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Pen Spinning Theory In the field of pen spinning, Theory refers to the ideas and observations made in attempts to better understand the hobby itself. Applied Theory Structural Analysis Zombo's Combo Transformation Method Released in 2012^[1] by Zombo, a theoretical method is described to allow transformations to be applied to existing combos to result in new derivative combos. Zombo's Structural Stability Theory First released in 2012^[2], Zombo proposed a system of analysing the structure of a combo by defining elements as 'stable' or 'unstable' and tracking the transitions between stability and instability. This would then be used to aid in constructing combos that created particular aesthetic feelings. Formal Notation Notation for analysis Hexbinmos' Elementary Notation A system proposed by Hexbinmos and collaborated on by Fel2Fram in 2012 for designating a series of 'elementary' tricks that can be combined to recreate all of the tricks and combinations in pen RPD's Simplified Elementary Notation An updated system chronicled in RPD's 2020 book "Pen Spinning History and Notation" further simplified and expanded upon Hexbinmos' work, fixing key problems that curbed the use of the original idea. Metaphysics within the context of pen spinning is concerned with defining the fundamental principles that differentiate pen spinning from other things. It aims to answer questions such as "What is Pen Spinning" and "What is Pen Spinning like?". Zombo's Discourse on the Metaphysics of Pen Spinning In 2010^[3] Zombo released an article detailing his observations made to answer the question "What is Pen Spinning, exactly?". It was considered one of the foundations of pen spinning philosophy. V 0 1 D's Anti-Aesthetic Amendment A suggested amendment to Zombo's Discourse on the Metaphysics of Pen Spinning, suggesting that 'aesthetic' need not be a required characteristic for an act to be considered pen spinning. Published March 2021.	{"url":"https://wiki.fenspinner.net/index.php/Theory","timestamp":"2024-11-10T18:45:28Z","content_type":"text/html","content_length":"36633","record_id":"<urn:uuid:4faca01a-ad2f-4cdb-9843-fa631ddf2ff6>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00164.warc.gz"}
An object with a mass of 18 kg is lying on a surface and is compressing a horizontal spring by 60 cm. If the spring's constant is 18 (kg)/s^2, what is the minimum value of the surface's coefficient of static friction? \| HIX Tutor An object with a mass of # 18 kg# is lying on a surface and is compressing a horizontal spring by #60 cm#. If the spring's constant is # 18 (kg)/s^2#, what is the minimum value of the surface's coefficient of static friction? Answer 1 The coefficient of static friction is $= 0.061$ The coefficient of static friction is The spring constant is #k=18kgs^-2# The compression is #x=0.6m# The coefficient of static friction is Sign up to view the whole answer By signing up, you agree to our Terms of Service and Privacy Policy Answer 2 To find the minimum value of the surface's coefficient of static friction, we can use the formula: μ_s = kx/mg • μ_s is the coefficient of static friction, • k is the spring constant (18 (kg)/s^2), • x is the compression of the spring (60 cm = 0.6 m), • m is the mass of the object (18 kg), • g is the acceleration due to gravity (9.8 m/s^2). Plugging in the values: μ_s = (18 (kg)/s^2 * 0.6 m) / (18 kg * 9.8 m/s^2) = 0.6 / 9.8 ≈ 0.061 Therefore, the minimum value of the surface's coefficient of static friction is approximately 0.061. Sign up to view the whole answer By signing up, you agree to our Terms of Service and Privacy Policy Answer from HIX Tutor When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some Not the question you need? HIX Tutor Solve ANY homework problem with a smart AI • 98% accuracy study help • Covers math, physics, chemistry, biology, and more • Step-by-step, in-depth guides • Readily available 24/7	{"url":"https://tutor.hix.ai/question/an-object-with-a-mass-of-18-kg-is-lying-on-a-surface-and-is-compressing-a-horizo-3-8f9af8ab0c","timestamp":"2024-11-10T12:33:27Z","content_type":"text/html","content_length":"583461","record_id":"<urn:uuid:63b0bbf2-820a-491b-9c4b-907b4e4f39c0>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00421.warc.gz"}
Higher Energy Derivatives in Hilbert Space Multi-Reference Coupled Cluster Theory : A Constrained Variational Approach Physical Chemistry Division, National Chemical Laboratory, Pune, India Author to whom correspondence should be addressed. Submission received: 18 December 2001 / Accepted: 10 April 2002 / Published: 30 June 2002 In this paper, we present formulation based on constrained variational approach to compute higher energy derivatives upto third order in Hilbert Space Multi-Reference Coupled Cluster (HSMRCC) Theory. This is done through the use of a functional with Lagrange multipliers corresponding to HSMRCC method, as done by Helgaker, Jorgensen and Szalay. We derive explicit expressions upto third order energy derivatives. Using (2n + 1) and (2n + 2) rules, the cancellation of higher order derivatives of functional parameters that are not necessary according to these rules, is explicitly demonstated. Simplified expressions are presented. We discuss several aspects of the functional used and its potential implications. In last two decades, coupled-cluster (CC) method [ ] has emerged as the most efficient and accurate method for calculation of electronic structure and spectra where inclusion of electron correlation is an important factor. For systems dominated by single determinant, the dynamical electron correlation is the most important part of correlation. The single-reference coupled-cluster (SRCC) method compactly sums up several infinite order perturbation terms through the solution of a non-linear equation leading to an accurate description of dynamical correlation. By its very method of formulation, the much required property of size-extensivity property is automatically included in CC method. Another reason for the emergence of SRCC as state-of-the-art method is development of efficient analytic energy derivative techniques for the calculation of molecular electronic properties [ ]. The linear response theory is the appropriate framework for this [ ]. In this theory, molecular energy gradients, hessians, dipole-moment, polarizability and other frequency dependent properties appear as energy response quantities of appropriate order in perturbation. Stationary electronic theories i.e., theories where some or all parameters are determined from stationarity of energy functional, enjoy an advantage for calculating energy derivatives. Such theories obey the Hellmann-Feynmann theorem and its generalization (2 + 1) rule, which drastically simplify the evaluation of energy derivatives [ ]. There have been several attempts to formulate a stationary theory based on coupled-cluster ansatz [ ]. All such attempts have resulted in theories much more complicated theories. As a result, the standard CC method is non-stationary. As a consequence of non-stationary nature of the standard CC method, the calculation of first order energy response requires that first order response of cluster amplitudes be calculated for every mode of perturbation. However, Bartlett and co-workers [ ] have shown that the requirement of first derivative of cluster amplitudes for each mode of perturbation can be replaced by a single perturbation-independent quantity, known as Z-vector. There have been further developments in obtaining compact expressions for first and second derivatives including orbital response [5, ?, 10]. By constructing an unconstrained Lagrangian functional, Helgaker, Jorgensen and coworkers have incorporated all such developments in a single formulation which is easily applicable to higher orders [ Description of molecular excited states and potential energy surfaces can not be easily done with single-reference theories such as SRCC. These are the cases which are marked by nearly equal domination of a number of determinants referred to as quasi-degenerate. This is essentially known as non-dynamical electron correlation, which can be efficiently introduced by using a multi-determinantal description of zeroth order situation [ ]. There have been several approaches developed in last two decades. Dominant among them is the effective Hamiltonian method [ ], where an effective Hamiltonian is diagonalized within a suitably chosen quasi-degenerate model space to approximately reproduce a part of the spectrum associated with the exact Hamiltonian. Pursuing coupled-cluster ideas have led to two different ansatz for wave operator to introduce dynamical correlation. First is the Fock-Space (FS) multi-reference coupled-cluster (MRCC) method [ ]. It is efficient for studying states associated with a model space of specified number of of valence electrons, while at the same time considering all lower valence model spaces with fewer number of electrons in them. This makes it suitable for studying spectroscopic quantities such as excitation energies. The second method, Hilbert-Space (HSMRCC) [ ], is useful for study of several interacting states with a fixed number of electrons and is suitable for potential energy surfaces. To overcome the intruder state problems associated with complete model spaces usually used in effective Hamiltonian theories several different directions are being pursued in recent years. Use of incomplete model space [ ] intermediate Hamiltonian [ ] and state-specific multi-reference approaches [ ] are major ones. It may be mentioned that the one-valence FSMRCC is equivalent to the Equation of Motion CC (EOM-CC) method. In the context of EOM-CC fully relaxed analytic response has been developed and implemented by Gauss and Stanton [ ]. However, FSMRCC method for one valence upward, and HSMRCC in particular, have no parallel in EOM-CC or any other SR based theories. Thus, it is important to formulate the linear response of effective Hamiltonian MRCC methods [ ]. Analytic linear response approach for a general effective Hamiltonian based on coupled-cluster method was originally formulated by Pal [ ]. It has been implemented to the FSMRCC [ ]. This has been recently used to calculate dipole-moments of open-shell molecules [ ]. Szalay has outlined an approach [ ] based on undetermined Lagrange multipliers method used by Hegaker, Jorgensen and coworkers in SRCC [ ], to outline the calculation of first-derivatives of MRCC methods, and analyze the cost of MRCC derivatives as compared to SRCC derivatives. The present authors have introduced Z-vector method for effective Hamiltonian MRCC methods and derived expressions for the first energy derivative for HSMRCC theory [ The current work focuses on deriving expressions for higher-order energy derivatives (specifically, upto third-derivative) for HSMRCC theory through appropriate higher-order generalization of -vector method. For this we propose an undetermined Lagrange multiplier functional which is similar to the one used by Szalay [ ]. The equations for model space coefficients, cluster amplitudes, Lagrange multipliers and their first derivatives are obtained. In section II, SRCC and MRCC linear response, -vector method and its relation to the method based on contrained Lagrange multipliers are briefly summarized. -vecror method for HSMRCC is also discussed, outlining its relation to the functional proposed by Szalay. In section III, using a functional with undetermined Lagrange multipliers corresponding to HSMRCC method leading to Z-vector equations, expressions upto third derivatives for a specific state are derived. Equations for the Lagrange multipliers and its derivatives are presented. Section IV summarizes the results with some general comments on the results of section III. In this section, key developments in SRCC and MRCC linear response theory are discussed. The relation of Z-vector to constrained variational method is examined in detail. The advantage of the latter method is highlighted for use in computing higher-order derivatives. 2.1 Single Reference Coupled Cluster Method The SRCC method has been thoroughly discussed in literature. It is obtained by using exponential ansatz acting on a determinant $\| Φ 〉$ in Schrodinger equation is the cluster operator with usual second quantized definition. $T = ∑ i , a t i a a a a i + 1 4 ∑ a , b , i , j t i j a b a a a b a j a i + ⋯$ In the above expression i,j,... refer to orbitals occupied in $\| Φ 〉$, and a,b,... refer to orbitals unoccupied in $\| Φ 〉$ and $a a$ and $a i$ are orbital creation and annihilation operators respectively. Although T is N-body operator, it is usually truncated. Truncation of T to one and two-body parts is quite common and referred to as CCSD, although upto four-body truncations have been reported in literature. By substituting ansatz into equation and premultiplication with e^−T followed by projection with hole-particle determinants $\| Φ i j … a b … 〉$ (henceforth denoted as $\| Φ q 〉$ with q denoting collective hole-particle excitations respect to $\| Φ 〉$), we get the CC equations. $〈 Φ \| e − T H e T \| Φ 〉 = E$ $〈 Φ q \| e − T H e T \| Φ 〉 = 0 , ∀ \| Φ q 〉$ The equations for cluster amplitudes are non-linear in nature and can be solved by iteration. 2.2 Analytic Linear Response for SRCC method. A computational framework to analytically calculate various properties with SRCC method was first outlined by Monkhrost using linear response approach [ ]. In this framework, various molecular properties are defined to be various order derivatives of energy with respect to a small external perturbation strength parameter . By expanding Energy , Hamiltonian and the cluster amplitudes as a Taylor series around zero perturbation strength, and collecting terms of same order in , expressions for various derivatives of CC Energy can be obtained. For first order, neglecting orbital response, the equations are, $E ( 1 ) = Y t T ( 1 ) + Q ( g )$ In above equations, superscript (1) on refers to first derivative of respective quantities with respect to external perturbation strength parameter . Terms within {} are used to denote column vectors and superscript denotes transpose. Similar equations can be derived for higher derivatives of . Inclusion of orbital response (which is important for geometric derivatives) was first done by Jorgensen and Simons [ ] who derivided detailed expressions upto second derivatives using the second-quantization framework. 2.3 Z-vector method and other developments As outlined, SRCC, being non-stationary theory, does not have the advantage of (2 + 1) rule. Thus, is required for calculating first derivative of energy . This has to be done for every mode of perturbation which is a disadvantage. A step towards eliminating this disadvantage was taken by Bartlett and coworkers [ ]. They make use of a technique based on Dalgarno’s interchange theorem [ ] used by Handy and Schaefer for configuration interaction (CI) energy derivatives [ ]. The technique, known as -vector technique, derives algebraic expressions for reducing the number of linear equations for orbital rotations and cluster amplitudes. By inverting equation (7) and substituting in (6), can be reorganized as, $E ( 1 ) = Z t B ( g ) + Q ( g )$ Equation (13) is a perturbation independent equation providing the vector. The advantage of such reorganization is, unlike earlier equation (7), equation (13) needs to solved only once. This -vector method is in some sense equivalent to (2 + 1) type of rule for non-stationary methods. Further simplifications have been carried out by introducing effective CC density matrices [ ] much akin to CI derivative developments. The technique of Rice and coworkers [ ] has also been applied to reduce the number of AO to MO transformations. First applications of CC analytic derivatives have been reported by Bartlett and coworkers [ ] and Scheiner and coworkers [ 2.4 Method of Undetermined Lagrange Multipliers The conceptually simple procedure of eliminating in favor of a -vector is somewhat cumbersome for higher orders and has been pursued by Salter and coworkers [ ]. However, Helgakar, Jorgensen and coworkers [ ], have pursued an attractive alternative formulation of CC derivative which automatically incorporates -vector technique to all orders. Such a formulation proceeds by constructing a Lagrange functional with undetermined multipliers $Λ = { λ q , ∀ q }$ corresponding to CC equations (5) as follows. $J ( T , Λ ) = 〈 Φ \| e − T H e T \| Φ 〉 + ∑ q λ q 〈 Φ q \| e − T H e T \| Φ 〉$ The functional may be viewed as approximation to the full Extended Coupled Cluster (ECC) functional of Arponen [ ]. The full ECC functional been used by Pal and coworkers to derive the first and higher order energy derivatives using a stationary approach [ ]. Optimization of functional (14) leads to same equations as cluster amplitude equations (5) and -vector equations (13) with Λ taking the role of -vector. In this formulation, it is transparant to derive expressions for higher order derivatives. While obey (2 + 1) rule, the undetermined multipliers obey (2 + 2) rule [ 2.5 Multi-Reference Coupled Cluster (MRCC) Theories. Extension of coupled-cluster method to quasi-degenerate situations demanding multi-reference description has been non-trivial. Different approaches were followed by different researchers. Among them, the approach of constructing an effective Hamiltonian in a complete model space has emerged as the standard one [ ]. We briefly summarize the approach in the following. A strongly interacting -dimensional complete model space is considered by distributing a given number of electrons among a conveniently chosen set of valence orbitals. This model space, , is assumed to approximate the quasi-degenerate target space , spanned by exact states of exact Hamiltonian . Hence, it contains the zeroth-order approximations for all the exact states under consideration. Dynamical correlation is brought in via wave-operator Ω mapping model and target space as, The wave-operator, Ω, through an appropriate parametrization, is constructed by solving generalized Bloch equation. Bloch equation is simpliy a restatement of the schrodinger equation for all targeted states. The exact energy and zeroth order approximations of target states within model space, $\| Ψ i ( 0 ) 〉$ , are obtained by diagonalizing the effective Hamiltonian. Two kinds of parametrizations for the wave-operator Ω have been widely discussed in literature. The first one involves the use of a common reference vaccum $\| Φ 〉$ for defining a single second-quantized cluster operator connecting model spaces with different number of valence electrons from zero to a specified number used to build the model space. This leads to Fock-Space MRCC (FSMRCC) [ is so constructed that the corresponding wave-operator, Ω = , is able to yield exact states for different model spaces connected by . FSMRCC has been successful in accurate computations of spectroscopic quantities such as ionization potential, electron affinity, excitation energies [ The second formulation does not use a single reference vaccum, but uses as many vacua as the number of states in the manifold, with one cluster operator for each vaccum. This is referred to as Hilbert-Space MRCC (HSMRCC) [ ]. Unlike FSMRCC, HSMRCC does not require the use of model spaces with different number of valence electrons. The wave-operator is written as, $Ω = ∑ μ e T μ \| Φ μ 〉〈 Φ μ \|$ $T μ$ is the cluster operator with respect to vaccum $\| Φ μ 〉$ similar in structure to SRCC cluster operator. $T μ$ does not contain excitations leading to states within model space. Using the above ansatz in equations (16) and (17) and projecting with determinants, $\| Φ q ( μ ) 〉$ , hole-particle excited with respect to $\| Φ μ 〉$ denotes collective orbital indices involved in the excitation), we get the following HSMRCC equations. $C ˜ i v$ $C μ i$ refer to left and right eigenvectors of the effective Hamiltonian . HSMRCC has been pursued by various researchers and has been found to be useful for description of potential energy surfaces [ ]. Spin adaptated formulations have also been pursued and successfully applied to various chemical systems [ There have been several other MRCC approaches such as Intermediate Hamiltonian approach [ ], use of incomplete model spaces [ ] and state-specific MRCC [ ]. However, the current work focuses on complete model space effective Hamiltonian based MRCC approaches with special emphasis on HSMRCC theory. 2.6 Analytic Linear Response for MRCC Theories Application of linear response to obtain expressions for MRCC derivatives was initiated by Pal, who, following Monkhrost’s approach outlined the formulation for the effective Hamiltonian based MRCC theory [ ]. Specific expressions were obtained for one hole, one particle and hole-particle sectors of FSMRCC theory [ ]. The structure of the original nonvariational formulation did not include prescription of -vector and thus were not applicable for higher energy derivatives and even gradients. Applications of the formalism to compute first order molecular properties have however been carried out in recent years [ ]. The formalism has also been extended to time-dependent perturbation case [ On the other hand, Szalay was the first to extend Helgaker and Jorgensen’s method by proposing a Lagrangian functional for a chosen state in the manifold which yields MRCC cluster amplitude and eeffective Hamiltonian diagonalization equations [ ]. His work mainly concerned with estimating the relative cost of MRCC first order response calculations as compared to SRCC response equations. First efforts towards directly introducing Handy-Schefer -vector type technique for the effective Hamiltonian, , were carried out by Pal and coworkers [ ]. Working on FSMRCC response equations for one hole model space, they concluded that only highest sector amplitude derivatives can be eliminated from under degenerate diagonal . Further work on HSMRCC theory has shown that it is not possible to eliminate cluster amplitude derivatives in effective Hamiltonian first derivative with a single -vector for all states, although number of independent -vectors would be sufficient. Recently, however, the present authors have showed that it is possible to eliminate the cluster amplitude derivatives for a chosen state energy derivative. They have derived detailed equations for HSMRCC -vector for a chosen state and developed efficient expressions for HSMRCC energy first derivative by extending the idea of effective CC density matrices [ Extension of algebraic -vector method is difficult for higher order energy derivatives, although it has been pursued in SRCC context by Salter and coworkers [ ] to obtain second derivative expressions. It is advantageous to go over to constrained variation approach [ ]. The approach generalizes the -vector method to all orders retaining the simplicity of HSMRCC method. In this section, we derive expressions for HSMRCC energy derivatives upto third-order for a chosen state. For this, we propose a functional with Lagrange multipliers corresponding using HSMRCC equations, (19)-(23), as constraints. The functional leads to equations for Lagrange multipliers corresponding to cluster amplitude equation (19) to be the same as -vector equations derived recently [ ]. Using (2 + 1) rule for cluster amplitudes and model space coefficients, (2 + 2) rule for Lagrange multipliers, the expressions for second and third order are written in a simple form. 3.1 The Lagrangian functional and obtaining linear response Before giving explicit expression for the Lagrangian functional which corresponds to HSMRCC method, we assume the following abbreviations. We denote the set of cluster amplitudes for all vacua by T, the set of all model space coefficients for the i-th state by C[i] and the set of all Lagrange multpliers corresponding to HSMRCC cluster amplitude equations (19) by Λ. We further collectively refer the set of quantities T,C[i],Λ and the energy of i-th state E[i] (the Lagrange multiplier corresponding to biorthogonal conditions (23)), by Θ. The following equations summarize the above Now, the Lagrangian functional which corresponds to HSMRCC equations (19)-(23) is constructed as, where HSMRCC equation for , (19), is denoted by ) = 0. Making stationary with respect to generates sufficient number of equations to determine these parameters. The above functional is similar to one used by Szalay [ ] in context with HSMRCC first derivatives. However, the present functional does not introduce dependence in second term. As a consequence, leads to -vector equations derived by eliminating the first order response of cluster amplitudes, from energy first derivative expression. Although current formulation results in slightly complicated expressions for effective CC density matrices, it will be advantageous as discussed in next To derive expressions for various order derivatives, linear response theory is used. A small perturbation with strength parameter is introduced into the Hamiltonian $H ( g ) = H + g H ( 1 )$ is determined at all strenghts of perturbation , the functional becomes a function of , denoted by $J ( g , Θ )$ . Response equations can be obtained in two different approaches. First approach, followed by Helgaker, Jorgensen and coworkers [ ] and Bartlett and coworkers [ ] is to expand the functional and the stationary equations obtained for as a Taylor series in strength parameter . Terms of same order in equations are collected and equated to obtain hierarchical equations for various response quantities Second approach, proposed by Pal [ ] in the context with stationary CC theories, is to derive response equations of any required order as a stationary equations. In this approach, $J ( g , Θ )$ is expanded as, $J ( g , Θ ) = J ( 0 ) + g J ( 1 ) + 1 2 ! g 2 J ( 2 ) + 1 3 ! g 3 J ( 3 ) + · · ·$ It should be noted that is a functional of quantities ${ Θ ( m ) m = 0 , n }$ . All response equations upto a required order can be derived by making the functionals ${ J ( k ) k = 0 , n }$ stationary with respect to ${ Θ ( m ) m = 0 , n ( m ≤ k ) }$ . This leads to the following equations. $∂ J ( k ) ∂ Θ ( m ) = 0 , ∀ k , ∀ m , ( m ≤ k ) , ( k = 0 , n )$ This includes the = 0 case corresponding to the unperturbed (zeroth order) equations as well. Henceforth, we drop the superscipt for zeroth-order quantities. It has been shown that there is a large amount of redundancy in the above equations. Hence it is sufficient to solve $∂ J ( m ) ∂ Θ = 0 ∀ m = 0 , n$ This leads to hierarchical equations for ${ Θ ( m ) ∀ m = 0 , n }$ . In this work, we follow the second approach in deriving response equations. According to (2 + 1) rule associated with stationary methods, we need to obtain expressions upto thrid order energy derivatives, and hence equations (32) need to be solved upto = 1. It should be noted that both approaches are entirely equivalent and lead to identical equations with a given functional. 3.2 Response response equations upto first order Detailed expressions for $J ( n )$ = 3 are given in Appendix A , equations (A-4)-(A-6). To derive response equations for upto first order, we use expression for along with equation (A-1). Zeroth-Order quantities, , can be obtained by When applied for different parameters , the above equation gives HSMRCC equations for cluster amplitudes (19) and the eigenvalue equations (20)-(21). In addition it gives equations for Λ as, $∑ η , q λ q ( η ) [ E q ( η ) ] τ i ( μ ) = − ∑ v C ˜ i v C μ i 〈 Φ v \| [ e − T μ , τ i ( μ ) ] \| Φ μ 〉 ∀ i , μ$ where subscript ) indicates differentiation with respect to specific cluster amplitude ) and ) is the hole-particle excitation operator associated with ). The quantity [ has been referred to CC Jacobian by Jorgensen and coworkers [ ] and in this case it is the HSMRCC Jacobian. Diagrammatic representation for the above equation can be easily obtained as outlined in [ ]. The above equation is the same as the -vector equations derived recently by the present authors [ ] using elimination technique of Handy and Schaefer [ The first order quantities, are necessary for calculating higher order energy derivatives and can be obtained by, The above equations lead to following equations for first order quantities . It should be noted that these equations depend on zeroth-order quantities All terms used here are defined in Appendix A . These equations should be solved in the same order. Equation (40) is the equation for Λ . It not only depends on first derivatives of , but also on zeroth order quantity Λ. The equations (36) and (40) reveal the same structure pointed out by Jorgensen and coworkers [ ] i.e., and Λ are related by the same HSMRCC Jacobian. The only difference between both equations is in the inhomogenous part. 3.3 Simplified expressions for Energy derivatives Energy derivative of n-th order, $E i ( n )$, is just the value of functional $𝒥$^(n), denoted by $J o p t ( n )$, when the stationary values of ${ Θ ( m ) m = 0 , n }$ are substituted in it. Hence, $J o p t ( n )$ can be considered as the required energy derivative and treat E[i] as another Lagrange multiplier. It has been shown that $C i ( n )$ and T^(n) obey (2n + 1) rule and Lagrange multipliers Λ^(n) and $E i ( n )$ obey (2n + 2) rule. However, the expressions obtained by simple substitution as above do not take advantage of the above rules. Hence the expressions must be simplified by explicit application of these rules. This eliminates any unnecessary higher order derivatives present in these expressions. Elimination is carried out by referring to appropriate response equations, including the zeroth-order response equations. For ${ J o p t ( n ) n = 1 , 3 }$ the quantities which need to be eliminated are given below, by the application of (2n + 1) and (2n + 2) rules. Appendix B , expressions for ${ J o p t ( n ) n = 1 , 3 }$ obtained from (A-4)-(A-6) are rearranged to indicate explicit cancellation of higher derivatives of , Λ and . Terms indicated in {} can be eliminated by application of response equations of appropriate order. This is discussed in following subsections. Elimination of higher derivatives of from the remaining expressions will be demonstrated seperately. 3.3.1 Simplified expression for $J o p t ( 1 )$ Expression for Energy first derivative $J o p t ( 1 )$ is given in (B-1). It can be noted that $C μ i ( 1 )$ is eliminated by zeroth-order equation for $C ˜ i v$, (21) and $C ˜ i v ( 1 )$ is eliminated by zeroth-order equation for C[µi], (20). The Lagrange multipliers Λ^(1) and $E i ( 1 )$ are eliminated by zeroth-order T equation (19), and biorthogonality equation (23), respectively. Since HSMRCC equations (20)-(21) contain E[i], presense of E[i] is automatically eliminated. Elimination of which is present in the surviving terms of (B-1), requires application of zeroth order Λ equations. For demonstate this, $E e f f νµ$^(1) $E q ( 1 )$ ) are expanded to seperate out terms containing . All such terms cancel precicely from Λ equations, (34). Hence, simplified expression for $J o p t ( 1 )$ is given by, $J o p t ( 1 ) = ∑ μ , v C ˜ i v C μ i [ H e f f v μ ( 1 ) ] T ( o ) + ∑ q , η λ q ( η ) [ E q ( 1 ) ( η ) ] T ( o )$ where subscript indicates retention of terms containing { T^(m) m = 0 }. This expression has been further simplified by making use of state-dependent effective CC density matrices [ 3.3.2 Simplified expression for $J o p t ( 2 )$ Expression for Energy second derivative $J o p t ( 2 )$ is given in (B-2) along with grouping necessary for elimination. Unlike $J o p t ( 1 )$, $J o p t ( 2 )$ depends on T^(1) and $C i ( 1 )$, which are determined by equations (36)-(39). $C i ( 2 )$ is eliminated by zeroth-order equations for C[i], (20)-(21). Similarly, Λ^(2), Λ^(1) and $E i ( 2 )$are trivially eliminated by T^(1) equation (36), T equation (19) and biorhogonality condition (23) respectively. Although, $E i ( 1 )$ can be easily eliminated from equation (39), we deliberately retain it to further simplify the expression. Since $C i ( 1 )$ depends on T^(1), terms containing both T^(1) and $C i ( 1 )$ can be eliminated using $C i ( 1 )$ response equations (37)- (38). For this some readjustments are necessary as indicated in (B-2). present in the remaining terms can be eliminated as done in section 3.3.1, by collecting terms containing , and making use of zeroth-order Λ equations. This leads to simplified expression for 3.3.3 Simplified expression for $J o p t ( 3 )$ Expression for Energy third derivative $J o p t ( 2 )$ is given in (B-3) along with groupings necessary for elimination. Equation (40) for Λ^(1) is the only additional equation to be solved in addition to T^(1) and $C i ( 1 )$ equations. $C i ( 3 )$ and $C i ( 2 )$ can be eliminated through C[i] and $C i ( 1 )$ equations respectively. Similary, Λ^(3), Λ^(2), $E i ( 3 )$, $E i ( 2 )$ can be trivially eliminated by applying appripriate response equations. Presence of can be eliminated by collecting terms containing these quantities from surviving terms. It can be easily seen that such terms arise with correct factor necessary for applying Λ and Λ equations. While cancels from Λ equation (34), cancellation of requires extraction of terms containing in remaining terms with a factor of 3. After collecting such terms, cancels by applying Λ equation (40). Final simplified expression for $J o p t ( 3 )$ can be given as, Results of the previous section demonstrate the advantage of constrained variational approach for non-stationary theories over elimination technique. In this formulation, it is particularly easy to derive expressions for higher derivatives and apply (2n + 1) and (2n + 2) rules to simplify them. The nature of cancellation of terms in $J o p t ( n )$ containing higher order derivatives of Θ, is also clear. Derivatives of Lagrange numtipliers Λ and E[i], which are not required by the (2n + 2) rule, cancel from lower order response equations of cluster amplitudes T and biorthogonality equations. Cancellation of higher order derivatives of C[i] that are not required by the (2n + 1) rule obeyed by it, occurs again by lower order response equation of its conjugate model space coefficient. Since the Lagrange multiplier E[i] appears in these equations, some derivatives of E[i] go into these cancellations. It should be noticed that while Lagrange multipliers Λ and Λ^(1) appear in $J o p t ( 3 )$, the same does not happen for E[i] and $E i ( 1 )$. In $𝒥$^(3), only $E i ( 1 )$ appears, while E[i] goes into cancellations involving higher order derivatives of C[i]. Cancellation of higher derivatives of T not required by (2n +1) rule, happen through lower order Λ response equations. Finally, for even order derivatives, as exemplified in $J o p t ( 2 )$, further simplifications are possible through the use of response equations for C[i], resulting in eliminating the terms where both $C i ( 1 )$ and T^(1) are present. In the context of SRCC linear response theory, it has been known that the constrained variational approach is identical to -vector method introduced much earlier, with the Lagrange multipliers Λ becoming identical to the -vector [ ]. However, this is not the case in MRCC, where several possibilities of functional open up. The functional proposed in this work leads to the same equations as the elimination method followed in -vector approach. There are several other possible functionals worth pursuing. One of them has been proposed by Szalay [ ]. In his functional, a factor of $C ˜ i µ$$C µ i$ is introduced in the second term of (28). This leads to expressions for effective CC density matrices which are closer in structure to SRCC counterparts [ ]. On the other hand, such a functional leads to Λ equations whose homogenous part depends on . As a result, and Λ do not share the same HSMRCC Jacobian, as it happens in SRCC. On the other hand, the functional in this work leads to same HSMRCC Jacobian for and Λ . Although this is a clear advantage, other possibile functionals should be explored. However, it should be noted that all such functional are equivalent in the sense that they provide same energies and derivatives differing only in the form of expressions. Another important aspect of constrained variational approach is the intrinsic state-dependency of the functional. It should be noted that the state-dependency of -vector is not special to MRCC methods, but it has been observed in Equation of Motion CC method [ ]. The present authors have investigated whether can be eliminated from in favor of a single -vector. It has been concluded to be not possible for a general case, because of matrix nature of . Hence, it is necessary to become state-selective and eliminate from the energy deriative of a specific state. It has been recognized that constrained variation approach for SRCC is related to bivariational CC methods investigated by Arponen [ ]. The Single-Reference Extended CC (SR-ECC) formulated by Arponen is attractive in many respects and has been investigated for its use in calculating molecular properties [ ]. Apart from being stationary, terminating and size-extensive nature of functional, with all their advantages, it is also known to sum much larger class of perturbation theory diagrams as compared to SRCC. Hence it is expected to be highly accurate and appropriate for use to calculate nonlinear molecular electronic properties. Generalization of SR-ECC to multi-reference situations (MR-ECC) has not been reported in literature. The nature of constrained variation approach for HSMRCC clearly indicates that MR-ECC has to be state-selective theory, much akin to the decontracted state-selective MRCC approach proposed by Mukherjee and coworkers [ ]. The non-uniqueness of the constrained variational functional indicates that many different formulations of MR-ECC may be possible. This is similar to the possibilty of having varieties of state-selective non-stationary MRCC theories. This line of research is worth pursuing. The authors acknowledge research grant from the Department of Science and Technology (DST), India towards the work. One of the authors, KRS, thanks Council of Scientific and Industrial Research (CSIR), India for research fellowship. Expressions for functionals in (30) can be obtained by expanding various quantities on the right hand side of (28) as a Taylor series in perturbation strength in g. In the following, superscript on zeroth-order quantities are Collecting and equating terms of same order in g on both sides after the expansion, we get, 1. Cizek, J. Adv. Quant. Chem. 1969, 14, 35. 2. Bartlett, R.J. Annu. Rev. Phys. Chem. 1981, 32, 359. 3. Paldus, J. Methods in Computational Molecular Physics; Wilson, S., Dierckson, G.H.F., Eds.; NATO ASI series B: Plenum, NY, 1992. [Google Scholar] 4. Helgaker, T.; Jorgensen, P. Adv. Quant. Chem. 1988, 19, 183. 5. Bartlett, R.J. 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Theor. Chim. Acc. 1998, 99, 100. Vaval, N.; Pal, S. J. Chem. Phys. 1999, 111, 4051. 48. Meissner, L.; Jankowski, K.; Wasilewski, J. Int. J. Quant. Chem 1988, 34, 535. Balkova, A.; Kucharski, S.A.; Meissner, L.; Bartlett, R.J. Theor. Chim. Acta 1991, 80, 335. ibid. J. Chem. Phys. 1991, 95, 4311. Balkova, A.; Kucharski, S.A.; Bartlett, R.J. Chem. Phys. Lett 1991, 182, 511. Kucharski, S.A.; Bartlett, R.J. J. Chem. Phys 1989, 95, 8227. Balkova, A.; Bartlett, R.J. Chem. Phys. Lett. 1992, 193, 364. 49. Paldus, J.; Piecuch, P.; Jeziorski, B.; Pylypow, L. Recent Progress in Many-Body Theories, Vol.3; Ainsworthy, T.L., Campbell, C.E., Clements, B.E., Krotschek, E., Eds.; Plenum Press: New York, 1992. [Google Scholar] Paldus, J.; Piecuch, P.; Pylypow, L.; Jeziorski, B. Phys. Rev. A. 1993, 47, 2738. Piecuch, P.; Toboła, R.; Paldus, J. Chem. Phys. Lett. 1993, 210, 243. Piecuch, P.; Paldus, J. Phys. Rev. A. 1994, 49, 3479. Kowalski, K.; Piecuch, P. Phys. Rev. A. 2000, 61. page number. 50. 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Higher Energy Derivatives in Hilbert Space Multi-Reference Coupled Cluster Theory : A Constrained Variational Approach. International Journal of Molecular Sciences. 2002; 3 (6):710-732. https://doi.org/10.3390/i3060710 Chicago/Turabian Style Shamasundar, K. R., and Sourav Pal. 2002. "Higher Energy Derivatives in Hilbert Space Multi-Reference Coupled Cluster Theory : A Constrained Variational Approach" International Journal of Molecular Sciences 3, no. 6: 710-732. https://doi.org/10.3390/i3060710 Article Metrics	{"url":"https://www.mdpi.com/1422-0067/3/6/710","timestamp":"2024-11-12T00:04:31Z","content_type":"text/html","content_length":"457600","record_id":"<urn:uuid:d6ac613b-47b2-45e5-8d4e-07c5b3d17fc0>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00744.warc.gz"}
Academic Requirements Students majoring in mathematics/computer science usually begin with a three-semester calculus sequence: Calculus I, II, and III. Students seeking placement beyond Calculus I should consult with a member of the faculty. Placement is determined by interviews and transcripts. Precalculus is offered for those lacking the necessary background for Calculus I. In addition to meeting General Education requirements and other degree requirements, students majoring in mathematics/computer science must complete each of the following requirements. A grade of C- or higher* is required in these courses, excluding the senior project: For Students Entering Fall 2024 and Later: • MAT 1500, 1510, and 3150/Calculus I, II, and III • MAT 1520 and 1540/Computer Science I and II • MAT3120, Discrete Mathematics. Note that this course is a prerequisite for some additional upper level courses taken later in the curriculum. • MAT 3170/Linear Algebra • Four upper-level electives (16 credits) in mathematics/computer science. One of the electives may be fulfilled by a tutorial, independent study, learning assistantship, or internship with the approval of the faculty advisor. • Two science courses (minimum 6–8 credits) • MAT 3880/Junior Seminar in Mathematics/Computer Science • MAT 4880/Mathematics Senior Seminar I (MAT 3880 Junior Seminar is prerequisite). • MAT 4890/Mathematics Senior Seminar II (MAT3880 Junior Seminar is Co-requisite OR MAT4880 Senior Seminar I is prerequisite). • SPJ 4990/Senior Project I • SPJ 4991/Senior Project II Note: A minimum grade of C is required in the prerequisite course(s). For example, the prerequisite for MAT 1510 is a minimum grade of C in MAT 1500. This grade minimum is stated in the prerequisite when applicable. For students in the program prior to fall 2024 follow these requirements: • MAT 1500, 1510, and 3150/Calculus I, II, and III • MAT 1520 and 1540/Computer Science I and II • MAT 3170/Linear Algebra • Five upper-level electives (20 credits) in mathematics/computer science. One of the five electives may be fulfilled by a tutorial, independent study, learning assistantship, or internship with the approval of the faculty advisor. • Two science courses (minimum 6–8 credits) • MAT 3880/Junior Seminar in Mathematics/Computer Science • MAT 4880/Mathematics Senior Seminar I (MAT 3880 Junior Seminar is prerequisite). • MAT 4890/Mathematics Senior Seminar II (MAT3880 Junior Seminar is Co-requisite OR MAT4880 Senior Seminar I is prerequisite). • SPJ 4990/Senior Project I • SPJ 4991/Senior Project II Note: A minimum grade of C is required in the prerequisite course(s). For example, the prerequisite for MAT 1510 is a minimum grade of C in MAT 1500. This grade minimum is stated in the prerequisite when applicable.	{"url":"https://www.purchase.edu/academics/mathematics-computer-science/requirements/","timestamp":"2024-11-05T06:34:00Z","content_type":"text/html","content_length":"26536","record_id":"<urn:uuid:7a00adef-688c-4db6-9897-3f3c542a8240>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00439.warc.gz"}
Navy Develops Mathematical Formula to Intercept Enemy Missile Attacks at Sea - Warrior Maven By Cameron Curtis, The answer is – at least two, probably three or more. Period. When an Aegis destroyer fires six Standard 3 ABMs at 300 incoming Iranian missiles, it is not going to shoot down six of the threats. It will be lucky to shoot down three. It might shoot down two. On many occasions, it might not shoot down any. The same goes for all our air defense missiles. Patriot PAC-3s, THAAD, Israeli Arrow 3. It is standard air defense doctrine to fire at least two, and often three air defense missiles, for each incoming ballistic missile threat. The purpose of this short article is to show the mathematics (don’t be scared, I promise it will be painless) behind the doctrine. I will also show that the doctrine is fundamentally flawed because of a basic statistical truth: The Law of Large Numbers. Think Las Vegas The mainstream media makes you think we fire a Patriot and bring down an enemy missile. The Pentagon doesn’t help if they say they fired twelve Standard 3 missiles and brought down twelve Iranian ballistic missiles on October 1. Of course they want their weapons to look good. I don’t believe it for one simple reason. They had to target specific threats, and they wouldn’t have targeted incoming threats one-to-one. They would have targeted threats two or three-to-one. That’s standard doctrine. That means at best they would have shot down six. At best. When you think about air defense, think about odds. It’s a game, like Las Vegas. Like the horse races. It’s gambling. How hard is shooting down an incoming missile? Our Anti-Ballistic Missiles (ABMs) are hit-to-kill. The experts say it’s like shooting a bullet with another bullet where both objects are racing toward each other at between six and ten miles per second closure. Can we do that every time? Really? Let’s take the Navy’s Standard 3 ABM as an example. It’s the same argument for all the other missiles. Suppose you do one thousand tests where you fire an incoming missile and one Standard 3. Suppose out of all those tests (just suppose) you score 800 hits. That means you have an estimated hit probability of 800/1000 = 0.8 or 80%. That’s pretty good, and that’s what standard doctrine assumes. Now that means if you have an incoming Iranian missile and you fire one Standard 3, you only have a 0.8 or 80% chance of hitting it. An 80% chance of killing it. We say the hit probability is 0.80 and the kill probability is 0.80 If that incoming Iranian missile is carrying an A-Bomb, that leaves a 1.0 – 0.8= 0.20 or 20% chance you will miss. That means there is a 20% chance an A-Bomb is going to blow up Tel Aviv. That’s not good enough. You need to fire more than one Standard 3 interceptor missile. This is Las Vegas, remember? You can never be 100% sure you’ll stop that incoming missile. Would you be happy with 95%? As it happens, there is some simple math that tells the Navy how many to fire to have a 95% chance of stopping that incoming missile. There are more realistic and complicated models, but I’m going to give you the simple Las Vegas one, because it’s actually pretty good at explaining air defense doctrine, and showing the weaknesses of air defense doctrine. Don’t be put off by the math, because I’ll do the math for you. Just try to understand the ideas. Let’s call N the number of missiles we have to fire to achieve a desired Probability(Kill). Put another way, first we decide how sure we want to be of killing the threat and then we work out how many missiles we need to fire. Then: N = —————————– So, if we want to be 95% sure of killing the missile, Probability(Kill) is 0.95 And we are assuming a chance of one Standard 3 hitting the target is 0.80 Let me do the math for you. This means the number of Standard 3s we have to fire at a single threat is: N = ————- = 1.86135 which we can round up to 2 Standard 3 missiles That’s where the 2 air defense missiles for one threat comes from. Is 95% kill probability good enough? Maybe we want to be more sure. For 99% kill probability, we use: N = ————- = 2.86135 which we can round up to 3 missiles That’s where standard air defense doctrine comes from. If you want a probability of kill between 95% and 99%, you fire two to three ABMs for each incoming threat. Notice that there is always a chance all three interceptors you fire will miss. If you don’t understand the ideas in this article, and somebody tells you the rule of thumb is “Fire two or three for every threat,” that is what you will do. If all you want to know is where standard doctrine comes from, read no further. But, as we’ve hinted, it is not that simple. If you want to know where standard doctrine goes wrong and why we are missing so many targets, read on. Why are we missing so much? The issue of Realized Hit Probability Remember our first experiment. We fire an incoming missile 1,000 times and fire an interceptor missile 1,000 times. We score 800 hits, so our probability of a hit is 800/1000 = 0.80 or 80%. With 1,000 trials, that’s a reasonably good estimate. Each of these Standard 3 missiles costs about $20 million. That’s a lot of cash. What Raytheon does is conduct one test, then another, then another. They conduct maybe half a dozen tests. Let’s be generous and say they do two dozen. That’s nowhere near 1,000. In statistics, there’s a principle called The Law of Large Numbers. In simple terms, it says that for your guess of the hit probability to be close to the real hit probability, you have to do lots and lots of tests. A thousand isn’t too many. Ten thousand is not too many. One thing is for sure. Two dozen is too little. What does that mean? It means your Estimated Probability [Hit] is not going to be very close to the Realized (True) Probability[Hit]. It means your 0.80 is going to be either too low, or too high. You don’t know. If there’s an A-Bomb coming, you don’t want it to be too high, but it probably is. From what we’ve observed in Iran missile attack overwhelms Israeli air defense, the Realized Probability [Hit] is more like 0.25. If that is true, how many missiles should we fire to take down each incoming threat? Let’s apply the Las Vegas model again with Probability[Hit] = 0.25 or 25% and a desired Probability[Kill] = 0.99 or 99%. N = ————- = 16.0078 which we can round up to 16 Standard 3 missiles That’s what it’s going to take to knock down one incoming threat. How much will it cost? $20 million x 16 missiles = $320 million. That’s a lot of “How are ya’s.” The more complicated model doesn’t make the results look any better. That’s why I say our simple Las Vegas model is good enough to understand the issues. If the incoming missile is carrying an A-Bomb, it’s worth it. If it’s carrying two thousand pounds of RDX, certainly not. But put yourself in the place of our officers. Our air defense was never meant to stop 300 or 1,000 Iranian ballistic missiles coming in at the same time. It was meant to stop a handful of Soviet ballistic missiles. Our officers, faced with a swarm attack, have no choice but to fall back on doctrine. They simply don’t have enough. It gets worse: The Problem of Decoys Suppose we assume our Estimated Probability[Hit] of 80% is close to what we would get if we did the 1,000 trial experiment. In other words, suppose it’s pretty good (which we know it is not). There’s another way for the enemy to reduce the Realized Probability[Hit] to 0.25 and it is not hard. In a previous article, Defeating Anti-Ballistic Missile Defenses we discussed how easy it is for the enemy to deploy decoys to throw off our ABM interceptors. They can either separate the warhead from the booster stage and slice the booster into pieces, or they can deploy balloons. In space, where there is no air resistance, balloons will fly along with the warhead at the same speed and on the same trajectory. The balloons can be of any size or reflectivity we desire. Let’s say we deploy three decoys. They appear to the seeker of the ABM’s kill vehicle exactly the same as the warhead. That means that when we fire one missile at one threat, and that threat deploys three decoys, our missile will suddenly be confronted with four threats, of which only one is real. It won’t know which is the real threat. Figure 4 shows what our ABM will see: Our interceptor can’t tell the warhead from the decoys. There is AI we can program to help, but there is a way to defeat every AI rule. For example, balloons can be made as reflective as we like or as large or small as we like compared to the size of the warhead. The chances of our interceptor hitting the correct threat are immediately reduced to one-in-four, or 0.25 or 25%. And it can get even worse. The last example assumed our Estimated Probability[Hit] of 0.80 was close to true. If, in fact, Realized Probability [Hit] is 0.25 and the enemy deploys three decoys as above, the Realized Probability [Hit With Decoys] = 0.25 * 0.25 = 0.0625 or 6.25%. How many missiles do we need to have a 99% chance of bringing down a threat that deploys 3 decoys? Let’s see what the Las Vegas model tells us: N = —————– = 71.3554 which we can round to 71 missiles In other words, forget about it. An Arleigh Burke has only 90 VLS cells. A THAAD battery has 48 missiles. If the Iranians fire 300 or 1,000 missiles, each deploying three decoys, it’s all over. If they slip half a dozen A-Bombs in with the conventional warheads, we face a catastrophe. Evidence for this is all around us because the wars in Ukraine and Israel are providing live experiments. Of course, the hit data collected by the military is classified, but we can follow the evidence of our eyes. We’ve seen video of Iranian missiles raining on Israeli targets – unopposed. Every day, we are treated to Ukraine’s requests for more air defense systems and missiles. They are obviously running out. Russia has been firing two thousand missiles a year at Ukraine, and they are getting through. Today, Ukraine has almost no air defense left. Thinking about these issues from first principles is rather sobering. Our ballistic missile defense was never designed to defend against enemy missile swarms. We need to be very selective about which threats we engage. For example, Iron Dome is useless against ballistic missiles. We should hold back Iron Dome missiles for use against drones. Patriot PAC-3 will not be able to cope with modern IRBMs, especially if they deploy decoys. Patriot PAC-3, however, is extremely effective against aircraft and drones. We should hold back Patriot PAC-3 interceptors to deal with cruise missiles, which are the Iranians’ most accurate weapons. Their cruise missiles are the biggest threat to our high-value targets. There isn’t much to say about their ballistic missiles. Russian ballistic missiles are highly accurate, Iranian missiles less so. We can do the best we can with THAAD, Arrow 3, and Standard 3. We have to be as selective as possible, and those decisions have to be left to our commanders in the field. They will be the ones with the best situational awareness. Interested readers with a bit of Excel skill can put the Las Vegas model on a spreadsheet and play with it. This is not just theory. It is the practical reason our air defenses are struggling with enemy missile attacks. This essay first appeared in SOFREP About the Author You can reach the author at: [email protected]	{"url":"https://warriormaven.com/sea/navy-develops-mathematical-formula-to-intercept-enemy-missile-attacks-at-sea","timestamp":"2024-11-09T04:43:22Z","content_type":"text/html","content_length":"87884","record_id":"<urn:uuid:33c69c84-fc3f-4f2a-80be-2dbd60cfa591>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00263.warc.gz"}
How do I find the percentage of two numbers without a calculator? If you need to find a percentage of a number, here’s what you do – for example, to find 35% of 240: Divide the number by 10 to find 10%. In this case, 10% is 24. Multiply this number by how many tens are in the percentage you’re looking for – in this case, that’s 3, so you work out 30% to be 24 x 3 = 72. How do you find a percentage on a calculator? In order to find the percentage of the marks, we divide the total scores with marks obtained in the examination and then multiply the result with 100. Example: Suppose if 1156 is the total score obtained by you in the examination out of 1200 marks, then you divide 1156 by 1200, and then multiply it by 100. How do you find 75%? How much is 15 percent off? 1. Divide your original number by 20 (halve it then divide by 10). 2. Multiply this new number by 3. 3. Subtract the number from step 2 off of your original number. 4. You’ve just found your percentage off! How do I calculate a percentage between two numbers? First: work out the difference (increase) between the two numbers you are comparing. Then: divide the increase by the original number and multiply the answer by 100. What is the formula for calculating percentage? To calculate the percentage, multiply this fraction by 100 and add a percent sign. 100 * numerator / denominator = percentage . In our example it’s 100 * 2/5 = 100 * 0.4 = 40 . Forty percent of the group are girls. How do you find a percentage using a calculator? How to Calculate Percentages with a Calculator 1. If your calculator has a “%” button. Let’s say you wanted to find 19 percent of 20. Press these buttons: 1 9 % * 2 0 = 2. If your calculator does not have a “%” button. Step 1: Remove the percent sign and add a couple of zeros after the decimal point. 19% becomes 19.00. How do you find the average percentage of 6 semesters? Answer. Take total of all marks ontained in all semesters and divide it by overall total marks of semesters to arrive at aggregate percentage. To arrive at aggregate marks simply in each semester simply add total marks in all semesters and divided by tital semester. How do I figure out area? The simplest (and most commonly used) area calculations are for squares and rectangles. To find the area of a rectangle, multiply its height by its width. For a square you only need to find the length of one of the sides (as each side is the same length) and then multiply this by itself to find the area. What number is 3 percent of 100? How do you take 20% off in Excel? If you want to calculate a percentage of a number in Excel, simply multiply the percentage value by the number that you want the percentage of. For example, if you want to calculate 20% of 500, multiply 20% by 500. – which gives the result 100. Note that the % operator tells Excel to divide the preceding number by 100. What jobs use excel the most? You may change your mind when you see this list of careers that require it. • Administrative Assistant. • Accountants. • Retail Manager. • Cost Estimator. • Financial Analyst. • Project Manager. • Business Analyst. • Data Journalist. How do I find out a percentage? 1. How to calculate percentage of a number. Use the percentage formula: P% * X = Y 1. Convert the problem to an equation using the percentage formula: P% * X = Y. 2. P is 10%, X is 150, so the equation is 10% * 150 = Y. 3. Convert 10% to a decimal by removing the percent sign and dividing by 100: 10/100 = 0.10. What is basic Excel skills? Write a formula with absolute and relative references. Create a drop down list of options in a cell, for easier data entry. Sort a list of text and/or numbers without messing up the data. Create a worksheet formula to look up a specific value in a table. Record and modify a simple Excel macro and use it to save time. How many Excel formulas are there? 475 formulas How many days it will take to learn Excel? So, it all depends on you. If you practice every day and dedicate around 2-3 hours every day to learn the concepts, then you can learn it within four weeks. But, to master the concepts in Excel, you need to use the tricks and formulas on a daily basis. Learning excel just 5 days but analysis take more time….. How do I find the difference between two numbers in Excel? Calculate the difference between two numbers by inputting a formula in a new, blank cell. If A1 and B1 are both numeric values, you can use the “=A1-B1” formula. Your cells don’t have to be in the same order as your formula. For example, you can also use the “=B1-A1” formula to calculate a different value. How do you find 25%? If you have to turn a percentage into a decimal, just divide by 100. For example, 25% = 25/100 = 0.25. To change a decimal into a percentage, multiply by 100. So 0.3 = 0.3 × 100 =30% . What Excel skills to employers value the most? Top 7 Excel Skills Employers Are Looking for (And How to Master Them While at Home) 1. VLOOKUP. Vlookup, the king of lookup data retrieval, is one of the most popular functions in Excel. 2. PivotTables. 3. BASIC MACROS. 4. IF Function. 5. Data Validation. 6. Graph/Charts. 7. Proper formatting of data. How do I do an Excel formula with a percentage? Format values as percentages. To show a number as a percent in Excel, you need to apply the Percentage format to the cells. Simply select the cells to format, and then click the Percent Style (%) button in the Number group on the ribbon’s Home tab. You can then increase (or decrease) the the decimical place as needed. What is a formula of percentage? Formula to Calculate Percentage The Percentage Formula is given as, Percentage = (Value ⁄ Total Value) × 100. How do you find 3%? Example 2. Find 3% of $4,000. First write it as 0.03 × $4,000. Then multiply 3 × $4,000 = $12,000. Lastly put the decimal point where it gives the answer two decimal digits: $120.00.	{"url":"https://indielullabies.com/how-do-i-find-the-percentage-of-two-numbers-without-a-calculator/","timestamp":"2024-11-12T09:00:25Z","content_type":"text/html","content_length":"125800","record_id":"<urn:uuid:5b6f759f-a9c8-4faa-8062-85e87f962820>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00368.warc.gz"}
Beta and Dirichlet distributions – The Dan MacKinlay stable of variably-well-consider’d enterprises Beta and Dirichlet distributions October 14, 2019 — April 4, 2022 Lévy processes stochastic processes time series \[\renewcommand{\var}{\operatorname{Var}} \renewcommand{\corr}{\operatorname{Corr}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\bb}[1]{\mathbb{#1}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \ renewcommand{\rv}[1]{\mathsf{#1}} \renewcommand{\vrv}[1]{\vv{\rv{#1}}} \renewcommand{\disteq}{\stackrel{d}{=}} \renewcommand{\gvn}{\mid} \renewcommand{\Ex}{\mathbb{E}} \renewcommand{\Pr}{\mathbb{P}} Suppose the joint pdf of $\rv{d}_{1}, \ldots, \rv{d}_{k-1}$ is \[\begin{aligned} f\left(y_{1}, \ldots, y_{k-1}\right) &=\frac{\alpha_{1}+\cdots+\alpha_{k}}{\Gamma\left(\alpha_{1}\right) \cdots \ Gamma\left(\alpha_{k}\right)} y_{1}^{\alpha_{1}-1} \cdots y_{k-1}^{\alpha_{k-1}-1}\left(1-y_{1}-\cdots-y_{k-1}\right)^{\alpha_{k}-1},\\ &=\frac{\Gamma(\alpha)}{\prod_{i=1}^k\Gamma(\alpha_i)}\prod_{i= 1}^k y_i^{\alpha_i-1} \end{aligned}\] where $y_{i}>0, y_{1}+\cdots+y_{k-1}<1, i=1, \ldots, k-1$ and $\alpha=\sum_i\alpha_i$. Then the random variables $\rv{d}_{1}, \ldots, \rv{d}_{k-1}$ follow the Dirichlet distribution with parameters $\alpha_{1}, \ldots, \alpha_{k}$. Usually, I write this as a vector random variate, with vector parameters, rather than a long list, \[\vrv{d}\sim\ The Beta distribution is a special case of the Dirichlet distribution with parameters $\vv{\alpha}=[\alpha_1,\alpha_2]$, i.e. the bivariate case. There is more information in Wikipedia, although these pages are IMO unusually uninspired and confusing. My prose is terrible because I rarely have time to revisit it. What is Wikipedia’s excuse? 1 A Beta RV is a ratio of Gamma RVS 2 A Dirichlet RV is a normalized sum of independent Gamma RVS 3 Beta as exponential family Beta distribution: $Y \sim \operatorname{Beta}(\alpha, \beta)$ \[ \begin{aligned} f_{Y}(y \mid \alpha, \beta)=& \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha) \Gamma(\beta)} y^{\alpha-1}(1-y)^{\beta-1} \\ =& {[y(1-y)]^{-1} \exp (\alpha \log (y)+\beta \log (1-y)} \\ +&\log \Gamma(\alpha+\beta)-\log \Gamma(\alpha)-\log \Gamma(\beta)) \end{aligned} \] with \[ \begin{aligned} \eta(\alpha, \beta) &=(\ alpha, \beta)^{\top} \\ T(y) &=(\log (y), \log (1-y))^{\top}. \end{aligned} \] 4 Dirichlet as exponential family The Dirichlet distribution is an exponential family and can be written in canonical form as \[ \operatorname{Dirichlet}(\boldsymbol{\theta} \mid \boldsymbol{\alpha})=f(\boldsymbol{\theta}) g(\ boldsymbol{\alpha}) e^{\phi(\boldsymbol{\alpha})^{T} u(\boldsymbol{\theta})} \] with \[ f(\boldsymbol{\theta})=1, g(\boldsymbol{\alpha})=1 / B(\boldsymbol{\alpha}) \] where \[ B(\boldsymbol{\alpha})= \prod_{t=1}^{D} \Gamma\left(\alpha_{t}\right) / \Gamma\left(\sum_{t=1}^{D} \alpha_{t}\right), \phi(\boldsymbol{\alpha})=\left(\begin{array}{c} \alpha_{1}-1 \\ \vdots \\ \alpha_{D}-1 \end{array}\ right) \] and \[ u(\boldsymbol{\theta})=\left(\begin{array}{c} \ln \theta_{1} \\ \vdots \\ \ln \theta_{D} \end{array}\right) \] 5 Conjugate prior for Dirichlet RVs Lefkimmiatis, Maragos, and Papandreou (2009) argue: Since for any member of the exponential family there exists a conjugate prior that can be written in the form \[ p(\boldsymbol{\alpha} \mid \mathbf{v}, \eta) \propto g(\boldsymbol{\alpha})^{\eta} e^{\phi(\boldsymbol{\alpha})^{T} \mathbf{v}} \] a suitable conjugate prior distribution for the parameters $\boldsymbol{\alpha}$ of the Dirichlet is \[ p(\boldsymbol{\alpha} \mid \mathbf{v}, \ eta) \propto \frac{1}{B(\boldsymbol{\alpha})^{\eta}} e^{-\sum_{t=1}^{D} v_{t} \alpha_{t}} \] Wikipedia claims that there is no efficient means for sampling from this, which is sad for MCMC. Generally this does not bother people because we rarely observe Dirichlet RVs directly; they are usually, e.g. a mixing probability for some other distribution. 6 Non-conjugate priors Anything that can be transformed to be an elementwise positive vector, presumably. multivariate gamma seems natural.	{"url":"https://danmackinlay.name/notebook/beta_dirichlet_distribution","timestamp":"2024-11-05T00:53:26Z","content_type":"application/xhtml+xml","content_length":"38218","record_id":"<urn:uuid:9ad56894-2078-4b6a-82bf-270052ef5e44>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00844.warc.gz"}
Review of X-443 The X-443 is a large sailboat designed by the maritime architect Niels Jeppesen in the mid 2010. The X-443 is built by the Danish yard X-Yachts A/S Here we would have liked to show you nice photos of the X-443. If you have a photo you would like to share: Upload Image Looking for a new boat? Find a X-443 or similar boat for sale The headroom is above average The boat equipped with a fractional rig. A fractional rig has smaller headsails which make tacking easier, which is an advantage for cruisers and racers, of course. The downside is that having the wind from behind often requires a genaker or a spinnaker for optimal speed. Sailing characteristics This section covers widely used rules of thumb to describe the sailing characteristics. Please note that even though the calculations are correct, the interpretation of the results might not be valid for extreme boats. Stability and Safety What is Capsize Screening Formula (CSF)? The capsize screening value for X-443 is 1.93, indicating that this boat could - if evaluated by this formula alone - be accepted to participate in ocean races. Theoretical Maximum Hull Speed What is Theoretical Maximum Hull Speed? The theoretical maximal speed of a displacement boat of this length is 8.2 knots. The term "Theoretical Maximum Hull Speed" is widely used even though a boat can sail faster. The term shall be interpreted as above the theoretical speed a great additional power is necessary for a small gain in speed. Immersion rate The immersion rate is defined as the weight required to sink the boat a certain level. The immersion rate for X-443 is about 299 kg/cm, alternatively 1676 lbs/inch. Meaning: if you load 299 kg cargo on the boat then it will sink 1 cm. Alternatively, if you load 1676 lbs cargo on the boat it will sink 1 inch. Sailing statistics This section is statistical comparison with similar boats of the same category. The basis of the following statistical computations is our unique database with more than 26,000 different boat types and 350,000 data points. Motion Comfort Ratio What is Motion Comfort Ratio (MCR)? The Motion Comfort Ratio for X-443 is 25.8. Comparing this ratio with similar sailboats show that it is more comfortable than 31% of all similar sailboat designs. This comfort value is just below average. L/B (Length Beam Ratio) What is L/B (Length Beam Ratio)? The l/b ratio for X-443 is 3.16. Compared with other similar sailboats it is more spacy than 66% of all other designs. It seems that the designer has chosen a slightly more spacy hull design. D/L (Displacement Length Ratio) What is Displacement Length Ratio? The DL-ratio for X-443 is 170 which categorizes this boat among 'light racers'. 82% of all similar sailboat designs are categorized as heavier. A light displacement requires less sailarea and has higher accellerations. SA/D (Sail Area Displacement ratio) SA/D (Sail Area Displacement ratio) Indicates how fast the boat is in light wind: - Cruising Boats have ratios 10-15 - Cruiser-Racers have ratios 16-20 - Racers have ratios above 20 - High-Performance Racers have ratios above 24 Sail-area/displacement ratio (SA/D ratio): 23.03 If you need to renew parts of your running rig and is not quite sure of the dimensions, you may find the estimates computed below useful. Guiding dimensions of running rig Usage Length Diameter Jib sheet 12.5 m (41.0 feet) 14 mm (0.55 inch) Genoa sheet 12.5 m (41.0 feet) 14 mm (0.55 inch) Mainsheet 31.2 m (102.5 feet) 14 mm (0.55 inch) Spinnaker sheet 27.5 m (90.2 feet) 14 mm (0.55 inch) Boat owner's ideas This section shown boat owner's changes, improvements, etc. Here you might find inspiration for your boat. Do you have changes/improvements you would like to share? Upload a photo and describe what to look for. We are always looking for new photos. If you can contribute with photos for X-443 it would be a great help. If you have any comments to the review, improvement suggestions, or the like, feel free to contact us. Criticism helps us to improve.	{"url":"https://www.udkik.dk/da/review.jsp?id=X-443","timestamp":"2024-11-15T02:49:36Z","content_type":"text/html","content_length":"24052","record_id":"<urn:uuid:f529fadb-cbbd-4300-9645-c235f4db9134>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00641.warc.gz"}
© 2005,2010,2012 John Abbott, Anna M. Bigatti GNU Free Documentation License, Version 1.2 CoCoALib Documentation Index User documentation The primary use for a variable of type ring is simply to specify the ring to which RingElem variables are associated. CoCoALib requires that the user specify first the rings in which to compute, then values in those rings can be created and manipulated. We believe that this explicit approach avoids any possible problem of ambiguity. The file ring.H introduces several classes used for representing rings and their elements. A normal user of the CoCoA library will use principally the classes ring and RingElem: an object of type ring represents a mathematical ring with unity, and objects of type RingElem represent values from some ring. To make the documentation more manageable it has been split into two: this file describes operations directly applicable to rings, whereas a separate file describes the operations on a RingElem. Documentation about the creation and use of homomorphisms between rings can be found in The documentation here is very general in nature: it applies to all rings which can be created in the CoCoA library. To find out how to create rings, and for more specific documentation about the various special types of ring CoCoALib offers, look at the relevant file: see the subsection below about Types of Ring (ring inheritance). While the CoCoA library was conceived primarily for computing with commutative rings, the possibility of creating and using certain non-commutative rings exists. The documentation for these rings is kept separately in RingWeyl. Here is a list of example programs (to be found in the examples/ subdirectory) illustrating the creation and use of various sorts of ring and their elements Types of ring (inheritance structure) The default initial value for a ring is the ring of integers (RingZZ). You can specify explicitly the initial value using one of the various ring pseudo-constructors: │ RingZZ() │ see RingZZ constructors │ │ RingQQ() │ see RingQQ constructors │ │ NewZZMod(n) │ see QuotientRing constructors │ │ NewRingTwinFloat(n) │ see RingTwinFloat constructors │ │ NewFractionField(R) │ see FractionField constructors │ │ NewQuotientRing(R,I) │ see QuotientRing constructors │ Operations on Rings Let R and R2 be two variable of type ring. • RingID(R) -- the identification of R (as a long) • characteristic(R) -- the characteristic of R (as a BigInt) • symbols(R) -- a std::vector of the symbols in R (e.g. Q(a)[x,y] contains the symbols a, x, and y) • R = R2 -- assign R2 to R (so they both refer to the same identical internal impl) • R == R2 -- test whether R and R2 are identical (i.e. they refer to the same internal impl) • R != R2 -- the logical negation of R == R2 │ zero(R) │ the zero element of R │ │ one(R) │ the one element of R │ │ BaseRing(R) │ the ring from which R was built │ In CoCoALib all rings are built starting from ZZ by applying various "constructors" (e.g. fraction field, quotient ring). The function BaseRing gives the immediate predecessor in the chain of In some cases the best algorithm to use may depend on whether the ring in which we are computing has certain properties or not; so CoCoALib offers some functions to ask a ring R about its properties: • IsCommutative(R) -- a boolean, true iff R is commutative • IsIntegralDomain(R) -- a boolean, true iff R has no zero divisors • IsIntegralDomain3(R) -- a 3-state boolean, like IsIntegralDomain but fast, gives uncertain3 if cannot determine proper answer quickly • IsTrueGCDDomain(R) -- a boolean, true iff R is a true GCD domain (note: fields are not true GCD domains) • IsOrderedDomain(R) -- a boolean, true iff R is arithmetically ordered • IsField(R) -- a boolean, true iff R is a field • IsFiniteField(R) -- a boolean, true iff R is a finite field • LogCardinality(R) -- the integer k such that card(R) = p^k where p is char(R); error if R is not a finite field. NOTE: a pragmatic approach is taken: e.g. IsOrderedDomain is true only if comparisons between elements can actually be performed using the CoCoA library. Queries and views It may also be important to discover practical structural details of a ring (e.g. some algorithms make sense only for a polynomial ring). The following query functions Is... tell you how the ring is implemented, and the view functions As... give access to the specific operations: In general the function "IsXYZ" should be read as "Is internally implemented as XYZ": for instance IsQuotientRing is true only if the internal implementation is as a quotient ring, so if ZZ denotes the ring of integers and R = ZZ[x]/ideal(x) then R and ZZ are obviously isomorphic but IsZZ(R) gives false and IsZZ(Z) gives true, while conversely IsQuotientRing(R) gives true and IsQuotientRing(ZZ) gives false. The rest of this section is for more advanced use of rings (e.g. by CoCoA library contributors). If you are new to CoCoA, you need not read this subsection. Writing C++ classes for new types of ring An important convention of the CoCoA library is that the class RingBase is to be used as an abstract base class for all rings. You are strongly urged to familiarize yourself with the maintainer documentation if you want to understand how and why rings are implemented as they are in the CoCoA library. The first decision to make when implementating a new ring class for CoCoALib is where to place it in the ring inheritance structure. This inheritance structure is a (currently) tree with all concrete classes at the leaves, and all abstract classes being internal nodes. Usually the new concrete ring class is attached to the structure by making it derive from one of the existing abstract ring classes. You may even decide that it is appropriate to add a new abstract ring class to this structure, and to make the new concrete class derive from this new abstract class. Note: I have note used multiple inheritance in the structure, largely because I do not trust multiple inheritance (not doubt due in part to my ignorance of the topic). Once you have decided where to attach the new concrete class to the structure, you will have to make sure that all pure virtual functions in the abstract class are implemented. Almost all instances of concrete rings are built through pseudo-constructors (the rings ZZ and QQ are exceptional cases). An important detail of the constructor for a new concrete ring is that the reference count of the new ring object must be incremented to 1 at the start of the constructor body (or more precisely, before any self references are created, e.g. when creating the zero and one elements of the ring); without this "trick" the constructor is not exception safe. NOTE Every concrete ring creates a copy of its zero and one elements (kept in auto_ptrs myZeroPtr and myOnePtr). This common implementation detail cannot (safely) be moved up into RingBase because during destruction by default the data members of RingBase are destroyed after the derived concrete ring. It seems much safer simply to duplicate the code for each ring implementation class. Maintainer documentation (NB consider consulting also QuotientRing, FractionField and PolyRing) The design underlying rings and their elements is more complex than I would have liked, but it is not as complex as the source code may make it appear. The guiding principles are that the implementation should be flexible and easy/pleasant to use while offering a good degree of safety; extreme speed of execution was not a goal (as it is usually contrary to good flexibility) though an interface offering slightly better run-time efficiency remains. Regarding flexibility: in CoCoALib we want to handle polynomials whose coefficients reside in (almost) any commutative ring. Furthermore, the actual rings to be used will be decided at run-time, and cannot restricted to a given finite set. We have chosen to use C++ inheritance to achieve the implementation: the abstract class RingBase defines the interface that every concrete ring class must Regarding ease of use: since C++ allows the common arithmetic operators to be overloaded, it is essential that these work as expected for elements of arbitrary rings -- with the caveat that / means exact division, as this is the only reasonable interpretation. Due to problems of ambiguity, arithmetic between elements of different rings is forbidden: e.g. let f be in QQ[x,y] and g in ZZ[y,x], where should f+g reside? The classes in the file ring.H are closely interrelated, and there is no obvious starting point for describing them -- you may find that you need to read the following more than once to comprehend it. Here is a list of the classes: │ ring │ value represents a ring; it is a smart pointer │ │ RingBase │ abstract class defining what a ring is │ │ RingElem │ value represents an element of a ring │ │ ConstRefRingElem │ const-reference to a RingElem │ │ RingElemConstRawPtr │ raw pointer to a const ring value │ │ RingElemRawPtr │ raw pointer to a ring value │ The class RingBase is of interest primarily to those wanting to implement new types of ring (see relevant section below); otherwise you probably don't need to know about it. Note that RingBase includes an intrusive reference counter -- so every concrete ring instance will have one. RingBase also includes a machine integer field containing a unique numerical ID -- this is so that distinct copies of otherwise identical rings can be distinguished when output (e.g. in OpenMath). The class ring is used to represent mathematical rings (e.g. possible values include ZZ, QQ, or QQ[x,y,z]). An object of type ring is just a reference counting smart pointer to a concrete ring implementation object -- so copying a ring is fairly cheap. In particular two rings are considered equal if and only if they refer to the same identical concrete ring implementation object. In other files you will find classes derived from ring which represent special subclasses of rings, for instance PolyRing is used to represent polynomial rings. The intrusive reference count, which must be present in every concrete ring implementation object, is defined as a data member of RingBase. For the other classes see RingElem. NOTE an earlier implemetation of rings memorized some RingHom values in data members of the ring object; this caused problems with circular reference counts, so was eliminated. Further comments about implementation aspects of the above classes. Recall that ring is essentially a smart pointer to a RingBase object, i.e. a concrete implementation of a ring. Access to the implementation is given via operator->. If necessary, the pointer may also be read using the member function myRingPtr: this is helpful for defining functions such as IsPolyRing where access to the pointer is required but operator-> cannot be used. The class RingBase declares a number of pure virtual functions for computing with ring elements. Since these functions are pure they must all be fully defined in any instantiable ring class (e.g. RingZZImpl or RingFpImpl). These member functions follow certain conventions: most arithmetic functions return no value, instead the result is placed in one of the arguments (normally the first argument is the one in which the result is placed), but functions which return particularly simple values (e.g. booleans or machine integers) do indeed return the values by the usual function return mechanism. ring element values are passed as raw pointers (i.e. a wrapped void* pointing to the actual value). A read-only arg is of type RingElemConstRawPtr, while a writable arg is of type RingElemRawPtr. When there are writable args they normally appear first. For brevity there are typedefs ConstRawPtr and RawPtr in the scope of RingBase or any derived class. sanity checks on the arguments are not conducted (e.g. the division function assumes the divisor is non-zero). These member functions are supposed to be fast rather than safe. However, if the compilation flag CoCoA_DEBUG was set then some checks may be performed. In a few cases there are non-pure virtual member functions in RingBase. They exist either because there is a simple universal definition or merely to avoid having to define inappropriate member functions (e.g. gcd functions when the ring cannot be a gcd domain). Here is a list of them: │ IamTrueGCDDomain() │ defaults to not IamField() │ │ IamOrderedDomain() │ defaults to false │ Bugs, Shortcomings and other ideas There is no description of what the various mem fns are supposed to do!! There is something incomplete in RingElem Printing rings is unsatisfactory. Need a mechanism for specifying a print name for a ring; and also a mechanism for printing out the full definition of the ring avoiding all/some print names. For instance, given the definitions R = QQ(x) and S = R[a,b] we see that S could be printed as S, R[a,b] or QQ(x)[a,b]. We should allow at least the first and the last of these possibilities. Should (some of) the query functions return bool3 values? What about properties which are hard to determine? The fn IsFiniteField is not very smart; it recognises only prime finite fields, and simple algebraic extensions of them.	{"url":"http://cocoa.altervista.org/cocoalib/doc/html/ring.html","timestamp":"2024-11-02T14:43:37Z","content_type":"text/html","content_length":"21761","record_id":"<urn:uuid:9818826f-a9d3-48c3-8f78-b5ee10c8de31>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00139.warc.gz"}
The forbidden tilings It is a surprising fact that 6 of the 21 possible vertex figures cannot be extended to any edge-to-edge regular polygon tiling. Since these "forbidden" vertex figures are the only ones that contain a pentagon (and a heptagon, nonagon and other larger polygons) this immediately implies that no edge-to-edge regular polygon plane tiling can contain any of these polygon types. The vertex figures from the three families analysed before this point have prototiles chosen from a set of only 5 polygon types: the triangle, square, hexagon, octagon and dodecagon and so this result implies that only these 5 prototiles can form any edge-to-edge tiling of regular polygons. Moreover, since the octagon occurs in only one legal vertex type, it can only occur in the unique 4.8.8 uniform tiling. So the result is actually even more restrictive and can be described in this remarkable theorem: Prototile theorem With the exception of the unique 4.8.8 uniform tiling, any edge-to-edge plane tiling of regular polygons must be constructed from equilateral triangle, square, regular hexagon and regular dodecagon This (to me at least) unexpected result is not difficult to prove using the Gap theorem mentioned before and so I will do so below. The forbidden vertex types Proof of the prototile theorem Let us start by showing that the 6 vertex figures above cannot occur in any edge-to-edge regular polygon tiling. There are really only two cases to consider. Case 1 is the vertex types 3.7.42, 3.8.24, 3.9.18 and 3.10.15 as the same argument works for all four types. Case 2 is the vertex types 4.5.20 and 5.5.10 as the same argument works for both types. Let us start by using the vertex type 3.10.15 as an example for Case 1 and assume for contradiction that it can be extended to a tiling of the plane by regular polygons. Start with the three polygons intersecting at the white vertex and ignore the purple 15-gon for now. If we consider the decagon and triangle intersecting at the red vertex, then the angle sum of these two polygons is (8π/10+π/3 = 34π/30). The gap angle is then 2π - 34π/30 = 26π/30. This is less than or equal to π and so the Gap theorem applies. The gap ratio g is 26/30 and m = 2/(1-g) = 15. By the Gap theorem, the only solution is a 15 sided polygon. But if we insert this polygon (shown in purple in the diagram), it overlaps the other 15-gon, contradicting our assumption that a tiling is possible. Exactly the same argument applies to the vertex types 3.7.42, 3.8.24, and 3.9.18 and shows that none of these types can be extended to a tiling by regular polygons. In essence, there is not enough room to fit the required large polygons. Let us start by using the vertex type 5.5.10 as an example for Case 2 and assume for contradiction that it can be extended to a tiling of the plane by regular polygons. Start with the three polygons intersecting at the white vertex and ignore the purple polygons for now. We will now add the purple polygons one by one, each the only option allowed by the Gap theorem and then as before derive a contradiction caused by an overlap. Consider the two pentagons intersecting at the lower green vertex. The angle sum for these two pentagons is 3π/5 + 3π/5 = 6π/5. The gap angle is 2π-6π/5 = 4π/5. This is less than or equal to π so the Gap theorem applies. The gap ratio is g = 4/5 and m = 2/ (1-(4/5)) = 10. By the Gap theorem, the only possible solution is a decagon. Add this as decagon a. Now consider decagon a and the green pentagon intersecting at the higher green vertex. The angle sum for these two polygons is 8π/10 + 3π/5 = 14π/10. The gap angle is 2π - 14π/10 = 6π/10. This is less than or equal to π so the Gap theorem applies. The gap ratio is 6/10 and m = 2/(1-(6/10)) = 5. By the Gap theorem, the only solution is a pentagon. Add this as pentagon b. Now consider pentagon b and the green pentagon intersecting at the red vertex. As before, the angle sum for these two pentagons is 3π/5 + 3π/5 = 6π/5. The gap angle is 2π-6π/5 = 4π/5. This is less than or equal to π so the Gap theorem applies. The gap ratio is g = 4/5 and m = 2/ (1-(4/5)) = 10. By the Gap theorem, the only possible solution is a decagon. Add this as decagon c. We can now derive our contradiction because decagon c intersects the original blue decagon in the vertex figure. This contradicts our original assumption that it is possible to extend the 5.5.10 vertex figure to a tiling of regular polygons. The same argument applies to the vertex figure 4.5.20 except that it involves a square and a pentagon instead of two pentagons and results in an even larger overlap between 2 20-gons. Having eliminated the 6 vertex figures in this family, we are almost finished proving the Prototile theorem. All that is left to prove the uniqueness of the 4.8.8 tiling. There are no other legal vertex types involving octagons, so it is clear that every octagon vertex in an edge-to-edge regular polygon tiling must have the 4.8.8 type, but why do all the square vertices in a tiling with octagons also have to have this type? See the 4.8.8 figure located around the white vertex in the illustration to the left. If we consider the two green vertices in the illustration, they also include octagons and so we are forced to add a second purple octagon for each green vertex in order to continue the tiling. But this completes the fourth red vertex as well, showing that octagons must be attached to every square vertex. The other 14 legal vertex types include only triangle, square, hexagon and dodecagon prototiles and so this completes the proof. This result is still more restrictive than it seems, because as you can see from the illustration to the right, a regular hexagon can be decomposed into triangles and a regular dodecagon into triangles and squares. Thus we have this central result: Catalog theorem To construct a catalog of all edge-to-edge regular polygon tilings, it is necessary only to start with the 4.8.8 uniform tiling and add all possible tilings with equilateral triangle and square The remaining tilings can be constructed by replacing some of the hexagon shaped patches of triangles with hexagons, and replacing some of the dodecagon shaped patches of triangles and squares with Brian Galebach catalogs hundreds of regular polygon tilings on his interesting website on n-uniform tilings. Many of these are quite beautiful. Nevertheless, it seems aesthetically disappointing that the range of polygons and patterns possible for edge-to-edge regular polygon tilings in the plane is so limited. It is not surprising that both mathematicians and artists have explored other options. As we will see in the next part of this site, adding additional prototiles introduces an enormous number of new and beautiful tiling possibilities. With a small change to the rules, it turns out that the forbidden tilings are not forbidden after all ...	{"url":"http://gruze.org/tilings/forbidden","timestamp":"2024-11-09T03:25:53Z","content_type":"application/xhtml+xml","content_length":"24057","record_id":"<urn:uuid:90560a53-b342-47f2-b0b3-8ff78a0482d1>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00589.warc.gz"}
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JavaScript Fundamentals: Math Object Today is the 4th day of my #100DaysOfCode journey with JavaScript. I write about my learnings in an explained way through my blogs and socials. If you want to join me on the learning journey, make sure to follow my blogs and social and share yours too. Let’s learn This Article is a part of the JavaScript Fundamentals series. Today, I learned about Math.random, Math.floor Functions and to call a function within our function. In JavaScript, there are many math utilities on the Math object. To get a random number we can call the Math.random function. const myRandomNumber = Math.random(); The above line will return some number between 0 and 1 (not including 1). Math.random can also be used to generate random numbers between the range. Example: Inside getRandom, get a random number from the Math.random() function. Then, return that number!👇🏼 function getRandom() { return Math.random(); A random number between 0 and 100 could be created by simply multiplying the output: // randomNumber will be between 0 and 100 const randomNumber = Math.random() * 100; We could multiply and then add to get a random number between 15 and 100: // randomNumber will be between 15 and 100 const randomNumber = (Math.random() * 85) + 15; Math.floor takes arguments. const two = Math.floor(2.2598223); Math.floor function will take 2.2598223 and return 2. A number will be rounded to the nearest integer using this function. For instance, if the input was 2.9999, the method would round it to 2. Example: Take the argument x and use Math.floor to turn it into an integer without the values after the decimal place. Once you have this floored value, return it! function getFloor(x) { return Math.floor(x); Ending with an extra bit of information about JavaScript functions&mldr; The JavaScript Math object allows us to perform mathematical tasks on numbers. There are various math object properties. Today I learned about Math.random, Math.floor Functions in JavaScript. If You ❤️ My Content! Connect Me on Twitter	{"url":"https://mranand.com/blogs/javascript-fundamentals-math-object/","timestamp":"2024-11-06T04:06:17Z","content_type":"text/html","content_length":"33570","record_id":"<urn:uuid:a910ade2-cb19-4384-b9e8-077ad92c3bb8>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00679.warc.gz"}
