Spectral shape analysis relies on the spectrum (eigenvalues and/or eigenfunctions) of the Laplace–Beltrami operator to compare and analyze geometric shapes. Since the spectrum of the Laplace–Beltrami operator is invariant under isometries, it is well suited for the analysis or retrieval of non-rigid shapes, i.e. bendable objects such as humans, animals, plants, etc. == Laplace == The Laplace–Beltrami operator is involved in many important differential equations, such as the heat equation and the wave equation. It can be defined on a Riemannian manifold as the divergence of the gradient of a real-valued function f: Δ f := div grad f . {\displaystyle \Delta f:=\operatorname {div} \operatorname {grad} f.} Its spectral components can be computed by solving the Helmholtz equation (or Laplacian eigenvalue problem): Δ φ i + λ i φ i = 0. {\displaystyle \Delta \varphi _{i}+\lambda _{i}\varphi _{i}=0.} The solutions are the eigenfunctions φ i {\displaystyle \varphi _{i}} (modes) and corresponding eigenvalues λ i {\displaystyle \lambda _{i}} , representing a diverging sequence of positive real numbers. The first eigenvalue is zero for closed domains or when using the Neumann boundary condition. For some shapes, the spectrum can be computed analytically (e.g. rectangle, flat torus, cylinder, disk or sphere). For the sphere, for example, the eigenfunctions are the spherical harmonics. The most important properties of the eigenvalues and eigenfunctions are that they are isometry invariants. In other words, if the shape is not stretched (e.g. a sheet of paper bent into the third dimension), the spectral values will not change. Bendable objects, like animals, plants and humans, can move into different body postures with only minimal stretching at the joints. The resulting shapes are called near-isometric and can be compared using spectral shape analysis. == Discretizations == Geometric shapes are often represented as 2D curved surfaces, 2D surface meshes (usually triangle meshes) or 3D solid objects (e.g. using voxels or tetrahedra meshes). The Helmholtz equation can be solved for all these cases. If a boundary exists, e.g. a square, or the volume of any 3D geometric shape, boundary conditions need to be specified. Several discretizations of the Laplace operator exist (see Discrete Laplace operator) for the different types of geometry representations. Many of these operators do not approximate well the underlying continuous operator. == Spectral shape descriptors == === ShapeDNA and its variants === The ShapeDNA is one of the first spectral shape descriptors. It is the normalized beginning sequence of the eigenvalues of the Laplace–Beltrami operator. Its main advantages are the simple representation (a vector of numbers) and comparison, scale invariance, and in spite of its simplicity a very good performance for shape retrieval of non-rigid shapes. Competitors of shapeDNA include singular values of Geodesic Distance Matrix (SD-GDM) and Reduced BiHarmonic Distance Matrix (R-BiHDM). However, the eigenvalues are global descriptors, therefore the shapeDNA and other global spectral descriptors cannot be used for local or partial shape analysis. === Global point signature (GPS) === The global point signature at a point x {\displaystyle x} is a vector of scaled eigenfunctions of the Laplace–Beltrami operator computed at x {\displaystyle x} (i.e. the spectral embedding of the shape). The GPS is a global feature in the sense that it cannot be used for partial shape matching. === Heat kernel signature (HKS) === The heat kernel signature makes use of the eigen-decomposition of the heat kernel: h t ( x , y ) = ∑ i = 0 ∞ exp ( − λ i t ) φ i ( x ) φ i ( y ) . {\displaystyle h_{t}(x,y)=\sum _{i=0}^{\infty }\exp(-\lambda _{i}t)\varphi _{i}(x)\varphi _{i}(y).} For each point on the surface the diagonal of the heat kernel h t ( x , x ) {\displaystyle h_{t}(x,x)} is sampled at specific time values t j {\displaystyle t_{j}} and yields a local signature that can also be used for partial matching or symmetry detection. === Wave kernel signature (WKS) === The WKS follows a similar idea to the HKS, replacing the heat equation with the Schrödinger wave equation. === Improved wave kernel signature (IWKS) === The IWKS improves the WKS for non-rigid shape retrieval by introducing a new scaling function to the eigenvalues and aggregating a new curvature term. === Spectral graph wavelet signature (SGWS) === SGWS is a local descriptor that is not only isometric invariant, but also compact, easy to compute and combines the advantages of both band-pass and low-pass filters. An important facet of SGWS is the ability to combine the advantages of WKS and HKS into a single signature, while allowing a multiresolution representation of shapes. == Spectral Matching == The spectral decomposition of the graph Laplacian associated with complex shapes (see Discrete Laplace operator) provides eigenfunctions (modes) which are invariant to isometries. Each vertex on the shape could be uniquely represented with a combinations of the eigenmodal values at each point, sometimes called spectral coordinates: s ( x ) = ( φ 1 ( x ) , φ 2 ( x ) , … , φ N ( x ) ) for vertex x . {\displaystyle s(x)=(\varphi _{1}(x),\varphi _{2}(x),\ldots ,\varphi _{N}(x)){\text{ for vertex }}x.} Spectral matching consists of establishing the point correspondences by pairing vertices on different shapes that have the most similar spectral coordinates. Early work focused on sparse correspondences for stereoscopy. Computational efficiency now enables dense correspondences on full meshes, for instance between cortical surfaces. Spectral matching could also be used for complex non-rigid image registration, which is notably difficult when images have very large deformations. Such image registration methods based on spectral eigenmodal values indeed capture global shape characteristics, and contrast with conventional non-rigid image registration methods which are often based on local shape characteristics (e.g., image gradients).
