengths as part of the space used. Often, the space complexity is given in terms of the number of indices or pointers needed, ignoring their length. In this article, we refer to total space complexity (DSPACE), counting pointer lengths. Therefore, the space requirements here have an extra log n factor compared to an analysis that ignores the lengths of indices and pointers. An algorithm may or may not count the output as part of its space usage. Since in-place algorithms usually overwrite their input with output, no additional space is needed. When writing the output to write-only memory or a stream, it may be more appropriate to only consider the working space of the algorithm. In theoretical applications such as log-space reductions, it is more typical to always ignore output space (in these cases it is more essential that the output is write-only). == Examples == Given an array a of n items, suppose we want an array that holds the same elements in reversed order and to dispose of the original. One seemingly simple way to do this is to create a new array of equal size, fill it with copies from a in the appropriate order and then delete a. function reverse(a[0..n - 1]) allocate b[0..n - 1] for i from 0 to n - 1 b[n − 1 − i] := a[i] return b Unfortunately, this requires O(n) extra space for having the arrays a and b available simultaneously. Also, allocation and deallocation are often slow operations. Since we no longer need a, we can instead overwrite it with its own reversal using this in-place algorithm which will only need constant number (2) of integers for the auxiliary variables i and tmp, no matter how large the array is. function reverse_in_place(a[0..n-1]) for i from 0 to floor((n-2)/2) tmp := a[i] a[i] := a[n − 1 − i] a[n − 1 − i] := tmp As another example, many sorting algorithms rearrange arrays into sorted order in-place, including: bubble sort, comb sort, selection sort, insertion sort, heapsort, and Shell sort. These algorithms require only a few pointers, so their space complexity is O(log n). Quicksort operates in-place on the data to be sorted. However, quicksort requires O(log n) stack space pointers to keep track of the subarrays in its divide and conquer strategy. Consequently, quicksort needs O(log2 n) additional space. Although this non-constant space technically takes quicksort out of the in-place category, quicksort and other algorithms needing only O(log n) additional pointers are usually considered in-place algorithms. Most selection algorithms are also in-place, although some considerably rearrange the input array in the process of finding the final, constant-sized result. Some text manipulation algorithms such as trim and reverse may be done in-place. == In computational complexity == In computational complexity theory, the strict definition of in-place algorithms includes all algorithms with O(1) space complexity, the class DSPACE(1). This class is very limited; it equals the regular languages. In fact, it does not even include any of the examples listed above. Algorithms are usually considered in L, the class of problems requiring O(log n) additional space, to be in-place. This class is more in line with the practical definition, as it allows numbers of size n as pointers or indices. This expanded definition still excludes quicksort, however, because of its recursive calls. Identifying the in-place algorithms with L has some interesting implications; for example, it means that there is a (rather complex) in-place algorithm to determine whether a path exists between two nodes in an undirected graph, a problem that requires O(n) extra space using typical algorithms such as depth-first search (a visited bit for each node). This in turn yields in-place algorithms for problems such as determining if a graph is bipartite or testing whether two graphs have the same number of connected components. == Role of randomness == In many cases, the space requirements of an algorithm can be drastically cut by using a randomized algorithm. For example, if one wishes to know if two vertices in a graph of n vertices are in the same connected component of the graph, there is no known simple, deterministic, in-place algorithm to determine this. However, if we simply start at one vertex and perform a random walk of about 20n3 steps, the chance that we will stumble across the other vertex provided that it is in the same component is very high. Similarly, there are simple randomized in-place algorithms for primality testing such as the Miller–Rabin primality test, and there are also simple in-place randomized factoring algorithms such as Pollard's rho algorithm. == In functional programming == Functional programming languages often discourage or do not support explicit in-place algorithms that overwrite data, since this is a type of side effect; instead, they only allow new data to be constructed. However, good functional language compilers will often recognize when an object very similar to an existing one is created and then the old one is thrown away, and will optimize this into a simple mutation "under the hood". Note that it is possible in principle to carefully construct in-place algorithms that do not modify data (unless the data is no longer being used), but this is rarely done in practice.