Divide into 2 congruent pieces • MHB • Thread starter I like Serena • Start date In summary, dividing into 2 congruent pieces means splitting an object, shape, or area into two equal parts or pieces that are identical in size, shape, and measurements. The purpose of this is to create equal, symmetrical parts for various purposes such as mathematical calculations, geometric constructions, or creating balanced designs. This is different from regular division as it focuses on creating equal parts rather than dividing numbers or quantities. Examples of this include cutting a pizza into two equal slices, folding a piece of paper in half, or splitting a rectangle into two identical triangles. The tools or methods used for this include rulers, compasses, protractors, scissors, or folding techniques, which help ensure precision and accuracy in creating congruent pieces. Last time I asked for 5 congruent pieces. This time I'm only asking for 2. \begin{tikzpicture}[ultra thick] \draw (0,0) -- (-5,0) -- (-5,5); \draw[rotate=-90] (0,0) -- (-5,0) -- (-5,5); \draw (-5,5) arc (135:45:{5sqrt(2)}); As before, that's a perfect solution maxkor. Thank you for your participation! For the record, this problem depends on rotational symmetry, while the https://mathhelpboards.com/challenge-questions-puzzles-28/divide-into-5-congruent-pieces-23597.html was about translational They are really a set. It also means that we can just as easily divide it into 5 congruent pieces: \begin{tikzpicture}[ultra thick] \draw (0,0) -- (-5,0) -- (-5,5); \draw[rotate=-18] (0,0) -- (-5,0) -- (-5,5); \draw[rotate=-36] (0,0) -- (-5,0) -- (-5,5); \draw[rotate=-54] (0,0) -- (-5,0) -- (-5,5); \draw[rotate=-72] (0,0) -- (-5,0) -- (-5,5); \draw[rotate=-90] (0,0) -- (-5,0) -- (-5,5); \draw (-5,5) arc (135:45:{5sqrt(2)}); I actually left a hint in the TikZ picture itself by using the rotate property, although I kind of doubt that anyone noticed. (Wink) FAQ: Divide into 2 congruent pieces 1. What does it mean to "divide into 2 congruent pieces"? Dividing into 2 congruent pieces means splitting an object, shape, or area into two equal parts or pieces that are identical in size, shape, and measurements. 2. What is the purpose of dividing into 2 congruent pieces? The purpose of dividing into 2 congruent pieces is to create equal, symmetrical parts for various purposes such as mathematical calculations, geometric constructions, or creating balanced designs. 3. How is dividing into 2 congruent pieces different from regular division? Dividing into 2 congruent pieces is different from regular division in that it focuses on creating equal parts that are identical in size and shape, rather than dividing numbers or quantities. 4. What are some examples of dividing into 2 congruent pieces? Examples of dividing into 2 congruent pieces include cutting a pizza into two equal slices, folding a piece of paper in half, or splitting a rectangle into two identical triangles. 5. What are the tools or methods used to divide into 2 congruent pieces? The tools or methods used to divide into 2 congruent pieces include rulers, compasses, protractors, scissors, or folding techniques. These tools help ensure precision and accuracy in creating congruent pieces.	{"url":"https://www.physicsforums.com/threads/divide-into-2-congruent-pieces.1039129/","timestamp":"2024-11-05T19:35:31Z","content_type":"text/html","content_length":"87974","record_id":"<urn:uuid:67fb9f2e-f5ed-41d5-8b6c-45f9a5bb9150>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00231.warc.gz"}
While the vpop function returns a VirtualPopulation ensemble of parameter values which correspond to relatively good fits to the data, in many cases this return is referred to as a "plausible patient population", i.e. a set of potentially good parameters which may or may not reflect the statistical effects of the population. For example, say that for the data series that is being fit, 50% of the population is known to be fast matabolizers and the other 50% is comprised of slow matabolizers (characterized by some parameter or measurement in the model). A virtual population method, vpop, will return a plausible population of N plausible patients but there is no guarentee that it captured the distribution of fast and slow metabolizers from the actual population. The purpose of a subsample algorithm is to downsample from N to M to find a subpopulation which captures important statistical quantities better than the plausible population. function subsample(alg::AbstractSubsamplingAlgorithm, vp::AbstractVirtualPopulation, trial::AbstractExperiment; kwargs) Subsamples a plausible patient population vp in order to create a virtual population which satisfies the population-level constraints as specified by the sampling algorithm alg defined on a given Keyword Arguments Any keyword arguments kwargs provided will be passed on as options to solving the optimization problem that is part of some subsampling methods, e.g. the Allen-Rieger-Musante (ARM) method. For more information about such arguments see the Optimization.jl documentation. MAPEL(; data::DataFrame, mechanistic_axes, nbins::Int, init_noise, maxiters=1000, optimizer = NelderMead()) The Mechanistic Axes Ensemble Population Linkage (MAPEL) algorithm. MAPEL assigns prevalence weights to each patient in a virtual population in order to match discretized (binned) measure derived from actual patient populations. The algorithm first maps plausible patients to the chosen mechanistic axes and discretizes these axes with the chosen number of bins. MAPEL then assigns a probability to each mechanistic axis bin, and by extent to all plausible patients that belong to that bin. MAPEL optimizes these bin probabilities so that the bin frequencies on the output measure axes match those in the population data. Finally, when calling subsample with MAPEL, the prevalence weights of plausible patients will be used to subsample the initial virtual population into a smaller population that better matches the data. Patients with higher prevalence weight are more likely to be sampled in the final population. Keyword Arguments • data: A 'DataFrame' containing all information about the discretized output measure. Each row of data is a different output measure including the name of the measure, the time at which the measure was taken, the number of patients that data is derived from and all bin probabilities and bin edges. Column names are not case sensitive. The required column names are : □ measure : Output measures that were taken from the actual population. □ t: Time at which a measure was taken. □ n: Number of patients that were part of each measure. □ p1, p2, p3, ... : Bin probabilities. There must be as many such columns as bins. The numbering must follow the natural numbers. ☆ e1, e2, e3, ... : Bin edges. Each edge is a Tuple of the form (left_edge, right_edge). There must be as many such columns as bins. The numbering must follow the natural numbers. • mechanistic_axes : A Vector of the mechanistic axes names. These should match the names provided in the search_space of the InverseProblem. • n: Number of plausible patients to subsample to the virtual population. • nbins: Number of bins to discretize each mechanistic axis into. • init_noise: Standard deviation of the noise that will be added to the initial bin probabilities before optimization. The noise is sampled from a Normal distribution centered at 0. If init_noise = 0 (default value) then no noise is added and the initial bin probabilities are uniform over each mechanistic axis. • optimizer: an Optimization.jl optimization method. This is used to find the optimal mechanistic axis bin probabilities. See the Optimization.jl documentation for more information on possible In the following example there are two output measures, s1(t) and s2(t) and two bins per measure. Hence, data must contain columns p1 and p2 for bin probabilities and e1 and e2 for bin edges. df = DataFrame( "measure" => ["s1(t)", "s2(t)"], "N" => [4000, 5000], "p1" => [0.7, 0.66], "p2" => [0.3, 0.34], "e1" => [(0.0, 0.35), (0.0, 0.3)], "e2" => [(0.35, 0.7), (0.3, 0.8)], "t" => [5, 4.8] alg = MAPEL( data = df, mechanistic_axes = ["k1", "c1"], nbins = 2, optimizer = NelderMead(), maxiters = 10_000, n = 1000, init_noise = 0.001 Schmidt BJ, Casey FP, Paterson T, Chan JR. Alternate virtual populations elucidate the type I interferon signature predictive of the response to rituximab in rheumatoid arthritis. BMC Bioinformatics. 2013 Jul 10;14:221. doi: 10.1186/1471-2105-14-221. PMID: 23841912; PMCID: PMC3717130. ARM(; data::DataFrame, bw=fill(0.5, length(save_names)), n_neighbors::Int=5, The Allen-Rieger-Musante (ARM) subsampling algorithm. Keyword Arguments • data: DataFrame containing data from the reference popultion, which is used to construct the probability density function and the cumulative density functions of the population. • bw: Real number or vector of length = length(data). Bandwidth of the kernel density function that will be fitted on data. This is a non-negative quantity that determines how much smoothing will be applied to each data dimension when producing the kernel density function. Higher values of bw correspond to greater smoothing. Defaults to 0.5 for all dimensions. See MultiKDE.jl for an • optimizer: an Optimization.jl optimization method. This is used to find the optimal prior probability of including a plausible patient into the virtual population. See the Optimization.jl documentation for more information on possible optimizers. • n_neighbors: Number of neighboring points around each point in model output space that are considered when estimating the probability density function. The density at each point is estimated as n_neighbors/V, where V is the volume of a hypersphere containing the point and n_neighbors around it. Allen RJ, Rieger TR, Musante CJ. Efficient Generation and Selection of Virtual Populations in Quantitative Systems Pharmacology Models. CPT Pharmacometrics Syst Pharmacol. 2016 Mar;5(3):140-6. doi: 10.1002/psp4.12063. Epub 2016 Mar 17. PMID: 27069777; PMCID: PMC4809626. DDS(; reference, n::Int, nbins::Int) Discretized Density Sampling (DDS) performs subsampling by matching the histogram of each model state under consideration to a reference histogram. The method matches the frequency of plausible patients that fall within each bin to the frequency of the same bin in the reference distribution. This way, the final Virtual Population has bin frequencies (or probabilities) equal to those of the reference distribution for each one of the considered model states. Keyword Arguments • reference: Vector of pairs. The first element of each pair is a model state and the second element is a NamedTuple containing two fields: □ dist: a reference distribution that the algorithm will match for the given state. This could be any distribution, see TransformedBeta or the Distributions.jl documentation) for more options. □ t: the timepoint along the state trajectory where the state should match the reference dist. This does not have to coincide with a saveat timepoint of the considered trial. • n: Number of plausible patients to subsample to the virtual population. • nbins: Number of bins to discretize the reference distributions into. State y1 at timepoint 5 [modelled units of time] should be distributed as a Beta(2,6) distribution within the bounds [1, 5], which is a TransformedBeta distribution, and state y2 should be distributed as an InverseGamma(2,2) at the 10 timepoint. The number of patients is set to 100 and number of bins for each state is 20. ref = [ y1 => (dist=TransformedBeta(Beta=Beta(2,6), lb=2, ub=6), t=5), y2 => (dist=InverseGamma(2,2), t=10) alg = DDS(reference=ref, n=100, nbins=20) Tip : Visualizing distributions One can easily visualize what a reference distribution looks like for a given bin size by running using StatsPlots reference_distribution = Normal(0,1) number_of_bins = 20 number_of_samples = 1000 histogram(rand(reference_distribution, number_of_samples), nbins=number_of_bins) RefWeights(; binning, reference_weights, n::Int) A reference-weight based algorithm that calculates assigns weigths to each plausible patient and then uses them to subsample the plausible population into a virtual population. Uses a binning function with reference weights for each bin to choose a subsample of the plausible patient population which bins with the same frequency as the reference. Keyword Arguments • binning: A function binning(sol) which returns an integer representing the bin the patient belongs to. • reference_weights: A vector containing the probability of a patient belonging to each bin in the actual population. These weights need to sum to 1. • n: Number of plausible patients to subsample to the virtual population. Schmidt BJ, Casey FP, Paterson T, Chan JR. Alternate virtual populations elucidate the type I interferon signature predictive of the response to rituximab in rheumatoid arthritis. BMC Bioinformatics. 2013 Jul 10;14:221. doi: 10.1186/1471-2105-14-221. PMID: 23841912; PMCID: PMC3717130. Each subsampling algorithm internally uses a Sampler object to perform subsampling on a plausible population vp. One can call the subsample method multiple times to rerun the subsampling process and produce a different population each time. However, this might be computationally expensive, as methods like ARM need to solve an optimization problem before producing a subsampled population. Users can access the internal Sampler object directly and call it multiple times to generate different populations. The Sampler is initialized with all the necessary information that the respective subsampling algorithm needs. Thus, by using this object, one can avoid the computational cost of running the entire subsample method multiple times, if multiple subsampled populations are required. function get_sampler(alg::AbstractSubsamplingAlgorithm, vp::AbstractVirtualPopulation, trial::AbstractExperiment; kwargs) Returns a sampler that, when called, produces a subset of indices from the plausible patient population vp. The sampled indices satisfy the population-level constraints as specified by the sampling algorithm alg defined on a given trial. Keyword Arguments Any keyword arguments kwargs provided will be passed on as options to solving the optimization problem that is part of some subsampling methods, e.g. the Allen-Rieger-Musante (ARM) method. For more information about such arguments see the Optimization.jl documentation. vp = vpop(prob, alg; population_size = 100) subsample_alg = ARM(data, save_names) sampler = get_sampler(subsample_alg, vp, trial) vp_subsampled = vp[sampler()]	{"url":"https://help.juliahub.com/pumasqsp/dev/manual/subsample/","timestamp":"2024-11-04T18:37:13Z","content_type":"text/html","content_length":"31216","record_id":"<urn:uuid:907d03df-4760-4568-9c09-f7bc472626fb>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00552.warc.gz"}
Summary Statistics Nutrient-load model results were used to compute a number of summary statistics. These statistics include mean annual loads, yields, and concentrations for each nutrient constituent at each site, with the following exceptions. Two sites in the San Joaquin Valley of California were on sloughs draining the same area; therefore, streamflows and estimated loads were combined to indicate output from a single "site." Eighteen other sites had insufficient data for model calibration; 12 of these had fewer than 18 samples during the high-intensity period, and another 6 had no complete water year of daily streamflow values. The resultant summary data set includes statistics for five nutrient constituents at 481 sites. These data are in the summary data set. For nutrients that could be fit to a regression model, mean annual load was estimated as the sum of the daily load values for the estimation period divided by the number of years. Mean annual yield was estimated by dividing the mean annual load by the upstream drainage-basin area. The flow-weighted mean concentration was calculated by dividing the total load over the estimation time period by the total streamflow. The time-weighted mean concentration was calculated as the average of the daily concentrations for the same time period. Time-weighted concentration is similar to flow-weighted concentration at many sites. However, at some sites where steamflow and load are affected by large storm events, the time-weighted concentration can differ from the flow-weighted concentration and represents the more common condition. For some nutrients, the calculated mean concentrations were less than the most common detection limit. In these cases, the yield and load estimates were retained, but the mean concentration was revised to less than detection. Censored values reported in the summary data set can be converted to uncensored values by calculating the ratio of mean annual load and streamflow and multiplying by 0.00112 to convert the units to milligrams per liter. In cases where a regression model could not be calibrated, the nutrient concentrations were plotted relative to streamflow and time. If there were no obvious curves or slopes in these relations, a mean concentration was computed. For nutrients with fewer than six uncensored concentrations, the flow-weighted and time-weighted mean concentrations were set to less than the most common detection limit. In these cases, load and yield were not calculated. For nutrients with more than six uncensored concentrations, a mean and standard error were computed using censored data techniques described by Helsel and Cohn (1988). If the standard error was less than 40 percent of the mean, the flow-weighted and time-weighted mean concentrations were set equal to the computed mean. Mean annual load was then calculated as the product of the flow-weighted mean, the total streamflow for the estimation period, and an appropriate units-conversion factor. Mean annual yield was estimated by dividing the load by the upstream drainage-basin area. After all possible mean concentrations were determined, sites were ranked from low to high for flow-weighted and time-weighted concentrations of each nutrient. Percentiles were calculated from these ranks. The percentiles indicate the relative magnitude of the mean concentration at a site in comparison to those at all other National Water-Quality Assessment Program sites. The median nutrient concentration for all sites is the 50th percentile value. One-half of the sites have concentrations between the 25th and 75th percentile values.	{"url":"https://pubs.usgs.gov/ds/2005/152/htdocs/data_report_sum_stat.htm","timestamp":"2024-11-12T15:19:17Z","content_type":"text/html","content_length":"7988","record_id":"<urn:uuid:d9682482-4cd1-4c7f-aeab-edee7203d24d>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00723.warc.gz"}
Lessons from SU(N) Seiberg-Witten Geometry Lessons from SU(N) Seiberg-Witten Geometry Nardoni, E. (2022). Lessons from SU(N) Seiberg-Witten Geometry. Perimeter Institute for Theoretical Physics. https://pirsa.org/22050069 Nardoni, Emily. Lessons from SU(N) Seiberg-Witten Geometry. Perimeter Institute for Theoretical Physics, Jun. 06, 2022, https://pirsa.org/22050069 @misc{ scivideos_PIRSA:22050069, doi = {10.48660/22050069}, url = {https://pirsa.org/22050069}, author = {Nardoni, Emily}, keywords = {Mathematical physics}, language = {en}, title = {Lessons from SU(N) Seiberg-Witten Geometry}, publisher = {Perimeter Institute for Theoretical Physics}, year = {2022}, month = {jun}, note = {PIRSA:22050069 see, \url{https://scivideos.org/pirsa/22050069}} Talk numberPIRSA:22050069 "Motivated by applications to soft supersymmetry breaking, we revisit the Seiberg-Witten solution for N=2 super Yang-Mills theory in four dimensions with gauge group SU(N). We present a simple exact Taylor series expansion for the periods obtained at the origin of moduli space, thereby generalizing earlier results for SU(2) and SU(3). With the help of these analytic results and others, we analyze the global structure of the Kahler potential, presenting evidence for a conjecture that the unique global minimum is the curve at the origin of moduli space. Two applications of these results are considered. Firstly, we analyze candidate walls of marginal stability of BPS states on special slices for which the expansions of the periods simplify. Secondly, we consider soft supersymmetry breaking of the N=2 theory to non-supersymmetric four-dimensional SU(N) gauge theory with two massless adjoint Weyl fermions (""adjoint QCD""). The Seiberg-Witten Kahler potential and strong coupling spectrum play a crucial role in this analysis, which ultimately leads to an exploration of the adjoint QCD phase diagram."	{"url":"https://scivideos.org/pirsa/22050069","timestamp":"2024-11-08T22:04:36Z","content_type":"text/html","content_length":"50809","record_id":"<urn:uuid:af652d06-63db-4d09-9b58-27fbab6e8982>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00036.warc.gz"}
Optimal actions and stopping in sequential learning Optimal actions and stopping in sequential learning Alexandra Carpentier and Markus Reiß One of the central models in mathematical statistics is the Gaussian linear model in which the least squares estimator (LSE) is efficient. For the nowadays common massive data sets, however, the LSE is computationally expensive. The seemingly attractive alternative of stopping an iterative numerical algorithm for its computation as soon as the approximation quality reaches the level of statistical resolution seriously fails at the missing knowledge of both of these quantities. In the general framework of sequential learning, this project develops methodology which covers this problem as a prominent particular case. Sequential learning copes with data or information that is not available at once, but comes in at different times. Two widespread situations of this kind are (a) iterative numerical algorithms that provide a stream of estimators and optimal regularization as well as feasible computational cost are achieved by stopping the algorithm early, and (b) when data comes in, the data flow may be actively steered in order to learn or estimate the unknowns near-optimally.	{"url":"https://for5381.uni-freiburg.de/en/projects/optimal-actions-and-stopping-in-sequential-learning-en/","timestamp":"2024-11-11T08:14:57Z","content_type":"text/html","content_length":"37831","record_id":"<urn:uuid:ed445aeb-f5ab-42be-917d-58bcecede333>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00479.warc.gz"}
Escalation Cost CalculatorEscalation Cost Calculator - Calculator Flares Escalation Cost Calculator Cost escalation is a critical factor in financial planning, especially in industries like construction, where costs can increase significantly over time. The Escalation Cost Calculator is a valuable tool for businesses and individuals to plan budgets effectively, accounting for future price increases due to inflation, labor costs, and other factors. Escalation cost calculator helps you determine cost escalation rates, allowing you to manage cost increases over time and make informed financial decisions, whether budgeting for a construction project, planning a contract, or managing long-term investments. What is Cost Escalation? Cost escalation refers to the increase in the cost of goods, services, or construction projects over time due to various factors such as inflation, labor costs, and material price increases. These increases can significantly impact budgets and financial planning, particularly for long-term projects or contracts. Understanding cost escalation is essential for anyone involved in budgeting, project management, or financial forecasting, as it allows for more accurate predictions of future expenses. In industries like construction, cost escalation can occur due to changes in the prices of raw materials, fluctuations in labor rates, or shifts in market demand. By accounting for these potential increases, businesses can ensure that they have adequate funds to complete projects without incurring unexpected expenses. the Escalation Cost Formula The escalation cost formula is used to calculate the future cost of an item or service after accounting for expected price increases over a specific time period. The formula is as follows: • EC is the escalated cost • IC is the initial cost • ER is the escalation rate, expressed as a decimal • TP is the time period in years This formula allows you to estimate the future cost of a product or service by considering the rate at which its price is expected to increase. For example, if the initial cost of a construction project is $10,000, with an escalation rate of 5% over three years, the future cost can be calculated as: Using an Escalation Rate Calculator An escalation rate calculator simplifies the process of determining the future cost of goods, services, or projects. By inputting the initial cost, escalation rate, and time period, the calculator automatically computes the escalated cost, saving time and reducing the likelihood of errors. To use an escalation rate calculator: 1. Enter the Initial Cost: Input the current cost of the item or service. 2. Input the Escalation Rate: Provide the expected annual rate of cost increase, expressed as a percentage. 3. Enter the Time Period: Specify the duration over which the cost is expected to escalate. 4. Calculate the Escalated Cost: The calculator will provide the future cost, accounting for the escalation rate over the specified time. The Role of Inflation in Cost Escalation Inflation is a key driver of cost escalation, influencing the price of goods and services over time. As inflation increases, the purchasing power of currency decreases, leading to higher costs for the same goods and services. This relationship is crucial to understanding cost escalation, as it affects nearly every aspect of the economy. To factor inflation into your cost escalation calculations, you can use an inflation calculator or reference a price index, such as the Consumer Price Index (CPI). These tools help determine how much prices are likely to rise due to inflation, allowing you to make more accurate financial forecasts. How to Calculate Escalation Rates Calculating escalation rates involves understanding the factors that contribute to price increases, such as inflation, labor costs, and material prices. The escalation rate is typically expressed as a percentage and represents the average annual increase in costs. To calculate the escalation rate: 1. Identify the Relevant Costs: Determine the costs that are likely to increase over time, such as labor, materials, or services. 2. Analyze Historical Data: Review historical price trends to estimate the average annual increase for each cost component. 3. Determine the Escalation Rate: Combine the average increases for each cost component to calculate the overall escalation rate. For example, if labor costs are expected to rise by 3% per year and material costs by 4%, the overall escalation rate might be calculated as a weighted average, depending on the proportion of each cost in the total budget. Understanding Cost Indices Cost indices, such as the Consumer Price Index (CPI) or the Producer Price Index (PPI), are tools used to measure the average change in prices over time. These indices are critical for calculating cost escalation, as they provide a benchmark for how much costs are expected to increase. Cost indices are typically published by government agencies and reflect changes in the prices of a basket of goods and services. By referencing these indices, you can adjust your cost estimates to account for expected price increases due to inflation and other economic factors. For instance, if the CPI indicates that prices have risen by 2% over the past year, you can use this data to adjust your cost estimates for the upcoming year, ensuring that your budget reflects current market conditions. Using Cost Indices in Escalation Calculations Incorporating cost indices into your escalation calculations allows you to account for expected changes in prices due to inflation and other factors. By applying the appropriate index to your cost estimates, you can ensure that your budget reflects the true cost of goods and services over time. To use cost indices in escalation calculations: 1. Select the Appropriate Index: Choose a cost index that reflects the type of costs you are estimating, such as the CPI for consumer goods or the PPI for industrial materials. 2. Apply the Index to Your Estimates: Multiply your initial cost estimates by the index to adjust for expected price increases. 3. Recalculate the Escalated Cost: Use the adjusted cost estimates in your escalation formula to determine the future cost of goods or services. For example, if youâ€™re estimating the cost of a construction project and the relevant cost index indicates a 3% annual increase, you can adjust your budget accordingly to account for this expected Managing Cost Escalation in Construction In the construction industry, managing cost escalation is essential for ensuring that projects stay within budget. Construction costs are particularly vulnerable to escalation due to factors such as material price fluctuations, labor shortages, and changes in market demand. To manage cost escalation in construction: 1. Use Accurate Cost Estimates: Start with a realistic estimate of initial costs, taking into account current market conditions. 2. Factor in Escalation Rates: Apply the appropriate escalation rate to your cost estimates to account for expected increases in material and labor costs. 3. Monitor Market Conditions: Regularly review market trends and adjust your cost estimates as needed to reflect changes in prices. By proactively managing cost escalation, construction companies can avoid budget overruns and ensure that projects are completed on time and within budget. The Impact of Labor Costs on Escalation Rates Labor costs are a significant factor in cost escalation, particularly in industries like construction and manufacturing. As labor costs rise due to factors such as wage increases, skill shortages, and changes in labor laws, the overall cost of projects and services can increase substantially. To account for rising labor costs in your escalation calculations: 1. Analyze Labor Market Trends: Review data on wage increases, labor demand, and workforce availability to estimate future labor costs. 2. Adjust Escalation Rates: Incorporate expected labor cost increases into your overall escalation rate, ensuring that your cost estimates reflect these additional expenses. 3. Plan for Contingencies: Include a buffer in your budget to account for unexpected labor cost increases, such as overtime or hiring additional workers. Advanced Tools: Inflation and Cost Calculators for Financial Planning Advanced financial tools, such as inflation calculators and cost escalation calculators, are essential for accurate financial planning. These tools allow you to account for expected price increases, ensuring that your budgets and forecasts are based on realistic assumptions. Features of advanced calculators include: 1. Customizable Inputs: Enter specific data, such as inflation rates, escalation factors, and time periods, to generate tailored cost estimates. 2. Scenario Analysis: Run multiple scenarios to see how different escalation rates and inflation levels impact your budget. 3. Real-Time Data Integration: Use up-to-date economic data to ensure that your cost estimates reflect current market conditions.	{"url":"https://calculatorflares.com/escalation-cost-calculator/","timestamp":"2024-11-03T05:36:11Z","content_type":"text/html","content_length":"197647","record_id":"<urn:uuid:c166f640-f69f-4f0e-8a69-cedba321ce4f>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00152.warc.gz"}
How do you determine the limit of 5^x/(3^x+2^x) as x approaches infinity? \| HIX Tutor How do you determine the limit of #5^x/(3^x+2^x)# as x approaches infinity? Answer 1 Use l'Hopital's Rule and some algebra. In the original form, l'Hopital's rule doesn't help, but note that: The limit as #xrarroo# is of the form #oo/oo#. L'Hopital tells us to consider the ratio of the derivatives: Note that this is simply #(5/2)^x/(3/2)^x# times the positive constant #ln(5/2)/ln(3/2)#. Furthermore, #(5/2)^x/(3/2)^x=(5/3)^x#. So, as #xrarroo#, #5^x/(3^x+2^x)# behaves like #((5/2)^xln(5/2))/((3/2)^xln(3/2)# which in turn behaves like #k(5/3)^x# with #k>0#. Which increases without bound (goes to #oo#) as #xrarroo#. Bonus: #lim_(xrarr-oo)(5/2)^x/((3/2)^x+1)=0/(0+1)# so #lim_(xrarr-oo)5^x/(3^x+2^x)=0# Sign up to view the whole answer By signing up, you agree to our Terms of Service and Privacy Policy Answer 2 To determine the limit of 5^x/(3^x+2^x) as x approaches infinity, we can use the concept of limits and properties of exponential functions. First, let's rewrite the expression as (5/3)^x / (1 + (2/3)^x). As x approaches infinity, (5/3)^x will tend to infinity since the base (5/3) is greater than 1. Similarly, (2/3)^x will tend to 0 as x approaches infinity since the base (2/3) is between 0 and 1. Therefore, the limit of (5/3)^x / (1 + (2/3)^x) as x approaches infinity is infinity. Sign up to view the whole answer By signing up, you agree to our Terms of Service and Privacy Policy Answer from HIX Tutor When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some Not the question you need? HIX Tutor Solve ANY homework problem with a smart AI • 98% accuracy study help • Covers math, physics, chemistry, biology, and more • Step-by-step, in-depth guides • Readily available 24/7	{"url":"https://tutor.hix.ai/question/how-do-you-determine-the-limit-of-5-x-3-x-2-x-as-x-approaches-infinity-8f9af9ccce","timestamp":"2024-11-02T11:36:36Z","content_type":"text/html","content_length":"573393","record_id":"<urn:uuid:c6aca3ca-3114-45da-81ec-4629d07114d1>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00185.warc.gz"}
DEtools[hyperode] - return the nth order ODE having a given hypergeometric pFq or MeijerG function as solution Calling Sequence hyperode(H, y(x), n, [S1, S2, ..., Sn]) H - hypergeometric function; can be or , or a product of the form y(x) - dependent variable of the ODE; can be any unknown function of one variable n - (optional) integer; differential order of the ODE to be returned [S1, ..., Sn] - (optional) list with expressions depending on x; solutions of the ODE to be returned • Hypergeometric functions , or their generalization known as MeijerG functions, can be defined in different ways. One possible way is to define them as the solutions to some linear ODEs. In connection with that approach, the hyperode routine receives as input a hypergeometric function H as a parameter, where H is constructed using one of hypergeom or MeijerG, as either – H([...], [...], f(x)), where is an arbitrary expression depending on x, or – the product of an arbitrary expression times H([...], [...], f(x)). Using this input, hyperode returns the general nth order ODE having as a solution that hypergeometric function H or that product of H times . • By identifying the general ODE underlying the definition of a hypergeometric function, this command can be of help when studying properties of these relevant functions and facilitate the understanding of related algorithms. In other cases, this ODE representation of pFq or MeijerG opens the way to computations (for example, differential elimination) that can only be performed with polynomial differential objects. See dpolyform, casesplit, DifferentialAlgebra, and rifsimp. • The ODE returned by hyperode, having a solution (where is expressed using hypergeom) is built by using the formula where D is the differential operator , followed by a change of variables and finally by the renaming of the variables according to The output in this case is a linear ODE where the differential order is the largest of and . • The ODE returned by hyperode, when the input is a MeijerG generalized hypergeometric function, is built using the same process, but the formula used in the first step is: where in above we assume, without loss of generality, that is less than or equal to , and and represent, respectively, the parameters entering the first and second lists of parameters in MeijerG. The output in this case is a th order linear ODE. • When the differential order of the ODE to be returned is not indicated, the one implied by the given pFq or MeijerG function is used. When the differential order is indicated as an extra argument, and provided it is greater than the one implied by the hypergeometric function, the ODE is constructed in the same way as just outlined except that before proceeding, the lists , are augmented by introducing into both of them the necessary number of additional parameters, all equal to zero. • Since the transformation shown above with arbitrary and represents the structure invariance group for the linear ODEs (that is, it is enough to map any linear ODE into any other one of the same differential order), then by keeping both and arbitrary, the ODE returned is "the most general linear ODE of a given order, written in such a way that its solution is expressed in terms of the product of a hypergeometric function (pFq or MeijerG) times an arbitrary function g(x)." Therefore, any linear ODE and its solution in terms of a hypergeometric function can be obtained from this general form by appropriately choosing f and g (see the last example). • When an optional list of additional ODE solutions is given, the returned ODE is built by first using the formula explained above, and then applying to it operators of the form for each , where the functions are adjusted so that in addition to is also a solution of the returned ODE. Note: This building process raises the differential order of the returned ODE by n. The general linear ODE family having as solution "2F1" (two indices in the first list and one index in the other list) is of second order since the number of operands of the first list is two, and the number of operands of the second list plus one is not greater than two. This ODE is known as the hypergeometric ODE (or Gauss ODE) and is given by The meaning of "1F0" can be determined by looking at the solution of the corresponding linear ODE (which is of order max( 1, 0+1 ) = 1). Hence, up to a constant (which may depend on a), 1F0 is equal to the right-hand side of the ODE solution above. An alternative approach to this result is to first convert 1F0 to an infinite sum and then perform the summation: Equivalently, but perhaps a more black-box approach, would be a one-step conversion. Consider the following statements in order to determine the meaning of the "0F0" hypergeometric function. In particular, consider the following "0F1", or use the one-step conversion routine. The general third order ODE equivalent to the second order hypergeometric ODE is obtained by departing from the general hypergeometric function: "3F2", in turn obtained from 2F1 by adding one arbitrary parameter to each of the two lists. Test these results by using odetest. The next third order ODE has as solutions both and . hyperode can be used in the same way with (generalized hypergeometric) MeijerG functions as input Comparing this output with the one for a pFq function it is apparent that, for some values of a, b, c, and d, there is a relation between the pFq and the more general MeijerG functions. This relation is actually given by: Finally, the most general second order linear ODE, can be written in terms of two functions such that the ODE solution is written in terms of and as: Such a form of the general 2nd order linear ODE is obtained as follows: Consider for instance some particular values of f and g. The following is the solution to this ODE. This is an example of an ODE family with radicals and its solution in terms of hypergeometric functions. The following is the solution to this ODE. See Also dchange, DEtools, DEtools[LCLM], dsolve, equinv, intfactor, odeadvisor, odetest, Ore_algebra/annihilators, PDEtools, PDEtools[dpolyform], redode, symgen; as well as the packages on the web HYPERG and gfun Marsden, J.E.; Sirovich, L.; and Antman, S.S. eds. Texts in Applied Mathematics. 56 vols. New York: Springer-Verlag, 1991. Vol. 8: Hypergeometric Functions and Their Applications. Mathai, A.M. A Handbook of Generalized Special Functions for Statistical and Physical Sciences. Oxford: Clarendon Press, 1993. Was this information helpful? Please add your Comment (Optional) E-mail Address (Optional) What is This question helps us to combat spam	{"url":"https://cn.maplesoft.com/support/helpjp/maple/view.aspx?path=DEtools/hyperode&L=C","timestamp":"2024-11-06T14:57:49Z","content_type":"application/xhtml+xml","content_length":"293904","record_id":"<urn:uuid:626ecd54-f912-4443-a759-707b99e3e9cd>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00151.warc.gz"}
Texas Go Math Kindergarten Module 15 Assessment Answer Key Refer to our Texas Go Math Kindergarten Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Kindergarten Module 15 Assessment Answer Key. Texas Go Math Kindergarten Module 15 Assessment Answer Key Concepts and Skills Question 1. 1 penny is equal to 1 cent There are 8 pennies. Question 2. 1 nickel is equal to 5 cents there are 7 nickels. Question 3. 1 dime is equal to ten cents there are 10 dimes. 1. Count the pennies. Write the number that shows how many pennies. TEKS K.4 2. Count the nickels. Write the number that shows how many nickels. TEKS K.4 3. Count the dimes. Write the number that shows how many dimes. TEKS K.4 Question 4. In the pouch there are nickels and pennies so, circled the pennies There are 4 pennies. Question 5. 1 quarter dollar = 25 cents In the bag there are quarter coins and nickels so, circled the quarter coins there are 6 quarter dollars Question 6. Texas test Prep Suzi needs 2 dimes to buy a snack. so, marked the 2 dimes that Suzi needs 4. Circle the nickels. Count and write the number of nickels. TEKS K.4 5. Circle the quarters. Count and write the number of quarters. TEKS K.4 6. Choose the correct answer, Suzi needs 2 dimes to buy a snack. Which set shows the coins Suzi needs? Leave a Comment You must be logged in to post a comment.	{"url":"https://gomathanswerkey.com/texas-go-math-kindergarten-module-15-assessment-answer-key/","timestamp":"2024-11-14T07:00:11Z","content_type":"text/html","content_length":"240179","record_id":"<urn:uuid:233d98e3-ee4f-4c8e-9015-364b1e7dd5e0>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00084.warc.gz"}
How a Health Diagnosis Made My Math Skills All Too Relevant (And Why Math Education Is Critical for Public Health) A huge international study of adult literacy and numeracy skills (Program for the International Assessment of Adult Competencies, known as PIACC) showed that in the U.S., 30% of adults had numeracy skills at or below level 1, which means they could only perform the most basic, single step, whole number operations.^1,2 This could impact the lives of adults in many ways, but one that has recently caught my attention is the role of numeracy in health care, and specifically diabetes management. I am a math educator and professional developer, and I am familiar with how often I use math and mathematical reasoning every day, but it has never been more explicit than when I was diagnosed with gestational diabetes (GDM) in the last trimester of my pregnancy. First, I needed to understand the probabilities. After the diagnosis, I was a little shocked since I think of myself as a generally healthy person and diabetes does not run in my family. How did this happen to me? I had been craving a lot of fruit to get me through the hot summer months, so part of my mind thought the extra apples must have thrown my sugar out of whack. I did a little research and learned that between 2-10% of pregnancies in the US develop gestational diabetes^3, with the risk going up for mothers over 25 years old (a benchmark I passed a while ago.) So to put things in perspective, the risk of someone my age getting GDM is probably between 1 in 10 and 1 in 20. Unlikely, but not rare. All throughout the pregnancy, I was faced with risk: 1 in 2500 chance of cystic fibrosis^4; 2% chance of heart defect; less than 1% chance of Downs Syndrome.^5 Almost an infinite number of other potential consequences to me and baby, although most of these risks are very small. I pictured each one in my head like a bathtub filled with marbles, with only a few bad marbles, and I had to keep drawing from the bathtub over and over. My chances for any one of these things happening was low, but the fact that after all these different draws I got one bad marble…not too hard to believe. Soon I learned about just how much proportional reasoning I was going to have to do every day. To manage the diabetes, I had to go on a very strict diet that called for eating a certain number of grams of carbohydrates at specific intervals six times a day. Each “meal” had a prescribed target of carbs I was supposed to hit (not too high or too low), and if I wanted to eat more than one type of food for the next three months, that meant a lot of math. Every meal involved planning. I had to consider the serving size for each carbohydrate-containing food (which is pretty much everything from the bread, fruit, or dairy food groups) and adjust it to get the number of carbs I wanted so that the meal would add up to the target. For example, in order to plan a reasonably normal lunch: • I first check the bread to see how many grams of carbs there are in each slice, after subtracting fiber. Two slices of wheat bread give me 23 grams of carbs. • Then I make a kale salad to eat all week, and add enough fruit so that if I divide it into four servings, each will have 6 grams of sugar. This amounts to most of an apple and 1/16 cup of low-sugar cranberries. • The soup I bought contained 11 grams of carbs for 6 ounces. I wanted to get that up to 15 grams. I mentally divided 11 grams by 3 (a little less than 4 grams, since 12 grams/3 = 4 grams), and reasoned that 2 ounces of soup contains about 4 grams of carbs, so adding two extra ounces would give me about the serving size I needed. Fortunately, I also knew that 8 ounces is one cup, so that was easy to measure. All of this involved knowledge of how to work with ratios, estimation and mental math, knowledge of measurement units and unit conversions, and comfort with fractions: all math taught in upper elementary and middle school, and precisely the math that most low-numeracy adults are lacking. In addition, I had to deal with elapsed time, another concept that some adults struggle with. My six meals had to be spaced 2-3 hours apart, with blood sugar readings to be taken 1 hour after each of my three main meals of the day. Depending on my agenda for the day (meetings, travel, errands, and other interruptions), I had to plan, sometimes down to 30 minute intervals, when each of these things had to happen. If I woke up late or missed a meal time (which sometimes happened), my last meal of the day might have to occur at 10 p.m. or later. A few times I had to set an alarm to wake up and eat that final snack. Skipping a meal would have put me at a deficit of carbs for the day (which could cause my blood sugar to fall too low) but eating too soon after dinner could have caused it to spike. Planning was essential. In 90% of cases, women with GDM go back to normal after giving birth, so I like my marble jar odds on that one, although I will be at increased risk of developing diabetes type 2 later in life. If I am lucky enough to avoid it, this will only be a short, three month foray into a mild form of the disease that an incredible 9.4% of the US population lives with.^6 The percentage goes up (12.6%, or about 1 in 8) for folks with less than a high school education,^7 who are also less likely to have the math (and other) skills to manage the disease. The math skills needed to manage diabetes are not advanced, but they need to be deep, flexible, and fluent to allow a person to use them on a day-to-day basis. This experience has brought home to me once again the importance of the work we do in adult numeracy for health equity and justice, and for giving people tools to improve their quality of life.	{"url":"https://www.terc.edu/adultnumeracycenter/how-a-health-diagnosis-made-my-math-skills-all-too-relevant-and-why-math-education-is-critical-for-public-health/","timestamp":"2024-11-09T11:11:44Z","content_type":"text/html","content_length":"92074","record_id":"<urn:uuid:002a6959-ebc6-444a-82c4-e5a92ca8e4dd>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00007.warc.gz"}
ESTIMATION of MASS of COMPACT OBJECT in H 1743-322 from 2010 and 2011 OUTBURSTS USING TCAF SOLUTION and SPECTRAL INDEX-QPO FREQUENCY CORRELATION The well-known black hole candidate (BHC) H 1743-322 exhibited temporal and spectral variabilities during several outbursts. The variation of the accretion rates and flow geometry that change on a daily basis during each of the outbursts can be very well understood using the recent implementation of the two-component advective flow solution of the viscous transonic flow equations as an additive table model in XSPEC. This has dramatically improved our understanding of accretion flow dynamics. Most interestingly, the solution allows us to treat the mass of the BHC as a free parameter and its mass could be estimated from spectral fits. In this paper, we fitted the data of two successive outbursts of H1743-322 in 2010 and 2011 and studied the evolution of accretion flow parameters, such as two-component (Keplerian and sub-Keplerian) accretion rates, shock location (i.e., size of the Compton cloud), etc. We assume that the model normalization remains the same across the states in both these outbursts. We used this to estimate the mass of the black hole and found that it comes out in the range of 9.2512.86 M o. For the sake of comparison, we also estimated mass using the Photon index versus Quasi Periodic Oscillation frequency correlation method, which turns out to be 11.65 ± 0.67 Mo using GRO J1655-40 as a reference source. Combining these two estimates, the most probable mass of the compact object becomes 11.21^+1.65 [-1.96] M o. • X-rays: binaries • accretion, accretion disks • radiation: dynamics • shock waves • stars: black holes • stars: individual (H 1743-322) ASJC Scopus subject areas • Astronomy and Astrophysics • Space and Planetary Science Dive into the research topics of 'ESTIMATION of MASS of COMPACT OBJECT in H 1743-322 from 2010 and 2011 OUTBURSTS USING TCAF SOLUTION and SPECTRAL INDEX-QPO FREQUENCY CORRELATION'. Together they form a unique fingerprint.	{"url":"https://cris.bgu.ac.il/en/publications/estimation-of-mass-of-compact-object-in-h-1743-322-from-2010-and-","timestamp":"2024-11-05T22:20:50Z","content_type":"text/html","content_length":"60772","record_id":"<urn:uuid:240630a3-3dbe-4e5f-a28d-a023736772bc>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00266.warc.gz"}
if / and / isblank formula If my status is closed, but I did not put a date in the date closed column, I need it flagged. Can someone help write this formula? I am using =IF(AND(ISBLANK([Date Closed]@row), Status@row, "Closed"), "High", "Low") and get the error #INVALID DATA TYPE Best Answer • Your formula should work, it sounds like the cell you are entering the formula into is not set to Text/Number The below formula would work as well =IF(AND([Date Closed]@row = "", Status@row = "Closed"), "high", "low") • Your formula should work, it sounds like the cell you are entering the formula into is not set to Text/Number The below formula would work as well =IF(AND([Date Closed]@row = "", Status@row = "Closed"), "high", "low") Help Article Resources	{"url":"https://community.smartsheet.com/discussion/76154/if-and-isblank-formula","timestamp":"2024-11-02T00:06:31Z","content_type":"text/html","content_length":"417887","record_id":"<urn:uuid:30e1308c-7b3b-4eca-adcb-81a156e0d654>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00645.warc.gz"}
Oblivious Transfers In this post we’re going to get to know one of the most fundamental constructions in cryptography known as Oblivious Transfer or OT for short. It has been used in wide spectrum of topics in cryptography, and as such I must admit that I’m very excited to finally to get write about it! Let’s start with a real life scenario^1! Say we have Alice and Bob, where Alice has two messages $m_0,m_1$ and Bob has a bit $b \in \lbrace 0,1 \rbrace$ such that Bob wants to retreive message $m_b$, how can Bob and Alice do this? Well, if Bob doesn’t care much about his privacy, he could send Alice the bit $b$ and Alice would send him $m_b$ in return. But what if Bob does care about his privacy? Of course Alice can send Bob both $m_0,m_1$ and Bob will simply pick $m_b$, but this also violates the privacy of Alice! So, if both parties care about their privacy we want Bob to learn $m_b$ without Alice learning $b$ and without Bob learning $m_{1-b}$. While this may sound impossible and paradoxical at this point, it is actually possible and if you finish reading this post you’ll probably be surprised by how simple it is. If Alice is tranferring $m_b$ to Bob while preserving all these privacy constraints, we say Alice Obliviously Transfers $m_b$ to Bob. Without getting into the the technical details we can already say that Oblivious-Transfer is a general name for protocols with which one party can obliviously transfer data to another party. By “general name” I mean that just like sorting is a “general name” for various algorithms that sort data, OT is a name for protocols that obliviously transfer data. This also means that OT protocols take place between two parties: A sender who offers messages to be sent and a receiver who selects to receive one of the offered messages. Let’s start with a short history of the problem. The first appearance of Oblivious-Transfer was in a paper by Michael (Oser) Rabin from 1981 and it didn’t even match the definition we have given here. In the original version the sender had a single message $m$ and at the end of the protocol the receiver may either learn $m$ with probability $1/2$ or learn nothing with probability $1/2$ without the sender knowing if $m$ was indeed learned or Cool Fact: The original paper was manually written! Tal Rabin noticed that the copies were becoming hard to find and decided to upload a scanner version. You can find the original manuscript as well as a typeset version on ePrint. (Paper link) Cool Fact 2: Tal is Michael’s daughter. A few years later, in 1985, Even, Goldreich and Lempel have given a construction to the OT as we have seen here, where one-of-two messages is being retreived. (Paper link) Since then, various constructions have been proposed to OT, one of them, titled “The Simplest Protocol for Oblivious Transfer” by Chou and Orlandi published in 2015 is what we will see today in greater detail. The OG OT In this section we’ll see the original construction of Michael Rabin for “OT”, we put the “OT” in quotes since it doesn’t really match our definition for OT. I start with it both as a nice historical lesson and because it gives a good sense on the underlying concepts from which OT can built! In this “OT” protocol we will have two parties, Alice and Bob. At the beginning of the protocol Alice holds a message $m$ while Bob has no input. At the end of the protocol we want Bob to learn $m$ with probability $1/2$ and learn nothing with probability $1/2$ without Alice knowing whether Bob did learn or didn’t learn $m$ eventually! The construction of the protocol relies on RSA groups, so first let’s recall what RSA groups are. RSA Groups Given two distinct primes $p,q$ we call $N=pq$ an RSA-modulus. We denote by $\mathbb{Z}_N^$ the set of numbers between $1$ and $N-1$ that are coprime to $N$. Notice that each number dividing $N$ has to divide at least one of it’s prime factors which are $p$ and $q$ so $\mathbb{Z}^_N$ is the set of all numbers between $1$ and $N-1$ that are neither a multiple of $p$ nor $q$. An RSA group is a group whose elements are $\mathbb{Z}^_N$ for some RSA-modulus $N$ and the group operation is modular multiplcation modulo $N$. Let’s have an example. Let $p=3$ and $q=5$ we have $N=15$ so $\mathbb{Z}^_N={1,2,4,7,8,11,13,14}$. Our RSA group operation will take two numbers from $\mathbb{Z}^_N$ and multiply them modulo $15$, for example $47=28 \equiv 13 \mod 15$. So if we apply the group operation on $4$ and $7$ we get $13$. One characteristic of groups is that each element should have an inverse. We say that $a,b$ are inverse if $ab=1$ according to the group operations. For example $28=16 \equiv 1 \mod 15$ so $2$ and $8$ are the inverse of each other! One question you may ask is: “Given a number $a$ in $\mathbb{Z}_N^*$, how fast can we find its inverse?”. According to a popular theorem called the Chinese Remainder Theorem finding the inverse of $a$ in the RSA group can be done very efficiently if we know $p,q$, the factors of $N$. But what if we don’t know them and instead we are just given $N$? Well, we don’t really know but we assume that it is very hard and for sufficiently large $p,q$ it can take very long time. This assumption is the RSA assumption and that is what Rabin has also assumed while constructing his original OT. Square Roots in RSA Groups 1. This is only cryptographers’ real lives ↩	{"url":"https://hamil.is/2022/04/14/Oblivious-Transfer.html","timestamp":"2024-11-13T07:24:09Z","content_type":"text/html","content_length":"51663","record_id":"<urn:uuid:24ca04b4-3a03-4caa-9fd6-d95795c30e65>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00171.warc.gz"}
What is a partial product? - California Learning Resource Network What is a partial product? What is a Partial Product? In the realm of mathematics, a partial product is a fundamental concept in multiplication, particularly when dealing with multi-digit numbers. The term "partial product" may seem complex, but don’t let the name intimidate you. In this article, we’ll break down the concept of partial products, explore its significance, and provide examples to help solidify your understanding. What is a Partial Product? A partial product is the result of multiplying a part of one number with another number. In other words, it’s the product of a subset of digits from one number multiplied with another number. For instance, consider the operation 456 × 7. To find the partial product, we would multiply the hundreds place (4) of 456 with 7, resulting in 28. This 28 is the partial product, which is then added to the next partial product (the tens place, which is 5, multiplied by 7, resulting in 35) to obtain the final product, 378. Why are Partial Products Important? Partial products play a crucial role in standard multiplication, especially when dealing with multi-digit numbers. By breaking down the multiplication process into smaller parts, partial products help us: • Simplify complex multiplication: By focusing on individual digits or groups of digits, we can reduce the complexity of the multiplication process, making it more manageable and less overwhelming. • Reduce mental math: Using partial products, you can quickly estimate the result of a multiplication problem, without necessitating lengthy calculations. • Build a deeper understanding of multiplication: Focusing on partial products encourages a deeper understanding of the underlying principles of multiplication, making it easier to solve problems and develop strategies for future math challenges. How to Calculate Partial Products Calculating partial products involves breaking down the numbers into individual digits or groups of digits and then multiplying each part by the other number. Here’s a step-by-step approach: 1. Identify the numbers: Isolate the numbers involved in the multiplication problem, in this case, 456 and 7. 2. Identify the digits: Break down each number into individual digits: 456 = 400 + 50 + 6, and 7 = 7. 3. Multiply each digit: Multiply each part (or digit) of the first number with the second number: □ 400 × 7 = 2800 □ 50 × 7 = 350 □ 6 × 7 = 42 4. Combine the partial products: Add the partial products obtained in step 3 to get the final result: 2800 + 350 + 42 = 3252 Using Partial Products with Different Numbers of Digits When dealing with numbers that have different numbers of digits, it’s essential to focus on the same number of digits when calculating partial products. For instance: 123 × 4 100 × 4 = 400 (hundreds place) 20 × 4 = 80 (tens place) 3 × 4 = 12 (ones place) Total: 492 Final product In conclusion, partial products are an essential concept in mathematics, particularly in standard multiplication. By understanding partial products, you’ll simplify complex multiplication, reduce mental math, and develop a deeper understanding of the underlying principles of multiplication. Practice calculating partial products to improve your math skills and become more confident in your ability to solve multiplication problems. Remember to always break down the numbers into individual digits or groups, and then multiply each part with the other number. With partial products, you’ll be well-equipped to conquer even the most complex multiplication challenges. Your friends have asked us these questions - Check out the answers! Leave a Comment	{"url":"https://www.clrn.org/what-is-a-partial-product/","timestamp":"2024-11-06T13:51:30Z","content_type":"text/html","content_length":"135104","record_id":"<urn:uuid:c099802f-c706-4cf3-a48b-1cd009ab51fc>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00443.warc.gz"}
Unit 'Graph' Package [Overview][Constants][Types][Procedures and functions][Variables][Index] [#rtl] Draw an ellipse. Source position: graphh.inc line 809 Ellipse draws part of an ellipse with center at (X,Y). XRadius and Yradius are the horizontal and vertical radii of the ellipse. Start and Stop are the starting and stopping angles of the part of the ellipse. They are measured counterclockwise from the X-axis (3 o'clock is equal to 0 degrees). Only positive angles can be specified. See also Arc Draw part of a circle. Circle Draw a complete circle. FillEllipse Draw and fill an ellipse.	{"url":"https://build.alb42.de/fpcbin/docu/html/rtl/graph/ellipse.html","timestamp":"2024-11-12T07:33:14Z","content_type":"text/html","content_length":"3852","record_id":"<urn:uuid:2cac466a-9c0f-4420-aa51-b9b47116c903>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00615.warc.gz"}
Why doesn't an argument of Vector{Vector{T} where T<:Real} accept a variable of Vector{Vector{Int64}}? Why does the function f3 below fail, while the functions f1 and f2 succeed? (Jupyter 0.6.2.2, JuliaPro 0.6.2.2) function f3(a::Vector{Vector{T} where T<:Real}) a = Vector{Vector}(3) for i in 1:3 a[i] = [i, i+ 1] MethodError: no method matching f3(::Array{Array{T,1} where T,1}) Closest candidates are: f3(::Array{Array{T,1} where T<:Real,1}) at In[7]:2 function f1(a::Vector{Vector}) function f2(a::Vector{Vector{T} where T}) Because your a is not a subtype of Vector{Vector{<:Real}} (which is a shorthand for what you write): julia> a = Vector{Vector}(3); julia> for i in 1:3 a[i] = [i, i+ 1] julia> a 3-element Array{Array{T,1} where T,1}: [1, 2] [2, 3] [3, 4] However, this does not work either: julia> a = Vector{Vector{Int}}(3); julia> for i in 1:3 a[i] = [i, i+ 1] julia> a 3-element Array{Array{Int64,1},1}: [1, 2] [2, 3] [3, 4] julia> f3(a) ERROR: MethodError: no method matching f3(::Array{Array{Int64,1},1}) Closest candidates are: f3(::Array{Array{T,1} where T<:Real,1}) at REPL[1]:2 This finally works: julia> a = Vector{Vector{<:Real}}(3); julia> for i in 1:3 a[i] = [i, i+ 1] julia> a 3-element Array{Array{T,1} where T<:Real,1}: [1, 2] [2, 3] [3, 4] julia> f3(a) 3-element Array{Array{Int64,1},1}: [2, 4] [4, 6] [6, 8] However, what you probably wanted to do was: function f3(a::Vector{Vector{T}}) where T<:Real then all of the above a will work. Search the docs or this forum for â€śinvarianceâ€ť to find enlightenment (although, it is a bit a tricky subject). PS: please quote your code snippets using backticks `: PSA: how to quote code with backticks 4 Likes Great! Thank you I tried it as follows, and it worked: function f4(a::Vector{Vector{T}}) where T<:Real a = Vector{Vector{Real}}(3) for i in 1:3 a[i] = [i, i+ 1] results in 3-element Array{Array{Int64,1},1}: [4, 8] [8, 12] [12, 16] Thanks for the tip, too. 1 Like	{"url":"https://discourse.julialang.org/t/why-doesnt-an-argument-of-vector-vector-t-where-t-real-accept-a-variable-of-vector-vector-int64/10792","timestamp":"2024-11-12T09:11:23Z","content_type":"text/html","content_length":"25734","record_id":"<urn:uuid:dd7e8db3-4fb3-4fad-8fe0-9b6f45158e98>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00749.warc.gz"}
Addition Concept \| Learn and Solve Questions The addition is a mathematical operation. They are methods and tasks related to numbers. It can be described by combining two or more numbers or adding two or more numbers. It is one of four arithmetic operations along with subtraction, multiplication, and division. How Can Addition be Performed? Let's start with an example. 7 + 4 means combining the numbers 7 and 4 to get the solution. The result, as you know, is 11. Sums can be divided into terms and sums. A term is a number added, and the answer to addition is a sum. To solve the sum, the terms must be joined together. So the expression 7 + 4 = 11 can be written as: Example of Addition Different Methods of Addition Now we will learn about different ways by which we can solve addition problems: • Go to Visualizations: Comprehension can be improved by using pictorial totals or having students draw countable objects Addition with Countable Objects • Addition Using a Number Line: A number line is a horizontal line that has numbers placed at equal intervals. • Number Lines help simplify the calculation process. • For example, if the sum is 3 + 4, the child can first count to three digits then by placing four fingers to get 7. You don't have to count to 3 first to find the solution. Addition Using Number Line • Counting: As your child gets used to the number strings, you will want to use the same strategy in their head. Counting your fingers out loud will help you gain. More about Addition Concept The addition operator can be divided into the following parts: • Addend: The numbers being added are called addition. • Plus Sign: There is a plus sign \[[ + ]\] between additions. • If the statement is written horizontally, place an equal sign \[[ = ]\] just before the sum item. • Sum: The final result after adding terms is called the sum. Addition Examples Worksheet Add the following examples: 1. \[\textbf{2 + 2 = }\] Ans: Here we have to add the terms which are 2 and 2 so on combining 2 and 2 together we get a sum of 4. Therefore, \[2 + 2 = 4\] 2. \[\textbf{9 + 0 = }\] Ans: Here we need to combine 9 and 0 so on combining 9 and 0 we get the sum as 9. Therefore, \[9 + 0 = 9\] 3. \[\textbf{8 + 2 =} \] Ans: As above problems , \[8 + 2 = 10\] 4. \[\textbf{4 + 4 = }\] Ans: Sum of \[4 + 4 = 8\]. 5. \[\textbf{2 + 9 = }\] Ans: Therefore, Sum will be 11. Addition is the mathematical operation that defines the sum or total of quantities. The addition operator is plus (+) sign. We have learnt about the concept of addition in this article and we have studied various methods by which addition can be done. You can now apply your understanding of addition in solving the worksheet given below. FAQs on Addition Concept 1. What are the four different properties of addition? The four different properties of addition include the following: • Closure Property: This property states that when we add two or more whole numbers, the result obtained is always a whole number. • Commutative Property: This property states that when we add two or more whole numbers, the result obtained is the same regardless of the order of their addends. • Associative Property: This property states that when we add three or more numbers, the result obtained is the same regardless of the order of their addend. • Identity Elements of Addition: When we add 0 to any number, the result obtained is the same as the original number. Adding 0 to any number does not change the value of the original number. 2. What are the two terms that are highly used while doing an addition? The two terms that are mostly used while the addition is addends, and the total value is called the sum. 3. What are the rules of addition? There are three basic rules of addition as given below: • When two positive numbers are added together, the result is a positive value. • When two negative numbers are added together, the outcome is the total of the two numbers with a negative sign. • Subtraction occurs when two numbers with opposite signs are added.	{"url":"https://www.vedantu.com/maths/addition-concept","timestamp":"2024-11-11T22:57:47Z","content_type":"text/html","content_length":"262504","record_id":"<urn:uuid:4e8fb8fa-bde1-4f51-a363-a3cd1c3618b2>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00135.warc.gz"}
Cracking the Code: Understanding the Mystery behind the Computational Tool Initial Publication Date: September 30, 2016 Cracking the Code: Understanding the Mystery behind the Computational Tool Marjorie Hubbard, Engineering and Technology, North Carolina School of Math and Science I have previously used MATLAB in my graduate work in computational electrophysiology and have found it to be an extremely useful platform for both simulation and for exploring complex science and engineering topics. Extending MATLAB as a computational teaching tool helps students to use a logical step-by-step approach to define and solve problems and to break technical theoretical concepts into concrete variables, constants, and conditions. When I am translating electrophysiology computational tools into the classroom, I follow several key steps to make sure students understand the underlying physical basis for the model, the applications and limitations of numerical models, the power of computation, and the utility of simulation for exploring how different variables impact the computational model. In the first step, the students are taught the physical basis for the equations and the foundation of the model. Many times when students learn computation in a stand-alone class they are often disconnected from the actual physical reality of the model. In the context of electrophysiology, students first learn about the biology of the excitable cell, then they learn the engineering model and its associated equations. This first step is especially important when using computation as a teaching tool in science and engineering because it helps to put the computation in context beyond simply programming equations. In the second step, the students learn how to perform numerical methods by hand and they learn the limitations of numerical methods. It is important for students to understand the underlying math principles that power the computational tool so that they can make informed decisions about the accuracy and reliability of the information that the tool provides. In the context of electrophysiology, students begin with a simple numerical method such as Forward Euler which uses data from a previous time step to calculate the next time step. Using small-scale examples, they explore how changing small variables such as the size of the time step can affect the answer, and solving the equations by hand also gives them an appreciation for how valuable computational tools become as the problems get more complex. Once the students understand the physical and mathematical basis for the problem, the third step is to translate the equations into MATLAB code to create the computational tool. In entry level courses, I will provide a skeleton code and have the students fill in the appropriate code to complete the required calculations. Students know how to complete the problems by hand, so they are also able verify that the output of the program makes physical sense. The process of applying the scientific or engineering knowledge in a logical format helps to reinforce the underlying principles and it also allows students to solve more complex problems than can be done by hand. The final step is to use simulation to explore the impact of different variables on the model. In electrophysiology, students are able to visualize the response of the action potential to threshold voltage and current stimulus and verify that the model response is correct based on their understanding of the biological and physical principles they learn in the first step. Being able to visualize the response of the model to many different parameters in real-time without actually doing physical experiments helps to reinforce understanding of the biological and physical concepts and makes computation a powerful and cost-effective tool for teaching science and engineering. Because I have experienced the power of computation for teaching complex ideas at the undergraduate level, I would also like to incorporate MATLAB into my engineering courses at the advanced high school where I teach. This workshop will allow me to get exposure to a wide range of experienced educators who can give advice on the best way to teach basic computational skills and can give a broad range of specific examples of how to use MATLAB to reinforce science and engineering principles. Downloadable version of this essay Cracking the Code: Understanding the Mystery behind the Computational Tool (Acrobat (PDF) 26kB Sep28 16)	{"url":"https://serc.carleton.edu/matlab_computation2016/essays/159557.html","timestamp":"2024-11-04T10:52:46Z","content_type":"text/html","content_length":"52527","record_id":"<urn:uuid:655608d7-9f9c-4312-a3a1-63d08133f916>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00431.warc.gz"}
This map displays the high level view of the map of risks: nodes represent large topics labelled by some of their most representative terms. Node size is a function of the number of publications for that thematic, node color is the indication of the growth rate of this size (red nodes like 'nanotechnology & nanomaterials' are the faster growing fields.) You can browse a more detailed semantic map (terms level) by loading one of the following maps from the top menu (they can take several seconds for loading, be patient): • - Size and topics map: size of node is a function of the number of publications on the terms, node color is related to its categorization into large topic. • - Growth and topics: size of node is a function of the growth of the number of publications mentionning the term over these last two years, node color is related to its categorization into large • - Size and growth: size of node is a function of the number of publications on the terms, color of node is related to the growth rate of that numnber (red node have the highest growth rate). - Double clic on a node to get more information - Double clic an empty area to erase current selection	{"url":"http://risk.iscpif.fr/maps2013/","timestamp":"2024-11-01T22:09:35Z","content_type":"text/html","content_length":"6459","record_id":"<urn:uuid:02e2ba6a-6e8a-4a50-a797-0e05a406f8d7>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00737.warc.gz"}
On Feasibly Solving NP-Complete Problems Download PDFOpen PDF in browser On Feasibly Solving NP-Complete Problems EasyChair Preprint 11063, version 2 5 pages•Date: October 23, 2023 ONE--IN--THREE 3SAT consists in knowing whether a Boolean formula $\phi$ in $3CNF$ has a truth assignment such that each clause contains exactly one true literal or exactly two true literals. $\ textit{ONE--IN--THREE 3SAT}$ remains $\textit{NP--complete}$ when all clauses are monotone. We create a polynomial time reduction which converts the monotone version into a bounded number of linear constraints on real numbers. Since the linear optimization on real numbers can be solved in polynomial time, then we can decide this $\textit{NP--complete}$ problem in polynomial time. Certainly, the problem of solving linear constraints on real numbers is equivalent to solve the particular case when there is a linear optimization without any objective to maximize or minimize. If any $\textit {NP--complete}$ can be solved in polynomial time, then we obtain that $P = NP$. Moreover, our polynomial reduction is feasible since it can be done in linear time. Keyphrases: Boolean formula, completeness, complexity classes, polynomial time Links: https://easychair.org/publications/preprint/vWzM Download PDFOpen PDF in browser	{"url":"https://easychair-www.easychair.org/publications/preprint/vWzM","timestamp":"2024-11-06T04:48:46Z","content_type":"text/html","content_length":"5177","record_id":"<urn:uuid:99bf6d7b-388f-4a73-90ed-7e8e307898b8>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00532.warc.gz"}
First principles determination of bubble wall velocity and local thermal equilibrium approximation Thursday, April 11, 2024 11:00 - 12:00 First-order phase transitions in the early Universe are well-motivated events predicted by several BSM models. In the first part of this talk, I will derive the fluid equations needed to compute the bubble wall velocity from first principles. By treating the background and out-of-equilibrium perturbations in a consistent way, the resulting equations are free of the discontinuity at v_w=c_s that was observed in previous studies. I will show that the solutions can naturally be classified as deflagration/hybrid walls (v_w ~ c_s) or ultrarelativistic detonations. In the second part, I will explain how this calculation can be significantly simplified when local thermal equilibrium (LTE) is maintained in the plasma. Using this LTE assumption, the fluid equations can be reexpressed in terms of only four parameters that completely characterize a particle physics model. I will present an efficient algorithm to solve these equations and discuss the properties of their solutions. Finally, I will compute the kinetic energy fraction which is essential for predicting the gravitational wave spectrum produced during the phase transition. Indico: https://indico.nikhef.nl/event/5409/ Zoom Link: https://nikhef.zoom.us/j/92535772755?pwd=aFZySWFtS1NXaEhZSmhKRzdtYXFIUT09 Meeting ID: 925 3577 2755 Passcode: nikh4f colloquium room Benoit Laurent (McGill U., Montreal, Canada)	{"url":"https://physicstalks.amsterdam/event/first-principles-determination-of-bubble-wall-velocity-and-local-thermal-equilibrium-approximation","timestamp":"2024-11-15T03:19:42Z","content_type":"text/html","content_length":"29511","record_id":"<urn:uuid:68674a0c-5072-4962-8a22-e47f4a2b27b5>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00457.warc.gz"}
Function-Oriented Networking and On-Demand Routing System in Network Using Ant Colony Optimization Algorithm Department of Computer Engineering, Kyung Hee University, Gyeonggi-do 17104, Korea Humanitas College, Kyung Hee University, Gyeonggi-do 17104, Korea Author to whom correspondence should be addressed. Submission received: 10 October 2017 / Revised: 2 November 2017 / Accepted: 3 November 2017 / Published: 10 November 2017 In this paper, we proposed and developed Function-Oriented Networking (FON), a platform for network users. It has a different philosophy as opposed to technologies for network managers of Software-Defined Networking technology, OpenFlow. It is a technology that can immediately reflect the demands of the network users in the network, unlike the existing OpenFlow and Network Functions Virtualization (NFV), which do not reflect directly the needs of the network users. It allows the network user to determine the policy of the direct network, so it can be applied more precisely than the policy applied by the network manager. This is expected to increase the satisfaction of the service users when the network users try to provide new services. We developed FON function that performs on-demand routing for Low-Delay Required service. We analyzed the characteristics of the Ant Colony Optimization (ACO) algorithm and found that the algorithm is suitable for low-delay required services. It was also the first in the world to implement the routing software using ACO Algorithm in the real Ethernet network. In order to improve the routing performance, several algorithms of the ACO Algorithm have been developed to enable faster path search-routing and path recovery. The relationship between the network performance index and the ACO routing parameters is derived, and the results are compared and analyzed. Through this, it was possible to develop the ACO algorithm. 1. Introduction As the era of new communication such as 5G era comes, it is expected that there will be a lot of services that cannot be compared with what existed before. Examples include Internet of Things (IoT) [ ], in which everything in everyday life is connected to the Internet, such as a refrigerator and a TV that can be remotely controlled, and services such as Tactile Internet. These services have network requirements [ The term Internet of Things was first introduced by Kevin Ashton in 1999 [ ]. IoT network means that things such as sensors are connected to the internet and is freely controllable from the remote. Internet of Things is regarded as one of the emerging technologies in IT. This will enable new services [ ]. It requires new network characteristics accessing for a high number of devices. Tactile Service or Tactile Internet is that providing real-time interactions with low delay that humans cannot recognize [ ]. If Tactile Service is realized, many novel services will appear: remote health care, virtual reality, robotics, etc. Each service needs a different delay. The network requirements for these new services are difficult to achieve at once. So, the core technology of future network proposed in 5G is to be able to dynamically and quickly provide the requirements of such a network [ ]. However, 5G’s core enabling technologies, SDN and Network Functions Virtualization (NFV) [ ], are designed exclusively for network administrators or managers. This is a powerful solution to many problems, but there are still problems that cannot be solved at the same time. In this paper, we investigate what these problems are and present functional-oriented networking as a solution and complement. Section 2 , related works are described. In Section 3 , Function-Oriented Networking is proposed and how to implement it is explained. In Section 4 , we illustrate how to implement Routing using ant colony optimization as FON Function. In Section 5 , the previous FON Function is evaluated and network performance analyzed. 2. Related Works 2.1. OpenFlow OpenFlow is the most well-known SDN project. It started with the paper [ ]. Their initial purpose is providing the testing platform for a new experimental network protocol. These days, many papers focus on network centralization characteristic of OpenFlow. The most popular SDN controller-related projects are ONOS and OpenDaylight [ 2.2. NFV NFV strands for Network Function Virtualization. Ref. [ ] states that “Network Functions Virtualization aims to transform the way that network operators architect networks by evolving standard IT virtualization technology to consolidate many network equipment types onto industry standard high volume servers, switches and storage, which could be located in Datacentres, Network Nodes and in the end user premises”. 2.3. Network Slicing Network slicing may be comprehended in many ways, but it can be accepted as the concept that the portion of all resources including network switches and routers in networks can be allocated for a specific purpose. Also, the allocated network slicing is designated for a specific service. Many papers and projects said that it is the key to make a network more flexible. One of the old projects related to network slicing is the GENI Project. According to [ ], “GENI (Global Environment for Network Innovations) provides a virtual laboratory for networking and distributed systems research and education”. The recent project NGMN [ ] defined network slicing in their final deliverable [ 2.4. Quagga According to [ ], “Quagga is a routing software suite, providing implementations of OSPFv2, OSPFv3, RIPv1 and v2, RIPng and BGP-4”. Zebra, which is the core module of Quagga, abstracts OS layers and provides many APIs for table-driven routing protocols. If some routing protocols like RIP or BGP is needed, it can be easily run. Also, if it is not needed, it can be stopped. From this point of view, Quagga is called as a “virtual routing system”. Quagga is an open source project distributed under the term of GPL; you can easily acquire the full source code from [ 3. Function-Oriented Networking Related-works above SDN and NFV are all researches for network managers. OpenFlow is a technology which separates date-plane from control-plane so that a network manager can easily control the entire network. NFV is a technology that is actively deploying a network function such as Deep Packet Inspection (DPI), Firewall, even mobile communication equipment-eNode B, Mobility Management Entity (MME), and Home Subscriber Service (HSS)—according to a network manager’s requirement. Because such technologies (SDN, NFV) are designed entirely for network managers, some requirements of network users are hard to meet. For instance, Quality of Service (QoS) policies are decided/set by network managers. Thus it could be different from network users’ requirements. In addition, if network users’ requirements are reflected only through network managers, immediate policy change is impossible when needed. In this context, existing technologies have unsolvable problems when these are taken into the network managers’ point of view. This paper proposes a novel user-centered network to solve the problems mentioned above: Function-Oriented Networking (FON). The network element supporting FON in a network is defined as FON Switch. Assume that the FON Switch can handle such a complex process including OpenFlow and classical functions (sort of Router, L2 Switch). In each FON Switch, FON is being processed as depicted in Figure 1 . Multiple FON Functions can be operated. FON Function is defined as a network function which processes something not specific. It can run classic routing protocols such as RIP and OSPF. This aspect is similar with NFV. However, FON enables network users to change network policy; FON Function can be dynamically deployed/deactivated on FON Switch by network users. Figure 2 shows this scenario. Assume that: (1) a network user begins a new service. This new service needs the network policy corresponding to it; (2) this service requires End-to-End Delay to be the lowest. Then, by using FON system, network users can apply new policy that is matched to each Service’s characteristic immediately by deploying New FON Function to each FON Switch. What follows is each Enabling technology needed to realize such FON. The first one is OpenFlow. As a matter of fact, the network manager is not a critical point in the first article of OpenFlow. Rather, Ref. [ ] proposed a 5-Tuple by analyzing common functions need to realize QoS, Firewall for each vendor. It tried to develop a network switch where various experimental policies can be applied by using the open-API OpenFlow. This paper focuses on the early OpenFlow philosophy that could be a useful tool to realize a policy for constructing a Programmable Network. The second one is a Network slicing technology that can fulfill FON. If network users can totally control the network resources on the network, a serious security problem can be caused. This problem can be solved if a certain amount of resources is assigned through network slicing. To realize the scenario in Figure 2 , FON must provide such technologies as below. (1) According to network user requirements, a new network function, FON , can be deployed/deleted on/from FON ; (2) FON should provide by making network resources abstractive so that FON can be easily implemented to apply new network policy. Especially, new policy is related directly to Routing policy . Because FON requests direct policy change, method is more well-matched than the In order to implement the FON proposed in this study to run on real network devices, and additionally to conduct experiments using ACO in FON, we built a physical device-based test bed. The test bed was tested by running a software image that implements the FON function on the Linux operating system on real Raspberry Pi Single Board Computer (SBC). The FON is designed to operate on Layer 3 IP protocols and layer 2 Ethernet, Bluetooth, and WLAN. For this purpose, a function to receive and forward FON messages in layer 2 upper layer and layer 3 upper layer has been developed. The reason for using Raspberry Pi in building a testbed is because you can construct a large-scale real network, and you can directly create network devices that send and receive traffic. However, it is relatively inexpensive and offers the advantage of being able to operate the same as real network equipment. In addition, it supports OVS to support OpenFlow, and it is possible to receive control of the actual SDN controller. In the case of the OpenFlow controller, ONOS was used, and the configuration and performance evaluation of the various shapes considered in the papers were performed through the corresponding test bed. In addition to wired Ethernet, wireless networking has been experimented with. 3.1. FON Identifier Each Node can have several physical interfaces. To represent these interfaces, the Unique Identifier is needed. So FON Identifier is defined as the Unique Identifier and assigned to each FON Switch. For the compatibility with existing OSPF for IPv4 and IPv6, FON ID have a length of 32 bits, which is the same length of router ID in OSPF. 3.2. FON Forwarding Table One of FON’s main purposes is to provide FON Function with an API abstracting system’s routing related-resource. Let us take the Linux machine as an example. For communication, ARP table and Routing table have to be controlled in a lump and also Flow table of Virtual Bridge has to be managed as well. In order to do this, FON constructs FON Forwarding Table by using FON ID. It manages various tables uniformly through this. FON manages these tables and provides APIs related with FON Forward Table, and FON Function can change easily network policy by modifying a FON Forwarding Table only; in other words, FON Function does not have to modify ARP, Routing, and Flow tables directly. 3.3. Design and Processing of FON Packet FON Packet is a packet used in a communication between FON switches. FON is not using IP protocol so FON Packet is encapsulated directly into Ethernet Frame. FON Packet is divided into FON data packet and FON ADV packet by ethertype field in Ethernet header. For FON Adv packet, its purpose is to synchronize and notify adjoining nodes, which is similar to LLDP. FON Adv packet uses ETH_P_FON_ADV(0xFF03) as a ethertype field. This type of packet is generated and received only through FON, not by FON Function. FON data packet is a packet for FON Function. FON Function exchanges each other using it. FON data packet uses ETH_P_FON(0xFF05) as a Figure 3 shows the design of packet. Each field is described in Table 1 3.4. Telecommunication Design between FON and FON Function There are two communication messages/channels between FON and FON Function. One is Synchronous message/channel, where FON Function asks for FON and FON sends a response. The other one is Asynchronous message/channel, where FON Function is notified when some event occurs. 3.4.1. For Synchronous Message All kind of messages in this type are messages that request to FON from FON Function. They have a typical Request-Response design. Figure 4 shows the format of Request message. Table 2 represents the definitions of every field. Table 3 lists all message types and describes them. Figure 5 shows the format of response message. Table 4 represents the definitions of response message’s every field. 3.4.2. For Asynchronous Message All kind of messages in this type are messages which notify FON Functions when some event occurs in FON. In this context, an event is such that: change of FON ID, FON Forward Table, etc. Typical Asynchronous message is passing FON data packets to a FON Function according to func_type filed in FON packet. 3.5. Processing and Exchanging FON Data Packets between FON Switches Figure 6 illustrates how FON switches process and exchange the FON Packet. (1) A network user sends a FON packet to FON Switch 1. (2) FON Switch1 sends it to proper FON Function through func_type of FON Packet. (3) After proper process, requests FON to send FON Packet. (4) Deliver packet to FON Switch 2. … Repeat (2)–(4). 4. FON Function—Ant Colony Optimization Routing in Network The previous section explains that OpenFlow is a novel technology and it can do many things such as QoS Policy. But it has a limit; many papers only focus on the perspective of central managing but not the needs of users such as datacenter, game server, etc. To overcome the limit of OpenFlow, this paper proposes and illustrates FON and its architecture. This section will discuss FON function and its implementation in a specific condition. Assume that a certain network service, such as a real-time game server, needs two requirements: (1) minimum End-To-End Delay; (2) Fast path recovery if a link failure occurs. In this case, it is reasonable to choose ACO algorithm as a routing, because the ACO algorithm has these characteristics. Actually, ACO routing does not minimize the delay of transmission but the distance (hops) of a source and destination [ ]. But, if a packet passes though the shortest path, then we can assume that the first sentence is true because if all network elements have same performance, the assume is true. In the rest of this section, the ACO algorithm background is explained. Then, how to implement ACO routing in the Ethernet network is illustrated. 