The Data-based Online Nonlinear Extremumseeker (DONE) algorithm is a black-box optimization algorithm. DONE models the unknown cost function and attempts to find an optimum of the underlying function. The DONE algorithm is suitable for optimizing costly and noisy functions and does not require derivatives. An advantage of DONE over similar algorithms, such as Bayesian optimization, is that the computational cost per iteration is independent of the number of function evaluations. == Methods == The DONE algorithm was first proposed by Hans Verstraete and Sander Wahls in 2015. The algorithm fits a surrogate model based on random Fourier features and then uses a well-known L-BFGS algorithm to find an optimum of the surrogate model. == Applications == DONE was first demonstrated for maximizing the signal in optical coherence tomography measurements, but has since then been applied to various other applications. For example, it was used to help extending the field of view in light sheet fluorescence microscopy.
Logistics automation
Logistics automation is the application of computer software or automated machinery to logistics operations in order to improve its efficiency. Typically this refers to operations within a warehouse or distribution center, with broader tasks undertaken by supply chain engineering systems and enterprise resource planning systems. Logistics automation systems can powerfully complement the facilities provided by these higher level computer systems. The focus on an individual node within a wider logistics network allows systems to be highly tailored to the requirements of that node. == Components == Logistics automation systems comprise a variety of hardware and software components: Fixed machinery Automated storage and retrieval systems, including: Cranes serve a rack of locations, allowing many levels of stock to be stacked vertically, and allowing for higher storage densities and better space utilization than alternatives. In systems produced by Amazon Robotics, automated guided vehicles move items to a human picker. Conveyors: Containers can enter automated conveyors in one area of the warehouse and, either through hard-coded rules or data input, be moved to a selected destination. Vertical carousels based on the paternoster lift system or using space optimization, similar to vending machines, but on a larger scale. Sortation systems: similar to conveyors but typically with higher capacity and able to divert containers more quickly. Typically used to distribute high volumes of small cartons to a large set of locations. Industrial robots: four- to six-axis industrial robots, e.g. palletizing robots, are used for palletizing, depalletizing, packaging, commissioning and order picking. Typically all of these will automatically identify and track containers using barcodes or, increasingly, RFID tags. Motion check weighers may be used to reject cases or individual products that are under or over their specified weight. They are often used in kitting conveyor lines to ensure all pieces belonging in the kit are present. Mobile technology Radio data terminals: these are handheld or truck-mounted terminals which connect by radio to logistics automation software and provide instructions to operators moving throughout the warehouse. Many also have barcode scanners to allow identification of containers more quickly and accurately than manual keyboard entry. Software Integration software: this provides overall control of the automation machinery and allows cranes to be connected to conveyors for seamless stock movements. Operational control software: provides low-level decision-making, such as where to store incoming containers, and where to retrieve them when requested. Business control software: provides higher-level functionality, such as identification of incoming deliveries/stock, scheduling order fulfillment, and assignment of stock to outgoing trailers. == Benefits == A typical warehouse or distribution center will receive stock of a variety of products from suppliers and store these until the receipt of orders from customers, whether individual buyers (e.g. mail order), retail branches (e.g. chain stores), or other companies (e.g. wholesalers). A logistics automation system may provide the following: Automated goods in processes: Incoming goods can be marked with barcodes and the automation system notified of the expected stock. On arrival, the goods can be scanned and thereby identified, and taken via conveyors, sortation systems, and automated cranes into an automatically assigned storage location. Automated goods retrieval for orders: On receipt of orders, the automation system is able to immediately locate goods and retrieve them to a pick-face location. Automated dispatch processing: Combining knowledge of all orders placed at the warehouse the automation system can assign picked goods into dispatch units and then into outbound loads. Sortation systems and conveyors can then move these onto the outgoing trailers. If needed, repackaging to ensure proper protection for further distribution or to change the package format for specific retailers/customers. A complete warehouse automation system can drastically reduce the workforce required to run a facility, with human input required only for a few tasks, such as picking units of product from a bulk packed case. Even here, assistance can be provided with equipment such as pick-to-light units. Smaller systems may only be required to handle part of the process. Examples include automated storage and retrieval systems, which simply use cranes to store and retrieve identified cases or pallets, typically into a high-bay storage system which would be unfeasible to access using fork-lift trucks or any other means. The use of Automatic Guided Vehicles maximizes the output compared to humans since they can do repetitive tasks for long hours and with