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  Visible (mobile app)

  Visible is a health tracking mobile app for people with long COVID and myalgic encephalomyelitis/chronic fatigue syndrome (ME/CFS). The company was founded by a Harry Leeming, an engineer from London living with long Covid since 2020, and Luke Martin-Fuller. In November 2022, Visible released an open beta of an app that aims to help people pace their activities to avoid post-exertional malaise. The app gathers data on exertion levels, symptom severity, and heart-rate variability. HRV is approximated using a smartphone's camera via a technique called photoplethysmography, and according to the app's developers, can indicate how much someone needs rest. The app is currently free, but is expected to be freemium in the future. Users can also opt to allow their data be used for research purposes. In July 2023, Visible and Imperial College London announced the start of the first two studies. One is on the effects of the menstrual cycle on long COVID symptoms, and the other is on the condition's epidemiology and economic impact. Visible has announced plans to couple the app with activity trackers for continuous monitoring of heart-rate and actimetry data, which the developers claim will be more effective. As of 2022, no clinical trials on Visible's effectiveness have been conducted.

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  Information architecture

  Information architecture is the structural design of shared information environments, in particular the organisation of websites and software to support usability and findability. The term information architecture was coined by Richard Saul Wurman. Since its inception, information architecture has become an emerging community of practice focused on applying principles of design, architecture and information science in digital spaces. Typically, a model or concept of information is used and applied to activities which require explicit details of complex information systems. These activities include library systems and database development. == Definition == The term information architecture has different meanings in different branches of information systems or information technology. === User experience === In user experience design, information architecture has been described as the structural design of shared information environments, comprising the study and practice of organising and labelling web sites, intranets, online communities, and software to support user experience, in particular, the findability and usability of information. It has also been described as an emerging community of practice focused on bringing principles of design and architecture to the digital landscape. === Information systems === Technically speaking, information architecture comprises the combination of organization, labeling, search and navigation systems within websites and intranets, serving as a navigational aid to the content of information-rich systems. === Data architecture === Information architecture can be described as a subset of data architecture where usable data is constructed, designed, and arranged in a fashion most useful to the users of data. === Systems design === In the field of systems design, for example, information architecture is a component of enterprise architecture that deals with the information component when describing the structure of an enterprise. Some system design practitioners regard information architecture as strictly the application of information science to web design, which considers such issues as classification and information retrieval, and not factors like user experience and information design. == Principles == Principles of information architecture include the following: The principle of objects The principle of choices The principle of disclosure The principle of exemplars The principle of front doors The principle of multiple classification The principle of focused navigation The principle of growth == History == Richard Saul Wurman is credited with coining the term information architecture in relation to the design of information. From 1998 to 2015, Peter Morville and Louis Rosenfeld were co-authors of Information Architecture for the World Wide Web. Other authors include Jesse James Garrett and Christina Wodtke.

