AI Data Labeling

AI Data Labeling — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

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  Cognitive computing

  Cognitive computing refers to technology platforms that, broadly speaking, are based on the scientific disciplines of artificial intelligence and signal processing. These platforms encompass machine learning, reasoning, natural language processing, speech recognition and vision (object recognition), human–computer interaction, dialog and narrative generation, among other technologies. == Definition == At present, there is no widely agreed upon definition for cognitive computing in either academia or industry. In general, the term cognitive computing has been used to refer to new hardware and/or software that mimics the functioning of the human brain (2004). In this sense, cognitive computing is a new type of computing with the goal of more accurate models of how the human brain/mind senses, reasons, and responds to stimulus. Cognitive computing applications link data analysis and adaptive page displays (AUI) to adjust content for a particular type of audience. As such, cognitive computing hardware and applications strive to be more affective and more influential by design. The term "cognitive system" also applies to any artificial construct able to perform a cognitive process where a cognitive process is the transformation of data, information, knowledge, or wisdom to a new level in the DIKW Pyramid. While many cognitive systems employ techniques having their origination in artificial intelligence research, cognitive systems, themselves, may not be artificially intelligent. For example, a neural network trained to recognize cancer on an MRI scan may achieve a higher success rate than a human doctor. This system is certainly a cognitive system but is not artificially intelligent. Cognitive systems may be engineered to feed on dynamic data in real-time, or near real-time, and may draw on multiple sources of information, including both structured and unstructured digital information, as well as sensory inputs (visual, gestural, auditory, or sensor-provided). == Cognitive analytics == Cognitive computing-branded technology platforms typically specialize in the processing and analysis of large, unstructured datasets. == Applications == Education Even if cognitive computing can not take the place of teachers, it can still be a heavy driving force in the education of students. Cognitive computing being used in the classroom is applied by essentially having an assistant that is personalized for each individual student. This cognitive assistant can relieve the stress that teachers face while teaching students, while also enhancing the student's learning experience over all. Teachers may not be able to pay each and every student individual attention, this being the place that cognitive computers fill the gap. Some students may need a little more help with a particular subject. For many students, Human interaction between student and teacher can cause anxiety and can be uncomfortable. With the help of Cognitive Computer tutors, students will not have to face their uneasiness and can gain the confidence to learn and do well in the classroom. While a student is in class with their personalized assistant, this assistant can develop various techniques, like creating lesson plans, to tailor and aid the student and their needs. Healthcare Numerous tech companies are in the process of developing technology that involves cognitive computing that can be used in the medical field. The ability to classify and identify is one of the main goals of these cognitive devices. This trait can be very helpful in the study of identifying carcinogens. This cognitive system that can detect would be able to assist the examiner in interpreting countless numbers of documents in a lesser amount of time than if they did not use Cognitive Computer technology. This technology can also evaluate information about the patient, looking through every medical record in depth, searching for indications that can be the source of their problems. Commerce Together with Artificial Intelligence, it has been used in warehouse management systems to collect, store, organize and analyze all related supplier data. All these aims at improving efficiency, enabling faster decision-making, monitoring inventory and fraud detection Human Cognitive Augmentation In situations where humans are using or working collaboratively with cognitive systems, called a human/cog ensemble, results achieved by the ensemble are superior to results obtainable by the human working alone. Therefore, the human is cognitively augmented. In cases where the human/cog ensemble achieves results at, or superior to, the level of a human expert then the ensemble has achieved synthetic expertise. In a human/cog ensemble, the "cog" is a cognitive system employing virtually any kind of cognitive computing technology. Other use cases Speech recognition Sentiment analysis Face detection Risk assessment Fraud detection Behavioral recommendations == Industry work == Cognitive computing in conjunction with big data and algorithms that comprehend customer needs, can be a major advantage in economic decision making. The powers of cognitive computing and artificial intelligence hold the potential to affect almost every task that humans are capable of performing. This can negatively affect employment for humans, as there would be no such need for human labor anymore. It would also increase the inequality of wealth; the people at the head of the cognitive computing industry would grow significantly richer, while workers without ongoing, reliable employment would become less well off. The more industries start to use cognitive computing, the more difficult it will be for humans to compete. Increased use of the technology will also increase the amount of work that AI-driven robots and machines can perform. The influence of competitive individuals in conjunction with artificial intelligence/cognitive computing has the potential to change the course of humankind.