Augmented Analytics
Augmented Analytics is an approach of data analytics that employs the use of machine learning and natural language processing to automate analysis processes normally done by a specialist or data scientist. The term was introduced in 2017 by Rita Sallam, Cindi Howson, and Carlie Idoine in a Gartner research paper. Augmented analytics is based on business intelligence and analytics. In the graph extraction step, data from different sources are investigated. == Defining Augmented Analytics == Machine Learning – a systematic computing method that uses algorithms to sift through data to identify relationships, trends, and patterns. It is a process that allows algorithms to dynamically learn from data instead of having a set base of programmed rules. Natural language generation (NLG) – a software capability that takes unstructured data and translates it into plain-English, readable, language. Automating Insights – using machine learning algorithms to automate data analysis processes. Natural Language Query – enabling users to query data using business terms that are either typed onto a search box or spoken. == Data Democratization == Data Democratization is the democratizing data access in order to relieve data congestion and get rid of any sense of data "gatekeepers". This process must be implemented alongside a method for users to make sense of the data. This process is used in hopes of speeding up company decision making and uncovering opportunities hidden in data. There are three aspects to democratising data: Data Parameterisation and Characterisation. Data Decentralisation using an OS of blockchain and DLT technologies, as well as an independently governed secure data exchange to enable trust. Consent Market-driven Data Monetisation. When it comes to connecting assets, there are two features that will accelerate the adoption and usage of data democratisation: decentralized identity management and business data object monetization of data ownership. It enables multiple individuals and organizations to identify, authenticate, and authorize participants and organizations, enabling them to access services, data or systems across multiple networks, organizations, environments, and use cases. It empowers users and enables a personalized, self-service digital onboarding system so that users can self-authenticate without relying on a central administration function to process their information. Simultaneously, decentralized identity management ensures the user is authorized to perform actions subject to the system’s policies based on their attributes (role, department, organization, etc.) and/ or physical location. == Use cases == Agriculture – Farmers collect data on water use, soil temperature, moisture content and crop growth, augmented analytics can be used to make sense of this data and possibly identify insights that the user can then use to make business decisions. Smart Cities – Many cities across the United States, known as Smart Cities collect large amounts of data on a daily basis. Augmented analytics can be used to simplify this data in order to increase effectiveness in city management (transportation, natural disasters, etc.). Analytic Dashboards – Augmented analytics has the ability to take large data sets and create highly interactive and informative analytical dashboards that assist in many organizational decisions. Augmented Data Discovery – Using an augmented analytics process can assist organizations in automatically finding, visualizing and narrating potentially important data correlations and trends. Data Preparation – Augmented analytics platforms have the ability to take large amounts of data and organize and "clean" the data in order for it to be usable for future analyses. Business – Businesses collect large amounts of data, daily. Some examples of types of data collected in business operations include; sales data, consumer behavior data, distribution data. An augmented analytics platform provides access to analysis of this data, which could be used in making business decisions.