4.1. Background ACO was first studied and established by M. Dorigo. It observes ants’ behavior and imitates it. The original study of ACO algorithm was to solve TSP or vehicle routing problems and network routing problems. Especially, ACO is expected to be a good solution for dynamic topology and ad-hoc networks such as mobile ad-hoc network (MANET) [ Compared to the existing network, MANET has different characteristics. So existing proactive routing protocol has disadvantages in MANET; dynamically forming network, no infrastructure, etc. To overcome these problems, the reactive routing protocol such as DSDV, TORA, and AODV was proposed [ ]. However, refs. [ ] showed that novel routing algorithms that applied the ACO algorithm can be more effective than AODV or DSR. According to our research [ ], most of the papers were studied using a simulator, and there is no ACO routing implementation for the real network. But this paper focuses on the real network routing problem, so our first objective is the implementation of ACO routing agent running in the complex enough but not dynamic topology real network. We will propose algorithms which are powerful when the assumption holds, analyze them, and evaluate their performance using our ACO routing agent. 4.1.1. Ant Density Ant Density model [ ] has the characteristic that the pheromone is updated without considering any distance or length concept like the length of the path that an ant passed. Therefore, the pheromones of all nodes an ant passed are increased by the same amount. The model is represented in is a constant, is an evaporation rate of the pheromone. 4.1.2. Ant Quality Ant Quality model [ ] considers the distance of neighbors when updating the pheromone. So unlike Ant Density model, the pheromone updating is influenced by the distance, and the pheromones of all nodes an ant passed are increased by each specific amount. But this model does not consider the total length of the path; the optimization is accomplished locally. So in some bad cases, this model cannot find the optimized path. The model is represented in: $τ ← ( 1 − ρ ) τ + Q d i , j$ is a constant, is an evaporation rate of the pheromone and $d i , j$ is a distance between neighbor nodes 4.1.3. Ant System Ant System model [ ] is an advanced model of Ant Quality. Unlike Ant Quality model, this model considers the total length of the path that an ant passed. So this model reflects the global information and updates the pheromones using this; the performance is better than the Ant Quality model’s. However, there is a disadvantage. If the pheromones of the nodes are far from a source node, they are very slowly converged. The model is represented in: is a constant and is the tour length of the ant, is an evaporation rate of the pheromone. 4.1.4. Ant-Q Ant-Q model [ ] is the Ant Colony Optimization Model applied to the Q-learning algorithm. The significant characteristic of this model is that it does not only consider neighbor nodes, but also next-next nodes. Because of this feature, non-adjacent nodes also must be taken into consideration and this non-adjacent node information should be exchanged to implement this model in real networks. For this, our previous work concluded that it is improper in the real network because exchanging the pheromone tables has a very high cost. The model is represented in: $τ ← ( 1 − ρ ) τ + α ( Δ τ D , i + γ ∑ z ∈ J j ( i ) τ i , z )$ $J j ( i )$ is to be visited nodes of the given ant at node 4.1.5. Ant Colony System Any Colony System [ ] is a very similar model with Ant System. But it updates the pheromone only when the ant passed the optimized path. With the simple principle, the performance is better than Ant System’s. $τ ← ( 1 − ρ ) τ + Q Q = { constant , if given ant came from the optimized path 0 , else$ Every ant knows that he passed the optimized path or not. So every ant must share the information and the colony (also called node) keeps information as a pheromone table. Ref. [ ] called this “communication”. 4.1.6. The Structure of the Pheromone Table To apply a ACO algorithm into a network routing problem, Ref. [ ] suggested a data structure to represent pheromone information— the pheromone table . According to [ ], the pheromone table consists of rows—destinations and columns—neighbors. The table structure is depicted in Figure 7 . In the rest of this paper, the concentration of a pheromone is represented as $τ D , i$ when given a destination and a neighbor node 4.1.7. Source Update Ref. [ ] proposed a Source Update algorithm which suggests new ways to update pheromones. Figure 8 a illustrates the way pheromone update in general ACO algorithm. In this paper, the way will be called as Destination Update Destination Update works this process: (1) select a neighbor node which ant will go through. The selected neighbor node will be called ; (2) increase $τ destination , neighbor _ out$ which is a concentration of pheromone in the pheromone table given a pair of . But Source Update works in slightly different ways. Figure 8 b illustrates Source Update : (1) when ant come in a node from a neighbor, the neighbor will be called ; (2) increase $τ source , neighbor _ in$ which is a concentration of pheromone in the pheromone table given a pair of (in fact, the way of Source Update is widely used already. Network routing protocols, such as RIP and OSPF, calculate the distance and update the routing table when receiving the routing information corresponding to the ant . But [ ] is meaningful that this method is applied for the first time). Source Update is superior to Destination Update; Source update prevents the wrong increment of pheromone concentration. Because Destination Update selects a next node (neighbor_out) and increases the pheromone concentration with no guarantee that an ant will reach a destination, so the ant may not find a destination. It leads to the wrong increment of pheromone concentration. But Source Update increases the pheromone concentration with the guarantee that ant can reach a source through the neighbor, neighbor_in. It makes a huge difference and source update–related experiment will be discussed in the next section. 4.2. Previous Works 4.2.1. Ant Local Our previous work [ ] presented a new perspective about Ant Colony Optimization. Ref. [ ] pointed out a problem about classic ACO algorithms: Ant System and Ant Colony System. These ACO models focus on the distance between source and a given node. But in fact, to optimize the path at a given node, the desirable consideration is the distance between a given node and destination. In this reason, we proposed Ant Local model in [ $τ ← ( 1 − ρ ) τ + Q l o c a l _ m i n D , i$ Also using (6), at our previous project [ ], proposed and implemented Ant Normalizing model $τ ← ( 1 − ρ ) τ + Q ( l o c a l _ m i n D , i − min k ∈ N e i g h b o r s ( l o c a l _ m i n D , k ) + 1 ) 2$ 4.2.2. Simple Backtracking Update The ACO algorithm is a kind of depth-first search (DFS) based on probability and also it is called a metaheuristic algorithm. If an ant enters into a wrong node that cannot lead to a destination, it does backtracking. Our previous work proved that if backtracking occurs, that information can be used for pheromone updates for a better search in [ All ants are divided into two types: Forward Ant and Backward Ant. Forward Ant is a request packet to find a destination. If it finds a destination, it changes into Backward Ant. Backward Ant is a response packet corresponding to the given Forward Ant. At a node, Forward Ant has three options; changing into Backward Ant, starving (exceeding the given TTL) and backtracking. If a Forward Ant finds a destination, backtracking never happens. Backtracking occurs only when a Forward Ant cannot find the destination D through a neighbor node i. In other words, there is no destination D in any nodes that can be reached through the node i, so we can assert that if a Forward Ant goes to the node i again then it will never find a destination D. So from this analysis, we propose the Simple Backtracking Update algorithm (Algorithm 1): Algorithm 1 Simple Backtracking Update 1: Ant packet has been received 2: if Ant is a backtracked from a neighbor node I then 3:$τ D , i ← 0$ Where $D$ is the destination of an ant. 4: else 5: Do nothing 6: end if 4.2.3. The Fast Path Recovery Algorithm The Fast Path Recovery Algorithm is our past proposed algorithm in [ ]. In fact, the ACO algorithm has a mechanism to recover the path even without this algorithm. But without this, it takes a long time to recognize an occurrence of link failure. So this algorithm is used to force to rediscover a new path between a source and destination if a network link failure occurs. This algorithm introduces a new variable called Endurance; this variable has the same concept of Timeout used in existing TCP. Figure 9 and Algorithm 2 explain how this algorithm works. Algorithm 2 The Fast Path Recovery Algorithm 1: For all neighbor nodes i 2:Visited ← False 3:Endurance ← MAX Endurance 4:Pheromone ← 0 5: End for 4.3. Design of ACO Packet The packet format is depicted in Figure 10 . Each field is described in Table 5 . These are similar in that these fields record an array of nodes but these fields have their own purpose. record all nodes so it is useful when debugging. field is used to know shortest path. field comes from [ ] and prevents revisit and the graph cycle. Whenever an ant packet visits a node, the size of the packet increases. So the maximum number of visited nodes depends on the value of having an upper bound as (8): $FON Header Size + ACO Fixed Field Size + n w a l k s ∗ 4 Bytes + n p a t h ∗ 4 Bytes + n v i s t e d ∗ 4 Bytes + n d i s t s ∗ 2 Bytes < = MTU$ In the Ethernet network, the default MTU is 1500 Bytes. But are not fixed so cannot be calculated. But in the worst case, can be calculated. Assume that the ant visits a new node always. Then (9) holds true. Finally, Equation (10) can be obtained: $n w a l k s = n p a t h = n v i s t e d = n d i s t s = i n i t _ t t l , When worst case$ $i n i t _ t t l < = 91.5$ So in the Ethernet network, the ant packet can visit at least 91 nodes. 4.4. Design of ACO Packet Packet Transmission Process Figure 11 illustrates how packets (ants) are handled on the ACO algorithm. (1) Packet in (2) If the ant is not registered in the node’s pheromone table, it registers in the pheromone table. (3) If the ant is backtracked, perform the Simple Backtracking algorithm. Then proceed to step (5). (4) Updates RX statistical information of the pheromone table. The received number, the byte, and the shortest distance to the transmission destination. (5) Update the pheromone concentration for the destination entry in the pheromone table using the Source Update algorithm and the Ant Colony System algorithm. (6) When you reach the destination, change the ant type from Forward (Request) to Backward (Response). (7) Decrease TTL. (8) If TTL is 0, go to step (11) and finish. (9-1) If it is a Forward type (9-2) If it is Backward type (10) And updates the TX information of the pheromone table. (11) End As shown in Figure 11 , not only ACS but various models such as AS, ACS, Ant Local, and Ant Normalizing are implemented. 4.5. Decision of Ant Model Before implementing ACO routing, ACO model must be determined because only one of them, ant system, ant colony system, ant local, ant normalizing, etc., can be used. In fact, the previous section illustrates how to process an ant packet implicitly assuming that ant colony system model and source update are chosen. In the next section, each ant model performance will be shown. It will be the answer to why we chose the combination of ant colony system with simple backtracking update, source update and fast path recovery; this combination showed the best performance. 5. Performance Evaluation and Analysis In this section, the AS and ACS models are set as the control group because these are typical and representative models of ACO algorithm. The Source Update, Simple Backtracking Update, and Fast Path Recovery algorithm are applied with AS and ACS and these cases were conducted and analyzed. A focus on the destination discovery time (as discrete time) shows how the theoretical performance In addition, when applying ACO algorithm as on-demand routing into an Ethernet network, two network performance measures are defined based on actual time rather than discrete time and the relationship between them is derived and verified from the experiment. 5.1. Performance Evaluation of Each ACO Model All experiments were conducted in the topology depicted in Figure 12 using Mininet [ ]. The topology came from [ ]. The node index was randomly assigned a value from 0 to 54. Common parameters in experiments are listed in Table 6 . The reason why we did not build the test bed based on the actual physical equipment mentioned above is that the structure of the network is composed of more than 50 nodes and the complexity of building a test bed based on actual equipment is taken into consideration. However, since all the nodes operating on Mininet are the software created in Section 3 above, they send and receive the actual Ethernet on the virtual machine. The performance of each case is measured using the frequency distribution and the mean distance. $Average n$ is defined as the average length of nth received ants when repeating the same experiment. $Average n$ is expressed in (11). $Average n = 1 J ∑ k = 1 J L e n g t h k , n$ is the number of repeated experiments. 5.1.1. Control Groups In the control groups, the performance of the pure Ant System and Ant Colony System algorithms with no additional algorithms was measured. First, this is the experimental result of Ant System. When the pheromone evaporation coefficient was increased from 0.1 to 0.9, it was confirmed that the optimized pheromone evaporation coefficient( ) was about 0.1. The average path length was 22.740940 when it was repeated 100 times. The time series graph of Figure 13 a shows that it is slowly approaching the optimal path. The histogram of Figure 13 b shows that the frequency of the path length between 21 and 23 is largest. Second, this is the experimental result of Ant Colony System. When the pheromone evaporation coefficient was increased from 0.1 to 0.9, it was confirmed that the optimized pheromone evaporation ) was about 0.1. The average path length was 24.748670 when it was repeated 100 times. The time series graph of Figure 14 a shows that it is not approaching the optimal path. This result is unexpected because the ACS algorithm is known to have the best performance [ ]. This result will be explained in a later section. The histogram of Figure 14 b shows that the frequency of the path length between 21 and 23 is largest. 5.1.2. Simple Backtracking Update In this section, it was measured that the performance of Ant System and Ant Colony System algorithms with Simple Backtracking Update. First, this is the experimental result of AS with Simple Backtracking Update . When the pheromone evaporation coefficient was increased from 0.1 to 0.9, it was confirmed that the optimized pheromone evaporation coefficient( ) was about 0.1. The average path length was 20.325750 which is better than the result of Ant System only. The time series graph of Figure 15 a shows a similar result but it is approaching the optimal path faster than Ant System only. The histogram of Figure 15 b shows a similar result of AS only. Second, this is the experimental result of ACS with Simple Backtracking Update . When the pheromone evaporation coefficient was increased from 0.1 to 0.9, it was confirmed that the optimized pheromone evaporation coefficient( ) was about 0.1. The average path length is 22.015080 which was a better result than ACS only. The time series graph of Figure 16 a shows that it is not approaching the optimal path and is similar to Figure 14 . But the histogram of Figure 16 b shows that the frequency of 14 (the optimal path) is 3-times than Figure 14 5.1.3. Source Update In this section, the performance of Ant System and Ant Colony System algorithms with Source Update was measured. First, this is the experimental result of AS with Source Update . When the pheromone evaporation coefficient was increased from 0.1 to 0.9, it was confirmed that the optimized pheromone evaporation coefficient( ) was about 0.3. This is a different coefficient( ) compared with AS only. The average path length was 14.998210. This performance overwhelmed the previous cases. The time series graph of Figure 17 a shows that it is very fatly approaching the optimal path than AS only. The histogram of Figure 17 b shows a totally different result compared with a result of AS only. Almost ant packets passed the optimized path. Second, this is the experimental result of ACS with Source Update . When the pheromone evaporation coefficient was increased from 0.1 to 0.9, it was confirmed that the optimized pheromone evaporation coefficient( ) was about 0.3. It is a different result compared with the previous cases. These results imply that if Source Update is applied, then the optimized pheromone evaporation coefficient( ) will be larger. The average path length is 14.219080 which was the best result than any other case. The time series graph of Figure 18 a shows that it is very fatly approaching the optimal path than ACS only. After about 2 evaporation cycles, the Optimized path was established. The histogram of Figure 18 b shows a totally different result compared with the result of ACS only. Almost ant packets passed the optimized path except for a very small number. 5.2. Network-Related Performance Measure In this paper, we apply the ACO algorithm to network routing. Therefore, it is important how the ACO algorithm affects network performance. One of the main concerns of this paper is the path recovery time in the end-to-end delay and link failure and the link overhead caused by the routing itself. In this section, we analyze how these two network performance indicators can be related to the ACO parameters and how there is a relation between the two values. 5.2.1. Definition of Network Performance Measure If transmission delay and processing delay can be ignored, the $R e c o v e r y T i m e ( R )$ can be expressed as a sum of $L i n k E r r o r D e t e c t i o n T i m e ( T e )$ $R e − S e a r c h i n g T i m e ( T s )$ . So it is written as (12): And $L i n k O v e r h e a d ( O )$ is defined as Load (bit-per-sec) caused by routing and satisfies these conditions: All ants travel only through a single path. There is no loss, so 100% of the response (backward ant) is received. These two conditions mean the worst case. In other words, $L i n k O v e r h e a d ( O )$ means the upper bound of the actual measured load. 5.2.2. Relation of Overhead and Recovery Time To figure out the relation of Recovery Time , we calculate how many messages can pass to a given link. According to the variables definition in Table 7 , N ants (packets or messages) occur per cycle. Considering the round trip of the packet and assuming that request packet size and response packet size are the same, the message flows twice in total. the number of messages per second $M$ can be written as (13): Link Overhead can be easily obtained by multiplying by the size of the packet as (14): It is not easy to calculate (14) because the size of the packet T is variable. But the upper bound of Link Overhead $O upper$ can be easily obtained by substituting MTU from T. Recovery Time R can be expressed using ACO parameters listed in Table 7 . First, Link-error detection time $T e$ Shortest Path Search time $T s$ are random variables. Both Link-error detection time Shortest Path Search time can be expressed in multiples . This is because the computation time of the ACO algorithm is proportional to the cycle. $T e$ $T s$ are expressed as Also (14) can be re-expressed as (17) using (15) and (16): are discrete random variables depending on the selected ACO algorithm. Finally, (18) can be obtained by multiplying the (14) by the (17): Even the values of $p$, $q$ are random variables, but they can be treated as constants since they are very small in the measurement. Therefore, the right side of Equation (4) can be treated as a constant. From Equation (4), we can see that the overhead increases exponentially with decreasing Recovery Time. 5.3. Network Performance Measure With Ant System; $N = 10 , E = 0.2$ , the average of was measured to be about 2 and 2, respectively. (Cycle) value is 50 ms, 100 ms, 200 ms $⋯ ⋯$ 2000 ms, etc., the upper bound value of the overhead is given by (19): Based on the experimental data, a regression curve was obtained by the regression analysis as (20): The curve of Equations (1) and (2) are depicted in Figure 19 . The -axis is the Recovery Time in sec. The -axis is expressed as overhead in Mbps and expressed in log-scale. Figure 19 shows that the upper bound and actual values are quite different because the packet size is calculated assuming MTU (1500 bytes) when deriving the theoretical value. As a result of the actual measurement, it is analyzed that the packet size is about 10% of the MTU and 20% of the MTU when the packet is transmitted. Also, it shows that if Recovery Time decreases, then Link Overhead increases exponentially. Therefore, there is a limitation in reducing the recovery time in real networks. 6. Conclusions In this paper, we proposed and developed Function-Oriented Networking, a platform for network users. It has a different philosophy as opposed to technologies for network managers of Software-Defined Networking technology, OpenFlow. It is the technology that can immediately reflect the demands of the network users in the network, unlike the existing OpenFlow and NFV, which do not reflect directly the needs of the network users. It allows the network user to determine the policy of the direct network, so it can be applied more precisely than the policy applied by the network manager. This is expected to increase the satisfaction of the service users when the network users try to provide new services. In addition, we developed a FON function as a method to demonstrate actual FON. This FON function performs on-demand routing for Low-Delay Required service. We analyzed the characteristics of the ACO algorithm and found that the algorithm is suitable for low-delay required services. It was also the first in the world to implement the routing software using Ant Colony Optimization Algorithm in the real Ethernet network. In order to improve the routing performance, several algorithms of the Ant Colony Optimization Algorithm have been developed to enable faster path search-routing and path recovery. The relationship between the network performance index and the ACO routing parameters is derived, and the results are compared and analyzed. Through this, it was possible to develop the ACO In this paper, we aimed to solve the problems where conventional biology-inspired networking such as ACO algorithm cannot be implemented in actual networks. Although there have been various researches for Biology-Inspired-Networking, it was the goal of this paper to solve the problem that it is almost impossible to send and receive real network traffic by applying technology such as ACO to network equipment. Through this paper, the proposed technology has made a great contribution to enabling traffic sending and receiving based on technologies such as ACO in real networks. Therefore, biology-inspired-networking research such as ACO is being carried out on real networks instead of simulations, and improvement of the Biology-Inspired-Networking algorithm is part of future research. In the future, based on the Biology-Inspired-Networking algorithm, we will analyze the difference between theory and practice in sending and receiving of the Internet traffic, and we will continue to research the improvement of Biology-Inspired-Networking based on this. This work was supported by Institute for Information & Communications Technology Promotion (IITP) grant funded by the Korea government (MSIT) (2014-0-00547, Development of Core Technology for Autonomous Network Control and Management). Author Contributions All the authors contributed equally to this work. All authors read and approved the final manuscript. Conflicts of Interest The authors declare no conflict of interest. Figure 16. Ant Colony System with simple backtracking algorithm. (a) Time series graph; (b) Histogram. Field Description sid Source FON identifier did Destination FON identifier type Field for identifying FON Function. 2 byte Paylen Payload length. 2 byte Payload Variable length Field Description func_type Represents Identifier in FON Function. FON can recognize in which FON function the message generated through this Field. msg_type Represents types of message. Refer [Table 1]. tot_len Represents the length of message including message’s header length in bytes. data Include Payload matched with given message type (func_type). Value of msg_type Description MSG_TYPE_REG = 0 For registering a new FON Function on FON MSG_TYPE_DEREG = 1 For deregistering a new FON Function on FON MSG_TYPE_SENDTO = 2 Request to Request FON for Transmitting FON Packet MSG_TYPE_TABLE_ADD = 3 Add (modify) Entry to FON’s Forward Table MSG_TYPE_TABLE_DEL = 4 Remove Entry in FON’s Forward Table MSG_TYPE_TABLE_GET = 5 Acquire All entries in FON’s Forward Table MSG_TYPE_HOST_GET = 6 Check current FON ID in FON Field Description result Inform success of failure to the request. 0 for success, error codes for failure tot_len Inform entire response length including header error_string Inform specific error type in text when occurring an error data notify all other information except success or failure to FON Function (filled in target field) Field Description source The node that generates an ant packet. destination The node that an ant packet should reach. direction It means the type of ant. There are two types of ant; forward is a request packet. backward is a response packet. init_ttl It is an initial time-to-live. Using this filed, each node can calculate the distance between source and current node. cur_ttl Current time-to-live. nwalks The length of walks field. npath The length of path field. nvisted The length of visited field. ndists The length of dists field. padding This field is a 2 bytes padding for a 4 bytes-align. walks The array of FON ID. This field records all nodes that ant (packet) dropped by. So, this field includes all nodes that backtracked nodes. path The array of FON ID. This field records all nodes that ant (packet) dropped by, but does not include backtracked node. Using this field, the shortest path from source to destination can be calculated. visited Visited field is the array of node IDs and where all visited nodes will be recorded in. It is used to confirm that duplicate visits have been made and to prevent the graph cycle. dists Visited field is the array of the distance. Each element has 2 byte length. And nth element is the distance between source and nth visited’ element. In other words, dists[i] == Dist (source, visited[i]) where i is the index of array. Parameter Value Source 0 Destination 15 The least distance of Source and Destination 14 hop Probability of packet loss 0% The number of packets per evaporation cycle 10 Total cycles per each experiment 100 Total packets (ants) per each experiment 1000 The number of repetitions of the same experiment 100 times Variable Description M Number of messages per second (Times/s) C Pheromone evaporation cycle (s/cycle) N Number of ant per each cycle (Times/cycle) T Size of the packet (Byte) O Link overhead (bps or Mbps) R Recovery time (s) $T e$ Link-error detection time (s) $T s$ Shortest Path Search time (s) © 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http:/ Share and Cite MDPI and ACS Style Sim, Y.-B.; Lee, S.; Lee, S. Function-Oriented Networking and On-Demand Routing System in Network Using Ant Colony Optimization Algorithm. Symmetry 2017, 9, 272. https://doi.org/10.3390/sym9110272 AMA Style Sim Y-B, Lee S, Lee S. Function-Oriented Networking and On-Demand Routing System in Network Using Ant Colony Optimization Algorithm. Symmetry. 2017; 9(11):272. https://doi.org/10.3390/sym9110272 Chicago/Turabian Style Sim, Young-Bo, SeungGwan Lee, and Sungwon Lee. 2017. "Function-Oriented Networking and On-Demand Routing System in Network Using Ant Colony Optimization Algorithm" Symmetry 9, no. 11: 272. https:// Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. 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How do I create a lesson sequence?How do I create a lesson sequence? How do I create a lesson sequence? This is a question I get asked a lot, both by pre-service and in-service primary teachers. You may feel you are great at designing single lesson ideas that are related or whole programs, but maybe the flow could be better – more conceptually than topically related. For example, the difference between designing a group of lessons for say addition and subtraction (topic-based), to designing a sequence of lessons for say a concept within that topic such as; understanding that combinations to 10 can be applied as the bridging to ten strategy. Lesson designing and sequencing is an art, it is the heart of teaching (something that recently has been suggested is better done by others outside the classroom context). It is time consuming yes, but it’s pretty much my favourite part of teaching, bar seeing when students have an ‘a-ha’ moment of discovery! With more than 20 years teaching experience, I am still learning this art form. And, like any art form, you learn by practice, you learn by studying others, and you learn by trial and error. Not every lesson or lesson sequence is going to work, not every lesson or lesson sequence is going to be ‘wow’, not every lesson sequence is going to be the same. But every lesson sequence in mathematics should have two goals – be open enough for change and flexibility, be focused on making connections for students. I’m sure I can think of more goals than these two – but they are at the top of my list. Open for change and flexibility I like routine, many students also like routine – this doesn’t mean every mathematics lesson needs to look the same, have the same elements, run for the same length of time or follow the same script. I’ve spoken in a previous blog about the Launch, Explore, Summarise model for individual lesson structure – even this structure is flexible. One the first day I introduce a concept, within the launch phase it might be more connecting to prior knowledge or mind mapping what my students already know and within the explore phase there may be a longer explicit (I prefer the word intentional) teaching to build some ground work. The follow day or days might be quite different, they might have an even shorter launch phase into a problem to solve or activity to investigate. If the students are deep in the investigation, we might stop, have a short summarise phase then start the next day straight into the explore phase again – continuing the investigation activity. Whatever the structure, summarising at the end of each lesson session is paramount. Being open for change also means my lesson sequence, although planned for, still has some wiggle room based on the students. Having your lessons student-driven doesn’t negate the role of the teacher nor does it mean the students are in the driver’s seat. What it means is that as the teacher I’m not always talking (I’m listening), I’m not always teaching (I’m observing), and I’m not always telling (well, I’m never telling!). Questioning plays a large role in leading great mathematics lessons – these questions can be planned for and should include both enabling and extending prompts to support student understanding and learning. Allow conversations to go where students take you, segues are good, but not too far that they derail your conceptual plan. For example, take my conceptual focus above: understanding that combinations to 10 can be applied as the bridging to ten strategy. When asking students for combinations to make 7, a student may suggest “what about zero and seven?” you may not have expected this response but now is a great time to divert the lesson and explore this concept together. If on the other hand, a students says “what about eight minus one?”. I would say, this is a great relationship to subtraction, let’s write that up on the board so we can explore it another time. You need to know when to reign students in. As the conceptual focus is on using combinations to add, although this student has made a great connection (between addition and subtraction), it doesn’t add to the learning intention for this sequence. It’s ok to come back to student ideas later. Viewing your lesson sequence as changeable and flexible, also means not getting stuck in 40 minute or one hour lots. A lesson you have planned may go over 2 or three days, “a lesson is never finished” (Charles Lovitt), there is always more to notice, wonder, investigate, and connect. Your conceptual focus like my example: understanding that combinations to 10 can be applied as the bridging to ten strategy, could be your learning intention for 3 days or 3 weeks. When your focus is small, you can dig deeper into the concept and you then know that what you plan will be connected. If I just had a focus on ‘addition’, the lessons might be on anything such as; addition as the inverse of subtraction, addition strategies, addition using place value groupings, addition using a number line, addition patterns in number bonds, addition number sentences, turn around facts (commutative), the list goes on … When we plan lesson sequences based on an overarching concept like addition, it is very hard to see how the lessons connect. But when we dig deeper to the smaller concept, our job of planning becomes easier, more directed by the choice we made for the learning intention. Always think about – what do I want my students to know. Making connections for students Now that you’ve started to think deeper about the smaller concepts–what you want your students to know–connections are the next thing to focus on. Mathematical connections to other mathematical concepts, to the students’ own knowledge, understanding, and contexts, and to real life, is similar to the idea of text-to-text, text-to-self, text-to-world from an English perspective (see Math Connections blog). Go slow, don’t rush to the next idea (see my blog on slow teaching). In NSW, apart from in Kindergarten, we have two years that span a stage of learning. There is plenty of time to revisit, revise and refine mathematical concepts. Also remember that we collectively have each students for seven years, mathematical connections build over time, don’t expect students to make every connect in one calendar year. Here are some of my top tips for making connections in mathematics lesson sequences: • teach the same lesson using different resources (making connections includes seeing the multiple ways a concept can be represented orally, in written words, in pictures, using diagrams, digitally, symbolically and numerically) • ask questions that lead students to make the connections (what do you already know that can help you? where else might we use this strategy? does this work all the time? when does it not work? Is that always true? what else could we find out?) • teach a number concept in conjunction with (or follow by) a measurement, geometry or data concept or another number concept (not always, but often, we apply number concepts when working on other mathematical tasks eg multiplication to find area, ordering numbers to create graphs, using decimals to represent accurate measurements) This idea can go both ways: number teaching then application, or application then number teaching – explicit (intentional) teaching can happen at any point in your sequence or lesson, it doesn’t always need to come first (see Russo & Hopkins, • create an ongoing map of mathematical connections on your classroom wall (or on a Jamboard). A bit like a map of the world or train map, draw lines of connection between big ‘mathematical cities’. If you regularly make anchor charts of concepts these can be connected over time to build an illustration of your class’ connections My advice would be to start small, plan a sequence of 3 or 4 lessons based on a concept within a topic/area of mathematics. I always head first to the syllabus (Australian Curriculum if you are perhaps in other states than NSW) this helps me locate the smaller concepts to expand. I’d make a mind map of the things I think the students might need time to explore or things they might need intentional teaching about to make the connections and understand the concept. Here’s an example based on my concept within addition and subtraction: Once I have this mind map, I know that the lessons I create for this concept (weather they take 3 days or 3 weeks) will be connected. They are all building on the same, small idea. I can then create further links from this mind map to other areas. For example patterning for the number sentences that show associative and/or commutative properties of addition, or a lesson sequence on applying the bridging strategy to solving problems related to length. Or a sequence on using combinations for subtraction (15 – 7 uses the same knowledge of combinations). I hope these ideas kick-start your thinking about how to ensure your lessons are sequential and build conceptual understanding. Remember to ask yourself, what comes next? where does this knowledge go? where do the students need to use this? why does this learning matter? Categories: Mathematics, Opinion, Pedagogy, SyllabusBy Katherin Cartwright Author: Katherin Cartwright Katherin Cartwright is a passionate mathematics educator and is currently a sessional lecturer and tutor at The University of Sydney teaching mathematics to pre-service teachers in primary education. She has just completed her PhD researching teacher noticing of mathematical fluency in primary students. 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Answers created by MetaPhysik • Back to user's profile • Next • Which pole of a compass needle points to a south pole of a magnet? • What is a motor pathway primarily concerned with volitional control of body parts called? • When the Hoover Dame was completed and the reservoir behind it filled with water, did the moment of inertia of the Earth increase, decrease, or stay the same? • Object displays from 5 metre height reaches at a ground distance of 10 metre what is the final velocity? • A boat pushes off from a pontoon and travels 5m/s East while the pontoon floats away at 2m/s south. Calculate the velocity of the boat relative to the pontoon????? . • A kit is 100 ft high. There are 260 ft of string out and it is being let out at a rate of 5 ft/sec. If this results in the kite being carried along horizontally, what is the horizontal speed of the kite? • On a surface I have two point charges Qa and Qb, 1C and 4C respectivly and are placed at A(-3,0) B(3,0). 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What are the object's average speed and velocity? • An astronaut moves on a circular orbit around the earth.what is the average speed for a whole revolution? • Why pair production and annihilation cannot occur in free space? • Calculate the height from the Earth's surface that a satellite must attain in order to be in geosynchronous orbit and the satellite's velocity? • Question #0c378 • What does the slope of an force vs. displacement time represent? • Question #84f65 • Question #ca63a • Question #7cd1e • Question #69513 • Question #1fce2 • Question #eb843 • What is the reference angle when theta is -1.8? and Why? • If an electron is placed in an electric field with a potential difference of (1.35x10^2) V, the maximum speed the electron can attain is? • Question #88df0 • Question #43ce0 • Can you explain in detail why the answer is D ? • Question #c79bd • How much work does it take to pump the oil to the rim of the tank if the conical tank from #y=2x# is filled to within 2 feet of the top of the olive oil weighing #57(lb)/(ft)^3#? • How do you find the critical numbers of #y = cos x - sin x#? • What is the antiderivative of #x(sinx)^2#? • Next	{"url":"https://api-project-1022638073839.appspot.com/users/mrphysik/answers","timestamp":"2024-11-02T21:59:15Z","content_type":"text/html","content_length":"31037","record_id":"<urn:uuid:b0216cae-3890-4b28-96ec-10e1669606f0>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00222.warc.gz"}
Find the Mean of a List in Python - Code Allow Find the Mean of a List in Python Finding the mean (average) of a list in Python can be done by using the statistics module or manually calculating the sum and dividing by the length of the list. Here are two examples: Example 1: import statistics numbers = [1, 2, 3, 4, 5] mean = statistics.mean(numbers) print(f"The mean of the list is: {mean}") The mean of the list is: 3 Example 2: grades = [85, 90, 92, 88, 95] mean = sum(grades) / len(grades) print(f"The mean grade is: {mean}") The mean grade is: 90.0	{"url":"https://codeallow.com/find-the-mean-of-a-list-in-python/","timestamp":"2024-11-15T02:50:22Z","content_type":"text/html","content_length":"61122","record_id":"<urn:uuid:6e264bba-4aed-49d8-b7a2-744a6aa62102>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00088.warc.gz"}
Fixed-Structure H-infinity Synthesis with Fixed-Structure H-infinity Synthesis with hinfstruct This example uses the hinfstruct command to tune a fixed-structure controller subject to ${H}_{\infty }$ constraints. The hinfstruct command extends classical ${H}_{\infty }$ synthesis (see hinfsyn) to fixed-structure control systems. This command is meant for users already comfortable with the hinfsyn workflow. If you are unfamiliar with ${H}_{\infty }$ synthesis or find augmented plants and weighting functions intimidating, use systune and looptune instead. See Tune Control Systems Using systune for the systune counterpart of this example. Plant Model This example uses a 9th-order model of the head-disk assembly (HDA) in a hard-disk drive. This model captures the first few flexible modes in the HDA. load hinfstruct_demo G bode(G), grid We use the feedback loop shown below to position the head on the correct track. This control structure consists of a PI controller and a low-pass filter in the return path. The head position y should track a step change r with a response time of about one millisecond, little or no overshoot, and no steady-state error. Figure 1: Control Structure Tunable Elements There are two tunable elements in the control structure of Figure 1: the PI controller $C\left(s\right)$ and the low-pass filter Use the tunablePID class to parameterize the PI block and specify the filter $F\left(s\right)$ as a transfer function depending on a tunable real parameter $a$. C0 = tunablePID('C','pi'); % tunable PI a = realp('a',1); % filter coefficient F0 = tf(a,[1 a]); % filter parameterized by a Loop Shaping Design Loop shaping is a frequency-domain technique for enforcing requirements on response speed, control bandwidth, roll-off, and steady state error. The idea is to specify a target gain profile or "loop shape" for the open-loop response $L\left(s\right)=F\left(s\right)G\left(s\right)C\left(s\right)$. A reasonable loop shape for this application should have integral action and a crossover frequency of about 1000 rad/s (the reciprocal of the desired response time of 0.001 seconds). This suggests the following loop shape: wc = 1000; % target crossover s = tf('s'); LS = (1+0.001s/wc)/(0.001+s/wc); bodemag(LS,{1e1,1e5}), grid, title('Target loop shape') Note that we chose a bi-proper, bi-stable realization to avoid technical difficulties with marginally stable poles and improper inverses. In order to tune $C\left(s\right)$ and $F\left(s\right)$ with hinfstruct, we must turn this target loop shape into constraints on the closed-loop gains. A systematic way to go about this is to instrument the feedback loop as follows: • Add a measurement noise signal n • Use the target loop shape LS and its reciprocal to filter the error signal e and the white noise source nw. Figure 2: Closed-Loop Formulation If $T\left(s\right)$ denotes the closed-loop transfer function from (r,nw) to (y,ew), the gain constraint secures the following desirable properties: • At low frequency (w<wc), the open-loop gain stays above the gain specified by the target loop shape LS • At high frequency (w>wc), the open-loop gain stays below the gain specified by LS • The closed-loop system has adequate stability margins • The closed-loop step response has small overshoot. We can therefore focus on tuning $C\left(s\right)$ and $F\left(s\right)$ to enforce $‖T{‖}_{\infty }<1$. Specifying the Control Structure in MATLAB In MATLAB®, you can use the connect command to model $T\left(s\right)$ by connecting the fixed and tunable components according to the block diagram of Figure 2: % Label the block I/Os Wn = 1/LS; Wn.u = 'nw'; Wn.y = 'n'; We = LS; We.u = 'e'; We.y = 'ew'; C0.u = 'e'; C0.y = 'u'; F0.u = 'yn'; F0.y = 'yf'; % Specify summing junctions Sum1 = sumblk('e = r - yf'); Sum2 = sumblk('yn = y + n'); % Connect the blocks together T0 = connect(G,Wn,We,C0,F0,Sum1,Sum2,{'r','nw'},{'y','ew'}); These commands construct a generalized state-space model T0 of $T\left(s\right)$. This model depends on the tunable blocks C and a: ans = struct with fields: C: [1x1 tunablePID] a: [1x1 realp] Note that T0 captures the following "Standard Form" of the block diagram of Figure 2 where the tunable components $C,F$ are separated from the fixed dynamics. Figure 3: Standard Form for Disk-Drive Loop Shaping Tuning the Controller Gains We are now ready to use hinfstruct to tune the PI controller $C$ and filter $F$ for the control architecture of Figure 1. To mitigate the risk of local minima, run three optimizations, two of which are started from randomized initial values for C0 and F0. opt = hinfstructOptions('Display','final','RandomStart',5); T = hinfstruct(T0,opt); Final: Peak gain = 3.88, Iterations = 67 Final: Peak gain = 597, Iterations = 190 Some closed-loop poles are marginally stable (decay rate near 1e-07) Final: Peak gain = 597, Iterations = 186 Some closed-loop poles are marginally stable (decay rate near 1e-07) Final: Peak gain = 1.56, Iterations = 123 Final: Peak gain = 1.56, Iterations = 98 Final: Peak gain = 1.56, Iterations = 98 The best closed-loop gain is 1.56, so the constraint $‖T{‖}_{\infty }<1$ is nearly satisfied. The hinfstruct command returns the tuned closed-loop transfer $T\left(s\right)$. Use showTunable to see the tuned values of $C$ and the filter coefficient $a$: C = Kp + Ki --- with Kp = 0.000846, Ki = 0.0103 Name: C Continuous-time PI controller in parallel form. a = 5.49e+03 Use getBlockValue to get the tuned value of $C\left(s\right)$ and use getValue to evaluate the filter $F\left(s\right)$ for the tuned value of $a$: C = getBlockValue(T,'C'); F = getValue(F0,T.Blocks); % propagate tuned parameters from T to F ans = From input "yn" to output "yf": s + 5486 Continuous-time transfer function. To validate the design, plot the open-loop response L=FGC and compare with the target loop shape LS: bode(LS,'r--',GCF,'b',{1e1,1e6}), grid, title('Open-loop response'), legend('Target','Actual') ans = Legend (Target, Actual) with properties: String: {'Target' 'Actual'} Location: 'northeast' Orientation: 'vertical' FontSize: 8.1000 Position: [0.7851 0.8257 0.1693 0.0884] Units: 'normalized' Use GET to show all properties The 0dB crossover frequency and overall loop shape are as expected. The stability margins can be read off the plot by right-clicking and selecting the Characteristics menu. This design has 24dB gain margin and 81 degrees phase margin. Plot the closed-loop step response from reference r to position y: step(feedback(GC,F)), grid, title('Closed-loop response') While the response has no overshoot, there is some residual wobble due to the first resonant peaks in G. You might consider adding a notch filter in the forward path to remove the influence of these Tuning the Controller Gains from Simulink Suppose you used this Simulink model to represent the control structure. If you have Simulink® Control Design™ installed, you can tune the controller gains from this Simulink model as follows. First mark the signals r,e,y,n as Linear Analysis points in the Simulink model. Then create an instance of the slTuner interface and mark the Simulink blocks C and F as tunable: ST0 = slTuner('rct_diskdrive',{'C','F'}); Since the filter $F\left(s\right)$ has a special structure, explicitly specify how to parameterize the F block: a = realp('a',1); % filter coefficient setBlockParam(ST0,'F',tf(a,[1 a])); Finally, use getIOTransfer to derive a tunable model of the closed-loop transfer function $T\left(s\right)$ (see Figure 2) % Compute tunable model of closed-loop transfer (r,n) -> (y,e) T0 = getIOTransfer(ST0,{'r','n'},{'y','e'}); % Add weighting functions in n and e channels T0 = blkdiag(1,LS) T0 * blkdiag(1,1/LS); You are now ready to tune the controller gains with hinfstruct: opt = hinfstructOptions('Display','final','RandomStart',5); T = hinfstruct(T0,opt); Final: Peak gain = 3.88, Iterations = 67 Final: Peak gain = 597, Iterations = 183 Some closed-loop poles are marginally stable (decay rate near 1e-07) Final: Peak gain = 597, Iterations = 173 Some closed-loop poles are marginally stable (decay rate near 1e-07) Final: Peak gain = 3.88, Iterations = 70 Final: Peak gain = 1.56, Iterations = 98 Final: Peak gain = 1.56, Iterations = 100 Verify that you obtain the same tuned values as with the MATLAB approach: C = Kp + Ki * --- with Kp = 0.000846, Ki = 0.0103 Name: C Continuous-time PI controller in parallel form. a = 5.49e+03 See Also Related Topics	{"url":"https://au.mathworks.com/help/robust/gs/fixed-structure-h-infinity-synthesis-with-hinfstruct.html","timestamp":"2024-11-12T06:17:40Z","content_type":"text/html","content_length":"93776","record_id":"<urn:uuid:fcb535d5-d05d-4b23-b4fa-ce245ff6c446>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00747.warc.gz"}
Turning Sources Off Turning off a source, which is usually used in solving circuits with superposition method, means setting its value equal to zero. For a voltage source, setting the voltage equal to zero means that it produces zero voltage between its terminals. Therefore, the voltage source must insure that the voltage across two terminals is zero. Replacing the source with a short circuit can do that. Thus, voltage sources become a short circuit when turned off. For a current source, setting the current equal to zero means that it produces zero current. Therefore, the current source must insure that no current flows through its branch. An open circuit can do that. Hence, to turn off a current source it should be replaced by an open circuit. How about dependent sources? The voltage/current of a dependent source is dependent on other variables of the circuit. Therefore, dependent sources cannot be turned off. Example I: Turn off sources one by one. Example 1 I) The voltage source: Turning off the voltage source II) The current source: Turning off the current source Example 2: For each source, leave the source on and turn off all other sources. Example 2 Contribution of V_1 Contribution of V2 Contribution of I1 Contribution of I2 Example 3: For each source, leave the source on and turn off all other sources. Example 3 Contribution of V1 Contribution of I1 Recall that dependent sources cannot be turned off. Join the Conversation 1. notes are very helpfull. Thank you. 2. my question is about a 2A current source in series with a 10 ohm resistor, and there is an unknown voltage across the branch. I understand the current source will supply 2 amps of current, regardless of the voltage across it. However, does the 10 ohm resistor also have 2 amps of current thru it (and 20 volts across it) always, completely independent of the voltage across the branch, or does the total current thru the branch (and current thru the resistor) depend on the voltage across the branch? (i may be answering my own question, that is if the voltage influenced the resistor, it would also have to pull more than 2 amps through the current source, and since that impedence is infinite, that can’t be. Is this correct, that the current through the 10 ohm resistor will always be 2 amps?) If this is true, is the voltage across the branch equal to the voltage across the resistor, i.e. 20 volts? 3. i think you are doing a great job sir. i respect you. 4. Sir you bush me forward in course of study thank you sir. 5. Sir. you bush me forward In my course of study thank you sir. 6. What if the current the dependent voltage source depends on is zero? Do we consider it as short circuit? This site uses Akismet to reduce spam. Learn how your comment data is processed.	{"url":"https://www.solved-problems.com/circuits/circuits-articles/839/turning-sources-off/","timestamp":"2024-11-06T20:46:03Z","content_type":"text/html","content_length":"96451","record_id":"<urn:uuid:22cbb7a6-f03c-45a4-8356-13e87b5b671d>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00555.warc.gz"}
Dude. How do you Uncross a cross product? Dude. How do you Uncross a cross product? • Thread starter Saladsamurai • Start date In summary, a cross product is a mathematical operation that takes two vectors as input and produces a third vector that is perpendicular to both of the input vectors. It is commonly used to calculate the area of a parallelogram or the direction of a torque in physics. The formula for calculating a cross product is A x B = (A_y * B_z - A_z * B_y, A_z * B_x - A_x * B_z, A_x * B_y - A_y * B_x), where A and B are the two input vectors. Uncrossing a cross product is not a common mathematical operation and is likely referring to finding the dot product, which is the opposite of the cross product. The result of a cross product Homework Statement If I have [itex]\vec{v}=\vec{\omega}\times\vec{r}[/itex] how do I solve for omega? How do I "cross divide"? Scalar operations are easier, I know. But how do you do this vectorially? FAQ: Dude. How do you Uncross a cross product? 1. What is a cross product? A cross product is a mathematical operation that takes two vectors as input and produces a third vector that is perpendicular to both of the input vectors. It is commonly used to calculate the area of a parallelogram or the direction of a torque in physics. 2. How do you calculate a cross product? The formula for calculating a cross product is: A x B = (A[y] * B[z] - A[z] * B[y], A[z] * B[x] - A[x] * B[z], A[x] * B[y] - A[y] * B[x]) where A and B are the two input vectors. 3. What is the purpose of uncrossing a cross product? Uncrossing a cross product is not a common mathematical operation. It is likely that the person means to find the dot product, which is the opposite of the cross product and is commonly used in vector algebra to calculate the angle between two vectors. 4. How do you interpret the result of a cross product? The result of a cross product is a vector that is perpendicular to both of the input vectors. The direction of the resulting vector can be determined using the right-hand rule, where the direction of the resulting vector is perpendicular to the plane formed by the two input vectors and follows the direction of the thumb when the fingers of the right hand are curled in the direction of the first vector and then the second vector. 5. Can a cross product be performed on more than two vectors? No, a cross product can only be performed on two vectors at a time. If you have more than two vectors, you can perform multiple cross products to find the final result. However, the number of vectors must always be even in order for the cross product to be valid.	{"url":"https://www.physicsforums.com/threads/dude-how-do-you-uncross-a-cross-product.287834/","timestamp":"2024-11-06T20:25:33Z","content_type":"text/html","content_length":"79802","record_id":"<urn:uuid:cc911390-3668-4835-ba9b-2adedbc3753a>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00789.warc.gz"}
Semianalytical estimates of scattering thresholds and gravitational radiation in ultrarelativistic black hole encounters Ultrarelativistic collisions of black holes are ideal gedanken experiments to study the nonlinearities of general relativity. In this paper we use semianalytical tools to better understand the nature of these collisions and the emitted gravitational radiation. We explain many features of the energy spectra extracted from numerical relativity simulations using two complementary semianalytical calculations. In the first calculation we estimate the radiation by a "zero-frequency limit" analysis of the collision of two point particles with finite impact parameter. In the second calculation we replace one of the black holes by a point particle plunging with arbitrary energy and impact parameter into a Schwarzschild black hole, and we explore the multipolar structure of the radiation paying particular attention to the near-critical regime. We also use a geodesic analogy to provide qualitative estimates of the dependence of the scattering threshold on the black hole spin and on the dimensionality of the spacetime. All Science Journal Classification (ASJC) codes • Nuclear and High Energy Physics • Physics and Astronomy (miscellaneous) Dive into the research topics of 'Semianalytical estimates of scattering thresholds and gravitational radiation in ultrarelativistic black hole encounters'. Together they form a unique fingerprint.	{"url":"https://collaborate.princeton.edu/en/publications/semianalytical-estimates-of-scattering-thresholds-and-gravitation","timestamp":"2024-11-07T02:57:32Z","content_type":"text/html","content_length":"53101","record_id":"<urn:uuid:94e6877e-1782-459f-afd0-a1a00d13a877>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00412.warc.gz"}
EViews Help: Background We begin with a standard multiple linear regression model with For the observations in regime Note that the regressors are divided into two groups. The Suppose that there is an observable threshold variable threshold values where we set j-th threshold value, but not as large as the For example, in the single threshold, two regime model, we have: Using an indicator function The identity of the threshold variable Equation (35.3) is a self-exciting (SE) model with Given the threshold variable and the regression specification in Equation (35.1) , we wish to find the coefficients Nonlinear least squares is an natural approach for estimating the parameters of the model. If we define the sum-of-squares objective function and we may obtain threshold regression estimates by minimizing Taking advantage of the fact that for a given all possible sets of This basic estimation setup is well known from the breakpoint testing and regression literature (see, for example, Hansen, 2001 and Perron, 2006), and indeed, by permuting the observation index so that the threshold variable is non-decreasing, one sees that estimation of the threshold and breakpoint models are fundamentally equivalent (Bai and Perron, 2003), In essence, threshold regressions can be thought of as breakpoint least squares regressions with data reordered with respect to the threshold variable. Alternately, breakpoint regressions may be thought of as threshold regressions with time as the threshold variable. Accordingly, the discussion of breakpoint testing ( “Multiple Breakpoint Tests” ) and estimation ( “Least Squares with Breakpoints” ) may generally be applied in the current context. We will assume for our purposes that you are familiar with, or can refer to this material, and in the interest of brevity, we will minimize the amount of repetition in our discussion below.	{"url":"https://help.eviews.com/content/tar-Background.html","timestamp":"2024-11-13T21:03:02Z","content_type":"application/xhtml+xml","content_length":"19365","record_id":"<urn:uuid:f1ddb1d7-4fa7-44d0-ba9d-2d4371d708c9>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00655.warc.gz"}
clinfun 1.1.5 (10/19/2023) • clarified one-sided vs two-sided futility boundary for gsdesign functions clinfun 1.1.4 (10/12/2023) • added check for n; since it is for stopping early at least two values should be given • addressed the warning in the example for calogrank; status had both death and transplant clinfun 1.1.3 (07/11/2023) • fixed the bug in ph2simon/twostage.admissible when minimax is also optimal • ph2simon and toxbdry help have been made more clear clinfun 1.1.2 (06/19/2023) • changed all dfloat in fortran to dble • print method for ph2simon now lists admissible designs New Functionality • plot and lines method for roc.curve includes the option for precision-recall curve • roc.curve returns PPV and NPV which are needed for precision-recall curve clinfun 1.1.1 (03/06/2023) • changed class check to use inherits instead of comparing strings New Functionality • Added out.ties option to coxphCPE for discrete risk score case clinfun 1.1.0 (02/22/2022) New Functionality • Added added admissible two stage design function • Added added futilbdry function for sequential futility stopping • added 1:r sample allocation in gsdesign functions • Fixed the warnings messages from deltaAUC.f New Documentation • Added a NEWS.md file to track changes to the package. • deposited development version on GitHub • created {pkgdown} website and github actions for building site clinfun 1.0.15 (04/13/2018) • Added new function deltaAUC to test for difference in the area under the ROC curves from nested binary regression models. • power.ladesign does exact null if number of combinations is small • Added to help of fedesign, ph2simon and ph2single functions clinfun 1.0.14 (04/25/2017) • added init.c to register native routines clinfun 1.0.13 (11/29/2016) • permlogrank bug fix - strata option gave an error. Names of elements in survfit.object changed. (strata instead of ntimes.strata and n a vector). clinfun 1.0.12 (08/03/2016) • added aucVardiTest to compare growth curves clinfun 1.0.11 (08/11/2015) • check for misssing values in jonckheere.test clinfun 1.0.10 (05/18/2015) • bug fix for pselect; nlen > 3 should have been nlen > 2 clinfun 1.0.9 (02/24/2015) • use requireNamespace to remove NOTES on functions using survival pkg clinfun 1.0.8 (02/19/2015) • fix name swap in the result of mdrr in v1.0.7 clinfun 1.0.7 (02/19/2015) • new function mdrr added to calculate minimum detectable difference in response rates for given average response rate and class proportion clinfun 1.0.6 (06/10/2014) • On rare occasions jonckheere.test gave a p-value bigger than 1. Sometimes 2min(iPVAL, dPVAL, 1) can be larger than 1. Replace with 2min(iPVAL, dPVAL, 0.5) (Thanks to Drs. Shterev and Owzar of clinfun 1.0.5 (04/16/2013) clinfun 1.0.4 (01/22/2013) • Added the option to calculate continuity corrected sample size in the function gsdesign.binomial. clinfun 1.0.3 (10/15/2012) • Fixed two-sided p-value > 1 bug when statistic is exactly its mean. • Fixed Rd file to address LaTeX warnings. clinfun 1.0.2 (09/25/2012) • integer overflow in djonck for the Jonckheere-Terpstra statistic. replace with pdf calculation using Mark van de Wiel convolution. • Create seperate help for functions permlogrank and jonckheere.test clinfun 1.0.1 (08/14/2012) • Fix integer overflow because of n0n1 == 0 in roc.curve and nnnd == 0 in roc.area.test • In coxphQuantile eliminate times for which survival probability is 0 or 1 from quantile computation. clinfun 1.0.0 (03/13/2012) • clinfun number changed to 1.0.0 in preparation for R 2.15. • toxbdry now does the entire Pocock to O’Brien-Fleming range of boundaries. Added references for the method. • Fixed linebreak in the help for coxphERR. clinfun 0.9.9 (01/18/2012) • Added coxphERR to calculate Heller’s explained relative risk. • Fixed NaN bug in toxbdry when priority=“alt” is used. clinfun 0.9.8 (09/13/2011) • fixed bug in print.gsdesign for binomial case (p1,p2 instead of pC,pE) • ROC functions now check that there are at least one each of status=0,1 clinfun 0.9.7 (04/27/2011) • fixed fortran code to address gfortran-4.6 warnings • added ktau a faster implementation of cor(x, y, method=“k”). not in NAMESPACE. Should be called using clinfun:::ktau clinfun 0.9.7 (04/25/2011) • bug fix: roc.area.test integer overflow for large nn*nd. • use sort function to speed up roc curve and area estimation. clinfun 0.9.6 (03/24/2011) • bug fix: roc.area.test gave NaN as the statistic and p-value when the markers are identical. Changed it to 0. clinfun 0.9.5 (03/09/2011) • bug fix: gsdesign funtions not returning the sample size / # events. clinfun 0.9.4 (02/24/2011) • twostage.inference for umvue, p-value and CI for 2 stage design. clinfun 0.9.3 (12/06/2010) • ph2simon was testing whether dim is null for feasible solution. Replaced with nrow == 0 since it is now possible to have 0 rows. clinfun 0.9.2 (11/05/2010) • Added functions to compute and plot the empirical ROC curve. clinfun 0.9.1 (11/03/2010) • Added functions for the area and permutation tests to compare ROC. • Checks that min.diff is greater than 0 in pselect. clinfun 0.9.0 (11/01/2010) • Added a non-binding futility boundary to gsdesign clinfun 0.8.10 (04/16/2010) • variable names for returned data.frame in coxphQuantile • examples in coxphQuantile & coxphCPE use status==2 clinfun 0.8.9 (04/09/2010) • check R# clinfun so that coxphCPE works for any # clinfun (see 0.8.9) clinfun 0.8.8 (04/07/2010) • Change coxphCPE to reflect the fact that model.matrix.coxph doesn’t have an intercept term. clinfun 0.8.8 (02/25/2010) • Added the function or2pcase clinfun 0.8.7 (11/20/2009) • Fixed the 0/0 bug in the revised pselect clinfun 0.8.6 (11/17/2009) • Changed Venkat’s affiliation to MSKCC. • Fixed pselect to calculate the selection probability correctly when only one treatment exceeds the min.resp threshold. clinfun 0.8.5 (07/10/2009) • Changed the Jonckheere-Terpstra statistic such that large value is indicative of increasing group locations and small for decreasing. Function warns that p-value is based on approximation for tied data. clinfun 0.8.4 (12/02/2008) • Added functionality to pselect. It can do unequal sample size for the case of two treatments. min.diff can be specified as a rate instead of number of responses. Output element names changed to be more descriptive. clinfun 0.8.3 (11/18/2008 & 09/18/2008) • Fixed the bug CPS.ssize. call inside used fixed alpha, power & r. • Fixed the bug in the approximate one-sided p-value of Jonckheere test. wrong tail was used. clinfun 0.8.2 (06/20/2008) • toxbdry allows the error threshold to prioritize when the sample size is too small to have both satisfied. clinfun 0.8.1 (06/17/2008) • Fixed gsdesign to allow for fixed sample designs. Help file fixed. clinfun 0.8.0 (05/23/2008) • New # clinfun with one new function power.lehmann.design	{"url":"https://archive.linux.duke.edu/cran/web/packages/clinfun/news/news.html","timestamp":"2024-11-09T09:27:12Z","content_type":"application/xhtml+xml","content_length":"11030","record_id":"<urn:uuid:7a641481-3a29-4db0-9b6f-5018476f0f47>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00518.warc.gz"}
Statistics - Department of Mathematics Probability & Statistics Faculty with research interests in this area: • Dr. Xin Dang • Dr. Hailin Sang • Dr. Martial Longla • Dr. Dao Nguyen • Dr. Jeremy Clark This group of faculty works on problems in various fields of statistics and probability including but not limited to the following: • Mathematical statistics • Applied statistics • Bayesian statistics • Nonparametric statistics • Robust statistics • Markov chains • Stochastic processes • Time series • Machine learning theory and application • Directed polymer in a random environment • Invariance principles More specifically, questions of interest include nonparametric and robust multivariate analysis, data mining, outlier identification, cluster analysis, applications of depth functions, analysis of dependent data, copula applications in estimation theory, theory of copulas, inference for time series, random fields, properties of self-normalized statistics, empirical processes and applications, deep learning models and theory, moderate or large deviations, survey sampling design and analysis, Markov and reversible Markov chains (limit theorems and copula approach), central limit theorems for dependent data, dependence modelling and applications, kernel estimation methods for dependent data, Bayesian analysis, survival analysis, large sample theory. These questions include their applications to various fields of research for interdisciplinary collaboration.	{"url":"https://math.olemiss.edu/statistics/","timestamp":"2024-11-10T12:55:25Z","content_type":"text/html","content_length":"111419","record_id":"<urn:uuid:ef96a9d1-df04-411c-bb3b-b0c81850b3aa>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00715.warc.gz"}
Definitions for exponential ˌɛk spəˈnɛn ʃəlex·po·nen·tial This dictionary definitions page includes all the possible meanings, example usage and translations of the word exponential. Princeton's WordNet 1. exponential, exponential functionadjective a function in which an independent variable appears as an exponent 2. exponentialadjective of or involving exponents "exponential growth" 1. Exponentialadjective changing over time in an exponential manner, i. e. increasing or decreasing by a fixed ratio for each unit of time; as, exponential growth; exponential decay. 1. exponentialnoun Any function that has an exponent as an independent variable. 2. exponentialadjective Relating to an exponent. 3. exponentialadjective Expressed in terms of a power of e. 4. exponentialadjective In modern English, used to describe a large quantity of an object or objects. Samuel Johnson's Dictionary 1. Exponentialadjective Exponential curves are such as partake both of the nature of algebraick and transcendental ones. They partake of the former, because they consist of a finite number of terms, though those terms themselves are indeterminate; and they are in some measure transcendental, because they cannot be algebraically constructed. John Harris Etymology: from exponent. 1. exponential Exponential refers to a growth or decay at a rate which is proportionate to the current state or value. In mathematics, it typically refers to a function or an equation, in which the variable is an exponent, or the usage of a constant base raised to a power or exponent. This term can be used in various contexts to describe a rapid increase or decrease or a phenomenon that is repeatedly Webster Dictionary 1. Exponentialadjective pertaining to exponents; involving variable exponents; as, an exponential expression; exponential calculus; an exponential function 2. Etymology: [Cf. F. exponentiel.] Usage in printed sourcesFrom: 1. [["1725","1"],["1730","2"],["1742","12"],["1754","1"],["1761","2"],["1766","3"],["1767","1"],["1770","1"],["1794","1"],["1798","1"],["1800","1"],["1801","4"],["1803","6"],["1808","1"], 1. Chaldean Numerology The numerical value of exponential in Chaldean Numerology is: 4 2. Pythagorean Numerology The numerical value of exponential in Pythagorean Numerology is: 9 Examples of exponential in a Sentence 1. Obviously it would mean an increase - an exponential increase. 2. The current environment represents a winding back of the overly bullish expectations of both commodity demand and Chinese growth – to a more balanced expectation of progressive, not exponential, 3. The potential goes way beyond automating repetitive tasks, but even in a contact center, the remaining humans don’t have to fear for their jobs just yet. Amelia is not replacing them right now. But it could be coming soon. Just a few short years ago, the experts at MIT were predicting we would never have a driverless car, and now Google has them on the road. The pace of change is so rapid—almost exponential—that even though Amelia is not yet really operational in too many situations, she’s learning very quickly and the landscape two years from now could be completely 4. We are picking segments where top line growth is exponential, large enterprises are not going to stop spending on security software. This spending growth is not going away. 5. We’ve had people call us afterwards or send us emails with pictures of their adopted dogs saying, ‘thank you for the referral,’ it’s definitely exponential in its reach of adoptive homes. Popularity rank by frequency of use Translations for exponential From our Multilingual Translation Dictionary Find a translation for the exponential definition in other languages: • - Select - • 简体中文 (Chinese - Simplified) • 繁體中文 (Chinese - Traditional) • Español (Spanish) • Esperanto (Esperanto) • 日本語 (Japanese) • Português (Portuguese) • Deutsch (German) • العربية (Arabic) • Français (French) • Русский (Russian) • ಕನ್ನಡ (Kannada) • 한국어 (Korean) • עברית (Hebrew) • Gaeilge (Irish) • Українська (Ukrainian) • اردو (Urdu) • Magyar (Hungarian) • मानक हिन्दी (Hindi) • Indonesia (Indonesian) • Italiano (Italian) • தமிழ் (Tamil) • Türkçe (Turkish) • తెలుగు (Telugu) • ภาษาไทย (Thai) • Tiếng Việt (Vietnamese) • Čeština (Czech) • Polski (Polish) • Bahasa Indonesia (Indonesian) • Românește (Romanian) • Nederlands (Dutch) • Ελληνικά (Greek) • Latinum (Latin) • Svenska (Swedish) • Dansk (Danish) • Suomi (Finnish) • فارسی (Persian) • ייִדיש (Yiddish) • հայերեն (Armenian) • Norsk (Norwegian) • English (English) Word of the Day Would you like us to send you a FREE new word definition delivered to your inbox daily? Use the citation below to add this definition to your bibliography: Are we missing a good definition for exponential? Don't keep it to yourself... A aberrate B abrade C excogitate D suffuse	{"url":"https://www.definitions.net/definition/exponential","timestamp":"2024-11-02T20:20:54Z","content_type":"text/html","content_length":"90905","record_id":"<urn:uuid:308ca98b-a4c3-49a9-9919-538cb1dea5c5>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00532.warc.gz"}
Design for algorithmic stablecoin backed by BTC The idea is to have an on-chain market maker that issues two tokens, STB (a stablecoin) whose value is automatically backed by BTC, and VOL which captures any excess value from the backing BTC. If successful, this allows a stablecoin to be purely on-chain, without requiring assets to be stored externally, allowing greater auditability and automation. The market is parameterized with two variables, p representing the current target value of 1 BTC in STB tokens, and m which gives a slowly changing prediction of the lowest expected future value for p. The market maker then aims to ensure that p STB tokens can be traded for 1 BTC, at least provided that p \ge m, and that m does not decrease. Further, if those assumptions are violated, failure should be graceful rather than catastrophic. At any point in time, we will denote the number of BTC held backing the issued STB and VOL tokens as b, the number of issued STB tokens as s and the number of issued VOL tokens as v. Call the price of 1 STB token \alpha BTC (thus the goal is \alpha = p^{-1}), and the price of 1 VOL token \beta BTC. This then gives the budget constraint: b = \alpha s + \beta v since selling all the issued STB and VOL tokens at the current price in BTC should give you exactly the value of all the backing BTC. We define two constraints: the ceiling for \alpha should be \frac{1}{p} and the floor for \beta should be some constant \bar{\beta} and at all times one or both of these conditions should be met. This gives two cases: \alpha, \beta = \begin{cases} p^{-1}, \frac{b-sp^{-1}}{v} & \text{if } b \ge sp^{-1} + v \bar{\beta} \\ \frac{b - v\bar{\beta}}{s}, \bar{\beta} & \text{otherwise} \end{cases} When b \ge sp^{-1} + v\bar{\beta} we say the peg holds, and when that is not the case, we say the peg is broken. Note that when the peg is broken, \alpha \lt p^{-1}, and hence selling STB will net a payment of less than the target price. Now that we know the prices at which tokens might be traded, to determine when a trade should be allowed we need to take into account possible changes in the price p. In particular, at this point we start to make use of m and set the goal b \ge sm^{-1} + v\bar{\beta}. We setup two scoring rules: \gamma_p &= b - sp^{-1} - v\bar{\beta} \\ \gamma_m &= b - sm^{-1} - v\bar{\beta} Note the following properties: • The peg holds precisely when \gamma_p \ge 0 • Provided m \lt p, \gamma_m \lt \gamma_p, and conversely. • When the peg holds: □ buying VOL with BTC increases \gamma_p and \gamma_m □ trading STB for BTC does not change \gamma_p, □ provided m \lt p, buying STB with BTC decreases \gamma_m • When the peg is broken: □ trading VOL for BTC does not change \gamma_p or \gamma_m. □ buying STB with BTC decreases \gamma_p □ if \gamma_m \lt 0, then buying STB with BTC decreases \gamma_m The peg can only be broken directly by external movements in p, but such movements are expected. In order to maintain the peg, we limit trades that may weaken the peg indirectly: • We block trades that would cause \gamma_m to become negative (some purchases of STB or sales of VOL while the peg holds) • We block trades that would cause \gamma_p to become more negative (purchases of STB while the peg is broken) This concept is mostly due to discussions with Rusty Russell, and is obviously similar to the Dai stablecoin. 2 Likes Isnâ€™t this what BitMatrix is on Liquid? BitMatrix is an automated market maker â€“ so given two assets, itâ€™ll let you swap one for the other. But you have to have the assets in the first place for this to work â€“ so you need a â€śUSDTrentâ€ť token thatâ€™s backed by USD in Trentâ€™s bank account, eg. An algorithmic stablecoin lets you establish a USDX token thatâ€™s backed by BTC instead; with the obvious caveat that if the real value of BTC plummets the token canâ€™t maintain par value. Ah yeah sorry, I confused with Fuji, I suppose.	{"url":"https://test.delvingbitcoin.org/t/design-for-algorithmic-stablecoin-backed-by-btc/20","timestamp":"2024-11-06T04:55:56Z","content_type":"text/html","content_length":"22972","record_id":"<urn:uuid:d1dcaff9-7407-42b0-9b29-2cdbe2d996d9>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00486.warc.gz"}
MATH 1P98 - Assignment #4 \| Complete Solution - CourseMerits Question DetailsNormal MATH 1P98 - Assignment #4 \| Complete Solution Question posted by • Rating : 24 • Grade : A+ • Questions : 0 • Solutions : 287 • Blog : 1 • Earned : $9883.30 MATH 1P98 - Assignment #4 Assignments must have a cover page (refer to the course outline). Please write on one side of the page only and show ALL your work. Answer questions with sentences. Include any printout for a question with the question and clearly label the printout with the question number and part. 1) You have been assigned to test the hypothesis that the average number of cars waiting in line for the drive-thru window during lunch hour differs between Chick-fil-A, Wendy's, and McDonald's. The following data show the number of cars in line during randomly selected times during the lunch hour at all three chains. (1) Wendy's (2) McDonald's Perform a one-way ANOVA using α = 0.05 to determine if a difference exists in the average number of cars waiting in line at the drive-thru during the lunch hour between these chains. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the mean of the data summarized in the given frequency distribution. 2) The test scores of 40 students are summarized in the frequency distribution below. Find the mean score. 2) _______ A) 66.6 B) 74.0 C) 70.3 D) 74.5 Find the indicated probability. 3) In a certain class of students, there are 11 boys from Wilmette, 4 girls from Winnetka, 7 girls from Wilmette, 4 boys from Glencoe, 5 boys from Winnetka and 4 girls from Glencoe. If the teacher calls upon a student to answer a question, what is the probability that the student will be a boy? 3) _______ A) 0.429 B) 0.571 C) 0.71 D) 0.314 Find the indicated complement. 4) The probability that Luis will pass his statistics test is 0.49. Find the probability that he will fail his statistics test. 4) _______ A) 0.96 B) 0.51 C) 0.25 D) 2.04 Find the indicated probability. 5) 100 employees of a company are asked how they get to work and whether they work full time or part time. The figure below shows the results. If one of the 100 employees is randomly selected, find the probability of getting someone who carpools or someone who works full time. 1. Public transportation: 10 full time, 7 part time 2. Bicycle: 3 full time, 3 part time 3. Drive alone: 30 full time, 35 part time 4. Carpool: 6 full time, 6 part time 5) _______ A) 0.55 B) 0.53 C) 0.61 D) 0.22 Find the indicated probability. Round to three decimal places. 6) In a study, 44% of adults questioned reported that their health was excellent. A researcher wishes to study the health of people living close to a nuclear power plant. Among 14 adults randomly selected from this area, only 3 reported that their health was excellent. Find the probability that when 14 adults are randomly selected, 3 or fewer are in excellent health. 6) _______ A) 0.020 B) 0.053 C) 0.046 D) 0.073 Solve the problem. 7) Human body temperatures are normally distributed with a mean of and a standard deviation of If 19 people are randomly selected, find the probability that their mean body temperature will be less than 7) _______ A) 0.4826 B) 0.0833 C) 0.3343 D) 0.9826 Use the given degree of confidence and sample data to construct a confidence interval for the population mean μ. Assume that the population has a normal distribution. 8) The amounts (in ounces) of juice in eight randomly selected juice bottles are: 15.4 15.8 15.4 15.1 15.8 15.9 15.8 15.7 Construct a 98% confidence interval for the mean amount of juice in all such bottles. 8) _______ A) 15.99 oz < μ < 15.23 oz B) 15.33 oz < μ < 15.89 oz C) 15.23 oz < μ < 15.99 oz D) 15.89 oz < μ < 15.33 oz Find the P-value for the indicated hypothesis test. 9) A random sample of 139 forty-year-old men contains 26% smokers. Find the P-value for a test of the claim that the percentage of forty-year-old men that smoke is 22%. 9) _______ A) 0.2542 B) 0.2802 C) 0.1271 D) 0.1401 State what the given confidence interval suggests about the two population means. 10) A researcher was interested in comparing the amount of time spent watching television by women and by men. Independent simple random samples of 14 women and 17 men were selected, and each person was asked how many hours he or she had watched television during the previous week. The summary statistics are as follows. The following 99% confidence interval was obtained for the difference between the mean amount of time spent watching television for women and the mean amount of time spent watching television for men: -5.73 hrs < μ1 - μ2 < 4.13 hrs. What does the confidence interval suggest about the population means? 10) ______ A) The confidence interval limits include 0 which suggests that the two population means are unlikely to be equal. There appears to be a significant difference between the mean amount of time spent watching television for women and the mean amount of time spent watching television for men. B) The confidence interval includes only negative values which suggests that the mean amount of time spent watching television for women is smaller than the mean amount of time spent watching television for men. C) The confidence interval limits include 0 which suggests that the two population means might be equal. There does not appear to be a significant difference between the mean amount of time spent watching television for women and the mean amount of time spent watching television for men. D) The confidence interval includes only positive values which suggests that the mean amount of time spent watching television for women is larger than the mean amount of time spent watching television for men. Determine which scatterplot shows the strongest linear correlation. 11) Which shows the strongest linear correlation? 11) ______ Available Answer [Solved] MATH 1P98 - Assignment #4 \| Complete Solution • This solution is not purchased yet. • Submitted On 11 Jul, 2016 03:35:59 Answer posted by • Rating : 24 • Grade : A+ • Questions : 0 • Solutions : 287 • Blog : 1 • Earned : $9883.30 We determine whether to accept or reject the null hypothesis by comparing the value of F and the value of Critical value of F also using the P... Other Similar Questions We determine whether to accept or reject the null hypothesis by comparing the value of F and the value of Critical value of F also using the P value and the level of significance The value of F= 1.112224 is < the critical valu... 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1 point of dosolved salt, runs inlo the tank per mimate. The mi... \| Filo Question asked by Filo student 1 point of dosolved salt, runs inlo the tank per mimate. The misture keje unifont by slining, rans col at the same rals. Then which of the mathematizal model gives the amoct ef sak \{n in the tank? Not the question you're searching for? + Ask your question Video solutions (1) Learn from their 1-to-1 discussion with Filo tutors. 6 mins Uploaded on: 9/28/2024 Was this solution helpful? Found 6 tutors discussing this question Discuss this question LIVE 12 mins ago One destination to cover all your homework and assignment needs Learn Practice Revision Succeed Instant 1:1 help, 24x7 60, 000+ Expert tutors Textbook solutions Big idea maths, McGraw-Hill Education etc Essay review Get expert feedback on your essay Schedule classes High dosage tutoring from Dedicated 3 experts Practice more questions on Differential Equations View more Students who ask this question also asked View more Stuck on the question or explanation? Connect with our Mathematics tutors online and get step by step solution of this question. 231 students are taking LIVE classes 1 point of dosolved salt, runs inlo the tank per mimate. The misture keje unifont by slining, rans col at the same rals. Then which of the mathematizal model gives the amoct ef sak Question Text \{n in the tank? Updated On Sep 28, 2024 Topic Differential Equations Subject Mathematics Class Class 12 Answer Type Video solution: 1 Upvotes 58 Avg. Video 6 min	{"url":"https://askfilo.com/user-question-answers-mathematics/1-point-of-dosolved-salt-runs-inlo-the-tank-per-mimate-the-3132353237373739","timestamp":"2024-11-14T09:08:44Z","content_type":"text/html","content_length":"335375","record_id":"<urn:uuid:c2f831d4-a5be-4685-b206-677ffaba0a34>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00101.warc.gz"}
ECE 476 POWER SYSTEM ANALYSIS Lecture 5 Power Download presentation ECE 476 POWER SYSTEM ANALYSIS Lecture 5 Power System Operation, Transmission Lines Professor Tom Overbye Department of Electrical and Computer Engineering Reading and Homework • • 1 st Exam moved to Oct 11 (in class) For lectures 4 through 6 please be reading Chapter 4 – • we will not be covering sections 4. 7, 4. 11, and 4. 12 in detail though you should still at least skim those sections. HW 1 is 2. 9, 22, 28, 32, 48; due Thursday 9/8 • For Problem 2. 32 you need to use the Power. World Software. You can download the software and cases at the below link; get version 15. http: //www. powerworld. com/gloversarma. asp Direct Power. World download page is http: //www. powerworld. com/Demo. Software/Glover. Sarma. Simdwnl dv 15. asp 1 Power Transactions l l l Power transactions are contracts between areas to do power transactions. Contracts can be for any amount of time at any price for any amount of power. Scheduled power transactions are implemented by modifying the area ACE: ACE = Pactual, tie-flow - Psched 3 100 MW Transaction Scheduled 100 MW Transaction from Left to Right Net tie-line flow is now 100 MW 4 Security Constrained ED Transmission constraints often limit system economics. l Such limits required a constrained dispatch in order to maintain system security. l In three bus case the generation at bus 3 must be constrained to avoid overloading the line from bus 2 to bus 3. l 5 Security Constrained Dispatch is no longer optimal due to need to keep line from bus 2 to bus 3 from overloading 6 Multi-Area Operation l l l If Areas have direct interconnections, then they may directly transact up to the capacity of their tie-lines. Actual power flows through the entire network according to the impedance of the transmission lines. Flow through other areas is known as “parallel path” or “loop flows. ” 7 Seven Bus Case: One-line System has three areas Area left has one bus Area top has five buses Area right has one bus 8 Seven Bus Case: Area View Actual flow between areas System has 40 MW of “Loop Flow” Scheduled flow Loop flow can result in higher losses 9 Seven Bus - Loop Flow? Note that Top’s Losses have increased from 7. 09 MW to 9. 44 MW 100 MW Transaction between Left and Right Transaction has actually decreased the loop flow 10 Pricing Electricity l l l Cost to supply electricity to bus is called the locational marginal price (LMP) Presently some electric makets post LMPs on the web In an ideal electricity market with no transmission limitations the LMPs are equal Transmission constraints can segment a market, resulting in differing LMP Determination of LMPs requires the solution on an Optimal Power Flow (OPF) 11 3 BUS LMPS - OVERLOAD IGNORED Gen 2’s cost is $12 per MWh Gen 1’s cost is $10 per MWh Line from Bus 1 to Bus 3 is over-loaded; all buses have same marginal cost 12 LINE OVERLOAD ENFORCED Line from 1 to 3 is no longer overloaded, but now the marginal cost of electricity at 3 is $14 / MWh 13 MISO and PJM are the reliability coordinators covering the electric grid in Illinois. Com. Ed is in PJM, and Ameren is in MISO. 14 MISO LMPs 8/31/11 at 11: 05 AM www. midwestmarket. org 15 Development of Line Models Goals of this section are 1) develop a simple model for transmission lines 2) gain an intuitive feel for how the geometry of the transmission line affects the model parameters 16 Primary Methods for Power Transfer l 1) 2) 3) 4) 5) The most common methods for transfer of electric power are Overhead ac Underground ac Overhead dc Underground dc other 17 345 k. V+ Transmission Growth at a Glance 18 18 345 k. V+ Transmission Growth at a Glance 19 19 345 k. V+ Transmission Growth at a Glance 20 20 345 k. V+ Transmission Growth at a Glance 21 21 345 k. V+ Transmission Growth at a Glance 22 22 Magnetics Review Ampere’s circuital law: 23 Line Integrals Line integrals are a generalization of traditional integration Integration along the x-axis Integration along a general path, which may be closed Ampere’s law is most useful in cases of symmetry, such as with an infinitely long line 24 Magnetic Flux Density Magnetic fields are usually measured in terms of flux density 25 Magnetic Fields from Single Wire Assume we have an infinitely long wire with current of 1000 A. How much magnetic flux passes through a 1 meter square, located between 4 and 5 meters from the wire? Direction of H is given by the “Right-hand” Rule Easiest way to solve the problem is to take advantage of symmetry. For an integration path we’ll choose a circle with a radius of x. 27 Single Line Example, cont’d For reference, the earth’s magnetic field is about 0. 6 Gauss (Central US) 28 Flux linkages and Faraday’s law 29 Inductance For a linear magnetic system, that is one where B =m. H we can define the inductance, L, to be the constant relating the current and the flux linkage l =Li where L has units of Henrys (H) Inductance Example Calculate the inductance of an N turn coil wound tightly on a torodial iron core that has a radius of R and a cross-sectional area of A. Assume 1) all flux is within the coil 2) all flux links each turn 31 Inductance Example, cont’d 32 Inductance of a Single Wire To development models of transmission lines, we first need to determine the inductance of a single, infinitely long wire. To do this we need to determine the wire’s total flux linkage, including 1. flux linkages outside of the wire 2. flux linkages within the wire We’ll assume that the current density within the wire is uniform and that the wire has a radius of r. 33 Flux Linkages outside of the wire 34 Flux Linkages outside, cont’d 35 Flux linkages inside of wire 36 Flux linkages inside, cont’d Wire cross section x r 37 Line Total Flux & Inductance 38 Inductance Simplification 39	{"url":"https://slidetodoc.com/ece-476-power-system-analysis-lecture-5-power-2/","timestamp":"2024-11-13T06:41:19Z","content_type":"text/html","content_length":"168079","record_id":"<urn:uuid:1b2c6d0a-01d8-47db-86a5-e6fa2bb78b3a>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00083.warc.gz"}
How long does it take to count to a billion in seconds? Counting to a billion in seconds would take 2 billion seconds or about 63 years counting every 2 seconds and if you count a number every second then it would take 1 billion seconds or around 31.5 We can count as high a we want too although counting up to certain numbers can take a long time. For example counting to 1 million can take 89 days or longer and counting to 1 billion could take 30 to 32 years. Trillion is one of the smallest numbers (along with million and billion) on our list. But do not forget that it is still an incredibly large number and if you were to count to a trillion, you would most likely take 31,709 years to do so! When Jeremy Harper counted to 1 million it took Jeremy Harper 89 days to count to 1 million during which he spent 16 hours counting. Jeremy Harper began counting to 1 million on June 18th 2007 and finished counting to 1 million on September 14th 2007. Jeremy Harper is an American entrant in the Guinness Book of World Records for counting aloud to 1,000,000, live-streaming the entire process. The count took Harper 89 days, during each of which he spent sixteen hours counting. He began on June 18, 2007, finishing on September 14. 1 million is a lot of numbers to count too and is hard to achieve but it can be done. Answerpail (305)	{"url":"https://answerpail.com/index.php/216392/how-long-does-it-take-to-count-to-a-billion-in-seconds","timestamp":"2024-11-08T05:00:32Z","content_type":"text/html","content_length":"29627","record_id":"<urn:uuid:8861880e-4fd8-412d-b620-fb45aef0df39>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00194.warc.gz"}