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  Exposure Notification

  The (Google/Apple) Exposure Notification System (GAEN) is a framework and protocol specification developed by Apple Inc. and Google to facilitate digital contact tracing during the COVID-19 pandemic. When used by health authorities, it augments more traditional contact tracing techniques by automatically logging close approaches among notification system users using Android or iOS smartphones. Exposure Notification is a decentralized reporting protocol built on a combination of Bluetooth Low Energy technology and privacy-preserving cryptography. It is an opt-in feature within COVID-19 apps developed and published by authorized health authorities. Unveiled on April 10, 2020, it was made available on iOS on May 20, 2020, as part of the iOS 13.5 update and on December 14, 2020, as part of the iOS 12.5 update for older iPhones. On Android, it was added to devices via a Google Play Services update, supporting all versions since Android Marshmallow. The Apple/Google protocol is similar to the Decentralized Privacy-Preserving Proximity Tracing (DP-3T) protocol created by the European DP-3T consortium and the Temporary Contact Number (TCN) protocol by Covid Watch, but is implemented at the operating system level, which allows for more efficient operation as a background process. Since May 2020, a variant of the DP-3T protocol is supported by the Exposure Notification Interface. Other protocols are constrained in operation because they are not privileged over normal apps. This leads to issues, particularly on iOS devices where digital contact tracing apps running in the background experience significantly degraded performance. The joint approach is also designed to maintain interoperability between Android and iOS devices, which constitute nearly all of the market. The ACLU stated the approach "appears to mitigate the worst privacy and centralization risks, but there is still room for improvement". In late April, Google and Apple shifted the emphasis of the naming of the system, describing it as an "exposure notification service", rather than "contact tracing" system. == Technical specification == Digital contact tracing protocols typically have two major responsibilities: encounter logging and infection reporting. Exposure Notification only involves encounter logging which is a decentralized architecture. The majority of infection reporting is centralized in individual app implementations. To handle encounter logging, the system uses Bluetooth Low Energy to send tracking messages to nearby devices running the protocol to discover encounters with other people. The tracking messages contain unique identifiers that are encrypted with a secret daily key held by the sending device. These identifiers change every 15–20 minutes as well as Bluetooth MAC address in order to prevent tracking of clients by malicious third parties through observing static identifiers over time. The sender's daily encryption keys are generated using a random number generator. Devices record received messages, retaining them locally for 14 days. If a user tests positive for infection, the last 14 days of their daily encryption keys can be uploaded to a central server, where it is then broadcast to all devices on the network. The method through which daily encryption keys are transmitted to the central server and broadcast is defined by individual app developers. The Google-developed reference implementation calls for a health official to request a one-time verification code (VC) from a verification server, which the user enters into the encounter logging app. This causes the app to obtain a cryptographically signed certificate, which is used to authorize the submission of keys to the central reporting server. The received keys are then provided to the protocol, where each client individually searches for matches in their local encounter history. If a match meeting certain risk parameters is found, the app notifies the user of potential exposure to the infection. Google and Apple intend to use the received signal strength (RSSI) of the beacon messages as a source to infer proximity. RSSI and other signal metadata will also be encrypted to resist deanonymization attacks. === Version 1.0 === To generate encounter identifiers, first a persistent 32-byte private Tracing Key ( t k {\displaystyle tk} ) is generated by a client. From this a 16 byte Daily Tracing Key is derived using the algorithm d t k i = H K D F ( t k , N U L L , 'CT-DTK' | | D i , 16 ) {\displaystyle dtk_{i}=HKDF(tk,NULL,{\text{'CT-DTK'}}||D_{i},16)} , where H K D F ( Key, Salt, Data, OutputLength ) {\displaystyle HKDF({\text{Key, Salt, Data, OutputLength}})} is a HKDF function using SHA-256, and D i {\displaystyle D_{i}} is the day number for the 24-hour window the broadcast is in starting from Unix Epoch Time. These generated keys are later sent to the central reporting server should a user become infected. From the daily tracing key a 16-byte temporary Rolling Proximity Identifier is generated every 10 minutes with the algorithm R P I i , j = Truncate ( H M A C ( d t k i , 'CT-RPI' | | T I N j ) , 16 ) {\displaystyle RPI_{i,j}={\text{Truncate}}(HMAC(dtk_{i},{\text{'CT-RPI'}}||TIN_{j}),16)} , where H M A C ( Key, Data ) {\displaystyle HMAC({\text{Key, Data}})} is a HMAC function using SHA-256, and T I N j {\displaystyle TIN_{j}} is the time interval number, representing a unique index for every 10 minute period in a 24-hour day. The Truncate function returns the first 16 bytes of the HMAC value. When two clients come within proximity of each other they exchange and locally store the current R P I i , j {\displaystyle RPI_{i,j}} as the encounter identifier. Once a registered health authority has confirmed the infection of a user, the user's Daily Tracing Key for the past 14 days is uploaded to the central reporting server. Clients then download this report and individually recalculate every Rolling Proximity Identifier used in the report period, matching it against the user's local encounter log. If a matching entry is found, then contact has been established and the app presents a notification to the user warning them of potential infection. === Version 1.1 === Unlike version 1.0 of the protocol, version 1.1 does not use a persistent tracing key, rather every day a new random 16-byte Temporary Exposure Key ( t e k i {\displaystyle tek_{i}} ) is generated. This is analogous to the daily tracing key from version 1.0. Here i {\displaystyle i} denotes the time is discretized in 10 minute intervals starting from Unix Epoch Time. From this two 128-bit keys are calculated, the Rolling Proximity Identifier Key ( R P I K i {\displaystyle RPIK_{i}} ) and the Associated Encrypted Metadata Key ( A E M K i {\displaystyle AEMK_{i}} ). R P I K i {\displaystyle RPIK_{i}} is calculated with the algorithm R P I K i = H K D F ( t e k i , N U L L , 'EN-RPIK' , 16 ) {\displaystyle RPIK_{i}=HKDF(tek_{i},NULL,{\text{'EN-RPIK'}},16)} , and A E M K i {\displaystyle AEMK_{i}} using the algorithm A E M K i = H K D F ( t e k i , N U L L , 'EN-AEMK' , 16 ) {\displaystyle AEMK_{i}=HKDF(tek_{i},NULL,{\text{'EN-AEMK'}},16)} . From these values a temporary Rolling Proximity Identifier ( R P I i , j {\displaystyle RPI_{i,j}} ) is generated every time the BLE MAC address changes, roughly every 15–20 minutes. The following algorithm is used: R P I i , j = A E S 128 ( R P I K i , 'EN-RPI' | | 0 x 000000000000 | | E N I N j ) {\displaystyle RPI_{i,j}=AES128(RPIK_{i},{\text{'EN-RPI'}}||{\mathtt {0x000000000000}}||ENIN_{j})} , where A E S 128 ( Key, Data ) {\displaystyle AES128({\text{Key, Data}})} is an AES cryptography function with a 128-bit key, the data is one 16-byte block, j {\displaystyle j} denotes the Unix Epoch Time at the moment the roll occurs, and E N I N j {\displaystyle ENIN_{j}} is the corresponding 10-minute interval number. Next, additional Associated Encrypted Metadata is encrypted. What the metadata represents is not specified, likely to allow the later expansion of the protocol. The following algorithm is used: Associated Encrypted Metadata i , j = A E S 128 _ C T R ( A E M K i , R P I i , j , Metadata ) {\displaystyle {\text{Associated Encrypted Metadata}}_{i,j}=AES128\_CTR(AEMK_{i},RPI_{i,j},{\text{Metadata}})} , where A E S 128 _ C T R ( Key, IV, Data ) {\displaystyle AES128\_CTR({\text{Key, IV, Data}})} denotes AES encryption with a 128-bit key in CTR mode. The Rolling Proximity Identifier and the Associated Encrypted Metadata are then combined and broadcast using BLE. Clients exchange and log these payloads. Once a registered health authority has confirmed the infection of a user, the user's Temporary Exposure Keys t e k i {\displaystyle tek_{i}} and their respective interval numbers i {\displaystyle i} for the past 14 days are uploaded to the central reporting server. Clients then download this report and individually recalculate every Rolling Proximity Identifier starting from interval number i {\displaystyle i} ,

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  CinePlayer

  CinePlayer is a software based media player used to review Digital Cinema Packages (DCP) without the need for a digital cinema server by Doremi Labs. CinePlayer can play back any DCP, not just those created by Doremi Mastering products. In addition to playing DCPs, CinePlayer can also playback JPEG2000 image sequences and many popular multimedia file types. There are two versions of CinePlayer available, standard and Pro. The standard version supports playback of non-encrypted, 2D DCP's up to 2K resolution. The Pro version supports playback of encrypted, 2D or 3D DCP's with subtitles up to 4K resolution. == Supported formats == === Containers === AVI MOV MXF MPG TS WMV M2TS MTS MP4 MKV === Video codecs === JPEG2000 ProRes 422 DNxHD YUV Uncompressed 8-10 bits DIVX XVID MPEG4 AVC / H-264 VC-1 MPEG2 === Supported image sequences === BMP TIFF TGA DPX JPG J2C === Supported audio files === WAV MP3 WMA MP2

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  Kernel-phase

  Kernel-phases are observable quantities used in high resolution astronomical imaging used for superresolution image creation. It can be seen as a generalization of closure phases for redundant arrays. For this reason, when the wavefront quality requirement are met, it is an alternative to aperture masking interferometry that can be executed without a mask while retaining phase error rejection properties. The observables are computed through linear algebra from the Fourier transform of direct images. They can then be used for statistical testing, model fitting, or image reconstruction. == Prerequisites == In order to extract kernel-phases from an image, some requirements must be met: Images are nyquist-sampled (at least 2 pixels per resolution element ( λ D {\displaystyle {\frac {\lambda }{D}}} )) Images are taken in near monochromatic light Exposure time is shorter than the timescale of aberrations Strehl ratio is high (good adaptive optics) Linearity of the pixel response (i.e. no saturation) Deviations from these requirements are known to be acceptable, but lead to observational bias that should be corrected by the observation of calibrators. == Definition == The method relies on a discrete model of the instrument's pupil plane and the corresponding list of baselines to provide corresponding vectors φ {\displaystyle \varphi } of pupil plane errors and Φ {\displaystyle \Phi } of image plane Fourier Phases. When the wavefront error in the pupil plane is small enough (i.e. when the Strehl ratio of the imaging system is sufficiently high), the complex amplitude associated to the instrumental phase in one point of the pupil φ k {\displaystyle \varphi _{k}} , can be approximated by e i φ k ≈ 1 + i φ k {\displaystyle e^{i\varphi _{k}}\approx 1+{\mathit {i}}\varphi _{k}} . This permits the expression of the pupil-plane phase aberrations φ {\displaystyle \varphi } to the image plane Fourier phase as a linear transformation described by the matrix A {\displaystyle A} : Φ = Φ 0 + A ⋅ φ {\displaystyle \Phi =\Phi _{0}+A\cdot \varphi } Where Φ 0 {\displaystyle \Phi _{0}} is the theoretical Fourier phase vector of the object. In this formalism, singular value decomposition can be used to find a matrix K {\displaystyle K} satisfying K ⋅ A = 0 {\displaystyle K\cdot A=0} . The rows of K {\displaystyle K} constitute a basis of the kernel of A T {\displaystyle A^{T}} . K ⋅ Φ = K ⋅ Φ 0 + K ⋅ A ⋅ φ {\displaystyle K\cdot \Phi =K\cdot \Phi _{0}+{\cancel {K\cdot A\cdot \varphi }}} The vector K . Φ {\displaystyle K.\Phi } is called the kernel-phase vector of observables. This equation can be used for model-fitting as it represents the interpretation of a sub-space of the Fourier phase that is immune to the instrumental phase errors to the first order. == Applications == The technique was first used in the re-analysis of archival images from the Hubble Space Telescope where it enabled the discovery of a number of brown dwarf in close binary systems. The technique is used as an alternative to aperture masking interferometry, especially for fainter stars because it does not require the use of masks that typically block 90% of the light, and therefore allows higher throughput. It is also considered to be an alternative to coronagraphy for direct detection of exoplanets at very small separations (below 2 λ D {\displaystyle 2{\frac {\lambda }{D}}} ) where coronagraphs are limited by the wavefront errors of adaptive optics. The same framework can be used for wavefront sensing. In the case of an asymmetric aperture, a pseudo-inverse of A {\displaystyle A} can be used to reconstruct the wavefront errors directly from the image. A Python library called xara is available on GitHub and maintained by Frantz Martinache to facilitate the extraction and interpretation of kernel-phases. The KERNEL project, has received funding from the European Research Council to explore the potential of these observables for a number of use-cases, including direct detection of exoplanets, image reconstruction, and image plane wavefront sensing for adaptive optics.