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  Higuchi dimension

  In fractal geometry, the Higuchi dimension (or Higuchi fractal dimension (HFD)) is an approximate value for the box-counting dimension of the graph of a real-valued function or time series. This value is obtained via an algorithmic approximation so one also talks about the Higuchi method. It has many applications in science and engineering and has been applied to subjects like characterizing primary waves in seismograms, clinical neurophysiology and analyzing changes in the electroencephalogram in Alzheimer's disease. == Formulation of the method == The original formulation of the method is due to T. Higuchi. Given a time series X : { 1 , … , N } → R {\displaystyle X:\{1,\dots ,N\}\to \mathbb {R} } consisting of N {\displaystyle N} data points and a parameter k m a x ≥ 2 {\displaystyle k_{\mathrm {max} }\geq 2} the Higuchi Fractal dimension (HFD) of X {\displaystyle X} is calculated in the following way: For each k ∈ { 1 , … , k m a x } {\displaystyle k\in \{1,\dots ,k_{\mathrm {max} }}\} and m ∈ { 1 , … , k } {\displaystyle m\in \{1,\dots ,k}\} define the length L m ( k ) {\displaystyle L_{m}(k)} by L m ( k ) = N − 1 ⌊ N − m k ⌋ k 2 ∑ i = 1 ⌊ N − m k ⌋ | X N ( m + i k ) − X N ( m + ( i − 1 ) k ) | . {\displaystyle L_{m}(k)={\frac {N-1}{\lfloor {\frac {N-m}{k}}\rfloor k^{2}}}\sum _{i=1}^{\lfloor {\frac {N-m}{k}}\rfloor }|X_{N}(m+ik)-X_{N}(m+(i-1)k)|.} The length L ( k ) {\displaystyle L(k)} is defined by the average value of the k {\displaystyle k} lengths L 1 ( k ) , … , L k ( k ) {\displaystyle L_{1}(k),\dots ,L_{k}(k)} , L ( k ) = 1 k ∑ m = 1 k L m ( k ) . {\displaystyle L(k)={\frac {1}{k}}\sum _{m=1}^{k}L_{m}(k).} The slope of the best-fitting linear function through the data points { ( log ⁡ 1 k , log ⁡ L ( k ) ) } {\displaystyle \left\{\left(\log {\frac {1}{k}},\log L(k)\right)\right\}} is defined to be the Higuchi fractal dimension of the time-series X {\displaystyle X} . == Application to functions == For a real-valued function f : [ 0 , 1 ] → R {\displaystyle f:[0,1]\to \mathbb {R} } one can partition the unit interval [ 0 , 1 ] {\displaystyle [0,1]} into N {\displaystyle N} equidistantly intervals [ t j , t j + 1 ) {\displaystyle [t_{j},t_{j+1})} and apply the Higuchi algorithm to the times series X ( j ) = f ( t j ) {\displaystyle X(j)=f(t_{j})} . This results into the Higuchi fractal dimension of the function f {\displaystyle f} . It was shown that in this case the Higuchi method yields an approximation for the box-counting dimension of the graph of f {\displaystyle f} as it follows a geometrical approach (see Liehr & Massopust 2020). == Robustness and stability == Applications to fractional Brownian functions and the Weierstrass function reveal that the Higuchi fractal dimension can be close to the box-dimension. On the other hand, the method can be unstable in the case where the data X ( 1 ) , … , X ( N ) {\displaystyle X(1),\dots ,X(N)} are periodic or if subsets of it lie on a horizontal line (see Liehr & Massopust 2020).

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  Irish logarithm

  The Irish logarithm was a system of number manipulation invented by Percy Ludgate for machine multiplication. The system used a combination of mechanical cams as lookup tables and mechanical addition to sum pseudo-logarithmic indices to produce partial products, which were then added to produce results. The technique is similar to Zech logarithms (also known as Jacobi logarithms), but uses a system of indices original to Ludgate. == Concept == Ludgate's algorithm compresses the multiplication of two single decimal numbers into two table lookups (to convert the digits into indices), the addition of the two indices to create a new index which is input to a second lookup table that generates the output product. Because both lookup tables are one-dimensional, and the addition of linear movements is simple to implement mechanically, this allows a less complex mechanism than would be needed to implement a two-dimensional 10×10 multiplication lookup table. Ludgate stated that he deliberately chose the values in his tables to be as small as he could make them; given this, Ludgate's tables can be simply constructed from first principles, either via pen-and-paper methods, or a systematic search using only a few tens of lines of program code. They do not correspond to either Zech logarithms, Remak indexes or Korn indexes. == Pseudocode == The following is an implementation of Ludgate's Irish logarithm algorithm in the Python programming language: Table 1 is taken from Ludgate's original paper; given the first table, the contents of Table 2 can be trivially derived from Table 1 and the definition of the algorithm. Note since that the last third of the second table is entirely zeros, this could be exploited to further simplify a mechanical implementation of the algorithm.