Whitehead's algorithm
Whitehead's algorithm is a mathematical algorithm in group theory for solving the automorphic equivalence problem in the finite rank free group Fn. The algorithm is based on a classic 1936 paper of J. H. C. Whitehead. It is still unknown (except for the case n = 2) if Whitehead's algorithm has polynomial time complexity. == Statement of the problem == Let F n = F ( x 1 , … , x n ) {\displaystyle F_{n}=F(x_{1},\dots ,x_{n})} be a free group of rank n ≥ 2 {\displaystyle n\geq 2} with a free basis X = { x 1 , … , x n } {\displaystyle X=\{x_{1},\dots ,x_{n}\}} . The automorphism problem, or the automorphic equivalence problem for F n {\displaystyle F_{n}} asks, given two freely reduced words w , w ′ ∈ F n {\displaystyle w,w'\in F_{n}} whether there exists an automorphism φ ∈ Aut ( F n ) {\displaystyle \varphi \in \operatorname {Aut} (F_{n})} such that φ ( w ) = w ′ {\displaystyle \varphi (w)=w'} . Thus the automorphism problem asks, for w , w ′ ∈ F n {\displaystyle w,w'\in F_{n}} whether Aut ( F n ) w = Aut ( F n ) w ′ {\displaystyle \operatorname {Aut} (F_{n})w=\operatorname {Aut} (F_{n})w'} . For w , w ′ ∈ F n {\displaystyle w,w'\in F_{n}} one has Aut ( F n ) w = Aut ( F n ) w ′ {\displaystyle \operatorname {Aut} (F_{n})w=\operatorname {Aut} (F_{n})w'} if and only if Out ( F n ) [ w ] = Out ( F n ) [ w ′ ] {\displaystyle \operatorname {Out} (F_{n})[w]=\operatorname {Out} (F_{n})[w']} , where [ w ] , [ w ′ ] {\displaystyle [w],[w']} are conjugacy classes in F n {\displaystyle F_{n}} of w , w ′ {\displaystyle w,w'} accordingly. Therefore, the automorphism problem for F n {\displaystyle F_{n}} is often formulated in terms of Out ( F n ) {\displaystyle \operatorname {Out} (F_{n})} -equivalence of conjugacy classes of elements of F n {\displaystyle F_{n}} . For an element w ∈ F n {\displaystyle w\in F_{n}} , | w | X {\displaystyle |w|_{X}} denotes the freely reduced length of w {\displaystyle w} with respect to X {\displaystyle X} , and ‖ w ‖ X {\displaystyle \|w\|_{X}} denotes the cyclically reduced length of w {\displaystyle w} with respect to X {\displaystyle X} . For the automorphism problem, the length of an input w {\displaystyle w} is measured as | w | X {\displaystyle |w|_{X}} or as ‖ w ‖ X {\displaystyle \|w\|_{X}} , depending on whether one views w {\displaystyle w} as an element of F n {\displaystyle F_{n}} or as defining the corresponding conjugacy class [ w ] {\displaystyle [w]} in F n {\displaystyle F_{n}} . == History == The automorphism problem for F n {\displaystyle F_{n}} was algorithmically solved by J. H. C. Whitehead in a classic 1936 paper, and his solution came to be known as Whitehead's algorithm. Whitehead used a topological approach in his paper. Namely, consider the 3-manifold M n = # i = 1 n S 2 × S 1 {\displaystyle M_{n}=\#_{i=1}^{n}\mathbb {S} ^{2}\times \mathbb {S} ^{1}} , the connected sum of n {\displaystyle n} copies of S 2 × S 1 {\displaystyle \mathbb {S} ^{2}\times \mathbb {S} ^{1}} . Then π 1 ( M n ) ≅ F n {\displaystyle \pi _{1}(M_{n})\cong F_{n}} , and, moreover, up to a quotient by a finite normal subgroup isomorphic to Z 2 n {\displaystyle \mathbb {Z} _{2}^{n}} , the mapping class group of M n {\displaystyle M_{n}} is equal to Out ( F n ) {\displaystyle \operatorname {Out} (F_{n})} ; see. Different free bases of F n {\displaystyle F_{n}} can be represented by isotopy classes of "sphere systems" in M n {\displaystyle M_{n}} , and the cyclically reduced form of an element w ∈ F n {\displaystyle w\in F_{n}} , as well as the Whitehead graph of [ w ] {\displaystyle [w]} , can be "read-off" from how a loop in general position representing [ w ] {\displaystyle [w]} intersects the spheres in the system. Whitehead moves can be represented by certain kinds of topological "swapping" moves modifying the sphere system. Subsequently, Rapaport, and later, based on her work, Higgins and Lyndon, gave a purely combinatorial and algebraic re-interpretation of Whitehead's work and of Whitehead's algorithm. The exposition of Whitehead's algorithm in the book of Lyndon and Schupp is based on this combinatorial approach. Culler and