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  PagedAttention

  PagedAttention is an attention algorithm for efficient serving of large language models (LLMs). It was introduced in 2023 by Woosuk Kwon and colleagues in the paper Efficient Memory Management for Large Language Model Serving with PagedAttention, alongside the vLLM serving engine. The method stores the key–value cache used during autoregressive decoding in fixed-size blocks that can be mapped to non-contiguous physical memory, borrowing ideas from virtual memory, paging, and operating system design. == Background == In transformer inference, the key–value cache grows with sequence length and the number of concurrent requests. Kwon et al. argued that earlier serving systems typically reserved contiguous cache regions in advance, which caused reserved space, internal fragmentation, and external fragmentation. In their experiments, the paper reported that the effective memory utilization of previous systems could fall as low as 20.4%. == Description == PagedAttention partitions the cache of each sequence into fixed-size KV blocks. A request's cache is represented as a sequence of logical blocks, while a block table maps those logical blocks to physical GPU-memory blocks. As a result, neighboring logical blocks do not need to be contiguous in physical memory, and new blocks can be allocated on demand as generation proceeds. The design also makes it easier to share cache state across related decoding paths. In vLLM, physical blocks can be reference-counted and shared among requests or branches, with block-granularity copy-on-write used when a shared block must be modified. The original paper applied this design to parallel sampling, beam search, and prompts with shared prefixes. == Mathematical formulation == For a query token i {\displaystyle i} in causal self-attention, the standard attention output can be written as a i j = exp ⁡ ( q i ⊤ k j / d ) ∑ t = 1 i exp ⁡ ( q i ⊤ k t / d ) , o i = ∑ j = 1 i a i j v j {\displaystyle a_{ij}={\frac {\exp(\mathbf {q} _{i}^{\top }\mathbf {k} _{j}/{\sqrt {d}})}{\sum _{t=1}^{i}\exp(\mathbf {q} _{i}^{\top }\mathbf {k} _{t}/{\sqrt {d}})}},\;\mathbf {o} _{i}=\sum _{j=1}^{i}a_{ij}\mathbf {v} _{j}} where q i {\displaystyle \mathbf {q} _{i}} , k j {\displaystyle \mathbf {k} _{j}} , and v j {\displaystyle \mathbf {v} _{j}} are the query, key, and value vectors, and d {\displaystyle d} is the attention dimension. If the cache is partitioned into blocks of size B {\displaystyle B} , the key and value blocks may be written as K j = ( k ( j − 1 ) B + 1 , … , k j B ) , V j = ( v ( j − 1 ) B + 1 , … , v j B ) {\displaystyle \mathbf {K} _{j}=(\mathbf {k} _{(j-1)B+1},\ldots ,\mathbf {k} _{jB}),\;\mathbf {V} _{j}=(\mathbf {v} _{(j-1)B+1},\ldots ,\mathbf {v} _{jB})} PagedAttention then performs the computation blockwise: A i j = exp ⁡ ( q i ⊤ K j / d ) ∑ t = 1 ⌈ i / B ⌉ exp ⁡ ( q i ⊤ K t / d ) , o i = ∑ j = 1 ⌈ i / B ⌉ V j A i j ⊤ {\displaystyle \mathbf {A} _{ij}={\frac {\exp(\mathbf {q} _{i}^{\top }\mathbf {K} _{j}/{\sqrt {d}})}{\sum _{t=1}^{\lceil i/B\rceil }\exp(\mathbf {q} _{i}^{\top }\mathbf {K} _{t}/{\sqrt {d}})}},\;\mathbf {o} _{i}=\sum _{j=1}^{\lceil i/B\rceil }\mathbf {V} _{j}\mathbf {A} _{ij}^{\top }} where A i j {\displaystyle \mathbf {A} _{ij}} is the vector of attention scores for the j {\displaystyle j} -th KV block. In the formulation given by Kwon et al., this preserves the causal attention calculation while allowing the key and value blocks to reside in non-contiguous physical memory. == Performance and use == The vLLM paper reported that, on its evaluated workloads, the use of PagedAttention and the associated memory-management design improved serving throughput by 2–4× over the compared baselines, including FasterTransformer and Orca, while preserving model outputs. In experiments on OPT-13B with the Alpaca trace, the paper also reported memory savings of 6.1–9.8% for parallel sampling and 37.6–55.2% for beam search through KV-block sharing. A 2024 survey of LLM serving systems described PagedAttention as having become an industry norm in LLM serving frameworks, citing support in TGI, vLLM, and TensorRT-LLM. == Limitations and alternatives == Subsequent work has described trade-offs in the approach. The 2025 vAttention paper argued that PagedAttention requires attention kernels to be rewritten to support paging and increases software complexity, portability issues, redundancy, and execution overhead, proposing instead a memory manager that keeps the cache contiguous in virtual memory while relying on demand paging for physical allocation. === vAttention === Unlike PagedAttention, vAttention does not introduce a different attention rule; it retains the standard attention computation Attention ⁡ ( q i , K , V ) = softmax ⁡ ( q i K ⊤ s c a l e ) V . {\displaystyle \operatorname {Attention} (q_{i},K,V)=\operatorname {softmax} \left({\frac {q_{i}K^{\top }}{\mathrm {scale} }}\right)V.} In the notation of Prabhu et al., the key and value tensors for a request seen so far are K , V ∈ R L ′ × ( H × D ) {\displaystyle K,V\in \mathbb {R} ^{L'\times (H\times D)}} , where L ′ {\displaystyle L'} is the context length seen so far, H {\displaystyle H} is the number of KV heads on a worker, and D {\displaystyle D} is the dimension of each KV head. In systems prior to PagedAttention, the K cache (or V cache) at each layer of a worker is typically allocated as a 4D tensor of shape [ B , L , H , D ] , {\displaystyle [B,L,H,D],} where B {\displaystyle B} is batch size and L {\displaystyle L} is the maximum context length supported by the model. vAttention preserves this contiguous virtual-memory view while deferring physical-memory allocation to runtime. A serving framework maintains separate K and V tensors for each layer, so vAttention reserves 2 N {\displaystyle 2N} virtual-memory buffers on a worker, where N {\displaystyle N} is the number of layers managed by that worker. The maximum size of one virtual-memory buffer is B S = B × S , {\displaystyle BS=B\times S,} where S {\displaystyle S} is the maximum size of a single request's per-layer K cache (or V cache) on a worker. The paper defines S = L × H × D × P , {\displaystyle S=L\times H\times D\times P,} where P {\displaystyle P} is the number of bytes needed to store one element. In this formulation, vAttention keeps the KV cache contiguous in virtual memory and relies on demand paging for physical allocation, rather than modifying the attention kernel to operate over non-contiguous KV-cache blocks.