least to no supervision. An AGV is built and programmed for precision and accuracy thereby reducing the chances of errors in a warehouse, especially when dealing with fragile goods. == Automation software == Software or cloud-based SaaS solutions are used for logistics automation which helps the supply chain industry in automating the workflow as well as management of the system. Knowledge @ Wharton staff writers noted in 2011 that some manufacturers and retailers were weathering the Great Recession "by signing up for pay-as-you-go logistics services available through the Internet 'cloud'". They identified the benefits and reduced costs which came from sharing information about shipments with suppliers, hauliers and end users. There is little generalized software available in this market. This is because there is no rule to generalize the system as well as work flow even though the practice is more or less the same. Most of the commercial companies do use one or the other of the custom solutions. But there are various software solutions that are being used within the departments of logistics. There are a few departments in Logistics, namely: Conventional Department, Container Department, Warehouse, Marine Engineering, Heavy Haulage, etc. Software used in these departments Conventional department : CVT software / CTMS software. Container Trucking: CTMS software Warehouse : WMS/WCS Improving Effectiveness of Logistics Management Logistical Network Information Transportation Sound Inventory Management Warehousing, Materials Handling & Packaging
CineAsset
CineAsset was a complete mastering software suite by Doremi Labs that could create and playback encrypted (Pro version) and unencrypted DCI compliant packages from virtually any source. CineAsset included a separate "Editor" application for generating Digital Cinema Packages (DCPs). CineAsset Pro added the ability to generate encrypted DCPs and Key Delivery Messages (KDMs) for any encrypted content in the database. It has since been discontinued, along with CineAsset Player. == Features == == Supported formats == === Input === Source: ==== 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 ==== Image Sequences ==== BMP TIFF TGA DPX JPG J2C ==== Audio Files ==== WAV MP3 WMA MP2 === Output === Source: ==== JPEG2000 ==== 2D and 3D at up to 4K resolution Bit Rate: 50–250 Mbit/s (500 Mbit/s for frame rates above 30 fps) Speed: Faster than real-time processing when using optional render nodes ==== MPEG2 ==== I-Only or Long GOP 1080p up to 80 Mbit/s ==== H264 ==== 1080p up to 50 Mbit/s ==== VC1 ==== DCP wrapping only (no transcode)
Level set (data structures)
In computer science, a level set is a data structure designed to represent discretely sampled dynamic level sets of functions. A common use of this form of data structure is in efficient image rendering. The underlying method constructs a signed distance field that extends from the boundary, and can be used to solve the motion of the boundary in this field. == Chronological developments == The powerful level-set method is due to Osher and Sethian 1988. However, the straightforward implementation via a dense d-dimensional array of values, results in both time and storage complexity of O ( n d ) {\displaystyle O(n^{d})} , where n {\displaystyle n} is the cross sectional resolution of the spatial extents of the domain and d {\displaystyle d} is the number of spatial dimensions of the domain. === Narrow band === The narrow band level set method, introduced in 1995 by Adalsteinsson and Sethian, restricted most computations to a thin band of active voxels immediately surrounding the interface, thus reducing the time complexity in three dimensions to O ( n 2 ) {\displaystyle O(n^{2})} for most operations. Periodic updates of the narrowband structure, to rebuild the list of active voxels, were required which entailed an O ( n 3 ) {\displaystyle O(n^{3})} operation in which voxels over the entire volume were accessed. The storage complexity for this narrowband scheme was still O ( n 3 ) . {\displaystyle O(n^{3}).} Differential constructions over the narrow band domain edge require careful interpolation and domain alteration schemes to stabilise the solution. === Sparse field === This O ( n 3 ) {\displaystyle O(n^{3})} time complexity was eliminated in the approximate "sparse field" level set method introduced by Whitaker in 1998. The sparse field level set method employs a set of linked lists to track the active voxels around the interface. This allows incremental extension of the active region as needed without incurring any significant overhead. While consistently O ( n 2 ) {\displaystyle O(n^{2})} efficient in time, O ( n 3 ) {\displaystyle O(n^{3})} storage space is still required by the sparse field level set method. See for implementation details. === Sparse block grid === The sparse block grid method, introduced by Bridson in 2003, divides the entire bounding volume of size n 3 {\displaystyle n^{3}} into small cubic blocks of m 3 {\displaystyle m^{3}} voxels each. A coarse grid of size ( n / m ) 3 {\displaystyle (n/m)^{3}} then stores pointers only to those blocks that intersect the narrow band of the level set. Block allocation and deallocation occur as the surface propagates to accommodate to the deformations. This method has a suboptimal storage complexity of O ( ( n m ) 3 + m 3 n 2 ) {\displaystyle O\left((nm)3+m^{3}n^{2}\right)} , but retains the constant time access inherent to dense grids. === Octree === The octree level set method, introduced by Strain in 1999 and refined by Losasso, Gibou and Fedkiw, and more recently by Min and Gibou uses a tree of nested cubes of which the leaf nodes contain signed distance values. Octree level sets currently require uniform refinement along the interface (i.e. the narrow band) in order to obtain sufficient precision. This representation is efficient in terms of storage, O ( n 2 ) , {\displaystyle O(n^{2}),} and relatively efficient in terms of access queries, O ( log n ) . {\displaystyle O(\log \,n).