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  Psychology in cybersecurity

  The psychology of cybersecurity (often intersecting with usable security and cyberpsychology) is an interdisciplinary field studying how human behavior, cognitive biases, and social dynamics influence information security. While traditional cybersecurity focuses on hardware and software vulnerabilities, this discipline addresses the "human factor," which is exploited in cyberattacks. Psychology in cybersecurity draws from cognitive psychology and human–computer interaction. == History and evolution == The challenge of human behavior in computing was noted as early as the 1960s with multi-user mainframes like the Compatible Time-Sharing System (CTSS). In 1966, a software error on CTSS caused the system's master password file to be displayed to every user upon login—one of the earliest documented security incidents attributable to a combination of system design and human factors. These behaviors gained broader significance in the 1990s as the Internet became widely accessible. High-profile incidents involving figures like Kevin Mitnick demonstrated how human trust could be exploited through social engineering such as pretexting over the phone. == Cognitive and behavioral factors == Much of the psychology of cybersecurity focuses on decision-making under stress or uncertainty. Researchers apply frameworks like dual process theory to explain why humans fall for phishing or business email compromise. Threat actors design malicious communications to trigger fast, emotional "System 1" thinking—using urgency, authority, or panic, which prompts users to click a link or wire funds before their analytical "System 2" can assess the situation's legitimacy. Industry research has consistently documented the effectiveness of these techniques at scale, pointing to several recurring psychological phenomena that influence daily security practices: Cognitive biases: The optimism bias leads users to believe they are unlikely to be targeted by cybercriminals, resulting in lax password practices or delayed software updates. The availability heuristic causes individuals to focus on highly publicized, sophisticated threats while ignoring common, statistically probable risks like credential reuse. Social influence: Attackers leverage established principles of persuasion, such as those categorized by Robert Cialdini. Impersonating a CEO leverages the psychological trigger of authority, while fake tech support scams use reciprocity (offering to fix a problem before asking for network credentials). == Neurological and pre-cognitive factors == Functional magnetic resonance imaging (fMRI) studies show that neural activation in visual and attentional regions decreases with repeated exposure to the same stimulus, a phenomenon termed repetition suppression. Experiments have confirmed this effect in the context of security warnings: static warning designs produce declines in user attention and adherence. Information processing research on phishing indicates that affective cues, such as artificial urgency or fear, increase cognitive load and elicit automatic heuristic processing, reducing the likelihood of analytical evaluation and facilitating compliance with malicious requests. == Security fatigue and organizational dynamics == Aggressive cybersecurity postures can sometimes lead to mental and emotional exhaustion, a phenomenon known as security fatigue. === Alert fatigue === One example is alert fatigue, which most frequently affects both end-users and security operations center analysts. Continuous exposure to browser warnings or antivirus pop-ups, particularly those that are false positives, conditions users to dismiss alerts automatically due to the volume of notifications rather than their repetitive appearance (see § Neurological and pre-cognitive factors). The scale of this problem is significant in enterprise: SOC teams in large organizations receive thousands of alerts daily, and a survey published in ACM Computer Surveys found that analysts spend over 25% of their time handling false positives, meaning that malicious indicators can be buried in the noise. === Password fatigue === Similarly, password fatigue is the feeling experienced by many people who are required to remember an excessive number of passwords as part of their daily routine, such as to log in to a computer at work. Users cope with the memory burden by making predictable, iterative changes to their passwords (such as updating "Password01!" to "Password02!"), which decreases password security.