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  Level-set method

  The Level-set method (LSM) is a conceptual framework for using level sets as a tool for numerical analysis of surfaces and shapes. LSM can perform numerical computations involving curves and surfaces on a fixed Cartesian grid without having to parameterize these objects. LSM makes it easier to perform computations on shapes with sharp corners and shapes that change topology (such as by splitting in two or developing holes). These characteristics make LSM effective for modeling objects that vary in time, such as an airbag inflating or a drop of oil floating in water. == Overview == The figure on the right illustrates several ideas about LSM. In the upper left corner is a bounded region with a well-behaved boundary. Below it, the red surface is the graph of a level set function φ {\displaystyle \varphi } determining this shape, and the flat blue region represents the X-Y plane. The boundary of the shape is then the zero-level set of φ {\displaystyle \varphi } , while the shape itself is the set of points in the plane for which φ {\displaystyle \varphi } is positive (interior of the shape) or zero (at the boundary). In the top row, the shape's topology changes as it is split in two. It is challenging to describe this transformation numerically by parameterizing the boundary of the shape and following its evolution. An algorithm can be used to detect the moment the shape splits in two and then construct parameterizations for the two newly obtained curves. On the bottom row, however, the plane at which the level set function is sampled is translated upwards, on which the shape's change in topology is described. It is less challenging to work with a shape through its level-set function rather than with itself directly, in which a method would need to consider all the possible deformations the shape might undergo. Thus, in two dimensions, the level-set method amounts to representing a closed curve Γ {\displaystyle \Gamma } (such as the shape boundary in our example) using an auxiliary function φ {\displaystyle \varphi } , called the level-set function. The curve Γ {\displaystyle \Gamma } is represented as the zero-level set of φ {\displaystyle \varphi } by Γ = { ( x , y ) ∣ φ ( x , y ) = 0 } , {\displaystyle \Gamma =\{(x,y)\mid \varphi (x,y)=0\},} and the level-set method manipulates Γ {\displaystyle \Gamma } implicitly through the function φ {\displaystyle \varphi } . This function φ {\displaystyle \varphi } is assumed to take positive values inside the region delimited by the curve Γ {\displaystyle \Gamma } and negative values outside. == The level-set equation == If the curve Γ {\displaystyle \Gamma } moves in the normal direction with a speed v {\displaystyle v} , then by chain rule and implicit differentiation, it can be determined that the level-set function φ {\displaystyle \varphi } satisfies the level-set equation ∂ φ ∂ t = v | ∇ φ | . {\displaystyle {\frac {\partial \varphi }{\partial t}}=v|\nabla \varphi |.} Here, | ⋅ | {\displaystyle |\cdot |} is the Euclidean norm (denoted customarily by single bars in partial differential equations), and t {\displaystyle t} is time. This is a partial differential equation, in particular a Hamilton–Jacobi equation, and can be solved numerically, for example, by using finite differences on a Cartesian grid. However, the numerical solution of the level set equation may require advanced techniques. Simple finite difference methods fail quickly. Upwinding methods such as the Godunov method are considered better; however, the level set method does not guarantee preservation of the volume and shape of the set level in an advection field that maintains shape and size, for example, a uniform or rotational velocity field. Instead, the shape of the level set may become distorted, and the level set may disappear over a few time steps. Therefore, high-order finite difference schemes, such as high-order essentially non-oscillatory (ENO) schemes, are often required, and even then, the feasibility of long-term simulations is questionable. More advanced methods have been developed to overcome this; for example, combinations of the leveling method with tracking marker particles suggested by the velocity field. == Example == Consider a unit circle in R 2 {\textstyle \mathbb {R} ^{2}} , shrinking in on itself at a constant rate, i.e. each point on the boundary of the circle moves along its inwards pointing normally at some fixed speed. The circle will shrink and eventually collapse down to a point. If an initial distance field is constructed (i.e. a function whose value is the signed Euclidean distance to the boundary, positive interior, negative exterior) on the initial circle, the normalized gradient of this field will be the circle normal. If the field has a constant value subtracted from it in time, the zero level (which was the initial boundary) of the new fields will also be circular and will similarly collapse to a point. This is due to this being effectively the temporal integration of the Eikonal equation with a fixed front velocity. == Applications == In mathematical modeling of combustion, LSM is used to describe the instantaneous flame surface, known as the G equation. Level-set data structures have been developed to facilitate the use of the level-set method in computer applications. Computational fluid dynamics Trajectory planning Optimization Image processing Computational biophysics Discrete complex dynamics (visualization of the parameter plane and the dynamic plane) == History == The level-set method was developed in 1979 by Alain Dervieux, and subsequently popularized by Stanley Osher and James Sethian. It has since become popular in many disciplines, such as image processing, computer graphics, computational geometry, optimization, computational fluid dynamics, and computational biology.

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  Softwarp

  Softwarp is a software technique to warp an image so that it can be projected on a curved screen. This can be done in real time by inserting the softwarp as a last step in the rendering cycle. The problem is to know how the image should be warped to look correct on the curved screen. There are several techniques to auto calibrate the warping by projecting a pattern and using cameras and/or sensors. The information from the sensors is sent to the software so that it can analyze the data and calculate the curvature of the projection screen. == Usage == The softwarp can be used to project virtual views on curved walls and domes. These are usually used in vehicle simulators, for instance boat-, car- and airplane simulators. To make it possible to cover a dome with a 360 degree view you need to use several projectors. A problem with using several projectors on the same screen is that the edges between the projected images get about twice the amount of light. This is solved by using a technique called edge blending. With this technique a “filter” is inserted on the edge that fades the image from 100% light strength (luminance) to 0% (the lowest luminance depends on the contrast ratio of the projector). == History == The first warping technologies used a hardware image processing unit to warp the image. This processing unit was inserted between the graphics card and the projector. The problem with this technique is that it depends on the type of signal and the quality of the signal from the graphics card to warp it correctly. The process unit also needs several lines of image information before it can start sending out the warped image. This adds a latency to the display system that could be a problem in simulators that need fast response time, for instance fighter jet simulators. Softwarping eliminates the latency.

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  Depop

  Depop Limited is a social e-commerce company based in London, with additional offices in Milan and New York City. The company allows users to buy and sell items, which are mostly used and vintage pieces of clothing. == History == Depop was founded in 2011 by entrepreneur Simon Beckerman at an Italian technological incubator and business start-up centre, H-Farm. Beckerman came up with the original outline of the application during his time working on PIG, a fashion magazine based in Italy that he co-founded. The idea was to create a platform where products shown in the magazine could be purchased by users online. This idea turned into a concept similar to a flea market but on the internet, where people could sell their items while also being in control of advertising, public relations, and the creative process behind their accounts. While being financially supported by H-Farm, Beckerman worked within a team to create and lay out the Depop application while exposing it to numerous investors. In 2013, Beckerman became a member of the company's board to help improve the application and business while concurrently ceding his role of CEO. Maria Raga, Depop's co-founder and former CEO, took on the role of vice president of operations in 2014, and in 2016, she became chief executive. According to Raga, the main goal while developing Depop was to become the next Airbnb or Spotify, but to make an impact on fashion. Paolo Barberis and Nana Bianca were two of the first investors in the platform in 2012 with a seed investment. Its headquarters were moved to London in 2012. Depop expanded and opened additional offices in Milan and New York City. Beckerman raised €1 million in funding in October 2013 from Red Circle Investment and brought on Faroese Runar Reistrup as new CEO. In 2015, Depop secured another investment of $8 million from Balderton Capital and HV Capital. In March 2016, former CEO, Runar Reistrup, stated that Depop's growth was achieved through word of mouth. During his time as CEO, this growth involved taking Depop as a startup and working to raise funds to eventually amass a significant user base within the United States. In June 2019, Depop raised $62 million in Series C from General Atlantic to fund its expansion. Previous investors HV Capital, Balderton Capital, Creandum, Octopus Ventures, TempoCap and Sebastian Siemiatkowski also participated. During this time, Depop held workshops and conversations as part of their Depop Live NY events, and the company also opened a London store through their partnership with Selfridges. In 2020, Depop's gross merchandise sales and revenue both more than doubled to $650 million and $70 million respectively. This may be attributed to Depop's responsiveness to user trends, its lack of issues regarding inventory management, and the increase in users looking to resell. As of 2024, Depop has over 35 million users, according to their website. Depop is popular for Gen Z and young millennials, it is the 10th most-visited shopping platform for Gen Z consumers in the US, and, in a poll conducted by The Strategist in 2019, Depop was voted by teenagers as their favorite resale website. === Acquisition by Etsy === In June 2021, Depop was acquired by Etsy for $1.6 billion in cash, making it Etsy's most expensive acquisition; however, Depop continues to operate as a standalone brand independent from Etsy. This means that in addition to Depop keeping its existing team, the company retained its London location. At the time of acquisition, Etsy CEO Josh Silverman’s goal was to counteract the influx of buyers starting to go back to physical shops for their purchases. He saw Depop for its potential as a platform supporting a variety of products and creating a greater community of users. According to Silverman, Depop may expand and improve its services for its significant Gen Z user base. For Etsy, this acquisition maintains the company's foothold in the clothing industry and allows the company to expand its customer base to a younger demographic; at the same time, Depop is now able to make use of Etsy's company operations. When Maria Raga relinquished her position as Depop's CEO in 2022, Etsy assigned the role to Kruti Patel Goyal, who was Etsy's former chief product officer and a leader there for eleven years. When Goyal was appointed president and chief growth officer for Etsy in May, Peter Semple, former chief marketing officer, was assigned CEO of Depop officially on August 1st. === Acquisition by eBay === In February 2026, Etsy announced a proposed sale of Depop to eBay for $1.2 billion that was estimated to close within the year. == Business model == === Selling === Depop operates as a marketplace and social platform, where users can follow friends and other influencers to view their buying and selling activities. Through the platform, users are able to sell branded and designer items, as well as vintage pieces. Depop users are also encouraged by the platform to use other social networking services such as Instagram to promote their shop profiles. Celebrities have resold their own items on Depop, with some donating proceeds to charitable causes. Depop's user interface is modeled after that of Instagram. According to Depop, users who list and sell items provide their own photos with item descriptions. Users also note their designer items' authenticity and if they include any labels, tags, and receipts. These listings will appear in users' feeds. The platform's "Explore" page features items picked out by Depop staff. According to Depop, purchases are made via Apple Pay, Google Pay, credit and debit cards, and Klarna. Depop payments stay in-app, allowing for the company to mediate disputes and process refunds. Depop payments allow sellers to directly receive their payments in their bank account. To get paid by Depop, a seller has to add a bank account and verify their identification by uploading an ID. On July 18, 2024, Depop CEO Kruti Patel Goyal announced the removal of selling fees for US sellers, while maintaining a payment processing fee. This policy adjustment aimed to enhance seller revenue and support the growth of the second-hand market. === Buying === A Depop transaction includes the agreed sale price of the item, shipping fees, VAT or other applicable taxes and duties, and the marketplace fee for buyers in the U.S. or U.K. For international deliveries, packages may be subject to import taxes, customs duties, or fees, payable upon arrival or at checkout if Depop collects the tax on behalf of the buyer. For domestic purchases, relevant taxes may be collected by the seller or charged by the platform at checkout, ensuring no additional taxes are due upon delivery. For users in Australia, the United Kingdom, and the United States, Depop allows users to receive a full refund if their item does not arrive, arrives damaged, or is considerably different from the original when the issue is reported within 30 days. === Competitors === As of June 2021, Depop's competitors include Vinted, a platform founded by Milda Mitkute and Justas Janauskas in 2008 and valued at €3.5 billion, as well as the U.S. resale site Poshmark, valued at $3.5 billion. Additional competitors include Grailed, a peer-to-peer e-commerce site founded in 2014 that is recognized for its high-end second-hand menswear and streetwear, and Vestiaire Collection, a European resale app established in 2009 which specializes in authenticated pre-owned luxury items. The popularity of Depop has negatively impacted traditional second-hand stores, which can struggle to compete due to high labor costs and quality demands. There is an oversupply of clothes with the rise of fast fashion; this has taken a toll on the revenue aspect of the second-hand clothing industry. == Criticism == In November 2019, Business of Fashion reported that users within the Depop app were receiving sexually suggestive messages. In February 2020, Jessica Hamilton, a Depop buyer, reported that she found many scammers on the platform. She noticed this issue after she attempted to purchase a Nintendo Switch from a seller who would suspiciously only accept payment through a direct bank transfer without buyer protection. Hamilton blamed the company for its lack of action and relaxed security measures compared to other e-commerce sites, which made the platform especially susceptible to hackers. Without a clear strategy for managing scams, Depop lost some users' trust because of its negligence. In October 2020, some Depop buyers were tricked into paying sellers directly to bypass Depop's buyer protections, and the Depop sellers then sold those users' information on the dark web. In response, Depop claimed that it would improve security through mandatory password updates and multi-factor authentication. Users have criticized Depop for belatedly taking action against this issue.