Vogtmann, in their 1986 paper that introduced the Outer space, gave a hybrid approach to Whitehead's algorithm, presented in combinatorial terms but closely following Whitehead's original ideas. == Whitehead's algorithm == Our exposition regarding Whitehead's algorithm mostly follows Ch.I.4 in the book of Lyndon and Schupp, as well as. === Overview === The automorphism group Aut ( F n ) {\displaystyle \operatorname {Aut} (F_{n})} has a particularly useful finite generating set W {\displaystyle {\mathcal {W}}} of Whitehead automorphisms or Whitehead moves. Given w , w ′ ∈ F n {\displaystyle w,w'\in F_{n}} the first part of Whitehead's algorithm consists of iteratively applying Whitehead moves to w , w ′ {\displaystyle w,w'} to take each of them to an "automorphically minimal" form, where the cyclically reduced length strictly decreases at each step. Once we find automorphically these minimal forms u , u ′ {\displaystyle u,u'} of w , w ′ {\displaystyle w,w'} , we check if ‖ u ‖ X = ‖ u ′ ‖ X {\displaystyle \|u\|_{X}=\|u'\|_{X}} . If ‖ u ‖ X ≠ ‖ u ′ ‖ X {\displaystyle \|u\|_{X}\neq \|u'\|_{X}} then w , w ′ {\displaystyle w,w'} are not automorphically equivalent in F n {\displaystyle F_{n}} . If ‖ u ‖ X = ‖ u ′ ‖ X {\displaystyle \|u\|_{X}=\|u'\|_{X}} , we check if there exists a finite chain of Whitehead moves taking u {\displaystyle u} to u ′ {\displaystyle u'} so that the cyclically reduced length remains constant throughout this chain. The elements w , w ′ {\displaystyle w,w'} are not automorphically equivalent in F n {\displaystyle F_{n}} if and only if such a chain exists. Whitehead's algorithm also solves the search automorphism problem for F n {\displaystyle F_{n}} . Namely, given w , w ′ ∈ F n {\displaystyle w,w'\in F_{n}} , if Whitehead's algorithm concludes that Aut ( F n ) w = Aut ( F n ) w ′ {\displaystyle \operatorname {Aut} (F_{n})w=\operatorname {Aut} (F_{n})w'} , the algorithm also outputs an automorphism φ ∈ Aut ( F n ) {\displaystyle \varphi \in \operatorname {Aut} (F_{n})} such that φ ( w ) = w ′ {\displaystyle \varphi (w)=w'} . Such an element φ ∈ Aut ( F n ) {\displaystyle \varphi \in \operatorname {Aut} (F_{n})} is produced as the composition of a chain of Whitehead moves arising from the above procedure and taking w {\displaystyle w} to w ′ {\displaystyle w'} . === Whitehead automorphisms === A Whitehead automorphism, or Whitehead move, of F n {\displaystyle F_{n}} is an automorphism τ ∈ Aut ( F n ) {\displaystyle \tau \in \operatorname {Aut} (F_{n})} of F n {\displaystyle F_{n}} of one of the following two types: There is a permutation σ ∈ S n {\displaystyle \sigma \in S_{n}} of { 1 , 2 , … , n } {\displaystyle \{1,2,\dots ,n\}} such that for i = 1 , … , n {\displaystyle i=1,\dots ,n} τ ( x i ) = x σ ( i ) ± 1 {\displaystyle \tau (x_{i})=x_{\sigma (i)}^{\pm 1}} Such τ {\displaystyle \tau } is called a Whitehead automorphism of the first kind. There is an element a ∈ X ± 1 {\displaystyle a\in X^{\pm 1}} , called the multiplier, such that for every x ∈ X ± 1 {\displaystyle x\in X^{\pm 1}} τ ( x ) ∈ { x , x a , a − 1 x , a − 1 x a } . {\displaystyle \tau (x)\in \{x,xa,a^{-1}x,a^{-1}xa\}.} Such τ {\displaystyle \tau } is called a Whitehead automorphism of the second kind. Since τ {\displaystyle \tau } is an automorphism of F n {\displaystyle F_{n}} , it follows that τ ( a ) = a {\displaystyle \tau (a)=a} in this case. Often, for a Whitehead automorphism τ ∈ Aut ( F n ) {\displaystyle \tau \in \operatorname {Aut} (F_{n})} , the corresponding outer automorphism in Out ( F n ) {\displaystyle \operatorname {Out} (F_{n})} is also called a Whitehead automorphism or a Whitehead move. ==== Examples ==== Let F 4 = F ( x 1 , x 2 , x 3 , x 4 ) {\displaystyle F_{4}=F(x_{1},x_{2},x_{3},x_{4})} . Let τ : F 4 → F 4 {\displaystyle \tau :F_{4}\to F_{4}} be a homomorphism such that τ ( x 1 ) = x 2 x 1 , τ ( x 2 ) = x 2 , τ ( x 3 ) = x 2 x 3 x 2 − 1 , τ ( x 4 ) = x 4 {\displaystyle \tau (x_{1})=x_{2}x_{1},\quad \tau (x_{2})=x_{2},\quad \tau (x_{3})=x_{2}x_{3}x_{2}^{-1},\quad \tau (x_{4})=x_{4}} Then τ {\displaystyle \tau } is actually an automorphism of F 4 {\displaystyle