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  Small data

  Small data is data that is 'small' enough for human comprehension. It is data in a volume and format that makes it accessible, informative and actionable. The term "big data" is about machines and "small data" is about people. This is to say that eyewitness observations or five pieces of related data could be small data. Small data is what we used to think of as data. The only way to comprehend Big data is to reduce the data into small, visually-appealing objects representing various aspects of large data sets (such as histogram, charts, and scatter plots). Big Data is all about finding correlations, but Small Data is all about finding the causation, the reason why. A formal definition of small data has been proposed by Allen Bonde, former vice-president of Innovation at Actuate - now part of OpenText: "Small data connects people with timely, meaningful insights (derived from big data and/or “local” sources), organized and packaged – often visually – to be accessible, understandable, and actionable for everyday tasks." Another definition of small data is: The small set of specific attributes produced by the Internet of Things. These are typically a small set of sensor data such as temperature, wind speed, vibration and status. It was estimated (2016) that “If one takes the top 100 biggest innovations of our time, perhaps around 60% to 65% percent are really based on Small Data.” as Martin Lindstrom puts it. Small data includes everything from Snapchat to simple objects such as the post-it note. Lindstrom believes we become so focused on Big-Data that we tend to forget about more basic concepts and creativity. Lindstrom defines Small Data "as seemingly insignificant observations you identify in consumers’ homes, is everything from how you place your shoes on how you hang your paintings". He thus considers that one should perfectly master the basic (Small Data) in order to mine and find correlations. == Academic Recognition and Methodology == The growing significance of "small data" as a distinct field of inquiry was highlighted by the 2024 Thematic Einstein Semester (TES) on Small Data Analysis, hosted by the Berlin Mathematics Research Center MATH+. A central focus of this semester was the transition from theoretical analysis to practical decision-making. Because small data sets are primarily used to drive specific actions, the presentation of results becomes an essential methodological step. The semester’s findings emphasized that while small data may lack volume, it often contains a high density of "many possible interpretations." Consequently, the final conference of the TES was structured around the pillars of interpretation, explanation, and knowledge gain. Participants sought to develop new mathematical and methodical representations that could accurately depict this wealth of interpretative possibilities. This work underscores that analyzing small data is not purely a computational task; it requires a robust interface between mathematics and diverse disciplines to ensure that insights are both contextually grounded and scientifically rigorous. == Uses in business == === Marketing === Bonde has written about the topic for Forbes, Direct Marketing News, CMO.com and other publications. According to Martin Lindstrom, in his book, Small Data: "{In customer research, small data is} Seemingly insignificant behavioural observations containing very specific attributes pointing towards an unmet customer need. Small data is the foundation for breakthrough ideas or completely new ways to turnaround brands." His approach is based on the combination of the observation of small samples with intuition. Marketers can obtain market insights from gathering Small Data by engaging with and observing people in their own environments. In comparison to Big Data, Small Data has the power to trigger emotions and to provide insights into the reasons behind the behaviours of customers. It may uncover detailed information on a person's extroversion or introversion, self-confidence, whether one is having problems in his/her relationship, etc. According to Lindstrom, relationships among people and customer segments are organized around four criteria: Climate: It reveals for example how a person's environment affects their diet. Rulership: The power or government in charge Religion: The prevalence of religion in a country, depending on its influence, indicates whether a person's decision making process is impacted by their belief system. Tradition: Cultural norms influence people's behaviors and interactions. Many companies underestimate the power of Small Data, using samples of millions of consumers instead of recognizing the value of closely observing small samples in their market research. In his book, Lindstrom defines "7Cs", which companies should consider in the attempt to derive meaningful customer insights and market trends through small data from their customers: Collecting: Understanding the manner in which observations are translated inside a home. Clues: Uncovering other distinctive emotional reflections that can be observed. Connecting: Identifying the consequences of emotional behaviour. Causation: Understanding what emotions are being evoked. Correlation: Identifying the initial date of appearance of the behaviour or emotion. Compensation: Identifying the unmet or unfulfilled desire. Concept: Defining the “big idea” compensation for the identified consumer need. Some of Lindstrom's clients such as Lowes Foods looked at data in a different way and actually chose to live with the customer. “As you enter their store, they have now created an amazing community where every staff member acts in a character mood, based on Small Data”. The supermarket made everything it can to make the customer feel at home. All the behaviours of employees are inspired by customer feedbacks gathered from interviews directly done at customer’s home. === Healthcare === Researchers at Cornell University started developing applications to monitor health problems in patients, based on small data. This is an initiative of Cornell's Small Data Lab, in close cooperation with Weill Cornell Medicine College, led by Deborah Estrin. The Small Data Lab developed a series of apps, focusing not only on gathering data from patients' pain but also tracking habits in areas such as grocery shopping. In the case of patients with rheumatoid arthritis for example, which has flares and remissions that do not follow a particular cycle, the app gathers information passively, thus allowing to forecast when a flare might be coming up based on small changes in behaviour. Other apps developed also include monitoring online grocery shopping, to use this information from every user to adapt their groceries to the recommendations of nutritionists, or monitoring email language to identify patterns that might indicate "fluctuations in cognitive performance, fatigue, side effects of medication or poor sleep, and other conditions and treatments that are typically self-reported and self-medicated". === Postal Service === The United States Postal Service (USPS) used optical character recognition (OCR) to automatically read and process 98% of all hand-addressed mail and 99.5% of machine-printed mail. By combining this technology with its small data sample of US zip codes, the USPS can now process more than 36,000 pieces of mail per hour. === Aerospace === In 2015, Boeing established the analytics lab for aerospace data in cooperation with the Carnegie Mellon University to leverage the university's leadership in machine learning, language technologies and data analytics. One of the initiatives projects aims to by standardize maintenance logs using AI to dramatically reduce costs. Currently, there is no standardized procedure to document maintenance logs leading to small but highly unstructured data sets. As a result, it becomes highly difficult for maintenance workers to translate these variations in maintenance logs within a short period of time. However, with AI and a narrow data set of common aircraft maintenance terminology, it becomes possible to dynamically translate these logs in real time. By using AI to enhance the speed and accuracy of the airline maintenance workflow, airlines stand to save billions according to the Harvard Business Review.