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  Percept (artificial intelligence)

  A percept is the input that an intelligent agent is perceiving at any given moment. It is essentially the same concept as a percept in psychology, except that it is being perceived not by the brain but by the agent. A percept is detected by a sensor, often a camera, processed accordingly, and acted upon by an actuator. Each percept is added to a "percept sequence", which is a complete history of each percept ever detected. The agent's action at any instant point may depend on the entire percept sequence up to that particular instant point. An intelligent agent chooses how to act not only based on the current percept, but the percept sequence. The next action is chosen by the agent function, which maps every percept to an action. For example, if a camera were to record a gesture, the agent would process the percepts, calculate the corresponding spatial vectors, examine its percept history, and use the agent program (the application of the agent function) to act accordingly. == Examples == Examples of percepts include inputs from touch sensors, cameras, infrared sensors, sonar, microphones, mice, and keyboards. A percept can also be a higher-level feature of the data, such as lines, depth, objects, faces, or gestures.

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  Sensory, Inc.

  Sensory, Inc. is an American company which develops software AI technologies for speech, sound and vision. It is based in Santa Clara, California. Sensory’s technologies have shipped in over three billion products from hundreds of leading consumer electronics manufacturers including AT&T, Hasbro, Huawei, Google, Amazon, Samsung, LG, Mattel, Motorola, Plantronics, GoPro, Sony, Tencent, Garmin, LG, Microsoft, Lenovo, and more. Sensory has over 60 issued patents covering speech recognition in consumer electronics, biometric authentication, sensor/speech combinations, wake word technology, and more. == History == Sensory, Inc. was founded in 1994, originally as Sensory Circuits, by Forrest Mozer, Mike Mozer and Todd Mozer. The three had also co-founded ESS Technology years earlier. In 1999 Sensory acquired Fluent Speech Technologies, which was formed and started by a group of professors out of the Oregon Graduate Institute (formerly OGI, now OHSU). Fluent Speech Technologies developed high performance embedded speech engines, the technology from this acquisition is now the core technology used throughout Sensory's chip and software line. === Company timeline === 1994 – Founded 1995 – Introduces the RSC 164 - first commercially successful speech recognition IC 1998 – Introduces first speaker verification IC 2000 – Acquires Oregon based Fluent-Speech Technologies 2002 – Acquires Texas Instruments line of speech output ICs (the SC series) 2007 – Introduces first Voice User Interface for Bluetooth silicon (CSR BC-5) - BlueGenie 2008 - Sensory and BlueAnt partner on the V1 - Revolutionary new Bluetooth headset with a voice user interface. First wearable to use a voice user interface for control and best-reviewed speech recognition product in history 2009 – Introduced world's smallest text to speech system (TTS) and Truly HandsfreeTM Triggers/ wake words. 2010 – Introduced the NLP-5x – First Natural Language Voice Processor and TrulyHandsfree wake words in SDKs for Android, iOS, Linux, and Windows. NLP5x used the first generation of TrulyHandsfree wake words with low power and enhanced accuracy. 2011 – Sensory partners with Google and Microsoft to enable TrulyHandsfree as a front end to Goog411 and Bing411 2012 – Partnered with Tensilica to offer ultra-low power TrulyHandsfree wake words; introduced Speaker Verification and Speaker Identification for mobile phones and other consumer electronics. 2012 - TrulyHandsfree released into Samsung's Galaxy S2 for "Hey Galaxy" wake word 2013 – TrulyHandsfree wake words migrated to many new platforms and began shipping as MotoVoice in the Google-owned MotoX. Sensory's TrulyHandsfree in mobile takes off with the Galaxy S3 and S4 and Galaxy Note and is licensed into wearables like Google Glass. 2014 – Announced new initiative in Vision; added LG and Motorola as customers; received the 2014 Global Mobile Award for Best Mobile Technology Breakthrough at the GSMA Mobile World Congress in Barcelona, Spain (judges commented, "A big advance for the wearables market, this offers many benefits for consumers, increasing uptake and usage of many mobile apps, driving revenue for operators and content providers.") 2015-2018 - Licensed Google, Amazon, MSFT, Baidu, Huawei, ZTE, and many others with TrulyHandsfree wake words. Sensory develops first wake words for OK Google, Hey Siri, and Hey Cortana. 2019 - Sensory launched two new solutions: SoundID, sound identification, and TrulyNatural, embedded large vocabulary speech recognition. Sensory also acquired Vocalize.ai, an independent testing lab. 2020 - Sensory introduced VoiceHub, which allows the automated generation of wake words. 2021 - Sensory expands VoiceHub with speech recognition and NLU capabilities. The company initiated a new cloud platform, SensoryCloud.ai. 2022-Sensory rolls out SensoryCloud.ai with speech to text, text to speech, face & voice biometrics 2024- Sensory Automotive & TrulyNatural Speech-to-text On-Device launched == Technology and products == Sensory originally developed both hardware (Integrated Circuit - IC or "chip") and software platforms but migrated to software only around 2005 and added cloud and hybrid computing capabilities in 2021. Sensory's RSC-164 IC (Integrated Circuit or "chip") was used on NASA's Mars Polar Lander in the Mars Microphone on the Lander. Speech Synthesis SC-6x chips – acquired some speech synthesis technology from Texas Instruments. Sensory’s embedded AI solutions include the following: TrulyHandsfree (THF) - wake word detection and phrase spotting. TrulyNatural (TNL) - large vocabulary continuous speech recognition with NLU. TrulySecure (TS) - face and voice biometrics. TrulySecureSpeakerVerification (TSSV) - speaker and sound identification. VoiceHub - Online portal for creating custom wake words and speech recognition models with NLU. Sensory Automotive- Sensory Automotive is a full voice and vision suite of AI technologies that operate efficiently in the car without connecting to a network. The cloud initiative, SensoryCloud.ai, is targeting Speech To Text (STT), Text To Speech (TTS), Wake Word verification, face and voice recognition, and sound identification.