F_{4}} , and, moreover, τ {\displaystyle \tau } is a Whitehead automorphism of the second kind, with the multiplier a = x 2 − 1 {\displaystyle a=x_{2}^{-1}} . Let τ ′ : F 4 → F 4 {\displaystyle \tau ':F_{4}\to F_{4}} be a homomorphism such that τ ′ ( x 1 ) = x 1 , τ ′ ( x 2 ) = x 1 − 1 x 2 x 1 , τ ′ ( x 3 ) = x 1 − 1 x 3 x 1 , τ ′ ( x 4 ) = x 1 − 1 x 4 x 1 {\displaystyle \tau '(x_{1})=x_{1},\quad \tau '(x_{2})=x_{1}^{-1}x_{2}x_{1},\quad \tau '(x_{3})=x_{1}^{-1}x_{3}x_{1},\quad \tau '(x_{4})=x_{1}^{-1}x_{4}x_{1}} Then τ ′ {\displaystyle \tau '} is actually an inner automorphism of F 4 {\displaystyle F_{4}} given by conjugation by x 1 {\displaystyle x_{1}} , and, moreover, τ ′ {\displaystyle \
Car–Parrinello molecular dynamics
Car–Parrinello molecular dynamics (CPMD) refers to either a method used in molecular dynamics (also known as the Car–Parrinello method) or the computational chemistry software package used to implement this method. The CPMD method is one of the major methods for calculating ab initio molecular dynamics (ab initio MD or AIMD). Ab initio molecular dynamics (AIMD) is a computational method that uses first principles through quantum mechanics to simulate the motion of atoms in a system. It is a type of molecular dynamics (MD) simulation that does not rely on empirical potentials or force fields to describe the interactions between atoms, but rather calculates these interactions entirely from the electronic structure of the system using quantum mechanics. In an ab initio MD simulation, the total energy of the system is calculated at each time step using density functional theory (DFT), Hartree-Fock (HF), or other electronic structure calculation methods. The forces acting on each atom are then determined from the gradient of the energy with respect to the atomic coordinates, and the equations of motion are solved to predict the trajectory of the atoms. AIMD permits chemical bond breaking and forming events to occur and accounts for electronic polarization effect. Therefore, Ab initio MD simulations can be used to study a wide range of phenomena, including the structural, thermodynamic, and dynamic properties of materials and chemical reactions. They are particularly useful for systems that are not well described by empirical potentials or force fields, such as systems with strong electronic correlation or systems with many degrees of freedom. However, ab initio MD simulations are computationally demanding and require significant computational resources. The CPMD method is related to the more common Born–Oppenheimer molecular dynamics (BOMD) method in that the quantum mechanical effect of the electrons is included in the calculation of energy and forces for the classical motion of the nuclei. CPMD and BOMD are different types of AIMD. However, whereas BOMD treats the electronic structure problem within the time-independent Schrödinger equation, CPMD explicitly includes the electrons as active degrees of freedom, via (fictitious) dynamical variables. The software is a parallelized plane wave / pseudopotential implementation of density functional theory, particularly designed for ab initio molecular dynamics. == Car–Parrinello method == The Car–Parrinello method is a type of molecular dynamics, usually employing periodic boundary conditions, planewave basis sets, and density functional theory, proposed by Roberto Car and Michele Parrinello in 1985 while working at SISSA, who were subsequently awarded the Dirac Medal by ICTP in 2009. In contrast to Born–Oppenheimer molecular dynamics wherein the nuclear (ions) degree of freedom are propagated using ionic forces which are calculated at each iteration by approximately solving the electronic problem with conventional matrix diagonalization methods, the Car–Parrinello method explicitly introduces the electronic degrees of freedom as (fictitious) dynamical variables, writing an extended Lagrangian for the system which leads to a system of coupled equations of motion for both ions and electrons. In this way, an explicit electronic minimization at each time step, as done in Born–Oppenheimer MD, is not needed: after an