Integrals: From Series to Accumulation - Expii Finite series tell you how much a (discrete) sequence accumulates. Areas or definite integrals tell you how much a (continuous) function accumulates. More precisely, the integral of f(t) from a to b gives you the accumulation of the "infinitesimal" quantity f(t) dt from t=a to t=b. This will all be clarified we discuss FTC and applications of integrals (such as accumulated change, probability, and slices) later.	{"url":"https://www.expii.com/t/integrals-from-series-to-accumulation-249","timestamp":"2024-11-10T19:31:57Z","content_type":"text/html","content_length":"5945","record_id":"<urn:uuid:ab39a747-3430-4fc9-85b9-1bf9d87506e8>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00859.warc.gz"}
TPTP Documents File: TPTP Documents File: SZSOntology The SZS ontologies (named after the authors of the original paper describing the success ontology [1]) provide status values to describe logical data. The SZS success ontology provides status values to describe what is known or has been successfully established about the relationship between the axioms and conjectures in logical data. The SZS no-success ontology provides status values to describe why a success ontology value has not been established. The SZS dataform ontology provides status values to describe the nature of logical data. All status values are expressed as "OneWord" to make system output parsing simple, and also have a three letter mnemonic. Commonly Used Ontology Values The ontologies are very fine grained ontology, which have more status values and dataforms than are commonly used by ATP systems. Suitable subsets for practical purposes are as follows: + For Success - For problems with conjectures - report Theorem or ContradictoryAxioms or CounterSatisfiable - For problems without conjectures - report Satisfiable or Unsatisfiable + For No-success - System stopped due to CPU limit - report Timeout - System gave up due to incompleteness - report GaveUp - System stopped due to an error - report Error + For Dataforms - A generic proof - report Proof - A CNF refutation - report CNFRefutation - A generic model - report Model - A finite model - report FiniteModel - An infinite model - report InfiniteModel - A saturation - report Saturation Success ontology values are also used in TPTP format proofs to record the relationship between the parents and inferred formula of each inference step. Commonly used values are: - The inferred formula is a theorem of the parents (logical consequences, e.g., resolvants, etc.) - report Theorem - The inferred and parent formulae are equisatisfiable (e.g., Skolemization) - report EquiSatisfiable - The negation of the inferred formula is a theorem of the parents (e.g., negating the conjecture in a proof by refutation) - report CounterTheorem Standard Presentation of Ontology Values The solution status should be reported exactly once, in a line starting "% SZS status" (the leading '%' makes the line into a TPTP language comment). For examples: % SZS status Unsatisfiable for SYN075+1 % SZS status GaveUp for SYN075+1 A success or no-success ontology value should be presented as early as possible, at least before any data output to justify the value. The justifying data should be reported exactly once, delimited by lines starting "% SZS output start" and "% SZS output end". For example: % SZS output start CNFRefutation for SYN075-1 % SZS output end CNFRefutation for SYN075-1 All "SZS" lines lines can optionally have software specific information appended, separated by a :. For examples: % SZS status GaveUp for SYN075+1 : Could not complete CNF conversion % SZS output end CNFRefutation for SYN075-1 : Completed in CNF conversion The SZS Success Ontology The ontology assumes that the input is a 2-tuple of the form <Ax, C>, where Ax is a set (conjunction) of axioms and C is a set (conjunction) of conjecture formulae. This is a common standard usage of ATP systems (often there is only a single conjecture formula). If the input is not of the form <Ax, C>, it is treated as a conjecture formula (even if it is a "set of axioms" from the user view point, e.g., a set of formulae all with the TPTP role "axiom"), and the 2-tuple is <TRUE, C>. The ontology values can also be interpreted in terms of the formula F, of the form Ax => C. The ontology values are based on the possible relationships between the sets of models of Ax and C. In the figure below many of the "OneWord" status values are abbreviated - see the list below for the official full "OneWord"s. The lines in the ontology can be followed up the hierarchy as isa links, e.g., an ETH isa EQV isa (SAT and a THM). \| \| \| \| \| \| UnsatPre SatPre \| Verified CtrSatPre CtrUnsatPre UNP SAP \| VER CSP CUP \|_______/ \| \| \| \_______\| \| \| \| \| \| EquSat \| FiniteThm \| EquCtrSat / ESA \| FTH \| ECS / \| \| / \| \| \| Sat'ble Theorem CtrThm CtrSat \| SAT THM CTH CSA \| \| \______.\|._____________________________________.\|.______/ \| \ \| \| \ \| \| \| \| \ ModPre \| FinSat \| NoConq \| FinUns \| FinCtrSat MPR \| FSA \| NOC \| FUN \| FCS \| \| \|_______________________________________\| \| \| \| \| \| \| \| \| \| \ \| SatAxThm CtraAx SatAxCth \| \| \ \| STH CAX SCT : \| \ _\|_________\|_ ____\|____ _\|_________\|_ \| \| \| \| \| \| \| : \| Eqvlnt TautC WeakC SatConCA SatCCoCA WkCC UnsCon\|CtrEqu EQV TAC WEC SCA SCC WCC UNC \| CEQ __\|__ _\|_ __\|__ __\|___ ___\|__ __\|__ _\|_ \|__\|__ \| \| / \ \| \| \| \ / \| \| \| / \\| \| Equiv Taut- Weaker Weaker TauCon WCon UnsCon Weaker Weaker Unsat Equiv Thm ology TautCo Thm CtraAx CtraAx CtraAx CtrThm UnsCon -able CtrTh ETH TAU WTC WTH TCA WCA UCA WCT WUC UNS ECT + Success (SUC): The logical data has been processed successfully. + UnsatisfiabilityPreserving (UNP): If there does not exist a model of Ax then there does not exist a model of C, i.e., if Ax is unsatisfiable then C is unsatisfiable. + SatisfiabilityPreserving (SAP): If there exists a model of Ax then there exists a model of C, i.e., if Ax is satisfiable then C is satisfiable. - F is satisfiable. + EquiSatisfiable (ESA): There exists a model of Ax iff there exists a model of C, i.e., Ax is (un)satisfiable iff C is (un)satisfiable. + Satisfiable (SAT): Some interpretations are models of Ax, and some models of Ax are models of C. - F is satisfiable, and ~F is not valid. - Possible dataforms are Models of Ax \| C. + FinitelySatisfiable (FSA): Some finite interpretations are finite models of Ax, and some finite models of Ax are finite models of C. - F is satisfiable, and ~F is not valid. - Possible dataforms are FiniteModels of Ax \| C. + FiniteTheorem (FTH): All finite models of Ax are finite models of C. - Any models of Ax \| ~C are infinite. + Theorem (THM): All models of Ax are models of C. - F is valid, and C is a theorem of Ax. - Possible dataforms are Proofs of C from Ax. + SatisfiableTheorem (STH): Some interpretations are models of Ax, and all models of Ax are models of C. - F is valid, and C is a theorem of Ax. - Possible dataforms are Models of Ax with Proofs of C from Ax. + Model Preserving (MPR): Some interpretations are models of Ax, and some interpretations are models of C, and all models of C are conservative extensions of models of Ax (which means that all models of C are models of Ax). + Equivalent (EQV): Some interpretations are models of Ax, all models of Ax are models of C, and all models of C are models of Ax. - F is valid, C is a theorem of Ax, and Ax is a theorem of C. - Possible dataforms are Proofs of C from Ax and of Ax from C. + TautologousConclusion (TAC): Some interpretations are models of Ax, and all interpretations are models of C. - F is valid, and C is a tautology. - Possible dataforms are Proofs of C. + WeakerConclusion (WEC): Some interpretations are models of Ax, all models of Ax are models of C, and some models of C are not models of Ax. - See Theorem and Satisfiable. + EquivalentTheorem (ETH): Some, but not all, interpretations are models of Ax, all models of Ax are models of C, and all models of C are models of Ax. - See Equivalent. + Tautology (TAU): All interpretations are models of Ax, and all interpretations are models of C. - F is valid, ~F is unsatisfiable, and C is a tautology. - Possible dataforms are Proofs of Ax and of C. + WeakerTautologousConclusion (WTC): Some, but not all, interpretations are models of Ax, and all interpretations are models of C. - F is valid, and C is a tautology. - See TautologousConclusion and WeakerConclusion. + WeakerTheorem (WTH): Some interpretations are models of Ax, all models of Ax are models of C, some models of C are not models of Ax, and some interpretations are not models of C. - See Theorem and Satisfiable. + ContradictoryAxioms (CAX): No interpretations are models of Ax. - F is valid, and anything is a theorem of Ax. - Possible dataforms are Refutations of Ax. + SatisfiableConclusionContradictoryAxioms (SCA): No interpretations are models of Ax, and some interpretations are models of C. - See ContradictoryAxioms. + TautologousConclusionContradictoryAxioms (TCA): No interpretations are models of Ax, and all interpretations are models of C. - See TautologousConclusion and SatisfiableConclusionContradictoryAxioms. + WeakerConclusionContradictoryAxioms (WCA): No interpretations are models of Ax, and some, but not all, interpretations are models of C. - See SatisfiableConclusionContradictoryAxioms and + UnsatisfiableConclusionContradictoryAxioms (UCA): No interpretations are models of Ax, and all interpretations are models of ~C, i.e., no interpretations are models of C. - See UnsatisfiableConclusion and - SatisfiableCounterConclusionContradictoryAxioms. + CounterSatisfiabilityPreserving (CSP): If there exists a model of Ax then there exists a model of ~C, i.e., if Ax is satisfiable then ~C is satisfiable. + CounterUnsatisfiabilityPreserving (CUP): If there does not exist a model of Ax then there does not exist a model of ~C, i.e., if Ax is unsatisfiable then ~C is unsatisfiable. + EquiCounterSatisfiable (ECS): There exists a model of Ax iff there exists a model of ~C, i.e., Ax is (un)satisfiable iff ~C is (un)satisfiable. + CounterTheorem (CTH): All models of Ax are models of ~C. - F is not valid, and ~C is a theorem of Ax. - Possible dataforms are Proofs of ~C from Ax. + CounterSatisfiable (CSA): Some interpretations are models of Ax, and some models of Ax are models of ~C. - F is not valid, ~F is satisfiable, and C is not a theorem of Ax. - Possible dataforms are Models of Ax \| ~C. + FinitelyCounterSatisfiable (FCS): Some finite interpretations are finite models of Ax, and some finite models of Ax are finite models of ~C. - F is not valid, ~F is satisfiable, and C is not a theorem of Ax. - Possible dataforms are FiniteModels of Ax \| ~C. + SatisfiableCounterTheorem (SCT): Some interpretations are models of Ax, and all models of Ax are models of ~C. - F is valid, and ~C is a theorem of Ax. - Possible dataforms are Models of Ax with Proofs of ~C from Ax. + CounterEquivalent (CEQ): Some interpretations are models of Ax, all models of Ax are models of ~C, and all models of ~C are models of Ax, i.e., all interpretations are models of Ax xor of C. - F is not valid, and ~C is a theorem of Ax. - Possible dataforms are Proofs of ~C from Ax and of Ax from ~C. + UnsatisfiableConclusion (UNC): Some interpretations are models of Ax, and all interpretations are models of ~C (i.e., no interpretations are models of C). - F is not valid, and ~C is a tautology. - Possible dataforms are Proofs of ~C. + WeakerCounterConclusion (WCC): Some interpretations are models of Ax, and all models of Ax are models of ~C, and some models of ~C are not models of Ax. - See CounterTheorem and CounterSatisfiable. + EquivalentCounterTheorem (ECT): Some, but not all, interpretations are models of Ax, all models of Ax are models of ~C, and all models of ~C are models of Ax. - See CounterEquivalent. + FinitelyUnsatisfiable (FUN): All finite interpretations are finite models of Ax, and all finite interpretations are finite models of ~C (i.e., no finite interpretations are finite models of C). + Unsatisfiable (UNS): All interpretations are models of Ax, and all interpretations are models of ~C, i.e., no interpretations are models of C. - F is unsatisfiable, ~F is valid, and ~C is a tautology. - Possible dataforms are Proofs of Ax and of C, and Refutations of F. + WeakerUnsatisfiableConclusion (WUC): Some, but not all, interpretations are models of Ax, and all interpretations are models of ~C. - See Unsatisfiable and WeakerCounterConclusion. + WeakerCounterTheorem (WCT): Some interpretations are models of Ax, all models of Ax are models of ~C, some models of ~C are not models of Ax, and some interpretations are not models of ~C. - See CounterSatisfiable. + SatisfiableCounterConclusionContradictoryAxioms (SCC): No interpretations are models of Ax, and some interpretations are models of ~C. - See ContradictoryAxioms. + Verified (VER): The solution output has been verified. + NoConsequence (NOC): Some interpretations are models of Ax, some models of Ax are models of C, and some models of Ax are models of ~C. - F is not valid, F is satisfiable, ~F is not valid, ~F is satisfiable, and C is not a theorem of Ax. - Possible dataforms are pairs of models, one Model of Ax \| C and one Model of Ax \| ~C. The NoSuccess Ontology In order to understand and make productive use of a lack of success, it is necessary to precisely specify the reason for and nature of the lack of success. The SZS no-success ontology provides status values for describing the reasons. Note that no-success is not the same as failure: failure means that the software has completed its attempt to process the logical data and could not establish a success ontology value. In contrast, no-success might be because the software is still running, or that it has not yet even started processing the logical data. In the figure below many of the "OneWord" status values are abbreviated - see the list below for the official full \| \| \| \| \| Open NotVer Assumed Unknown Incorrect OPN NVE ASS(UNK,SUC) UNK INC \| _________________\|_________________ FailVer \| \| \| FVE Stopped InProgress NotTried STP INP NTT ____________________\|________________ ____\|____ \| \| \| \| \| Error Forced GaveUp \| NotTriedYet ERR FOR GUP \| NTY ____\|____ ____\|____ _________\|__________ \| \| \| \| \| \| \| \| \| \| OSError InputEr User ResourceOut Incompl \| Inappro OSE INE USR RSO INC \| IAP ___\|___ ___\|___ v \| \| \| \| \| ERR UseEr SynEr SemEr Timeout MemOut USE SYE SEE TMO MMO ____\|____ ____\|____ \| \| \| \| TypeError Unsemantic CPUTimeout WCTimeout TYE USM CTO WTO + NoSuccess (NOS): The logical data has not been processed successfully (yet). + Open (OPN): A success value for the abstract problem has never been established. + NotVerified (NVE): The solution output has not been verified. + FailedVerified (FVE): The solution output failed verification. + Unknown (UNK): A success value for the ATP problem has never been established. + Assumed (ASS(U,S)): The success ontology value S has been assumed because the actual value is unknown for the no-success ontology reason U. U is taken from the subontology starting at Unknown in the no-success ontology. + Stopped (STP): Software attempted to process the data, and stopped without a success status. + Error (ERR): Software stopped due to an error. + OSError (OSE): Software stopped due to an operating system error. + InputError (INE): Software stopped due to an input error. + UsageError (USE): Software stopped due to an ATP system usage error. + SyntaxError (SYE): Software stopped due to an input syntax error. + SemanticError (SEE): Software stopped due to an input semantic error. + TypeError (TYE): Software stopped due to an input type error (for typed logical data). + Unsemantic (USM): The semantics makes no sense (for semantics specifications). + Forced (FOR): Software was forced to stop by an external force. + User (USR): Software was forced to stop by the user. + ResourceOut (RSO): Software stopped because some resource ran out. + Timeout (TMO): Software stopped because a time limit ran out. + CPUTimeout (CTO): Software stopped because the CPU time limit ran out. + WCTimeout (WTO): Software stopped because the wall clock time limit ran out. + MemoryOut (MMO): Software stopped because the memory limit ran out. + GaveUp (GUP): Software gave up of its own accord. + Incomplete (INC): Software gave up because it's incomplete. + Inappropriate (IAP): Software gave up because it cannot process this type of data. + InProgress (INP): Software is still running. + NotTried (NTT): Software has not tried to process the data. + NotTriedYet (NTY): Software has not tried to process the data yet, but might in the future. The Dataform Ontology The dataform ontology provides suitable values for describing the form of logical data. The ontology values are commonly used to describe data provided to justify a success ontology value, e.g., if an ATP system reports the success ontology value Theorem it might output a proof to justify that. In the figure below many of the "OneWord" status values are abbreviated - see the list below for the official full "OneWord"s. \| \| LogicalData NonLogicalData LDa NLd __________\|___________________________ ____\|____ \| \| \| \| \| None Solution NotSoln Comment FreeText Non Sol NSo Com FTx __________\|____________ ______\|______ \| \| \| \| \| \| Proof Interpretation ListFrm Assure IncPrf IncInt Prf Int Lof Ass IPr IIn ___\|___ \|\ \|___________ \| \| \| Model \| \| \| \| Derivn Refutn \| Mod LiTHF/TFF/FOF/CNF Der Ref \|/ Lth/Ltf/Lfo/Lcn \| \|________ \|___\|___\|___\| CNFRef \|\ \ \| CRf \| Partial Strictly \| \| PIn/PMo SIn/SMo \| \|/_______/ \| __________\|___________ _____\| \| \|/ Domain Int/Mod Herbrand Int/Mod DIn/DMo HIn/HMo DPI/DPM/DSI/DSM HPI/HPM/HSI/HSM ________\|________ ____\|____ \| \| \| \| \| Finite Integer Real Formula Saturation FIn/FMo IIn/IMo RIn/RMo TIn/TMo Sat FPI/FPM IPI/IPM RPI/RPM TPI/TPM FSI/FSM ISI/ISM RSI/RSM TSI/TSM + Data (Dat): Data output. + LogicalData (LDa): Logical data. + Solution (Sln): A solution. + Proof (Prf): A proof. + Derivation (Der): A derivation (inference steps ending in the theorem, in the Hilbert style). + Refutation (Ref): A refutation (starting with Ax U ~C and ending in FALSE). + CNFRefutation (CRf): A refutation in clause normal form, including, for FOF Ax or C, the translation from FOF to CNF (without the FOF to CNF translation it's an IncompleteProof). + Interpretation (Int): An interpretation. + Model (Mod): A model. + PartialInterpretation (Pin): A partial interpretation. + PartialModel (PMo): A partial model. + StrictlyPartialInterpretation (SIn): A strictly partial interpretation. + StrictlyPartialModel (SMo): A strictly partial model. + DomainInterpretation (DIn): An interpretation whose domain is not the Herbrand universe. + DomainModel (DMo): A model whose domain is not the Herbrand universe. + DomainPartialInterpretation (DPI): A domain interpretation that is partial. + DomainPartialModel (DPM): A domain model that is partial. + DomainStrictlyPartialInterpretation (DSI): A domain interpretation that is strictly partial. + DomainStrictlyPartialModel (DSM): A domain model that is strictly partial. + FiniteInterpretation: A domain interpretation with a finite domain. + FiniteModel (FMo): A domain model with a finite domain. + FinitePartialInterpretation (FPI): A domain partial interpretation with a finite domain. + FinitePartialModel (FPM): A domain partial model with a finite domain. + FiniteStrictlyPartialInterpretation (FSI): A domain strictly partial interpretation with a finite domain. + FiniteStrictlyPartialModel (FSM): A domain strictly partial model with a finite domain. + IntegerInterpretation (IIn): An integer domain interpretation. + IntegerModel (IMo): An integer domain model. + IntegerPartialInterpretation (IPI): An integer domain partial interpretation. + IntegerPartialModel (IPM): An integer domain partial model. + IntegerStrictlyPartialInterpretation (ISI): An integer domain strictly partial interpretation. + IntegerStrictlyPartialModel (ISM): An integer domain strictly partial model. + RealInterpretation (RIn): A real domain interpretation. + RealModel (RMo): A real domain model. + RealPartialInterpretation (RPI): A real domain partial interpretation. + RealPartialModel (RPM): A real domain partial model. + RealStrictlyPartialInterpretation (RSI): A real domain strictly partial interpretation. + RealStrictlyPartialModel (RSM): A real domain strictly partial model. + HerbrandInterpretation (HIn): A Herbrand interpretation. + HerbrandModel (HMo): A Herbrand model. + FormulaInterpretation (TIn): A Herbrand interpretation defined by a set of TPTP formulae. + FormulaModel (TMo): A Herbrand model defined by a set of TPTP formulae. + FormulaPartialInterpretation (TPI): A Herbrand partial interpretation defined by a set of TPTP formulae. + FormulaPartialModel (TPM): A Herbrand partial model defined by a set of TPTP formulae. + FormulaStrictlyPartialInterpretation (TSI): A Herbrand strictly partial interpretation defined by a set of TPTP formulae. + FormulaStrictlyPartialModel (TSM): A Herbrand strictly partial model defined by a set of TPTP formulae. + Saturation (Sat): A Herbrand model expressed as a saturated set of formulae. + ListOfFormulae (Lof): A list of formulae. + ListOfTHF (Lth): A list of THF formulae. + ListOfTFF (Ltf): A list of TFF formulae. + ListOfFOF (Lfo): A list of FOF formulae. + ListOfCNF (Lcn): A list of CNF formulae. + NotASolution (NSo): Something that is not a well formed solution. + Assurance (Ass): Only an assurance of the success ontology value. + IncompleteProof (IPr): A proof with some part missing. + IncompleteInterpretation (IIn): An interpretation with some part missing. + NonLogicalData (NLd): Non-logical output. + Comment (Com): TPTP format comments (starting with %). + FreeText (FTx): Anything you want. + None (Non): [1] Sutcliffe G., Zimmer J., Schulz S. (2003), Communication Formalisms for Automated Theorem Proving Tools, Sorge V. Colton S. Fisher M. Gow J., Proceedings of the Workshop on Agents and Automated Reasoning, 18th International Joint Conference on Artificial Intelligence, (Acapulco, Mexico), 52-57. [2] Sutcliffe G., Zimmer J., Schulz S. (2004), TSTP Data-Exchange Formats for Automated Theorem Proving Tools, Zhang W., Sorge V., Distributed Constraint Problem Solving and Reasoning in Multi-Agent Systems, Frontiers in Artificial Intelligence and Applications 112, 201-215. [3] Sutcliffe G. (2008), The SZS Ontologies for Automated Reasoning Software, Rudnicki P., Sutcliffe G., Proceedings of the LPAR Workshops: Knowledge Exchange: Automated Provers and Proof Assistants, and The 7th International Workshop on the Implementation of Logics (Doha, Qattar), CEUR Workshop Proceedings 418, 38-49.	{"url":"https://tptp.org/cgi-bin/SeeTPTP?Category=Documents&File=SZSOntology","timestamp":"2024-11-10T10:40:32Z","content_type":"application/xhtml+xml","content_length":"27832","record_id":"<urn:uuid:18b0c76b-d589-4ba5-908a-825bcf2c10b2>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00786.warc.gz"}
( UPLO, TRANS, DIAG, N, A, LDA, X, INCX ) INTEGER INCX, LDA, N CHARACTER1 DIAG, TRANS, UPLO DOUBLE PRECISION A( LDA, ), X( * ) DTRSV solves one of the systems of equations where b and x are n element vectors and A is an n by n unit, or non-unit, upper or lower triangular matrix. No test for singularity or near-singularity is included in this routine. Such tests must be performed before calling this routine. UPLO - CHARACTER1. On entry, UPLO specifies whether the matrix is an upper or lower triangular matrix as follows: UPLO = 'U' or 'u' A is an upper triangular matrix. UPLO = 'L' or 'l' A is a lower triangular matrix. Unchanged on exit. TRANS - CHARACTER1. On entry, TRANS specifies the equations to be solved as follows: TRANS = 'N' or 'n' Ax = b. TRANS = 'T' or 't' A'x = b. TRANS = 'C' or 'c' A'x = b. Unchanged on exit. DIAG - CHARACTER1. On entry, DIAG specifies whether or not A is unit triangular as follows: DIAG = 'U' or 'u' A is assumed to be unit triangular. DIAG = 'N' or 'n' A is not assumed to be unit triangular. Unchanged on exit. N - INTEGER. On entry, N specifies the order of the matrix A. N must be at least zero. Unchanged on exit. A - DOUBLE PRECISION array of DIMENSION ( LDA, n ). Before entry with UPLO = 'U' or 'u', the leading n by n upper triangular part of the array A must contain the upper triangular matrix and the strictly lower triangular part of A is not referenced. Before entry with UPLO = 'L' or 'l', the leading n by n lower triangular part of the array A must contain the lower triangular matrix and the strictly upper triangular part of A is not referenced. Note that when DIAG = 'U' or 'u', the diagonal elements of A are not referenced either, but are assumed to be unity. Unchanged on exit. LDA - INTEGER. On entry, LDA specifies the first dimension of A as declared in the calling (sub) program. LDA must be at least max( 1, n ). Unchanged on exit. X - DOUBLE PRECISION array of dimension at least ( 1 + ( n - 1 )*abs( INCX ) ). Before entry, the incremented array X must contain the n element right-hand side vector b. On exit, X is overwritten with the solution vector x. INCX - INTEGER. On entry, INCX specifies the increment for the elements of X. INCX must not be zero. Unchanged on exit. Level 2 Blas routine. -- Written on 22-October-1986. Jack Dongarra, Argonne National Lab. Jeremy Du Croz, Nag Central Office. Sven Hammarling, Nag Central Office. Richard Hanson, Sandia National Labs.	{"url":"https://manpages.org/dtrsv/3","timestamp":"2024-11-03T17:10:29Z","content_type":"text/html","content_length":"33415","record_id":"<urn:uuid:1ca79e3f-5afd-41c0-8a9a-632e9b9e2ee3>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00082.warc.gz"}
Methods for computing the maximum performance of computational models of fMRI responses Computational neuroimaging methods aim to predict brain responses (measured e.g. with functional magnetic resonance imaging [fMRI]) on the basis of stimulus features obtained through computational models. The accuracy of such prediction is used as an indicator of how well the model describes the computations underlying the brain function that is being considered. However, the prediction accuracy is bounded by the proportion of the variance of the brain response which is related to the measurement noise and not to the stimuli (or cognitive functions). This bound to the performance of a computational model has been referred to as the noise ceiling. In previous fMRI applications two methods have been proposed to estimate the noise ceiling based on either a split-half procedure or Monte Carlo simulations. These methods make different assumptions over the nature of the effects underlying the data, and, importantly, their relation has not been clarified yet. Here, we derive an analytical form for the noise ceiling that does not require computationally expensive simulations or a splitting procedure that reduce the amount of data. The validity of this analytical definition is proved in simulations, we show that the analytical solution results in the same estimate of the noise ceiling as the Monte Carlo method. Considering different simulated noise structure, we evaluate different estimators of the variance of the responses and their impact on the estimation of the noise ceiling. We furthermore evaluate the effect the interplay between regularization (often used to estimate model fits to the data when the number of computational features in the model is large) and model complexity on the performance with respect to the noise ceiling. Our results indicate that when considering the variance of the responses across runs, the computing the noise ceiling analytically results in similar estimates as the split half estimator and approaches the true noise ceiling under a variety of simulated noise scenarios. Finally, the methods are tested on real fMRI data acquired at 7 Tesla. • GENERAL LINEAR-MODEL • SINGLE-SUBJECT • ORGANIZATION • RELIABILITY • PATTERNS Dive into the research topics of 'Methods for computing the maximum performance of computational models of fMRI responses'. Together they form a unique fingerprint.	{"url":"https://cris.maastrichtuniversity.nl/en/publications/methods-for-computing-the-maximum-performance-of-computational-mo","timestamp":"2024-11-13T02:07:05Z","content_type":"text/html","content_length":"64567","record_id":"<urn:uuid:a81b3a48-d0e6-4b4c-8c9a-9fe4d7c06b83>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00859.warc.gz"}
class compas.geometry.Bezier(points)[source] Bases: Primitive A Bezier curve is defined by control points and a degree. A Bezier curve of degree n is a linear combination of n + 1 Bernstein basis polynomials of degree n. points (sequence[point]) – A sequence of control points, represented by their location in 3D space. ☆ points (list[Point]) – The control points. ☆ degree (int, read-only) – The degree of the curve. >>> curve = Bezier([[0.0, 0.0, 0.0], [0.5, 1.0, 0.0], [1.0, 0.0, 0.0]]) >>> curve.degree from_data Construct a curve from its data representation. locus Compute the locus of all points on the curve. point Compute a point on the curve. tangent Compute the tangent vector at a point on the curve. transform Transform this curve. Inherited Methods ToString Converts the instance to a string. copy Make an independent copy of the data object. from_json Construct an object from serialized data contained in a JSON file. from_jsonstring Construct an object from serialized data contained in a JSON string. sha256 Compute a hash of the data for comparison during version control using the sha256 algorithm. to_data Convert an object to its native data representation. to_json Serialize the data representation of an object to a JSON file. to_jsonstring Serialize the data representation of an object to a JSON string. transformed Returns a transformed copy of this geometry. validate_data Validate the object's data against its data schema. validate_json Validate the object's data against its json schema.	{"url":"https://compas.dev/compas/1.16.0/api/generated/compas.geometry.Bezier.html","timestamp":"2024-11-11T07:37:32Z","content_type":"text/html","content_length":"25753","record_id":"<urn:uuid:432da825-760c-4208-bb0a-71a4e428c6b8>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00878.warc.gz"}
Tanya Khovanova's Math Blog Whenever I am under stress, I turn to jokes. My recent problems with spam attacks on my blog led me to surf the web for new math jokes. Here are some of my recent translations from Russian. * * * Two is the same thing as eight, to some degree. * * * A girl to her mathematician boyfriend: — Let’s do something that is forbidden tonight. — Divide by zero? * * * If thoughts converge, they are bounded. * * * A mathematician’s son: — Dad, how do I write the number 8? — That’s easy: rotate the infinity symbol by pi over 2. * * * My student couldn’t take an integral from my book. So he took the book together with all the integrals there. * * * Archimedes, Pascal and Newton play hide and seek. Archimedes is the seeker. Pascal hides, but Newton draws a 1-meter square around himself. Archimedes opens his eyes and shouts: — I see Newton! — Oh, no! One newton per square meter is the pascal. * * * What a pleasure to smoke an e-cigarette after cybersex… * * * Russians were the first in the world to create a computer program that passes the Turing test. Scientists tested the program using several Russians with a variety of questions, and each time the program gave the same answer as the people. The reply to every question was, “Go fck yourself!” * * There are two types of people: those who know nothing about fractals and those who think that there are two types of people: those who know nothing about fractals and those who think that there are two types people… 5 Comments 1. Robb Seaton: Sorry to hear about your spam attacks and consequent stress. Here are a couple of my favorites (I have also collected a few more in a math jokes post I made a while back): Theorem. All positive integers are interesting. Proof. Assume the contrary. Then there is a lowest noninteresting positive integer. But, hey, that’s pretty interesting! A contradiction. One day a farmer called up an engineer, a physicist, and a mathematician and asked them to fence in the largest possible area with the least amount of fence. The engineer made the fence in a circle and proclaimed that he had the most efficient design. The physicist made a long, straight line and proclaimed “We can assume the length is infinite…” and pointed out that fencing off half of the Earth was certainly a more efficient way to do it. The mathematician just laughed at them. He built a tiny fence around himself and said, “I declare myself to be on the outside.” 10 October 2014, 11:48 pm 2. George R.: Dear Tanya, as far as your anti-spam hunting…ΜΗΔΕΝ ΆΓΑΝ! (miden agan) 🙂 Speaking about hunting… Me(an engineer) and two of my friends (Bob: a theoretical physicist and Rob:a statistician) went hunting the other day. A magnificent deer at 300 yards. Bob worked out relevant equations and calculations (speed of bullet,trajectories, angles etc.) assumed zero wind resistance 🙂 and shot! The bullet was too short. 5 yards short. I took of course Bob’s calculation for granted, added a corrective coefficient for air-resistance (which I found in “Beton-Kalender” of course..) raised the gun at a slightly bigger angle ,and shot! Bullet was 5 yards too long. Rob the statistician cheered: WE’VE GOT IT!! And a “Joke” from real mathematical life. Check out the Latin inscription/moto on Fields medal (on the side where it is written “AΡΧΙΜΗΔΟΥΣ” (by Archimedes) ”Transire suum pectus mundoque potiri” (“Rise above oneself and grasp the world”..or something like this ,anyway..) Various well respected sources (such as Wolfram Alpha, among others) attribute this quote to Archimedes. But this is not the case. The quote is an abstract of a 1st century ASTROLOGICAL poem by the roman poet Manilius. “Multum’ inquis ‘tenuemque iubes me ferre laborem, rursus et in magna mergis caligine mentem, cernere cum facili lucem ratione viderer.’ quod quaeris, deus est: conaris scandere caelum fataque fatali genitus cognoscere lege et transire tuum pectus mundoque potiri. pro pretio labor est nec sunt immunia tanta, ne mirere viae flexus rerumque catenas” As a friend (a mathematician) observed : “Of all quotes in the World, they ve’ chosen a false quote of Archimedes and moreover a quote related to Astrology?? The black sheep of the “family”? 🙂 PS. I solved quite easily 8-?=2 stupid captcha! ha! 🙂 11 October 2014, 6:06 am 3. Bruce: This year the first woman was awarded a Fields Medal. Oddly, no post has been placed yet to celebrate this achievement. 15 October 2014, 2:48 pm 4. tanyakh: Everyone wrote about it. I try to write things that are not available elsewhere. If I find something original to say on the subject I will. 18 October 2014, 5:58 am 5. Edward Starr: Tanya Khovanova, Your jokes are fabulous and your humor contagious. I wish you many more jokes, but without the stress. Thanks. 21 October 2014, 2:28 pm	{"url":"https://blog.tanyakhovanova.com/2014/10/de-stressing-jokes/","timestamp":"2024-11-10T22:40:59Z","content_type":"text/html","content_length":"63769","record_id":"<urn:uuid:de5ff16b-ef6b-4858-ab7c-d13e833d347f>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00272.warc.gz"}
M E T A L vs S K I N -based, except race and class are separate, non-fighters progress (slowly) in combat, and magic is this system . Players are new to D&D or haven't played in a long time, so I'm keeping things simple. I'm compiling this for me, but hey, why not share? 1. 3d6 in order (Str, Dex, Con, Int, Wis, Cha); you may swap one. Then derive Power score (highest two of Int, Wis, Cha -28). Ability Score Modifier 3 –3 4–5 –2 6–8 –1 9–12 0 13–15 +1 16–17 +2 18 +3 2. Choose race: (no adjustments) (humans with ancestors who interbred with magical beings, requires +200 times level XP to progress in level over normal humans in class; IE +200 for level 2, +400 for level 3, etc) (+1 Con, -1 Dex, +1 on rolls related to Earth Elementals, LotFP Dwarf Architecture bonus) (+1 Dex, -1 Str or Con, +1 to rolls related to Air Elementals, 1/2 falling damage) (+1 Str or Cha, -1 Wis, +1 rolls w/Fire Elementals, 1 Dmg Resistance & +3 saves VS heat) (+1 Wis or Dex, -1 Int or Str, +1 rolls w/Water Elementals, double breathing & movement in water) (+1 Cha or Wis, -1 Cha or Wis, +1 rolls w/Demons/Devils, sense magic/spirits by concentrating 1 round and rolling under Int, +1 Stealth) (~3 ft tall anthropomorphic cat or rabbit with tiny bat wings; +1 Wis, +1 Dex, -2 Str, +1 Pow, requires +500 times level XP to advance to next) (~7 ft tall steamwork automaton; +2 Str, +1 Con, -2 Dex, -1 Cha, -1 Pow, LotFP Dwarf encumberance bonus, +1 Int Check to repair self with other steamworks or generally operate them) (more races will be available if PCs make friends with them as they adventure) 3. Choose Class: Zak's Random (as LotFP; I may allow Zak's Random one later if people get antsy) Zak's Random (as LotFP but +half level to hit round down, or Zak's Random (+one third level to hit IE +1 at level 3; edited to state cannot wear metal armor) Class determines HP, Saves, level progression, etc, modified by race as above 4. Choose Equipment: Everyone in the first session is starting as a teen (age 12+1d6) in a mining town near the edge of a steam powered magitek empire. People who start later will get more adventurey equipment options. To start: Spend 1d6x10 SP in LotFP, GP in AD&D, or roll 1d20 below until you get the same number. You can reroll duplicates that are not the same number if you want (IE if you roll a 3, then a 5, you could reroll the 5). 1. candle 2. hammer and 2d6 nails 3-5. knife 6. leather apron and goggles +2 AC 7. sturdy hat or protective mask +1 AC 8-10. Backpack 11. hatchet 12. flint and steel 13. mirror 14. 10 foot pole 15. marbles or jacks 16. 25 ft rope 17. lantern +1d4 flasks of oil 18. lockpick 19-20. 1 day's rations EDIT: Then roll 1d10 on this table: 1. wrench 2. bow and 10 arrows 3. a ranged weapon that is not a bow 4. bolt gun and 2d4 bolts 5. bolt of cloth 6. jug of water 7. compass/pocketwatch combo 8. spyglass 9. key to parent's Rouncer (a 10 ft tall two legged tank-like thing that fits 2 people. All weapons have been tripped from them in this setting, but they are useful for overland travel) 10. 1d4 jars of lard I may use some tables to determine parent's occupation and resulting starting equipment or bonuses; if so I'll use the Hill Cantons Compendium or DCCRPG. The premise is that spirits do magic, spirits are everywhere, but not many people can talk to them or get them to do things. The more you want them to do, the more risk you assume. It's mashing LotFP Summon with (1-3e) Stormbringer's magic system, trying to make it a little more accessible and broadly useful, but not losing too much flavor. Note: the equation to get one's POW score below assumes "3d6 in order, swap one" for Ability Scores. If your Ability Score method is more forgiving (i.e. 4d6 drop lowest), use 30 instead of 28. Magic is primarily done by elementals, ghosts, and demons. Humans are not inherently magical creatures. Some gifted men (and beasts) can communicate with the spirits and find minor spirits bored enough to fulfill simple requests. For more powerful magics, more powerful entities must be summoned and bound by a Sorcerer or Witch. POWer: this Ability Score is found by adding the two highest of INT, WIS and CHA, then subtracting 28. If the result is positive, your character is one of the gifted individuals with some instinct for communicating with Elementals, Spirits, and Demons. How POW is expended is detailed below. A character recovers 1 POW by sleeping 4 hours uninterrupted. Another point is recovered for every 2 hours' continuous sleep beyond the initial 4. Vespers are the most minor of spirits, immature and playful, like small children. They can be asked to do minor favors easily, since they are often bored. The favors they do are called "Cantrips." This list is not exhaustive, but covers most basics. To get a Vesper to do a Cantrip for you, spend 1 POW attracting their attention, then make a successful roll under the relevant ability score to see if you were convincing. If the character wishes the Vesper to bluntly enact their will, the roll is straight or at a penalty. If the player enacts the conversation with the DM or does a little ritual, the roll may have a bonus. │ Cantrip Name │ Roll │ Effect │ │ (Vesper type) │ Under │ │ │ Light │ INT │ Produce one small globe of light (40 │ │ (Fire) │ │ foot radiance) for 1 hour per level │ │ │ │ of caster. Produces heat if the │ │ │ │ requestor wishes, enough to light │ │ │ │ torches from. │ │ Control of Body │ WIS │ Total control over body (breath, │ │ (Earth, Water, or │ │ heart beat, grip etc) for 10 minutes │ │ Air) │ │ per level of the caster. │ │ Clairvoyance │ INT │ Can see through walls and other │ │ (Ghosts, Demons) │ │ obstructions up to 50 feet from │ │ │ │ caster for 10 minutes times the │ │ │ │ caster's level. │ │ Telekinesis │ INT │ Objects within site-distance of the │ │ (Air) │ │ caster may be moved up to 50 feet. │ │ │ │ Object can weigh no more than one │ │ │ │ ounce per level of the caster. │ │ Cure Wounds │ WIS │ Cure 1d6 hit points damage. Also │ │ (Earth) │ │ stops fatal major or critical wounds │ │ │ │ from bleeding out. │ │ Comprehend │ CHA │ Can read/write/speak any language for │ │ Language │ │ 5 minutes duration per caster level. │ │ (Ghosts, Demons) │ │ │ (table adapted from +Chris Kutalik 's Stormbringer Hack with ideas from Beyond the Wall.) Sorcerers and Witches These are individuals who, through a teacher and intense study of magical tomes, has discovered shortcuts and systems by which they may more easily contact spirits, demons, and elementals. Even if they are not naturally gifted in such matters (IE not high enough INT/WIS/CHA to have POW) they can contact spirits. They have a number of POW equal to their level, plus any gained as figured above or through other methods that come up during play. Eedited to add: Sorcerers and Witches cannot use their POW while wearing metal armor. The studied form of this magic cannot be channeled and focused while in contact with lots of metal. Sorcerer and Witch are interchangeable terms, and used for both sexes depending upon the region. These individuals are able to summon and bind beings more powerful than Vespers. Summoning and Binding There are many types of spirits more powerful than Vespers, but the three who most commonly traffic with mortals are Elementals, Demons, and Ghosts. One can attempt to summon Elementals of Earth, Air, Fire, or Water, and Demons, at any time. Ghosts tend to be bound to specific locations, and thus can only be contacted in those locations. Step 1: Attract Attention Determine the type of creature you would like to summon, its number of Hit Dice, and number of powers. A creature's Hit Dice is always equal to or more than the Power spent to summon it. Spend the appropriate amount of Power. Minor Elemental or Ghost: 1 POW Minor Demon: 2 POW Major Elemental: 3 POW Major Demon: 4 POW Step 2: Make an Ability check to determine the entity's disposition toward you; you are trying to roll under the relevant ability... plus your Power score after spending the above and any points from Garmonbozia or Rituals minus the creature's number of Hit Dice and powers. •If you have no previous experience with this type of creature roll under your Charisma, modified by Intelligence modifiers. •If you have studied this creature from a book or another Sorcerer's anecdotes, you may choose to roll under Intelligence, modified by any Charisma modifiers. •If you have previous experience with this type of creature, you may choose to roll under your Wisdom, modified by any INT and CHA modifiers. Success: the creature will do the first thing you ask of it, then accompany you for a number of rounds equal to your success. On a critical success (roll a 1), add 1 to your POW for a day. If you roll your current max Power or less on a d100, the bonus is permanent. Failure: The creature is angry at being bothered, and will manifest for a number of rounds equal to your failure. Your companions may be able to reason with it, depending, and it will probably not sacrifice it's life to destroy you, but it's first instinct will be to attack. Elementals are similar to the descriptions in Stormbringer, editions 1-3 by Chaosium Demons are as the LotFP Summon spell, except they have one Spell from the Magic User list or the reverse of one spell from the Cleric list per Hit Point. These are chosen by the summoner prior to summoning ("I call forth a Demon with the power of...") Ghosts are each unique and what powers they have are unpredictable. They are not so much summoned as stumbled upon and communicated with. They can be powerful allies at little cost, or terrible enemies, possessing the summoner's body. It is not wise to meddle with ghosts one knows nothing about. ...is the energy produced by strong human emotions. Spirits of all types are attracted to it, and Demons especially feed upon it. Sex and death reliably produce it. It is the reason virgin sacrifice is associated with demon summoning. Not only do you get the fear of the sacrifice, but the melancholy of the cultist thinking on the virgin's wasted potential in life. The producer of Garmonbozia does not have to even be aware the summoning is taking place, however, only be nearby. Several couples have been surprised when their conjugal tryst helped some Sorcerer in the next room. The methods of attempting to create Garmonbozia near the summoning are left to the player's imagination and the DM's discretion; when in doubt, use INT modifiers. ...let the players have fun making these up, but they cost 500 silver pieces worth of components for every +1 bonus desired. Alternatively, a ritual circle can be constructed from materials in the natural world over a continuous period of 36-INT hours for a +1 bonus, but the summoner cannot sleep, rest, or carry out any other meaningful work while constructing the circle. After a spirit is summoned, instead of requesting an immediate favor, the summoner may choose to Bind the spirit instead. An item must be chosen for this purpose beforehand, and the same amount of Power spent to bind that it took to summon. Make the same Ability Check in Step 2 above. The creature is bound to the object for 1 day for every point of success, and can be summoned forth from the object at no additional cost during that time. At the end of the time, the creature is freed from the object, and will treat the summoner and his companions appropriately for how it was treated. A Sorcerer/Witch can only have one creature summoned or bound per level, except... At the expense of one Power point permanently, a spirit can be Bound to an object for a year per level of the summoner. At the end of this time, the spirit may choose to remain in the object. A spirit bound so does not count toward the Sorcerer's maximum. A Witch can try to give the object some effect of the spirit by making a check as detailed in Step 2 above, with the difficulty adjusted appropriate to the effect. There are also less predictable events that can cause Bound spirits to affect their objects. ...is the binding of a spirit to a corpse. The spirit is caught in the corpse for a number of months equal to the amount of bonuses gained from Garmonbozia and/or Rituals in the binding. It can also be freed if it's "body" is destroyed by violence. This process forces the spirit to use whatever brains the body has left, or only be able to carry out simple one sentence commands. Decay is arrested by the binding. (Necromancy will be fleshed out - har har - by the demon "construction" rules from Stormbringer if need be. I think it'll be an NPC thing mostly, though). Click Santicore above to get the Secret Santicore 2013 Volume 3 PDF! Click the thumbnail below to get an illustrated Secret Santicore 2013 cover by Scrap Princess! Volume 1 is here Volume 2 is here Edits, corrections, etc can be emailed to metalvsskin at gmail dot com I'm starting a background, play-by-post/email game based on Zak S's Slow War rules for my Kalak-Nur setting. I was trying to be coy about it, but it's not gonna get done right without a big blog post everyone can refer to, so here is that. Players, remember, email me your moves. If you put them here or G+ you're giving info away to your potential enemies. If you're not playing, it's full up at the moment, sorry. Comment here and if I remember I'll let you know if a spot opens up and there are unclaimed factions. If you're in my every-other Friday games, hey, we're all adults here; either don't read this or separate player knowledge from character knowledge. Some of this your guy might even know, as history or whatever. There's no functional difference between light and dark grey hexes; they're all dunes of metal filings. The whitish hexes in the frozen south are, uh, frozen. Movement is generally 4 Hexes on the Map per turn. Special Movement notes: -The following forces move at half the above, 2 hexes per turn, barring special circumstances: Digrot, Trepan, Rebel Droids -EMFLWEM's forces must Occupy a hex for a full turn before moving on to properly freeze the ground for movement. Hella slow, but see bonuses to fight in those hexes. -Occupying a hex: at least 20 STR worth of your forces are there unopposed (or winning a conflict) for a turn. This lets you take resources if there's a town there. Only the major cities are listed above; there are smaller settlements that can be found by exploration. -The Goblin Coalition forces can move at double-speed below ground, IE not exploring just getting troops into position; there's less NPC stuff underground to encounter, but more of it is hostile. A lot of this is adapted from Zak S.'s Slow War Rules. Direct and relevant quotes from his blog: Make a move at any time by telling me. Do whatever you want. I'll inform you what needs to be done to resolve your action. There'll be a lot of rulings-not-rules and Common Law game design here. Most of the rest of this is just nitpicking and I-dotting, so don't let it intimidate you.... Timing and Speed: I'll try to get back to y'all at least once per day. Actions where timing is immediately important will be resolved in order of fastest action first. Nominally, each day represents at least a day, so put everything you want done (including if/thens) in your orders. Strength: Basically you start with 100 Strength. This is the currency you use for a lot of things representing wealth, power, influence, etc. 1 strength point represents 1% of your army (usually about 1000 hd worth of regulars) or 10 levels/HD worth of special types (including whatever is needed to train, transport and feed them)--like if you need an assassin, you can spend a Strength point to get a 10th level assassin or spend a strength point to make a little force of 2 3hd fighters and one 4th level wizard, etc. (At the start, some armies (like [Amaya Neow's or potentially Digrot's] army) are big; some, like [The Black Veil] are small, but the total strength of those forces is equal. One [Sister of the Veil] is effectively worth [several crazy biotech mutants or mindless undead].) When you win fights or take important territory you win Strength. A lot of things cost no Strength, you just do 'em. I'll keep track of how much Strength you have. Exceptions: Town armies start at 50 STR, but a player may have two of them as their Forces. Digrot's army's STR depends on his first move. Every space on the map has something on it... Just rolling around purposefully searching territory without fighting anyone else is kind of like pulling cards off the Chance pile in Monopoly--you might get something good, you might get If you find anything good, likely at least one other faction will hear about it. Each side rolls d100 plus or minus modifiers plus the total Strength of the troops you sent. High roll wins, disparity equals the percent of the strength gambled that the losing side lost. You can send as many troops as you want, but a maximum of 30 Strength worth of troops can be "gambled" on one roll. Battles larger than that go in phases, and there is time for forces to maneuver after the first engagement/roll. Many factions have a headquarters at their starting location. If you lose that, you lose half your Strength. [Digrot, Rebel Droids, Amaya Neow, and The Elves of Trepan have no headquarters to begin with.] Winning and Losing: When you're out of Strength, you gotta lick your wounds and take your army home. Maybe make a new faction. Last man, woman or salient entity standing is the winner or, if the game ends early, whoever has the most Strength when the game ends wins. The players will be ranked, and their FLAILSNAILS D&D PCs will be awarded bonus xp based on that rank. When you do secret stuff there's a pretty good chance at least one other faction will find out. This is the table I'll roll on (note [Digrot] isn't on there, he only notices the obvious): Who hears about your plans? Roll d20 1. Mocata (I know, I know, no one's playing him...) 2. Mocata 3. Victarion 4. Victarion 5. Chchcchchctktktktktk 6. Chchcchchctktktktktk 7. Chchcchchctktktktktk 8. Goblins 9. Goblins 10. Rebel Droids 11. EMFLWEM 12. Black Veil 13. Black Veil 14. Amaya Neow 15. The Elves of East Trepan 16. Targenmoor 17. Baksheesh 18. Vance 19. Cortuth 20. roll 1d4 on 16-19 Temporary Defense Coalition of Independent Goblin Working CollectivesCan move quickly underground to position troops. -1/4 STR if they attack Victarion's forces (they are traditional allies). -10 in open battle; +10% to success of tricks and traps. The Droid Army of Victarion the Brave Occasionally, something draws Victarion's attentions beyond his plantation and the delivery of food from it to the settlements of the Grey Waste. Once his droids get out of the Westlands, whatever that something is usually doesn't last long. -1/4 STR if his forces attack Goblins (they are traditionally allies.) Rebel Droids for a Free Future A faction of Droids have freed themselves from Victarion's will and formed their own force. Through means both physical and otherwise, they have bound some of the mindless zombies infesting the Westlands to use as footsoldiers. 1/2 move, +10 to open warfare, 1 in 10 chance of fuckup (zombies, mindless, etc); -10 to attack zombies except in self-defense. EMFLWEM The Ice Demon Since being released by adventurers some time ago, EMFLWEM has taken over in the dark, frozen South. Ice Goblins are his footsoldiers, and if he freezes enough ground he can call forth Frost Giants. (Must occupy a hex one entire turn before moving to the next; +20 to open combat in frozen squares) Sisterhood of the Black Veil Look, poetry fails you sometimes, so here: Psychic Jedi Nuns whose mission is to keep anyone else from getting too much magical strength or using big items. Low in numbers, so -10 STR to open warfare each turn after the first in sustained fights. They act first in battle. Amayawjkqd neow3j0100101010 - Amaya Neow for short An insane mutant vocaloid who has turned most of the isolated cults in The Grey Waste into human/android/insect hybrids. Lots of weird abilities, but 1 in 6 chance her forces get distracted by something shiny for a turn. "We have lived among them, even loved them, but their time is over. The pretender moves through the waste, and we must end her and her tortured mutants." Starts at 25 STR in any settlement, must get to any other to gain another 25 STR. The one who had bound me is destroyed; I awake, and soon, all will sleep. Forever. (His forces are at -25 STR to attack Droids; they hunger instead for living flesh. Possibility to raise the slain fleshy types to increase his strength.) The Elves of East Trepan Our spies in many settlements have stopped reporting; perhaps our inferiors are finally catching up. Time to crush them. Even separated from our homeland, we will triumph. 1/2 move (unfamiliar territory), Sorcery, -10 STR if they have not occupied a settlement in the last 5 turns. The Armies of the Free Cities Each starts at 50 STR; a player may choose 2 of them to begin with. The grand bazaar is soon. (modeled on fantasy arabia, but just one sprawling city; allied with genies) Built on crashed ships. Has some weird tech. Closely allied with Mocata. Why hasn't be communicated recently? Grew around a remote trading post. Defiantly independent. Relies on trade to survive though. Protected by the Ludo, giant furry beasts that can control stone by singing. Ancient, dark forces flow through here, causing the people to keep secrets, conspire against one another, and get in touch with dark powers.	{"url":"https://metalvsskin.blogspot.com/2014/03/","timestamp":"2024-11-06T21:39:56Z","content_type":"text/html","content_length":"159925","record_id":"<urn:uuid:35173fca-a83d-4812-8ec2-2ac764ca8a59>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00421.warc.gz"}
Modular Arithmetic \| CS70 Guide What is Modular Arithmetic? Modular arithmetic is "clock math" - that is, when numbers wrap around back to 0 if they get too big. You could think about it like a remainder: $21 \pmod{10}$ for example can be read as "what is the remainder of 21 when it is divided by 10?" (it's 1, by the way.) This is an important concept in many aspects of computer science, namely cryptography and error correction among many others. Key Ideas If d divides x and d divides y, then d divides (y-x). $d \mid x, d \mid y \implies d \mid (y-x)$ (Reminder: $a \mid b \iff (\exists q \in \mathbb{Z})(a = qb)$) Modular Equivalence: If you're looking at a clock and it becomes 25:00, you know it's actually the same as 1:00. Even if they're technically not the same number, they can be treated the same way. More formally: x is congruent to y modulo m: $x \equiv y \pmod{m}$**** Important Notation Distinction: $x \pmod{m}$ is the class of numbers that follow the mod rule $m$. It can be used to write equivalences ($10 \equiv 21 \pmod{11}$). However, $\mod(x, m)$is just a number (the remainder when dividing x by m). $\mod(x,m) = x - \lfloor{\frac{x}{y}}\rfloor \cdot y$ Greatest Common Denominator (GCD Mod Corollary): Modular arithmetic can be used to identify an important property of the GCD, which is that $GCD(x,y) = GCD(x \mod y, y)$. Given that $a \equiv b \pmod{m}$, $a+c \equiv b+c \pmod{m}$. This result shouldn't be too surprising, and suggests that adding a constant value to both sides won't change the congruence, just like any other equation. Division and Inverses Now, let's explore three famous algorithms for computing useful information using modular arithmetic: Euclid's Algorithm for GCD and inverses, the Chinese Remainder Theorem, and Fermat's Little Euclid's Algorithm Base Case: If y is 0, then any value for x is the GCD since everything can divide 0 to get 0. Inductive Case: Proof of the GCD Mod Corollary. For an example of a computation, check out the Extended Algorithm section below (the computation is extremely similar). Using Euclid's Extended Algorithm for Inverses Great! We got the GCD. So what? Remember that if the GCD of x and m is 1, then there is an inverse of x. In more concrete terms, we can state Euclid's Extended GCD Theorem (Bezout's Theorem) as such: The process can be tedious to compute by hand, but here's a nice video that walks through that process: The Chinese Remainder Theorem The Chinese Remainder Theorem (CRT) guarantees existence and uniqueness of a solution to a system of modular congruences. More formally stated: Formal Statement of CRT: Alternative Statement of CRT: Uniqueness of CRT Solution Fermat's Little Theorem The Formal Definition An Alternative Definition For an application of Fermat's Little Theorem, head over to RSA Cryptography!	{"url":"https://cs70.bencuan.me/discrete-math/modular-arithmetic","timestamp":"2024-11-09T07:16:46Z","content_type":"text/html","content_length":"684307","record_id":"<urn:uuid:56704928-4488-477d-92e9-0b504589a121>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00126.warc.gz"}
Alexander Pilz: Katalogdaten im Herbstsemester 2016 Name Herr Dr. Alexander Pilz Departement Informatik Beziehung Dozent Nummer Titel ECTS Umfang Dozierende 252-1425-00L Geometry: Combinatorics and Algorithms 6 KP 2V + 2U B. Gärtner, E. Welzl, M. Hoffmann, A. Pilz + 1A Geometric structures are useful in many areas, and there is a need to understand their structural properties, and to work with them algorithmically. The lecture addresses theoretical Kurzbeschreibung foundations concerning geometric structures. Central objects of interest are triangulations. We study combinatorial (Does a certain object exist?) and algorithmic questions (Can we find a certain object efficiently?) The goal is to make students familiar with fundamental concepts, techniques and results in combinatorial and computational geometry, so as to enable them to model, analyze, and solve Lernziel theoretical and practical problems in the area and in various application domains. In particular, we want to prepare students for conducting independent research, for instance, within the scope of a thesis project. Planar and geometric graphs, embeddings and their representation (Whitney's Theorem, canonical orderings, DCEL), polygon triangulations and the art gallery theorem, convexity in R^d, Inhalt planar convex hull algorithms (Jarvis Wrap, Graham Scan, Chan's Algorithm), point set triangulations, Delaunay triangulations (Lawson flips, lifting map, randomized incremental construction), Voronoi diagrams, the Crossing Lemma and incidence bounds, line arrangements (duality, Zone Theorem, ham-sandwich cuts), 3-SUM hardness, counting planar Skript yes Mark de Berg, Marc van Kreveld, Mark Overmars, Otfried Cheong, Computational Geometry: Algorithms and Applications, Springer, 3rd ed., 2008. Satyan Devadoss, Joseph O'Rourke, Discrete and Computational Geometry, Princeton University Press, 2011. Literatur Stefan Felsner, Geometric Graphs and Arrangements: Some Chapters from Combinatorial Geometry, Teubner, 2004. Jiri Matousek, Lectures on Discrete Geometry, Springer, 2002. Takao Nishizeki, Md. Saidur Rahman, Planar Graph Drawing, World Scientific, 2004. Voraussetzungen Prerequisites: The course assumes basic knowledge of discrete mathematics and algorithms, as supplied in the first semesters of Bachelor Studies at ETH. / Besonderes Outlook: In the following spring semester there is a seminar "Geometry: Combinatorics and Algorithms" that builds on this course. There are ample possibilities for Semester-, Bachelor- and Master Thesis projects in the area.	{"url":"https://www.vorlesungen.ethz.ch/Vorlesungsverzeichnis/dozent.view?lang=de&dozide=10048005&semkez=2016W&ansicht=2&","timestamp":"2024-11-06T17:33:58Z","content_type":"text/html","content_length":"9087","record_id":"<urn:uuid:714af556-2698-4e5a-8564-841f520d3aec>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00216.warc.gz"}
Calculate the cell potential, ΔG°, and equilibrium constant K for each reaction - Chemistry Steps General Chemistry Calculate the cell potential, Gibbs free energy change ΔG°, and equilibrium constant, K for each redox reaction: a) Zn(s) + Pb^2+(aq) → Zn^2+(aq) + Pb(s) b) Sn^2^+(aq) + Mg(s) → Sn(s) + Mg^2^+(aq) c) Br[2](l) + 2I^–(aq) → 2Br^–(aq) + I[2](s) d) 2Al(s) + 3Cu^2^+(aq) → 2Al^3^+(aq) + 3Cu(s) e) O[2](g) + 4H^+(aq) + Cu(s) → Cu^2+(aq) + 2H[2]O(l) f) MnO[2](s) + 4H^+(aq) + Cd(s) → Mn^2^+(aq) + 2H[2]O(l ) + Cd^2^+(aq) ΔG° is correlated to E^o[cell] with the following formula: ΔG° = –nFE^o[cell] Where n is the moles of electrons and F is the Faraday’s constant which is equal to 96,485 C. Therefore, the plan is to first calculate the cell potential and then use it to determine the ΔG°. There are two equations we can use to calculate the equilibrium constant. One is linked through the E^o[cell, ]and the other, through the ΔG° which is also calculated through the E^o[cell. ]Yes, we can calculate the ΔG° using the standard free energies of formations, however, this is not related to electrochemistry and we won’t focus on that here. \[{E^o}_{{\rm{cell}}}\, = \,\frac{{0.0257\;{\rm{V}}}}{n}\,\ln \,K\] \[K\; = \;{e^{\frac{{nE}}{{0.0257}}}}\] For reaction (a), E^o[cell ]= 0.63 V, therefore, K would be: \[K\; = \;{e^{\frac{{2\, \cdot \,0.63}}{{0.0257}}}}\, = \,{e^{49.0}}\, = \;1.91\, \times \,{10^{21}}\] The second approach: The equilibrium constant is correlated with the standard free energy change (ΔG°) by this formula: ΔG° = –RT ln K Therefore, the equilibrium constant would be equal to: \[\ln \,K\; = \;\frac{{ – \Delta G^\circ }}{{RT}}\] \[K\; = \;{e^{\frac{{ – \Delta G^\circ }}{{RT}}}}\] For the next problems, we will use the first approach to calculate the K. This content is available to registered users only. By joining Chemistry Steps, you will gain instant access to the Answers and Solutions for all the Practice Problems, Quizzes, and the powerful set of General Chemistry 1 and 2 Summary Study Guides. Check Also General Chemistry Leave a Comment	{"url":"https://general.chemistrysteps.com/calculate-cell-potential-%CE%B4g-k-for-each-reaction/","timestamp":"2024-11-06T18:06:17Z","content_type":"text/html","content_length":"191136","record_id":"<urn:uuid:143a681d-7f9a-4a67-bb9a-fa60537f88d5>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00342.warc.gz"}
Protrudes 4674 - math word problem (4674) Protrudes 4674 The pole has a base 1/3 to the bottom of the pond. 1/4 is submerged. One meter protrudes above the surface. How long does the stake have to be cut? Correct answer: Did you find an error or inaccuracy? Feel free to write us . Thank you! Tips for related online calculators Need help calculating sum, simplifying, or multiplying fractions? Try our fraction calculator Do you have a linear equation or system of equations and are looking for its ? Or do you have a quadratic equation Do you want to convert length units You need to know the following knowledge to solve this word math problem: Units of physical quantities: Grade of the word problem: Related math problems and questions:	{"url":"https://www.hackmath.net/en/math-problem/4674","timestamp":"2024-11-02T07:51:36Z","content_type":"text/html","content_length":"46579","record_id":"<urn:uuid:12aefadb-03af-4e00-921d-eb9d8514820e>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00242.warc.gz"}
1.5 ounces to grams Convert 1.5 Ounces to Grams (oz to gm) with our conversion calculator. 1.5 ounces to grams equals 42.5242870378035 oz. Enter ounces to convert to grams. Formula for Converting Ounces to Grams (Oz to Gm): grams = ounces * 28.3495 By multiplying the number of grams by 28.3495, you can easily obtain the equivalent weight in grams from ounces. Understanding the Conversion from Ounces to Grams When it comes to converting ounces to grams, it’s essential to know the conversion factor. One ounce is equivalent to approximately 28.3495 grams. This means that to convert ounces to grams, you simply multiply the number of ounces by this conversion factor. This conversion is particularly important for those who work with both the imperial and metric systems, as it helps bridge the gap between these two measurement systems. Formula for Converting Ounces to Grams The formula to convert ounces to grams is straightforward: Grams = Ounces × 28.3495 Step-by-Step Calculation: Converting 1.5 Ounces to Grams Let’s walk through the calculation of converting 1.5 ounces to grams: 1. Start with the number of ounces you want to convert: 1.5 ounces. 2. Use the conversion factor: 28.3495 grams per ounce. 3. Multiply the number of ounces by the conversion factor: 1.5 × 28.3495. 4. Perform the multiplication: 1.5 × 28.3495 = 42.52425. 5. Round the result to two decimal places: 42.52 grams. Therefore, 1.5 ounces is equal to 42.52 grams. The Importance of Ounce to Gram Conversion This conversion is crucial for various applications, especially in cooking, where precise measurements can make a significant difference in the outcome of a recipe. For instance, if a recipe calls for 1.5 ounces of an ingredient, knowing that this is equivalent to 42.52 grams allows you to measure accurately using a kitchen scale that may only display metric units. In scientific measurements, converting ounces to grams is equally vital. Many laboratory protocols require precise measurements in grams, and understanding how to convert from ounces ensures accuracy in experiments and results. Everyday use also benefits from this conversion. Whether you’re tracking your food intake, calculating nutritional values, or simply trying to understand product labels, knowing how to convert ounces to grams can enhance your understanding and help you make informed decisions. In summary, converting 1.5 ounces to grams is a simple yet essential skill that can aid in cooking, scientific endeavors, and daily life. With the conversion factor of 28.3495, you can easily navigate between the imperial and metric systems, ensuring accuracy and precision in your measurements. Here are 10 items that weigh close to 1.5 ounces to grams – • Standard AA Battery Shape: Cylindrical Dimensions: 1.99 inches long, 0.57 inches in diameter Usage: Commonly used in remote controls, flashlights, and toys. Fact: An AA battery can power a small LED light for up to 30 hours! • Medium-Sized Apple Shape: Round Dimensions: Approximately 3 inches in diameter Usage: Eaten raw, used in salads, or baked in pies. Fact: Apples float in water because 25% of their volume is air! • Deck of Playing Cards Shape: Rectangular Dimensions: 2.5 inches by 3.5 inches Usage: Used for various card games and magic tricks. Fact: A standard deck has 52 cards, but there are also jokers! • Small Bar of Soap Shape: Rectangular or oval Dimensions: Approximately 3 inches long, 1.5 inches wide Usage: Used for personal hygiene and cleaning. Fact: The world’s largest bar of soap weighed over 1,000 pounds! • Single Cup of Coffee Shape: Cylindrical Dimensions: About 4 inches tall, 3 inches in diameter Usage: Consumed as a beverage for energy and enjoyment. Fact: Coffee is the second most traded commodity in the world after oil! • Small Notebook Shape: Rectangular Dimensions: 5 inches by 7 inches Usage: Used for jotting down notes, sketches, or ideas. Fact: The first notebooks were made from papyrus in ancient Egypt! • Plastic Water Bottle Shape: Cylindrical Dimensions: 8 inches tall, 2.5 inches in diameter Usage: Used for carrying and drinking water on the go. Fact: Over 60 million plastic water bottles are used every day in the U.S.! • Small Bag of Chips Shape: Rectangular Dimensions: Approximately 6 inches by 4 inches Usage: Snack food enjoyed during parties, movies, or as a quick bite. Fact: The first potato chips were invented in 1853 by George Crum! • USB Flash Drive Shape: Rectangular Dimensions: About 2.5 inches long, 0.75 inches wide Usage: Used for storing and transferring digital data. Fact: The first USB flash drive was released in 1998 and had a capacity of 8 MB! • Small Toy Car Shape: Rectangular with rounded edges Dimensions: Approximately 3 inches long, 1.5 inches wide Usage: Used for play and collection by children and adults alike. Fact: The first toy car was made in the early 1900s and was a simple wooden model! Other Oz <-> Gm Conversions –	{"url":"https://www.gptpromptshub.com/grams-ounce-converter/1-5-ounces-to-grams","timestamp":"2024-11-14T17:23:02Z","content_type":"text/html","content_length":"185299","record_id":"<urn:uuid:83066f60-4b53-43a6-abe4-d923b839350d>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00265.warc.gz"}
Proposed in [29]. Other people include the sparse PCA and PCA that is \| AMPK Inhibitor Proposed in [29]. Others involve the sparse PCA and PCA that’s constrained to certain subsets. We adopt the typical PCA simply because of its simplicity, representativeness, in depth applications and satisfactory empirical performance. Partial least squares Partial least squares (PLS) is also a dimension-reduction strategy. In contrast to PCA, when constructing linear combinations of the original measurements, it utilizes details in the survival outcome for the weight at the same time. The common PLS process is often carried out by constructing orthogonal directions Zm’s working with X’s weighted by the strength of SART.S23503 their effects on the outcome and then orthogonalized with respect to the former directions. Extra detailed discussions and also the algorithm are provided in [28]. Within the context of Tariquidar mechanism of action high-dimensional genomic data, Nguyen and Rocke [30] proposed to apply PLS inside a two-stage manner. They utilised linear regression for survival information to ascertain the PLS elements and then applied Cox regression around the resulted elements. Bastien [31] later replaced the linear regression step by Cox regression. The comparison of unique solutions is often discovered in Lambert-Lacroix S and Letue F, unpublished information. Considering the computational burden, we pick the system that replaces the survival instances by the deviance residuals in extracting the PLS directions, which has been shown to possess a good approximation overall performance [32]. We implement it utilizing R package plsRcox. Least absolute shrinkage and selection operator Least absolute shrinkage and selection operator (Lasso) is a penalized `variable selection’ process. As described in [33], Lasso applies model selection to decide on a compact variety of `important’ covariates and achieves parsimony by generating coefficientsthat are MGCD516 msds exactly zero. The penalized estimate under the Cox proportional hazard model [34, 35] may be written as^ b ?argmaxb ` ? topic to X b s?P Pn ? exactly where ` ??n di bT Xi ?log i? j? Tj ! Ti ‘! T exp Xj ?denotes the log-partial-likelihood ands > 0 is often a tuning parameter. The approach is implemented using R package glmnet in this write-up. The tuning parameter is chosen by cross validation. We take a couple of (say P) crucial covariates with nonzero effects and use them in survival model fitting. You will discover a large quantity of variable selection solutions. We pick penalization, due to the fact it has been attracting many interest inside the statistics and bioinformatics literature. Comprehensive reviews can be identified in [36, 37]. Among all the accessible penalization approaches, Lasso is maybe by far the most extensively studied and adopted. We note that other penalties including adaptive Lasso, bridge, SCAD, MCP and other folks are potentially applicable right here. It truly is not our intention to apply and examine multiple penalization strategies. Beneath the Cox model, the hazard function h jZ?with all the chosen attributes Z ? 1 , . . . ,ZP ?is of the kind h jZ??h0 xp T Z? exactly where h0 ?is an unspecified baseline-hazard function, and b ? 1 , . . . ,bP ?could be the unknown vector of regression coefficients. The chosen features Z ? 1 , . . . ,ZP ?is often the very first couple of PCs from PCA, the very first couple of directions from PLS, or the handful of covariates with nonzero effects from Lasso.Model evaluationIn the area of clinical medicine, it’s of excellent interest to evaluate the journal.pone.0169185 predictive energy of an individual or composite marker. We focus on evaluating the prediction accuracy within the idea of discrimination, that is typically known as the `C-statistic’. For binary outcome, well-liked measu.Proposed in [29]. Other individuals include things like the sparse PCA and PCA that’s constrained to particular subsets. We adopt the standard PCA mainly because of its simplicity, representativeness, substantial applications and satisfactory empirical efficiency. Partial least squares Partial least squares (PLS) can also be a dimension-reduction approach. In contrast to PCA, when constructing linear combinations on the original measurements, it utilizes facts from the survival outcome for the weight too. The normal PLS process might be carried out by constructing orthogonal directions Zm’s working with X’s weighted by the strength of SART.S23503 their effects around the outcome and then orthogonalized with respect towards the former directions. Much more detailed discussions and the algorithm are offered in [28]. Within the context of high-dimensional genomic information, Nguyen and Rocke [30] proposed to apply PLS within a two-stage manner. They utilised linear regression for survival information to determine the PLS elements after which applied Cox regression on the resulted components. Bastien [31] later replaced the linear regression step by Cox regression. The comparison of distinctive methods could be located in Lambert-Lacroix S and Letue F, unpublished data. Thinking of the computational burden, we pick the system that replaces the survival instances by the deviance residuals in extracting the PLS directions, which has been shown to possess a fantastic approximation efficiency [32]. We implement it applying R package plsRcox. Least absolute shrinkage and choice operator Least absolute shrinkage and selection operator (Lasso) is usually a penalized `variable selection’ technique. As described in [33], Lasso applies model selection to select a tiny variety of `important’ covariates and achieves parsimony by creating coefficientsthat are exactly zero. The penalized estimate under the Cox proportional hazard model [34, 35] could be written as^ b ?argmaxb ` ? topic to X b s?P Pn ? exactly where ` ??n di bT Xi ?log i? j? Tj ! Ti ‘! T exp Xj ?denotes the log-partial-likelihood ands > 0 is a tuning parameter. The approach is implemented using R package glmnet in this write-up. The tuning parameter is selected by cross validation. We take several (say P) important covariates with nonzero effects and use them in survival model fitting. There are actually a big variety of variable selection procedures. We decide on penalization, due to the fact it has been attracting plenty of consideration within the statistics and bioinformatics literature. Extensive testimonials can be located in [36, 37]. Among each of the obtainable penalization techniques, Lasso is perhaps by far the most extensively studied and adopted. We note that other penalties like adaptive Lasso, bridge, SCAD, MCP and others are potentially applicable right here. It’s not our intention to apply and evaluate multiple penalization methods. Beneath the Cox model, the hazard function h jZ?with all the selected functions Z ? 1 , . . . ,ZP ?is with the kind h jZ?? h0 xp T Z? exactly where h0 ?is definitely an unspecified baseline-hazard function, and b ? 1 , . . . ,bP ?could be the unknown vector of regression coefficients. The chosen attributes Z ? 1 , . . . ,ZP ?can be the first handful of PCs from PCA, the first few directions from PLS, or the couple of covariates with nonzero effects from Lasso.Model evaluationIn the area of clinical medicine, it truly is of wonderful interest to evaluate the journal.pone.0169185 predictive energy of a person or composite marker. We concentrate on evaluating the prediction accuracy in the idea of discrimination, that is normally known as the `C-statistic’. For binary outcome, well known measu.	{"url":"https://www.ampkinhibitor.com/2018/02/02/proposed-in-29-other-people-include-the-sparse-pca-and-pca-that-is/","timestamp":"2024-11-02T03:04:21Z","content_type":"text/html","content_length":"89191","record_id":"<urn:uuid:4c06e133-5e6d-4d09-8132-ab58d1de8ff1>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00729.warc.gz"}
531 mph to knot - How fast is 531 miles per hour in knots? [CONVERT] ✔ 531 miles per hour in knots Conversion in the opposite direction The inverse of the conversion factor is that 1 knot is equal to 0.00216719293413096 times 531 miles per hour. It can also be expressed as: 531 miles per hour is equal to $1 0.00216719293413096$ knots. An approximate numerical result would be: five hundred and thirty-one miles per hour is about four hundred and sixty-one point four three knots, or alternatively, a knot is about zero times five hundred and thirty-one miles per hour. [1] The precision is 15 significant digits (fourteen digits to the right of the decimal point). Results may contain small errors due to the use of floating point arithmetic.	{"url":"https://converter.ninja/velocity/miles-per-hour-to-knots/531-mph-to-knot/","timestamp":"2024-11-09T00:28:36Z","content_type":"text/html","content_length":"17706","record_id":"<urn:uuid:7aadf859-f17b-4376-9590-3de58dc3ca63>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00070.warc.gz"}
Electronic Structure in a Fixed Basis is QMA-complete We have shown the electronic structure problem is QMA-hard without magnetic field but restricted to a fixed orbital basis. Our main result shows consider electronic structure Hamiltonians restricted to a fixed number of electrons and projected into a fixed single-particle basis. We conclusively demonstrate that these properties do not add enough structure to enable the existence of an efficient quantum simulation algorithm to approximate the ground state energy of such systems. For details see “Electronic Structure in a Fixed Basis is QMA-complete”, B. O’Gorman, S. Irani, J. Whitfield, B. Fefferman. arXiv:2103.08215	{"url":"https://overqc.sandia.gov/2021/04/22/electronic-structure-in-a-fixed-basis-is-qma-complete/","timestamp":"2024-11-13T09:35:38Z","content_type":"text/html","content_length":"31592","record_id":"<urn:uuid:34c747ac-7acd-42fa-8d79-303c35b555c7>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00279.warc.gz"}