} An advantage of the level method on octree data structures is that one can solve the partial differential equations associated with typical free boundary problems that use the level set method. The CASL research group has developed this line of work in computational materials, computational fluid dynamics, electrokinetics, image-guided surgery and controls. === Run-length encoded === The run-length encoding (RLE) level set method, introduced in 2004, applies the RLE scheme to compress regions away from the narrow band to just their sign representation while storing with full precision the narrow band. The sequential traversal of the narrow band is optimal and storage efficiency is further improved over the octree level set. The addition of an acceleration lookup table allows for fast O ( log r ) {\displaystyle O(\log r)} random access, where r is the number of runs per cross section. Additional efficiency is gained by applying the RLE scheme in a dimensional recursive fashion, a technique introduced by Nielsen & Museth's similar DT-Grid. === Hash Table Local Level Set === The Hash Table Local Level Set method was introduced in 2011 by Eyiyurekli and Breen and extended in 2012 by Brun, Guittet, and Gibou, only computes the level set data in a band around the interface, as in the Narrow Band Level-Set Method, but also only stores the data in that same band. A hash table data structure is used, which provides an O ( 1 ) {\displaystyle O(1)} access to the data. However, Brun et al. conclude that their method, while being easier to implement, performs worse than a quadtree implementation. They find that as it is, [...] a quadtree data structure seems more adapted than the hash table data structure for level-set algorithms. Three main reasons for worse efficiency are listed: to obtain accurate results, a rather large band is required close to the interface, which counterbalances the absence of grid nodes far from the interface; the performances are deteriorated by extrapolation procedures on the outer edges of the local grid and the width of the band restricts the time step and slows down the method. === Point-based === Corbett in 2005 introduced the point-based level set method. Instead of using a uniform sampling of the level set, the continuous level set function is reconstructed from a set of unorganized point samples via moving least squares.
Acoustic model
An acoustic model is used in automatic speech recognition to represent the relationship between an audio signal and the phonemes or other linguistic units that make up speech. The model is learned from a set of audio recordings and their corresponding transcripts. It is created by taking audio recordings of speech, and their text transcriptions, and using software to create statistical representations of the sounds that make up each word. == Background == Modern speech recognition systems use both an acoustic model and a language model to represent the statistical properties of speech. The acoustic model models the relationship between the audio signal and the phonetic units in the language. The language model is responsible for modeling the word sequences in the language. These two models are combined to get the top-ranked word sequences corresponding to a given audio segment. Most modern speech recognition systems operate on the audio in small chunks known as frames with an approximate duration of 10ms per frame. The raw audio signal from each frame can be transformed by applying the mel-frequency cepstrum. The coefficients from this transformation are commonly known as mel-frequency cepstral coefficients (MFCCs) and are used as an input to the acoustic model along with other features. Recently, the use of convolutional neural networks has led to major improvements in acoustic modeling. == Speech audio characteristics == Audio can be encoded at different sampling rates (i.e. samples per second – the most common being: 8, 16, 32, 44.1, 48, and 96 kHz), and different bits per sample (the most common being: 8-bits, 16-bits, 24-bits or 32-bits). Speech recognition engines work best if the acoustic model they use was trained with speech audio which was recorded at the same sampling rate/bits per sample as the speech being recognized. == Telephony-based speech recognition == The limiting factor for telephony based speech recognition is the bandwidth at which speech can be transmitted. For example, a standard land-line telephone only has a bandwidth of 64 kbit/s at a sampling rate of 8 kHz and 8-bits per sample (8000 samples per second 8-bits per sample = 64000 bit/s). Therefore, for telephony based speech recognition, acoustic models should be trained with 8 kHz/8-bit speech audio files. In the case of voice over IP, the codec determines the sampling rate/bits per sample of speech transmission. Codecs with a higher sampling rate/bits per sample for speech transmission (which improve the sound quality) necessitate acoustic models trained with audio data that matches that sampling rate/bits per sample. == Desktop-based speech recognition == For speech recognition on a standard desktop PC, the limiting factor is the sound card. Most sound cards today can record at sampling rates of between 16–48 kHz of audio, with bit rates of 8- to 16-bits per sample, and playback at up to 96 kHz. As a general rule, a speech recognition engine works better with acoustic models trained with speech audio data recorded at higher sampling rates/bits per sample. But using audio with too high a sampling rate/bits per sample can slow the recognition engine down. A compromise is needed. Thus for desktop speech recognition, the current standard is acoustic models trained with speech audio data recorded at sampling rates of 16 kHz/16 bits per sample.