initial standard electronic minimization, the fictitious dynamics of the electrons keeps them on the electronic ground state corresponding to each new ionic configuration visited along the dynamics, thus yielding accurate ionic forces. In order to maintain this adiabaticity condition, it is necessary that the fictitious mass of the electrons is chosen small enough to avoid a significant energy transfer from the ionic to the electronic degrees of freedom. This small fictitious mass in turn requires that the equations of motion are integrated using a smaller time step than the one (1–10 fs) commonly used in Born–Oppenheimer molecular dynamics. Currently, the CPMD method can be applied to systems that consist of a few tens or hundreds of atoms and access timescales on the order of tens of picoseconds. == General approach == In CPMD the core electrons are usually described by a pseudopotential and the wavefunction of the valence electrons are approximated by a plane wave basis set. The ground state electronic density (for fixed nuclei) is calculated self-consistently, usually using the density functional theory method. Kohn-Sham equations are often used to calculate the electronic structure, where electronic orbitals are expanded in a plane-wave basis set. Then, using that density, forces on the nuclei can be computed, to update the trajectories (using, e.g. the Verlet integration algorithm). In addition, however, the coefficients used to obtain the electronic orbital functions can be treated as a set of extra spatial dimensions, and trajectories for the orbitals can be calculated in this context. == Fictitious dynamics == CPMD is an approximation of the Born–Oppenheimer MD (BOMD) method. In BOMD, the electrons' wave function must be minimized via matrix diagonalization at every step in the trajectory. CPMD uses fictitious dynamics to keep the electrons close to the ground state, preventing the need for a costly self-consistent iterative minimization at each time step. The fictitious dynamics relies on the use of a fictitious electron mass (usually in the range of 400 – 800 a.u.) to ensure that there is very little energy transfer from nuclei to electrons, i.e. to ensure adiabaticity. Any increase in the fictitious electron mass resulting in energy transfer would cause the system to leave the ground-state BOMD surface. === Lagrangian === L = 1 2 ( ∑ I n u c l e i M I R ˙ I 2 + μ ∑ i o r b i t a l s ∫ d r | ψ ˙ i ( r , t ) | 2 ) − E [ { ψ i } , { R I } ] + ∑ i j Λ i j ( ∫ d r ψ i ψ j − δ i j ) , {\displaystyle {\mathcal {L}}={\frac {1}{2}}\left(\sum _{I}^{\mathrm {nuclei} }\ M_{I}{\dot {\mathbf {R} }}_{I}^{2}+\mu \sum _{i}^{\mathrm {orbitals} }\int d\mathbf {r} \ |{\dot {\psi }}_{i}(\mathbf {r} ,t)|^{2}\right)-E\left[\{\psi _{i}\},\{\mathbf {R} _{I}\}\right]+\sum _{ij}\Lambda _{ij}\left(\int d\mathbf {r} \ \psi _{i}\psi _{j}-\delta _{ij}\right),} where μ {\displaystyle \mu } is the fictitious mass parameter; E[{ψi},{RI}] is the Kohn–Sham energy density functional, which outputs energy values when given Kohn–Sham orbitals and nuclear positions. === Orthogonality constraint === ∫ d r ψ i ∗ ( r , t ) ψ j ( r , t ) = δ i j , {\displaystyle \int d\mathbf {r} \ \psi _{i}^{}(\mathbf {r} ,t)\psi _{j}(\mathbf {r} ,t)=\delta _{ij},} where δij is the Kronecker delta. === Equations of motion === The equations of motion are obtained by finding the stationary point of the Lagrangian under variations of ψi and RI, with the orthogonality constraint. M I R ¨ I = − ∇ I E [ { ψ i } , { R I } ] {\displaystyle M_{I}{\ddot {\mathbf {R} }}_{I}=-\nabla _{I}\,E\left[\{\psi _{i}\},\{\mathbf {R} _{I}\}\right]} μ ψ ¨ i ( r , t ) = − δ E δ ψ i ∗ ( r , t ) + ∑ j Λ i j ψ j ( r , t ) , {\displaystyle \mu {\ddot {\psi }}_{i}(\mathbf {r} ,t)=-{\frac {\delta E}{\delta \psi _{i}^{}(\mathbf {r} ,t)}}+\sum _{j}\Lambda _{ij}\psi _{j}(\mathbf {r} ,t),} where Λij is a Lagrangian multiplier matrix to comply with the orthonormality constraint. === Born–Oppenheimer limit === In the formal limit where μ → 0, the equations of motion approach Born–Oppenheimer molecular dynamics. == Software packages == There are a number