Understanding the Fundamental Principles of Vector Network Analysis - E-DoodlesUnderstanding the Fundamental Principles of Vector Network Analysis Understanding the Fundamental Principles of Vector Network Analysis In this application note, the fundamental principles of vector network analysis will be reviewed. The discussion includes the common parameters that can be measured, including the concept of scattering parameters (S-parameters). RF fundamentals such as transmission lines and the Smith chart will also be reviewed. Agilent Technologies offers a wide range of both scalar and vector network analyzers for characterizing components from DC to 110 GHz. These instruments are available with a wide range of options to simplify testing in both laboratory and production environments. Measurements in Communications Systems In any communications system, the effect of signal distortion must be considered. While we generally think of the distortion caused by nonlinear effects (for example, when intermodulation products are produced from desired carrier signals), purely linear systems can also introduce signal distortion. Linear systems can change the time waveform of signals passing through them by altering the amplitude or phase relationships of the spectral components that make up the signal. Let’s examine the difference between linear and nonlinear behavior more closely. Linear devices impose magnitude and phase changes on input signals (Figure 1). Any sinusoid appearing at the input will also appear at the output, and at the same frequency. No new signals are created. Both active and passive nonlinear devices can shift an input signal in frequency or add other frequency components, such as harmonic and spurious signals. Large input signals can drive normally linear devices into compression or saturation, causing nonlinear operation. Linear behavior: input and output frequencies are the same (no additional frequencies created) output frequency only undergoes magnitude and phase change Time A to f1 Frequency Time Sin 360* f * t Frequency A phase shift = to * 360* f 1 f DUT A * Sin 360* f ( t – t) Input Output Time Frequency Nonlinear behavior: output frequency may undergo frequency shift (e.g. with mixers) additional frequencies created (harmonics, intermodulation) f1 Figure 1. Linear versus Nonlinear Behavior 3 For linear distortion-free transmission, the amplitude response of the device under test (DUT) must be flat and the phase response must be linear over the desired bandwidth. As an example, consider a square-wave signal rich in highfrequency components passing through a bandpass filter that passes selected frequencies with little attenuation while attenuating frequencies outside of the passband by varying amounts. Even if the filter has linear phase performance, the out-of-band components of the square wave will be attenuated, leaving an output signal that, in this example, is more sinusoidal in nature (Figure 2). If the same square-wave input signal is passed through a filter that only inverts the phase of the third harmonic, but leaves the harmonic amplitudes the same, the output will be more impulse-like in nature (Figure 3). While this is true for the example filter, in general, the output waveform will appear with arbitrary distortion, depending on the amplitude and phase nonlinearities. Frequency Frequency Frequency Magnitude Time Linear Network Time F(t) = sin wt + 1/3 sin 3wt + 1/5 sin 5wt Figure 2. Magnitude Variation with Frequency Frequency Magnitude Linear Network Frequency Frequency Time 0 –360 –180Time F(t) = sin wt + 1/3 sin 3wt + 1/5 sin 5wt Figure 3. Phase Variation with Frequency 4 Figure 4. Nonlinear Induced Distortion Nonlinear devices also introduce distortion (Figure 4). For example, if an amplifier is overdriven, the output signal clips because the amplifier is saturated. The output signal is no longer a pure sinusoid, and harmonics are present at multiples of the input frequency. Passive devices may also exhibit nonlinear behavior at high power levels, a good example of which is an L-C filter that uses inductors with magnetic cores. Magnetic materials often exhibit hysteresis effects that are highly nonlinear. Efficient transfer of power is another fundamental concern in communications systems. In order to efficiently convey, transmit or receive RF power, devices such as transmissions lines, antennas and amplifiers must present the proper impedance match to the signal source. Impedance mismatches occur when the real and imaginary parts of input and output impedances are not ideal between two connecting devices. Importance of Vector Measurements Measuring both magnitude and phase of components is important for several reasons. First, both measurements are required to fully characterize a linear network and ensure distortion-free transmission. To design efficient matching networks, complex impedance must be measured. Engineers developing models for computer-aided-engineering (CAE) circuit simulation programs require magnitude and phase data for accurate models. In addition, time-domain characterization requires magnitude and phase information in order to perform an inverse-fourier transform. Vector error correction, which improves measurement accuracy by removing the effects of inherent measurement-system errors, requires both magnitude and phase data to build an effective error model. Phase-measurement capability is very important even for scalar measurements such as return loss, in order to achieve a high level of accuracy (see Applying Error Correction to Network Analyzer Measurements, Agilent application note 1287-3). Nonlinear Networks Frequency Frequency Time Time Saturation, crossover, intermodulation, and other nonlinear effects can cause signal distortion 5 The Basis of Incident and Reflected Power In its fundamental form, network analysis involves the measurement of incident, reflected, and transmitted waves that travel along transmission lines. Using optical wavelengths as an analogy, when light strikes a clear lens (the incident energy), some of the light is reflected from the lens surface, but most of it continues through the lens (the transmitted energy) (Figure 5). If the lens has mirrored surfaces, most of the light will be reflected and little or none will pass through it. While the wavelengths are different for RF and microwave signals, the principle is the same. Network analyzers accurately measure the incident, reflected, and transmitted energy, e.g., the energy that is launched onto a transmission line, reflected back down the transmission line toward the source (due to impedence mismatch), and successfully transmitted to the terminating device (such as an antenna). Figure 5. Lightwave Analogy to High-Frequency Device Characterization The Smith Chart The amount of reflection that occurs when characterizing a device depends on the impedance that the incident signal “sees.” Since any impedance can be represented with real and imaginary parts (R+jX or G+jB), they can be plotted on a rectilinear grid known as the complex impedance plane. Unfortunately, an open circuit (a common RF impedence) appears at infinity on the real axis, and therefore cannot be shown. The polar plot is useful because the entire impedance plane is covered. However, instead of plotting impedance directly, the complex reflection coefficient is displayed in vector form. The magnitude of the vector is the distance from the center of the display, and phase is displayed as the angle of vector referenced to a flat line from the center to the right-most edge. The drawback of polar plots is that impedance values cannot be read directly from the display. Incident Reflected Transmitted Lightwave Analogy 6 Since there is a one-to-one correspondence between complex impedance and reflection coefficient, the positive real half of the complex impedance plane can be mapped onto the polar display. The result is the Smith chart. All values of reactance and all positive values of resistance from 0 to infinity fall within the outer circle of the Smith chart (Figure 6). On the Smith chart, loci of constant resistance appear as circles, while loci of constant reactance appear as arcs. Impedances on the Smith chart are always normalized to the characteristic impedance of the component or system of interest, usually 50 ohms for RF and microwave systems and 75 ohms for broadcast and cable-television systems. A perfect termination appears in the center of the Smith chart. Figure 6. Smith Chart Review Power Transfer Conditions A perfectly matched condition must exist at a connection between two devices for maximum power transfer into a load, given a source resistance of RS and a load resistance of RL. This condition occurs when RL = RS, and is true whether the stimulus is a DC voltage source or a source of RF sine waves (Figure 7). When the source impedance is not purely resistive, maximum power transfer occurs when the load impedance is equal to the complex conjugate of the source impedance. This condition is met by reversing the sign of the imaginary part of the impedance. For example, if RS = 0.6 + j 0.3, then the complex conjugate is RS* = 0.6 – j 0.3. The need for efficient power transfer is one of the main reasons for the use of transmission lines at higher frequencies. At very low frequencies (with much larger wavelengths), a simple wire is adequate for conducting power. The resistance of the wire is relatively low and has little effect on low-frequency signals. The voltage and current are the same no matter where a measurement is made on the wire. –90o 0o 180o +– .2 .4 .6 .8 1.0 90o 0 +R +jX –jX Smith chart maps rectilinear impedance plane onto polar plane Rectilinear impedance plane Polar plane ZL = Zo = 0 Constant X Constant R Z = L =1 0O Smith chart (open) L Z = 0 = 1 180O (short) 7 At higher frequencies, wavelengths are comparable to or smaller than the length of the conductors in a high-frequency circuit, and power transmission can be thought of in terms of traveling waves. When the transmission line is terminated in its characteristic impedance, maximum power is transferred to the load. When the termination is not equal to the characteristic impedance, that part of the signal that is not absorbed by the load is reflected back to the source. If a transmission line is terminated in its characteristic impedance, no reflected signal occurs since all of the transmitted power is absorbed by the load (Figure 8). Looking at the envelope of the RF signal versus distance along the transmission line shows no standing waves because without reflections, energy flows in only one direction. 0 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 1.2 Load Power (normalized) RL / RS RS RL Maximum power is transferred when RL = RS For complex impedances, maximum power transfer occurs when ZL = ZS* (conjugate match) Zs = R + jX ZL = Zs* = R – jX Figure 7. Power Transfer For reflection, a transmission line terminated in Zo behaves like an infinitely long transmission line Zs = Zo Zo Vrefl = 0 (all the incident power is absorbed in the load) Vinc Zo = characteristic impedance of transmission line Figure 8. Transmission Line Terminated with Z0 8 When the transmission line is terminated in a short circuit (which can sustain no voltage and therefore dissipates zero power), a reflected wave is launched back along the line toward the source (Figure 9). The reflected voltage wave must be equal in magnitude to the incident voltage wave and be 180 degrees out of phase with it at the plane of the load. The reflected and incident waves are equal in magnitude but traveling in the opposite directions. If the transmission line is terminated in an open-circuit condition (which can sustain no current), the reflected current wave will be 180 degrees out of phase with the incident current wave, while the reflected voltage wave will be in phase with the incident voltage wave at the plane of the load. This guarantees that the current at the open will be zero. The reflected and incident current waves are equal in magnitude, but traveling in the opposite directions. For both the short and open cases, a standing wave pattern is set up on the transmission line. The voltage valleys will be zero and the voltage peaks will be twice the incident voltage level. If the transmission line is terminated with say a 25-ohm resistor, resulting in a condition between full absorption and full reflection, part of the incident power is absorbed and part is reflected. The amplitude of the reflected voltage wave will be one-third that of the incident wave, and the two waves will be 180 degrees out of phase at the plane of the load. The valleys of the standing-wave pattern will no longer be zero, and the peaks will be less than those of the short and open cases. The ratio of the peaks to valleys will be 2:1. The traditional way of determining RF impedance was to measure VSWR using an RF probe/detector, a length of slotted transmission line, and a VSWR meter. As the probe was moved along the transmission line, the relative position and values of the peaks and valleys were noted on the meter. From these measurements, impedance could be derived. The procedure was repeated at different frequencies. Modern network analyzers measure the incident and reflected waves directly during a frequency sweep, and impedance results can be displayed in any number of formats (including VSWR). Zs = Zo Vrefl Vinc For reflection, a transmission line terminated in a short or open reflects all power back to source In phase (0 ) for open Out of phase (180 ) for short o o Figure 9. Transmission Line Terminated with Short, Open 9 Network Analysis Terminology Now that we understand the fundamentals of electromagnetic waves, we must learn the common terms used for measuring them. Network analyzer terminology generally denotes measurements of the incident wave with the R or reference channel. The reflected wave is measured with the A channel, and the transmitted wave is measured with the B channel (Figure 10). With the amplitude and phase information in these waves, it is possible to quantify the reflection and transmission characteristics of a DUT. The reflection and transmission characteristics can be expressed as vector (magnitude and phase), scalar (magnitude only), or phase-only quantities. For example, return loss is a scalar measurement of reflection, while impedance is a vector reflection measurement. Ratioed measurements allow us to make reflection and transmission measurements that are independent of both absolute power and variations in source power versus frequency. Ratioed reflection is often shown as A/R and ratioed transmission as B/R, relating to the measurement channels in the instrument. Figure 10. Common Terms for High-Frequency Device Characterization The most general term for ratioed reflection is the complex reflection coefficient, or gamma (Figure 11). The magnitude portion of is called or rho. The reflection coefficient is the ratio of the reflected signal voltage level to the incident signal voltage level. For example, a transmission line terminated in its characteristic impedance Zo,will have all energy transferred to the load so Vrefl = 0 and = 0. When the impedance of the load, ZL is not equal to the characteristic impedance, energy is reflected and is greater than zero. When the load impedance is equal to a short or open circuit, all energy is reflected and = 1. As a result, the range of possible values for is 0 to 1. TRANSMISSION Gain / Loss S-Parameters S21,S12 Group Delay Transmission Coefficient Insertion Phase REFLECTION SWR S-Parameters S11,S22 Reflection Coefficient Impedance, Admittance R+jX, G+jB Return Loss Incident Reflected R Transmitted B A Reflected Incident A R = Transmitted Incident B R = 10 Figure 11. Reflection Parameters Return loss is a way to express the reflection coefficient in logarithmic terms (decibels). Return loss is the number of decibels that the reflected signal is below the incident signal. Return loss is always expressed as a positive number and varies between infinity for a load at the characteristic impedance and 0 dB for an open or short circuit. Another common term used to express reflection is voltage standing wave ratio (VSWR), which is defined as the maximum value of the RF envelope over the minimum value of the RF envelope. It is related to as (1 + )/(1 – ). VSWR ranges from 1 (no reflection) to infinity (full reflection). The transmission coefficient is defined as the transmitted voltage divided by the incident voltage (Figure 12). If the absolute value of the transmitted voltage is greater than the absolute value of the incident voltage, a DUT or system is said to have gain. If the absolute value of the transmitted voltage is less than the absolute value of the incident voltage, the DUT or system is said to have attenuation or insertion loss. The phase portion of the transmission coefficient is called insertion phase. = ZL ZO ZL + O Z Reflection Coefficient = Vreflected Vincident = = dB No reflection (ZL = Zo) RL VSWR 0 1 Full reflection (ZL = open, short) 0 dB 1 Return loss = –20 log(), VSWR = Emax Emin = 1 + 1 – Voltage Standing Wave Ratio Emax Emin VTransmitted VIncident Transmission Coefficient = = VTransmitted VIncident = DUT Gain (dB) = 20 Log VTrans VInc = 20 log Insertion Loss (dB) = –20 Log VTrans VInc = –20 log Figure 12. Transmission Parameters 11 Direct examination of insertion phase usually does not provide useful information. This is because the insertion phase has a large (negative) slope with respect to frequency due to the electrical length of the DUT. The slope is proportional to the length of the DUT. Since it is only deviation from linear phase that causes distortion in communications systems, it is desirable to remove the linear portion of the phase response to analyze the remaining nonlinear portion. This can be done by using the electrical delay feature of a network analyzer to mathematically cancel the average electrical length of the DUT. The result is a high-resolution display of phase distortion or deviation from linear phase (Figure 13). Figure 13. Deviation from Linear Phase Measuring Group Delay Another useful measure of phase distortion is group delay (Figure 14). This parameter is a measure of the transit time of a signal through a DUT versus frequency. Group delay can be calculated by differentiating the DUT’s phase response versus frequency. It reduces the linear portion of the phase response to a constant value, and transforms the deviations from linear phase into deviations from constant group delay, (which causes phase distortion in communications systems). The average delay represents the average signal transit time through a DUT. Use electrical delay to remove linear portion of phase response Linear electrical length added + yields Frequency (Electrical delay function) Frequency RF filter response Deviation from linear phase Phase 1o /Div Phase 45 /Div o Frequency Low resolution High resolution Deviation from constant group delay indicates distortion Average delay indicates transit time Group Delay Frequency Group Delay Average Delay to tg Group Delay (t g ) = 360o = d d df in radians in radians/sec in degrees f in Hz (=2f ) Phase* FrequencyFigure 14. What is group Delay? 12 Depending on the device, both deviation from linear phase and group delay may be measured, since both can be important. Specifying a maximum peakto- peak phase ripple in a device may not be sufficient to completely characterize it, since the slope of the phase ripple depends on the number of ripples that occur per unit of frequency. Group delay takes this into account because it is the differentiated phase response. Group delay is often a more easily interpreted indication of phase distortion (Figure 15). Figure 15. Why measure Group Delay? Network Characterization In order to completely characterize an unknown linear two-port device, we must make measurements under various conditions and compute a set of parameters. These parameters can be used to completely describe the electrical behavior of our device (or network), even under source and load conditions other than when we made our measurements. Low-frequency device or network characterization is usually based on measurement of H, Y, and Z parameters. To do this, the total voltage and current at the input or output ports of a device or nodes of a network must be measured. Furthermore, measurements must be made with open-circuit and short-circuit conditions. Since it is difficult to measure total current or voltage at higher frequencies, S-parameters are generally measured instead (Figure 16). These parameters relate to familiar measurements such as gain, loss, and reflection coefficient. They are relatively simple to measure, and do not require connection of undesirable loads to the DUT. The measured S-parameters of multiple devices can be cascaded to predict overall system performance. S-parameters are readily used in both linear and nonlinear CAE circuit simulation tools, and H, Y, and Z parameters can be derived from S-parameters when necessary. The number of S-parameters for a given device is equal to the square of the number of ports. For example, a two-port device has four S-parameters. The numbering convention for S-parameters is that the first number following the S is the port at which energy emerges, and the second number is the port at which energy enters. So S21 is a measure of power emerging from Port 2 as a result of applying an RF stimulus to Port 1. When the numbers are the same (e.g. S11), a reflection measurement is indicated. Same peak-to-peak phase ripple can result in different group delay Phase Phase Group Delay Group Delay ddf f f f dd13 Forward S-parameters are determined by measuring the magnitude and phase of the incident, reflected, and transmitted signals when the output is terminated in a load that is precisely equal to the characteristic impedance of the test system. In the case of a simple two-port network, S11 is equivalent to the input complex reflection coefficient or impedance of the DUT, while S21 is the forward complex transmission coefficient. By placing the source at the output port of the DUT and terminating the input port in a perfect load, it is possible to measure the other two (reverse) S-parameters. Parameter S22 is equivalent to the output complex reflection coefficient or output impedance of the DUT while S12 is the reverse complex transmission coefficient (Figure 17). 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On This Day in Math - January 21 The whole of the developments and operations of analysis are now capable of being executed by machinery. ... As soon as an Analytical Engine exists, it will necessarily guide the future course of C. Babbage The 21st day of the year; To tile a square out of integer sided squares requires a minimum of 21 squares. (technically, this is true for what are called "simple" squared squares, one where no subset of the squares forms a rectangle or square. See the solution here ) (btw: There are no cubed cubes!) There are 21 possible ways to draw 5 circle s that touch all the points on a 5x5 lattice. gotmath.com 21 repeated twenty-one times, following 1, forms a smoothly undulating palindromic prime 121212121212121212121212121212121212121 is prime Blackjack primes are separated by exactly 21 consecutive composite numbers. Note that the pair {1129, 1151} is the smallest example.(Can you find more?) Prime Curios , the great daylight comet of 1472 passed within 10.5 million km of earth.TIS (Johannes Müller von Königsberg (Regiomontanus) is said to have observed this comet) Modern astronomy dates all astronomical events using the Julian Day Count a system of dating that was first conceived by a Renaissance historian and Bible chronologist, Joseph Justus Scalier, who died on this day. The Julian Day Number (JDN) is the integer assigned to a whole solar day in the Julian day count starting from noon Greenwich Mean Time, with Julian day number 0 assigned to the day starting at noon on January 1, 4713 BC. At noon on the date of his death, the Julian Day 2308756 would have began. Wik For a few details about his life see this post at the Renaissance Mathematicus.1665 Samuel Pepys, having acquired a copy of Hooke’s Micrographia the day before, stays up to read it, “Before I went to bed I sat up till two o'clock in my chamber reading of Mr. Hooke's Microscopicall Observations, the most ingenious book that ever I read in my life.” Pepy’s Diary , the London Institution received a royal charter signed by King George III, to "promote the diffusion of Science, Literature, and the Arts, by means of Lectures and Experiments, and by easy access to an extensive collection of books, both ancient and modern, in all languages." The full name in the charter was the "London Institution for the Advancement of Literature and The Diffusion of Useful Knowledge." The first president was Sir Francis Baring. Its incorporation came after the Royal Society (1663) and Royal Institution (1800). The institution had an extensive lecture program. Instruction in practical chemistry was given in its laboratory, and significant chemistry research was done there through the 19th century. TIS , Charles Wheatstone and William F. Cooke were granted the earliest English alphabetic telegraph patent. Wheatstone made contributions to a broad range of fields in the mid 19th century. The ABC telegraph was popular in England and Europe because it did not require a trained telegraphist to read or send the messages. The operator simply rotates a wheel to the desired letter. During rotation the instrument sends out the proper number of electric pulses to an electromagnetically controlled pointer on a remote synchronized slave receiver with a similarly lettered wheel which moves to the sender's letter. Electric telegraphs of the 1840-50's are of special historic importance as the earliest practical application of serial binary coded digital communication. TIS Babbage's Analytical Engine Passes the First Test The Analytical Engine of Charles Babbage was never completed in his lifetime, but his son Henry Provost Babbage built the mill portion of the machine from his father's drawings, and on January 21, 1888 computed multiples of pi to prove the adequacy of the design. Perhaps this represents the first successful test of a portion of a modern computer. Recently a portion of his earlier machine, the Difference Engine, was sold at auction by Christies of London to the Powerhouse Museum in Sydney, Australia.CHM the first atomic submarine, the U.S.S. Nautilus, was launched at Groton, Connecticut. Nautilus' nuclear propulsion system was a landmark in the history of naval engineering and submersible craft. All vessels previously known as "submarines" were in fact only submersible craft. Because of the nuclear power plant, the Nautilus could stay submerged for months at a time, unlike diesel-fueled subs, whose engines required vast amounts of oxygen. Nautilus demonstrated her capabilities in 1958 when she sailed beneath the Arctic icepack to the North Pole. Scores of nuclear submarines followed Nautilus, replacing the United States' diesel boat fleet. After patrolling the seas until 1980, the Nautilus is back home at Groton. TIS Pluto moves closer to the sun than Neptune. VFR Pluto is usually farther from the Sun. However, its orbit "crosses" inside of Neptune's orbit for 20 years out of every 248 years. Pluto last crossed inside Neptune's orbit on February 7, 1979, and temporarily became the 8th planet from the Sun. Pluto crossed back over Neptune's orbit again on February 11, 1999 to resume its place as the 9th planet from the Sun for the next 228 years (well, now it is now one of five known "dwarf planets"). 1793 Théodore Olivier (21 Jan 1793 in France - 5 Aug 1853 in France) From the 1840's Olivier wrote textbooks. His greatest fame, however, is as a result of the mathematical models which he created to assist in his teaching of geometry. Some of the models were of ruled surfaces, with moving parts to illustrate to students how the ruled surfaces were generated. Others were designed to illustrate the curves of intersection of certain surfaces. In fact Olivier earned quite a good income from selling these models, particularly in the United States. The United States Military Academy at West Point had 23 mathematical models made for them by Olivier to use as teaching aids:= These models are built on wooden boxes as bases, have metal supports, and consist of strings suspended from movable arms and arranged to form a variety of geometrical figures. The strings are held in place by lead weights that are concealed by the bases. The models illustrate such things as the intersection of two half cones, the intersection of a plane, hyperbolic paraboloid and a hyperboloid of one sheet, and the intersection of two half cylinders. Other institutions in the United States such as the Columbia School of Mines also purchased models from Olivier while Princeton had copies of Olivier's models made for them. In 1849 Olivier presented a full set of the range of models he had created to the Conservatoire National des Arts et Métiers. The models had been manufactured by the firm of Pixii, Père et Fils, and later by the firm of Fabre de Lagrange which took over their manufacture. In 1857, four years after Olivier died, Harvard University purchased 24 of Olivier's models from Fabre de Lagrange and after the university received the order Benjamin Peirce gave a series of lectures on the mathematics which they illustrated. These models are still in Harvard's collection of scientific instruments. Even after giving a complete set of his models to the Conservatoire National des Arts et Métiers, forty models were still in Olivier's possession at the time of his death. These were sold in 1869 to William Gillispie from Union College in Schenectady, east-central New York, United States. Gillispie exhibited the models at Union College which was appropriate since, twenty years earlier, Union College had became one of the first liberal arts colleges in the United States to give engineering courses. When Gillispie died Olivier's models were sold to the college. SAU Thomas Jefferson writes his long time Frenimy, John Adams, "I have given up Newspapers in exchange for Tacitus and Thucydides, for Newton and Euclid; and I find myself much the happier." L. J. Cappon, The Adams-Jefferson letters. 1846 Pieter Hendrik Schoute (January 21, 1846, Wormerveer–April 18, 1923, Groningen) was a Dutch mathematician known for his work on regular polytopes and Euclidean geometry. Wik Schoute was a typical geometer. In his early work he investigated quadrics, algebraic curves, complexes, and congruences in the spirit of nineteenth-century projective, metrical, and enumerative geometry. Schläfli's work of the 1850's was brought to the Netherlands by Schoute who, in three papers beginning in 1893 and in his elegant two-volume textbook on many-dimensional geometry 'Mehrdimensionale Geometrie' (2 volumes 1902, 1905), studied the sections and projections of regular polytopes and compound polyhedra. ... Alicia Boole Stott (1870-1940), George Boole's third daughter (of five), ... studied sections of four- and higher-dimensional polytopes after her husband showed her Schoute's 1893 paper, and Schoute later (in his last papers) gave an analytic treatment of her constructions. SAU 1860 David Eugene Smith, Ph.D., LL.D. (January 21, 1860 in Cortland, New York – July 29, 1944 in New York) was an American mathematician, educator, and editor. David Eugene Smith attended Syracuse University, graduating in 1881 (Ph. D., 1887; LL.D., 1905). He studied to be a lawyer concentrating in arts and humanities, but accepted an instructorship in mathematics at the Cortland Normal School in 1884. He also knew Latin, Greek, and Hebrew. He became a professor at the Michigan State Normal College in 1891, the principal at the State Normal School in Brockport, New York (1898), and a professor of mathematics at Teachers College, Columbia University (1901). Smith became president of the Mathematical Association of America in 1920.[3] He also wrote a large number of publications of various types. He was editor of the Bulletin of the American Mathematical Society; contributed to other mathematical journals; published a series of textbooks; translated Klein's Famous Problems of Geometry, Fink's History of Mathematics, and the Treviso Arithmetic. He edited Augustus De Morgan's Budget of Paradoxes (1915) and wrote many books on Mathematics and Mathematics History. Wik 1874 René-Louis Baire (21 Jan 1874; 5 Jul 1932) French mathematician whose study of irrational numbers and whose concept to divide the notion of continuity into upper and lower semi-continuity greatly influenced the French School of Mathematics. His doctoral thesis led to the solution of the problem of the characteristic property of limited functions of continuous functions and helped establish the theory of functions of real variables.TIS 1897 Alexander Weinstein (21 Jan 1897 in Saratov, Russia - 6 Nov 1979 in Washington DC, USA) is famed for solving a variety of boundary value problems which have been used in a wide range of applications. Weinstein's method was developed to give accurate bounds for eigenvalues of plates and membranes. In examining singular partial differential equations he introduced a new branch of potential theory and applied the results to many different situations including flow about a wedge, flow around lenses and flow around spindles. SAU 1908 Bengt Strömgren (21 Jan 1908; 4 Jul 1987) Bengt (Georg Daniel) Strömgren was a Danish astrophysicist who pioneered the present-day knowledge of the gas clouds in space. Researching for his theory of the ionized gas clouds around hot stars, he found relations between the gas density, the luminosity of the star, and the size of the "Strömgren sphere" of ionized hydrogen around it. He surveyed such H II regions in the Galaxy, and he also did important work on stellar atmospheres and ionization in stars. TIS 1915 Yuri Vladimirovich Linnik (January 8, 1915 – June 30, 1972) was a Soviet mathematician active in number theory, probability theory and mathematical statistics.Wik 1923 Daniel E. Gorenstein (January 1, 1923 – August 26, 1992) was an American mathematician. He earned his undergraduate and graduate degrees at Harvard University, where he earned his Ph.D. in 1950 under Oscar Zariski, introducing in his dissertation a duality principle for plane curves that motivated Grothendieck's introduction of Gorenstein rings. He was a major influence on the classification of finite simple After teaching mathematics to military personnel at Harvard before earning his doctorate, Gorenstein held posts at Clark University and Northeastern University before he began teaching at Rutgers University in 1969, where he remained for the rest of his life. He was the founding director of DIMACS in 1989, and remained as its director until his death.[1] Gorenstein was awarded many honors for his work on finite simple groups. He was recognised, in addition to his own research contributions such as work on signalizer functors, as a leader in directing the classification proof, the largest collaborative piece of pure mathematics ever attempted. In 1972 he was a Guggenheim Fellow and a Fulbright Scholar; in 1978 he gained membership in the National Academy of Sciences and the American Academy of Arts and Sciences, and in 1989 won the Steele Prize for mathematical exposition. Wik 1609 Joseph Justus Scaliger (5 Aug 1540, 21 Jan 1609) French scholar who was one of the founders of the science of chronology. Like Roger de Losinga, Bishop of Hereford, centuries before, Scaliger recognized that combining the three cycles of the 28-year solar cycle (S), the 19-year cycle of Golden Numbers (G) and the 15-year indiction cycle (I) produced one greater cycle of 7980 years (28×9×15). Scalinger applied this fact, called a Julian cycle, in his attempt to resolve a patchwork of historical eras and he used notation (S, G, I) to characterize years. The year of Christ's birth had been determined by Dionysius Exigus to be the number 9 on the solar cycle, by Golden Number 1, and by 3 of the indiction cycle, thus (9, 1, 3), which was 4713 of his chronological era. Hence, the year (1, 1, 1) was 4713 B.C. (later adopted as the initial epoch for the Julian day numbers).TIS A formula for converting days to Julian day numbers is 1800 Jean-Baptiste Le Roy (15 August 1720;Paris, France - 21 January 1800, Paris) Son of the renowned clockmaker Julien Le Roy, Jean-Baptiste Le Roy was one of four brothers to achieve scientific prominence in Enlightenment France; the others were Charles Le Roy (medicine and chemistry), Julien-David Le Roy (architecture), and Pierre Le Roy(chronometry). Elected to the Académie Royale des Sciences in 1751 as adjoint géomètre, Le Roy played an active role in technical as well as administrative aspects of French science for the next half-century. He was elected pensionnaire mécanicies in 1770 and director of the Academy for 1773 and 1778, and became both a fellow of the Royal Society and a member of the American Philosophical Society in 1773. Le Roy’s major field of enquiry was electricity, a subject on which European opinion was much divided at mid-century. The most prominent controversy engaged the proponents of the Abbé Nollet’s doctrine of two distinct streams of electric fluids (outflowing and inflowing) and the partisans of Benjamin Franklin’s concept of a single electric fluid. This debate intensified in France in 1753 with an attack on Franklin’s views by Nollet. Le Roy, later a friend and correspondent of Franklin, defended his single-fluid theory and offered considerable experimental evidence in support thereof. He played an important role in the dissemination of Franklin’s ideas, stressing particularly their practical applications, and published many memoirs on electrical machines and theory in the annual Histoires and Mémoires of the Academy and in the Journal de Physique. A regular contributor to the Encyclopédie, Le Roy wrote articles dealing with scientific instruments. The most important of these included comprehensive treatments of “Horlogerie,” “Télescope,” and “Électrométre” (in which Le Roy claimed priority for the invention of the electrometer). He also promoted the use of lightning rods in France, urged that the Academy support technical education, and was active in hospital and prison reform. After the Revolutionary suppression of royal academies, Le Roy was appointed to the first class of the Institut National (section de mécanique) at its formation in 1795. Encycopedia.com 1892 John Couch Adams (5 Jun 1819, 21 Jan 1892) In 1878 he published his calculation of Euler’s constant (Euler-Mascheronie constant) to 263 decimal places. (he also calculated the Bernoulli numbers up to the 62 nd) VFR The Euler-Mascheronie constant is the limiting value of the difference between the sum of the first n values in the harmonic series and the natural log of n. (not 263 places, but the approximate value is 0.5772156649015328606065...) He also predicted the location of the then unknown planet of Neptune, but it seems he failed to convince Airy to search for the planet. Independently, Urbanne LeVerrier predicted its locatin in Germany, and then assisted Galle in the Berlin Observatory in locating the planet on 23 September 1846. As a side note, when he was appointed to a Regius position at St. Andrews in Scotland, he was the last professor ever to have to swear and oath of “abjuration and allegience”, swearing fealty to Queen Victoria, and abjuring the Jacobite succession. The need for the oath was removed by the 1858 Universities Scotland Act. Adams made many other contributions to astronomy, notably his studies of the Leonid meteor shower (1866) where he showed that the orbit of the meteor shower was very similar to that of a comet. He was able to correctly conclude that the meteor shower was associated with the comet. Wik 1930 H(ugh) L(ongbourne) Callendar (18 Apr 1863, 21 Jan 1930) was an English physicist famous for work in calorimetry, thermometry and especially, the thermodynamic properties of steam. He published the first steam tables (1915). In 1886, he invented the platinum resistance thermometer using the electrical resistivity of platinum, enabling the precise measurement of temperatures. He also invented the electrical continuous-flow calorimeter, the compensated air thermometer (1891), a radio balance (1910) and a rolling-chart thermometer (1897) that enabled long-duration collection of climatic temperature data. His son, Guy S. Callendar linked climatic change with increases in carbon dioxide (CO2) resulting from mankind's burning of carbon fuels (1938), known as the Callendar effect, part of the greenhouse effect.TIS 1931 Cesare Burali-Forti (13 August 1861 – 21 January 1931) was an Italian mathematician. He was born in Arezzo, and was an assistant of Giuseppe Peano in Turin from 1894 to 1896, during which time he discovered what came to be called the Burali-Forti paradox of Cantorian set theory.Wik 1946 Harry Bateman FRS (29 May 1882 – 21 January 1946) was an English mathematician. He first grew to love mathematics at Manchester Grammar School, and in his final year, won a scholarship to Trinity College, Cambridge. There he distinguished himself in 1903 as Senior Wrangler (tied with P.E. Marrack) and by winning the Smith's Prize (1905). He studied in Göttingen and Paris, taught at the University of Liverpool and University of Manchester before moving to the US in 1910. First he taught at Bryn Mawr College and then Johns Hopkins University. There, working with Frank Morley in geometry, he achieved the Ph.D. In 1917 he took up his permanent position at California Institute of Technology, then still called Throop Polytechnic Institute. Eric Temple Bell says, "Like his contemporaries and immediate predecessors among Cambridge mathematicians of the first decade of this century [1901–1910]... Bateman was thoroughly trained in both pure analysis and mathematical physics, and retained an equal interest in both throughout his scientific career."Wik 1974 Arnaud Denjoy (5 January 1884 – 21 January 1974 in Paris) a French mathematician born in Auch, Gers. His contributions include work in harmonic analysis and differential equations. His integral was the first to be able to integrate all derivatives. Among his students is Gustave Choquet.Wik Credits : CHM=Computer History Museum FFF=Kane, Famous First Facts NSEC= NASA Solar Eclipse Calendar RMAT= The Renaissance Mathematicus, Thony Christie SAU=St Andrews Univ. Math History TIA = Today in Astronomy TIS= Today in Science History VFR = V Frederick Rickey, USMA Wik = Wikipedia WM = Women of Mathematics, Grinstein & Campbell	{"url":"https://pballew.blogspot.com/2023/01/on-this-day-in-math-january-21.html","timestamp":"2024-11-05T00:57:48Z","content_type":"application/xhtml+xml","content_length":"158224","record_id":"<urn:uuid:27c66ee6-fcbc-44c4-9ed0-ddd7013233b5>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00295.warc.gz"}
From MediaWiki Glossary of terms used in this Wiki affine as it relates to Jwidfire, the affines are the triangles. The term as it applies to math is well defined here. Math World a method to reduce unwanted visual effects like jaggy edges, which may also occur in flame-fractals chaos game an often-used algorithm to compute flame-fractals short notion of flame-fractal a special kind of ifs, invented by Scott Draves with various esthetic improvements in mind Iteration means to repeat a process or function. iterated function system, a method to describe a certain families of fractals. A flame-fractal, invented by Scott Draves, is a special kind of ifs when used in digital image creation noise refers to random variations in brightness and color that can appear in an image. Static on a television image is one example of noise. a measurement of number of iterations inside the chaos game producing a visualisation of the flame-fractal. Usually, the higher number of iterations, the higher the visual quality is one member-function of the iterated function system. All the transforms together must meat certain conditions in order to form a flame-fractal which is actually visible and visually appealing	{"url":"http://wiki.jwildfire.org/index.php?title=Glossary&printable=yes","timestamp":"2024-11-09T10:18:02Z","content_type":"text/html","content_length":"14282","record_id":"<urn:uuid:728a1fd7-3628-4c93-acc7-b85f41cfcbae>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00825.warc.gz"}
Fine, Oronce \| Encyclopedia.com Fine, Oronce (b. Briancon, France, 20 December 1494; d. Paris, France, 6 October 1555) astronomy, mathematics, cosmography. Fine (Orontius Finaeus Delphinatus) was born in the Dauphiné but spent his scientific career at Paris.^1 His father, François Fine,^2 had attended the University of Paris^3 and practiced medicine in Briançon. Upon his fater’s death Fine was sent to Paris and was confided to the care of Antoine Silvestre, regent of the Collège de Montaigu and later of the Collège de Navarre. Although he earned his Bachelor of Medicine degree in 1522,^4 his career developed outside the university; in 1531 he was appointed to the chair of mathematics at the recently founded Collége Royal, where he taught until his death. From 1515 Fine edited astronomical and mathematical writings for printers in Paris and abroad. Among them were Peuerbach’s Theorkae planetarunu Sacrobosco’s De sphaera (1516), and Gregor Reisch’s Margarita philosophiea (1535), as well as a tract by his grandfather, Michel Fine, on the plague (1522). He also was responsible for an edition of Euclid’s Elements, of which he published only the first six books (the manuscript of the seventh book, prepared for the printer, is extant). Fine’s first book (1526) was a treatise on the equatorium. an instrument designed to determine the true positions of the planets. In this work Fine exploited the possibilities of curves traced by points (the diagrams of the equations of center), used to facilitate the placement, with respect to the equant. of the mean apsidal line (auge)on the epicycle. These curves, drawn on the basis of lists of the equations of center and of the proportional minutes furnished by the Alphonsine Tables, were a very ingenious innovation^5. At the same time Francisco Sarzosa composed a treatise on the equatorium with the same innovations.* It is difficult to believe that their research was independent, but it is now impossible to establish the proper priority. Fine wrote four other treatises on the equatorium that are extant in manuscript at the library of the University of Paris (Univ. 149); three treatises are little more than outlines, and the fourth treatise describes an instrument similar to Apian’s Astronomicum Caesarewn, with the planetary instruments bound as a book (each of them simply reproduces the geometric decomposition of the Ptolemaic theory of epicycles). Fine’s further works on astronomical instruments include trealises on the new quadrant (1527, 1534) and on the astrolabe (incomplete manuscript layouts are in Paris tat 7415 and Univ. 149). These are not innovative and offer only the standard university account. Fine also inserted a treatise on the new quadrant at the end of his work on gnomonics, De solaribus horologiis et quadrantibus, which first appeared in 1532 as the concluding section of his Protomathesis. The latter consists of four parts, each with its own title page and each separately reprinted: De solaribus horologiis has a separate title page dated 1531, but it is not known to exist separately, and it is unlikely that it was distributed by itself. Among the many types of sundials described in this book are a multiple dial and a navicular.^6 A very rare ivory navicula signed “Opus Orontii F. 1524” —the only scientific instrument certainly attributable to Fine, and perhaps the only one he ever constructed—is in the private Portaluppi collection at Milan. Besides treatises on instruments. Fine’s astronomical work included theoretical writings of a popular nature. These were presented at the two levels of traditional instruction: the elementary one represented by Sacrobosco’s De sphaera, and the higher one of epicyclic astronomy. The Cosmogra phia, an elementary manual, was first published in Latin as the third part of the Protvmathesis (1532), with a separate title page dated 1530, and was reprinted several times, both in Latin and French. It includes the description of the fixed celestial sphere used for reference, the essential ideas concerning the astronomy of the primum mobile (right and oblique ascensions and the duration of diurnal arcs), and a few brief notions of astronomical geography (climates and terrestrial longitudes and latitudes); but it contains no information on the motions of the planets. The latter were discussed in the Théorique de cieux (published anonymously in 1528), which gives a detailed exposition of the Alphonsine epicyclic theory, the first one in French. The brief Canons… touchatu Vusage… des communs almanacks (1543) is a succinct explanation of an almanac computed for the meridian of Tübingen (undoubtedly by Johann Stöffler. which exists in editions dated 1531 and 1533). Although Fine’s interest in astronomy extended to astrology, he wrote only minor works on that subject. The A Imanach of 1529, actually a calendar giving the dates and hours of the new moons in the nineteen-year cycle and the duration of the diurnal arcs, included a short commentary on medical astrology. De XII coeli domiciliis (1553) was a complete theory of the celestial houses, important for casting horoscopes. In this work Fine adopted the definition of the houses advocated by Campanus, for whom the divisions of the celestial vault, for a given horizon, are constructed on the equal divisions of its first azimuth and converge to the south and to the north of the horizon; following this definition, the one usually employed on astrolabes, the lines of the celestial houses were projected onto the astrolabe in accordance with circles passing through the points of intersection of the horizon and of the meridian. Fine also gave an original definition of the unequal hours, however, which were no longer the equal divisions of the diurnal arc and of the nocturnal arc, but the equal divisions of the ecliptic computed, at each moment, from its intersection with the The result, for the construction of the tympana of the astrolabes, was a highly original plotting of the hourly lines -a geometric locus of these equal divisions when the ecliptic turns about the axis of the earth according to the daily movement. Although no astrolabe is known to have been constructed on the basis of this definition, it did find an application in the astrolabic dial of the planetary clock at the Bibliothéque Ste. Genevieve. The dial expressly refers to Fine, and it has therefore been assumed since the seventeenth century that the clock itself was his work. This is highly improbable.^7 It is virtually certain that the clock dates from the fifteenth century and that about 1553 one of its panels, containing the dial of the hours and the astrolabic dial, was replaced by Fine with a panel designed to illustrate his new conception of the unequal hours. The level of technical competence displayed by the mechanism of this dial is very low, for the araignée (the stereographic projection of the celestial vault) completes its revolution in a mean solar day and not in a sidereal day. While the third and fourth parts of the Proto-mathesis dealt with astronomy, the first two treated arithmetic and geometry. De arithmetic practicathe first part, is Fine’s only work on arithmetic. In accordance with the traditional schema of medieval arithmetic, the various operations carried out on the numbers were enumerated and described following a plan that distinguished whole numbers, common fractions (fractiones vulgares), and natural or sexagesimal fractions. The latter were of particular interest to practitioners of Alphonsine astronomy, since they were the basis of their preferred mathematical tool. Fine made it easier to work with these fractions by providing a tabula proportions (so called because of its aid in computing proportional parts of the equation of the argument). or multiplication table in sexagesimal numeration, similar to the same table by John of Murs or Bian-chini. The last book of the De arithmetica, on ratios and proportions, developed theorems established by Euclid and Ptolemy. The two books on geometry (dated 1530 in the Protomathesis) treated the subject at a more elementary level. After stating the definitions of plane and solid figures, borrowed from the Elements, as well as the Euclidean postulates, Fine discussed the measurement of length, height, and width in the tradition of the treatises on practical geometry, of which one of the most popular aspects was geometrical canons for the use of the astrolabe. To this end he treated the geometric square, the quadrant, the cross-staff (Jacob’s staff), and the mirror. The calculation of surfaces and volumes, which was the complement of the measurement of lengths, included that of circular surfaces and volumes. For the latter, Fine computed the ratio of the circumference of the circle to its diameter as Returning to the ratio of circumference to diameter in De quadratura circuli (1544), Fine offered what he believed to be a more precise value of π: In De circuit mensura, which follows Dequadratura, he reduced that ratio to 47/15. Finally, in the posthumous De rebus mathematicis (1556), he increased the value slightly to one between the two preceding ones: These attempts to determine the true figure were but one aspect of his efforts to solve the quadrature of the circle, for which he examined several solutions. None of them was satisfactory; and Fine was vehemently attacked by some of his contemporaries, notably Pedro Nuńez Salaciense, in De erratis Orontii Finaei (1546), and Johannes Buteo (Jean Borrel), in De quadratura circuli (1559). It must be acknowledged that Fine’s arrogance about his own accomplishments undoubtedly made his errors of logic all the more intolerable to his opponents. Fine’s work in trigonometry scarcely went beyond what was necessary to establish a table of sines: three chapters of book II of De geometria, included in the Protomathesis, and De rectis in circuli quadrante subtensis and De universali quadrante, both published in 1542 as appendixes to the first reprinting of De mundi sphaera (which had been included in the Protomatlwsis). Although the works of Regiomontanus and Copernicus on this subject were printed during Fine’s lifetime, his writings fell entirely within the Ptolemaic tradition. For example, he limited himself to demonstrating the properties that allow successive evaluations of the half chords of arcs starting from the half chords of some other noteworthy arcs. Also, the table of sines that he constructed for intervals of fifteen minutes and a radius of sixty units is very similar to that (for example) of Fusoris.^8 Nevertheless, Fine indicated how his sines, expressed in sexagesimal notation, may be transformed into those given by Regiomontanus, which were calculated with a radius of 60,000 units. The universal quadrant described in 1542 was the trigonometric quadrant deriving from the eleventh-century quadrans vetustissimus. This earlier instrument had been described and commented upon by Apian in his Instrumentum… primi mobilis (1534). Fine dealt only with the strictly trigonometric uses of the quadrant, determination of the right and versed sines of a given arc—or vice versa-and the products or ratios of two sines. Virtually ignoring the application of its properties to astronomical calculations—a task carried out by J. Bonie and by B. de Solliolis in works that Fine owned^9 -Fine did no more than enumerate these possibilities. In his Latin thesis of 1890, L. Gallois dealt only with Fine’s cartography: a large map of France on four sheets and two cordiform world maps, one of the eastern hemisphere and the other, doubly cordiform, of the northern and southern hemispheres. Gallois held that the world maps were original creations and provided the source of the similar maps executed by Schoner and Apian.^10 This hypothesis is unlikely: and in the absence of an established chronology of these maps, it may be supposed that the relations of dependence were in fact the reverse, for Fine’s usual procedure was to elaborate his astronomical works on the basis of the writings of others. This was undoubtedly the case with his map of France, but the scarcity of the surviving documents does not allow its genesis to be reconstructed. Fine’s map of France does not truly comprise the grid of the parallels and meridians but, rather, transfers the schema to the margins; the longitudes are computed there from l’extremité occidentale du mande as in the Alphonsine Tables. Fine’s scientific work may be briefly characterized as encyclopedic, elementary, and unoriginal. It appears that the goal of his publications, which ranged in subject from astronomy to instrumental music, was to popularize the university science that he himself had been taught. In this perspective, it is perhaps his works in French (such as Théorique des vieux) or the French translations of his works first published in Latin (for instance, Canons et documents très amples touchant l’usage des communs almanacks and the Sphére du monde) that best illustrate his scientific career. 1. There is disagreement as to whether the last letter of his name should be accented. Citing the Latin form Finaeus, bookkeeping records in which the name is spelled Finee, and the bad rhyme of Finé with Dauphiné and affiné made by André Thevet, L. Gallois (De Orontio Finaeo, 2) opted for the pronunciation Finé. This is the one that has generally been accepted, despite the objections of Dauphinois scholars, who, citing local usage —which ought to decide the question—prefer Fine. The form Finée probably resulted from rendering the Latin form into French. As for Thevets rhymes, which are very late (1584). their significance is diminished by the fact that a contemporary and close friend of Fine’s. Antoine Mizauid, rhymed Fine with doctrine in his verses. (See MS Paris fr. 1334. fol. 17.) The date of his birth is specified in an autograph note in MS Paris lat. 7147. fol. ii. 2. See E. Wickersheimer, Dictiannatrc biouraphique des médecins en France au moyen âge (Paris, 1936). 553 and 154 3. There are two MSS of a course on Aristotle given by Jean Hannon and copied by François Fine in 1472- 1473 (Paris lat. 6436, 6529); see Catalogue des manuscriu en écriturc latine portant des indications de date, de lieu ou de copiste II (Paris. 1962). 341,353; pi. cli. 4.Commenta’tres de la Fuculté de médecine de I’Université de Paris,II, 1516-1560, M.-L. Concasty. ed. (Paris, 1964). 50b. 54a. This is in the series Collection de Documents Inédits sur I His to ire de France. 5. E. Poulle and Fr. Maddison. “Un équatoire de Franciscus Sarzosius.” in Physis,5 (1963), 43-64. 6. D. J. de Soila Price. “The Little Ship of Venice, a Middle English Instrument Tract,” in Journal of the History of Medicine and Allied Sciences.15 (1960), 399-407: and Fr. Maddison, Medieval Scientific Instruments and the Development of Navigational Instruments in the XVth and XVlth Centuries (Coimbia, 1969). 14, This book is Agrupa-mento de Estudos de Cartografia Antiga, XXX. 7. D. Hillard and E. Poulle, “Oronee Fine et I’horloge planétaire de la Bibliothèque Sainte-Geneviève” and E. Poulle, “l..es mécanisations de rastronomie des épicycles, I’horloge d’Oronce Fine.” in Comptes rendus des séances de Académie des inscriptions et belles-lettres (1974). 59-79. 8. E. Poulle. Un constructed d’instruments astronomiques au X’i c siècle.Jcan Fusam (Paris, 1963). 75-80. This work is Bibliothèque dc “I École Pratique des Hautes Études, IV’” sect.. Ease. 318. 9. E. Poulle, “Théorie des planètes et trigonométric au XVe siècle d’après un équatoire inédit. Ic sexagenarium.” in Journal des savants (1966). 129- 161.esp. 131- 132. 10. L. Gallois, Les géographes allemands de la renaissance (Paris, 1890), 92-97. This work is Bibliothèque de ia Faculté des Lettresde Lyon. XIII. I. Original Works. The list of books published by Fine is difficult to establish, for it involves sorting out many reprintings, some of them only partial, and a number of translations. There are four contemporary lists, three of them inserted in his eds. of Euclid’s Elements of 1536, 1544, and 1551; the fourth was included by An-toine Mizauld in his ed. of Fine’s De rebus mathematics in 1556. All of these lists, however, pose problems. That drawn up by L. Gallois in De Orontio Finaeo gallico geographo (Paris. 1890). 71 -79. is incomplete and has been superseded by those of R. P. Ross, in his unpublished doctoral dissertation, “Studies on Oronce Fine (1494-1555)” (Columbia University. 1971); and of D. Hiilard and E. Poulle, in “Oronee Fine et I’horloge planetaire de la Bibliothéque Sainte-Geneviève,” in Bi-hliotheque d’humanisme et renaissance,33 (1971), 311-351, see 335-351. The latter list is numbered and indexed, and includes MSS. One should consult the latest findings concerning the bibliography in R. P. Ross, “Oronce Fine’s Printed Works: Additions to Hillard and Poulle’s Bibliography.” in Bihliotheque d’humanisme et renaissance. 36 (1974), 83-85. II. Secondary Literature. Gallous’sDe Oromio Finaeohas become quite dated and presents an extremely limited picture of Fine’s work. Ross’s “Studies on Oronce Fine (1494-1555)” deals only with the mathematical works, among which those on astronomy are not included. It does, however, contain a recent bibliography. An overall account of Fine’s work is in the exposition catalog of the Bibliotheque Ste.-Genevieve, Science et astrologie an XVIe Siècle. Or Once Fine et son harloge planétaire (Paris, 1971).See also Richard P. Ross, “Oronce Fine’s De minibus libri II: The First Printed Trigonometric Treatise of the French Renaissance,” in Isis, 66 (1975), 378-386. Emmanuel Poulle More From encyclopedia.com About this article Fine, Oronce	{"url":"https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/fine-oronce","timestamp":"2024-11-09T00:30:44Z","content_type":"text/html","content_length":"59967","record_id":"<urn:uuid:67de6fbf-3fa2-4840-b5ec-61227b36d5db>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00274.warc.gz"}
How To Measure Diameter With A Tape Measure I love creating free content full of tips for my readers, you. I don't accept paid sponsorships, my opinion is my own, but if you find my recommendations helpful and you end up buying something you like through one of my links, I could earn a commission at no extra cost to you. Learn more It’s pretty simple to determine an object’s length or height. You can accomplish it with the help of a ruler. But when it comes to determining the diameter of a hollow cylinder or circle it appears to be difficult somewhat. I am pretty sure that many of us have tried to measure the diameter with a simple ruler at least once in our lives. I have been in that scenario numerous times myself. However, measuring the diameter of a hollow cylinder or circle is not as difficult as it looks. You can do that easily if you know the basic procedure for it. In this article, I’ll show you how to measure diameter with a tape measure. Continue reading this article if you don’t wish to be bothered by the question anymore. What Is A Tape Measure A tape measure or measuring tape is a lengthy, thinner, malleable strip of plastic, cloth, or metal having measurement units printed on it (such as inches, centimeters, or meters). It is made up of a variety of components, including case length, spring and brake, blade/tape, hook, connector hole, finger lock, and belt buckle. You can measure the length, height, width of an object using this tool. You can also use it to calculate the diameter of a circle. Measure Diameter With A Tape Measure Before we measure the diameter of a circle, we must first understand what a circle is and what exactly is a diameter. A circle is a curved line with all points at the same distance from the center. And the diameter is the distance between two points (a point on one side and a point on the other side) of the circle that passes through the center. As we know what a circle is and what its diameter is, now we are ready to measure the diameter of a circle with a tape measure. You must take specific procedures to accomplish this, which I will detail in this portion of the post. • Find the circle’s center. • Attach the tape to any point on the circle. • Calculate the circle’s radius. • Determine the circumference. • Calculate the diameter. Step 1: Find The Circle’s Center The first step is to locate the center of the hollow cylinder or circular object whose diameter you want to determine. You can easily find the center with a compass, so don’t be concerned. Step 2: Attach The Tape To Any Point On The Circle In this stage attach one end of the tape measure somewhere on the circle. Now drag the tape measure’s other end to a position on the other side of the circle. You must ensure that the straight line connecting two points (one end and the other end of the measuring tape) goes through the circle’s center. Now, using a color marker, mark these two points on the scale and take a reading. Note that you should keep your readings in a notepad. Step 3: Calculate The Circle’s Radius Now you have to measure the radius of the circle. The radius of a circle is the distance between the circle’s center and any point on it. It’s extremely simple to calculate and you can do that with the help of a measuring tame or a compass. Place one end of the measuring tape in the middle and the other end on any point of the curved line to do this. Take note of the number; it’s the radius of a circle or a hollow cylinder. Step 4: Determine The Circumference Now measure the circumference of the circle, which equals the length around the circle. In other terms, it’s the circle’s perimeter. To determine the perimeter of the circle you have to use a formula which is C = 2πr. Where r is the radius of the circle(r= radius) and π is a constant whose value is 3.1416(π=3.1416). Step 5: Calculate The Diameter We’ve gathered all of the information we’ll need to figure out the diameter of the circle. We’ll be able to figure out the diameter now. To do so, divide the circumference by 3.141592,( C = 2πr/ 3.1416) which is the value of pi. For example, if you wish to find the diameter of a circle with a radius of r=4, the circumference of the circle will be C=23.14164=25.1322 ( using the formula C = 2πr). And the diameter of the circle will be D=(25.1328/3.1416)=8. Frequently Asked Questions (FAQs) Q: Is it possible to use a ruler to measure diameter? Answer: Yes it is possible to measure the diameter of a circle using a ruler. In this situation, the calculations will be the same as before, but instead of using a measuring tape, you will need to use a ruler to take your measurements. Q: What is the most effective instrument for measuring the diameter of a circle? Answer: Respectively measuring tape, calipers and micrometers are the most effective instrument for measuring diameter. A long time ago, the diameter measurement method was discovered. Despite a long period passed, calculating diameter is still useful in several fields, including mathematics, physics, geometry, astronomy, and more. And it won’t change in the future. So, don’t ignore the importance of buying a good-quality tape measure. You’ll find all the information you need about measuring a circle’s diameter in this article. Please scroll up to the article and read it without further delay, if you have not done so already. Also read: how to read a tape measure in meters I'm Joost Nusselder, the founder of Tools Doctor, content marketer, and dad. I love trying out new equipment, and together with my team I've been creating in-depth blog articles since 2016 to help loyal readers with tools & crafting tips.	{"url":"https://toolsdoctor.com/how-to-measure-diameter-with-a-tape-measure/","timestamp":"2024-11-04T16:43:55Z","content_type":"text/html","content_length":"76065","record_id":"<urn:uuid:6f8aefd5-f73c-4b05-9e4f-deeb108e6657>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00647.warc.gz"}
Riesel number base 5 Currently there may be errors shown on top of a page, because of a missing Wiki update (PHP version and extension DPL3). Topics Help • Register • News • History • How to • Sequences statistics • Template prototypes A Riesel number base 5 is a value of k such that k•5n-1 is always a composite number. Using the same method presented in the Riesel problem article, it was found that 346802•5n-1 is multiple of 3, 7, 13, 31 or 601 (the covering set) depending on the value of n, so it is always In order to demonstrate whether 346,802 is the smallest Riesel number base 5 or not, a distributed computing project was created named Sierpiński-Riesel Base 5.	{"url":"https://www.rieselprime.de/ziki/Riesel_number_base_5","timestamp":"2024-11-13T02:36:40Z","content_type":"text/html","content_length":"20241","record_id":"<urn:uuid:f166accb-008b-4b67-bc80-3c572b2f9281>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00596.warc.gz"}
Total Harmonic Distortion - Electrical Equipment Viewing 5 posts - 1 through 5 (of 5 total) • Author • Is there any value per standard to use as reference for Total Harmonic in the system. What is the value we can say it is safe for a sensitive instrument. Need to educate me, how the hamonic and distortion is appearing in the electrical system?please need your expert advice, thanks paule said: Is there any value per standard to use as reference for Total Harmonic in the system. What is the value we can say it is safe for a sensitive instrument. Need to educate me, how the hamonic and distortion is appearing in the electrical system?please need your expert advice, thanks Dear Paule ; First, the harmonics currents are generating by the Elctronics Loads like : Computers, Printers, Faxes machines, Variable Speed Drives, DC Drives, Electrnic transformers for lighting, ext. These loads generates the Harmonics Currents, and all distortion of currents we called ” THD I – Total Harmonic Distortion of Current ” that affacte the voltage sine wave ( becuase we have the Harmonic Impedances ), so, we have that we called the distortion of Voltage ” THD U or THD V “. Noting that the most importatnt value is ” THD U or THD V “. The ” IEC 61000-2-2,3,… ” defines the maximim percantages value of Voltage Distrotion of each harmonic for ” HV, MV, and LV networks, for exp. for LV networks : for n=3, the maximum is ” 5 % “, for ” n=5 ” the max is ” 6 % “, and also this IEC… defines these percantages for Odd & Even Harmonics. In France, the EDF defines the max percantage of ” THD V ” on the LV networks by ” 5 % “. Thanks, if its exceeds the maximum value of 5 % for n=3 and 6 % for n=5, what will be the impact in the connected instrument or devices, does it will damage the instrument?what precaution or protection we can apply. You can download “IEEE-519-1992” standard about THD, it's a good referenace. paule said: Thanks, if its exceeds the maximum value of 5 % for n=3 and 6 % for n=5, what will be the impact in the connected instrument or devices, does it will damage the instrument?what precaution or protection we can apply. Dear Mr. Paule ; If the Voltage Harmonic Distortion for each harmonics exceeds the value defined by standard, that means the Total Harmonic Distortion of Voltage ” THD U or THD V ” will be globaly more than standardrized valeus, that can creats some disturbances in working of instruments, and even the currents for lieanr Loads will be distorted. By the way, when we have these high percentages of THD U, that means we have big percentages of current distortion ” THD I “, that leads to increase the losses in transformers, generators, cables, and also the heating of these equipments will be more. In some cases when we have a big value of ” THD I ” but the ” THD U ” is less than the stadard value, we should in this case dereat the power of Transformes and Generators, and also over-size the section of cables. In the same time, the most dengarous harmonics are the Double Third Harmonics, that means : 3, 9, 15, …, because these harmonics have the same vector,so, the sum of them aren't ” 0 ” and they run in the neutral cables, for this reason once we have this kind of harmonics in the network, we should calculate the value of them and do correctly the choice of the neutral cable's section. But most of the maufacters of some elctronic equipments like : Virable Speed Drives, UPS, try to minimize the percentages of them in their euipments. Noting that in some equipments of serious manufacters the percentages of this kind of harmonics are ” 0 “. • Author Viewing 5 posts - 1 through 5 (of 5 total) • You must be logged in to reply to this topic.	{"url":"https://engineering.electrical-equipment.org/forum/general-discussion/total-harmonic-distortion","timestamp":"2024-11-14T17:59:43Z","content_type":"text/html","content_length":"69485","record_id":"<urn:uuid:bd114635-eeca-4be9-ae76-e0d403beab9d>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00885.warc.gz"}
(Help) CD error simulating laminar steady-state flow over a sphere [OpenFOAM] June 13, (Help) CD error simulating laminar steady-state flow over a sphere [OpenFOAM] #1 2017, 10:36 Member Hello, Join Date: I’m having some trouble simulating laminar steady-state flow over a sphere (OpenFOAM). The solution is converged but I’m having about 15% error on the drag coefficient (CD) for 10 < Re < May 2017 100, and about 20% for Re <= 1. Please, see some pictures of my simulation in this website Posts: 47 http://imgur.com/a/AxIWE Rep Power: . 9 1. Mesh I’ve tested finer meshes, and the mesh is converged. (The CD of this one differs only 1.5% from a 15.4% more refined mesh) 2. Solution convergence and CD 3. Post-processing From picture "Pressure over the sphere and U streamlines for Re = 80", it seems like the flow isn’t steady (because of the different size of the recirculation zone), but shouldn’t it be for Re = 80? The separation zone is increasing the pressure in the back of the sphere, is that right? 4. Results I think that there’s something strange with the pressure field, because for very low Reynolds numbers (<= 1) the drag should be predominantly viscous (my guess would be viscous CD > 5. Simulation parameters I’m using simpleFoam solver (SIMPLE algorithm), GAMG for pressure (p) field and smoothSolver for velocity field (U) (see pictures for fvSoultion and fvSchemes). I’ve already changed those field solvers (PGC, GAMG, …), refined the mesh, reduced the tolerance of p, checked the boundary conditions and checked the size of the fluid domain, but the results barely Does anyone have any clue what’s wrong, please? Thanks in advance.	{"url":"https://www.cfd-online.com/Forums/openfoam-solving/189107-help-cd-error-simulating-laminar-steady-state-flow-over-sphere-openfoam.html","timestamp":"2024-11-05T12:42:05Z","content_type":"application/xhtml+xml","content_length":"95131","record_id":"<urn:uuid:d92ee155-65d0-488b-b9d9-1b4d06cf9e38>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00166.warc.gz"}
Linear-Scaling Open-Shell MP2 Approach: Algorithm, Benchmarks, and Large-Scale Applications Quantum Electronic Structure Linear-Scaling Open-Shell MP2 Approach: Algorithm, Benchmarks, and Large-Scale Applications Click to copy article linkArticle link copied! Click to copy section linkSection link copied! A linear-scaling local second-order Møller–Plesset (MP2) method is presented for high-spin open-shell molecules based on restricted open-shell (RO) reference functions. The open-shell local MP2 (LMP2) approach inherits the iteration- and redundancy-free formulation and the completely integral-direct, OpenMP-parallel, and memory and disk use economic algorithms of our closed-shell LMP2 implementation. By utilizing restricted local molecular orbitals for the demanding integral transformation step and by introducing a novel long-range spin-polarization approximation, the computational cost of RO-LMP2 approaches that of closed-shell LMP2. Extensive benchmarks were performed for reactions of radicals, ionization potentials, as well as spin-state splittings of carbenes and transition-metal complexes. Compared to the conventional MP2 reference for systems of up to 175 atoms, local errors of at most 0.1 kcal/mol were found, which are well below the intrinsic accuracy of MP2. RO-LMP2 computations are presented for challenging protein models of up to 601 atoms and 11000 basis functions, which involve either spin states of a complexed iron ion or a highly delocalized singly occupied orbital. The corresponding runtimes of 9–15 h obtained with a single, many-core CPU demonstrate that MP2, as well as spin-scaled MP2 and double-hybrid density functional methods, become widely accessible for open-shell systems of unprecedented size and complexity. This publication is licensed under CC-BY 4.0 License Summary* You are free to share(copy and redistribute) this article in any medium or format and to adapt(remix, transform, and build upon) the material for any purpose, even commercially within the parameters • Creative Commons (CC): This is a Creative Commons license. • Attribution (BY): Credit must be given to the creator. View full license • License Summary* You are free to share(copy and redistribute) this article in any medium or format and to adapt(remix, transform, and build upon) the material for any purpose, even commercially within the parameters below: □ Creative Commons (CC): This is a Creative Commons license. □ Attribution (BY): Credit must be given to the creator. View full license • License Summary* You are free to share(copy and redistribute) this article in any medium or format and to adapt(remix, transform, and build upon) the material for any purpose, even commercially within the parameters below: □ Creative Commons (CC): This is a Creative Commons license. □ Attribution (BY): Credit must be given to the creator. View full license ACS Publications Copyright © 2021 The Authors. Published by American Chemical Society 1. Introduction Click to copy section linkSection link copied! While open-shell species are ubiquitous in chemistry, their investigation remains challenging from both the experimental and the theoretical perspective. Here, we focus on systems of a high-spin open-shell electronic structure, for which single-reference quantum chemical methods can provide an accurate description comparable to what is expected for closed-shell molecules. The variety of such systems includes the ionized and electron-attached states of closed-shell molecules; species relevant in combustion, polymer, atmospheric, interstellar, electro-, and redox chemistry; and radicals appearing as intermediates or transition states of reactions. The importance and difficulties of the explicit treatment of electron correlation for such systems and the related processes are also well understood. The second-order Møller–Plesset approach (MP2) is one of the standard tools for that purpose. While the accuracy of conventional MP2 does not reach that of the “gold standard” coupled-cluster (CC) model with single, double, and perturbative triple excitations [CCSD(T)], its favorable computational cost motivated the development of various improved MP2-based methods. Among those, the spin-component-scaled (SCS) MP2 schemes and the double-hybrid (DH) density functionals, both proposed by Grimme, have gained wide popularity. In the SCS-MP2 methods, the opposite- and same-spin contributions to the correlation energy are scaled by different empirical factors, whereas for the DH functionals, the energy is augmented with an MP2-like second-order perturbation theory (PT2) correction evaluated on Kohn–Sham (KS) orbitals. The introduction of spin-scaled PT2 expressions into the formulation of DH functionals turned out to be particularly successful to raise the accuracy of DH functionals above that of conventional density functionals. Especially when combined with the resolution-of-identity or density-fitting (DF) technique, MP2-based methods can target systems of more than 100 atoms, thereby extending the about 30-atom applicability limit of CCSD(T) considerably. The Laplace transform (LT) technique proposed by Almlöf to eliminate the energy denominator of MP2 has also become fundamental to reduce the fifth-power-scaling computational complexity of MP2. Aiming toward the same goal, the particularly simple form of MP2 was also utilized in a number of creative developments on the basis of, for instance, Cholesky-decomposed pseudo-density matrices; and pseudospectral approaches; nonorthogonal or Slater-type orbitals; tensor hypercontraction; as well as large-scale parallelization. Parallel to such reformulations, the group of local correlation approaches exploits the rapid decay of electron correlation with distance, especially in combination with localized molecular orbitals (LMOs). Following the pioneering work of Pulay and Saebø, the correlation energy contribution of distant LMO pairs can be approximated via a more cost-efficient level of theory (pair approximation), often using a restricted, spatially close list of correlating orbitals (domain approximation). Particular methods also compute the first-order MP (MP1) amplitudes, required for the MP2 energy, directly in the LMO basis, for which the solution of coupled amplitude equations is frequently accelerated using some sort of MP2-based natural orbitals (NOs). Alternatively, the coupling of MP1 amplitudes with distant LMO indices can be neglected using fragmentation approximations and fragment-specific semicanonical basis sets. Our previous developments combine the above benefits of decoupled MP1 amplitude expressions with the sparsity provided by the LMO basis via an LT or Cholesky-decomposition (CD)-based MP2 formulation. We also utilize a form of MP2-based local NOs (LNOs) in our LNO-CCSD(T) and higher-order LNO-CC schemes. However, as the computational cost of the MP2-based LNO construction is comparable to that of the MP2 correlation energy, our local MP2 (LMP2) approach employs only the pair and domain approximations in combination with an LT/CD-based energy expression written in the LMO basis but does not require NOs. Compared to the variety of local correlation methods targeting closed-shell systems, the application of local approximations for open-shell cases is much less explored. Open-shell extensions of the incremental scheme were developed by Dolg, Tew, Friedrich, and co-workers based on unrestricted Hartree–Fock (UHF) as well as restricted open-shell HF (ROHF) references. Most recently, the high-spin open-shell variants of the pair NO (PNO)-based method of Werner, Ma, and co-workers, as well as the domain-based local PNO (DLPNO) method of Neese, Valeev, Hansen, Saitow, Guo, Kumar, and co-workers were also published up to the CCSD(T) level of theory. The DLPNO family of methods employs the multireference second-order Ansatz of the -electron valence state perturbation theory (NEVPT2) for the PNO generation, while the PNO methods of Werner and Ma utilize a spin-adapted MP2 formulation (PNO-RMP2). Both approaches share the benefit of spin-free amplitudes useful to obtain a spin-restricted set of PNOs at the price of a somewhat more complicated second-order treatment. Since neither NOs nor iterative amplitude equations are required for the efficient computation of MP1 amplitudes in our LMP2 approach, we prioritized simplicity and chose the ROHF-based but unrestricted MP2 Ansatz proposed by Lauderdale and Bartlett and Knowles et al. However, the most demanding integral transformation step is carried out in a restricted, intermediate MO basis; thus, the computational cost remains comparable to that of the parent closed-shell LMP2 method. To that end, the use of ROHF or restricted open-shell KS (ROKS) reference is required, but UHF and unrestricted KS (UKS) orbitals are also supported by the construction of quasi-restricted orbitals (QROs). For the generalization of our local correlation methods to the high-spin open-shell case, here, we identify and resolve a number of technical subtleties emerging already at the LMP2 level of theory. Special attention is devoted to the treatment of singly occupied MOs (SOMOs) in the pair and domain approximations as well as in the pair and domain correlation energy contributions and to the energy contribution of single excitations. The independent evaluation of the MP1 amplitudes also allows for the introduction of a novel cost-reduction approach: we show that up to 50–90% of the correlation energy contributions can be computed relying on closed-shell algorithms by approximating long-range spin-polarization effects far away from the localized SOMOs. The resulting open-shell LMP2 correlation energies are equivalent to the closed-shell ones for systems with only doubly occupied orbitals in the zeroth-order wave function. The open-shell LMP2 approach inherits the beneficial properties of our previous algorithms, which are the iteration- and redundancy-free amplitude evaluation, and the operation-count and memory-efficient, integral-direct, practically disk I/O-free, and OpenMP-parallel implementation. The present local approximations are free from empirical parameters, manual fragment definitions, real-space cutoffs, etc. often associated with local correlation methods. All approximations are systematically improvable and automatically adapt to the electronic structure because of the employed energy and orbital coefficient-based threshold definitions. Additional unique properties include the treatment of near-linear-dependent AO basis sets, integration to multilevel local correlation methods, the utilization of general point group symmetry, and frequent checkpointing. The accuracy of the open-shell LMP2 method is benchmarked for radical stabilization energies (RSEs), ionization potentials (IPs), and spin-state energy differences of a large set of open-shell species. Mean (maximum) absolute errors against canonical DF-MP2 references are well below the intrinsic accuracy of MP2 being 0.01–0.06 (0.04–0.13) kcal/mol for all three types of quantities with various basis sets. The same errors remain in the 0.01–0.12 kcal/mol range for a smaller set of systems including 37–175 atoms, while the corresponding LMP2 calculations take only up to 3–4 h with an 8-core CPU. The capabilities of the present open-shell LMP2 code are demonstrated on three-dimensional, real-life protein models including 565 and 601 atoms and about 11000 atomic orbitals. Both examples represent current challenges of large-scale correlated calculations (see Section 4 ): the lowest-lying quintet and triplet states of the first system were especially complicated to find at the self-consistent field (SCF) level, and one of the SOMOs of the other molecule is delocalized over a large fragment, leading to extremely large domains to handle. In spite of these difficulties, it was feasible to perform LMP2 computations for four species in this size range, each taking about 9–15 h on a single, 20-core CPU. Thus, the present implementation is highly capable of extending the reach of open-shell (spin-scaled) MP2 and DH density functional computations to systems of unprecedented size. The paper is organized as follows. Sections 2 provide the theoretical details of the LMP2 Ansatz and the corresponding algorithms, respectively. The employed technical details and test systems are introduced in Section 4 . The accuracy of the individual and combined local approximations is assessed in Sections 5 . Finally, Section 7 presents large-scale applications and analyzes the corresponding computational requirements. 2. Theoretical Background Click to copy section linkSection link copied! A restricted open-shell reference determinant of doubly and singly occupied molecular orbitals (DOMOs and SOMOs, respectively) is assumed. These orbitals will be subjected to various orbital transformations, and the notations distinguishing the different orbital sets are collected in Table 1 . The correlation energy expressions of the conventional theory will be introduced in terms of unrestricted, semicanonical (also known as pseudocanonical) MOs denoted by ( , ..., , ...) and ( , ..., , ...) indices for the occupied and virtual subsets, respectively. Lower (upper) case indices label the spin-up (spin-down) set of semicanonical orbitals. Local approximations will rely on localized molecular orbitals (LMOs) obtained from a restricted open-shell reference , while these LMOs will be labeled as ′, ... ( ′, ...), respectively, when occupied by spin-up (spin-down) electrons. i, j, k, ... (semi-)canonical occupied orbitals (spin-up) I, J, K, ... (semi-)canonical occupied orbitals (spin-down) a, b, c, ... (semi-)canonical virtual orbitals (spin-up) A, B, C, ... (semi-)canonical virtual orbitals (spin-down) i′, j′, k′, ... localized restricted occupied orbitals (spin-up) I′, J′, K′, ... localized restricted occupied orbitals (spin-down) localized restricted occupied orbitals (spatial) ĩ, ..., ã, ... (semi-)canonical orbitals in the primary or extended domain (spin-up) Ĩ, ..., Ã, ... (semi-)canonical orbitals in the primary or extended domain (spin-down) μ, ν, λ, ... atomic orbitals P, Q, ... auxiliary functions for the DF approximation 2.1. Canonical Open-Shell MP2 Ansatz Here, we briefly summarize the MP2 approach introduced by Lauderdale and Bartlett and Knowles et al. for restricted open-shell reference determinants. Starting from a set of restricted orbitals, spin-up and spin-down Fock matrices are constructed using the DOMOs and SOMOs for the former and only the DOMOs for the latter. The spin-up (spin-down) MOs of the restricted open-shell determinant are then canonicalized separately using the respective spin-up (spin-down) Fock matrices, yielding the semicanonicalized MO sets. The corresponding second-order correlation energy ( ) is calculated relying on an unrestricted formalism Here, quantities ... denote MP1 amplitudes corresponding to single and double excitations. Moreover, and ⟨ ⟩ stand for Fock-matrix elements and electron repulsion integrals (ERIs) in the Dirac notation, respectively, while ⟨ ⟩ = ⟨ ⟩ – ⟨ ⟩. The beneficial properties of this correlation energy expression include the invariance to the separate unitary transformation of the occupied and unoccupied MOs. This opens the possibility of introducing local correlation approximations exploiting LMOs. Naturally, eq 1 is equivalent to the closed-shell MP2 correlation energy in the special case of closed-shell systems. 2.2. Open-Shell Local MP2 Ansatz The present open-shell local MP2 Ansatz is constructed analogously to our highly efficient closed-shell local MP2 (LMP2) implementation. The LMP2 approach employs ideas from fragmentation approaches, such as the incremental expansion up to orbital pairs, which can also be interpreted as pair approximations for distant orbital pairs as employed frequently in direct local correlation approaches. The main correlation energy contribution is obtained using orbital-specific basis sets reminiscent of the cluster-in-molecule, as well as the divide-expand-consolidate models. Moreover, a Laplace-transform (LT) or Cholesky-decomposition (CD)-based MP2 formulation is employed to circumvent redundant amplitude evaluations and the need for the iterative solution of MP1 amplitude equations expressed in a noncanonical basis. By exploiting the unitary invariance of eq 1 can be rewritten in terms of restricted orbitals. Then, is expressed in terms of correlation energy contributions of occupied orbitals by separating one occupied index in the summations of eq 1 It is important to emphasize that denotes a spatial orbital occupied by either one or two electrons, while ′ ( ′) refers to orbitals with the same spatial component as but occupied by at most one spin-up (spin-down) electron. Here, we also assume that the restricted orbitals of eq 2 are LMOs; hence, the orbital indices are primed. The equivalence of eqs 1 can be utilized to define the correlation energy contributions of individual LMOs occupied by spin-up and spin-down electrons Note that the last term of eq 1 including both spin-up and spin-down occupied indices results in both the spin-up and spin-down correlation energy contributions (cf. the last terms of eqs 3 ) because both of its occupied indices can be transformed to the LMO basis. Furthermore, because of the unitary transformation of ′ ( ′) and the restriction of the summations over this index, the complete permutational symmetry of the MP1 amplitudes and two-electron integrals is lost. Consequently, the final terms of eqs 3 cannot be combined into a single term like in the conventional theory, which explains the appearance of the additional, sixth type of term (third one of eq 4 ). Note also that the δ contribution of a singly occupied (SO) restricted LMO is zero by definition; therefore, the correlation energy contribution of the SO LMOs contains only half as many terms as for a doubly occupied (DO) LMO. To achieve asymptotically linear scaling with system size, the summations of eqs 3 are restricted. Therefore, the correlation energy contribution of each LMO can be computed with asymptotically constant cost, at least for nonmetallic systems, where the pair correlation energy of all orbital pairs decays with distance. First, a list of occupied and virtual orbitals called the extended domain (ED) is assembled around each LMO, in which the selected LMO serves as the central MO (CMO) of the ED. The occupied subspace, that is, the strong pair LMO list, of the ED consists of those restricted LMOs that have a sufficiently high pair correlation energy with the CMO. Then, the virtual subspace of the ED is constructed from such restricted projected atomic orbitals (PAOs, as defined by eq 6 ) that are important for the correlation energy contributions of any of the CMO’s strong pairs. For the accurate and efficient estimation of the MP2 pair correlation energies required for the strong pair list construction, we combine domain approximations with the multipole expansion of the two-electron integrals. Specifically, the distant and strong pairs are characterized using multipole approximated pair energies evaluated in the primary domains (PDs) of the LMOs forming the pair. Moreover, the pair correlation energies of the distant pairs are used to estimate their contribution to the total MP2 correlation energy indicate that the corresponding energy contribution is evaluated only in the orbital spaces of the pair’s PDs or that of the ED, respectively. For closed-shell systems, the resulting expression is equivalent to our spin-restricted LMP2 formulation. 3. Local MP2 Algorithm Click to copy section linkSection link copied! The major steps of the present restricted open-shell LMP2 (RO-LMP2) algorithm are collected in Figure 1 and discussed in this section in detail. Figure 1 3.1. Self-Consistent Field Calculation First, a restricted or quasi-restricted open-shell HF or KS reference determinant is obtained for the entire molecule. Then, semicanonical Fock matrices are computed in the AO basis using the (quasi-)restricted electron densities. If any core orbitals are kept frozen, that is, left out of the correlation calculation, then the mixing of those orbitals with the correlated orbitals is avoided during both the semicanonicalization and orbital localization steps. For the cases where the convergence of ROHF/ROKS is problematic, we also implemented quasi-restricted orbitals (QROs) as proposed by Neese. Here, the starting point can be an unrestricted HF/KS (UHF/UKS) solution, which is often simpler to find than the ROHF/ROKS one. However, such UHF/UKS solutions may exhibit considerable spin-contamination, that is, the single determinant wave function does not provide the appropriate + 1) eigenvalue for the square of the spin operator, , with as the spin quantum number. To circumvent this, the QROs are constructed as the eigenvectors of the total density matrix of the unrestricted computation. The orbitals obtained in this way with occupation numbers close to 2, 1, and 0 are selected to be DO, SO, and unoccupied in the QRO determinant, respectively, which becomes an eigenfunction of by construction. Our numerical experience to date is in line with the findings of Neese and co-workers that the QRO determinant provides a reliable reference (when the RO solution is unavailable) with a somewhat higher energy than the corresponding ROHF/ROKS determinant. At the few-hundred-atom range, the fourth-power-scaling HF computation can become a computational bottleneck even in combination with the DF approach. However, it is possible to accelerate the evaluation of the exchange term in DF-HF via local DF (LDF) domains, that is, using a restricted list of auxiliary functions for each LMO lying in its proximity. This LDF approach was extended to both restricted open-shell and unrestricted HF- and KS-SCF in the present work. Additionally, the third-power-scaling of exchange evaluation in the LDF-HF approach can be brought down to asymptotically linear scaling by also restricting the list of AOs appearing in the exchange matrix contribution of each LMO. Most recently, our (L)DF-HF algorithms were further sped up by multipole approximations as well as approximate SCF iterations followed by first-order corrections; however, these improved schemes were not yet employed here. 3.2. Orbital Localization The localization of the reference (quasi-)restricted occupied orbitals can be carried out by the Boys or the Pipek–Mezey scheme, although we found the Boys orbitals to be considerably more suitable for our algorithm. The highly demanding localization of the unoccupied orbitals is not required. Additionally, the localization can be carried out in a spin-unrestricted or a spin-restricted manner. To take advantage of the computational savings offered by the above RO-LMP2 Ansatz, here we localize the (restricted and correlated) DO and SO orbitals separately. A drawback of this approach may emerge when there is only one SOMO in the entire molecule (or when SOMOs cannot mix due to symmetry) because in this case the unchanged SOMO(s) of the canonical basis may remain considerably delocalized. Moreover, the number of SOMOs is anyway smaller than that of the DOMOs leading to fewer degrees of freedom for their localization and consequently a somewhat larger spread of the localized SOMOs. Alternatively, the spin-up and spin-down orbitals can be localized separately, leading to potentially better localized but unrestricted LMOs and twice as many integrals to transform. Thus, in agreement with previous studies, restricted LMOs are employed. 3.3. PAO Construction The LMO optimization is followed by the construction of PAOs, which span the virtual subspace of the entire system defined by Here, \| ⟩ denotes the PAO projected from AO \|μ⟩, and the summation runs over both DO and SO MOs. The PAOs of eq 6 span only the virtual subspace of spin-up electrons because the projection makes the PAOs orthogonal also to the SO subspace. Therefore, the unoccupied subspace of the spin-down electrons is spanned by the union of the above PAOs and all SOMOs. In other words, the SOMOs have a dual role: they are occupied in the spin-up and unoccupied in the spin-down orbital set. 3.4. Pair Energy Calculation The approximate MP2 pair correlation energies are evaluated for each LMO using nearby virtual orbitals residing in the PDs of the LMOs. The PDs are built using our modified Boughton–Pulay (BP) domain construction scheme similar to our closed-shell implementations. Briefly, the BP algorithm selects a compact list of atoms and corresponding AOs so that the overlap of the projection of the input MO onto the selected AOs with the input MO will be larger than the specified threshold ( ). In other words, the MOs projected onto their BP AO list exhibit a well-controlled truncation error of 1 – For the PD construction, the BP atom lists are obtained for each LMO and PAO with completeness criteria of = 0.999 and = 0.98, respectively. The SOMOs are part of both the occupied and virtual subspaces; thus, both of these BP atom lists are assembled for them. The PAO list of the PD contains the PAOs derived from the AOs in the BP domain of the PD’s LMO. Additionally, if a BP list of a SOMO obtained with the criterion overlaps with the BP list of the LMO, then this SOMO is appended to the spin-down PAO list of the PD. The atom and AO lists of the PD are obtained as the union of the BP lists of all MOs (LMO, PAOs, SOMOs) of the PD. The MOs of the PD are projected onto the AO basis of the PD, and the PAOs (and potential spin-down SOMOs) are orthogonalized within this truncated AO basis, leading to different spin-up and spin-down MOs. Then, for the noniterative evaluation of the MP2 pair energies, the PD’s virtual space is canonicalized separately for the spin-up and spin-down MOs. Finally, the multipole approximated opposite-spin MP2 pair correlation energies are evaluated as Here, subscript of the virtual orbitals indicates that the virtual indices run over the virtual subspace of the PD of the corresponding LMO. Furthermore, denotes the pseudocanonical orbital energy of orbital , and ) is the diagonal element of the spin-up (spin-down) Fock matrix. The ERIs of written in the Mulliken notation are obtained using the multipole expansion up to the fourth order, that is, including terms with dipole–dipole, dipole–quadrupole, quadrupole–quadrupole, and dipole–octopole moments. With that, the LMO pair of is classified as a strong pair if ; otherwise, the pair is treated as a distant pair, and its contribution is added to the final MP2 correlation energy (see eq 5 ). Here, ε is the same strong pair threshold employed in our methods previously, is a scaling factor introduced for the following reasons. Let us consider the case when one LMO of the pair, say , is SO. Then, the second and third terms of eq 7 vanish, and therefore, all such pair correlation energies contain only half as many terms compared to the pair energy of two DO LMOs. Furthermore, if both are SO, then only the first term of eq 7 survives, leading to 4 times less terms contributing to an SO–SO pair than to a DO–DO pair. In accord with this consideration, on average, we find the SO–DO (SO–SO) pair correlation energies to be twice (four times) as small as those of DO–DO pairs. To handle the strong/distant pair characterization of all pair types on an equal footing, factors of 1, (1/2), and (1/4) are employed for the DO–DO, DO–SO, and SO–SO pairs, respectively. The numerical properties of this scaling are analyzed in Section 5.1 . We note that a similar scaling factor of (1/3) is introduced in ref in the DLPNO context for pairs involving at least one SOMO. On systems with unusually large numbers of SOMOs, the factor of (1/3) provided better numerical performance than 0.1 or 0.5. This could be explained by the fact that, for the systems explored in ref , (1/3) is the closest to the weighted average of (1/2) and (1/4) recommended here. The closed-shell limit of this pair energy expression matches the formulae used in our closed-shell LMP2 method. However, due to the fact that the MP2 pair energy is evaluated on an unrestricted basis, the computational requirement is somewhat higher. In practice, this does not pose a computational bottleneck as the multipole-based MP2 pair energy calculation is very efficient compared to the remaining steps of the algorithm. 3.5. Extended Domain Construction The main correlation energy contribution (first term on the right side of eq 5 ) of each LMO is evaluated in LMO-specific EDs of asymptotically constant size to ensure the linear scaling of this step. The ED construction scheme closely follows our algorithm developed for closed-shell systems; thus, only a brief summary is provided here focusing on the modifications required for open-shell systems. The occupied space of the ED consists of the CMO and all of its strong pair LMOs. The atom list of the ED is the union of the BP atom lists obtained with a BP completeness criterion of = 0.9999 for all LMOs of the ED. The AOs on these atoms form the AO basis of the ED. The LMOs are projected onto the AOs of their respective BP atom lists ensuring at most 1 – truncation error and are then reorthogonalized. The virtual space of the ED is spanned by restricted PAOs originating from atoms of the PAO center domain (PCD) of the ED. The PCD is the union of the more compact BP atom lists of all LMOs in the ED obtained with = 0.985. Since the PAOs tend to be more delocalized than the LMOs, they are projected onto the whole AO basis of the ED. Analogously to the case of the PD construction, the SO LMOs of the ED are appended to the spin-down unoccupied MOs of the ED. The specific combination of the Gram–Schmidt and Löwdin algorithms is employed for the orthogonalization of the virtual space of the ED analogously to our previous approach. Finally, pseudocanonical and hence unrestricted occupied and virtual orbitals are obtained for the iteration free MP2 energy formulae of the ED. 3.6. Integral Transformation in the Extended Domain The correlation energy computation in the ED is accelerated using the density-fitting (DF) approximation. The required antisymmetrized two-electron integrals are denoted as and the ERI tensors are factorized as = ( ) denotes three-center two-electron integrals, and refers to the auxiliary basis functions. The two-center integral matrix = ( ) is subjected to Cholesky decomposition ( ) yielding the tensor. We showed in ref that the auxiliary basis functions residing on the atoms of the PCD can accurately expand all LMO–PAO orbital product densities of the ED; thus, the auxiliary function list of the ED is chosen The integral-direct ERI transformation algorithm proceeds as follows collect the occupied and virtual MO coefficients discussed in Section 3.5 . First, the ( ) AO integrals are evaluated for a shell triplet at a time using a highly optimized three-center two-electron AO integral code only for the AOs and auxiliary functions of the ED. These batches are immediately subjected to the first transformation of scheme eq 10 , leading to half-transformed integrals with one index in the restricted LMO basis and then discarded. This integral-direct approach effectively makes use of the available memory and data traffic bandwidth between the lower levels of cache and the CPU. The evaluation and first transformation of the three-center ERIs are the most computationally demanding operations in our LMP2 scheme and can be performed at a similar cost as in the closed-shell implementation because restricted LMOs are employed. The introduction of this intermediate step transforming to the restricted LMO basis is thus more effective than transforming from the AO basis directly to the semicanonical occupied basis. The latter, restricted LMO to semicanonical MO transformation is performed much more efficiently as the final step of scheme eq 10 . Before that, however, it is beneficial to decrease the number of operations by performing the AO-to PAO transformations (second step of scheme eq 10 ). Note that the number of integrals entering the second half-transformation is considerably lower than in the first step. Consequently, there is no motivation to perform the AO-to-PAO transformation in two steps by making use of the restricted PAO basis unlike in the case of the first half-transformation. In conclusion, the three-center ERIs are thus transformed to the spin-up and spin-down ED MO bases in a cost comparable to that of the closed-shell alternative. 