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  Synthetic data

  Synthetic data are artificially generated data not produced by real-world events. Typically created using algorithms, synthetic data can be deployed to validate mathematical models and to train machine learning models. Data generated by a computer simulation can be seen as synthetic data. This encompasses most applications of physical modeling, such as music synthesizers or flight simulators. The output of such systems approximates the real thing, but is fully algorithmically generated. Synthetic data is used in a variety of fields as a filter for information that would otherwise compromise the confidentiality of particular aspects of the data. In many sensitive applications, datasets theoretically exist but cannot be released to the general public; synthetic data sidesteps the privacy issues that arise from using real consumer information without permission or compensation. == Usefulness == Synthetic data is generated to meet specific needs or certain conditions that may not be found in the original, real data. One of the hurdles in applying up-to-date machine learning approaches for complex scientific tasks is the scarcity of labeled data, a gap effectively bridged by the use of synthetic data, which closely replicates real experimental data. This can be useful when designing many systems, from simulations based on theoretical value, to database processors, etc. This helps detect and solve unexpected issues such as information processing limitations. Synthetic data are often generated to represent the authentic data and allows a baseline to be set. Another benefit of synthetic data is to protect the privacy and confidentiality of authentic data, while still allowing for use in testing systems. Computer security experts claim generated synthetic data "... enables us to create realistic behavior profiles for users and attackers. The data is used to train the fraud detection system itself, thus creating the necessary adaptation of the system to a specific environment." In defense and military contexts, synthetic data is seen as a potentially valuable tool to develop and improve complex AI systems, particularly in contexts where high-quality real-world data is scarce. At the same time, synthetic data together with the testing approach can give the ability to model real-world scenarios. == History == Scientific modelling of physical systems has a long history that runs concurrent with the history of physics. For example, research into synthesis of audio and voice can be traced back to the 1930s and before, driven forward by the developments of the telephone and audio recording technologies. Digitization gave rise to software synthesizers from the 1970s onwards. In the context of privacy-preserving statistical analysis, in 1993, the idea of original fully synthetic data was created by Donald Rubin. Rubin originally designed this to synthesize the Decennial Census long form responses for the short form households. He then released samples that did not include any actual long form records - in this he preserved anonymity of the household. Later that year, the idea of original partially synthetic data was created by Little. Little used this idea to synthesize the sensitive values on the public use file. A 1993 work fitted a statistical model to 60,000 MNIST digits, then it was used to generate over 1 million examples. Those were used to train a LeNet-4 to reach state of the art performance. In 1994, Stephen Fienberg introduced 'critical refinement', in which a parametric posterior predictive distribution (instead of a Bayes bootstrap) is used to do the sampling. Later, other important contributors to the development of synthetic data generation were Trivellore Raghunathan, Jerry Reiter, Donald Rubin, John M. Abowd, and Jim Woodcock. Collectively they came up with a solution for how to treat partially synthetic data with missing data. Similarly, they developed the technique of Sequential Regression Multivariate Imputation. == Calculations == Researchers test the framework on synthetic data, which is "the only source of ground truth on which they can objectively assess the performance of their algorithms". Synthetic data can be generated through the use of random lines, having different orientations and starting positions. Datasets can get fairly complicated. A more complicated dataset can be generated by using a synthesizer build. To create a synthesizer build, first use the original data to create a model or equation