of software packages available for performing AIMD simulations. Some of the most widely used packages include: CP2K: an open-source software package for AIMD. Quantum Espresso: an open-source package for performing DFT calculations. It includes a module for AIMD. VASP: a commercial software package for performing DFT calculations. It includes a module for AIMD. Gaussian: a commercial software package that can perform AIMD. NWChem: an open-source software package for AIMD. LAMMPS: an open-source software package for performing classical and ab initio MD simulations. SIESTA: an open-source software package for AIMD. ORCA: a general-purpose quantum chemistry package. == Applications == Studying the behavior of water across different environments, such as near a hydrophobic graphene sheet. Investigating the structure and dynamics of liquid water at ambient temperature. Solving the heat transfer problems (heat conduction and thermal radiation), such as in Si/Ge superlattices. Probing the proton transfer along hydrogen-bonds in different environments, such as in 1D water chains inside carbon nanotubes. Evaluating the critical point of crystals, composites, and solid-state materials, such as aluminum. Predicting and modelling different phases and phase transitions, such as in the amorphous phase of the phase-change memory material GeSbTe. Studying the combustion of combustibles, such as lignite-water systems. Measuring th
Lancichinetti–Fortunato–Radicchi benchmark
Lancichinetti–Fortunato–Radicchi benchmark is an algorithm that generates benchmark networks (artificial networks that resemble real-world networks). They have a priori known communities and are used to compare different community detection methods. The advantage of the benchmark over other methods is that it accounts for the heterogeneity in the distributions of node degrees and of community sizes. == The algorithm == The node degrees and the community sizes are distributed according to a power law, with different exponents. The benchmark assumes that both the degree and the community size have power law distributions with different exponents, γ {\displaystyle \gamma } and β {\displaystyle \beta } , respectively. N {\displaystyle N} is the number of nodes and the average degree is ⟨ k ⟩ {\displaystyle \langle k\rangle } . There is a mixing parameter μ {\displaystyle \mu } , which is the average fraction of neighboring nodes of a node that do not belong to any community that the benchmark node belongs to. This parameter controls the fraction of edges that are between communities. Thus, it reflects the amount of noise in the network. At the extremes, when μ = 0 {\displaystyle \mu =0} all links are within community links, if μ = 1 {\displaystyle \mu =1} all links are between nodes belonging to different communities. One can generate the benchmark network using the following steps. Step 1: Generate a network with nodes following a power law distribution with exponent γ {\displaystyle \gamma } and choose extremes of the distribution k min {\displaystyle k_{\min }} and k max {\displaystyle k_{\max }} to get desired average degree is ⟨ k ⟩ {\displaystyle \langle k\rangle } . Step 2: ( 1 − μ ) {\displaystyle (1-\mu )} fraction of links of every node is with nodes of the same community, while fraction μ {\displaystyle \mu } is with the other nodes. Step 3: Generate community sizes from a power law distribution with exponent β {\displaystyle \beta } . The sum of all sizes must be equal to N {\displaystyle N} . The minimal and maximal community sizes s min {\displaystyle s_{\min }} and s max {\displaystyle s_{\max }} must satisfy the definition of community so that every non-isolated node is in at least in one community: s min > k min {\displaystyle s_{\min }>k_{\min }} s max > k max {\displaystyle s_{\max }>k_{\max }} Step 4: Initially, no nodes are assigned to communities. Then, each node is randomly assigned to a community. As long as the number of neighboring nodes within the community does not exceed the community size a new node is added to the community, otherwise stays out. In the following iterations the “homeless” node