3.7. Energy Contribution in the Extended Domain Let us first note that the MP1 amplitudes appearing in the ED’s correlation energy expressions of eqs 3 are required only for a fixed ′ or ′ index. Thus, we recommended circumventing the redundant evaluation of MP1 amplitudes via CD or LT techniques. The benefit is that, by factorizing the energy denominators, we can directly evaluate the amplitudes with mixed restricted LMO and semicanonical ED MO indices, e.g., Here, ω labels the summation index over the Cholesky vectors or integration quadrature weights used in the LT. Since the doubles amplitudes can have both spin-up and spin-down indices, it is more beneficial to obtain spin-independent Cholesky vectors or quadrature points. For instance, in the case of LT, this is achieved using the range [min(ε , ε ), max(ε , ε )] to determine the weights ( ) and quadrature points ( ). Then, the integrals of eq 11 can be constructed, e.g., as denotes the elements of the Cholesky vector, or in the case of LT Utilizing these integrals, the energy denominator free expression of eq 11 can be directly written down with obtained from via the unitary transformation of the occupied MO index. Let us note that our original closed-shell LMP2 algorithm employed an additional, so-called natural auxiliary function (NAF) approximation to compress the auxiliary function space of the EDs. To simplify the Ansatz, the NAF approximation is not invoked in the EDs in our most recent closed-shell LMP2 approach. NAFs are only employed in combination with natural orbitals for our LNO-CC methods. For the sake of compatibility, the open-shell extension of the NAF approach is not employed here at the MP2 level. The remaining amplitudes with the other three spin cases ( , and ) are evaluated analogously using the appropriate spin cases of the tensors. Finally, the RO-LMP2 energy contribution of orbital can be evaluated in its ED as Here, we exploit the permutational symmetries of in the second and fifth terms; thus, the evaluation of eq 14 takes about three times more operations than its closed-shell analogue. 3.8. Contribution of Single Excitations Special attention has to be devoted also to the energy contribution of single excitations (first and fourth terms of eq 14 ). These contributions appear because the presented Ansatz assumes a reference determinant with unrestricted orbitals, but an ROHF/ROKS/QRO reference is employed instead of UHF. Consequently, the occupied-virtual block of the complete molecule Fock matrix written in the basis of the semicanonical MOs is nonzero even without any of the above local approximations. Additionally, the truncation of the MOs in the EDs would result in a second contribution to eq 14 . The reason for that is a small contamination of the projected occupied (virtual) orbitals of the ED from the virtual (occupied) subspace spanned by the untruncated MOs of the entire system. In the closed-shell context, we found this source of error small and well-controlled by the BP completeness criteria governing the truncation of the ED’s LMOs. Previously, it was found best not to include these artificial off-diagonal Fock-matrix contributions into the ED’s correlation energy contribution. However, this strategy is more challenging to follow for the open-shell case because one cannot simply discard the correlation energy contributions of the single excitations. To maintain the exact MP2 energy as the approximation-free limit of the present local scheme and to handle the off-diagonal Fock-matrix contributions comparably to the closed-shell case, the two effects are separated as follows. Let us recognize that, if we consider the Fock matrix in the MO representation, the majority of the above nonorthogonality error would originate from its dominating diagonal elements. However, only the off-diagonal occupied-virtual block is required for the correlation energy expression. Therefore, we build the quantities in each ED only from the off-diagonal part of the original semicanonical Fock matrices. The latter are computed in the AO basis at the end of the complete molecule SCF computation as are the complete Fock matrix and its off-diagonal part in the AO basis, respectively, holds the unrestricted MO coefficients, and is a matrix with the corresponding orbital energies on its diagonal. The benefits of storing the additional (spin-up and spin-down) matrices are illustrated with the example of vitamin E succinate (see Section 4.2 ). Using the complete to compute the first and fourth terms of eq 14 would result in a 124% relative error in the singles contribution or in a 0.1% relative error with respect to the total correlation energy. Compared to that, replacing in the calculation of the matrices, the error in the singles contribution reduces to 0.01%, which is negligible from the perspective of the total correlation energy. For clarity, the complete Fock matrices are employed everywhere else in the algorithm, such as for the semicanonicalization of PD or ED orbitals. The use of is limited to the energy contribution of single excitations. Let us also note that the energy contribution of single excitations is omitted from the second-order contribution of DH density functionals to ensure compatibility with the conventional 3.9. Approximate Long-Range Spin Polarization Here, we analyze the spin-polarization effect of the SOMOs on the MP1 amplitudes of the EDs. The present Ansatz employs unrestricted amplitudes where the contributions of spin-up and spin-down MOs to the correlation energy differ because of their different interactions with the spin-up SOMOs. The reason for that is the construction of unrestricted semicanonical MOs in the EDs even if the ED’s amplitudes are otherwise computed independent of each other, that is, they are not coupled. This spin-polarization effect takes place in all EDs, which include at least one SOMO, and thus we take this effect into account in its full extent. The other case when the ED does not contain any SOMO is of more interest here. In these “doubly occupied MO-only” EDs, the equivalence of the spin-up and spin-down MOs originating from the restricted LMOs and PAOs of the ED is split only because their semicanonicalization in the ED is performed with the respective spin-dependent Fock matrices. In other words, in these EDs, there is no direct mixing between the SOMOs and the spin-up MOs of the doubly occupied space of the ED upon canonicalization. Note that the most important second-order contribution to the long-range and spin-polarized interaction of the SOMOs and the CMOs of such doubly occupied MO-only EDs is already taken into account via the distant pair correlation energy terms. What remains in such domains is a secondary spin-polarization effect caused by the interaction with the SOMOs through the spin-dependent Fock matrices resulting in the splitting of the orbital energies of the ED’s MOs. However, when such CMOs are distant pairs with all SOMOs, we expect that the magnitude of this long-range effect decreases rapidly. An option has been implemented to exploit the long-range decay of spin polarization. When this is activated, the EDs without any SOMOs are treated as closed-shell subsystems, and their LMP2 energy contributions are calculated using the closed-shell formulae. This requires the introduction of an approximation: in these doubly occupied MO-only domains, the canonicalization step is performed with the average of the spin-up and spin-down Fock matrices projected onto the ED. We note in passing that alternatively, the MP1 wave function could also be spin-adapted, leading to a different Ansatz, but the applicability thereof in combination with the present local CD/LT techniques is yet to be explored. The benefit of the introduced approximation is that the ED’s canonical MOs remain restricted, and the complete ED computation can be performed using the closed-shell algorithm. Consequently, the memory requirements of such EDs can be cut in half, and the operation count needed for the doubles amplitude evaluation can be decreased by about a factor of three. Moreover, our numerical experience presented in Section 5.4 shows that this long-range spin-polarization effect can indeed be approximated with negligible loss of accuracy. 3.10. Scaling of the Algorithm The computational requirement of the presented open-shell approach is only moderately higher than that of the analogous closed-shell one, achieving for the rate-determining steps asymptotically linear scaling with system size. To verify this, the runtimes of the fifth-power scaling DF-MP2 and the present LMP2 methods were measured for quasi-linear [Th-(CH diradicals, where Th denotes thiophene rings attached to the end of the alkane chains. Detailed timing data are presented in Section S1 of the Supporting Information. In these measurements, canonical DF-MP2 exhibited an -scaling, which is somewhat lower than its formal -scaling. This can be understood as the most time-consuming step is still the -scaling integral transformation even for the largest chain. In comparison, the LMP2 algorithm exhibits clear linear scaling, which sets in already for the smallest systems. Because of the redundancy-free evaluation of the LMP2 amplitudes, the DF-MP2 and LMP2 calculations take comparable time only up to about 50 atoms followed by the clearly superior performance of LMP2 for larger For the sake of completeness, we note that the PAO construction and the multipole-based pair energy evaluation currently scale with the third and second powers of the system size, respectively, but constitute only a few percent of the total runtime even for our largest examples. The same observation can be made for the cubic-scaling localization of the occupied orbitals. The most computationally demanding part of an open-shell LMP2 calculation is thus the SCF calculation, which can also be accelerated to cubic scaling via local approximations as discussed in Section 3.1 The memory requirement of the algorithm is even closer to its closed-shell analogue. The open-shell LMP2 program requires the storage of six matrices with dimensions equal to the number of basis functions. In comparison, the preceding SCF procedure needs eight such matrices. Moreover, all arrays related to the EDs are asymptotically constant in size, and thus the memory requirement of the open-shell LMP2 algorithm is again lower than that of the preceding SCF calculation, just as for our closed-shell implementation. 4. Computational Details and Test Systems Click to copy section linkSection link copied! 4.1. Technicalities The presented RO-LMP2 approach is implemented in the suite of quantum chemical programs and will be made available in a forthcoming release of the package. The default or threshold values controlling the accuracy of the local approximations are collected in Table S1 of the Supporting Information. These settings correspond to the threshold combination employed currently in the closed-shell LMP2 approach, which are the tighter settings introduced in ref The performed calculations utilize the split valence and triple-ζ valence basis sets including polarization functions (def2-SVP and def2-TZVP) developed by Weigend and Ahlrichs, Dunning’s (augmented) correlation-consistent polarized valence basis sets [(aug-)cc-pVXZ, X = D, T, and Q], and for third-row atoms, the revised (aug-)cc-pV(X + d)Z basis sets were also employed. The corresponding auxiliary basis sets of Weigend et al. were used for all AO bases. Extrapolations toward the complete basis set (CBS) limit were performed according to standard formulae for both the HF and correlation energies. The DF approximation was employed in all HF and reference canonical MP2 calculations. The evaluation of the exchange contribution in the HF calculations was accelerated by utilizing local fitting domains as implemented in the package (see Section 3.1 ) for systems containing more than 500 atoms. The Boys localization scheme was chosen for the construction of the LMOs in each presented LMP2 calculation. The core electrons, including the subvalence electrons for the iron and cobalt atoms, were kept frozen in the correlation calculations. The energy denominators of the EDs were factorized via Cholesky decomposition, with an automatically determined number of Cholesky vectors such that the diagonal elements of the residual matrix were less than 10 The statistical measures utilized for accuracy characterization are the maximum absolute error (MAX), mean absolute error (MAE), and the standard deviation of the absolute error (STD), the latter measuring the consistency of the errors. Relative energy differences with respect to a reference energy (E[DF-MP2]^c) are obtained as (100%)·(E[LMP2]^c – E[DF-MP2]^c)/E[DF-MP2]^c. The presented wall-clock times were measured with an 8-core 3.0 GHz Intel Xeon E5-1660 and a 20-core 1.3 GHz Intel Xeon Gold 6138 CPU. 4.2. Benchmark Sets and Test Systems The RO-LMP2 correlation energies are benchmarked on three test sets composed of small to medium-sized open-shell molecules with an average (maximum) system size of 11 (23) atoms. The first test set collects 30 radical stabilization energies (RSE30) and is a 30-species selection from the RSE43 compilation as defined in ref and reoptimized in ref . Furthermore, 21 adiabatic ionization potentials of organic species (IP21) are considered for systems of ref . The structures of the neutral systems were taken from ref , while the geometries of the ions were optimized using unrestricted B3LYP with the cc-pV(T + d)Z basis (see the Supporting Information). Finally, a set of 12 singlet–triplet energy gaps of aryl (AC12) was also investigated. Five processes involving larger open-shell systems of 42–81 atoms were also selected for the accuracy assessment. These are the radical stabilization of vitamin E succinate, the singlet–triplet energy gap of artemisinin (structures taken from ref ), and the vertical ionization potential of testosterone, borrelidin, and glutathione (taken from ref ). The corresponding structures are depicted in Figure 2 Figure 2 Large-scale calculations were carried out for a three-dimensional iron(II) complex of 175 atoms in its quintet and triplet spin state (see Figure 3 ). Additionally, a homolytic bond-breaking reaction involving the coenzyme B (5′-deoxyadenosylcobalamin, dAdoCbl) with open-shell systems of up to 179 atoms [the Cob alamin (Cbl) radical] was also considered Figure 3 Figure 3 To demonstrate the current capabilities of our LMP2 method, calculations for even larger systems were carried out for a 565-atom model of bicarbonate in photosystem II (PSII) Figure 9 ) and for a 601-atom model of the -amino acid oxidase (DAAO) Figure 4 Figure 4 Following the recent mechanistic study of Kiss and Ferenczy, two steps are taken from the DAAO-catalyzed oxidation of -alanine along the oxidative half-reaction. As illustrated in Figure 5 , the reduced form of the flavin moiety of the flavin adenine dinucleotide (FAD) cofactor is reoxidized by O . The diradical reactant state of Figure 5 results from a single electron transfer from reduced FAD to O , leading finally to the oxidized form of FAD and H . Models of the corresponding triplet and singlet states of the structures labeled by O1 and O3 in ref are provided in the Supporting Information Figure 5 The bicarbonate system of PSII contains an iron(II) center for which the SCF computations were found complicated for both the quintet and triplet spin states (see Section S3 of the Supporting Information). Satisfactory UHF-based QRO references were obtained using the def2-TZVP basis set, as well as a mixed basis set labeled by def2-SVP’, which includes def2-SVP for all atoms, except for the def2-TZVP basis used for the Fe atom. 5. Accuracy of the Local Approximations Click to copy section linkSection link copied! The truncation threshold dependence of the RO-LMP2 approach is documented in this section compared to approximation-free DF-MP2 references showing the systematic convergence of the introduced local approximations. The majority of the approximations have been extensively benchmarked in our related studies on closed-shell systems. Therefore, convergence tests illustrating individual approximations focus on the two parameters (ε ) responsible for the bulk of the local error. Open-shell-specific approximations, which did not appear before, are also thoroughly benchmarked. For the remaining truncation parameters, which affect the closed- and open-shell systems similarly, such as the BP parameters of the PDs or the order of multipole expansion, the previously assessed values are adopted. Note that such approximations are also active and hence tested in the benchmarks of Section 6 5.1. Strong Pair Classification As discussed in Section 3.4 , the pair correlation energy expression of eq 7 does not contain an equal number of nonzero terms for orbital pairs involving different numbers of SOMOs. To handle the strong/distant pair classification of the DOMO–SOMO and SOMO–SOMO pairs on an equal footing with that of the DOMO–DOMO pairs, we propose to scale the pair energy threshold (ε ) by factors of (1/2) and (1/4) for the pairs including one or two SOMO(s), respectively. The numerical behavior of this approach is illustrated in Figure 6 , which plots pair correlation energy contributions as a function of the real-space distance between the centers of LMOs . The values are collected from multiple systems containing two methyl carbene species placed at varying distances from each other, with both methyl carbene subsystems being in their local triplet state. The left panel, collecting unscaled pair energies, illustrates that pairs involving different numbers of SOMOs gather into three distinct clusters of points. This verifies our expectation that for pairs with comparable orbital center distances, smaller pair correlation energies are obtained for SO–SO or DO–SO pairs than for DO–DO pairs. Consequently, the curves of the three groups of unscaled pair energies intersect the default pair energy threshold (dashed horizontal line) at different distances. This reveals a potential bias in the strong/distant pair classification of pairs involving SOMOs. However, our goal is to ensure comparable classification for all pairs exhibiting a similar pair distance or interaction strength regardless of their occupation. To that end, we examine the distance dependence of the same pair correlation energies scaled by , that is, by 2 and 4 for the DO–SO and SO–SO pairs, respectively. This emulates the use of the strong pair threshold instead of ε . The resulting scaled pair energies collected in the right panel of Figure 6 indeed exhibit the same trend for all three types of pairs independent of the occupation. Another beneficial consequence of using the scaled pair threshold is that the chance of including the SOMOs in the EDs increases. These SOMOs often play an important role in the chemical processes of open-shell species, and therefore, their improved description is advantageous. Figure 6 5.2. Strong Pair Selection Here, we assess the convergence of the LMP2 correlation energy toward the canonical DF-MP2 reference as a function of the pair energy threshold (ε[w]). To that end, LMP2 calculations are performed in which all local approximations are turned off except for the strong pair criterion of the ED construction. The approximations governed by this threshold are negligible for small systems and start to operate to a considerable extent for larger molecules. Besides the correlation energies of such extended systems (42–81 atoms), the accuracy of three different kinds of relative energies is also assessed: the vertical ionization potential (VIP) of testosterone, the radical stabilization reaction energy (RSE) of vitamin E succinate, and the singlet–triplet (S–T) gap for artemisinin. The basis set of aug-cc-pVTZ is used for all species so that the tests will be performed with a large basis set including diffuse functions sufficient for realistic applications. Diffuse AOs are more challenging to handle for local approximations, and consequently, such AOs cannot be omitted in representative convergence tests. The relative errors of LMP2 correlation energies obtained for the open-shell species (left panel) and the corresponding energy difference deviations (right panel) are depicted in Figure 7 as a function of ε . Rapid convergence is observed for all cases, similar to previous findings on closed-shell systems. The energy differences are practically converged already at the default ε = 10 setting, and the largest error of 0.05 kcal/mol is negligible compared to the 217 kcal/mol VIP of testosterone. The corresponding correlation energies are also accurate up to 0.03% relative errors with this default threshold. Figure 7 Note that this default value of ε = 10 corresponds to the tighter settings employed in ref , and it has been employed also as default in the context of our LNO-CC approaches and also with the LMP2 scheme since 2018. The strong pair selection and ED construction controlled by ε = 10 were found to be similarly accurate previously for a number of alternative systems containing up to 260 atoms and for various reaction and interaction energies involving closed-shell systems. 5.3. Representation of the LMOs The second most important threshold determining the tightness of the local approximations is the BP criterion governing the completeness of the LMOs in the ED (T[EDo]). Together with ε[w], these two thresholds also determine the number of atoms, AOs, and the truncation errors of the MOs in the ED. The convergence tests for the parameter are performed for the same open-shell species and energy differences as used in Section 5.2 for ε . Again, only the local approximation corresponding to was active, and all other approximations were turned off to separate the effect of The relative correlation energy (left panel) and energy difference (right panel) deviations of Figure 8 again reveal rapid convergence with increasing toward the DF-MP2 reference. Both the correlation energies and the energy differences are converged already at = 0.9999 (1 – = 10 Figure 8 ), which is chosen as default. We note again that this value corresponds to the tighter setting introduced in ref , and it is chosen as default also in our recent closed-shell LMP2 as well as LNO-CC methods. Figure 8 5.4. Assessment of the Long-Range Spin-Polarization Approximation The long-range spin-polarization approximation of Section 3.9 is evaluated both on correlation energies and on energy differences with respect to LMP2 references obtained without this approximation. The approach is only active in EDs, which do not contain any SOMOs as strong pairs of the ED’s CMO. Thus, reasonably large systems have to be considered for this test to properly activate the long-range spin-polarization approximation. Accordingly, seven correlation energies and five energy differences (reaction energies, spin-state splittings, and one RSE) are benchmarked in Table 2 for systems containing 81–601 atoms. The test cases include reactions that also involve closed-shell species. For such cases, error compensation between the reactants and products cannot occur for this particular source of error because the long-range spin-polarization approximation affects only the open-shell species. error in energy difference atoms LMOs E[LMP2]^c error [%] [cal/mol] [%] EDs without SOMOs [%] vitamin E succinate 81 89 7.8 × 10^–7 0.027 2.8 × 10^–4 54 FeC[72]N[2]H[100] ^5A 175 205 1.7 × 10^–5 0.66 1.4 × 10^–3 54 ^3A 204 8.5 × 10^–6 54 Cbl radical 179 250 7.7 × 10^–6 0.81 1.6 × 10^–3 68 bicarbonate ^5A 565 789 1.2 × 10^–4 3.5 8.7 × 10^–3 91 ^3A 788 1.1 × 10^–4 92 DAAO 601 838 2.3 × 10^–7 0.078 2.6 × 10^–4 76 See the text for explanation. The last column of Table 2 collects the ratio of EDs without SOMOs, that is, the ratio of EDs affected by the approximation. Even for the smaller vitamin E succinate system, 54% of the EDs can be treated with the more efficient closed-shell formulation, while for the spin state of bicarbonate, more than 90% of the EDs are built without SOMOs. In light of the relatively large number of EDs where the approximation is activated, the relative correlation energy errors of about 10 % for all cases are surprisingly small. This error range is comparable to or even better than that of any other employed approximation, including the DF approach. Consequently, most of the energy differences are also practically unaffected by this approximation being below 1 cal/mol for all but one example. One should also note the key role of the SOMOs in the considered reactions, ionizations, and spin-state splittings as opposed to different possible processes occurring far from the SOMOs. This suggests that any severe approximation to the spin-polarization effects would be indicated by the investigated energy differences. Interestingly, the quality of the approximated energy differences is similar for the systems of Table 2 even if closed-shell species are also involved (cf., vitamin E succinate RSE, the formation of dAdoCbl from the Cbl radical, and the DAAO reaction). The case of bicarbonate is somewhat an outlier in Table 2 ; however, the error of 3.5 cal/mol (or 0.0087%) observed in the spin-state splitting is still satisfactory, especially if considering that more than 90% of the ED contributions are approximated. It is also interesting to point out that the bicarbonate system is the only one where we had to rely on QROs due to lack of a converged ROHF reference. While the QRO approach also provides an eigenfunction as the reference, the QRO reference energy, and potentially also the corresponding unrestricted Fock-matrix elements, may differ from the completely variationally optimized UHF solution more than the analogous ROHF-based quantities. Therefore, the approximation of QRO-based unrestricted Fock-matrix elements by spin-averaged ones may affect the interaction of the bicarbonate’s SOMOs with the rest of the DOMOs in a somewhat more pronounced manner. The spatial distribution of the EDs in which the approximation is active is visualized for the quintet state of bicarbonate in Figure 9 . Green spheres denote the centers of LMOs without a strong SOMO pair (that is, without any spin-dependent interaction in their EDs), whereas purple spheres label the centers of LMOs having at least one strong pair involving a SOMO. Clearly, the EDs including at least one SOMO, in which complete open-shell treatment is required, are clustered around the Fe(II) ion, where all four SOMOs are localized. In this particular case, the SOMOs are located near the edge of the protein system; thus, the long-range spin-polarization approximation can be employed for over 90% of the EDs. Figure 9 6. Benchmarks for Small and Medium-Sized Systems Click to copy section linkSection link copied! The accuracy of RO-LMP2 correlation energies and energy differences is also benchmarked against canonical DF-MP2 references. The corresponding reference data is provided in the Supporting Information. First, statistical measures are presented for three test sets containing IPs, RSEs, and spin-state energy differences for molecules of small to medium size. Next, the accuracy is assessed also on a set of larger systems of up to 175 atoms to explore the behavior of the employed approximations with increasing system size. 6.1. Accuracy of Correlation Energies The accuracy of the open-shell LMP2 correlation energies using the default settings were benchmarked on the RSE30, IP21, and AC12 test sets, containing 128 species of up to 23 atoms, thereby allowing for the statistical analysis of the correlation energies compared to the approximation-free DF-MP2 reference. The accuracies of the LMP2 correlation energies for the RSE30, IP21, and AC12 compilations are highly satisfactory (see Tables 3 , respectively). The relative deviations in the correlation energies (third column of these tables) are in all cases below 0.05% and are lower than 0.02% for all species in the RSE30 and IP21 sets using basis sets of various qualities as well as CBS extrapolation. The corresponding MAEs of at most 0.004% for the RSE30 and IP21 and the MAE of 0.03% for the AC12 set are also excellent. The largest errors are found for the AC12 test set with the cc-pVDZ basis set in accordance with the observation that the employed local approximations perform best for sufficiently flexible, at least triple-ζ-quality basis sets. Furthermore, these somewhat larger deviations in the correlation energies are similar for both the triplet and singlet states of the AC12 set, leading to highly accurate singlet–triplet energy gaps Section 6.4 ). The observed STD values being comparable to or even smaller than the MAE measures for all three test sets also indicate well-balanced correlation energy errors, which is beneficial for reliable energy differences. We also find that the CBS-extrapolated LMP2 energies of Table 3 maintain the accuracy of the LMP2 energies obtained with triple- and quadruple-ζ basis sets similar to our experience with closed-shell systems. basis error measure error in E[LMP2]^c [%] error in RSE [kcal/mol] aug-cc-pV(T + d)Z MAX 0.014 0.041 MAE 0.003 0.010 STD 0.003 0.011 aug-cc-pV(Q + d)Z MAX 0.016 0.065 MAE 0.003 0.029 STD 0.004 0.012 CBS(T,Q) MAX 0.017 0.097 MAE 0.004 0.055 STD 0.005 0.020 basis error measure error in E[LMP2]^c [%] error in IP [meV] aug-cc-pV(T + d)Z MAX 0.016 2.03 MAE 0.004 0.47 STD 0.005 0.60 basis error measure error in E[LMP2]^c [%] error in S–T gap [kcal/mol] cc-pVDZ MAX 0.04 0.13 MAE 0.03 0.06 STD 0.01 0.04 cc-pVTZ MAX 0.04 0.13 MAE 0.02 0.05 STD 0.01 0.04 The relative LMP2 correlation energy deviations are collected in Table 6 for larger systems of 37–175 atoms. The relative deviation remains around the 0.05% mark for almost all entries of Table 6 , matching the largest errors obtained for the smaller and simpler systems. The maximum error of 0.11% is obtained for both the quintet and triplet states of the largest FeC complex, but again this consistency leads to a negligible error in the spin-state splitting. Considering that the average system size increases by about 10 times when stepping from smaller to larger systems, the size dependence of the relative accuracy also appears excellent well above the size range where all approximations start to operate to their full extent. molecule atoms no. of AOs E[LMP2]^c error [%] ΔE error [kcal/mol] time^b [min] glutathione ion 37 1320 0.05 –0.05 21 artemisinin ^3A 42 1426 0.04 –0.05 70 testosterone ion 49 1610 0.05 –0.01 77 borrelidin ion 78 2599 0.08 0.12 256 vitamin E succinate 81 2553 0.05 0.01 89 [Th–(CH[2])[50]–Th]^2+ 166 2508^c 0.05 5 FeC[72]N[2]H[100] ^5A 175 2939^c 0.11 0.005 180 ^3A 0.11 186 Unless otherwise noted, the calculations were carried out with the aug-cc-pV(T + d)Z basis set. Wall-clock times measured on an 8-core 3 GHz Intel Xeon E5-1660 processor. The def2-TZVP basis set was utilized. All in all, the accuracy of the RO-LMP2 correlation energies closely matches that of closed-shell LMP2 correlation energies presented previously for a large number of closed-shell systems both in the smaller (<36-atom) and in the larger (up to 260-atom) size range. The benchmarks presented here and in ref for the entire size range accessible for efficient DF-MP2 implementations indicate that highly reliable LMP2 correlation energies can be expected consistently for both open- and closed-shell systems. To illustrate the accuracy along a full potential energy surface (PES), an example was adopted from ref , where the rotational barrier of ethane-1,2-diphenyl was studied using our closed-shell LMP2 approach (see Figure 2 of the Supporting Information). Here, a single hydrogen atom is removed from one of the phenyl rings to make the comparison to our previous test feasible (see Section S5 of the Supporting Information for more details). Structures at the two edges of the PES differ significantly: the phenyl groups interact weakly in the trans conformation but exhibit stronger π–π interaction in the cis arrangement. In agreement with our experience on the closed-shell analogue, we find the deviations with respect to the exact ROMP2 reference on the PES comparable to the error established above. Explicitly, the relative error varies in the narrow range of 0.014–0.029% across the PES with an MAE of 0.021%. 6.2. Radical Stabilization Energies The radical stabilization reactions investigated in this section are taken from the RSE30 compilation and can be written as where R denotes various radicals containing C, N, O, F, P, and S atoms. The MAEs of the LMP2 RSEs collected in Table 3 are below 0.03 kcal/mol for the aug-cc-pV(X + d)Z basis set with both X = T and X = Q, while the CBS extrapolation slightly increases the MAE to 0.05 kcal/mol. The corresponding MAX errors of 0.04, 0.06, and 0.10 kcal/mol at the triple-ζ, quadruple-ζ, and CBS(T,Q) levels, respectively, are still well within the intrinsic accuracy of MP2. The STD values of 0.01–0.02 kcal/mol underline the reproducibility of the excellent accuracy. One can also compare the accuracy of the present LMP2 results to those obtained with PNO-ROMP2 in ref for the same structures and with the same aug-cc-pV(T + d)Z basis set. The two approaches perform similarly well; in terms of the MAX and MAE measures compared to the respective references, LMP2 is somewhat more accurate than PNO-ROMP2 and slightly worse than the explicitly correlated PNO-ROMP2 variant. For this test set, the SCS-LMP2 energies were also assessed (see Table S4 of the Supporting Information) to demonstrate that the accuracy of the local approximations is consistent also for spin-scaled MP2 methods. As expected, the accuracy of both the SCS-LMP2 correlation and reaction energies matches that of LMP2; in fact, the SCS-LMP2 results are slightly but consistently better. The same trend was also observed for closed-shell systems and can be understood from the theoretical perspective because the approximations do not distinguish between the same and opposite spin terms of the SCS scheme. 6.3. Ionization Potentials The accuracy measures of ionization potentials are collected in Table 4 . Compared to the RSE30 compilation in Table 3 , both the correlation energies and the IPs obtained for the IP21 set are almost identically accurate. This excellent performance can partly be attributed to the fact that the reactants and products of the radical stabilization reactions, as well as the natural and ionized structures of the ionization processes are relatively similar and therefore some cancellation of local errors can occur. One major difference is, however, that the IPs lying in the range of about 8–14 eV (184–323 kcal/mol) are significantly larger than the RSEs. Thus, the relative deviations of the IPs compared to those of the RSEs are considerably better. Similar to the case of the RSEs, the LMP2 IP deviations are again almost twice as small as the corresponding PNO-ROMP2 errors of ref with the same basis set, and LMP2 performs almost as well as the explicitly correlated PNO-ROMP2 method. 6.4. Singlet–Triplet Energy Gaps The energy gaps between the singlet and triplet spin states are also benchmarked for 12 aryl carbenes of 13–23 atoms taken from the AC12 compilation. Inspecting the numerical data of Table 5 , the accuracy of LMP2 for S–T gaps is found similarly gratifying as for the RSEs and IPs. The MAE (MAX) measures of 0.06 (0.13) kcal/mol corresponding to the S–T gaps are again well within both the chemical accuracy and the intrinsic accuracy of MP2. It is worth noting the small improvement in accuracy observed for the more suitable cc-pVTZ basis set. Benchmark calculations were also performed for the AC12 set using the B2PLYP functional to demonstrate the accuracy of DH density functionals approximated via our LMP2 scheme. Here, we denote the resulting method as LB2PLYP, highlighting that the second-order correlation energy contribution of B2PLYP is replaced by a corresponding LMP2 term evaluated with Kohn–Sham orbitals. Since the weight of the second-order correlation energy contribution is 0.27 for the B2PLYP functional, the accuracy of the LB2PLYP gaps is expected to be even better than that of the LMP2 gaps because the local errors are also scaled by 0.27. The numerical data of Table S9 of the Supporting Information verifies this expectation. The LB2PLYP S–T gaps are indeed found to be at least times more accurate, exhibiting 0.01 (0.04) kcal/mol MAE (MAX) values for both basis sets. In other words, the presented local approximations operate similarly well with HF and KS orbitals in accordance with our experience for the closed-shell local DH density functional theory (DFT) variants utilizing the LMP2 method. Consequently, the LMP2 algorithm may greatly accelerate the most demanding steps in many DH density functionals with a negligible loss of accuracy. 6.5. Energy Differences for Larger Systems Large-scale benchmark calculations are also presented for systems of 37–175 atoms using sufficiently large AO basis sets [aug-cc-pV(T + d)Z and def2-TZVP]. These molecules represent more faithfully the expected targets of LMP2 in practice. Furthermore, by observing potential trends in accuracy with increasing system size, one can reasonably estimate the expected deviations for even larger systems for which a reference DF-MP2 calculation becomes unfeasible. The test cases are selected so that both the pair and the domain approximations can take effect, and the domain sizes already saturate for the largest two examples. The six energy differences collected in Table 6 include three IPs of the three ions, an RSE for vitamin E succinate, and two spin-state energy differences for artemisinin and the FeC complex. It is reassuring that none of the RSE or spin-state gap errors exceed the corresponding MAEs obtained for the same properties but with much smaller systems. Regarding the IPs, only the still highly acceptable 0.12 kcal/mol error of borrelidin exceeds the inaccuracies obtained for the IP test compilation. Thus, as expected from the underlying accurate LMP2 correlation energies, we do not find any increase in the inaccuracy of the inspected energy differences in spite of the considerable growth in system size. Moreover, except for the Q–T gap of the FeC[72]N[2]H[100] complex, the remaining energy differences involve both open- and closed-shell species. Consequently, the performance of LMP2 is balanced irrespective of the presence of SOMOs, allowing for the investigation of chemical processes involving both open- and closed-shell species. Finally, representative timings are also given in the last column of Table 6 using a six-year-old, 8-core CPU. The measured runtimes of 3–4 h or less prove that RO-LMP2 is routinely applicable merely as a laptop calculation up to a few hundred atoms while maintaining the intrinsic accuracy of MP2. 7. Representative Applications and Computational Requirements Click to copy section linkSection link copied! The capabilities and detailed computational requirements of the open-shell LMP2 algorithm are also illustrated for even larger molecules. The four systems collected in Table 7 can be arranged into two groups. The FeC complex and the Cbl radical of 175–179 atoms and of about 3000 AOs constitute the first group, as these systems are close to the capability limits of efficient open-shell DF-MP2 implementations. An additional similarity is that both systems contain a transition-metal atom, and the corresponding SOMO(s) are located close to the center of the molecule, resulting in a large number of strong pairs involving SOMO(s). The 21–25% strong pair ratio is indeed noticeably higher than the 16% obtained for the closed-shell vancomycin molecule of the same size (176 atoms) with the same settings. The corresponding EDs containing on an average (at most) about 115 (167) atoms are also significantly larger than the EDs of vancomycin built with 72 (129) atoms. molecule FeC[72]N[2]H[100] Cbl radical bicarbonate DAAO atoms 175 179 565 601 LMOs 205 250 788 837 838 SOMOs 4 1 4 0 2 AO basis def2-TZVP def2-TZVP def2-SVP’ def2-TZVP def2-TZVP basis functions 2939 3369 5434 10 560 11 006 auxiliary functions 7306 8379 17 782 26 064 27 071 strong pairs [%] 25 21 6.3 6.8 5.9 5.9 [%] 0.19 0.19 0.28 0.25 0.25 0.25 atoms in ED 114 (165) 116 (169) 138 (317) 132 (295) 124 (268) 137 (353) AOs in ED 2086 (2854) 2342 (3278) 1376 (3195) 2634 (6015) 2449 (5408) 2693 (6943) PAOs in ED 1020 (1812) 1017 (1828) 481 (1052) 973 (2037) 864 (1884) 902 (2930) type of reference ROHF ROHF QRO (UHF) RHF ROHF DF-HF energy –4156.159945 –5878.796625 –15182.8673^c –15197.9344^c –14740.9398 –14740.9040 LMP2 energy –12.3329 –16.4723 –43.2442 –52.3847 –55.3431 –55.3319 HF (1 iteration) 28 43 29^b 183^b 152^b 157^b localization 0.1 0.3 4.8 4.3 2.8 3.4 pair energies 1.2 8.7 38 4.8 11 48 integral trf. 56 157 188 451 374 639 amplitudes & E[LMP2]^c 38 98 18 100 54 213 total LMP2 95 264 245 557 439 900 memory req. 9.8 10 4.6 17 6.7 45 Using a 20-core 1.3 GHz Intel Xeon Gold 6138 CPU. Using the default local fitting domain size. The final iteration with larger fitting domains took about 3.5–4.8 times longer. DF-HF energies calculated with semicanonical QRO orbitals. Considering the wall-time measurements, it is reassuring that the complete LMP2 calculation took less than the time required for three to six HF iterations; thus, the LMP2 correlation energy computation is clearly not the bottleneck in these cases. It is worth noting that compared to DF-ROHF and LMP2, the formally cubic-scaling orbital localization takes negligible time even for the largest systems of Table 7 . The nonlinear-scaling steps of the LMP2 computation (see the “pair energies” line of Table 7 measuring the time of the PAO construction and the pair energy computation) are similarly efficient. As expected from the measurements performed for the closed-shell algorithm, the integral transformation and the amplitude evaluation steps dominate the time requirement of RO-LMP2 too. While the operation count required for the former is comparable to that of the closed-shell algorithm because of the use of restricted intermediate bases, the relative cost of the latter is somewhat higher for the open-shell case. The bicarbonate and DAAO species represent the second group of examples in Table 7 consisting of 565 and 601 atoms and 10–11 thousand AOs with the def2-TZVP basis set. To the best of our knowledge, these are currently the largest three-dimensional open-shell systems for which correlated quantum chemical computations have ever been presented, at least on a single CPU. The 6–7% strong pair ratio obtained for both systems appears to be representative for protein systems of a similar size (cf. the 6% strong pair ratio for the crambin protein of 644 atoms ). While the average (maximum) ED sizes of bicarbonate and the closed-shell DAAO species in columns 4–6 of Table 7 are similar to or only slightly larger than the EDs of crambin holding 128 (270) atoms, the largest domain of the triplet DAAO system reaches an unprecedented size of 353 atoms. A closer inspection reveals that the CMO of this ED is a SO LMO, which expands over the entire flavin moiety close to the center of the protein. In comparison, the closed-shell singlet DAAO system has well-localized MOs and at most 268 atoms in its largest domain. The fact that the ED computation with 353 atoms and almost 7000 AO takes only about 40 min highlights the importance and efficiency of the elaborate local approximations employed within each ED. Without exploiting the locality of the LMOs, local DF domains, the restriction of the external space to ED PAOs, the redundancy-free MP1 amplitude computation, etc., it would not be possible to compute the correlation energy contribution of several EDs reaching over 300 atoms and 6000 AOs. However, as a consequence of the delocalized SO LMO, the open-shell calculation took twice as long as its closed-shell counterpart for the analogous singlet DAAO structure of the same size. In this case, the time of integral transformation for the triplet species is longer because of the larger EDs, which would be even worse without the efficiency provided by the restricted intermediate basis. In parallel with our experience with the closed-shell LMP2 scheme, the relative cost of the integral transformation compared to that of the amplitude evaluation increases with system size. Since the operation-count requirement of the integral transformation is expected to be similar for open- and closed-shell systems with comparable domain sizes, we anticipate that the SCF iterations remain the bottleneck also for RO-LMP2. It is important to be aware of the potential cost increase with highly delocalized SO LMOs, but we think that most of the practical applications will behave considerably better in this respect than in the challenging case of DAAO with its LMO spreading over the entire flavin moiety. All in all, the RO-LMP2 computations of the largest systems required only about the time of three to six HF iterations, even if local DF is used to accelerate the HF step, and thus LMP2 is not rate-determining. Unfortunately, for such open-shell systems, the SCF procedure could take a considerably higher number of iterations than for closed-shell molecules, especially if transition-metal atoms are also involved. In many cases, one has to explore a number of options including ROHF, UHF, various density functionals, basis sets, SCF algorithms, and convergence accelerators to find a qualitatively satisfactory SCF solution. In the present study, the optimization of the quintet and especially the triplet state of the bicarbonate proved to be particularly challenging. All of our attempts for the two states accumulate into several hundreds of SCF iterations. In comparison, the triplet ROHF computation of DAAO can be considered relatively routine if accelerated with local DF It is also important to point out the benefits of the completely integral-direct and hence practically disk I/O-free LMP2 algorithm. The corresponding minimal memory requirements collected in the last row of Table 7 are also exceptionally low, being in the range of 10–20 GB for all cases except for the 45 GB allocation needed for the largest DAAO calculation. The minimal memory needed for our SCF program with local DF is also at most about 10 GB for the systems considered, but it is always beneficial to allow more memory to speed up the SCF iteration. Consequently, at least for systems accessible by current HF implementations, we do not foresee severe data bottlenecks up to the LMP2 level. Finally, the LMP2 energy differences of the four largest examples are collected in Table 8 . The quintet–triplet gap of 2.030 eV obtained for the FeC complex with both RO-LMP2 and the corresponding DF-MP2 reference is in good agreement with the 2.018, 1.852, and 2.120 eV values reported with PNO-RMP2, and CASPT2, respectively. It is interesting to realize that the LMP2/def2-TZVP value of 1.759 eV obtained for the quintet–triplet gap of bicarbonate is considerably lower because of the markedly different ligand field of its Fe(II) center. While the slow basis set convergence issue of electron correlation calculations is well-known, the insufficient level of AO basis completeness provided by double-ζ-quality basis sets should still be pointed out as frequently as possible. def2-SVP def2-TZVP HF ΔE[LMP2]^c ΔE[LMP2]^total HF ΔE[LMP2]^c ΔE[LMP2]^total FeC[72]N[2]H[100]^5A–^3A gap 57.52 –8.92 48.60 57.56 –10.73 46.82 bicarbonate ^5A–^3A gap 52.62 –12.35 40.26 52.67 –12.11 40.56 Cbl + Ado → dAdoCbl –43.38 99.84 56.46 –50.41 102.13 51.73 DAAO 20.46 8.71 29.17 22.44 7.08 29.52 8. Summary and Conclusions Click to copy section linkSection link copied! A high-spin open-shell local MP2 (RO-LMP2) method is presented using restricted open-shell Hartree–Fock (ROHF) or Kohn–Sham (ROKS) reference determinants. The efficiency of the open-shell LMP2 approach matches that of our previous closed-shell LMP2 algorithm because restricted orbitals are used for the most demanding integral transformation step. The amplitudes and correlation energy contributions are evaluated using a relatively simple, unrestricted formulation, but the corresponding computational overhead is largely mitigated by a novel approximation of long-range spin-polarization effects in the correlation energy. For closed-shell systems, the present method is identical to our closed-shell LMP2 approach. The RO-LMP2 algorithm is also especially operation-count and memory-efficient, integral-direct, OpenMP-parallel, and requires negligible hard disk use. Spatial symmetry, checkpointing, and near-linear-dependent basis sets can also be utilized. Usually, the entire RO-LMP2 computation takes the time of about three to six ROHF iterations; thus, even if accelerated with local approximations, the SCF optimization remains the main bottleneck, especially for large systems and/or with transition-metal atoms. The errors caused by the local approximations are mostly below 0.1 kcal/mol and thus negligible compared to the intrinsic accuracy of MP2 as demonstrated for reactions of radicals, spin-state energy gaps, and ionization potentials. The accuracy of local spin-scaled MP2 variants is similarly excellent, while even better performance is found for double-hybrid (DH) functionals because their second-order energy contribution is usually downscaled. As an additional use case, local MP2-based corrections are often suggested to decrease the basis set incompleteness of (local) CC methods, such as local CCSD(T). The RO-LMP2 algorithm also provides important components to our high-spin open-shell LNO-CCSD(T) and higher-order LNO-CC implementations, which are currently under extensive benchmarking. The capabilities of the RO-LMP2 implementation are illustrated on three-dimensional protein models containing up to 601 atoms and 11000 atomic orbitals with triple-ζ basis sets. The quintet–triplet gap in the bicarbonate protein of photosystem II is relatively complicated because of the nontrivial electronic structure around the Fe(II) ion in the triplet state. The second large-scale example involving the reduction of O -amino acid oxidase is also challenging because of a poorly localized SOMO spreading over an entire flavin moiety. We anticipate that the common target applications of RO-LMP2 will be significantly simpler. However, it is satisfactory that such complicated systems have also become routinely available, especially if the single-node (20-core) RO-LMP2 runtimes of 9–15 h are considered. Consequently, the presented local approximations extend the reach of open-shell MP2 as well as of spin-scaled MP2 and DH DFT methods to systems of 500–600 atoms with reasonable basis sets. Except for potential bottlenecks in the ROHF/ROKS optimization, RO-LMP2 should also be applicable for even larger molecules approaching the limit of our closed-shell LMP2 and LNO-CCSD(T) codes, which is currently about 1000–2000 atoms and 45000 atomic orbitals. Supporting Information Click to copy section linkSection link copied! The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.1c00093. • Linear-scaling time measurements, list of default settings for the local approximations, PES of the ethane-1,2-diphenyl radical, reference energies for the RSE30, IP21, and AC12 test sets and the larger molecules considered as well as the accuracy assessment of the SCS-LMP2 and LB2PLYP methods; structures of the IP21 ions and DAAO models are also provided (PDF) Terms & Conditions Most electronic Supporting Information files are available without a subscription to ACS Web Editions. Such files may be downloaded by article for research use (if there is a public use license linked to the relevant article, that license may permit other uses). Permission may be obtained from ACS for other uses through requests via the RightsLink permission system: http://pubs.acs.org/page Click to copy section linkSection link copied! Helpful discussions with Qianli Ma regarding the structures used for the RSE test set, with Masaaki Saitow and Ashutosh Kumar regarding the bicarbonate SCF computations of refs (63) and (98), and with Dóra J. Kiss regarding the DAAO structures are gratefully acknowledged. The authors are grateful for the financial support from the National Research, Development, and Innovation Office (NKFIH, Grant No. KKP126451). The research reported in this paper and carried out at BME has been supported by the NRDI Fund (TKP2020 IES, Grant No. BME-IE-BIO) based on the charter of bolster issued by the NRDI Office under the auspices of the Ministry for Innovation and Technology. The work of PRN is supported by the ÚNKP-19-4 and ÚNKP-20-5 New National Excellence Program of the Ministry for Innovation and Technology from the source of the National Research, Development and Innovation Fund and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. The computing time granted on the Hungarian HPC Infrastructure at NIIF Institute, Hungary, and the DECI resource Saga based in Norway at Trondheim with support from the PRACE aisbl (NN9914K) are gratefully Click to copy section linkSection link copied! This article references 137 other publications. 1. 1 Zhang, J.; Head-Gordon, M. Electronic structures and reaction dynamics of open-shell species. Phys. Chem. Chem. Phys. 2009, 11, 4699, DOI: 10.1039/b909815c Google Scholar Electronic structures and reaction dynamics of open-shell species Zhang, Jingsong; Head-Gordon, Martin Physical Chemistry Chemical Physics (2009), 11 (23), 4699-4700CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry) There is no expanded citation for this reference. 2. 2 Bally, T.; Borden, W. T. Reviews in Computational Chemistry; John Wiley & Sons, Ltd, 1999; pp 1– 97. 3. 3 Helgaker, T.; Jørgensen, P.; Olsen, J. Molecular Electronic Structure Theory; Wiley: Chichester, 2000. 4. 4 Krylov, A. I. Reviews in Computational Chemistry; John Wiley & Sons, Ltd, 2017; Chapter 4, pp 151– 224. 5. 5 Stanton, J. F.; Gauss, J. Advances in Chemical Physics; John Wiley & Sons, Ltd, 2003; pp 101– 146. 6. 6 Møller, C.; Plesset, M. S. Note on an Approximation Treatment for Many-Electron Systems. Phys. Rev. 1934, 46, 618, DOI: 10.1103/PhysRev.46.618 Google Scholar Note on the approximation treatment for many-electron systems Moller, Chr.; Plesset, M. S. Physical Review (1934), 46 (), 618-22CODEN: PHRVAO; ISSN:0031-899X. 7. 7 Raghavachari, K.; Trucks, G. W.; Pople, J. A.; Head-Gordon, M. A fifth-order perturbation comparison of electron correlation theories. Chem. Phys. Lett. 1989, 157, 479, DOI: 10.1016/S0009-2614 Google Scholar A fifth-order perturbation comparison of electron correlation theories Raghavachari, Krishnan; Trucks, Gary W.; Pople, John A.; Head-Gordon, Martin Chemical Physics Letters (1989), 157 (6), 479-83CODEN: CHPLBC; ISSN:0009-2614. Electron correlation theories such as CI (CI), coupled-cluster theory (CC), and quadratic CI (QCI) are assessed by means of a Moller-Plesset perturbation expansion of the correlation energy up to fifth order. The computational efficiencies and relative merits of the different techniques are outlined. A new augmented version of coupled-cluster theory, denoted as CCSD(T), is proposed to remedy some of the deficiencies of previous augmented coupled-cluster models. 8. 8 Bartlett, R. J.; Musiał, M. Coupled-cluster theory in quantum chemistry. Rev. Mod. Phys. 2007, 79, 291, DOI: 10.1103/RevModPhys.79.291 Google Scholar Coupled-cluster theory in quantum chemistry Bartlett, Rodney J.; Musial, Monika Reviews of Modern Physics (2007), 79 (1), 291-352CODEN: RMPHAT; ISSN:0034-6861. (American Physical Society) A review. Today, coupled-cluster theory offers the most accurate results among the practical ab initio electronic-structure theories applicable to moderate-sized mols. Though it was originally proposed for problems in physics, it has seen its greatest development in chem., enabling an extensive range of applications to mol. structure, excited states, properties, and all kinds of spectroscopy. In this review, the essential aspects of the theory are explained and illustrated with informative numerical results. 9. 9 Cremer, D. M. Møller-Plesset perturbation theory: from small molecule methods to methods for thousands of atoms. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2011, 1, 509, DOI: 10.1002/wcms.58 Google Scholar Moller-Plesset perturbation theory: From small molecule methods to methods for thousands of atoms Cremer, Dieter Wiley Interdisciplinary Reviews: Computational Molecular Science (2011), 1 (4), 509-530CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell) A review. The development of Mueller-Plesset perturbation theory (MPPT) has seen four different periods in almost 80 years. In the first 40 years (period 1), MPPT was largely ignored because the focus of quantum chemists was on variational methods. After the development of many-body perturbation theory by theor. physicists in the 1950s and 1960s, a second 20-yr long period started, during which MPn methods up to order n = 6 were developed and computer-programed. In the late 1980s and in the 1990s (period 3), shortcomings of MPPT became obvious, esp. the sometimes erratic or even divergent behavior of the MPn series. The phys. usefulness of MPPT was questioned and it was suggested to abandon the theory. Since the 1990s (period 4), the focus of method development work has been almost exclusively on MP2. A wealth of techniques and approaches has been put forward to convert MP2 from a O(M5) computational problem into a low-order or linear-scaling task that can handle mols. with thousands of atoms. In addn., the accuracy of MP2 has been systematically improved by introducing spin scaling, dispersion corrections, orbital optimization, or explicit correlation. The coming years will see a continuously strong development in MPPT that will have an essential impact on other quantum chem. methods. 10. 10 Szabados, Á. Reference Module in Chemistry, Molecular Sciences and Chemical Engineering; Elsevier, 2017. 11. 11 Grimme, S. Improved second-order Møller-Plesset perturbation theory by separate scaling of parallel- and antiparallel-spin pair correlation energies. J. Chem. Phys. 2003, 118, 9095, DOI: 10.1063/ Google Scholar Improved second-order Moller-Plesset perturbation theory by separate scaling of parallel- and antiparallel-spin pair correlation energies Grimme, Stefan Journal of Chemical Physics (2003), 118 (20), 9095-9102CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) A simple modification of the second-order Moller-Plesset perturbation theory (MP2) to improve the description of mol. ground state energies is proposed. The total MP2 correlation energy is partitioned into parallel- and antiparallel-spin components which are sep. scaled. The two parameters (scaling factors), whose values can be justified by basic theor. arguments, were optimized on a benchmark set of 51 reaction energies composed of 74 first-row mols. The new method performs significantly better than std. MP2: the rms [mean abs. error (MAE)] deviation drops from 4.6 (3.3) to 2.3 (1.8) kcal/mol. The max. error is reduced from 13.3 to 5.1 kcal/mol. Significant improvements are esp. obsd. for cases which are usually known as MP2 pitfalls while cases already described well with MP2 remain almost unchanged. Even for 11 atomization energies not considered in the fit, uniform improvements [MAE: 8.1 kcal/mol (MP2) vs. 3.2 kcal/mol (new)] were found. The results are furthermore compared with those from d. functional theory (DFT/B3LYP) and quadratic CI [QCISD/QCISD(T)] calcns. Also for difficult systems including strong (nondynamical) correlation effects, the improved MP2 method clearly outperforms DFT/B3LYP and yields results of QCISD or sometimes QCISD(T) quality. Preliminary calcns. of the equil. bond lengths and harmonic vibrational frequencies for ten diat. mols. also show consistent enhancements. The uniformity with which the new method improves upon MP2, thereby rectifying many of its problems, indicates significant robustness and suggests it as a valuable quantum chem. method of general use. 12. 12 Jung, Y.; Lochan, R. C.; Dutoi, A. D.; Head-Gordon, M. Scaled opposite-spin second order Møller-Plesset correlation energy: An economical electronic structure method. J. Chem. Phys. 2004, 121, 9793, DOI: 10.1063/1.1809602 Google Scholar Scaled opposite-spin second order Moller-Plesset correlation energy: An economical electronic structure method Jung, Yousung; Lochan, Rohini C.; Dutoi, Anthony D.; Head-Gordon, Martin Journal of Chemical Physics (2004), 121 (20), 9793-9802CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) A simplified approach to treating the electron correlation energy is suggested in which only the α-β component of the second order Moller-Plesset energy is evaluated, and then scaled by an empirical factor which is suggested to be 1.3. This scaled opposite-spin second order energy (SOS-MP2), where MP2 is Moller-Plesset theory, yields results for relative energies and deriv. properties that are statistically improved over the conventional MP2 method. Furthermore, the SOS-MP2 energy can be evaluated without the fifth order computational steps assocd. with MP2 theory, even without exploiting any spatial locality. A fourth order algorithm is given for evaluating the opposite spin MP2 energy using auxiliary basis expansions, and a Laplace approach, and timing comparisons are given. 13. 13 Janesko, B. G.; Scuseria, G. E. Coulomb-only second-order perturbation theory in long-range-corrected hybrid density functionals. Phys. Chem. Chem. Phys. 2009, 11, 9677, DOI: 10.1039/b910905f Google Scholar Coulomb-only second-order perturbation theory in long-range-corrected hybrid density functionals Janesko, Benjamin G.; Scuseria, Gustavo E. Physical Chemistry Chemical Physics (2009), 11 (42), 9677-9686CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry) We have been investigating the combination of a short-range d. functional approxn. with long-range RPA correlation, where the direct RPA correlation is constructed using only Coulomb (i.e., not antisymmetrized) two-electron integrals. Our group's recently demonstrated connection between RPA and coupled cluster theory suggests investigating a related method: second-order Moller-Plesset perturbation theory correlation (MP2) constructed using only Coulomb integrals. This new "JMP2" method is related to the scaled-opposite-spin SOS-MP2 approxn., which is also constructed using only Coulomb integrals. While JMP2 and SOS-MP2 yield identical results for closed shell systems, they have important differences for open shells. We show here that both JMP2 and SOS-MP2 provide a reasonable treatment of long-range correlation when combined with a short-range exchange-correlation functional. Remarkably, JMP2's explicit inclusion of (approx.) like-spin correlation effects provides significant improvements over SOS-MP2 for thermochem. 14. 14 Szabados, Á.; Nagy, P. Spin component scaling in multiconfiguration perturbation theory. J. Phys. Chem. A 2011, 115, 523, DOI: 10.1021/jp108575a Google Scholar Spin Component Scaling in Multiconfiguration Perturbation Theory Szabados, Agnes; Nagy, Peter Journal of Physical Chemistry A (2011), 115 (4), 523-534CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society) We investigate a term-by-term scaling of the second-order energy correction obtained by perturbation theory (PT) starting from a multiconfiguration wave function. The total second-order correction is decompd. into several terms, based on the level and the spin pattern of the excitations. To define individual terms, we extend the same spin/different spin categorization of spin component scaling in various ways. When needed, identification of the excitation level is facilitated by the pivot determinant underlying the multiconfiguration PT framework. Scaling factors are detd. from the stationary condition of the total energy calcd. up to order 3. The decompn. schemes are tested numerically on the example of bond dissocn. profiles and energy differences. We conclude that Grimme's parameters detd. for single-ref. Moller-Plesset theory may give a modest error redn. along the entire potential surface, if adopting a multireference based PT formulation. Scaling factors obtained from the stationary condition show relatively large variation with mol. geometry, at the same time they are more efficient in reducing the error when following a bond dissocn. process. 15. 15 Grimme, S.; Goerigk, L.; Fink, R. F. Spin-component-scaled electron correlation methods. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2012, 2, 886– 906, DOI: 10.1002/wcms.1110 Google Scholar Spin-component-scaled electron correlation methods Grimme, Stefan; Goerigk, Lars; Fink, Reinhold F. Wiley Interdisciplinary Reviews: Computational Molecular Science (2012), 2 (6), 886-906CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell) A review. Spin-component-scaled (SCS) electron correlation methods for electronic structure theory are reviewed. The methods can be derived theor. by applying special conditions to the underlying wave functions in perturbation theory. They are based on the insight that low-order wave function expansions treat the correlation effects of electron pairs with opposite spin (OS) and same spin (SS) differently because of their different treatment at the underlying Hartree-Fock level. Phys., this is related to the different av. inter-electronic distances in the SS and OS electron pairs. The overview starts with the original SCS-MP2 method and discusses its strengths and weaknesses and various ways to parameterize the scaling factors. Extensions to coupled-cluster and excited state methods as well the connection to virtual-orbital dependent d. functional approaches are highlighted. The performance of various SCS methods in large thermochem. benchmarks and for excitation energies is discussed in comparison with other common electronic structure methods. 16. 16 Grimme, S. Semiempirical hybrid density functional with perturbative second-order correlation. J. Chem. Phys. 2006, 124, 034108 DOI: 10.1063/1.2148954 Google Scholar Semiempirical hybrid density functional with perturbative second-order correlation Grimme, Stefan Journal of Chemical Physics (2006), 124 (3), 034108/1-034108/16CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) A new hybrid d. functional for general chem. applications is proposed. It is based on a mixing of std. generalized gradient approxns. (GGAs) for exchange by Becke (B) and for correlation by Lee, Yang, and Parr (LYP) with Hartree-Fock (HF) exchange and a perturbative second-order correlation part (PT2) that is obtained from the Kohn-Sham (GGA) orbitals and eigenvalues. This virtual orbital-dependent functional contains only two global parameters that describe the mixt. of HF and GGA exchange (ax) and of the PT2 and GGA correlation (c), resp. The parameters are obtained in a least-squares-fit procedure to the G2/97 set of heat of formations. Opposed to conventional hybrid functionals, the optimum ax is found to be quite large (53% with c = 27%) which at least in part explains the success for many problematic mol. systems compared to conventional approaches. The performance of the new functional termed B2-PLYP is assessed by the G2/97 std. benchmark set, a second test suite of atoms, mols., and reactions that are considered as electronically very difficult (including transition-metal compds., weakly bonded complexes, and reaction barriers) and comparisons with other hybrid functionals of GGA and meta-GGA types. According to many realistic tests, B2-PLYP can be regarded as the best general purpose d. functional for mols. (e.g., a mean abs. deviation for the two test sets of only 1.8 and 3.2 kcal/mol compared to about 3 and 5 kcal/mol, resp., for the best other d. functionals). Very importantly, also the max. and minium errors (outliers) are strongly reduced (by about 10-20 kcal/mol). Furthermore, very good results are obtained for transition state barriers but unlike previous attempts at such a good description, this definitely comes not at the expense of equil. properties. Preliminary calcns. of the equil. bond lengths and harmonic vibrational frequencies for diat. mols. and transition-metal complexes also show very promising results. The uniformity with which B2-PLYP improves for a wide range of chem. systems emphasizes the need of (virtual) orbital-dependent terms that describe nonlocal electron correlation in accurate exchange-correlation functionals. From a practical point of view, the new functional seems to be very robust and it is thus suggested as an efficient quantum chem. method of general purpose. 17. 17 Sancho-García, J. C.; Adamo, C. Double-hybrid density functionals: Merging wavefunction and density approaches to get the best of both worlds. Phys. Chem. Chem. Phys. 2013, 15, 14581, DOI: Google Scholar Double-hybrid density functionals: merging wavefunction and density approaches to get the best of both worlds Sancho-Garcia, J. C.; Adamo, C. Physical Chemistry Chemical Physics (2013), 15 (35), 14581-14594CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry) A review. We review why and how double-hybrid d. functionals have become new leading actors in the field of computational chem., thanks to the combination of an unprecedented accuracy together with large robustness and reliability. Similar to their predecessors, the widely employed hybrid d. functionals, they are rooted in the Adiabatic Connection Method from which they emerge in a natural way. We present recent achievements concerning applications to chem. systems of the most interest, and current extensions to deal with challenging issues such as non-covalent interactions and excitation energies. These promising methods, despite a slightly higher computational cost than other typical d.-based models, are called to play a key role in the near future and can thus pave the way towards new discoveries or advances. 18. 18 Goerigk, L.; Grimme, S. Double-hybrid density functionals. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2014, 4, 576, DOI: 10.1002/wcms.1193 Google Scholar Double-hybrid density functionals Goerigk, Lars; Grimme, Stefan Wiley Interdisciplinary Reviews: Computational Molecular Science (2014), 4 (6), 576-600CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell) Double-hybrid d. functionals (DHDFs) are reviewed in this study. In DHDFs parts of conventional d. functional theory (DFT) exchange and correlation are replaced by contributions from nonlocal Fock-exchange and second-order perturbative correlation. The latter portion is based on the well-known MP2 wave-function approach in which, however, Kohn-Sham orbitals are used to calc. its contribution. First, related methods preceding this idea are reviewed, followed by a thorough discussion of the first modern double-hybrid B2-PLYP. Parallels and differences between B2-PLYP and its various successors are then outlined. This discussion is rounded off with representative thermochem. examples demonstrating that DHDFs belong to the most robust and accurate DFT approaches currently available. This anal. also presents hitherto unpublished results for recently developed DHDFs. Finally, how double-hybrids can be combined with linear-response time-dependent DFT is also outlined and the value of this approach for electronically excited states is shown. WIREs Comput Mol Sci 2014, 4:576-600. doi: 10.1002/wcms.1193 For further resources related to this article, please visit the . Conflict of interest: The authors have declared no conflicts of interest for this article. 19. 19 Martin, J. M. L.; Santra, G. Empirical Double-Hybrid Density Functional Theory: A ‘Third Way’ in Between WFT and DFT. Isr. J. Chem. 2020, 60, 787, DOI: 10.1002/ijch.201900114 Google Scholar Empirical Double-Hybrid Density Functional Theory: A 'Third Way' in Between WFT and DFT Martin, Jan M. L.; Santra, Golokesh Israel Journal of Chemistry (2020), 60 (8-9), 787-804CODEN: ISJCAT; ISSN:0021-2148. (Wiley-VCH Verlag GmbH & Co. KGaA) A review. Double hybrid d. functional theory arguably sits on the seamline between wavefunction methods and DFT: it represents a special case of Rung 5 on the "Jacob's Ladder" of John P. Perdew. For large and chem. diverse benchmarks such as GMTKN55, empirical double hybrid functionals with dispersion corrections can achieve accuracies approaching wavefunction methods at a cost not greatly dissimilar to hybrid DFT approaches, provided RI-MP2 and/or another MP2 acceleration techniques are available in the electronic structure code. Only a half-dozen or fewer empirical parameters are required. For vibrational frequencies, accuracies intermediate between CCSD and CCSD(T) can be achieved, and performance for other properties is encouraging as well. Organometallic reactions can likewise be treated well, provided static correlation is not too strong. Further prospects are discussed, including range-sepd. and RPA-based approaches. 20. 20 Chai, J.-D.; Head-Gordon, M. Long-range corrected double-hybrid density functionals. J. Chem. Phys. 2009, 131, 174105 DOI: 10.1063/1.3244209 Google Scholar Long-range corrected double-hybrid density functionals Chai, Jeng-Da; Head-Gordon, Martin Journal of Chemical Physics (2009), 131 (17), 174105/1-174105/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) We extend the range of applicability of our previous long-range cor. (LC) hybrid functional, ωB97X, with a nonlocal description of electron correlation, inspired by second-order Moller-Plesset (many-body) perturbation theory. This LC "double-hybrid" d. functional, denoted as ωB97X-2, is fully optimized both at the complete basis set limit (using 2-point extrapolation from calcns. using triple and quadruple zeta basis sets), and also sep. using the somewhat less expensive 6-311++G(3df,3pd) basis. On independent test calcns. (as well as training set results), ωB97X-2 yields high accuracy for thermochem., kinetics, and noncovalent interactions. In addn., owing to its high fraction of exact Hartree-Fock exchange, ωB97X-2 shows significant improvement for the systems where self-interaction errors are severe, such as sym. homonuclear radical cations. (c) 2009 American Institute of Physics. 21. 21 Goerigk, L.; Grimme, S. Efficient and Accurate Double-Hybrid-Meta-GGA Density Functionals—Evaluation with the Extended GMTKN30 Database for General Main Group Thermochemistry, Kinetics, and Noncovalent Interactions. J. Chem. Theory Comput. 2011, 7, 291, DOI: 10.1021/ct100466k Google Scholar Efficient and Accurate Double-Hybrid-Meta-GGA Density Functionals-Evaluation with the Extended GMTKN30 Database for General Main Group Thermochemistry, Kinetics, and Noncovalent Interactions Goerigk, Lars; Grimme, Stefan Journal of Chemical Theory and Computation (2011), 7 (2), 291-309CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society) We present an extended and improved version of our recently published database for general main group thermochem., kinetics, and noncovalent interactions, which is dubbed GMTKN30. Furthermore, we suggest and investigate two new double-hybrid-meta-GGA d. functionals called PTPSS-D3 and PWPB95-D3. PTPSS-D3 is based on reparameterized TPSS exchange and correlation contributions; PWPB95-D3 contains reparameterized PW exchange and B95 parts. Both functionals contain fixed amts. of 50% Fock-exchange. Furthermore, they include a spin-opposite scaled perturbative contribution and are combined with our latest atom-pairwise London-dispersion correction. When evaluated with the help of the Laplace transformation algorithm, both methods scale as N4 with system size. The functionals are compared with the double hybrids B2PLYP-D3, B2GPPLYP-D3, DSD-BLYP-D3, and XYG3 for GMTKN30 with a quadruple-ζ basis set. PWPB95-D3 and DSD-BLYP-D3 are the best functionals in our study and turned out to be more robust than B2PLYP-D3 and XYG3. Furthermore, PWPB95-D3 is the least basis set dependent and the best functional at the triple-ζ level. For the example of transition metal carbonyls, it is shown that, mainly due to the lower amt. of Fock-exchange, PWPB95-D3 and PTPSS-D3 are better applicable than the other double hybrids. Finally, we discuss in some detail the XYG3 functional, which makes use of B3LYP orbitals and electron densities. We show that it is basically a highly nonlocal variant of B2PLYP and that its partially good performance is mainly due to a larger effective amt. of perturbative correlation compared to other double hybrids. We finally recommend the PWPB95-D3 functional in general chem. applications. 22. 22 Zhang, I. Y.; Xu, X.; Jung, Y.; Goddard, W. A. A fast doubly hybrid density functional method close to chemical accuracy using a local opposite spin ansatz. Proc. Natl. Acad. Sci. U.S.A. 2011, 108, 19896, DOI: 10.1073/pnas.1115123108 Google Scholar A fast doubly hybrid density functional method close to chemical accuracy using a local opposite spin ansatz Zhang, Igor Ying; Xu, Xin; Jung, Yousung; Goddard, William A., III Proceedings of the National Academy of Sciences of the United States of America (2011), 108 (50), 19896-19900, S19896/1-S19896/10CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences) We develop and validate the XYGJ-OS functional, based on the adiabatic connection formalism and Girling-Levy perturbation theory to second order and using the opposite-spin (OS) ansatz combined with locality of electron correlation. XYGJ-OS with local implementation scales as N3 with an overall accuracy of 1.28 kcal/mol for thermochem., bond dissocn. energies, reaction barrier heights, and nonbonded interactions, comparable to that of 1.06 kcal/mol for the accurate coupled-cluster based G3 method (scales as N7) and much better than many popular d. functional theory methods: B3LYP (4.98), PBEO (4.36), and PBE (12.10). 23. 23 Zhang, I. Y.; Su, N. Q.; Brémond, É. A. G.; Adamo, C.; Xu, X. Doubly hybrid density functional xDH-PBE0 from a parameter-free global hybrid model PBE0. J. Chem. Phys. 2012, 136, 174103 DOI: Google Scholar Doubly hybrid density functional xDH-PBE0 from a parameter-free global hybrid model PBE0 Zhang, Igor Ying; Su, Neil Qiang; Bremond, Eric A. G.; Adamo, Carlo; Xu, Xin Journal of Chemical Physics (2012), 136 (17), 174103/1-174103/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) Following the XYG3 model which uses orbitals and d. from B3LYP, an empirical doubly hybrid (DH) functional is developed by using inputs from PBE0. This new functional, named xDH-PBE0, has been tested on a no. of different mol. properties, including atomization energies, bond dissocn. enthalpies, reaction barrier heights, and nonbonded interactions. From the results obtained, xDH-PBE0 not only displays a significant improvement with respect to the parent PBE0, but also shows a performance that is comparable to XYG3. Arguably, while PBE0 is a parameter-free global hybrid (GH) functional, the B3LYP GH functional contains eight fit parameters. From a more general point of view, the present work points out that reliable and general-purpose DHs can be obtained with a limited no. of fit parameters. (c) 2012 American Institute of Physics. 24. 24 Kozuch, S.; Martin, J. M. L. Spin-Component-Scaled Double Hybrids: An Extensive Search for the Best Fifth-Rung Functionals Blending DFT and Perturbation Theory. J. Comput. Chem. 2013, 34, 2327, DOI: 10.1002/jcc.23391 Google Scholar Spin-component-scaled double hybrids: An extensive search for the best fifth-rung functionals blending DFT and perturbation theory Kozuch, Sebastian; Martin, Jan M. L. Journal of Computational Chemistry (2013), 34 (27), 2327-2344CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.) Following up on an earlier preliminary communication (Kozuch and Martin, Phys. Chem. Chem. Phys. 2011, 13, 20104), we report here in detail on an extensive search for the most accurate spin-component-scaled double hybrid functionals [of which conventional double hybrids (DHs) are a special case]. Such fifth-rung functionals approach the performance of composite ab initio methods such as G3 theory at a fraction of their computational cost, and with anal. derivs. available. In this article, we provide a crit. anal. of the variables and components that maximize the accuracy of DHs. These include the selection of the exchange and correlation functionals, the coeffs. of each component [d. functional theory (DFT), exact exchange, and perturbative correlation in both the same spin and opposite spin terms], and the addn. of an ad-hoc dispersion correction; we have termed these parametrizations "DSD-DFT" (Dispersion cor., Spin-component scaled, Double-hybrid DFT). Somewhat surprisingly, the quality of DSD-DFT is only mildly dependent on the underlying DFT exchange and correlation components, with even DSD-LDA yielding respectable performance. Simple, nonempirical GGAs appear to work best, whereas meta-GGAs offer no advantage (with the notable exception of B95c). The best correlation components appear to be, in that order, B95c, P86, and PBEc, while essentially any good GGA exchange yields nearly identical results. On further validation with a wider variety of thermochem., weak interaction, kinetic, and spectroscopic benchmarks, we find that the best functionals are, roughly in that order, DSD-PBEhB95, DSD-PBEP86, DSD-PBEPW91, and DSD-PBEPBE. In addn., DSD-PBEP86 and DSD-PBEPBE can be used without source code modifications in a wider variety of electronic structure codes. Sample job decks for several commonly used such codes are supplied as electronic Supporting Information. Copyright © 2013 Wiley Periodicals, Inc. 25. 25 Brémond, É.; Ciofini, I.; Sancho-García, J. C.; Adamo, C. Nonempirical Double-Hybrid Functionals: An Effective Tool for Chemists. Acc. Chem. Res. 2016, 49, 1503, DOI: 10.1021/acs.accounts.6b00232 Google Scholar Nonempirical Double-Hybrid Functionals: An Effective Tool for Chemists Bremond, Eric; Ciofini, Ilaria; Sancho-Garcia, Juan Carlos; Adamo, Carlo Accounts of Chemical Research (2016), 49 (8), 1503-1513CODEN: ACHRE4; ISSN:0001-4842. (American Chemical Society) A review. D. functional theory (DFT) emerged in the last two decades as the most reliable tool for the description and prediction of properties of mol. systems and extended materials, coupling in an unprecedented way high accuracy and reasonable computational cost. This success rests also on the development of more and more performing d. functional approxns. (DFAs). Indeed, the Achilles' heel of DFT is represented by the exchange-correlation contribution to the total energy, which, being unknown, must be approximated. Since the beginning of the 1990s, global hybrids (GH) functionals, where an explicit dependence of the exchange-correlation energy on occupied Kohn-Sham orbitals is introduced thanks to a fraction of Hartree-Fock-like exchange, imposed themselves as the most reliable DFAs for chem. applications. However, if these functionals normally provide results of sufficient accuracy for most of the cases analyzed, some properties, such as thermochem. or dispersive interactions, can still be significantly improved. A possible way out is represented by the inclusion, into the exchange-correlation functional, of an explicit dependence on virtual Kohn-Sham orbitals via perturbation theory. This leads to a new class of functionals, called double-hybrids (DHs). In this Account, we describe our nonempirical approach to DHs, which, following the line traced by the Perdew-Burke-Ernzerhof approach, allows for the definition of a GH (PBE0) and a DH (QIDH) model. In such a way, a whole family of nonempirical functionals, spanning on the highest rungs of the Perdew's quality scale, is now available and competitive with other-more empirical-DFAs. Discussion of selected cases, ranging from thermochem. and reactions to weak interactions and excitation energies, not only show the large range of applicability of nonempirical DFAs, but also underline how increasing the no. of theor. constraints parallels with an improvement of the DFA's numerical performances. This fact further consolidates the strong theor. framework of nonempirical DFAs.Finally, even if nonempirical DH approaches are still computationally expensive, relying on the fact that they can benefit of all tech. enhancements developed for speeding up post-Hartree-Fock methods, there is substantial hope for their near future routine application to the description and prediction of complex chem. systems and reactions. 26. 26 Su, N. Q.; Xu, X. The XYG3 Type of Doubly Hybrid Density Functionals. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2016, 6, 721, DOI: 10.1002/wcms.1274 Google Scholar The XYG3 type of doubly hybrid density functionals Su, Neil Qiang; Xu, Xin Wiley Interdisciplinary Reviews: Computational Molecular Science (2016), 6 (6), 721-747CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell) Doubly hybrid (DH) functionals have emerged as a new class of d. functional approxns. (DFAs), which not only have a nonlocal orbital-dependent component in the exchange part, but also incorporate the information of unoccupied orbitals in the correlation part, being at the top rung of Perdew's view of Jacob's ladder in DFAs. This review article focuses on the XYG3 type of DH (xDH) functionals, which use a low rung functional to perform the self-consistent-field calcn. to generate orbitals and densities, with which a top rung DH functional is used for final energy evaluation. We will discuss the theor. background of the xDH functionals, briefly reviewing the adiabatic connection formalism, coordinate scaling relations, and Goerling-Levy perturbation theory. General performance of the xDH functionals will be presented for both energies and structures. In particular, we will present the fractional charge behaviors of the xDH functionals, examg. the self-interaction errors, the delocalization errors and the deviation from the linearity condition, as well as their effects on the predicted ionization potentials, electron affinities and fundamental gaps. This provides a theor. rationale for the obsd. good performance of the xDH functionals. WIREs Comput Mol Sci 2016, 6:721-747. doi: 10.1002/wcms.1274 For further resources related to this article, please visit the . 27. 27 Feyereisen, M.; Fitzgerald, G.; Komornicki, A. Use of approximate integrals in ab initio theory. An application in MP2 energy calculations. Chem. Phys. Lett. 1993, 208, 359, DOI: 10.1016/ Google Scholar Use of approximate integrals in ab initio theory. An application in MP2 energy calculations Feyereisen, Martin; Fitzgerald, George; Komornicki, Andrew Chemical Physics Letters (1993), 208 (5-6), 359-63CODEN: CHPLBC; ISSN:0009-2614. Authors use the resoln. of the identity (RI) as a convenient way to replace the use of four-index two-electron integrals with linear combinations of three-index integrals. The method is broadly applicable to a wide range of problems in quantum chem. Authors demonstrate the effectiveness of RI for the calcn. of MP2 energies. For the water dimer, agreement within 0.1 kcal/mol is obtained with respect to exact MP2 calcns. The RI-MP2 energies require only about 10% of the time required by conventional MP2. 28. 28 Gyevi-Nagy, L.; Kállay, M.; Nagy, P. R. Integral-direct and parallel implementation of the CCSD(T) method: Algorithmic developments and large-scale applications. J. Chem. Theory Comput. 2020, 16, 366, DOI: 10.1021/acs.jctc.9b00957 Google Scholar Integral-Direct and Parallel Implementation of the CCSD(T) Method: Algorithmic Developments and Large-Scale Applications Gyevi-Nagy Laszlo; Kallay Mihaly; Nagy Peter R Journal of chemical theory and computation (2020), 16 (1), 366-384 ISSN:. A completely integral-direct, disk I/O, and network traffic economic coupled-cluster singles, doubles, and perturbative triples [CCSD(T)] implementation has been developed relying on the density-fitting approximation. By fully exploiting the permutational symmetry, the presented algorithm is highly operation count and memory-efficient. Our measurements demonstrate excellent strong scaling achieved via hybrid MPI/OpenMP parallelization and a highly competitive, 60-70% utilization of the theoretical peak performance on up to hundreds of cores. The terms whose evaluation time becomes significant only for small- to medium-sized examples have also been extensively optimized. Consequently, high performance is also expected for systems appearing in extensive data sets used, e.g., for density functional or machine learning parametrizations, and in calculations required for certain reduced-cost or local approximations of CCSD(T), such as in our local natural orbital scheme [LNO-CCSD(T)]. The efficiency of this implementation allowed us to perform some of the largest CCSD(T) calculations ever presented for systems of 31-43 atoms and 1037-1569 orbitals using only four to eight many-core CPUs and 1-3 days of wall time. The resulting 13 correlation energies and the 12 corresponding reaction energies and barrier heights are added to our previous benchmark set collecting reference CCSD(T) results of molecules at the applicability limit of current implementations. 29. 29 Almlöf, J. Elimination of energy denominators in Møller-Plesset perturbation theory by a Laplace transform approach. Chem. Phys. Lett. 1991, 181, 319, DOI: 10.1016/0009-2614(91)80078-C Google Scholar Elimination of energy denominators in Moeller-Plesset perturbation theory by a Laplace transform approach Almlof, Jan Chemical Physics Letters (1991), 181 (4), 319-20CODEN: CHPLBC; ISSN:0009-2614. It is shown how the energy denominators encountered in various schemes for electronic structure calcn. can be removed by a Laplace transform technique. The method is applicable to a wide variety of electronic structure calcns. 30. 30 Häser, M.; Almlöf, J. Laplace transform techniques in Møller-Plesset perturbation theory. J. Chem. Phys. 1992, 96, 489, DOI: 10.1063/1.462485 31. 31 Ayala, P. Y.; Scuseria, G. E. Linear scaling second-order Møller-Plesset theory in the atomic orbital basis for large molecular systems. J. Chem. Phys. 1999, 110, 3660, DOI: 10.1063/1.478256 Google Scholar Linear scaling second-order Moeller-Plesset theory in the atomic orbital basis for large molecular systems Ayala, Philippe Y.; Scuseria, Gustavo E. Journal of Chemical Physics (1999), 110 (8), 3660-3671CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) We have used Almlof and Haser's Laplace transform idea to eliminate the energy denominator in second-order perturbation theory (MP2) and obtain an energy expression in the AO basis. We show that the asymptotic computational cost of this method scales quadratically with mol. size. We then define AO domains such that selective pairwise interactions can be neglected using well-defined thresholding criteria based on the power law decay properties of the long-range contributions. For large mols., our scheme yields linear scaling computational cost as a function of mol. size. The errors can be controlled in a precise manner and our method reproduces canonical MP2 energies. We present benchmark calcns. of polyglycine chains and water clusters contg. up to 3040 basis 32. 32 Surján, P. R. The MP2 energy as a functional of the Hartree-Fock density matrix. Chem. Phys. Lett. 2005, 406, 318– 320, DOI: 10.1016/j.cplett.2005.03.024 Google Scholar The MP2 energy as a functional of the Hartree-Fock density matrix Surjan, Peter R. Chemical Physics Letters (2005), 406 (4-6), 318-320CODEN: CHPLBC; ISSN:0009-2614. (Elsevier B.V.) The explicit E[2][P] functional is presented, where E [2] is the second order Moller-Plesset correlation energy and P is the std. Hartree-Fock d. matrix. The ideas leading to this functional are implicit in previous studies, but the significance of its existence has not yet been sufficiently emphasized and its simple explicit form has not been presented. With the proposed functional one may obtain the correlation energy in the absence of MOs, knowing merely the d. matrix. This may further facilitate linear scaling computation of the correlation energy. 