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  Tensor operator

  In pure and applied mathematics, quantum mechanics and computer graphics, a tensor operator generalizes the notion of operators which are scalars and vectors. A special class of these are spherical tensor operators which apply the notion of the spherical basis and spherical harmonics. The spherical basis closely relates to the description of angular momentum in quantum mechanics and spherical harmonic functions. The coordinate-free generalization of a tensor operator is known as a representation operator. == The general notion of scalar, vector, and tensor operators == In quantum mechanics, physical observables that are scalars, vectors, and tensors, must be represented by scalar, vector, and tensor operators, respectively. Whether something is a scalar, vector, or tensor depends on how it is viewed by two observers whose coordinate frames are related to each other by a rotation. Alternatively, one may ask how, for a single observer, a physical quantity transforms if the state of the system is rotated. Consider, for example, a system consisting of a molecule of mass M {\displaystyle M} , traveling with a definite center of mass momentum, p z ^ {\displaystyle p{\mathbf {\hat {z}} }} , in the z {\displaystyle z} direction. If we rotate the system by 90 ∘ {\displaystyle 90^{\circ }} about the y {\displaystyle y} axis, the momentum will change to p x ^ {\displaystyle p{\mathbf {\hat {x}} }} , which is in the x {\displaystyle x} direction. The center-of-mass kinetic energy of the molecule will, however, be unchanged at p 2 / 2 M {\displaystyle p^{2}/2M} . The kinetic energy is a scalar and the momentum is a vector, and these two quantities must be represented by a scalar and a vector operator, respectively. By the latter in particular, we mean an operator whose expected values in the initial and the rotated states are p z ^ {\displaystyle p{\mathbf {\hat {z}} }} and p x ^ {\displaystyle p{\mathbf {\hat {x}} }} . The kinetic energy on the other hand must be represented by a scalar operator, whose expected value must be the same in the initial and the rotated states. In the same way, tensor quantities must be represented by tensor operators. An example of a tensor quantity (of rank two) is the electrical quadrupole moment of the above molecule. Likewise, the octupole and hexadecapole moments would be tensors of rank three and four, respectively. Other examples of scalar operators are the total energy operator (more commonly called the Hamiltonian), the potential energy, and the dipole-dipole interaction energy of two atoms. Examples of vector operators are the momentum, the position, the orbital angular momentum, L {\displaystyle {\mathbf {L} }} , and the spin angular momentum, S {\displaystyle {\mathbf {S} }} . (Fine print: Angular momentum is a vector as far as rotations are concerned, but unlike position or momentum it does not change sign under space inversion, and when one wishes to provide this information, it is said to be a pseudovector.) Scalar, vector and tensor operators can also be formed by products of operators. For example, the scalar product L ⋅ S {\displaystyle {\mathbf {L} }\cdot {\mathbf {S} }} of the two vector operators, L {\displaystyle {\mathbf {L} }} and S {\displaystyle {\mathbf {S} }} , is a scalar operator, which figures prominently in discussions of the spin–orbit interaction. Similarly, the quadrupole moment tensor of our example molecule has the nine components Q i j = ∑ α q α ( 3 r α , i r α , j − r α 2 δ i j ) . {\displaystyle Q_{ij}=\sum _{\alpha }q_{\alpha }\left(3r_{\alpha ,i}r_{\alpha ,j}-r_{\alpha }^{2}\delta _{ij}\right).} Here, the indices i {\displaystyle i} and j {\displaystyle j} can independently take on the values 1, 2, and 3 (or x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyle z} ) corresponding to the three Cartesian axes, the index α {\displaystyle \alpha } runs over all particles (electrons and nuclei) in the molecule, q α {\displaystyle q_{\alpha }} is the charge on particle α {\displaystyle \alpha } , and r α , i {\displaystyle r_{\alpha ,i}} is the i {\displaystyle i} -th component of the position of this particle. Each term in the sum is a tensor operator. In particular, the nine products r α , i r α , j {\displaystyle r_{\alpha ,i}r_{\alpha ,j}} together form a second rank tensor, formed by taking the outer product of the vector operator r α {\displaystyle {\mathbf {r} }_{\alpha }} with itself. == Rotations of quantum states == === Quantum rotation operator === The rotation operator about the unit vector n (defining the axis of rotation) through angle θ is U [ R ( θ , n ^ ) ] = exp ⁡ ( − i θ ℏ n ^ ⋅ J ) {\displaystyle U[R(\theta ,{\hat {\mathbf {n} }})]=\exp \left(-{\frac {i\theta }{\hbar }}{\hat {\mathbf {n} }}\cdot \mathbf {J} \right)} where J = (Jx, Jy, Jz) are the rotation generators (also the angular momentum matrices): J x = ℏ 2 ( 0 1 0 1 0 1 0 1 0 ) J y = ℏ 2 ( 0 i 0 − i 0 i 0 − i 0 ) J z = ℏ ( − 1 0 0 0 0 0 0 0 1 ) {\displaystyle J_{x}={\frac {\hbar }{\sqrt {2}}}{\begin{pmatrix}0&1&0\\1&0&1\\0&1&0\end{pmatrix}}\,\quad J_{y}={\frac {\hbar }{\sqrt {2}}}{\begin{pmatrix}0&i&0\\-i&0&i\\0&-i&0\end{pmatrix}}\,\quad J_{z}=\hbar {\begin{pmatrix}-1&0&0\\0&0&0\\0&0&1\end{pmatrix}}} and let R ^ = R ^ ( θ , n ^ ) {\displaystyle {\widehat {R}}={\widehat {R}}(\theta ,{\hat {\mathbf {n} }})} be a rotation matrix. According to the Rodrigues' rotation formula, the rotation operator then amounts to U [ R ( θ , n ^ ) ] = 1 1 − i sin ⁡ θ ℏ n ^ ⋅ J − 1 − cos ⁡ θ ℏ 2 ( n ^ ⋅ J ) 2 . {\displaystyle U[R(\theta ,{\hat {\mathbf {n} }})]=1\!\!1-{\frac {i\sin \theta }{\hbar }}{\hat {\mathbf {n} }}\cdot \mathbf {J} -{\frac {1-\cos \theta }{\hbar ^{2}}}({\hat {\mathbf {n} }}\cdot \mathbf {J} )^{2}.} An operator Ω ^ {\displaystyle {\widehat {\Omega }}} is invariant under a unitary transformation U if Ω ^ = U † Ω ^ U ; {\displaystyle {\widehat {\Omega }}={U}^{\dagger }{\widehat {\Omega }}U;} in this case for the rotation U ^ ( R ) {\displaystyle {\widehat {U}}(R)} , Ω ^ = U ( R ) † Ω ^ U ( R ) = exp ⁡ ( i θ ℏ n ^ ⋅ J ) Ω ^ exp ⁡ ( − i θ ℏ n ^ ⋅ J ) . {\displaystyle {\widehat {\Omega }}={U(R)}^{\dagger }{\widehat {\Omega }}U(R)=\exp \left({\frac {i\theta }{\hbar }}{\hat {\mathbf {n} }}\cdot \mathbf {J} \right){\widehat {\Omega }}\exp \left(-{\frac {i\theta }{\hbar }}{\hat {\mathbf {n} }}\cdot \mathbf {J} \right).} === Angular momentum eigenkets === The orthonormal basis set for total angular momentum is | j , m ⟩ {\displaystyle |j,m\rangle } , where j is the total angular momentum quantum number and m is the magnetic angular momentum quantum number, which takes values −j, −j + 1, ..., j − 1, j. A general state within the j subspace | ψ ⟩ = ∑ m c j m | j , m ⟩ {\displaystyle |\psi \rangle =\sum _{m}c_{jm}|j,m\rangle } rotates to a new state by: | ψ ¯ ⟩ = U ( R ) | ψ ⟩ = ∑ m c j m U ( R ) | j , m ⟩ {\displaystyle |{\bar {\psi }}\rangle =U(R)|\psi \rangle =\sum _{m}c_{jm}U(R)|j,m\rangle } Using the completeness condition: I = ∑ m ′ | j , m ′ ⟩ ⟨ j , m ′ | {\displaystyle I=\sum _{m'}|j,m'\rangle \langle j,m'|} we have | ψ ¯ ⟩ = I U ( R ) | ψ ⟩ = ∑ m m ′ c j m | j , m ′ ⟩ ⟨ j , m ′ | U ( R ) | j , m ⟩ {\displaystyle |{\bar {\psi }}\rangle =IU(R)|\psi \rangle =\sum _{mm'}c_{jm}|j,m'\rangle \langle j,m'|U(R)|j,m\rangle } Introducing the Wigner D matrix elements: D ( R ) m ′ m ( j ) = ⟨ j , m ′ | U ( R ) | j , m ⟩ {\displaystyle {D(R)}_{m'm}^{(j)}=\langle j,m'|U(R)|j,m\rangle } gives the matrix multiplication: | ψ ¯ ⟩ = ∑ m m ′ c j m D m ′ m ( j ) | j , m ′ ⟩ ⇒ | ψ ¯ ⟩ = D ( j ) | ψ ⟩ {\displaystyle |{\bar {\psi }}\rangle =\sum _{mm'}c_{jm}D_{m'm}^{(j)}|j,m'\rangle \quad \Rightarrow \quad |{\bar {\psi }}\rangle =D^{(j)}|\psi \rangle } For one basis ket: | j , m ¯ ⟩ = ∑ m ′ D ( R ) m ′ m ( j ) | j , m ′ ⟩ {\displaystyle |{\overline {j,m}}\rangle =\sum _{m'}{D(R)}_{m'm}^{(j)}|j,m'\rangle } For the case of orbital angular momentum, the eigenstates | ℓ , m ⟩ {\displaystyle |\ell ,m\rangle } of the orbital angular momentum operator L and solutions of Laplace's equation on a 3d sphere are spherical harmonics: Y ℓ m ( θ , ϕ ) = ⟨ θ , ϕ | ℓ , m ⟩ = ( 2 ℓ + 1 ) 4 π ( ℓ − m ) ! ( ℓ + m ) ! P ℓ m ( cos ⁡ θ ) e i m ϕ {\displaystyle Y_{\ell }^{m}(\theta ,\phi )=\langle \theta ,\phi |\ell ,m\rangle ={\sqrt {{(2\ell +1) \over 4\pi }{(\ell -m)! \over (\ell +m)!}}}\,P_{\ell }^{m}(\cos {\theta })\,e^{im\phi }} where Pℓm is an associated Legendre polynomial, ℓ is the orbital angular momentum quantum number, and m is the orbital magnetic quantum number which takes the values −ℓ, −ℓ + 1, ... ℓ − 1, ℓ The formalism of spherical harmonics have wide applications in applied mathematics, and are closely related to the formalism of spherical tensors, as shown below. Spherical harmonics are functions of the polar and azimuthal angles, ϕ and θ respectively, which can be conveniently collected into a unit vector n(θ, ϕ) pointing in the direction of those angles, in the Cartesian basis it is: n ^ ( θ , ϕ ) = cos ⁡ ϕ sin ⁡ θ e x + s