Multimodal representation learning
Multimodal representation learning is a subfield of representation learning focused on integrating and interpreting information from different modalities, such as text, images, audio, or video, by projecting them into a shared latent space. This allows for semantically similar content across modalities to be mapped to nearby points within that space, facilitating a unified understanding of diverse data types. By automatically learning meaningful features from each modality and capturing their inter-modal relationships, multimodal representation learning enables a unified representation that enhances performance in cross-media analysis tasks such as video classification, event detection, and sentiment analysis. It also supports cross-modal retrieval and translation, including image captioning, video description, and text-to-image synthesis. == Motivation == The primary motivations for multimodal representation learning arise from the inherent nature of real-world data and the limitations of unimodal approaches. Since multimodal data offers complementary and supplementary information about an object or event from different perspectives, it is more informative than relying on a single modality. A key motivation is to narrow the heterogeneity gap that exists between different modalities by projecting their features into a shared semantic subspace. This allows semantically similar content across modalities to be represented by similar vectors, facilitating the understanding of relationships and correlations between them. Multimodal representation learning aims to leverage the unique information provided by each modality to achieve a more comprehensive and accurate understanding of concepts. These unified representations are crucial for improving performance in various cross-media analysis tasks such as video classification, event detection, and sentiment analysis. They also enable cross-modal retrieval, allowing users to search and retrieve content across different modalities. Additionally, it facilitates cross-modal translation, where information can be converted from one modality to another, as seen in applications like image captioning and text-to-image synthesis. The abundance of ubiquitous multimodal data in real-world applications, including understudied areas like healthcare, finance, and human-computer interaction (HCI), further motivates the development of effective multimodal representation learning techniques. == Approaches and methods == === Canonical-correlation analysis based methods === Canonical-correlation analysis (CCA) was first introduced in 1936 by Harold Hotelling and is a fundamental approach for multimodal learning. CCA aims to find linear relationships between two sets of variables. Given two data matrices X ∈ R n × p {\displaystyle X\in \mathbb {R} ^{n\times p}} and Y ∈ R n × q {\displaystyle Y\in \mathbb {R} ^{n\times q}} representing different modalities, CCA finds projection vectors w x ∈ R p {\displaystyle w_{x}\in \mathbb {R} ^{p}} and w y ∈ R q {\displaystyle w_{y}\in \mathbb {R} ^{q}} that maximizes the correlation between the projected variables: ρ = max w x , w y w x ⊤ Σ x y w y w x ⊤ Σ x x w x w y ⊤ Σ y y w y {\displaystyle \rho =\max _{w_{x},w_{y}}{\frac {w_{x}^{\top }\Sigma _{xy}w_{y}}{{\sqrt {w_{x}^{\top }\Sigma _{xx}w_{x}}}{\sqrt {w_{y}^{\top }\Sigma _{yy}w_{y}}}}}} such that Σ x x {\displaystyle \Sigma _{xx}} and Σ y y {\displaystyle \Sigma _{yy}} are the within-modality covariance matrices, and Σ x y {\displaystyle \Sigma _{xy}} is the between-modality covariance matrix. However, standard CCA is limited by its linearity, which led to the development of nonlinear extensions, such as kernel CCA and deep CCA. ==== Kernel CCA ==== Kernel canonical correlation analysis (KCCA) extends traditional CCA to capture nonlinear relationships between modalities by implicitly mapping the data into high dimensional feature spaces using kernel functions. Given kernel functions K x {\displaystyle K_{x}} and K y {\displaystyle K_{y}} with corresponding Gram matrices K x ∈ R n × n {\displaystyle K_{x}\in \mathbb {R} ^{n\times n}} and K y ∈ R n × n {\displaystyle K_{y}\in \mathbb {R} ^{n\times n}} , KCCA seeks coefficients α {\displaystyle \alpha } and β {\displaystyle \beta } that maximize: ρ = max α , β α ⊤ K x K y β α ⊤ K x 2 α β ⊤ K y 2 β {\displaystyle \rho =\max _{\alpha ,\beta }{\frac {\alpha ^{\top }K_{x}Ky\beta }{{\sqrt {\alpha ^{\top }K_{x}^{2}\alpha }}{\sqrt {\beta ^{\top }K_{y}^{2}\beta }}}}} To