33. 33 Kobayashi, M.; Nakai, H. Implementation of Surján’s density matrix formulae for calculating second-order Møller-Plesset energy. Chem. Phys. Lett. 2006, 420, 250– 255, DOI: 10.1016/ Google Scholar Implementation of Surjan's density matrix formulae for calculating second-order Moller-Plesset energy Kobayashi, Masato; Nakai, Hiromi Chemical Physics Letters (2006), 420 (1-3), 250-255CODEN: CHPLBC; ISSN:0009-2614. (Elsevier B.V.) We numerically assess the method for obtaining second-order Moller-Plesset (MP2) energy from the Hartree-Fock d. matrix (DM) recently proposed by Surjan [Surjan, Chem. Phys. Lett. 406 (2005) 318]. It is confirmed that Surjan's method, referred to as DM-Laplace MP2, can obtain MP2 energy accurately by means of appropriate integral quadrature and a matrix exponential evaluation scheme. Numerical tests reveal that the Euler-Maclaurin and the Romberg numerical integration schemes can achieve milli-hartree accuracy with small quadrature points. This Letter also indicates the possibility of the application of DM-Laplace MP2 to linear-scaling SCF techniques, which give approx. DM. 34. 34 Doser, B.; Lambrecht, D. S.; Kussmann, J.; Ochsenfeld, C. Linear-scaling atomic orbital-based second-order Møller-Plesset perturbation theory by rigorous integral screening criteria. J. Chem. Phys. 2009, 130, 064107 DOI: 10.1063/1.3072903 Google Scholar Linear-scaling atomic orbital-based second-order Moller-Plesset perturbation theory by rigorous integral screening criteria Doser, Bernd; Lambrecht, Daniel S.; Kussmann, Joerg; Ochsenfeld, Christian Journal of Chemical Physics (2009), 130 (6), 064107/1-064107/14CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) A Laplace-transformed second-order Moller-Plesset perturbation theory (MP2) method is presented, which allows to achieve linear scaling of the computational effort with mol. size for electronically local structures. Also for systems with a delocalized electronic structure, a cubic or even quadratic scaling behavior is achieved. Numerically significant contributions to the AO (AO)-MP2 energy are preselected using the so-called multipole-based integral ests. (MBIE) introduced earlier by us. Since MBIE provides rigorous upper bounds, numerical accuracy is fully controlled and the exact MP2 result is attained. While the choice of thresholds for a specific accuracy is only weakly dependent upon the mol. system, our AO-MP2 scheme offers the possibility for incremental thresholding: for only little addnl. computational expense, the numerical accuracy can be systematically converged. We illustrate this dependence upon numerical thresholds for the calcn. of intermol. interaction energies for the S22 test set. The efficiency and accuracy of our AO-MP2 method is demonstrated for linear alkanes, stacked DNA base pairs, and carbon nanotubes: e.g., for DNA systems the crossover toward conventional MP2 schemes occurs between one and two base pairs. In this way, it is for the first time possible to compute wave function-based correlation energies for systems contg. more than 1000 atoms with 10 000 basis functions as illustrated for a 16 base pair DNA system on a single-core computer, where no empirical restrictions are introduced and numerical accuracy is fully preserved. (c) 2009 American Institute of Physics. 35. 35 Schäfer, T.; Ramberger, B.; Kresse, G. Quartic scaling MP2 for solids: A highly parallelized algorithm in the plane wave basis. J. Chem. Phys. 2017, 146, 104101 DOI: 10.1063/1.4976937 Google Scholar Quartic scaling MP2 for solids: A highly parallelized algorithm in the plane wave basis Schafer Tobias; Ramberger Benjamin; Kresse Georg The Journal of chemical physics (2017), 146 (10), 104101 ISSN:. We present a low-complexity algorithm to calculate the correlation energy of periodic systems in second-order Moller-Plesset (MP2) perturbation theory. In contrast to previous approximation-free MP2 codes, our implementation possesses a quartic scaling, O(N(4)), with respect to the system size N and offers an almost ideal parallelization efficiency. The general issue that the correlation energy converges slowly with the number of basis functions is eased by an internal basis set extrapolation. The key concept to reduce the scaling is to eliminate all summations over virtual orbitals which can be elegantly achieved in the Laplace transformed MP2 formulation using plane wave basis sets and fast Fourier transforms. Analogously, this approach could allow us to calculate second order screened exchange as well as particle-hole ladder diagrams with a similar low complexity. Hence, the presented method can be considered as a step towards systematically improved correlation energies. 36. 36 Pulay, P.; Saebø, S. Orbital-invariant formulation and second-order gradient evaluation in Møller-Plesset perturbation theory. Theor. Chem. Acc. 1986, 69, 357, DOI: 10.1007/BF00526697 Google Scholar Orbital-invariant formulation and second-order gradient evaluation in Moeller-Plesset perturbation theory Pulay, Peter; Saeboe, Svein Theoretica Chimica Acta (1986), 69 (5-6), 357-68CODEN: TCHAAM; ISSN:0040-5744. Based on the Hylleraas functional form, the second and third orders of the Moeller-Plesset (MP) perturbation theory were reformulated in terms of arbitrary (e.g., localized) internal orbitals, and AOs in the virtual space. The results are strictly equiv. to the canonical formulation if no further approxns. are introduced. The new formalism permits the extension of the local correlation method to MP theory. It also facilitates the treatment of weak pairs at a lower (e.g., second-order) level of theory in CI and coupled-cluster methods. Based on the formalism, an MP2 gradient algorithm is outlined, which does not require the storage of deriv. integrals, integrals with three external MO indexes, and, using the method of N. C. Handy and H. F. Schaefer III (1984), the repeated soln. of the coupled-perturbed SCF equations. 37. 37 Kats, D.; Usvyat, D.; Schütz, M. On the use of the Laplace transform in local correlation methods. Phys. Chem. Chem. Phys. 2008, 10, 3430, DOI: 10.1039/b802993h Google Scholar On the use of the Laplace transform in local correlation methods Kats, Danylo; Usvyat, Denis; Schuetz, Martin Physical Chemistry Chemical Physics (2008), 10 (23), 3430-3439CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry) The applicability of the Laplace transform ansatz of Almlof in the context of local correlation methods with a priori restricted sets of wavefunction parameters is explored. A new local MP2 method based on the Laplace transform ansatz is described, its relation to the local MP2 method based on the Pulay ansatz is elucidated, and its accuracy and efficiency are compared to the 38. 38 Nagy, P. R.; Samu, G.; Kállay, M. An integral-direct linear-scaling second-order Møller-Plesset approach. J. Chem. Theory Comput. 2016, 12, 4897, DOI: 10.1021/acs.jctc.6b00732 Google Scholar An Integral-Direct Linear-Scaling Second-Order Moller-Plesset Approach Nagy, Peter R.; Samu, Gyula; Kallay, Mihaly Journal of Chemical Theory and Computation (2016), 12 (10), 4897-4914CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society) An integral-direct, iteration-free, linear-scaling, local second-order Moller-Plesset (MP2) approach is presented, which is also useful for spin-scaled MP2 calcns. as well as for the efficient evaluation of the perturbative terms of double-hybrid d. functionals. The method is based on a fragmentation approxn.: the correlation contributions of the individual electron pairs are evaluated in domains constructed for the corresponding localized orbitals, and the correlation energies of distant electron pairs are computed with multipole expansions. The required electron repulsion integrals are calcd. directly invoking the d. fitting approxn.; the storage of integrals and intermediates is avoided. The approach also utilizes natural auxiliary functions to reduce the size of the auxiliary basis of the domains and thereby the operation count and memory requirement. Our test calcns. show that the approach recovers 99.9% of the canonical MP2 correlation energy and reproduces reaction energies with an av. (max.) error below 1 kJ/mol (4 kJ/mol). Our benchmark calcns. demonstrate that the new method enables MP2 calcns. for mols. with more than 2300 atoms and 26000 basis functions on a single processor. 39. 39 Saebø, S. Linear-Scaling Techniques in Computational Chemistry and Physics: Methods and Applications; Springer: Netherlands, 2011; pp 65– 82. 40. 40 Zienau, J.; Clin, L.; Doser, B.; Ochsenfeld, C. Cholesky-decomposed densities in Laplace-based second-order Møller-Plesset perturbation theory. J. Chem. Phys. 2009, 130, 204112 DOI: 10.1063/ Google Scholar Cholesky-decomposed densities in Laplace-based second-order Moller-Plesset perturbation theory Zienau, Jan; Clin, Lucien; Doser, Bernd; Ochsenfeld, Christian Journal of Chemical Physics (2009), 130 (20), 204112/1-204112/4CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) Based on our linear-scaling AO second-order Moller-Plesset perturbation theory (AO-MP2) method , we explore the use of Cholesky-decompd. pseudodensity (CDD) matrixes within the Laplace formulation. Numerically significant contributions are preselected using our multipole-based integral ests. as upper bounds to two-electron integrals so that the 1/R6 decay behavior of transformed Coulomb-type products is exploited. In addn., we combine our new CDD-MP2 method with the resoln. of the identity (RI) approach. Even though the use of RI results in a method that shows a quadratic scaling behavior in the dominant steps, gains of up to one or two orders of magnitude vs. our original AO-MP2 method are obsd. in particular for larger basis sets. (c) 2009 American Institute of Physics. 41. 41 Maurer, S. A.; Clin, L.; Ochsenfeld, C. Cholesky-decomposed density MP2 with density fitting: Accurate MP2 and double-hybrid DFT energies for large systems. J. Chem. Phys. 2014, 140, 224112 DOI: Google Scholar Cholesky-decomposed density MP2 with density fitting: Accurate MP2 and double-hybrid DFT energies for large systems Maurer, Simon A.; Clin, Lucien; Ochsenfeld, Christian Journal of Chemical Physics (2014), 140 (22), 224112/1-224112/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) Our recently developed QQR-type integral screening is introduced in our Cholesky-decompd. pseudo-densities Moller-Plesset perturbation theory of second order (CDD-MP2) method. We use the resoln.-of-the-identity (RI) approxn. in combination with efficient integral transformations employing sparse matrix multiplications. The RI-CDD-MP2 method shows an asymptotic cubic scaling behavior with system size and a small prefactor that results in an early crossover to conventional methods for both small and large basis sets. We also explore the use of local fitting approxns. which allow to further reduce the scaling behavior for very large systems. The reliability of our method is demonstrated on test sets for interaction and reaction energies of medium sized systems and on a diverse selection from our own benchmark set for total energies of larger systems. Timings on DNA systems show that fast calcns. for systems with more than 500 atoms are feasible using a single processor core. Parallelization extends the range of accessible system sizes on one computing node with multiple cores to more than 1000 atoms in a double-zeta basis and more than 500 atoms in a triple-zeta basis. (c) 2014 American Institute of Physics. 42. 42 Helmich-Paris, B.; Repisky, M.; Visscher, L. Relativistic Cholesky-decomposed density matrix MP2. Chem. Phys. 2019, 518, 38, DOI: 10.1016/j.chemphys.2018.11.009 Google Scholar Relativistic Cholesky-decomposed density matrix MP2 Helmich-Paris, Benjamin; Repisky, Michal; Visscher, Lucas Chemical Physics (2019), 518 (), 38-46CODEN: CMPHC2; ISSN:0301-0104. (Elsevier B.V.) We introduce the relativistic Cholesky-decompd. d. (CDD) matrix second-order Moller-Plesset perturbation theory (MP2) energies. The working equations are formulated in terms of the usual intermediates of MP2 when employing the resoln.-of-the-identity approxn. (RI) for two-electron integrals. Those intermediates are obtained by substituting the occupied and virtual quaternion pseudo-d. matrixes of our previously proposed two-component (2C) AO-based MP2 (Helmich-Paris et al., 2016) by the corresponding pivoted quaternion Cholesky factors. While working within the Kramers-restricted formalism, we obtain a formal spin-orbit overhead of 16 and 28 for the Coulomb and exchange contribution to the 2C MP2 correlation energy, resp., compared to a non-relativistic (NR) spin-free CDD-MP2 implementation. This compact quaternion formulation could also be easily explored in any other algorithm to compute the 2C MP2 energy. The quaternion Cholesky factors become sparse for large mols. and, with a block-wise screening, block sparse-matrix multiplication algorithm, we obsd. an effective quadratic scaling of the total wall time for heavy-element contg. linear mols. with increasing system size. The total run time for both NR and 2C calcns. was dominated by the contraction to the exchange energy. We have also investigated a bulky Te-contg. supramol. complex. For such bulky, three-dimensionally extended mols. the present screening scheme has a much larger prefactor and is less effective. 43. 43 Glasbrenner, M.; Graf, D.; Ochsenfeld, C. Efficient Reduced-Scaling Second-Order Møller-Plesset Perturbation Theory with Cholesky-Decomposed Densities and an Attenuated Coulomb Metric. J. Chem. Theory Comput. 2020, 16, 6856, DOI: 10.1021/acs.jctc.0c00600 Google Scholar Efficient Reduced-Scaling Second-Order Moller-Plesset Perturbation Theory with Cholesky-Decomposed Densities and an Attenuated Coulomb Metric Glasbrenner, Michael; Graf, Daniel; Ochsenfeld, Christian Journal of Chemical Theory and Computation (2020), 16 (11), 6856-6868CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society) We present a novel, highly efficient method for the computation of second-order Moller-Plesset perturbation theory (MP2) correlation energies, which uses the resoln. of the identity (RI) approxn. and local MOs obtained from a Cholesky decompn. of pseudodensity matrixes (CDD), as in the RI-CDD-MP2 method developed previously in our group [Maurer, S.A. et al., J. Chem. Phys., 2014, 140, 224112]. In addn., we introduce an attenuated Coulomb metric and subsequently redesign the RI-CDD-MP2 method in order to exploit the resulting sparsity in the three-center integrals. Coulomb and exchange energy contributions are computed sep. using specialized algorithms. A simple, yet effective integral screening protocol based on Schwarz ests. is used for the MP2 exchange energy. The Coulomb energy computation and the preceding transformations of the three-center integrals are accelerated using a modified version of the natural blocking approach [Jung, Y., Head-Gordon, M., Phys. Chem. Chem. Phys., 2006, 8, 2831]. Effective subquadratic scaling for a wide range of mol. sizes is demonstrated in test calcns. in conjunction with a low prefactor. The method is shown to enable cost-efficient MP2 calcns. on large mol. systems with several thousand basis functions. 44. 44 Neuhauser, D.; Rabani, E.; Baer, R. Expeditious Stochastic Approach for MP2 Energies in Large Electronic Systems. J. Chem. Theory Comput. 2013, 9, 24, DOI: 10.1021/ct300946j Google Scholar Expeditious Stochastic Approach for MP2 Energies in Large Electronic Systems Neuhauser, Daniel; Rabani, Eran; Baer, Roi Journal of Chemical Theory and Computation (2013), 9 (1), 24-27CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society) A fast stochastic method for calcg. the second order Moller-Plesset (MP2) correction to the correlation energy of large systems of electrons is presented. The approach is based on reducing the exact summation over occupied and unoccupied states to a time-dependent trace formula amenable to stochastic sampling. We demonstrate the abilities of the method to treat systems with thousands of electrons using hydrogen passivated silicon spherical nanocrystals represented on a real space grid, much beyond the capabilities of present day MP2 implementations. 45. 45 Willow, S. Y.; Kim, K. S.; Hirata, S. Stochastic evaluation of second-order many-body perturbation energies. J. Chem. Phys. 2012, 137, 204122 DOI: 10.1063/1.4768697 Google Scholar Stochastic evaluation of second-order many-body perturbation energies Willow, Soohaeng Yoo; Kim, Kwang S.; Hirata, So Journal of Chemical Physics (2012), 137 (20), 204122/1-204122/5CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) With the aid of the Laplace transform, the canonical expression of the second-order many-body perturbation correction to electronic energy is converted into a sum of two 13-dimensional integrals, the 12-dimensional parts of which are evaluated by Monte Carlo integration. Wt. functions are identified that are anal. normalizable, finite, non-neg. everywhere, and share the same singularities as the integrands. They generate appropriate distributions of four-electron walkers via the Metropolis algorithm, yielding correlation energies of small mols. within a few mEh of the correct values after 108 Monte Carlo steps. This algorithm does away with the integral transformation as the hotspot of the usual algorithms, has a far superior size dependence of cost, does not suffer from the sign problem of some quantum Monte Carlo methods, and potentially easily parallelizable and extensible to other more complex electron-correlation theories. (c) 2012 American Institute of 46. 46 Barca, G. M. J.; McKenzie, S. C.; Bloomfield, N. J.; Gilbert, A. T. B.; Gill, P. M. W. Q-MP2-OS: Møller-Plesset Correlation Energy by Quadrature. J. Chem. Theory Comput. 2020, 16, 1568, DOI: Google Scholar Q-MP2-OS: Moller-Plesset Correlation Energy by Quadrature Barca, Giuseppe M. J.; McKenzie, Simon C.; Bloomfield, Nathaniel J.; Gilbert, Andrew T. B.; Gill, Peter M. W. Journal of Chemical Theory and Computation (2020), 16 (3), 1568-1577CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society) We present a quadrature-based algorithm for computing the opposite-spin component of the MP2 correlation energy which scales quadratically with basis set size and is well-suited to large-scale parallelization. The key ideas, which are rooted in the earlier work of Hirata and co-workers, are to abandon all two-electron integrals, recast the energy as a seven-dimensional integral, approx. that integral by quadrature, and employ a cutoff strategy to minimize the no. of intermediate quantities. We discuss our implementation in detail and show that it parallelizes almost perfectly on 840 cores for cyclosporine (a mol. with roughly 200 atoms), exhibits scaling for a sequence of polyglycines, and is principally limited by the accuracy of its quadrature. 47. 47 Martínez, T. J.; Carter, E. A. Pseudospectral Møller-Plesset perturbation theory through third order. J. Chem. Phys. 1994, 100, 3631, DOI: 10.1063/1.466350 Google Scholar Pseudospectral Moeller-Plesset perturbation theory through third order Martinez, Todd J.; Carter, Emily A. Journal of Chemical Physics (1994), 100 (5), 3631-8CODEN: JCPSA6; ISSN:0021-9606. The authors present a formulation and implementation of Moeller-Plesset perturbation theory in a pseudospectral framework. At the second-order level, the pseudospectral formulation is a formally a factor of N/n faster than conventional approaches, while the third order is formally faster by a factor of n, where N is the no. of AOs and n is the no. of occupied orbitals. The accuracy of the resulting energies is probed for a no. of test cases. Practical timings are presented and show conclusively that the pseudospectral formulation is faster than conventional ones. 48. 48 Kossmann, S.; Neese, F. Efficient Structure Optimization with Second-Order Many-Body Perturbation Theory: The RIJCOSX-MP2 Method. J. Chem. Theory Comput. 2010, 6, 2325, DOI: 10.1021/ct100199k Google Scholar Efficient Structure Optimization with Second-Order Many-Body Perturbation Theory: The RIJCOSX-MP2 Method Kossmann, Simone; Neese, Frank Journal of Chemical Theory and Computation (2010), 6 (8), 2325-2338CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society) Efficient energy calcns. and structure optimizations employing second-order Moller-Plesset perturbation theory (MP2) are presented. The application of the RIJCOSX approxn., which involves different approxns. for the formation of the Coulomb- and exchange-type matrixes, to MP2 theory is demonstrated. The RIJCOSX approxn. incorporates the resoln. of the identity' approxn. in terms of a Split-RI-J variant for the evaluation of the Coulomb matrixes and a seminumeric exchange treatment via the chain-of-spheres' algorithm for the formation of the exchange-type matrixes. Beside the derivation of the working equations, the RIJCOSX-MP2 method is benchmarked against the original MP2 and the already highly efficient RI-MP2 method. Energies as well as gradients are computed employing various basis sets and are compared to the conventional MP2 results concerning accuracy and total wall clock times. Speedups of typically a factor of 5-7 in comparison to MP2 can be obsd. for the largest basis set employed in our study. Total energies are reproduced with an av. error of ≤0.8 kcal/mol and min. energy geometries differ by ∼0.1 pm in bond lengths and typically ∼0.2 degrees in bond angles. The RIJCOSX-MP2 gradient parallelizes with a speedup of 8.2 on 10 processors. The algorithms are implemented into the ORCA electronic structure package. 49. 49 Maslen, P. E.; Head-Gordon, M. Non-iterative local second order Møller-Plesset theory. Chem. Phys. Lett. 1998, 283, 102, DOI: 10.1016/S0009-2614(97)01333-X Google Scholar Non-iterative local second order Moller-Plesset theory Maslen, P. E.; Head-Gordon, M. Chemical Physics Letters (1998), 283 (1,2), 102-108CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.) Second order Moller-Plesset perturbation theory (MP2) is formulated in terms of atom-centered occupied and virtual orbitals. Both the occupied and the virtual orbitals are non-orthogonal. A new parameter-free atoms-in-mols. local approxn. is employed to reduce the cost of the calcn. to cubic scaling, and a quasi-canonical two-particle basis is introduced to enable the soln. of the local MP2 equations via explicit matrix diagonalization rather than iteration. 50. 50 Jung, Y.; Shao, Y.; Head-Gordon, M. Fast evaluation of scaled opposite-spin second-order Møller-Plesset correlation energies using auxiliary basis expansions and exploiting sparsity. J. Comput. Chem. 2007, 28, 1953, DOI: 10.1002/jcc.20590 Google Scholar Fast evaluation of scaled opposite spin second-order Moller-Plesset correlation energies using auxiliary basis expansions and exploiting sparsity Jung, Yousung; Shao, Yihan; Head-Gordon, Martin Journal of Computational Chemistry (2007), 28 (12), 1953-1964CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.) The scaled opposite spin Moller-Plesset method (SOS-MP2) is an economical way of obtaining correlation energies that are computationally cheaper, and yet, in a statistical sense, of higher quality than std. MP2 theory, by introducing one empirical parameter. But SOS-MP2 still has a fourth-order scaling step that makes the method inapplicable to very large mol. systems. We reduce the scaling of SOS-MP2 by exploiting the sparsity of expansion coeffs. and local integral matrixes, by performing local auxiliary basis expansions for the occupied-virtual product distributions. To exploit sparsity of 3-index local quantities, we use a blocking scheme in which entire zero-rows and columns, for a given third global index, are deleted by comparison against a numerical threshold. This approach minimizes sparse matrix book-keeping overhead, and also provides sufficiently large submatrixes after blocking, to allow efficient matrix-matrix multiplies. The resulting algorithm is formally cubic scaling, and requires only moderate computational resources (quadratic memory and disk space) and, in favorable cases, is shown to yield effective quadratic scaling behavior in the size regime we can apply it to. Errors assocd. with local fitting using the attenuated Coulomb metric and numerical thresholds in the blocking procedure are found to be insignificant in terms of the predicted relative energies. A diverse set of test calcns. shows that the size of system where significant computational savings can be achieved depends strongly on the dimensionality of the system, and the extent of localizability of the MOs. 51. 51 Förster, A.; Franchini, M.; van Lenthe, E.; Visscher, L. A Quadratic Pair Atomic Resolution of the Identity Based SOS-AO-MP2 Algorithm Using Slater Type Orbitals. J. Chem. Theory Comput. 2020, 16 , 875, DOI: 10.1021/acs.jctc.9b00854 Google Scholar A Quadratic Pair Atomic Resolution of the Identity Based SOS-AO-MP2 Algorithm Using Slater Type Orbitals Forster Arno; Franchini Mirko; Visscher Lucas; Franchini Mirko; van Lenthe Erik Journal of chemical theory and computation (2020), 16 (2), 875-891 ISSN:. We report a production level implementation of pair atomic resolution of the identity (PARI) based second-order Moller-Plesset perturbation theory (MP2) in the Slater type orbital (STO) based Amsterdam Density Functional (ADF) code. As demonstrated by systematic benchmarks, dimerization and isomerization energies obtained with our code using STO basis sets of triple-ζ-quality show mean absolute deviations from Gaussian type orbital, canonical, basis set limit extrapolated, global density fitting (DF)-MP2 results of less than 1 kcal/mol. Furthermore, we introduce a quadratic scaling atomic orbital based spin-opposite-scaled (SOS)-MP2 approach with a very small prefactor. Due to a worst-case scaling of [Formula: see text], our implementation is very fast already for small systems and shows an exceptionally early crossover to canonical SOS-PARI-MP2. We report computational wall time results for linear as well as for realistic three-dimensional molecules and show that triple-ζ quality calculations on molecules of several hundreds of atoms are only a matter of a few hours on a single compute node, the bottleneck of the computations being the SCF rather than the post-SCF energy correction. 52. 52 Förster, A.; Visscher, L. Double hybrid DFT calculations with Slater type orbitals. J. Comput. Chem. 2020, 41, 1660, DOI: 10.1002/jcc.26209 Google Scholar Double hybrid DFT calculations with Slater type orbitals Forster Arno; Visscher Lucas Journal of computational chemistry (2020), 41 (18), 1660-1684 ISSN:. On a comprehensive database with 1,644 datapoints, covering several aspects of main-group as well as of transition metal chemistry, we assess the performance of 60 density functional approximations (DFA), among them 36 double hybrids (DH). All calculations are performed using a Slater type orbital (STO) basis set of triple-ζ (TZ) quality and the highly efficient pair atomic resolution of the identity approach for the exchange- and Coulomb-term of the KS matrix (PARI-K and PARI-J, respectively) and for the evaluation of the MP2 energy correction (PARI-MP2). Employing the quadratic scaling SOS-AO-PARI-MP2 algorithm, DHs based on the spin-opposite-scaled (SOS) MP2 approximation are benchmarked against a database of large molecules. We evaluate the accuracy of STO/PARI calculations for B3LYP as well as for the DH B2GP-PLYP and show that the combined basis set and PARI-error is comparable to the one obtained using the well-known def2-TZVPP Gaussian-type basis set in conjunction with global density fitting. While quadruple-ζ (QZ) calculations are currently not feasible for PARI-MP2 due to numerical issues, we show that, on the TZ level, Jacob's ladder for classifying DFAs is reproduced. However, while the best DHs are more accurate than the best hybrids, the improvements are less pronounced than the ones commonly found on the QZ level. For conformers of organic molecules and noncovalent interactions where very high accuracy is required for qualitatively correct results, DHs provide only small improvements over hybrids, while they still excel in thermochemistry, kinetics, transition metal chemistry and the description of strained organic systems. 53. 53 Hohenstein, E. G.; Parrish, R. M.; Martínez, T. J. Tensor hypercontraction density fitting. I. Quartic scaling second- and third-order Møller-Plesset perturbation theory. J. Chem. Phys. 2012, 137 , 044103 DOI: 10.1063/1.4732310 Google Scholar Tensor hypercontraction density fitting. I. Quartic scaling second- and third-order Moller-Plesset perturbation theory Hohenstein, Edward G.; Parrish, Robert M.; Martinez, Todd J. Journal of Chemical Physics (2012), 137 (4), 044103/1-044103/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) Many approxns. have been developed to help deal with the O(N4) growth of the electron repulsion integral (ERI) tensor, where N is the no. of one-electron basis functions used to represent the electronic wavefunction. Of these, the d. fitting (DF) approxn. is currently the most widely used despite the fact that it is often incapable of altering the underlying scaling of computational effort with respect to mol. size. We present a method for exploiting sparsity in three-center overlap integrals through tensor decompn. to obtain a low-rank approxn. to d. fitting (tensor hypercontraction d. fitting or THC-DF). This new approxn. reduces the 4th-order ERI tensor to a product of five matrixes, simultaneously reducing the storage requirement as well as increasing the flexibility to regroup terms and reduce scaling behavior. As an example, we demonstrate such a scaling redn. for second- and third-order perturbation theory (MP2 and MP3), showing that both can be carried out in O(N4) operations. This should be compared to the usual scaling behavior of O(N5) and O(N6) for MP2 and MP3, resp. The THC-DF technique can also be applied to other methods in electronic structure theory, such as coupled-cluster and CI, promising significant gains in computational efficiency and storage redn. (c) 2012 American Institute of Physics. 54. 54 Bangerter, F. H.; Glasbrenner, M.; Ochsenfeld, C. Low-Scaling Tensor Hypercontraction in the Cholesky Molecular Orbital Basis Applied to Second-Order Møller-Plesset Perturbation Theory. J. Chem. Theory Comput. 2021, 17, 211, DOI: 10.1021/acs.jctc.0c00934 Google Scholar Low-Scaling Tensor Hypercontraction in the Cholesky Molecular Orbital Basis Applied to Second-Order Moller-Plesset Perturbation Theory Bangerter, Felix H.; Glasbrenner, Michael; Ochsenfeld, Christian Journal of Chemical Theory and Computation (2021), 17 (1), 211-221CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society) We employ various reduced scaling techniques to accelerate the recently developed least-squares tensor hypercontraction (LS-THC) approxn. [Parrish, R.M. et al., J. Chem. Phys. 2012, 137, 224106] for electron repulsion integrals (ERIs) and apply it to second-order Moller-Plesset perturbation theory (MP2). The grid-projected ERI tensors are efficiently constructed using a localized Cholesky MO basis from d.-fitted integrals with an attenuated Coulomb metric. Addnl., rigorous integral screening and the natural blocking matrix format are applied to reduce the complexity of this step. By recasting the equations to form the quantized representation of the 1/r operator Z into the form of a system of linear equations, the bottleneck of inverting the grid metric via pseudoinversion is removed. This leads to a reduced scaling THC algorithm and application to MP2 yields the (sub-)quadratically scaling THC-ω-RI-CDD-SOS-MP2 method. The efficiency of this method is assessed for various systems including DNA fragments with over 8000 basis functions and the subquadratic scaling is illustrated. 55. 55 Del Ben, M.; Hutter, J.; VandeVondele, J. Second-Order Møller-Plesset Perturbation Theory in the Condensed Phase: An Efficient and Massively Parallel Gaussian and Plane Waves Approach. J. Chem. Theory Comput. 2012, 8, 4177, DOI: 10.1021/ct300531w Google Scholar Second-Order Moller-Plesset Perturbation Theory in the Condensed Phase: An Efficient and Massively Parallel Gaussian and Plane Waves Approach Del Ben, Mauro; Hutter, Jurg; VandeVondele, Joost Journal of Chemical Theory and Computation (2012), 8 (11), 4177-4188CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society) A novel algorithm, based on a hybrid Gaussian and plane waves (GPW) approach, is developed for the canonical second-order Moller-Plesset perturbation energy (MP2) of finite and extended systems. The key aspect of the method is that the electron repulsion integrals (ia\|λσ) are computed by direct integration between the products of Gaussian basis functions λσ and the electrostatic potential arising from a given occupied-virtual pair d. ia. The electrostatic potential is obtained in a plane waves basis set after solving the Poisson equation in Fourier space. In particular, for condensed phase systems, this scheme is highly efficient. Furthermore, our implementation has low memory requirements and displays excellent parallel scalability up to 100 000 processes. In this way, canonical MP2 calcns. for condensed phase systems contg. hundreds of atoms or more than 5000 basis functions can be performed within minutes, while systems up to 1000 atoms and 10 000 basis functions remain feasible. Solid LiH has been employed as a benchmark to study basis set and system size convergence. Lattice consts. and cohesive energies of various mol. crystals have been studied with MP2 and double-hybrid functionals. 56. 56 Katouda, M.; Naruse, A.; Hirano, Y.; Nakajima, T. Massively parallel algorithm and implementation of RI-MP2 energy calculation for peta-scale many-core supercomputers. J. Comput. Chem. 2016, 37, 2623, DOI: 10.1002/jcc.24491 Google Scholar Massively parallel algorithm and implementation of RI-MP2 energy calculation for peta-scale many-core supercomputers Katouda, Michio; Naruse, Akira; Hirano, Yukihiko; Nakajima, Takahito Journal of Computational Chemistry (2016), 37 (30), 2623-2633CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.) A new parallel algorithm and its implementation for the RI-MP2 energy calcn. utilizing peta-flop-class many-core supercomputers are presented. Some improvements from the previous algorithm have been performed: (1) a dual-level hierarchical parallelization scheme that enables the use of more than 10,000 Message Passing Interface (MPI) processes and (2) a new data communication scheme that reduces network communication overhead. A multi-node and multi-GPU implementation of the present algorithm is presented for calcns. on a central processing unit (CPU)/graphics processing unit (GPU) hybrid supercomputer. Benchmark results of the new algorithm and its implementation using the K computer (CPU clustering system) and TSUBAME 2.5 (CPU/GPU hybrid system) demonstrate high efficiency. The peak performance of 3.1 PFLOPS is attained using 80,199 nodes of the K computer. The peak performance of the multi-node and multi-GPU implementation is 514 TFLOPS using 1349 nodes and 4047 GPUs of TSUBAME 2.5. 57. 57 Zaleśny, R.; Papadopoulos, M.; Mezey, P.; Leszczynski, J. Linear-Scaling Techniques in Computational Chemistry and Physics: Methods and Applications; Challenges and Advances in Computational Chemistry and Physics; Springer: Netherlands, 2011. 58. 58 Herbert, J. M. Fantasy versus reality in fragment-based quantum chemistry. J. Chem. Phys. 2019, 151, 170901 DOI: 10.1063/1.5126216 Google Scholar Fantasy versus reality in fragment-based quantum chemistry Herbert, John M. Journal of Chemical Physics (2019), 151 (17), 170901/1-170901/38CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) A review. Since the introduction of the fragment MO method 20 years ago, fragment-based approaches have occupied a small but growing niche in quantum chem. These methods decomp. a large mol. system into subsystems small enough to be amenable to electronic structure calcns., following which the subsystem information is reassembled in order to approx. an otherwise intractable supersystem calcn. Fragmentation sidesteps the steep rise (with respect to system size) in the cost of ab initio calcns., replacing it with a distributed cost across numerous computer processors. Such methods are attractive, in part, because they are easily parallelizable and therefore readily amenable to exascale computing. As such, there has been hope that distributed computing might offer the proverbial "free lunch" in quantum chem., with the entre´e being high-level calcns. on very large systems. While fragment-based quantum chem. can count many success stories, there also exists a seedy underbelly of rarely acknowledged problems. As these methods begin to mature, it is time to have a serious conversation about what they can and cannot be expected to accomplish in the near future. Both successes and challenges are highlighted in this Perspective. (c) 2019 American Institute of Physics. 59. 59 Fragmentation: Toward Accurate Calculations on Complex Molecular Systems; Gordon, M., Ed.; Wiley: New York, 2017. 60. 60 Pulay, P. Localizability of dynamic electron correlation. Chem. Phys. Lett. 1983, 100, 151, DOI: 10.1016/0009-2614(83)80703-9 Google Scholar Localizability of dynamic electron correlation Pulay, Peter Chemical Physics Letters (1983), 100 (2), 151-4CODEN: CHPLBC; ISSN:0009-2614. The convergence of the intrapair correlation energy for a localized internal orbital was studied as the virtual subspace was enlarged. At variance with previous investigation of this kind, the virtual subspace was represented in AO's, which allowed definition of the spatial relations between the orbitals involved. Typically, over 98% of the pair correlation energy was recovered with a small local basis set consisting of the valence orbitals of the atoms with which the electron pair was assocd. This opens the possibility of an efficient CI procedure based on localized pairs. The mols. considered are H2O2, butadiene, and propane. 61. 61 Kurashige, Y.; Yang, J.; Chan, G. K.-L.; Manby, F. R. Optimization of orbital-specific virtuals in local Møller-Plesset perturbation theory. J. Chem. Phys. 2012, 136, 124106 DOI: 10.1063/ Google Scholar Optimization of orbital-specific virtuals in local Moller-Plesset perturbation theory Kurashige, Yuki; Yang, Jun; Chan, Garnet K.-L.; Manby, Frederick R. Journal of Chemical Physics (2012), 136 (12), 124106/1-124106/7CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) We present an orbital-optimized version of our orbital-specific-virtuals second-order Moller-Plesset perturbation theory (OSV-MP2). The OSV model is a local correlation ansatz with a small basis of virtual functions for each occupied orbital. It is related to the Pulay-Saebo approach, in which domains of virtual orbitals are drawn from a single set of projected AOs; but here the virtual functions assocd. with a particular occupied orbital are specifically tailored to the correlation effects in which that orbital participates. In this study, the shapes of the OSVs are optimized simultaneously with the OSV-MP2 amplitudes by minimizing the Hylleraas functional or approxns. to it. It is found that optimized OSVs are considerably more accurate than the OSVs obtained through singular value decompn. of diagonal blocks of MP2 amplitudes, as used in our earlier work. Orbital-optimized OSV-MP2 recovers smooth potential energy surfaces regardless of the no. of virtuals. Full optimization is still computationally demanding, but orbital optimization in a diagonal or Kapuy-type MP2 approxn. provides an attractive scheme for detg. accurate OSVs. (c) 2012 American Institute of Physics. 62. 62 Riplinger, C.; Neese, F. An efficient and near linear scaling pair natural orbital based local coupled cluster method. J. Chem. Phys. 2013, 138, 034106 DOI: 10.1063/1.4773581 Google Scholar An efficient and near linear scaling pair natural orbital based local coupled cluster method Riplinger, Christoph; Neese, Frank Journal of Chemical Physics (2013), 138 (3), 034106/1-034106/18CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) In previous publications, it was shown that an efficient local coupled cluster method with single- and double excitations can be based on the concept of pair natural orbitals (PNOs) . The resulting local pair natural orbital-coupled-cluster single double (LPNO-CCSD) method has since been proven to be highly reliable and efficient. For large mols., the no. of amplitudes to be detd. is reduced by a factor of 105-106 relative to a canonical CCSD calcn. on the same system with the same basis set. In the original method, the PNOs were expanded in the set of canonical virtual orbitals and single excitations were not truncated. This led to a no. of fifth order scaling steps that eventually rendered the method computationally expensive for large mols. (e.g., >100 atoms). In the present work, these limitations are overcome by a complete redesign of the LPNO-CCSD method. The new method is based on the combination of the concepts of PNOs and projected AOs (PAOs). Thus, each PNO is expanded in a set of PAOs that in turn belong to a given electron pair specific domain. In this way, it is possible to fully exploit locality while maintaining the extremely high compactness of the original LPNO-CCSD wavefunction. No terms are dropped from the CCSD equations and domains are chosen conservatively. The correlation energy loss due to the domains remains below <0.05%, which implies typically 15-20 but occasionally up to 30 atoms per domain on av. The new method has been given the acronym DLPNO-CCSD ("domain based LPNO-CCSD"). The method is nearly linear scaling with respect to system size. The original LPNO-CCSD method had three adjustable truncation thresholds that were chosen conservatively and do not need to be changed for actual applications. In the present treatment, no addnl. truncation parameters have been introduced. Any addnl. truncation is performed on the basis of the three original thresholds. There are no real-space cutoffs. Single excitations are truncated using singles-specific natural orbitals. Pairs are prescreened according to a multipole expansion of a pair correlation energy est. based on local orbital specific virtual orbitals (LOSVs). Like its LPNO-CCSD predecessor, the method is completely of black box character and does not require any user adjustments. It is shown here that DLPNO-CCSD is as accurate as LPNO-CCSD while leading to computational savings exceeding one order of magnitude for larger systems. The largest calcns. reported here featured >8800 basis functions and >450 atoms. In all larger test calcns. done so far, the LPNO-CCSD step took less time than the preceding Hartree-Fock calcn., provided no approxns. have been introduced in the latter. Thus, based on the present development reliable CCSD calcns. on large mols. with unprecedented efficiency and accuracy are realized. (c) 2013 American Institute of Physics. 63. 63 Saitow, M.; Becker, U.; Riplinger, C.; Valeev, E. F.; Neese, F. A new near-linear scaling, efficient and accurate, open-shell domain-based local pair natural orbital coupled cluster singles and doubles theory. J. Chem. Phys. 2017, 146, 164105 DOI: 10.1063/1.4981521 Google Scholar A new near-linear scaling, efficient and accurate, open-shell domain-based local pair natural orbital coupled cluster singles and doubles theory Saitow, Masaaki; Becker, Ute; Riplinger, Christoph; Valeev, Edward F.; Neese, Frank Journal of Chemical Physics (2017), 146 (16), 164105/1-164105/31CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) The Coupled-Cluster expansion, truncated after single and double excitations (CCSD), provides accurate and reliable mol. electronic wave functions and energies for many mol. systems around their equil. geometries. However, the high computational cost, which is well-known to scale as O(N6) with system size N, has limited its practical application to small systems consisting of not more than approx. 20-30 atoms. To overcome these limitations, low-order scaling approxns. to CCSD have been intensively investigated over the past few years. In our previous work, we have shown that by combining the pair natural orbital (PNO) approach and the concept of orbital domains it is possible to achieve fully linear scaling CC implementations (DLPNO-CCSD and DLPNO-CCSD(T)) that recover around 99.9% of the total correlation energy [C. Riplinger et al., J. Chem. Phys. 144, 024109 (2016)]. The prodn. level implementations of the DLPNO-CCSD and DLPNO-CCSD(T) methods were shown to be applicable to realistic systems composed of a few hundred atoms in a routine, black-box fashion on relatively modest hardware. In 2011, a reduced-scaling CCSD approach for high-spin open-shell UHF ref. wave functions was proposed (UHF-LPNO-CCSD) [A. Hansen et al., J. Chem. Phys. 135, 214102 (2011)]. After a few years of experience with this method, a few shortcomings of UHF-LPNO-CCSD were noticed that required a redesign of the method, which is the subject of this paper. To this end, we employ the high-spin open-shell variant of the N-electron valence perturbation theory formalism to define the initial guess wave function, and consequently also the open-shell PNOs. The new PNO ansatz properly converges to the closed-shell limit since all truncations and approxns. have been made in strict analogy to the closed-shell case. Furthermore, given the fact that the formalism uses a single set of orbitals, only a single PNO integral transformation is necessary, which offers large computational savings. We show that, with the default PNO truncation parameters, approx. 99.9% of the total CCSD correlation energy is recovered for open-shell species, which is comparable to the performance of the method for closed-shells. UHF-DLPNO-CCSD shows a linear scaling behavior for closed-shell systems, while linear to quadratic scaling is obtained for open-shell systems. The largest systems we have considered contain more than 500 atoms and feature more than 10 000 basis functions with a triple-ζ quality basis set. (c) 2017 American Institute of Physics. 64. 64 Ma, Q.; Werner, H.-J. Explicitly correlated local coupled-cluster methods using pair natural orbitals. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2018, 8, e1371, DOI: 10.1002/wcms.1371 65. 65 Krause, C.; Werner, H.-J. Scalable Electron Correlation Methods. 6. Local Spin-Restricted Open-Shell Second-Order Møller-Plesset Perturbation Theory Using Pair Natural Orbitals: PNO-RMP2. J. Chem. Theory Comput. 2019, 15, 987, DOI: 10.1021/acs.jctc.8b01012 66. 66 Hättig, C.; Tew, D. P.; Helmich, B. Local explicitly correlated second- and third-order Møller-Plesset perturbation theory with pair natural orbitals. J. Chem. Phys. 2012, 136, 204105 DOI: Google Scholar Local explicitly correlated second- and third-order Moller-Plesset perturbation theory with pair natural orbitals Hattig Christof; Tew David P; Helmich Benjamin The Journal of chemical physics (2012), 136 (20), 204105 ISSN:. We present an algorithm for computing explicitly correlated second- and third-order Moller-Plesset energies near the basis set limit for large molecules with a cost that scales formally as N(4) with system size N. This is achieved through a hybrid approach where locality is exploited first through orbital specific virtuals (OSVs) and subsequently through pair natural orbitals (PNOs) and integrals are approximated using density fitting. Our method combines the low orbital transformation costs of the OSVs with the compactness of the PNO representation of the doubles amplitude vector. The N(4) scaling does not rely upon the a priori definition of domains, enforced truncation of pair lists, or even screening and the energies converge smoothly to the canonical values with decreasing occupation number thresholds, used in the selection of the PNO basis. For MP2.5 intermolecular interaction energies, we find that 99% of benchmark basis set limit correlation energy contributions are recovered using an aug-cc-pVTZ basis and that on average only 50 PNOs are required to correlate the significant orbital pairs. 67. 67 Schmitz, G.; Hättig, C. Perturbative triples correction for local pair natural orbital based explicitly correlated CCSD(F12) using Laplace transformation techniques. J. Chem. Phys. 2016, 145, 234107 DOI: 10.1063/1.4972001 Google Scholar Perturbative triples correction for local pair natural orbital based explicitly correlated CCSD(F12) using Laplace transformation techniques Schmitz, Gunnar; Haettig, Christof Journal of Chemical Physics (2016), 145 (23), 234107/1-234107/15CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) We present an implementation of pair natural orbital coupled cluster singles and doubles with perturbative triples, PNO-CCSD(T), which avoids the quasi-canonical triples approxn. (T0) where couplings due to off-diagonal Fock matrix elements are neglected. A numerical Laplace transformation of the canonical expression for the perturbative (T) triples correction is used to avoid an I/ O and storage bottleneck for the triples amplitudes. Results for a test set of reaction energies show that only very few Laplace grid points are needed to obtain converged energy differences and that PNO-CCSD(T) is a more robust approxn. than PNO-CCSD(T0) with a reduced mean abs. deviation from canonical CCSD(T) results. We combine the PNO-based (T) triples correction with the explicitly correlated PNO-CCSD(F12) method and investigate the use of specialized F12-PNOs in the conventional triples correction. We find that no significant addnl. errors are introduced and that PNO-CCSD (F12)(T) can be applied in a black box manner. (c) 2016 American Institute of Physics. 68. 68 Collins, M. A.; Bettens, R. P. A. Energy-Based Molecular Fragmentation Methods. Chem. Rev. 2015, 115, 5607, DOI: 10.1021/cr500455b Google Scholar Energy-Based Molecular Fragmentation Methods Collins, Michael A.; Bettens, Ryan P. A. Chemical Reviews (Washington, DC, United States) (2015), 115 (12), 5607-5642CODEN: CHREAY; ISSN:0009-2665. (American Chemical Society) A review including the following topics: methods and principles, applications and examples, and speculations and future developments etc. 69. 69 Raghavachari, K.; Saha, A. Accurate Composite and Fragment-Based Quantum Chemical Models for Large Molecules. Chem. Rev. 2015, 115, 5643– 5677, DOI: 10.1021/cr500606e Google Scholar Accurate Composite and Fragment-Based Quantum Chemical Models for Large Molecules Raghavachari, Krishnan; Saha, Arjun Chemical Reviews (Washington, DC, United States) (2015), 115 (12), 5643-5677CODEN: CHREAY; ISSN:0009-2665. (American Chemical Society) A review. A range of composite methods have been reviewed with the ultimate goal of obtaining accurate energies on large mols. Direct calcns. are possible on small mols. using extrapolated coupled cluster approaches to obtain results within chem. accuracy. Medium-sized mols. can be treated with composite models such as Gn. Error-cancellation strategies are discussed for larger mols. using a hierarchy of chem. based ideas. Finally, a variety of fragment-based methods are discussed as important tools to remove the steep computational bottleneck for large mols. A generalized view of classification of all the major fragment-based methods has also been provided. The potential of fragment-based methods in understanding complex chem. and phys. phenomena in large mols., like DNA, proteins, clusters, and crystals, can be definitely seen. With the combination of new methods, algorithms, and rapid developments in high performance computing, fragment-based quantum chem. clearly will have high impact in the next decade. 70. 70 Friedrich, J.; Dolg, M. Fully Automated Incremental Evaluation of MP2 and CCSD(T) Energies: Application to Water Clusters. J. Chem. Theory Comput. 2009, 5, 287, DOI: 10.1021/ct800355e Google Scholar Fully Automated Incremental Evaluation of MP2 and CCSD(T) Energies: Application to Water Clusters Friedrich, Joachim; Dolg, Michael Journal of Chemical Theory and Computation (2009), 5 (2), 287-294CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society) A fully automated implementation of the incremental scheme for CCSD energies has been extended to treat MP2 and CCSD(T) energies. It is shown in applications on water clusters that the error of the incremental expansion for the total energy is below 1 kcal/mol already at second or third order. It is demonstrated that the approach saves CPU time, RAM, and disk space. Finally it is shown that the calcns. can be run in parallel on up to 50 CPUs, without significant loss of computer time. 71. 71 Fiedler, B.; Schmitz, G.; Hättig, C.; Friedrich, J. Combining Accuracy and Efficiency: An Incremental Focal-Point Method Based on Pair Natural Orbitals. J. Chem. Theory Comput. 2017, 13, 6023, DOI: 10.1021/acs.jctc.7b00654 Google Scholar Combining Accuracy and Efficiency: An Incremental Focal-Point Method Based on Pair Natural Orbitals Fiedler, Benjamin; Schmitz, Gunnar; Haettig, Christof; Friedrich, Joachim Journal of Chemical Theory and Computation (2017), 13 (12), 6023-6042CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society) In this work we present a new PNO-based incremental scheme to calc. CCSD(T) and CCSD(T0) reaction, interaction and binding energies. We perform an extensive anal., which shows small incremental errors similar to previous non-PNO calcns. Furthermore, slight PNO errors are obtained by using TPNO = TTNO with appropriate values of 10-7 to 10-8 for reactions and 10-8 for interaction or binding energies. The combination with the efficient MP2 focal-point approach yields chem. accuracy relative to the complete basis-set (CBS) limit. In this method small basis sets (cc-pVDZ, def2-TZVP) for the CCSD(T) part are sufficient in case of reactions or interactions, while some larger ones (e.g. (aug)-cc-pVTZ) are necessary for mol. clusters. For these larger basis sets we show the very high efficiency of our scheme. We obtain not only tremendous decreases of the wall times (i.e. factors > 102) due to the parallelization of the increment calcns. as well as of the total times due to the application of PNOs (i.e. compared to the normal incremental scheme) but also smaller total times with respect to the std. PNO method. That way, our new method features a perfect applicability by combining an excellent accuracy with a very high efficiency as well as the accessibility to larger systems due to the sepn. of the full computation into several small 72. 72 Kobayashi, M.; Nakai, H. Divide-and-conquer-based linear-scaling approach for traditional and renormalized coupled cluster methods with single, double, and noniterative triple excitations. J. Chem. Phys. 2009, 131, 114108 DOI: 10.1063/1.3211119 Google Scholar Divide-and-conquer-based linear-scaling approach for traditional and renormalized coupled cluster methods with single, double, and noniterative triple excitations Kobayashi, Masato; Nakai, Hiromi Journal of Chemical Physics (2009), 131 (11), 114108/1-114108/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) We have reported the divide-and-conquer (DC)-based linear-scaling correlation treatment of coupled-cluster method with single and double excitations (CCSD). In the DC-CCSD method, the CCSD equations derived from subsystem orbitals are solved for each subsystem in order to obtain the total correlation energy by summing up subsystem contributions using energy d. anal. In this study, we extend the DC-CCSD method for treating noniterative perturbative triple excitations using CCSD T1 and T2 amplitudes, namely, CCSD(T). In the DC-CCSD(T) method, the so-called (T) corrections are also computed for each subsystem. Numerical assessments indicate that DC-CCSD(T) reproduces the CCSD(T) results with high accuracy and significantly less computational cost. We further extend the DC-based correlation method to renormalized CCSD(T) for avoiding the divergence that occurs in multireference problems such as bond dissocn. (c) 2009 American Institute of Physics. 73. 73 Nakano, M.; Yoshikawa, T.; Hirata, S.; Seino, J.; Nakai, H. Computerized implementation of higher-order electron-correlation methods and their linear-scaling divide-and-conquer extensions. J. Comput. Chem. 2017, 38, 2520– 2527, DOI: 10.1002/jcc.24912 Google Scholar Computerized implementation of higher-order electron-correlation methods and their linear-scaling divide-and-conquer extensions Nakano, Masahiko; Yoshikawa, Takeshi; Hirata, So; Seino, Junji; Nakai, Hiromi Journal of Computational Chemistry (2017), 38 (29), 2520-2527CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.) We have implemented a linear-scaling divide-and-conquer (DC)-based higher-order coupled-cluster (CC) and Moller-Plesset perturbation theories (MPPT) as well as their combinations automatically by means of the tensor contraction engine, which is a computerized symbolic algebra system. The DC-based energy expressions of the std. CC and MPPT methods and the CC methods augmented with a perturbation correction were proposed for up to high excitation orders [e.g., CCSDTQ, MP4, and CCSD(2)TQ]. The numerical assessment for hydrogen halide chains, polyene chains, and first coordination sphere (C1) model of photoactive yellow protein has revealed that the DC-based correlation methods provide reliable correlation energies with significantly less computational cost than that of the conventional implementations. © 2017 Wiley Periodicals, Inc. 74. 74 Mochizuki, Y.; Yamashita, K.; Nakano, T.; Okiyama, Y.; Fukuzawa, K.; Taguchi, N.; Tanaka, S. Higher-order correlated calculations based on fragment molecular orbital scheme. Theor. Chem. Acc. 2011, 130, 515– 530, DOI: 10.1007/s00214-011-1036-3 Google Scholar Higher-order correlated calculations based on fragment molecular orbital scheme Mochizuki, Yuji; Yamashita, Katsumi; Nakano, Tatsuya; Okiyama, Yoshio; Fukuzawa, Kaori; Taguchi, Naoki; Tanaka, Shigenori Theoretical Chemistry Accounts (2011), 130 (2-3), 515-530CODEN: TCACFW; ISSN:1432-2234. (Springer) We have developed a new module for higher-order correlated methods up to coupled-cluster singles and doubles with perturbative triples (CCSD(T)). The matrix-matrix operations through the routine were pursued for a no. of contractions. This code was then incorporated into the program for the fragment MO (FMO) calcns. Intra-fragment processings were parallelized with OpenMP in a node-wise fashion, whereas the message passing interface (MPI) was used for the fragment-wise parallelization over nodes. Our new implementation made the FMO-based higher-order calcns. applicable to realistic proteins. We have performed several benchmark tests on the Earth Simulator (ES2), a massively parallel computer. For example, the FMO-CCSD(T)/6-31G job for the HIV-1 protease (198 amino acid residues)-lopinavir complex was completed in 9.8 h with 512 processors (or 64 nodes). Another example was the influenza neuraminidase (386 residues) with oseltamivir calcd. at the full fourth-order Moller-Plesset perturbation level (MP4), of which job timing was 10.3 h with 1024 processors. The applicability of the methods to commodity cluster computers was tested as well. 75. 75 Yuan, D.; Li, Y.; Ni, Z.; Pulay, P.; Li, W.; Li, S. Benchmark Relative Energies for Large Water Clusters with the Generalized Energy-Based Fragmentation Method. J. Chem. Theory Comput. 2017, 13, 2696– 2704, DOI: 10.1021/acs.jctc.7b00284 Google Scholar Benchmark Relative Energies for Large Water Clusters with the Generalized Energy-Based Fragmentation Method Yuan, Dandan; Li, Yunzhi; Ni, Zhigang; Pulay, Peter; Li, Wei; Li, Shuhua Journal of Chemical Theory and Computation (2017), 13 (6), 2696-2704CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society) The generalized energy-based fragmentation (GEBF) method has been applied to investigate relative energies of large water clusters (H2O)n (n = 32, 64) with the coupled-cluster singles and doubles with noniterative triple excitations (CCSD(T)) and second-order Moller-Plesset perturbation theory (MP2) at the complete basis set (CBS) limit. Here large water clusters are chosen to be representative structures sampled from mol. dynamics (MD) simulations of liq. water. Our calcns. show that the GEBF method is capable of providing highly accurate relative energies for these water clusters in a cost-effective way. We demonstrate that the relative energies from GEBF-MP2/CBS are in excellent agreement with those from GEBF-CCSD(T)/CBS for these water clusters. With the GEBF-CCSD(T)/CBS relative energies as the benchmark results, we have assessed the performance of several theor. methods widely used for ab initio MD simulations of liqs. and aq. solns. These methods include d. functional theory (DFT) with a no. of different functionals, MP2, and d. functional tight-binding (the third generation, DFTB3 in short). We find that MP2/aug-cc-pVDZ and several DFT methods (such as LC-ωPBE-D3 and ωB97XD) with the aug-cc-pVTZ basis set can provide satisfactory descriptions for these water clusters. Some widely used functionals (such as B3LYP, PBE0) and DFTB3 are not accurate enough for describing the relative energies of large water clusters. Although the basis set dependence of DFT is less than that of ab initio electron correlation methods, we recommend the combination of a few best functionals and large basis sets (at least aug-cc-pVTZ) in theor. studies on water clusters or aq. solns. 76. 76 Li, W.; Ni, Z.; Li, S. Cluster-in-molecule local correlation method for post-Hartree-Fock calculations of large systems. Mol. Phys. 2016, 114, 1447, DOI: 10.1080/00268976.2016.1139755 Google Scholar Cluster-in-molecule local correlation method for post-Hartree-Fock calculations of large systems Li, Wei; Ni, Zhigang; Li, Shuhua Molecular Physics (2016), 114 (9), 1447-1460CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.) Our recent developments on cluster-in-mol. (CIM) local correlation method are reviewed in this paper. In the CIM method, the correlation energy of a large system can be approx. obtained from electron correlation calcns. on a series of clusters, each of which contains a subset of occupied and virtual localised MOs in a certain region. The CIM method is a linear scaling method and its inherent parallelisation allows electron correlation calcns. of very large systems to be feasible at ordinary workstations. In the illustrative applications, this approach is applied to investigate the conformational energy differences, reaction barriers, and binding energies of large systems at the levels of Moller-Plesset perturbation theory and coupled-cluster theory. 77. 77 Findlater, A. D.; Zahariev, F.; Gordon, M. S. Combined Fragment Molecular Orbital Cluster in Molecule Approach to Massively Parallel Electron Correlation Calculations for Large Systems. J. Phys. Chem. A 2015, 119, 3587, DOI: 10.1021/jp509266g Google Scholar Combined Fragment Molecular Orbital Cluster in Molecule Approach to Massively Parallel Electron Correlation Calculations for Large Systems. Findlater, Alexander D.; Zahariev, Federico; Gordon, Mark S. Journal of Physical Chemistry A (2015), 119 (15), 3587-3593CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society) The local correlation "cluster-in-mol." (CIM) method is combined with the fragment MO (FMO) method, providing a flexible, massively parallel, and near-linear scaling approach to the calcn. of electron correlation energies for large mol. systems. Although the computational scaling of the CIM algorithm is already formally linear, previous knowledge of the Hartree-Fock (HF) ref. wave function and subsequent localized orbitals is required; therefore, extending the CIM method to arbitrarily large systems requires the aid of low-scaling/linear-scaling approaches to HF and orbital localization. Through fragmentation, the combined FMO-CIM method linearizes the scaling, with respect to system size, of the HF ref. and orbital localization calcns., achieving near-linear scaling at both the ref. and electron correlation levels. For the 20-residue alanine α helix, the preliminary implementation of the FMO-CIM method captures 99.6% of the MP2 correlation energy, requiring 21% of the MP2 wall time. The new method is also applied to solvated adamantine to illustrate the multilevel capability of the FMO-CIM method. 78. 78 Stoll, H. Correlation energy of diamond. Phys. Rev. B 1992, 46, 6700, DOI: 10.1103/PhysRevB.46.6700 Google Scholar Correlation energy of diamond Stoll, Hermann Physical Review B: Condensed Matter and Materials Physics (1992), 46 (11), 6700-4CODEN: PRBMDO; ISSN:0163-1829. The correlation energy of diamond is detd. by means of increments obtained in ab initio calcns. for localized C-C bond orbitals and for pairs and triples of such bonds. The resulting correlation contribution to the cohesive energy is -0.129 a.u., which is approx. 85% of the exptl. value. 79. 79 Li, W.; Li, S. Divide-and-conquer local correlation approach to the correlation energy of large molecules. J. Chem. Phys. 2004, 121, 6649, DOI: 10.1063/1.1792051 Google Scholar Divide-and-conquer local correlation approach to the correlation energy of large molecules Li, Wei; Li, Shuhua Journal of Chemical Physics (2004), 121 (14), 6649-6657CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) A divide-and-conquer local correlation approach for correlation energy calcns. on large mols. is proposed for any post-Hartree-Fock correlation method. The main idea of this approach is to decomp. a large system into various fragments capped by their local environments. The total correlation energy of the whole system can be approx. obtained as the summation of correlation energies from all capped fragments, from which correlation energies from all adjacent caps are removed. This approach computationally achieves linear scaling even for medium-sized systems. Our test calcns. for a wide range of mols. using the 6-31G or 6-31G** basis set demonstrate that this simple approach recovers more than 99.0% of the conventional second-order Moller-Plesset perturbation theory and coupled cluster with single and double excitations correlation energies. 80. 80 Li, W.; Piecuch, P.; Gour, J. R.; Li, S. Local correlation calculations using standard and renormalized coupled-cluster approaches. J. Chem. Phys. 2009, 131, 114109 DOI: 10.1063/1.3218842 Google Scholar Local correlation calculations using standard and renormalized coupled-cluster approaches Li, Wei; Piecuch, Piotr; Gour, Jeffrey R.; Li, Shuhua Journal of Chemical Physics (2009), 131 (11), 114109/1-114109/30CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) The linear scaling local correlation approach, termed "cluster-in-mol." (CIM), is extended to the coupled-cluster (CC) theory with singles and doubles (CCSD) and CC methods with singles, doubles, and noniterative triples, including CCSD(T) and the completely renormalized CR-CC(2,3) approach. The resulting CIM-CCSD, CIM-CCSD(T), and CIM-CR-CC(2,3) methods are characterized by (i) the linear scaling of the CPU time with the system size, (ii) the use of orthonormal orbitals in the CC subsystem calcns., (iii) the natural parallelism, (iv) the high computational efficiency, enabling calcns. for much larger systems and at higher levels of CC theory than previously possible, and (v) the purely noniterative character of local triples corrections. By comparing the results of the canonical and CIM-CC calcns. for normal alkanes and water clusters, it is shown that the CIM-CCSD, CIM-CCSD(T), and CIM-CR-CC(2,3) approaches accurately reproduce the corresponding canonical CC correlation and relative energies, while offering savings in the computer effort by orders of magnitude. (c) 2009 American Institute of Physics. 81. 81 Fedorov, D. G.; Kitaura, K. Second order Møller-Plesset perturbation theory based upon the fragment molecular orbital method. J. Chem. Phys. 2004, 121, 2483, DOI: 10.1063/1.1769362 Google Scholar Second order Moller-Plesset perturbation theory based upon the fragment molecular orbital method Fedorov, Dmitri G.; Kitaura, Kazuo Journal of Chemical Physics (2004), 121 (6), 2483-2490CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) The fragment MO (FMO) method was combined with the second order Moller-Plesset (MP2) perturbation theory. The accuracy of the method using the 6-31G* basis set was tested on (H2O)n, n = 16,32,64; α-helixes and β-strands of alanine n-mers, n = 10,20,40; as well as on (H2O)n, n = 16,32,64 using the 6-31++G basis set. Relative to the regular MP2 results that could be afforded, the FMO2-MP2 error in the correlation energy did not exceed 0.003 a.u., the error in the correlation energy gradient did not exceed 0.000 05 a.u./bohr and the error in the correlation contribution to dipole moment did not exceed 0.03 debye. An approxn. reducing computational load based on fragment sepn. was introduced and tested. The FMO2-MP2 method demonstrated nearly linear scaling and drastically reduced the memory requirements of the regular MP2, making possible calcns. with several thousands basis functions using small Pentium clusters. As an example, (H2O)64 with the 6-31++G basis set (1920 basis functions) can be run in 1 Gbyte RAM and it took 136 s on a 40-node Pentium4 cluster. 82. 82 Guo, Y.; Li, W.; Li, S. Improved Cluster-in-Molecule Local Correlation Approach for Electron Correlation Calculation of Large Systems. J. Phys. Chem. A 2014, 118, 8996, DOI: 10.1021/jp501976x Google Scholar Improved Cluster-in-Molecule Local Correlation Approach for Electron Correlation Calculation of Large Systems Guo, Yang; Li, Wei; Li, Shuhua Journal of Physical Chemistry A (2014), 118 (39), 8996-9004CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society) An improved cluster-in-mol. (CIM) local correlation approach is developed to allow electron correlation calcns. of large systems more accurate and faster. We have proposed a refined strategy of constructing virtual LMOs of various clusters, which is suitable for basis sets of various types. To recover medium-range electron correlation, which is important for quant. descriptions of large systems, we find that a larger distance threshold (ξ) is necessary for highly accurate results. Our illustrative calcns. show that the present CIM-MP2 (second-order Moller-Plesset perturbation theory, MP2) or CIM-CCSD (coupled cluster singles and doubles, CCSD) scheme with a suitable ξ value is capable of recovering more than 99.8% correlation energies for a wide range of systems at different basis sets. The present CIM-MP2 scheme can provide reliable relative energy differences as the conventional MP2 method for secondary structures of polypeptides. 83. 83 Kobayashi, M.; Imamura, Y.; Nakai, H. Alternative linear-scaling methodology for the second-order Møller-Plesset perturbation calculation based on the divide-and-conquer method. J. Chem. Phys. 2007, 127, 074103 DOI: 10.1063/1.2761878 Google Scholar Alternative linear-scaling methodology for the second-order Moeller-Plesset perturbation calculation based on the divide-and-conquer method Kobayashi, Masato; Imamura, Yutaka; Nakai, Hiromi Journal of Chemical Physics (2007), 127 (7), 074103/1-074103/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) A new scheme for obtaining the approx. correlation energy in the divide-and-conquer (DC) method of Yang [Phys. Rev. Lett. 66, 1438 (1991)] is presented. In this method, the correlation energy of the total system is evaluated by summing up subsystem contributions, which are calcd. from subsystem orbitals based on a scheme for partitioning the correlation energy. We applied this method to the second-order Moeller-Plesset perturbation theory (MP2), which we call DC-MP2. Numerical assessment revealed that this scheme provides a reliable correlation energy with significantly less computational cost than the conventional MP2 calcn. 84. 84 Ziółkowski, M.; Jansík, B.; Kjærgaard, T.; Jørgensen, P. Linear scaling coupled cluster method with correlation energy based error control. J. Chem. Phys. 2010, 133, 014107 DOI: 10.1063/1.3456535 Google Scholar Linear scaling coupled cluster method with correlation energy based error control Ziolkowski, Marcin; Jansik, Branislav; Kjaergaard, Thomas; Jorgensen, Poul Journal of Chemical Physics (2010), 133 (1), 014107/1-014107/5CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) Coupled cluster calcns. can be carried out for large mol. systems via a set of calcns. that use small orbital fragments of the full MO space. The error in the correlation energy of the full mol. system is controlled by the precision in the small fragment calcns. The detn. of the orbital spaces for the small orbital fragments is black box in the sense that it does not depend on any user-provided mol. fragmentation, rather orbital spaces are carefully selected and extended during the calcn. to give fragment energies of a specified precision. The computational method scales linearly with the size of the mol. system and is massively parallel. (c) 2010 American Institute of Physics. 85. 85 Kjærgaard, T. The Laplace transformed divide-expand-consolidate resolution of the identity second-order Møller-Plesset perturbation (DEC-LT-RIMP2) theory method. J. Chem. Phys. 2017, 146, 044103 DOI: 10.1063/1.4973710 Google Scholar The Laplace transformed divide-expand-consolidate resolution of the identity second-order Moller-Plesset perturbation (DEC-LT-RIMP2) theory method Kjaergaard, Thomas Journal of Chemical Physics (2017), 146 (4), 044103/1-044103/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) The divide-expand-consolidate resoln. of the identity second-order Moller-Plesset perturbation (DEC-RI-MP2) theory method introduced in Baudin et al. [J. Chem. Phys. 144, 054102 (2016)] is significantly improved by introducing the Laplace transform of the orbital energy denominator in order to construct the double amplitudes directly in the local basis. Furthermore, this paper introduces the auxiliary redn. procedure, which reduces the set of the auxiliary functions employed in the individual fragments. The resulting Laplace transformed divide-expand-consolidate resoln. of the identity second-order Moller-Plesset perturbation method is applied to the insulin mol. where we obtain a factor 9.5 speedup compared to the DEC-RI-MP2 method. (c) 2017 American Institute of Physics. 86. 86 Anacker, T.; Tew, D. P.; Friedrich, J. First UHF Implementation of the Incremental Scheme for Open-Shell Systems. J. Chem. Theory Comput. 2016, 12, 65, DOI: 10.1021/acs.jctc.5b00933 Google Scholar First UHF Implementation of the Incremental Scheme for Open-Shell Systems Anacker, Tony; Tew, David P.; Friedrich, Joachim Journal of Chemical Theory and Computation (2016), 12 (1), 65-78CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society) The incremental scheme makes it possible to compute CCSD(T) correlation energies to high accuracy for large systems. We present the first extension of this fully automated black-box approach to open-shell systems using an UHF (UHF) wave function, extending the efficient domain-specific basis set approach to handle open-shell refs. We test our approach on a set of org. and metal org. structures and mol. clusters and demonstrate std. deviations from canonical CCSD(T) values of only 1.35 kJ/mol using a triple ζ basis set. We find that the incremental scheme is significantly more cost-effective than the canonical implementation even for relatively small systems and that the ease of parallelization makes it possible to perform high-level calcns. on large systems in a few hours on inexpensive computers. We show that the approxns. that make our approach widely applicable are significantly smaller than both the basis set incompleteness error and the intrinsic error of the CCSD(T) method, and we further demonstrate that incremental energies can be reliably used in extrapolation schemes to obtain near complete basis set limit CCSD(T) reaction energies for large systems. 87. 87 Zhang, J.; Dolg, M. Third-Order Incremental Dual-Basis Set Zero-Buffer Approach for Large High-Spin Open-Shell Systems. J. Chem. Theory Comput. 2015, 11, 962, DOI: 10.1021/ct501052e Google Scholar Third-Order Incremental Dual-Basis Set Zero-Buffer Approach for Large High-Spin Open-Shell Systems Zhang, Jun; Dolg, Michael Journal of Chemical Theory and Computation (2015), 11 (3), 962-968CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society) The third-order incremental dual-basis set zero-buffer approach (inc3-db-B0) is an efficient, accurate, and black-box quantum chem. method for obtaining correlation energies of large systems, and it has been successfully applied to many real chem. problems. In this work, we extend this approach to high-spin open-shell systems. In the open-shell approach, we will first decomp. the occupied orbitals of a system into several domains by a K-means clustering algorithm. The essential part is that we preserve the active (singly occupied) orbitals in all the calcns. of the domain correlation energies. The duplicated contributions of the active orbitals to the correlation energy are subtracted from the incremental expansion. All techniques of truncating the virtual space such as the B0 approxn. can be applied. This open-shell inc3-db-B0 approach is combined with the CCSD and CCSD(T) methods and applied to the computations of a singlet-triplet gap and an electron detachment process. Our approach exhibits an accuracy better than 0.6 kcal/mol or 0.3 eV compared with the std. implementation, while it saves a large amt. of the computational time and can be efficiently parallelized. 88. 88 Kállay, M. Linear-scaling implementation of the direct random-phase approximation. J. Chem. Phys. 2015, 142, 204105 DOI: 10.1063/1.4921542 Google Scholar Linear-scaling implementation of the direct random-phase approximation Kallay, Mihaly Journal of Chemical Physics (2015), 142 (20), 204105/1-204105/16CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) We report the linear-scaling implementation of the direct RPA (dRPA) for closed-shell mol. systems. As a bonus, linear-scaling algorithms are also presented for the second-order screened exchange extension of dRPA as well as for the second-order Moller-Plesset (MP2) method and its spin-scaled variants. Our approach is based on an incremental scheme which is an extension of our previous local correlation method [Rolik et al., J. Chem. Phys. 139, 094105 (2013)]. The approach extensively uses local natural orbitals to reduce the size of the MO basis of local correlation domains. In addn., we also demonstrate that using natural auxiliary functions [M. Kallay, J. Chem. Phys. 141, 244113 (2014)], the size of the auxiliary basis of the domains and thus that of the three-center Coulomb integral lists can be reduced by an order of magnitude, which results in significant savings in computation time. The new approach is validated by extensive test calcns. for energies and energy differences. Our benchmark calcns. also demonstrate that the new method enables dRPA calcns. for mols. with more than 1000 atoms and 10 000 basis functions on a single processor. (c) 2015 American Institute of Physics. 89. 89 Nagy, P. R.; Samu, G.; Kállay, M. Optimization of the linear-scaling local natural orbital CCSD(T) method: Improved algorithm and benchmark applications. J. Chem. Theory Comput. 2018, 14, 4193, DOI: 10.1021/acs.jctc.8b00442 Google Scholar Optimization of the Linear-Scaling Local Natural Orbital CCSD(T) Method: Improved Algorithm and Benchmark Applications Nagy, Peter R.; Samu, Gyula; Kallay, Mihaly Journal of Chemical Theory and Computation (2018), 14 (8), 4193-4215CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society) An optimized implementation of the local natural orbital (LNO) coupled-cluster (CC) with single-, double-, and perturbative triple excitations [LNO-CCSD(T)] method is presented. The integral-direct, in-core, highly efficient domain construction technique of the authors' local second-order Moller-Plesset (LMP2) scheme is extended to the CC level. The resulting scheme, which is also suitable for general-order LNO-CC calcns., inherits the beneficial properties of the LMP2 approach, such as the asymptotically linear-scaling operation count, the asymptotically const. data storage requirement, and the completely independent domain calcns. In addn. to integrating the authors' recent redundancy-free LMP2 and Laplace-transformed (T) algorithms with the LNO-CCSD (T) code, the memory demand, the domain and LNO construction, the auxiliary basis compression, and the previously rate-detg. two-external integral transformation have been significantly improved. The accuracy of all of the approxns. is carefully examd. on medium-sized to large systems to det. reasonably tight default truncation thresholds. The authors' benchmark calcns., performed on mols. of up to 63 atoms, show that the optimized method with the default settings provides av. correlation and reaction energy errors of <0.07% and 0.34 kcal/mol, resp., compared to the canonical CCSD(T) ref. The efficiency of the present LNO-CCSD(T) implementation is demonstrated on realistic, three-dimensional examples. Using the new code, an LNO-CCSD(T) correlation energy calcn. with a triple-ζ basis set is feasible on a single processor for a protein mol. including 2380 atoms and >44000 AOs. 90. 90 Nagy, P. R.; Kállay, M. Approaching the basis set limit of CCSD(T) energies for large molecules with local natural orbital coupled-cluster methods. J. Chem. Theory Comput. 2019, 15, 5275, DOI: Google Scholar Approaching the Basis Set Limit of CCSD(T) Energies for Large Molecules with Local Natural Orbital Coupled-Cluster Methods Nagy, Peter R.; Kallay, Mihaly Journal of Chemical Theory and Computation (2019), 15 (10), 5275-5298CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society) Recent optimization efforts and extensive benchmark applications are presented illustrating the accuracy and efficiency of the linear-scaling local natural orbital (LNO) coupled-cluster single-, double-, and perturbative triple-excitations [CCSD(T)] method. A composite threshold combination hierarchy (Loose, Normal, Tight, etc.) is introduced, which enables black box convergence tests and is useful to est. the accuracy of the LNO-CCSD(T) energies with respect to CCSD(T). We also demonstrate that the complete basis set limit (CBS) of LNO-CCSD(T) energies can be reliably approached via basis set extrapolation using large basis sets including diffuse functions. Where ref. CCSD(T) results are available, the mean (max.) abs. errors of the LNO-CCSD(T) reaction and intermol. interaction energies with the default Normal threshold combination are below 0.2-0.3 (0.6-1.0) kcal/mol, while the same measures with the Tight setting are 0.1 (0.2-0.5) kcal/mol for all the tested systems including highly complicated cases. The performance of LNO-CCSD(T) is also compared with that of other popular local CCSD(T) schemes. The exceptionally low hardware requirements of the present scheme enables the routine calcn. of benchmark-quality energy differences within chem. accuracy of CCSD(T)/CBS for systems including a few hundred atoms. LNO-CCSD(T)/ CBS calcns. can also be performed for more than 1000 atoms with 45,000 AOs using a single, six-core CPU, about 100 GB memory, and comparable disk space. 91. 91 Koch, H.; Sánchez de Merás, A. M. Size-intensive decomposition of orbital energy denominators. J. Chem. Phys. 2000, 113, 508, DOI: 10.1063/1.481910 Google Scholar Size-intensive decomposition of orbital energy denominators Koch, Henrik; Sanchez de Meras, Alfredo Journal of Chemical Physics (2000), 113 (2), 508-513CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) We introduce an alternative to Almlof and Haser's Laplace transform decompn. of orbital energy denominators used in obtaining reduced scaling algorithms in perturbation theory based methods. The new decompn. is based on the Cholesky decompn. of pos. semidefinite matrixes. Orbital denominators have a particular short and size-intensive Cholesky decompn. The main advantage in using the Cholesky decompn., besides the shorter expansion, is the systematic improvement of the results without the penalties encountered in the Laplace transform decompn. when changing the no. of integration points in order to control the convergence. Applications will focus on the coupled-cluster singles and doubles model including connected triples corrections [CCSD(T)], and several numerical examples are discussed. 92. 92 Rolik, Z.; Szegedy, L.; Ladjánszki, I.; Ladóczki, B.; Kállay, M. An efficient linear-scaling CCSD(T) method based on local natural orbitals. J. Chem. Phys. 2013, 139, 094105 DOI: 10.1063/ Google Scholar An efficient linear-scaling CCSD(T) method based on local natural orbitals Rolik, Zoltan; Szegedy, Lorant; Ladjanszki, Istvan; Ladoczki, Bence; Kallay, Mihaly Journal of Chemical Physics (2013), 139 (9), 094105/1-094105/17CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) An improved version of our general-order local coupled-cluster (CC) approach and its efficient implementation at the CC singles and doubles with perturbative triples CCSD(T) level is presented. The method combines the cluster-in-mol. approach of with frozen natural orbital (NO) techniques. To break down the unfavorable fifth-power scaling of our original approach a two-level domain construction algorithm has been developed. First, an extended domain of localized MOs (LMOs) is assembled based on the spatial distance of the orbitals. The necessary integrals are evaluated and transformed in these domains invoking the d. fitting approxn. In the second step, for each occupied LMO of the extended domain a local subspace of occupied and virtual orbitals is constructed including approx. second-order Moller-Plesset NOs. The CC equations are solved and the perturbative corrections are calcd. in the local subspace for each occupied LMO using a highly-efficient CCSD(T) code, which was optimized for the typical sizes of the local subspaces. The total correlation energy is evaluated as the sum of the individual contributions. The computation time of our approach scales linearly with the system size, while its memory and disk space requirements are independent thereof. Test calcns. demonstrate that currently our method is one of the most efficient local CCSD(T) approaches and can be routinely applied to mols. of up to 100 atoms with reasonable basis sets. (c) 2013 American Institute of Physics. 93. 93 Rolik, Z.; Kállay, M. A general-order local coupled-cluster method based on the cluster-in-molecule approach. J. Chem. Phys. 2011, 135, 104111 DOI: 10.1063/1.3632085 Google Scholar A general-order local coupled-cluster method based on the cluster-in-molecule approach Rolik, Zoltan; Kallay, Mihaly Journal of Chemical Physics (2011), 135 (10), 104111/1-104111/18CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) A general-order local coupled-cluster (CC) method is presented which has the potential to provide accurate correlation energies for extended systems. Our method combines the cluster-in-mol. approach of with the frozen natural orbital (NO) techniques widely used for the cost redn. of correlation methods. The occupied MOs are localized, and for each occupied MO a local subspace of occupied and virtual orbitals is constructed using approx. Moller-Plesset NOs. The CC equations are solved and the correlation energies are calcd. in the local subspace for each occupied MO, while the total correlation energy is evaluated as the sum of the individual contributions. The size of the local subspaces and the accuracy of the results can be controlled by varying only one parameter, the threshold for the occupation no. of NOs which are included in the subspaces. Though our local CC method in its present form scales as the fifth power of the system size, our benchmark calcns. show that it is still competitive for the CC singles and doubles (CCSD) and the CCSD with perturbative triples CCSD(T) approaches. For higher order CC methods, the redn. in computation time is more pronounced, and the new method enables calcns. for considerably bigger mols. than before with a reasonable loss in accuracy. We also demonstrate that the independent calcn. of the correlation contributions allows for a higher order description of the chem. more important segments of the mol. and a lower level treatment of the rest delivering further significant savings in computer time. (c) 2011 American Institute of Physics. 94. 94 Ma, Q.; Werner, H.-J. Scalable Electron Correlation Methods. 7. Local Open-Shell Coupled-Cluster Methods Using Pair Natural Orbitals: PNO-RCCSD and PNO-UCCSD. J. Chem. Theory Comput. 2020, 16, 3135, DOI: 10.1021/acs.jctc.0c00192 Google Scholar Scalable Electron Correlation Methods. 7. Local Open-Shell Coupled-Cluster Methods Using Pair Natural Orbitals: PNO-RCCSD and PNO-UCCSD Ma, Qianli; Werner, Hans-Joachim Journal of Chemical Theory and Computation (2020), 16 (5), 3135-3151CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society) We present well-parallelized local implementations of high-spin open-shell coupled cluster methods with single and double excitations (CCSD) using pair natural orbitals (PNOs). The methods are based on the spin-orbital coupled cluster theory using restricted open-shell Hartree-Fock (ROHF) ref. functions. Two variants, namely, PNO-UCCSD and PNO-RCCSD are implemented and compared. In PNO-UCCSD, the coupled cluster amplitudes are spin-unrestricted, while in PNO-RCCSD the linear terms are spin-adapted by a spin-projection approach as described in. Near linear scaling of the computational cost with the no. of correlated electrons is achieved by applying domain and pair approxns. The PNOs are spin-independent and obtained using a semicanonical spin-restricted MP2 approxn. with large domains of projected AOs (PAOs). The pair approxns. of our previously described closed-shell PNO-LCCSD method are carefully revised so that they are compatible to the UCCSD theory, and PNO-UCCSD or PNO-RCCSD calcns. for closed-shell mols. yield exactly the same results as corresponding spin-free closed-shell PNO-LCCSD calcns. The convergence of the results with respect to the thresholds and options that control the domain and pair approxns. is demonstrated. It is found that large domains are required for the single excitations in open-shell calcns. in order to obtain converged results. In general, the errors of relative energies caused by the local approxns. can be reduced to below 1 kcal mol-1, even for difficult cases. Presently, PNO-RCCSD and PNO-UCCSD calcns. for mols. with 100-200 atoms and augmented triple-ζ basis sets can be carried out in a few hours of elapsed time using ∼ 100 CPU cores. In addn., the program is also capable of performing distinguishable cluster (PNO-RDCSD and PNO-UDCSC) calcns. The present work is a crit. step in developing fully local open-shell PNO-RCCSD(T)-F12 methods. 95. 95 Ma, Q.; Werner, H.-J. Scalable Electron Correlation Methods. 8. Explicitly Correlated Open-Shell Coupled-Cluster with Pair Natural Orbitals PNO-RCCSD(T)-F12 and PNO-UCCSD(T)-F12. J. Chem. Theory Comput. 2021, 17, 902, DOI: 10.1021/acs.jctc.0c01129 Google Scholar Scalable Electron Correlation Methods. 8. Explicitly Correlated Open-Shell Coupled-Cluster with Pair Natural Orbitals PNO-RCCSD(T)-F12 and PNO-UCCSD(T)-F12 Ma, Qianli; Werner, Hans-Joachim Journal of Chemical Theory and Computation (2021), 17 (2), 902-926CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society) We present explicitly correlated open-shell pair natural orbital local coupled-cluster methods, PNO-RCCSD(T)-F12 and PNO-UCCSD(T)-F12. The methods are extensions of our previously reported PNO-R/ UCCSD methods (J. Chem. Theory Comput., 2020, 16, 3135-3151, https://pubs.acs.org/doi/10.1021/acs.jctc.0c00192) with addns. of explicit correlation and perturbative triples corrections. The explicit correlation treatment follows the spin-orbital CCSD-F12b theory using Ansatz 3A, which is found to yield comparable or better basis set convergence than the more rigorous Ansatz 3C in computed ionization potentials and reaction energies using double- to quaduple-ζ basis sets. The perturbative triples correction is adapted from the spin-orbital (T) theory to use triples natural orbitals (TNOs). To address the coupling due to off-diagonal Fock matrix elements, the local triples amplitudes are iteratively solved using small domains of TNOs, and a semicanonical (T0) domain correction with larger domains is applied to reduce the domain errors. The performance of the methods is demonstrated through benchmark calcns. on ionization potentials, radical stabilization energies, reaction energies of fragmentations and rearrangements in radical cations, and spin-state energy differences of iron complexes. For a few test sets where canonical calcns. are feasible, PNO-RCCSD(T)-F12 results agree with the canonical ones to within 0.4 kcal mol-1, and this max. error is reduced to below 0.2 kcal mol-1 when large local domains are used. For larger systems, results using different thresholds for the local approxns. are compared to demonstrate that 1 kcal mol-1 level of accuracy can be achieved using our default settings. For a couple of difficult cases, it is demonstrated that the errors from individual approxns. are only a fraction of 1 kcal mol-1, and the overall accuracy of the method does not rely on error compensations. In contrast to canonical calcns., the use of spin-orbitals does not lead to a significant increase of computational time and memory usage in the most expensive steps of PNO-R/UCCSD(T)-F12 calcns. The only exception is the iterative soln. of the (T) amplitudes, which can be avoided without significant errors by using a perturbative treatment of the off-diagonal coupling, known as (T1) approxn. For most systems, even the semicanonical approxn. (T0) leads only to small errors in relative energies. Our program is well parallelized and capable of computing accurate correlation energies for mols. with 100-200 atoms using augmented triple-ζ basis sets in less than a day of elapsed time on a small computer cluster. 