that fits the data the best. This model or equation will be called a synthesizer build. This build can be used to generate more data. Constructing a synthesizer build involves constructing a statistical model. In a linear regression line example, the original data can be plotted, and a best fit linear line can be created from the data. This line is a synthesizer created from the original data. The next step will be generating more synthetic data from the synthesizer build or from this linear line equation. In this way, the new data can be used for studies and research, and it protects the confidentiality of the original data. David Jensen from the Knowledge Discovery Laboratory explains how to generate synthetic data: "Researchers frequently need to explore the effects of certain data characteristics on their data model." To help construct datasets exhibiting specific properties, such as auto-correlation or degree disparity, proximity can generate synthetic data having one of several types of graph structure: random graphs that are generated by some random process; lattice graphs having a ring structure; lattice graphs having a grid structure, etc. In all cases, the data generation process follows the same process: Generate the empty graph structure. Generate attribute values based on user-supplied prior probabilities. Since the attribute values of one object may depend on the attribute values of related objects, the attribute generation process assigns values collectively. == Applications == === Fraud detection and confidentiality systems === Testing and training fraud detection and confidentiality systems are devised using synthetic data. Specific algorithms and generators are designed to create realistic data, which then assists in teaching a system how to react to certain situations or criteria. For example, intrusion detection software is tested using synthetic data. This data is a representation of the authentic data and may include intrusion instances that are not found in the authentic data. The synthetic data allows the software to recognize these situations and react accordingly. If synthetic data was not used, the software would only be trained to react to the situations provided by the authentic data and it may not recognize another type of intrusion. === Scientific research === Researchers doing clinical trials or any other research may generate synthetic data to aid in creating a baseline for future studies and testing. Real data can contain information that researchers may not want released, so synthetic data is sometimes used to protect the privacy and confidentiality of a dataset. Using synthetic data reduces confidentiality and privacy issues since it holds no personal information and cannot be traced back to any individual. Beyond privacy protection, synthetic data is also being explored for methodological innovation in drug development. For instance, synthetic data may be used to construct synthetic control arms as an alternative to conventional external control arms based on real-world data (RWD) or randomized controlled trials (RCTs). Collectively, regulatory agencies such as the FDA and EMA appear to be at various stages of recognizing and integrating AI-generated synthetic data into their methodologies. While there is growing consensus on the potential of such data to support model development and the broader lifecycle of medicinal products, to date no drug or medical device has been approved using solely or predominantly synthetic data—particularly not as a comparator arm generated entirely via data-driven algorithms. The quality and statistical handling of synthetic data are expected to become more prominent in future regulatory discussions, particularly in contexts such as predictive modeling (e.g., digital twins), where innovative approaches have already been referenced. === Machine learning === Synthetic data is increasingly being used for machine learning applications: a model is trained on a synthetically generated dataset with the intention of transfer learning to real data. Efforts have been made to enable more data science experiments via the construction of general-purpose synthetic data generators, such as the Synthetic Data Vault. In general, synthetic data has several natural advantages: once the synthetic environment is ready, it is fast and cheap to produce as much data as needed; synthetic data can have perfectly accurate labels, including labeling that may be very expensive or impo

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