prevent overfitting, regularization terms are typically added, resulting in: ρ = max α , β α T K x K y β α T ( K x 2 + λ x K x ) α β T ( K y 2 + λ y K y ) β {\displaystyle \rho =\max _{\alpha ,\beta }{\frac {\alpha ^{T}K_{x}K_{y}\beta }{{\sqrt {\alpha ^{T}\left(K_{x}^{2}+\lambda _{x}K_{x}\right)\alpha }}{\sqrt {\;\beta ^{T}\left(K_{y}^{2}+\lambda _{y}K_{y}\right)\beta }}}}} where λ x {\displaystyle \lambda _{x}} and λ y {\displaystyle \lambda _{y}} are regularization parameters. KCCA has proven effective for tasks such as cross-modal retrieval and semantic analysis, though it faces computational challenges with large datasets due to its O ( n 2 ) {\displaystyle O(n^{2})} memory requirement for sorting kernel matrices. KCCA was proposed independently by several researchers. ==== Deep CCA ==== Deep canonical correlation analysis (DCCA), introduced in 2013, employs neural networks to learn nonlinear transformations for maximizing the correlation between modalities. DCCA uses separate neural networks f x {\displaystyle f_{x}} and f y {\displaystyle f_{y}} for each modality to transform the original data before applying CCA: max W x , W y , θ x , θ y corr ( f x ( X ; θ x ) , f y ( Y ; θ y ) ) {\displaystyle \max _{W_{x},W_{y},\theta _{x},\theta _{y}}\operatorname {corr} \left(f_{x}(X;\theta _{x}),f_{y}(Y;\theta _{y})\right)} where θ x {\displaystyle \theta _{x}} and θ y {\displaystyle \theta _{y}} represent the parameters of the neural networks, and W x {\displaystyle W_{x}} and W y {\displaystyle W_{y}} are the CCA projection matrices. The correlation objective is computed as: corr ( H x , H y ) = tr ( T − 1 / 2 H x T H y S − 1 / 2 ) {\displaystyle \operatorname {corr} (H_{x},H_{y})=\operatorname {tr} \left(T^{-1/2}H_{x}^{T}H_{y}S^{-1/2}\right)} where H x = f x ( X ) {\displaystyle H_{x}=f_{x}(X)} and H y = f y ( Y ) {\displaystyle H_{y}=f_{y}(Y)} are the network outputs, T = H x T H x + r x I {\displaystyle T=H_{x}^{T}H_{x}+r_{x}I} , S = H y T H y + r y I {\displaystyle S=H_{y}^{T}H_{y}+r_{y}I} and r x , r y {\displaystyle r_{x},r_{y}} are the regularization parameters. DCCA overcomes the limitations of linear CCA and kernel CCA by learning complex nonlinear relationships while maintaining computational efficiency for large datasets through mini-batch optimization. === Graph-based methods === Graph-based approaches for multimodal representation learning leverage graph structure to model relationships between entities across different modalities. These methods typically represent each modality as a graph and then learn embedding that preserve cross-modal similarities, enabling more effective joint representation of heterogeneous data. One such method is cross-modal graph neural networks (CMGNNs) that extend traditional graph neural networks (GNNs) to handle data from multiple modalities by constructing graphs that capture both intra-modal and inter-modal relationships. These networks model interactions across modalities by representing them as nodes and their relationships as edges. Other graph-based methods include Probabilistic Graphical Models (PGMs) such as deep belief networks (DBN) and deep Boltzmann machines (DBM). These models can learn a joint representation across modalities, for instance, a multimodal DBN achieves this by adding a shared restricted Boltzmann Machine (RBM) hidden layer on top of modality-specific DBNs. Additionally, the structure of data in some domains like Human-Computer Interaction (HCI), such as the view hierarchy of app screens, can potentially be modeled using graph-like structures. The field of graph representation learning is also relevant, with ongoing progress in developing evaluation benchmarks. === Diffusion maps === Another set of methods relevant to multimodal representation learning are based on diffusion maps and their extensions to handle multiple modalities. ==== Multi-view diffusion maps ==== Multi-view diffusion maps address the challenge of achieving multi-view dimensionality reduction by effectively utilizing the availability of multiple views to extract a coherent low-dimensional representation of the data. The core idea is to exploit both the intrinsic relations within each view and the mutual relations between the different views, defining a cross-view model where a random walk process implicitly hops between objects in different views. A multi-view kernel matrix is constructed by combining these relations, defining a cross-view diffusion process and associ
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