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  Firefox Lockwise

  Firefox Lockwise (formerly Lockbox) is a deprecated password manager for the Firefox web browser, as well as the mobile operating systems iOS and Android. On desktop, Lockwise was simply part of Firefox, whereas on iOS and Android it was available as a standalone app. If Firefox Sync was activated (with a Firefox account), then Lockwise synced passwords between Firefox installations across devices. It also featured a built-in random password generator. The application and branding have since been "phased out." == History == Developed by Mozilla, it was originally named Firefox Lockbox in 2018. It was renamed "Lockwise" in May 2019. It was introduced for iOS on 10 July 2018 as part of the Test Pilot program. On 26 March 2019, it was released for Android. On desktop, Lockwise started out as a browser addon. Alphas were released between March and August 2019. Since Firefox version 70, Lockwise has been integrated into the browser (accessible at about:logins), having replaced a basic password manager presented in a popup window. Mozilla ended support for Firefox Lockwise on December 13, 2021. As of January 2026, Lockwise is still fully functional on Android to this day.

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  Hit-testing

  In computer graphics programming, hit-testing (hit detection, picking, or pick correlation) is the process of determining whether a user-controlled cursor (such as a mouse cursor or touch-point on a touch-screen interface) intersects a given graphical object (such as a shape, line, or curve) drawn on the screen. Hit-testing may be performed on the movement or activation of a mouse or other pointing device. Hit-testing is used by GUI environments to respond to user actions, such as selecting a menu item or a target in a game based on its visual location. In web programming languages such as HTML, SVG, and CSS, this is associated with the concept of pointer-events (e.g. user-initiated cursor movement or object selection). Collision detection is a related concept for detecting intersections of two or more different graphical objects, rather than intersection of a cursor with one or more graphical objects. == Algorithm == There are many different algorithms that may be used to perform hit-testing, with different performance or accuracy outcomes. One common hit-test algorithm for axis aligned bounding boxes. A key idea is that the box being tested must be either entirely above, entirely below, entirely to the right or left of the current box. If this is not possible, they are colliding. Example logic is presented in the pseudo-code below: In Python:

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  Tensor operator

  In pure and applied mathematics, quantum mechanics and computer graphics, a tensor operator generalizes the notion of operators which are scalars and vectors. A special class of these are spherical tensor operators which apply the notion of the spherical basis and spherical harmonics. The spherical basis closely relates to the description of angular momentum in quantum mechanics and spherical harmonic functions. The coordinate-free generalization of a tensor operator is known as a representation operator. == The general notion of scalar, vector, and tensor operators == In quantum mechanics, physical observables that are scalars, vectors, and tensors, must be represented by scalar, vector, and tensor operators, respectively. Whether something is a scalar, vector, or tensor depends on how it is viewed by two observers whose coordinate frames are related to each other by a rotation. Alternatively, one may ask how, for a single observer, a physical quantity transforms if the state of the system is rotated. Consider, for example, a system consisting of a molecule of mass M {\displaystyle M} , traveling with a definite center of mass momentum, p z ^ {\displaystyle p{\mathbf {\hat {z}} }} , in the z {\displaystyle z} direction. If we rotate the system by 90 ∘ {\displaystyle 90^{\circ }} about the y {\displaystyle y} axis, the momentum will change to p x ^ {\displaystyle p{\mathbf {\hat {x}} }} , which is in the x {\displaystyle x} direction. The center-of-mass kinetic energy of the molecule will, however, be unchanged at p 2 / 2 M {\displaystyle p^{2}/2M} . The kinetic energy is a scalar and the momentum is a vector, and these two quantities must be represented by a scalar and a vector operator, respectively. By the latter in particular, we mean an operator whose expected values in the initial and the rotated states are p z ^ {\displaystyle p{\mathbf {\hat {z}} }} and p x ^ {\displaystyle p{\mathbf {\hat {x}} }} . The kinetic energy on the other hand must be represented by a scalar operator, whose expected value must be the same in the initial and the rotated states. In the same way, tensor quantities must be represented by tensor operators. An example of a tensor quantity (of rank two) is the electrical quadrupole moment of the above molecule. Likewise, the octupole and hexadecapole moments would be tensors of rank three and four, respectively. Other examples of scalar operators are the total energy operator (more commonly called the Hamiltonian), the potential energy, and the dipole-dipole interaction energy of two atoms. Examples of vector operators are the momentum, the position, the orbital angular momentum, L {\displaystyle {\mathbf {L} }} , and the spin angular momentum, S {\displaystyle {\mathbf {S} }} . (Fine print: Angular momentum is a vector as far as rotations are concerned, but unlike position or momentum it does not change sign under space inversion, and when one wishes to provide this information, it is said to be a pseudovector.) Scalar, vector and tensor operators can also be formed by products of operators. For example, the scalar product L ⋅ S {\displaystyle {\mathbf {L} }\cdot {\mathbf {S} }} of the two vector operators, L {\displaystyle {\mathbf {L} }} and S {\displaystyle {\mathbf {S} }} , is a scalar operator, which figures prominently in discussions of the spin–orbit interaction. Similarly, the quadrupole moment tensor of our example molecule has the nine components Q i j = ∑ α q α ( 3 r α , i r α , j − r α 2 δ i j ) . {\displaystyle Q_{ij}=\sum _{\alpha }q_{\alpha }\left(3r_{\alpha ,i}r_{\alpha ,j}-r_{\alpha }^{2}\delta _{ij}\right).} Here, the indices i {\displaystyle i} and j {\displaystyle j} can independently take on the values 1, 2, and 3 (or x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyle z} ) corresponding to the three Cartesian axes, the index α {\displaystyle \alpha } runs over all particles (electrons and nuclei) in the molecule, q α {\displaystyle q_{\alpha }} is the charge on particle α {\displaystyle \alpha } , and r α , i {\displaystyle r_{\alpha ,i}} is the i {\displaystyle i} -th component of the position of this particle. Each term in the sum is a tensor operator. In particular, the nine products r α , i r α , j {\displaystyle r_{\alpha ,i}r_{\alpha ,j}} together form a second rank tensor, formed by taking the outer product of the vector operator r α {\displaystyle {\mathbf {r} }_{\alpha }} with itself. == Rotations of quantum states == === Quantum rotation operator === The rotation operator about the unit vector n (defining the axis of rotation) through angle θ is U [ R ( θ , n ^ ) ] = exp ⁡ ( − i θ ℏ n ^ ⋅ J ) {\displaystyle U[R(\theta ,{\hat {\mathbf {n} }})]=\exp \left(-{\frac {i\theta }{\hbar }}{\hat {\mathbf {n} }}\cdot \mathbf {J} \right)} where J = (Jx, Jy, Jz) are the rotation generators (also the angular momentum matrices): J x = ℏ 2 ( 0 1 0 1 0 1 0 1 0 ) J y = ℏ 2 ( 0 i 0 − i 0 i 0 − i 0 ) J z = ℏ ( − 1 0 0 0 0 0 0 0 1 ) {\displaystyle J_{x}={\frac {\hbar }{\sqrt {2}}}{\begin{pmatrix}0&1&0\\1&0&1\\0&1&0\end{pmatrix}}\,\quad J_{y}={\frac {\hbar }{\sqrt {2}}}{\begin{pmatrix}0&i&0\\-i&0&i\\0&-i&0\end{pmatrix}}\,\quad J_{z}=\hbar {\begin{pmatrix}-1&0&0\\0&0&0\\0&0&1\end{pmatrix}}} and let R ^ = R ^ ( θ , n ^ ) {\displaystyle {\widehat {R}}={\widehat {R}}(\theta ,{\hat {\mathbf {n} }})} be a rotation matrix. According to the Rodrigues' rotation formula, the rotation operator then amounts to U [ R ( θ , n ^ ) ] = 1 1 − i sin ⁡ θ ℏ n ^ ⋅ J − 1 − cos ⁡ θ ℏ 2 ( n ^ ⋅ J ) 2 . {\displaystyle U[R(\theta ,{\hat {\mathbf {n} }})]=1\!\!1-{\frac {i\sin \theta }{\hbar }}{\hat {\mathbf {n} }}\cdot \mathbf {J} -{\frac {1-\cos \theta }{\hbar ^{2}}}({\hat {\mathbf {n} }}\cdot \mathbf {J} )^{2}.} An operator Ω ^ {\displaystyle {\widehat {\Omega }}} is invariant under a unitary transformation U if Ω ^ = U † Ω ^ U ; {\displaystyle {\widehat {\Omega }}={U}^{\dagger }{\widehat {\Omega }}U;} in this case for the rotation U ^ ( R ) {\displaystyle {\widehat {U}}(R)} , Ω ^ = U ( R ) † Ω ^ U ( R ) = exp ⁡ ( i θ ℏ n ^ ⋅ J ) Ω ^ exp ⁡ ( − i θ ℏ n ^ ⋅ J ) . {\displaystyle {\widehat {\Omega }}={U(R)}^{\dagger }{\widehat {\Omega }}U(R)=\exp \left({\frac {i\theta }{\hbar }}{\hat {\mathbf {n} }}\cdot \mathbf {J} \right){\widehat {\Omega }}\exp \left(-{\frac {i\theta }{\hbar }}{\hat {\mathbf {n} }}\cdot \mathbf {J} \right).} === Angular momentum eigenkets === The orthonormal basis set for total angular momentum is | j , m ⟩ {\displaystyle |j,m\rangle } , where j is the total angular momentum quantum number and m is the magnetic angular momentum quantum number, which takes values −j, −j + 1, ..., j − 1, j. A general state within the j subspace | ψ ⟩ = ∑ m c j m | j , m ⟩ {\displaystyle |\psi \rangle =\sum _{m}c_{jm}|j,m\rangle } rotates to a new state by: | ψ ¯ ⟩ = U ( R ) | ψ ⟩ = ∑ m c j m U ( R ) | j , m ⟩ {\displaystyle |{\bar {\psi }}\rangle =U(R)|\psi \rangle =\sum _{m}c_{jm}U(R)|j,m\rangle } Using the completeness condition: I = ∑ m ′ | j , m ′ ⟩ ⟨ j , m ′ | {\displaystyle I=\sum _{m'}|j,m'\rangle \langle j,m'|} we have | ψ ¯ ⟩ = I U ( R ) | ψ ⟩ = ∑ m m ′ c j m | j , m ′ ⟩ ⟨ j , m ′ | U ( R ) | j , m ⟩ {\displaystyle |{\bar {\psi }}\rangle =IU(R)|\psi \rangle =\sum _{mm'}c_{jm}|j,m'\rangle \langle j,m'|U(R)|j,m\rangle } Introducing the Wigner D matrix elements: D ( R ) m ′ m ( j ) = ⟨ j , m ′ | U ( R ) | j , m ⟩ {\displaystyle {D(R)}_{m'm}^{(j)}=\langle j,m'|U(R)|j,m\rangle } gives the matrix multiplication: | ψ ¯ ⟩ = ∑ m m ′ c j m D m ′ m ( j ) | j , m ′ ⟩ ⇒ | ψ ¯ ⟩ = D ( j ) | ψ ⟩ {\displaystyle |{\bar {\psi }}\rangle =\sum _{mm'}c_{jm}D_{m'm}^{(j)}|j,m'\rangle \quad \Rightarrow \quad |{\bar {\psi }}\rangle =D^{(j)}|\psi \rangle } For one basis ket: | j , m ¯ ⟩ = ∑ m ′ D ( R ) m ′ m ( j ) | j , m ′ ⟩ {\displaystyle |{\overline {j,m}}\rangle =\sum _{m'}{D(R)}_{m'm}^{(j)}|j,m'\rangle } For the case of orbital angular momentum, the eigenstates | ℓ , m ⟩ {\displaystyle |\ell ,m\rangle } of the orbital angular momentum operator L and solutions of Laplace's equation on a 3d sphere are spherical harmonics: Y ℓ m ( θ , ϕ ) = ⟨ θ , ϕ | ℓ , m ⟩ = ( 2 ℓ + 1 ) 4 π ( ℓ − m ) ! ( ℓ + m ) ! P ℓ m ( cos ⁡ θ ) e i m ϕ {\displaystyle Y_{\ell }^{m}(\theta ,\phi )=\langle \theta ,\phi |\ell ,m\rangle ={\sqrt {{(2\ell +1) \over 4\pi }{(\ell -m)! \over (\ell +m)!}}}\,P_{\ell }^{m}(\cos {\theta })\,e^{im\phi }} where Pℓm is an associated Legendre polynomial, ℓ is the orbital angular momentum quantum number, and m is the orbital magnetic quantum number which takes the values −ℓ, −ℓ + 1, ... ℓ − 1, ℓ The formalism of spherical harmonics have wide applications in applied mathematics, and are closely related to the formalism of spherical tensors, as shown below. Spherical harmonics are functions of the polar and azimuthal angles, ϕ and θ respectively, which can be conveniently collected into a unit vector n(θ, ϕ) pointing in the direction of those angles, in the Cartesian basis it is: n ^ ( θ , ϕ ) = cos ⁡ ϕ sin ⁡ θ e x + s

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