is randomly assigned to some community. If that community is complete, i.e. the size is exhausted, a randomly selected node of that community must be unlinked. Stop the iteration when all the communities are complete and all the nodes belong to at least one community. Step 5: Implement rewiring of nodes keeping the same node degrees but only affecting the fraction of internal and external links such that the number of links outside the community for each node is approximately equal to the mixing parameter μ {\displaystyle \mu } . == Testing == Consider a partition into communities that do not overlap. The communities of randomly chosen nodes in each iteration follow a p ( C ) {\displaystyle p(C)} distribution that represents the probability that a randomly picked node is from the community C {\displaystyle C} . Consider a partition of the same network that was predicted by some community finding algorithm and has p ( C 2 ) {\displaystyle p(C_{2})} distribution. The benchmark partition has p ( C 1 ) {\displaystyle p(C_{1})} distribution. The joint distribution is p ( C 1 , C 2 ) {\displaystyle p(C_{1},C_{2})} . The similarity of these two partitions is captured by the normalized mutual information. I n = ∑ C 1 , C 2 p ( C 1 , C 2 ) log 2 p ( C 1 , C 2 ) p ( C 1 ) p ( C 2 ) 1 2 H ( { p ( C 1 ) } ) + 1 2 H ( { p ( C 2 ) } ) {\displaystyle I_{n}={\frac {\sum _{C_{1},C_{2}}p(C_{1},C_{2})\log _{2}{\frac {p(C_{1},C_{2})}{p(C_{1})p(C_{2})}}}{{\frac {1}{2}}H(\{p(C_{1})\})+{\frac {1}{2}}H(\{p(C_{2})\})}}} If I n = 1 {\displaystyle I_{n}=1} the benchmark and the detected partitions are identical, and if I n = 0 {\displaystyle I_{n}=0} then they are independent of each other.
Conversational user interface
A conversational user interface (CUI) is a user interface for computers that emulates a conversation with a human. Historically, computers have relied on text-based user interfaces and graphical user interfaces (GUIs) (such as the user pressing a "back" button) to translate the user's desired action into commands the computer understands. While an effective mechanism of completing computing actions, there is a learning curve for the user associated with GUI. Instead, CUIs provide opportunity for the user to communicate with the computer in their natural language rather than in a syntax specific commands.
Irish logarithm
The Irish logarithm was a system of number manipulation invented by Percy Ludgate for machine multiplication. The system used a combination of mechanical cams as lookup tables and mechanical addition to sum pseudo-logarithmic indices to produce partial products, which were then added to produce results. The technique is similar to Zech logarithms (also known as Jacobi logarithms), but uses a system of indices original to Ludgate. == Concept == Ludgate's algorithm compresses the multiplication of two single decimal numbers into two table lookups (to convert the digits into indices), the addition of the two indices to create a new index which is input to a second lookup table that generates the output product. Because both lookup tables are one-dimensional, and the addition of linear movements is simple to implement mechanically, this allows a less complex mechanism than would be needed to implement a two-dimensional 10×10 multiplication lookup table. Ludgate stated that he deliberately chose the values in his tables to be as small as he could make them; given this, Ludgate's tables can be simply constructed from first principles, either via pen-and-paper methods, or a systematic search using only a few tens of lines of program code. They do not correspond to either Zech logarithms, Remak indexes or Korn indexes. == Pseudocode == The following is an implementation of Ludgate's Irish logarithm algorithm in the Python programming language: Table 1 is taken from Ludgate's original paper; given the first table, the contents of Table 2 can be trivially derived from Table 1 and the definition of the algorithm. Note since that the last third of the second table is entirely zeros, this could be exploited to further simplify a mechanical implementation of the algorithm.