96. 96 Hansen, A.; Liakos, D. G.; Neese, F. Efficient and accurate local single reference correlation methods for high-spin open-shell molecules using pair natural orbitals. J. Chem. Phys. 2011, 135, 214102 DOI: 10.1063/1.3663855 Google Scholar Efficient and accurate local single reference correlation methods for high-spin open-shell molecules using pair natural orbitals Hansen, Andreas; Liakos, Dimitrios G.; Neese, Frank Journal of Chemical Physics (2011), 135 (21), 214102/1-214102/20CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) A prodn. level implementation of the high-spin open-shell (spin unrestricted) single ref. coupled pair, quadratic CI and coupled cluster methods with up to doubly excited determinants in the framework of the local pair natural orbital (LPNO) concept is reported. This work is an extension of the closed-shell LPNO methods developed earlier. The internal space is spanned by localized orbitals, while the external space for each electron pair is represented by a truncated PNO expansion. The laborious integral transformation assocd. with the large no. of PNOs becomes feasible through the extensive use of d. fitting (resoln. of the identity (RI)) techniques. Tech. complications arising for the open-shell case and the use of quasi-restricted orbitals for the construction of the ref. determinant are discussed in detail. As in the closed-shell case, only three cutoff parameters control the av. no. of PNOs per electron pair, the size of the significant pair list, and the no. of contributing auxiliary basis functions per PNO. The chosen threshold default values ensure robustness and the results of the parent canonical methods are reproduced to high accuracy. Comprehensive numerical tests on abs. and relative energies as well as timings consistently show that the outstanding performance of the LPNO methods carries over to the open-shell case with minor modifications. Finally, hyperfine couplings calcd. with the variational LPNO-CEPA/1 method, for which a well-defined expectation value type d. exists, indicate the great potential of the LPNO approach for the efficient calcn. of mol. properties. (c) 2011 American Institute of Physics. 97. 97 Guo, Y.; Riplinger, C.; Liakos, D. G.; Becker, U.; Saitow, M.; Neese, F. Linear scaling perturbative triples correction approximations for open-shell domain-based local pair natural orbital coupled cluster singles and doubles theory [DLPNO-CCSD(T0/T)]. J. Chem. Phys. 2020, 152, 024116 DOI: 10.1063/1.5127550 Google Scholar Linear scaling perturbative triples correction approximations for open-shell domain-based local pair natural orbital coupled cluster singles and doubles theory [DLPNO-CCSD(T0/T)] Guo, Yang; Riplinger, Christoph; Liakos, Dimitrios G.; Becker, Ute; Saitow, Masaaki; Neese, Frank Journal of Chemical Physics (2020), 152 (2), 024116CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) The coupled cluster method with single-, double-, and perturbative triple excitations [CCSD(T)] is considered to be one of the most reliable quantum chem. theories. However, the steep scaling of CCSD(T) has limited its application to small or medium-sized systems for a long time. In our previous work, the linear scaling domain based local pair natural orbital CCSD variant (DLPNO-CCSD) has been developed for closed-shell and open-shell. However, it is known from extensive benchmark studies that triple-excitation contributions are important to reach chem. accuracy. In the present work, two linear scaling (T) approxns. for open-shell DLPNO-CCSD are implemented and compared: (a) an algorithm based on the semicanonical approxn., in which off-diagonal Fock matrix elements in the occupied space are neglected [referred to as DLPNO-(T0)]; and (b) an improved algorithm in which the triples amplitudes are computed iteratively [referred to as DLPNO-(T)]. This work is based on the previous open-shell DLPNO-CCSD algorithm [M. Saitow et al., J. Chem. Phys. 146, 164105 (2017)] as well as the iterative (T) correction for closed-shell systems [Y. Guo et al., J. Chem. Phys. 148, 011101 (2018)]. Our results show that the new open-shell perturbative corrections, DLPNO-(T0/T), can predict accurate abs. and relative correlation energies relative to the canonical ref. calcns. with the same basis set. The abs. energies from DLPNO-(T) are significantly more accurate than those of DLPNO-(T0). The addnl. computational effort of DLPNO-(T) relative to DLPNO-(T0) is a factor of 4 on av. We report calcns. on systems with more than 4000 basis functions. (c) 2020 American Institute of Physics. 98. 98 Kumar, A.; Neese, F.; Valeev, E. F. Explicitly correlated coupled cluster method for accurate treatment of open-shell molecules with hundreds of atoms. J. Chem. Phys. 2020, 153, 094105 DOI: Google Scholar Explicitly correlated coupled cluster method for accurate treatment of open-shell molecules with hundreds of atoms Kumar, Ashutosh; Neese, Frank; Valeev, Edward F. Journal of Chemical Physics (2020), 153 (9), 094105CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) We present a near-linear scaling formulation of the explicitly correlated coupled-cluster singles and doubles with the perturbative triples method [CCSD(T)F12] for high-spin states of open-shell species. The approach is based on the conventional open-shell CCSD formalism [M. Saitow et al., J. Chem. Phys. 146, 164105 (2017)] utilizing the domain local pair-natural orbitals (DLPNO) framework. The use of spin-independent set of pair-natural orbitals ensures exact agreement with the closed-shell formalism reported previously, with only marginally impact on the cost (e.g., the open-shell formalism is only 1.5 times slower than the closed-shell counterpart for the C160H322 n-alkane, with the measured size complexity of ≈ 1.2). Evaluation of coupled-cluster energies near the complete-basis-set (CBS) limit for open-shell systems with more than 550 atoms and 5000 basis functions is feasible on a single multi-core computer in less than 3 days. The aug-cc-pVTZ DLPNO-CCSD(T)F12 contribution to the heat of formation for the 50 largest mols. among the 348 core combustion species benchmark set [J. Klippenstein et al., J. Phys. Chem. A 121, 6580-6602 (2017)] had root-mean-square deviation (RMSD) from the extrapolated CBS CCSD(T) ref. values of 0.3 kcal/mol. For a more challenging set of 50 reactions involving small closed- and open-shell mols. [G. Knizia et al., J. Chem. Phys. 130, 054104 (2009)], the aug-cc-pVQ( + d)Z DLPNO-CCSD(T)F12 yielded a RMSD of ∼0.4 kcal/mol with respect to the CBS CCSD(T) est. (c) 2020 American Institute of Physics. 99. 99 Angeli, C.; Cimiraglia, R.; Evangelisti, S.; Leininger, T.; Malrieu, J.-P. Introduction of n-electron valence states for multireference perturbation theory. J. Chem. Phys. 2001, 114, 10252, DOI: Google Scholar Introduction of n-electron valence states for multireference perturbation theory Angeli, C.; Cimiraglia, R.; Evangelisti, S.; Leininger, T.; Malrieu, J.-P. Journal of Chemical Physics (2001), 114 (23), 10252-10264CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) The present work presents three second-order perturbative developments from a complete active space (CAS) zero-order wave function, which are strictly additive with respect to mol. dissocn. and intruder state free. They differ by the degree of contraction of the outer-space perturbers. Two types of zero-order Hamiltonians are proposed, both are bielectronic, incorporating the interactions between electrons in the active orbitals, therefore introducing a rational balance between the zero-order wave function and the outer-space. The use of Dyall's Hamiltonian, which puts the active electrons in a fixed core field, and of a partially contracted formalism seems a promising compromise. The formalism is generalizable to multireference spaces which are parts of a CAS. A few test applications of the simplest variant developed in this paper illustrate its potentialities. 100. 100 Lauderdale, W. J.; Stanton, J. F.; Gauss, J.; Watts, J. D.; Bartlett, R. J. Many-body perturbation theory with a restricted open-shell Hartree-Fock reference. Chem. Phys. Lett. 1991, 187, 21, DOI: 10.1016/0009-2614(91)90478-R Google Scholar Many-body perturbation theory with a restricted open-shell Hartree-Fock reference Lauderdale, Walter J.; Stanton, John F.; Gauss, Jurgen; Watts, John D.; Bartlett, Rodney J. Chemical Physics Letters (1991), 187 (1-2), 21-8CODEN: CHPLBC; ISSN:0009-2614. A new, efficient ROHF-based MBPT method is presented. The method, which is noniterative, invariant to transformations among occupied or virtual orbitals, and generalizable to any order, is illustrated by application to the UHF spin-contaminated CN radical and the H + OCH2 transition state. 101. 101 Knowles, P. J.; Andrews, J. S.; Amos, R. D.; Handy, N. C.; Pople, J. A. Restricted Møller-Plesset theory for open-shell molecules. Chem. Phys. Lett. 1991, 186, 130, DOI: 10.1016/S0009-2614(91) Google Scholar Restricted Moeller-Plesset theory for open-shell molecules Knowles, Peter J.; Andrews, Jamie S.; Amos, Roger D.; Handy, Nicholas C.; Pople, John A. Chemical Physics Letters (1991), 186 (2-3), 130-6CODEN: CHPLBC; ISSN:0009-2614. Moeller-Plesset perturbation-theory calcns. are examd. for open-shell mols. based on a spin-RHF ref. wavefunction through the development of a new prescription for obtaining unique semi-canonical orbitals. These orbitals, which are different for α and β spins while avoiding the spin contamination present in UHF ref. functions, satisfy the criteria on which Koopmans's theorem is based, lending justification to their use in defining a zeroth-order Hamiltonian for perturbation theory. This new and straightforward Moeller-Plesset theory for open-shell mols. is called RMP theory. The convergence of the RMP series is examd. to high order, and shows the greatly improved convergence characteristics also found with the authors' alternative ROMP theory. For the mols. NH2(re, 1.5re, 2re) and CN, RMP2 energies are substantially lower than UMP2 energies. 102. 102 Neese, F. Importance of Direct Spin-Spin Coupling and Spin-Flip Excitations for the Zero-Field Splittings of Transition Metal Complexes: A Case Study. J. Am. Chem. Soc. 2006, 128, 10213, DOI: Google Scholar Importance of Direct Spin-Spin Coupling and Spin-Flip Excitations for the Zero-Field Splittings of Transition Metal Complexes: A Case Study Neese, Frank Journal of the American Chemical Society (2006), 128 (31), 10213-10222CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society) This work reports the evaluation of several theor. approaches to the zero-field splitting (ZFS) in transition metal complexes. The exptl. well-known complex [Mn(acac)3] is taken as an example. The direct spin-spin contributions to the ZFS have been calcd. on the basis of d. functional theory (DFT) or complete active space SCF (CASSCF) wave functions and have been found to be much more important than previously assumed. The contributions of the direct term may exceed ∼1 cm-1 in magnitude and therefore cannot be neglected in any treatment that aims at a realistic quant. modeling of the ZFS. In the DFT framework, two different variants to treat the spin-orbit coupling (SOC) term have been evaluated. The first approach is based on previous work by Pederson, Khanna, and Kortus, and the second is based on a "quasi-restricted" DFT treatment which is rooted in our previous work on ZFS. Both approaches provide very similar results and underestimate the SOC contribution to the ZFS by a factor of 2 or more. The SOC is represented by an accurate multicenter spin-orbit mean-field (SOMF) approxn. which is compared to the popular effective DFT potential-derived SOC operator. In addn. to the DFT results, direct "infinite order" ab initio calcns. of the SOC contribution to the ZFS based on CASSCF wave functions, the spectroscopy-oriented CI (SORCI), and the difference-dedicated CI (DDCI) approach are reported. In general, the multireference ab initio results provide a more realistic description of the ZFS in [Mn(acac)3]. The conclusions likely carry over to many other systems. This is attributed to the explicit treatment of the multiplet effects which are of dominant importance, since the calcns. demonstrate that, even in the high-spin d4 system Mn(III), the spin-flip excitations make the largest contribution to the SOC. It is demonstrated that the ab initio methods can be used even for somewhat larger mols. (the present calcns. were done with more than 500 basis functions) in a reasonable time frame. Much more economical but still fairly reasonable results have been achieved with the INDO/S treatment based on CASSCF and SOC-CI wave functions. 103. 103 Hégely, B.; Nagy, P. R.; Kállay, M. Dual basis set approach for density functional and wave function embedding schemes. J. Chem. Theory Comput. 2018, 14, 4600, DOI: 10.1021/acs.jctc.8b00350 Google Scholar Dual Basis Set Approach for Density Functional and Wave Function Embedding Schemes Hegely, Bence; Nagy, Peter R.; Kallay, Mihaly Journal of Chemical Theory and Computation (2018), 14 (9), 4600-4615CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society) A dual basis (DB) approach is proposed which is suitable for the redn. of the computational expenses of the Hartree-Fock, Kohn-Sham, and wave function-based correlation methods. The approach is closely related to the DB approxn. of Head-Gordon and co-workers [ J. Chem. Phys. 2006, 125, 074108] but specifically designed for embedding calcns. The new approach is applied to our variant of the projector-based embedding theory utilizing the Huzinaga-equation, multilevel local correlation methods, and combined d. functional-multilevel local correlation approxns. The performance of the resulting DB d. functional and wave function embedding methods is evaluated in extensive benchmark calcns. and also compared to that of the corresponding embedding schemes exploiting the mixed-basis approxn. Our results show that, with an appropriate combination of basis sets, the DB approach significantly speeds up the embedding calcns., and, for chem. processes where the electronic structure considerably changes, it is clearly superior to the mixed-basis approxn. The results also demonstrate that the DB approach, if integrated with the mixed-basis approxn., efficiently eliminates the major weakness of the latter, and the combination of the DB and mixed-basis schemes is the most efficient strategy to accelerate embedding calcns. 104. 104 Hégely, B.; Nagy, P. R.; Ferenczy, G. G.; Kállay, M. Exact density functional and wave function embedding schemes based on orbital localization. J. Chem. Phys. 2016, 145, 064107 DOI: 10.1063/ Google Scholar Exact density functional and wave function embedding schemes based on orbital localization Hegely, Bence; Nagy, Peter R.; Ferenczy, Gyorgy G.; Kallay, Mihaly Journal of Chemical Physics (2016), 145 (6), 064107/1-064107/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) Exact schemes for the embedding of d. functional theory (DFT) and wave function theory (WFT) methods into lower-level DFT or WFT approaches are introduced utilizing orbital localization. First, a simple modification of the projector-based embedding scheme of Manby and co-workers [J. Chem. Phys. 140, 18A507 (2014)] is proposed. We also use localized orbitals to partition the system, but instead of augmenting the Fock operator with a somewhat arbitrary level-shift projector we solve the Huzinaga-equation, which strictly enforces the Pauli exclusion principle. Second, the embedding of WFT methods in local correlation approaches is studied. Since the latter methods split up the system into local domains, very simple embedding theories can be defined if the domains of the active subsystem and the environment are treated at a different level. The considered embedding schemes are benchmarked for reaction energies and compared to quantum mechanics (QM)/mol. mechanics (MM) and vacuum embedding. We conclude that for DFT-in-DFT embedding, the Huzinaga-equation-based scheme is more efficient than the other approaches, but QM/MM or even simple vacuum embedding is still competitive in particular cases. Concerning the embedding of wave function methods, the clear winner is the embedding of WFT into low-level local correlation approaches, and WFT-in-DFT embedding can only be more advantageous if a non-hybrid d. functional is employed. (c) 2016 American Institute of Physics. 105. 105 Polly, R.; Werner, H.-J.; Manby, F. R.; Knowles, P. J. Fast Hartree-Fock theory using local fitting approximations. Mol. Phys. 2004, 102, 2311, DOI: 10.1080/0026897042000274801 Google Scholar Fast Hartree-Fock theory using local density fitting approximations Polly, Robert; Werner, Hans-Joachim; Manby, Frederick R.; Knowles, Peter J. Molecular Physics (2004), 102 (21-22), 2311-2321CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.) D. fitting approxns. are applied to generate the Fock matrix in Hartree-Fock calcns. By localizing the orbitals in each iteration and performing sep. fits for each orbital the O(N4) scaling of the computational effort for the exchange can be reduced to O(N). We also use the Poisson method to replace almost all Coulomb integrals with simple overlaps, an efficient alternative to diagonalization, and dual basis sets such that the Hartree-Fock calcn. is performed in a smaller basis than the subsequent treatment of electron correlation. The accuracy and efficiency of the method is demonstrated in calcns. with almost 4000 basis functions. The errors introduced by the local approxns. on HF and MP2 energies are small compared to those that arise from the d. fitting, and the fitting errors themselves (typically 1-10 microhartree per atom) are very small compared, for example, to the effect of basis set variations. 106. 106 Köppl, C.; Werner, H.-J. Parallel and Low-Order Scaling Implementation of Hartree-Fock Exchange Using Local Density Fitting. J. Chem. Theory Comput. 2016, 12, 3122, DOI: 10.1021/acs.jctc.6b00251 Google Scholar Parallel and Low-Order Scaling Implementation of Hartree-Fock Exchange Using Local Density Fitting Koppl Christoph; Werner Hans-Joachim Journal of chemical theory and computation (2016), 12 (7), 3122-34 ISSN:. Calculations using modern linear-scaling electron-correlation methods are often much faster than the necessary reference Hartree-Fock (HF) calculations. We report a newly implemented HF program that speeds up the most time-consuming step, namely, the evaluation of the exchange contributions to the Fock matrix. Using localized orbitals and their sparsity, local density fitting (LDF), and atomic orbital domains, we demonstrate that the calculation of the exchange matrix scales asymptotically linearly with molecular size. The remaining parts of the HF calculation scale cubically but become dominant only for very large molecular sizes or with many processing cores. The method is well parallelized, and the speedup scales well with up to about 100 CPU cores on multiple compute nodes. The effect of the local approximations on the accuracy of computed HF and local second-order Moller-Plesset perturbation theory energies is systematically investigated, and default values are established for the parameters that determine the domain sizes. Using these values, calculations for molecules with hundreds of atoms in combination with triple-ζ basis sets can be carried out in less than 1 h, with just a few compute nodes. The method can also be used to speed up density functional theory calculations with hybrid functionals that contain HF exchange. 107. 107 Csóka, J.; Kállay, M. Speeding up density fitting Hartree-Fock calculations with multipole approximations. Mol. Phys. 2020, 118, e1769213 DOI: 10.1080/00268976.2020.1769213 Google Scholar Speeding up density fitting Hartree-Fock calculations with multipole approximations Csoka, Jozsef; Kallay, Mihaly Molecular Physics (2020), 118 (19-20), e1769213/1-e1769213/16CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.) The multipole approxn. is utilized for reducing the computational expenses of the exchange contribution in d. fitting Hartree-Fock (DF-HF) calcns. Strategies for approximating the relevant three-center Coulomb integrals with the multipole expansion are discussed. Based on the factorised form of the integrals, an algorithm is proposed for the evaluation of the exchange term for both conventional and local DF-HF methods. The accuracy of the resulting energies, the numerical stability of the algorithm, and the speedups achieved are benchmarked with respect to the order of the multipole expansion for various mol. systems. Our results suggest that computation times for a conventional DF-HF calcn. can be reduced roughly by a factor of 1.5 for mols. of a couple of hundreds of atoms without any loss of accuracy, while the speedups are somewhat more moderate if local d. fitting approxns. are also deployed. 108. 108 Csóka, J.; Kállay, M. Speeding up Hartree-Fock and Kohn-Sham calculations with first-order corrections J. Chem. Phys. 2021, 154. submitted. 109. 109 Foster, J. M.; Boys, S. F. Canonical Configurational Interaction Procedure. Rev. Mod. Phys. 1960, 32, 300, DOI: 10.1103/RevModPhys.32.300 Google Scholar Canonical configurational interaction procedure Foster, J.M.; Boys, S. F. Reviews of Modern Physics (1960), 32 (), 300-2CODEN: RMPHAT; ISSN:0034-6861. A method of choosing predetor functions to express wave functions in their briefest form is proposed. Besides facilitating calcns., this choice of functions appears to approx. chem. invariant orbitals. The method is restricted to electronic states in which a crude approxn. can be obtained in the form of a single Slater determinant. 110. 110 Pipek, J.; Mezey, P. A fast intrinsic localization procedure applicable for ab initio and semiempirical linear combination of atomic orbital wave functions. J. Chem. Phys. 1989, 90, 4916, DOI: Google Scholar A fast intrinsic localization procedure applicable for ab initio and semiempirical linear combination of atomic orbital wave functions Pipek, Janos; Mezey, Paul G. Journal of Chemical Physics (1989), 90 (9), 4916-26CODEN: JCPSA6; ISSN:0021-9606. A new intrinsic localization algorithm is suggested based on a recently developed math. measure of localization. No external criteria are used to define a priori bonds, lone pairs, and core orbitals. The method similarly to Edmiston-Ruedenberg's localization prefers the well established chem. concept of σ-π sepn., while on the other hand, works as economically as Boys' procedure. For the applications of the new localization algorithm, no addnl. quantities are to be calcd., the knowledge of at. overlap integrals is sufficient. This feature allows a unique formulation of the theory, adaptable for both ab initio and semiempirical methods, even in those cases where the exact form of the at. basis functions is not defined (line in the EHT and PPP calcns). The implementation of the procedure in already existing program systems is particularly easy. The Emiston-Ruedenberg and Boys localized orbitals are compared with those calcd. by the method suggested here, within both the CNDO/2 and ab initio frameworks (using STO-3G and 6-31G* basis sets) for several mols. (CO, H2CO, B2H6, and N2O4). 111. 111 Boughton, J. W.; Pulay, P. Comparison of the Boys and Pipek-Mezey Localizations in the Local Correlation Approach and Automatic Virtual Basis Selection. J. Comput. Chem. 1993, 14, 736, DOI: Google Scholar Comparisons of the Boys and Pipek-Mezey localizations in the local correlation approach and automatic virtual basis selection Boughton, James W.; Pulay, Peter Journal of Computational Chemistry (1993), 14 (6), 736-40CODEN: JCCHDD; ISSN:0192-8651. The authors' implementation of Pipek-Mezey electron population localization is described. It is compared with other localization schemes, and its use in the framework of the local-correlation method is discussed. For such use, this localization is shown to be clearly superior to the Boys localization method in the case of phys. well-localized systems. The authors' current algorithm for selection of local virtual spaces is also described. 112. 112 Nagy, P. R.; Surján, P. R.; Szabados, Á. Mayer’s orthogonalization: relation to the Gram-Schmidt and Löwdin’s symmetrical scheme. Theor. Chem. Acc. 2012, 131, 1109, DOI: 10.1007/s00214-012-1109-y 113. 113 Tóth, Z.; Nagy, P. R.; Jeszenszki, P.; Szabados, Á. Novel orthogonalization and biorthogonalization algorithms. Theor. Chem. Acc. 2015, 134, 100, DOI: 10.1007/s00214-015-1703-x 114. 114 Boys, S. F.; Cook, G. B.; Reeves, C. M.; Shavitt, I. Automatic Fundamental Calculations of Molecular Structure. Nature 1956, 178, 1207, DOI: 10.1038/1781207a0 115. 115 Whitten, J. L. Coulombic potential energy integrals and approximations. J. Chem. Phys. 1973, 58, 4496, DOI: 10.1063/1.1679012 Google Scholar Coulombic potential energy integrals and approximations Whitten, J. L. Journal of Chemical Physics (1973), 58 (10), 4496-501CODEN: JCPSA6; ISSN:0021-9606. Theorems are derived which establish a method of approxg. 2-particle Coulombic potential energy integrals, [.vphi.a(1)\|r12-1\|.vphi.b-(2)], in terms of approx. charge ds. .vphi.a' and .vphi.b'. Rigorous error bounds, \|[.vphi.a(1)\|r12-1\|.vphi.b(2)] - [.vphi.a'(1)\|r12-1\|.vphi.b'(2)]\| ≤ δ, are simply expressed in terms of information calcd. sep. for the pair of ds. .vphi.a and .vphi.b' and the pair .vphi.b and .vphi.b'. From the structure of the bound, a simple method of optimizing charge d. approxns. such that δ is minimized is derived. The framework of the theory appears to be well suited for application to the approxn. of electron repulsion integrals which occur in mol. structure theory, and applications to the approxn. of integrals over Slater orbitals or grouped Gaussian functions are discussed. 116. 116 Samu, G.; Kállay, M. Efficient evaluation of three-center Coulomb intergrals. J. Chem. Phys. 2017, 146, 204101 DOI: 10.1063/1.4983393 Google Scholar Efficient evaluation of three-center Coulomb integrals Samu, Gyula; Kallay, Mihaly Journal of Chemical Physics (2017), 146 (20), 204101/1-204101/19CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) In this study we pursue the most efficient paths for the evaluation of three-center electron repulsion integrals (ERIs) over solid harmonic Gaussian functions of various angular momenta. First, the adaptation of the well-established techniques developed for four-center ERIs, such as the Obara-Saika, McMurchie-Davidson, Gill-Head-Gordon-Pople, and Rys quadrature schemes, and the combinations thereof for three-center ERIs is discussed. Several algorithmic aspects, such as the order of the various operations and primitive loops as well as pre-screening strategies, are analyzed. Second, the no. of floating point operations (FLOPs) is estd. for the various algorithms derived, and based on these results the most promising ones are selected. We report the efficient implementation of the latter algorithms invoking automated programming techniques and also evaluate their practical performance. We conclude that the simplified Obara-Saika scheme of Ahlrichs is the most cost-effective one in the majority of cases, but the modified Gill-Head-Gordon-Pople and Rys algorithms proposed herein are preferred for particular shell triplets. Our numerical expts. also show that even though the solid harmonic transformation and the horizontal recurrence require significantly fewer FLOPs if performed at the contracted level, this approach does not improve the efficiency in practical cases. Instead, it is more advantageous to carry out these operations at the primitive level, which allows for more efficient integral pre-screening and memory layout. (c) 2017 American Institute of Physics. 117. 117 Kállay, M. A systematic way for the cost reduction of density fitting methods. J. Chem. Phys. 2014, 141, 244113 DOI: 10.1063/1.4905005 Google Scholar A systematic way for the cost reduction of density fitting methods Kallay, Mihaly Journal of Chemical Physics (2014), 141 (24), 244113/1-244113/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) We present a simple approach for the redn. of the size of auxiliary basis sets used in methods exploiting the d. fitting (resoln. of identity) approxn. for electron repulsion integrals. Starting out of the singular value decompn. of three-center two-electron integrals, new auxiliary functions are constructed as linear combinations of the original fitting functions. The new functions, which we term natural auxiliary functions (NAFs), are analogous to the natural orbitals widely used for the cost redn. of correlation methods. The use of the NAF basis enables the systematic truncation of the fitting basis, and thereby potentially the redn. of the computational expenses of the methods, though the scaling with the system size is not altered. The performance of the new approach has been tested for several quantum chem. methods. It is demonstrated that the most pronounced gain in computational efficiency can be expected for iterative models which scale quadratically with the size of the fitting basis set, such as the direct RPA. The approach also has the promise of accelerating local correlation methods, for which the processing of three-center Coulomb integrals is a bottleneck. (c) 2014 American Institute of Physics. 118. 118 Gyevi-Nagy, L.; Kállay, M.; Nagy, P. R. Accurate reduced-cost CCSD(T) energies: parallel implementation, benchmarks, and large-scale applications. J. Chem. Theory Comput. 2021, 17, 860, DOI: Google Scholar Accurate Reduced-Cost CCSD(T) Energies: Parallel Implementation, Benchmarks, and Large-Scale Applications Gyevi-Nagy, Laszlo; Kallay, Mihaly; Nagy, Peter R. Journal of Chemical Theory and Computation (2021), 17 (2), 860-878CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society) The accurate and systematically improvable frozen natural orbital (FNO) and natural auxiliary function (NAF) cost-reducing approaches are combined with our recent coupled-cluster singles, doubles, and perturbative triples [CCSD(T)] implementations. Both of the closed- and open-shell FNO-CCSD(T) codes benefit from OpenMP parallelism, completely or partially integral-direct d.-fitting algorithms, checkpointing, and hand-optimized, memory- and operation count effective implementations exploiting all permutational symmetries. The closed-shell CCSD(T) code requires negligible disk I/O and network bandwidth, is MPI/OpenMP parallel, and exhibits outstanding peak performance utilization of 50-70% up to hundreds of cores. Conservative FNO and NAF truncation thresholds benchmarked for challenging reaction, atomization, and ionization energies of both closed- and open-shell species are shown to maintain 1 kJ/mol accuracy against canonical CCSD(T) for systems of 31-43 atoms even with large basis sets. The cost redn. of up to an order of magnitude achieved extends the reach of FNO-CCSD(T) to systems of 50-75 atoms (up to 2124 AOs) with triple- and quadruple-ζ basis sets, which is unprecedented without local approxns. Consequently, a considerably larger portion of the chem. compd. space can now be covered by the practically "gold std." quality FNO-CCSD(T) method using affordable resources and about a week of wall time. Large-scale applications are presented for organo-catalytic and transition-metal reactions as well as noncovalent interactions. Possible applications for benchmarking local CCSD(T) methods, as well as for the accuracy assessment or parametrization of less complete models, for example, d. functional approxns. or machine learning potentials, are also outlined. 119. 119 Graham, D. C.; Menon, A. S.; Goerigk, L.; Grimme, S.; Radom, L. Optimization and Basis-Set Dependence of a Restricted-Open-Shell Form of B2-PLYP Double-Hybrid Density Functional Theory. J. Phys. Chem. A 2009, 113, 9861, DOI: 10.1021/jp9042864 Google Scholar Optimization and Basis-Set Dependence of a Restricted-Open-Shell Form of B2-PLYP Double-Hybrid Density Functional Theory Graham, David C.; Menon, Ambili S.; Goerigk, Lars; Grimme, Stefan; Radom, Leo Journal of Physical Chemistry A (2009), 113 (36), 9861-9873CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society) The performance of the restricted-open-shell form of the double-hybrid d. functional theory (DHDFT) B2-PLYP procedure has been compared with that of its unrestricted counterpart using the G3/05 test set. Addnl., the influence of basis set on the parametrization and performance of ROB2-PLYP, and the further improvement of ROB2-PLYP through augmentation with a long-range dispersion function, have been investigated. We find that, after optimization of the two empirical DHDFT parameters, the ROB2-PLYP method (HF exchange = 59% and MP2 correlation = 28%) performs slightly better than the corresponding UB2-PLYP method (HF exchange = 62% and MP2 correlation = 35%), with mean abs. deviations (MADs) from the exptl. energies in the G3/05 test set of 9.1 and 9.9 kJ mol-1, resp., when the cc-pVQZ basis set is employed. Sep. optimizations of the parameters for the RO and U procedures are crucial for a fair comparison. For example, for the G2/97 test set, ROB2-PLYP(53,27) and ROB2-PLYP(62,35) show MADs of 12.2 and 13.5 kJ mol-1, resp., compared with the 6.6 kJ mol-1 for (the optimized) ROB2-PLYP(59,28). The performance of ROB2-PLYP deteriorates significantly as the basis-set size is decreased, reflecting the enhanced basis-set dependence of the MP2 contribution compared with std. DFT. We find that this deficiency can be partly overcome through reparametrization. However, when the basis set drops below triple-ζ, the improvements made on reoptimizing the ROB2-PLYP parameters are not sufficient to warrant their general use. We find that the dispersion- and BSSE-cor. ROB2-PLYP(59,28)-D HCP procedure performs significantly better than ROB2-PLYP(59,28) for the S22 test set of interaction energies in which dispersion interactions are particularly important, with the MAD falling from 6.1 to 1.6 kJ mol-1. However, when the same D correction is applied to the G3/05 test set, the performance of ROB2-PLYP(59,28)-D deteriorates slightly compared with ROB2-PLYP(59,28), with the MAD increasing from 9.1 to 9.5 kJ mol-1. 120. 120 Guo, Y.; Sivalingam, K.; Valeev, E. F.; Neese, F. SparseMaps—A systematic infrastructure for reduced-scaling electronic structure methods. III. Linear-scaling multireference domain-based pair natural orbital N-electron valence perturbation theory. J. Chem. Phys. 2016, 144, 094111 DOI: 10.1063/1.4942769 Google Scholar SparseMaps-A systematic infrastructure for reduced-scaling electronic structure methods. III. Linear-scaling multireference domain-based pair natural orbital N-electron valence perturbation Guo, Yang; Sivalingam, Kantharuban; Valeev, Edward F.; Neese, Frank Journal of Chemical Physics (2016), 144 (9), 094111/1-094111/16CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) Multi-ref. (MR) electronic structure methods, such as MR CI or MR perturbation theory, can provide reliable energies and properties for many mol. phenomena like bond breaking, excited states, transition states or magnetic properties of transition metal complexes and clusters. However, owing to their inherent complexity, most MR methods are still too computationally expensive for large systems. Therefore the development of more computationally attractive MR approaches is necessary to enable routine application for large-scale chem. systems. Among the state-of-the-art MR methods, second-order N-electron valence state perturbation theory (NEVPT2) is an efficient, size-consistent, and intruder-state-free method. However, there are still two important bottlenecks in practical applications of NEVPT2 to large systems: (a) the high computational cost of NEVPT2 for large mols., even with moderate active spaces and (b) the prohibitive cost for treating large active spaces. In this work, we address problem (a) by developing a linear scaling "partially contracted" NEVPT2 method. This development uses the idea of domain-based local pair natural orbitals (DLPNOs) to form a highly efficient algorithm. As shown previously in the framework of single-ref. methods, the DLPNO concept leads to an enormous redn. in computational effort while at the same time providing high accuracy (approaching 99.9% of the correlation energy), robustness, and black-box character. In the DLPNO approach, the virtual space is spanned by pair natural orbitals that are expanded in terms of projected AOs in large orbital domains, while the inactive space is spanned by localized orbitals. The active orbitals are left untouched. Our implementation features a highly efficient "electron pair prescreening" that skips the negligible inactive pairs. The surviving pairs are treated using the partially contracted NEVPT2 formalism. A detailed comparison between the partial and strong contraction schemes is made, with conclusions that discourage the strong contraction scheme as a basis for local correlation methods due to its non-invariance with respect to rotations in the inactive and external subspaces. A minimal set of conservatively chosen truncation thresholds controls the accuracy of the method. With the default thresholds, about 99.9% of the canonical partially contracted NEVPT2 correlation energy is recovered while the crossover of the computational cost with the already very efficient canonical method occurs reasonably early; in linear chain type compds. at a chain length of around 80 atoms. Calcns. are reported for systems with more than 300 atoms and 5400 basis functions. (c) 2016 American Institute of 121. 121 Kállay, M.; Nagy, P. R.; Mester, D.; Rolik, Z.; Samu, G.; Csontos, J.; Csóka, J.; Szabó, P. B.; Gyevi-Nagy, L.; Hégely, B.; Ladjánszki, I.; Szegedy, L.; Ladóczki, B.; Petrov, K.; Farkas, M.; Mezei, P. D.; Ganyecz, Á. The MRCC program system: Accurate quantum chemistry from water to proteins. J. Chem. Phys. 2020, 152, 074107 DOI: 10.1063/1.5142048 Google Scholar The MRCC program system: Accurate quantum chemistry from water to proteins Kallay, Mihaly; Nagy, Peter R.; Mester, David; Rolik, Zoltan; Samu, Gyula; Csontos, Jozsef; Csoka, Jozsef; Szabo, P. Bernat; Gyevi-Nagy, Laszlo; Hegely, Bence; Ladjanszki, Istvan; Szegedy, Lorant; Ladoczki, Bence; Petrov, Klara; Farkas, Mate; Mezei, Pal D.; Ganyecz, Adam Journal of Chemical Physics (2020), 152 (7), 074107CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) MRCC is a package of ab initio and d. functional quantum chem. programs for accurate electronic structure calcns. The suite has efficient implementations of both low- and high-level correlation methods, such as second-order Moller-Plesset (MP2), RPA, second-order algebraic-diagrammatic construction [ADC(2)], coupled-cluster (CC), CI, and related techniques. It has a state-of-the-art CC singles and doubles with perturbative triples [CCSD(T)] code, and its specialties, the arbitrary-order iterative and perturbative CC methods developed by automated programming tools, enable achieving convergence with regard to the level of correlation. The package also offers a collection of multi-ref. CC and CI approaches. Efficient implementations of d. functional theory (DFT) and more advanced combined DFT-wave function approaches are also available. Its other special features, the highly competitive linear-scaling local correlation schemes, allow for MP2, RPA, ADC(2), CCSD(T), and higher-order CC calcns. for extended systems. Local correlation calcns. can be considerably accelerated by multi-level approxns. and DFT-embedding techniques, and an interface to mol. dynamics software is provided for quantum mechanics/mol. mechanics calcns. All components of MRCC support shared-memory parallelism, and multi-node parallelization is also available for various methods. For academic purposes, the package is available free of charge. (c) 2020 American Institute of Physics. 122. 122 Kállay, M.; Nagy, P. R.; Rolik, Z.; Mester, D.; Samu, G.; Csontos, J.; Csóka, J.; Szabó, P. B.; Gyevi-Nagy, L.; Ladjánszki, I.; Szegedy, L.; Ladóczki, B.; Petrov, K.; Farkas, M.; Mezei, P. D.; Hégely, B. MRCC: A Quantum Chemical Program Suite. https://www.mrcc.hu/ (accessed Jan 1, 2021). 123. 123 Weigend, F.; Ahlrichs, R. Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy integrals over Gaussian functions. Phys. Chem. Chem. Phys. 2005, 7, 3297, DOI: 10.1039/b508541a Google Scholar Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy Weigend, Florian; Ahlrichs, Reinhart Physical Chemistry Chemical Physics (2005), 7 (18), 3297-3305CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry) Gaussian basis sets of quadruple zeta valence quality for Rb-Rn are presented, as well as bases of split valence and triple zeta valence quality for H-Rn. The latter were obtained by (partly) modifying bases developed previously. A large set of more than 300 mols. representing (nearly) all elements-except lanthanides-in their common oxidn. states was used to assess the quality of the bases all across the periodic table. Quantities investigated were atomization energies, dipole moments and structure parameters for Hartree-Fock, d. functional theory and correlated methods, for which we had chosen Moller-Plesset perturbation theory as an example. Finally recommendations are given which type of basis set is used best for a certain level of theory and a desired quality of 124. 124 Dunning, T. H., Jr. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. J. Chem. Phys. 1989, 90, 1007, DOI: 10.1063/1.456153 Google Scholar Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen Dunning, Thom H., Jr. Journal of Chemical Physics (1989), 90 (2), 1007-23CODEN: JCPSA6; ISSN:0021-9606. Guided by the calcns. on oxygen in the literature, basis sets for use in correlated at. and mol. calcns. were developed for all of the first row atoms from boron through neon, and for hydrogen. As in the oxygen atom calcns., the incremental energy lowerings, due to the addn. of correlating functions, fall into distinct groups. This leads to the concept of correlation-consistent basis sets, i.e., sets which include all functions in a given group as well as all functions in any higher groups. Correlation-consistent sets are given for all of the atoms considered. The most accurate sets detd. in this way, [5s4p3d2f1g], consistently yield 99% of the correlation energy obtained with the corresponding at.-natural-orbital sets, even though the latter contains 50% more primitive functions and twice as many primitive polarization functions. It is estd. that this set yields 94-97% of the total (HF + 1 + 2) correlation energy for the atoms neon through boron. 125. 125 Dunning, T. H., Jr.; Peterson, K. A.; Wilson, A. K. Gaussian basis sets for use in correlated molecular calculations. X. The atoms aluminum through argon revisited. J. Chem. Phys. 2001, 114, 9244 , DOI: 10.1063/1.1367373 Google Scholar Gaussian basis sets for use in correlated molecular calculations. X. The atoms aluminum through argon revisited Dunning, Thom H., Jr.; Peterson, Kirk A.; Wilson, Angela K. Journal of Chemical Physics (2001), 114 (21), 9244-9253CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) For mols. contg. second row atoms, unacceptable errors have been found in extrapolating dissocn. energies calcd. with the std. correlation consistent basis sets to the complete basis set limit. By carefully comparing the convergence behavior of De(O2) and De(SO), we show that the cause of these errors is a result of two inter-related problems: near duplication of the exponents in two of the d sets and a lack of high-exponent functions in the early members of the sets. Similar problems exist for the f sets (and probably in higher angular momentum sets), but have only a minor effect on the calcd. dissocn. energies. A no. of approaches to address the problems in the d sets were investigated. Well behaved convergence was obtained by augmenting the (1d) and (2d) sets with a high-exponent function and by replacing the (3d) set by the (4d) set and the (4d) set by the (5d) set and so on. To ensure satisfactory coverage of both the L and M shell regions, the exponents of the new d sets were re-optimized. Benchmark calcns. on Si2, PN, SO, and AlCl with the new cc-pV(n + d)Z sets show greatly improved convergence behavior not only for De but for many other properties as well. 126. 126 Weigend, F.; Köhn, A.; Hättig, C. Efficient use of the correlation consistent basis sets in resolution of the identity MP2 calculations. J. Chem. Phys. 2002, 116, 3175, DOI: 10.1063/1.1445115 Google Scholar Efficient use of the correlation consistent basis sets in resolution of the identity MP2 calculations Weigend, Florian; Kohn, Andreas; Hattig, Christof Journal of Chemical Physics (2002), 116 (8), 3175-3183CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) The convergence of the second-order Moller-Plesset perturbation theory (MP2) correlation energy with the cardinal no. X is investigated for the correlation consistent basis-set series cc-pVXZ and cc-pV(X+d)Z. For the aug-cc-pVXZ and aug-cc-pV(X+d)Z series the convergence of the MP2 correlation contribution to the dipole moment is studied. It is found that, when d-shell electrons cannot be frozen, the cc-pVXZ and aug-cc-pVXZ basis sets converge much slower for third-row elements then they do for first- and second-row elements. Based on the results of these studies criteria are deduced for the accuracy of auxiliary basis sets used in the resoln. of the identity (RI) approxn. for electron repulsion integrals. Optimized auxiliary basis sets for RI-MP2 calcns. fulfilling these criteria are reported for the sets cc-pVXZ, cc-pV(X+d)Z, aug-cc-pVXZ, and aug-cc-pV(X+d)Z with X=D, T, and Q. For all basis sets the RI error in the MP2 correlation energy is more than two orders of magnitude smaller than the usual basis-set error. For the auxiliary aug-cc-pVXZ and aug-cc-pV(X+d)Z sets the RI error in the MP2 correlation contribution to the dipole moment is one order of magnitude smaller than the usual basis set error. Therefore extrapolations towards the basis-set limit are possible within the RI approxn. for both energies and properties. The redn. in CPU time obtained with the RI approxn. increases rapidly with basis set size. For the cc-pVQZ basis an acceleration by a factor of up to 170 is obsd. 127. 127 Karton, A.; Martin, J. M. L. Comment on: “Estimating the Hartree-Fock limit from finite basis set calculations”. Theor. Chem. Acc. 2006, 115, 330, DOI: 10.1007/s00214-005-0028-6 Google Scholar Comment on: "Estimating the Hartree-Fock limit from finite basis set calculations" [Jensen F (2005) Theor Chem Acc 113:267] Karton, Amir; Martin, Jan M. L. Theoretical Chemistry Accounts (2006), 115 (4), 330-333CODEN: TCACFW; ISSN:1432-881X. (Springer GmbH) We demonstrate that a minor modification of the extrapolation proposed by Jensen [(2005): Theor Chem Acc 113: 267] yields very reliable ests. of the Hartree-Fock limit in conjunction with correlation consistent basis sets. Specifically, a two-point extrapolation of the form yields HF limits E HF,∞ with an RMS error of 0.1 millihartree using aug-cc-pVQZ and aug-cc-pV5Z basis sets, and of 0.01 millihartree using aug-cc-pV5Z and aug-cc-pV6Z basis sets. 128. 128 Helgaker, T.; Klopper, W.; Koch, H.; Noga, J. Basis-set convergence of correlated calculations on water. J. Chem. Phys. 1997, 106, 9639, DOI: 10.1063/1.473863 Google Scholar Basis-set convergence of correlated calculations on water Helgaker, Trygve; Klopper, Wim; Koch, Henrik; Noga, Jozef Journal of Chemical Physics (1997), 106 (23), 9639-9646CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) The basis-set convergence of the electronic correlation energy in the water mol. is investigated at the second-order Moller-Plesset level and at the coupled-cluster singles-and-doubles level with and without perturbative triples corrections applied. The basis-set limits of the correlation energy are established to within 2mEh by means of (1) extrapolations from sequences of calcns. using correlation-consistent basis sets and (2) from explicitly correlated calcns. employing terms linear in the inter-electronic distances rij. For the extrapolations to the basis-set limit of the correlation energies, fits of the form a + bX-3 (where X is two for double-zeta sets, three for triple-zeta sets, etc.) are found to be useful. CCSD(T) calcns. involving as many as 492 AOs are 129. 129 Goerigk, L.; Grimme, S. A general database for main group thermochemistry, kinetics, and noncovalent interactions—Assessment of common and reparameterized (meta-)GGA density functionals. J. Chem. Theory Comput. 2010, 6, 107, DOI: 10.1021/ct900489g Google Scholar A General Database for Main Group Thermochemistry, Kinetics, and Noncovalent Interactions - Assessment of Common and Reparameterized (meta-)GGA Density Functionals Goerigk, Lars; Grimme, Stefan Journal of Chemical Theory and Computation (2010), 6 (1), 107-126CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society) We present a quantum chem. benchmark database for general main group thermochem., kinetics, and noncovalent interactions (GMTKN24). It is an unprecedented compilation of 24 different, chem. relevant subsets that either are taken from already existing databases or are presented here for the first time. The complete set involves a total of 1.049 at. and mol. single point calcns. and comprises 731 data points (relative chem. energies) based on accurate theor. or exptl. ref. values. The usefulness of the GMTKN24 database is shown by applying common d. functionals on the (meta-)generalized gradient approxn. (GGA), hybrid-GGA, and double-hybrid-GGA levels to it, including an empirical London dispersion correction. Furthermore, we refitted the functional parameters of four (meta-)GGA functionals based on a fit set contg. 143 systems, comprising seven chem. different problems. Validation against the GMTKN24 and the mol. structure (bond lengths) databases shows that the reparameterization does not change bond lengths much, whereas the description of energetic properties is more prone to the parameters' values. The empirical dispersion correction also often improves for conventional thermodn. problems and makes a functional's performance more uniform over the entire database. The refitted functionals typically have a lower mean abs. deviation for the majority of subsets in the proposed GMTKN24 set. This, however, is also often accompanied at the expense of poor performance for a few other important subsets. Thus, creating a broadly applicable (and overall better) functional by just reparameterizing existing ones seems to be difficult. Nevertheless, this benchmark study reveals that a reoptimized (i.e., empirical) version of the TPSS-D functional (oTPSS-D) performs well for a variety of problems and may meet the stds. of an improved functional. We propose validation against this new compilation of benchmark sets as a definitive way to evaluate a new quantum chem. method's true performance. 130. 130 Liu, Y. Linear Scaling High-spin Open-shell Local Correlation Methods. Ph.D. Thesis, Institut für Theoretische Chemie der Universität Stuttgart, 2011. 131. 131 Ghafarian Shirazi, R.; Neese, F.; Pantazis, D. A. Accurate Spin-State Energetics for Aryl Carbenes. J. Chem. Theory Comput. 2018, 14, 4733, DOI: 10.1021/acs.jctc.8b00587 Google Scholar Accurate Spin-State Energetics for Aryl Carbenes Ghafarian Shirazi, Reza; Neese, Frank; Pantazis, Dimitrios A. Journal of Chemical Theory and Computation (2018), 14 (9), 4733-4746CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society) A test set of 12 aryl carbenes (AC12) is compiled with the purpose of establishing their adiabatic singlet-triplet energy splittings using correlated wave function based methods. The set covers both singlet and triplet ground state aryl carbenes, as well as a range of magnitudes for the ground state to excited state gap. The performance of coupled cluster methods is examd. with respect to the ref. wave function, the basis set, and a no. of addnl. methodol. parameters that enter the calcn. Inclusion of perturbative triples and basis set extrapolation with a combination of triple and quadruple-ζ basis sets are both required to ensure high accuracy. When canonical coupled cluster calcns. become too expensive, the domain-based local pair natural orbital approach DLPNO-CCSD (T) can be used as a reliable method for larger systems, as it achieves a mean abs. error of only 0.2 kcal/mol for the singlet-triplet gaps in the present test set. Other first-principles wave function methods and selected d. functional methods are also evaluated. Second-order Moller-Plesset perturbation theory approaches are only applicable in conjunction with orbital optimization (OO-MP2). Among the representative d. functional methods tested, only double hybrid functionals perform sufficiently accurately to be considered useful for systems with small singlet-triplet gaps. On the basis of the ref. coupled cluster results, projected gas-phase free energies are reported for all aryl carbenes. Finally, the treatment of singlet-triplet gaps in soln. is discussed in terms of both implicit and explicit solvation. 132. 132 Wick, C. R.; Smith, D. M. Modeling the Reactions Catalyzed by Coenzyme B12 Dependent Enzymes: Accuracy and Cost-Quality Balance. J. Phys. Chem. A 2018, 122, 1747, DOI: 10.1021/acs.jpca.7b11798 Google Scholar Modeling the Reactions Catalyzed by Coenzyme B12 Dependent Enzymes: Accuracy and Cost-Quality Balance Wick, Christian R.; Smith, David M. Journal of Physical Chemistry A (2018), 122 (6), 1747-1755CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society) The reactions catalyzed by coenzyme B12 dependent enzymes are formally initiated by the homolytic cleavage of a carbon-cobalt bond and a subsequent or concerted H-atom-transfer reaction. A reasonable model chem. for describing those reactions should, therefore, account for an accurate description of both reactions. The inherent limitation due to the necessary system size renders the coenzyme B12 system a suitable candidate for DFT or hybrid QM/MM methods; however, the accurate description of both homolytic Co-C cleavage and H-atom-transfer reactions within this framework is challenging and can lead to controversial results with varying accuracy. We present an assessment study of 16 common d. functionals applied to prototypical model systems for both reactions. H-abstraction reactions were modeled on the basis of four ref. reactions designed to resemble a broad range of coenzyme B12 reactions. The Co-C cleavage reaction is treated by an ONIOM(QM/MM) setup that is in excellent agreement with soln.-phase exptl. data and is as accurate as full DFT calcns. on the complete model system. We find that the meta-GGAs TPSS-D3 and M06L-D3 and the meta-hybrid M06-D3 give the best overall performance with MUEs for both types of reactions below 10 kJ mol-1. Our recommended model chem. allows for a fast and accurate description of coenzyme B12 chem. that is readily applicable to study the reactions in an enzymic framework. 133. 133 Kiss, D. J.; Ferenczy, G. G. A detailed mechanism of the oxidative half-reaction of D-amino acid oxidase: another route for flavin oxidation. Org. Biomol. Chem. 2019, 17, 7973, DOI: 10.1039/ Google Scholar A detailed mechanism of the oxidative half-reaction of D-amino acid oxidase: another route for flavin oxidation Kiss, Dora Judit; Ferenczy, Gyorgy G. Organic & Biomolecular Chemistry (2019), 17 (34), 7973-7984CODEN: OBCRAK; ISSN:1477-0520. (Royal Society of Chemistry) D-Amino acid oxidase (DAAO) is a flavoenzyme whose inhibition is expected to have therapeutic potential in schizophrenia. DAAO catalyzes hydride transfer from the substrate to the flavin in the reductive half-reaction, and the flavin is reoxidized by O2 in the oxidative half-reaction. Quantum mech./mol. mech. calcns. were performed and their results together with available exptl. information were used to elucidate the detailed mechanism of the oxidative half-reaction. The reaction starts with a single electron transfer from FAD to O2, followed by triplet-singlet transition. FAD oxidn. is completed by a proton coupled electron transfer to the oxygen species and the reaction terminates with H2O2 formation by proton transfer from the oxidized substrate to the oxygen species via a chain of water mols. The substrate plays a double role by facilitating the first electron transfer and by providing a proton in the last step. The mechanism differs from the oxidative half-reaction of other oxidases. 134. 134 Paulechka, E.; Kazakov, A. Efficient Estimation of Formation Enthalpies for Closed-Shell Organic Compounds with Local Coupled-Cluster Methods. J. Chem. Theory Comput. 2018, 14, 5920, DOI: 10.1021 Google Scholar Efficient Estimation of Formation Enthalpies for Closed-Shell Organic Compounds with Local Coupled-Cluster Methods Paulechka, Eugene; Kazakov, Andrei Journal of Chemical Theory and Computation (2018), 14 (11), 5920-5932CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society) Efficient estn. of the enthalpies of formation for closed-shell org. compds. via atom-equiv.-type computational schemes and with the use of different local coupled-cluster with single, double, and perturbative triple excitation (CCSD(T)) approxns. was investigated. Detailed anal. of established sources of uncertainty, inclusive of contributions beyond frozen-core CCSD(T) and errors due to local CCSD(T) approxns. and zero-point energy anharmonicity, suggests the lower limit of about 2 kJ·mol-1 for the expanded uncertainty of the proposed estn. framework. Among the tested computational schemes, the best-performing cases demonstrate expanded uncertainty of about 2.5 kJ·mol-1, based on the anal. against 44 critically evaluated exptl. values. Computational efficiency, accuracy commensurable with that of a typical expt., and absence of the need for auxiliary reactions and addnl. exptl. data offer unprecedented advantages for practical use, such as prompt validation of existing measurements and estn. of missing values, as well as resoln. of exptl. conflicts. The utility of the proposed methodol. was demonstrated using a representative sample of the most recent exptl. measurements. 135. 135 Sylvetsky, N.; Banerjee, A.; Alonso, M.; Martin, J. M. L. Performance of Localized Coupled Cluster Methods in a Moderately Strong Correlation Regime: Hückel-Möbius Interconversions in Expanded Porphyrins. J. Chem. Theory Comput. 2020, 16, 3641, DOI: 10.1021/acs.jctc.0c00297 Google Scholar Performance of Localized Coupled Cluster Methods in a Moderately Strong Correlation Regime: Hueckel-Mobius Interconversions in Expanded Porphyrins Sylvetsky, Nitai; Banerjee, Ambar; Alonso, Mercedes; Martin, Jan M. L. Journal of Chemical Theory and Computation (2020), 16 (6), 3641-3653CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society) Localized orbital coupled cluster theory has recently emerged as a nonempirical alternative to DFT for large systems. Intuitively, one might expect such methods to perform less well for highly delocalized systems. In the present work, we apply both canonical CCSD(T) approxns. and a variety of localized approxns. to a set of flexible expanded porphyrins-macrocycles that can switch between Huckel, figure-eight, and Mobius topologies under external stimuli. Both min. and isomerization transition states are considered. We find that Mobius(-like) structures have much stronger static correlation character than the remaining structures, and that this causes significant errors in DLPNO-CCSD(T) and even DLPNO-CCSD(T1) approaches, unless TightPNO cutoffs are employed. If sub-kcal mol-1 accuracy with respect to canonical relative energies is required even for Mobius-type systems (or other systems plagued by strong static correlation), then Nagy and Kallay's LNO-CCSD(T) method with "tight" settings is the suitable localized approach. We propose the present POLYPYR21 data set as a benchmark for localized orbital methods or, more broadly, for the ability of lower-level methods to handle energetics with strongly varying degrees of static correlation. 136. 136 Menezes, F.; Kats, D.; Werner, H.-J. Local complete active space second-order perturbation theory using pair natural orbitals (PNO-CASPT2). J. Chem. Phys. 2016, 145, 124115 DOI: 10.1063/1.4963019 Google Scholar Local complete active space second-order perturbation theory using pair natural orbitals (PNO-CASPT2) Menezes, Filipe; Kats, Daniel; Werner, Hans-Joachim Journal of Chemical Physics (2016), 145 (12), 124115/1-124115/20CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) We present a CASPT2 method which exploits local approxns. to achieve linear scaling of the computational effort with the mol. size, provided the active space is small and local. The inactive orbitals are localized, and the virtual space for each electron pair is spanned by a domain of pair-natural orbitals (PNOs). The configuration space is internally contracted, and the PNOs are defined for uniquely defined orthogonal pairs. Distant pair energies are obtained by multipole approxns., so that the no. of configurations that are explicitly treated in the CASPT2 scales linearly with mol. size (assuming a const. active space). The PNOs are generated using approx. amplitudes obtained in a pair-specific semi-canonical basis of projected AOs (PAOs). The evaluation and transformation of the two-electron integrals use the same parallel local d. fitting techniques as recently described for linear-scaling PNO-LMP2 (local second-order Moller-Plesset perturbation theory). The implementation of the amplitude equations, which are solved iteratively, employs the local integrated tensor framework. The efficiency and accuracy of the method are tested for excitation energies and correlation energies. It is demonstrated that the errors introduced by the local approxns. are very small. They can be well controlled by few parameters for the distant pair approxn., initial PAO domains, and the PNO domains. (c) 2016 American Institute of Physics. 137. 137 Liakos, D. G.; Neese, F. Is It Possible To Obtain Coupled Cluster Quality Energies at near Density Functional Theory Cost? Domain-Based Local Pair Natural Orbital Coupled Cluster vs Modern Density Functional Theory. J. Chem. Theory Comput. 2015, 11, 4054, DOI: 10.1021/acs.jctc.5b00359 Google Scholar Is It Possible To Obtain Coupled Cluster Quality Energies at near Density Functional Theory Cost? Domain-Based Local Pair Natural Orbital Coupled Cluster vs Modern Density Functional Theory Liakos, Dimitrios G.; Neese, Frank Journal of Chemical Theory and Computation (2015), 11 (9), 4054-4063CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society) The recently developed domain-based local pair natural orbital coupled cluster theory with single, double, and perturbative triple excitations (DLPNO-CCSD(T)) delivers results that are closely approaching those of the parent canonical coupled cluster method at a small fraction of the computational cost. A recent extended benchmark study established that, depending on the three main truncation thresholds, it is possible to approach the canonical CCSD(T) results within 1 kJ (default setting, TightPNO), 1 kcal/mol (default setting, NormalPNO), and 2-3 kcal (default setting, LoosePNO). Although thresholds for calcns. with TightPNO are 2-4 times slower than those based on NormalPNO thresholds, they are still many orders of magnitude faster than canonical CCSD(T) calcns., even for small and medium sized mols. where there is little locality. The computational effort for the coupled cluster step scales nearly linearly with system size. Since, in many instances, the coupled cluster step in DLPNO-CCSD(T) is cheaper or at least not much more expensive than the preceding Hartree-Fock calcn., it is useful to compare the method against modern d. functional theory (DFT), which requires an effort comparable to that of Hartree-Fock theory (at least if Hartree-Fock exchange is part of the functional definition). Double hybrid d. functionals (DHDF's) even require a MP2-like step. The purpose of this article is to evaluate the cost vs accuracy ratio of DLPNO-CCSD(T) against modern DFT (including the PBE, B3LYP, M06-2X, B2PLYP, and B2GP-PLYP functionals and, where applicable, their van der Waals cor. counterparts). To eliminate any possible bias in favor of DLPNO-CCSD(T), we have chosen established benchmark sets that were specifically proposed for evaluating DFT functionals. It is demonstrated that DLPNO-CCSD(T) with any of the three default thresholds is more accurate than any of the DFT functionals. Furthermore, using the aug-cc-pVTZ basis set and the LoosePNO default settings, DLPNO-CCSD(T) is only about 1.2 times slower than B3LYP. With NormalPNO thresholds, DLPNO-CCSD(T) is about a factor of 2 slower than B3LYP and shows a mean abs. deviation of less than 1 kcal/mol to the ref. values for the four different data sets used. Our conclusion is that coupled cluster energies can indeed be obtained at near DFT cost. Cited By Click to copy section linkSection link copied! This article is cited by 18 publications. Learn about these metrics Article Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days. Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts. The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. • This article references 137 other publications. 1. 1 Zhang, J.; Head-Gordon, M. Electronic structures and reaction dynamics of open-shell species. Phys. Chem. Chem. Phys. 2009, 11, 4699, DOI: 10.1039/b909815c Electronic structures and reaction dynamics of open-shell species Zhang, Jingsong; Head-Gordon, Martin Physical Chemistry Chemical Physics (2009), 11 (23), 4699-4700CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry) There is no expanded citation for this reference. 2. 2 Bally, T.; Borden, W. T. Reviews in Computational Chemistry; John Wiley & Sons, Ltd, 1999; pp 1– 97. There is no corresponding record for this reference. 3. 3 Helgaker, T.; Jørgensen, P.; Olsen, J. Molecular Electronic Structure Theory; Wiley: Chichester, 2000. There is no corresponding record for this reference. 4. 4 Krylov, A. I. Reviews in Computational Chemistry; John Wiley & Sons, Ltd, 2017; Chapter 4, pp 151– 224. There is no corresponding record for this reference. 5. 5 Stanton, J. F.; Gauss, J. Advances in Chemical Physics; John Wiley & Sons, Ltd, 2003; pp 101– 146. There is no corresponding record for this reference. 6. 6 Møller, C.; Plesset, M. S. Note on an Approximation Treatment for Many-Electron Systems. Phys. Rev. 1934, 46, 618, DOI: 10.1103/PhysRev.46.618 Note on the approximation treatment for many-electron systems Moller, Chr.; Plesset, M. S. Physical Review (1934), 46 (), 618-22CODEN: PHRVAO; ISSN:0031-899X. 7. 7 Raghavachari, K.; Trucks, G. W.; Pople, J. A.; Head-Gordon, M. A fifth-order perturbation comparison of electron correlation theories. Chem. Phys. Lett. 1989, 157, 479, DOI: 10.1016/ A fifth-order perturbation comparison of electron correlation theories Raghavachari, Krishnan; Trucks, Gary W.; Pople, John A.; Head-Gordon, Martin Chemical Physics Letters (1989), 157 (6), 479-83CODEN: CHPLBC; ISSN:0009-2614. Electron correlation theories such as CI (CI), coupled-cluster theory (CC), and quadratic CI (QCI) are assessed by means of a Moller-Plesset perturbation expansion of the correlation energy up to fifth order. The computational efficiencies and relative merits of the different techniques are outlined. A new augmented version of coupled-cluster theory, denoted as CCSD(T), is proposed to remedy some of the deficiencies of previous augmented coupled-cluster models. 8. 8 Bartlett, R. J.; Musiał, M. Coupled-cluster theory in quantum chemistry. Rev. Mod. Phys. 2007, 79, 291, DOI: 10.1103/RevModPhys.79.291 Coupled-cluster theory in quantum chemistry Bartlett, Rodney J.; Musial, Monika Reviews of Modern Physics (2007), 79 (1), 291-352CODEN: RMPHAT; ISSN:0034-6861. (American Physical Society) A review. Today, coupled-cluster theory offers the most accurate results among the practical ab initio electronic-structure theories applicable to moderate-sized mols. Though it was originally proposed for problems in physics, it has seen its greatest development in chem., enabling an extensive range of applications to mol. structure, excited states, properties, and all kinds of spectroscopy. In this review, the essential aspects of the theory are explained and illustrated with informative numerical results. 9. 9 Cremer, D. M. Møller-Plesset perturbation theory: from small molecule methods to methods for thousands of atoms. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2011, 1, 509, DOI: 10.1002/wcms.58 Moller-Plesset perturbation theory: From small molecule methods to methods for thousands of atoms Cremer, Dieter Wiley Interdisciplinary Reviews: Computational Molecular Science (2011), 1 (4), 509-530CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell) A review. The development of Mueller-Plesset perturbation theory (MPPT) has seen four different periods in almost 80 years. In the first 40 years (period 1), MPPT was largely ignored because the focus of quantum chemists was on variational methods. After the development of many-body perturbation theory by theor. physicists in the 1950s and 1960s, a second 20-yr long period started, during which MPn methods up to order n = 6 were developed and computer-programed. In the late 1980s and in the 1990s (period 3), shortcomings of MPPT became obvious, esp. the sometimes erratic or even divergent behavior of the MPn series. The phys. usefulness of MPPT was questioned and it was suggested to abandon the theory. Since the 1990s (period 4), the focus of method development work has been almost exclusively on MP2. A wealth of techniques and approaches has been put forward to convert MP2 from a O(M5) computational problem into a low-order or linear-scaling task that can handle mols. with thousands of atoms. In addn., the accuracy of MP2 has been systematically improved by introducing spin scaling, dispersion corrections, orbital optimization, or explicit correlation. The coming years will see a continuously strong development in MPPT that will have an essential impact on other quantum chem. methods. 10. 10 Szabados, Á. Reference Module in Chemistry, Molecular Sciences and Chemical Engineering; Elsevier, 2017. There is no corresponding record for this reference. 11. 11 Grimme, S. Improved second-order Møller-Plesset perturbation theory by separate scaling of parallel- and antiparallel-spin pair correlation energies. J. Chem. Phys. 2003, 118, 9095, DOI: Improved second-order Moller-Plesset perturbation theory by separate scaling of parallel- and antiparallel-spin pair correlation energies Grimme, Stefan Journal of Chemical Physics (2003), 118 (20), 9095-9102CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) A simple modification of the second-order Moller-Plesset perturbation theory (MP2) to improve the description of mol. ground state energies is proposed. The total MP2 correlation energy is partitioned into parallel- and antiparallel-spin components which are sep. scaled. The two parameters (scaling factors), whose values can be justified by basic theor. arguments, were optimized on a benchmark set of 51 reaction energies composed of 74 first-row mols. The new method performs significantly better than std. MP2: the rms [mean abs. error (MAE)] deviation drops from 4.6 (3.3) to 2.3 (1.8) kcal/mol. The max. error is reduced from 13.3 to 5.1 kcal/mol. Significant improvements are esp. obsd. for cases which are usually known as MP2 pitfalls while cases already described well with MP2 remain almost unchanged. Even for 11 atomization energies not considered in the fit, uniform improvements [MAE: 8.1 kcal/mol (MP2) vs. 3.2 kcal/mol (new)] were found. The results are furthermore compared with those from d. functional theory (DFT/B3LYP) and quadratic CI [QCISD/QCISD(T)] calcns. Also for difficult systems including strong (nondynamical) correlation effects, the improved MP2 method clearly outperforms DFT/B3LYP and yields results of QCISD or sometimes QCISD(T) quality. Preliminary calcns. of the equil. bond lengths and harmonic vibrational frequencies for ten diat. mols. also show consistent enhancements. The uniformity with which the new method improves upon MP2, thereby rectifying many of its problems, indicates significant robustness and suggests it as a valuable quantum chem. method of general use. 12. 12 Jung, Y.; Lochan, R. C.; Dutoi, A. D.; Head-Gordon, M. Scaled opposite-spin second order Møller-Plesset correlation energy: An economical electronic structure method. J. Chem. Phys. 2004, 121 , 9793, DOI: 10.1063/1.1809602 Scaled opposite-spin second order Moller-Plesset correlation energy: An economical electronic structure method Jung, Yousung; Lochan, Rohini C.; Dutoi, Anthony D.; Head-Gordon, Martin Journal of Chemical Physics (2004), 121 (20), 9793-9802CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) A simplified approach to treating the electron correlation energy is suggested in which only the α-β component of the second order Moller-Plesset energy is evaluated, and then scaled by an empirical factor which is suggested to be 1.3. This scaled opposite-spin second order energy (SOS-MP2), where MP2 is Moller-Plesset theory, yields results for relative energies and deriv. properties that are statistically improved over the conventional MP2 method. Furthermore, the SOS-MP2 energy can be evaluated without the fifth order computational steps assocd. with MP2 theory, even without exploiting any spatial locality. A fourth order algorithm is given for evaluating the opposite spin MP2 energy using auxiliary basis expansions, and a Laplace approach, and timing comparisons are given. 13. 13 Janesko, B. G.; Scuseria, G. E. Coulomb-only second-order perturbation theory in long-range-corrected hybrid density functionals. Phys. Chem. Chem. Phys. 2009, 11, 9677, DOI: 10.1039/b910905f Coulomb-only second-order perturbation theory in long-range-corrected hybrid density functionals Janesko, Benjamin G.; Scuseria, Gustavo E. Physical Chemistry Chemical Physics (2009), 11 (42), 9677-9686CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry) We have been investigating the combination of a short-range d. functional approxn. with long-range RPA correlation, where the direct RPA correlation is constructed using only Coulomb (i.e., not antisymmetrized) two-electron integrals. Our group's recently demonstrated connection between RPA and coupled cluster theory suggests investigating a related method: second-order Moller-Plesset perturbation theory correlation (MP2) constructed using only Coulomb integrals. This new "JMP2" method is related to the scaled-opposite-spin SOS-MP2 approxn., which is also constructed using only Coulomb integrals. While JMP2 and SOS-MP2 yield identical results for closed shell systems, they have important differences for open shells. We show here that both JMP2 and SOS-MP2 provide a reasonable treatment of long-range correlation when combined with a short-range exchange-correlation functional. Remarkably, JMP2's explicit inclusion of (approx.) like-spin correlation effects provides significant improvements over SOS-MP2 for thermochem. 14. 14 Szabados, Á.; Nagy, P. Spin component scaling in multiconfiguration perturbation theory. J. Phys. Chem. A 2011, 115, 523, DOI: 10.1021/jp108575a Spin Component Scaling in Multiconfiguration Perturbation Theory Szabados, Agnes; Nagy, Peter Journal of Physical Chemistry A (2011), 115 (4), 523-534CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society) We investigate a term-by-term scaling of the second-order energy correction obtained by perturbation theory (PT) starting from a multiconfiguration wave function. The total second-order correction is decompd. into several terms, based on the level and the spin pattern of the excitations. To define individual terms, we extend the same spin/different spin categorization of spin component scaling in various ways. When needed, identification of the excitation level is facilitated by the pivot determinant underlying the multiconfiguration PT framework. Scaling factors are detd. from the stationary condition of the total energy calcd. up to order 3. The decompn. schemes are tested numerically on the example of bond dissocn. profiles and energy differences. We conclude that Grimme's parameters detd. for single-ref. Moller-Plesset theory may give a modest error redn. along the entire potential surface, if adopting a multireference based PT formulation. Scaling factors obtained from the stationary condition show relatively large variation with mol. geometry, at the same time they are more efficient in reducing the error when following a bond dissocn. process. 15. 15 Grimme, S.; Goerigk, L.; Fink, R. F. Spin-component-scaled electron correlation methods. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2012, 2, 886– 906, DOI: 10.1002/wcms.1110 Spin-component-scaled electron correlation methods Grimme, Stefan; Goerigk, Lars; Fink, Reinhold F. Wiley Interdisciplinary Reviews: Computational Molecular Science (2012), 2 (6), 886-906CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell) A review. Spin-component-scaled (SCS) electron correlation methods for electronic structure theory are reviewed. The methods can be derived theor. by applying special conditions to the underlying wave functions in perturbation theory. They are based on the insight that low-order wave function expansions treat the correlation effects of electron pairs with opposite spin (OS) and same spin (SS) differently because of their different treatment at the underlying Hartree-Fock level. Phys., this is related to the different av. inter-electronic distances in the SS and OS electron pairs. The overview starts with the original SCS-MP2 method and discusses its strengths and weaknesses and various ways to parameterize the scaling factors. Extensions to coupled-cluster and excited state methods as well the connection to virtual-orbital dependent d. functional approaches are highlighted. The performance of various SCS methods in large thermochem. benchmarks and for excitation energies is discussed in comparison with other common electronic structure methods. 16. 16 Grimme, S. Semiempirical hybrid density functional with perturbative second-order correlation. J. Chem. Phys. 2006, 124, 034108 DOI: 10.1063/1.2148954 Semiempirical hybrid density functional with perturbative second-order correlation Grimme, Stefan Journal of Chemical Physics (2006), 124 (3), 034108/1-034108/16CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) A new hybrid d. functional for general chem. applications is proposed. It is based on a mixing of std. generalized gradient approxns. (GGAs) for exchange by Becke (B) and for correlation by Lee, Yang, and Parr (LYP) with Hartree-Fock (HF) exchange and a perturbative second-order correlation part (PT2) that is obtained from the Kohn-Sham (GGA) orbitals and eigenvalues. This virtual orbital-dependent functional contains only two global parameters that describe the mixt. of HF and GGA exchange (ax) and of the PT2 and GGA correlation (c), resp. The parameters are obtained in a least-squares-fit procedure to the G2/97 set of heat of formations. Opposed to conventional hybrid functionals, the optimum ax is found to be quite large (53% with c = 27%) which at least in part explains the success for many problematic mol. systems compared to conventional approaches. The performance of the new functional termed B2-PLYP is assessed by the G2/ 97 std. benchmark set, a second test suite of atoms, mols., and reactions that are considered as electronically very difficult (including transition-metal compds., weakly bonded complexes, and reaction barriers) and comparisons with other hybrid functionals of GGA and meta-GGA types. According to many realistic tests, B2-PLYP can be regarded as the best general purpose d. functional for mols. (e.g., a mean abs. deviation for the two test sets of only 1.8 and 3.2 kcal/mol compared to about 3 and 5 kcal/mol, resp., for the best other d. functionals). Very importantly, also the max. and minium errors (outliers) are strongly reduced (by about 10-20 kcal/mol). Furthermore, very good results are obtained for transition state barriers but unlike previous attempts at such a good description, this definitely comes not at the expense of equil. properties. Preliminary calcns. of the equil. bond lengths and harmonic vibrational frequencies for diat. mols. and transition-metal complexes also show very promising results. The uniformity with which B2-PLYP improves for a wide range of chem. systems emphasizes the need of (virtual) orbital-dependent terms that describe nonlocal electron correlation in accurate exchange-correlation functionals. From a practical point of view, the new functional seems to be very robust and it is thus suggested as an efficient quantum chem. method of general purpose. 17. 17 Sancho-García, J. C.; Adamo, C. Double-hybrid density functionals: Merging wavefunction and density approaches to get the best of both worlds. Phys. Chem. Chem. Phys. 2013, 15, 14581, DOI: Double-hybrid density functionals: merging wavefunction and density approaches to get the best of both worlds Sancho-Garcia, J. C.; Adamo, C. Physical Chemistry Chemical Physics (2013), 15 (35), 14581-14594CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry) A review. We review why and how double-hybrid d. functionals have become new leading actors in the field of computational chem., thanks to the combination of an unprecedented accuracy together with large robustness and reliability. Similar to their predecessors, the widely employed hybrid d. functionals, they are rooted in the Adiabatic Connection Method from which they emerge in a natural way. We present recent achievements concerning applications to chem. systems of the most interest, and current extensions to deal with challenging issues such as non-covalent interactions and excitation energies. These promising methods, despite a slightly higher computational cost than other typical d.-based models, are called to play a key role in the near future and can thus pave the way towards new discoveries or advances. 18. 18 Goerigk, L.; Grimme, S. Double-hybrid density functionals. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2014, 4, 576, DOI: 10.1002/wcms.1193 Double-hybrid density functionals Goerigk, Lars; Grimme, Stefan Wiley Interdisciplinary Reviews: Computational Molecular Science (2014), 4 (6), 576-600CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell) Double-hybrid d. functionals (DHDFs) are reviewed in this study. In DHDFs parts of conventional d. functional theory (DFT) exchange and correlation are replaced by contributions from nonlocal Fock-exchange and second-order perturbative correlation. The latter portion is based on the well-known MP2 wave-function approach in which, however, Kohn-Sham orbitals are used to calc. its contribution. First, related methods preceding this idea are reviewed, followed by a thorough discussion of the first modern double-hybrid B2-PLYP. Parallels and differences between B2-PLYP and its various successors are then outlined. This discussion is rounded off with representative thermochem. examples demonstrating that DHDFs belong to the most robust and accurate DFT approaches currently available. This anal. also presents hitherto unpublished results for recently developed DHDFs. Finally, how double-hybrids can be combined with linear-response time-dependent DFT is also outlined and the value of this approach for electronically excited states is shown. WIREs Comput Mol Sci 2014, 4:576-600. doi: 10.1002/wcms.1193 For further resources related to this article, please visit the . Conflict of interest: The authors have declared no conflicts of interest for this article. 19. 19 Martin, J. M. L.; Santra, G. Empirical Double-Hybrid Density Functional Theory: A ‘Third Way’ in Between WFT and DFT. Isr. J. Chem. 2020, 60, 787, DOI: 10.1002/ijch.201900114 Empirical Double-Hybrid Density Functional Theory: A 'Third Way' in Between WFT and DFT Martin, Jan M. L.; Santra, Golokesh Israel Journal of Chemistry (2020), 60 (8-9), 787-804CODEN: ISJCAT; ISSN:0021-2148. (Wiley-VCH Verlag GmbH & Co. KGaA) A review. Double hybrid d. functional theory arguably sits on the seamline between wavefunction methods and DFT: it represents a special case of Rung 5 on the "Jacob's Ladder" of John P. Perdew. For large and chem. diverse benchmarks such as GMTKN55, empirical double hybrid functionals with dispersion corrections can achieve accuracies approaching wavefunction methods at a cost not greatly dissimilar to hybrid DFT approaches, provided RI-MP2 and/or another MP2 acceleration techniques are available in the electronic structure code. Only a half-dozen or fewer empirical parameters are required. For vibrational frequencies, accuracies intermediate between CCSD and CCSD(T) can be achieved, and performance for other properties is encouraging as well. Organometallic reactions can likewise be treated well, provided static correlation is not too strong. Further prospects are discussed, including range-sepd. and RPA-based approaches. 20. 20 Chai, J.-D.; Head-Gordon, M. Long-range corrected double-hybrid density functionals. J. Chem. Phys. 2009, 131, 174105 DOI: 10.1063/1.3244209 Long-range corrected double-hybrid density functionals Chai, Jeng-Da; Head-Gordon, Martin Journal of Chemical Physics (2009), 131 (17), 174105/1-174105/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) We extend the range of applicability of our previous long-range cor. (LC) hybrid functional, ωB97X, with a nonlocal description of electron correlation, inspired by second-order Moller-Plesset (many-body) perturbation theory. This LC "double-hybrid" d. functional, denoted as ωB97X-2, is fully optimized both at the complete basis set limit (using 2-point extrapolation from calcns. using triple and quadruple zeta basis sets), and also sep. using the somewhat less expensive 6-311++G(3df,3pd) basis. On independent test calcns. (as well as training set results), ωB97X-2 yields high accuracy for thermochem., kinetics, and noncovalent interactions. In addn., owing to its high fraction of exact Hartree-Fock exchange, ωB97X-2 shows significant improvement for the systems where self-interaction errors are severe, such as sym. homonuclear radical cations. (c) 2009 American Institute of Physics. 21. 21 Goerigk, L.; Grimme, S. Efficient and Accurate Double-Hybrid-Meta-GGA Density Functionals—Evaluation with the Extended GMTKN30 Database for General Main Group Thermochemistry, Kinetics, and Noncovalent Interactions. J. Chem. Theory Comput. 2011, 7, 291, DOI: 10.1021/ct100466k Efficient and Accurate Double-Hybrid-Meta-GGA Density Functionals-Evaluation with the Extended GMTKN30 Database for General Main Group Thermochemistry, Kinetics, and Noncovalent Interactions Goerigk, Lars; Grimme, Stefan Journal of Chemical Theory and Computation (2011), 7 (2), 291-309CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society) We present an extended and improved version of our recently published database for general main group thermochem., kinetics, and noncovalent interactions, which is dubbed GMTKN30. Furthermore, we suggest and investigate two new double-hybrid-meta-GGA d. functionals called PTPSS-D3 and PWPB95-D3. PTPSS-D3 is based on reparameterized TPSS exchange and correlation contributions; PWPB95-D3 contains reparameterized PW exchange and B95 parts. Both functionals contain fixed amts. of 50% Fock-exchange. Furthermore, they include a spin-opposite scaled perturbative contribution and are combined with our latest atom-pairwise London-dispersion correction. When evaluated with the help of the Laplace transformation algorithm, both methods scale as N4 with system size. The functionals are compared with the double hybrids B2PLYP-D3, B2GPPLYP-D3, DSD-BLYP-D3, and XYG3 for GMTKN30 with a quadruple-ζ basis set. PWPB95-D3 and DSD-BLYP-D3 are the best functionals in our study and turned out to be more robust than B2PLYP-D3 and XYG3. Furthermore, PWPB95-D3 is the least basis set dependent and the best functional at the triple-ζ level. For the example of transition metal carbonyls, it is shown that, mainly due to the lower amt. of Fock-exchange, PWPB95-D3 and PTPSS-D3 are better applicable than the other double hybrids. Finally, we discuss in some detail the XYG3 functional, which makes use of B3LYP orbitals and electron densities. We show that it is basically a highly nonlocal variant of B2PLYP and that its partially good performance is mainly due to a larger effective amt. of perturbative correlation compared to other double hybrids. We finally recommend the PWPB95-D3 functional in general chem. applications. 22. 22 Zhang, I. Y.; Xu, X.; Jung, Y.; Goddard, W. A. A fast doubly hybrid density functional method close to chemical accuracy using a local opposite spin ansatz. Proc. Natl. Acad. Sci. U.S.A. 2011 , 108, 19896, DOI: 10.1073/pnas.1115123108 A fast doubly hybrid density functional method close to chemical accuracy using a local opposite spin ansatz Zhang, Igor Ying; Xu, Xin; Jung, Yousung; Goddard, William A., III Proceedings of the National Academy of Sciences of the United States of America (2011), 108 (50), 19896-19900, S19896/1-S19896/10CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences) We develop and validate the XYGJ-OS functional, based on the adiabatic connection formalism and Girling-Levy perturbation theory to second order and using the opposite-spin (OS) ansatz combined with locality of electron correlation. XYGJ-OS with local implementation scales as N3 with an overall accuracy of 1.28 kcal/mol for thermochem., bond dissocn. energies, reaction barrier heights, and nonbonded interactions, comparable to that of 1.06 kcal/mol for the accurate coupled-cluster based G3 method (scales as N7) and much better than many popular d. functional theory methods: B3LYP (4.98), PBEO (4.36), and PBE (12.10). 23. 23 Zhang, I. Y.; Su, N. Q.; Brémond, É. A. G.; Adamo, C.; Xu, X. Doubly hybrid density functional xDH-PBE0 from a parameter-free global hybrid model PBE0. J. Chem. Phys. 2012, 136, 174103 DOI: Doubly hybrid density functional xDH-PBE0 from a parameter-free global hybrid model PBE0 Zhang, Igor Ying; Su, Neil Qiang; Bremond, Eric A. G.; Adamo, Carlo; Xu, Xin Journal of Chemical Physics (2012), 136 (17), 174103/1-174103/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) Following the XYG3 model which uses orbitals and d. from B3LYP, an empirical doubly hybrid (DH) functional is developed by using inputs from PBE0. This new functional, named xDH-PBE0, has been tested on a no. of different mol. properties, including atomization energies, bond dissocn. enthalpies, reaction barrier heights, and nonbonded interactions. From the results obtained, xDH-PBE0 not only displays a significant improvement with respect to the parent PBE0, but also shows a performance that is comparable to XYG3. Arguably, while PBE0 is a parameter-free global hybrid (GH) functional, the B3LYP GH functional contains eight fit parameters. From a more general point of view, the present work points out that reliable and general-purpose DHs can be obtained with a limited no. of fit parameters. (c) 2012 American Institute of Physics. 24. 24 Kozuch, S.; Martin, J. M. L. Spin-Component-Scaled Double Hybrids: An Extensive Search for the Best Fifth-Rung Functionals Blending DFT and Perturbation Theory. J. Comput. Chem. 2013, 34, 2327, DOI: 10.1002/jcc.23391 Spin-component-scaled double hybrids: An extensive search for the best fifth-rung functionals blending DFT and perturbation theory Kozuch, Sebastian; Martin, Jan M. L. Journal of Computational Chemistry (2013), 34 (27), 2327-2344CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.) Following up on an earlier preliminary communication (Kozuch and Martin, Phys. Chem. Chem. Phys. 2011, 13, 20104), we report here in detail on an extensive search for the most accurate spin-component-scaled double hybrid functionals [of which conventional double hybrids (DHs) are a special case]. Such fifth-rung functionals approach the performance of composite ab initio methods such as G3 theory at a fraction of their computational cost, and with anal. derivs. available. In this article, we provide a crit. anal. of the variables and components that maximize the accuracy of DHs. These include the selection of the exchange and correlation functionals, the coeffs. of each component [d. functional theory (DFT), exact exchange, and perturbative correlation in both the same spin and opposite spin terms], and the addn. of an ad-hoc dispersion correction; we have termed these parametrizations "DSD-DFT" (Dispersion cor., Spin-component scaled, Double-hybrid DFT). Somewhat surprisingly, the quality of DSD-DFT is only mildly dependent on the underlying DFT exchange and correlation components, with even DSD-LDA yielding respectable performance. Simple, nonempirical GGAs appear to work best, whereas meta-GGAs offer no advantage (with the notable exception of B95c). The best correlation components appear to be, in that order, B95c, P86, and PBEc, while essentially any good GGA exchange yields nearly identical results. On further validation with a wider variety of thermochem., weak interaction, kinetic, and spectroscopic benchmarks, we find that the best functionals are, roughly in that order, DSD-PBEhB95, DSD-PBEP86, DSD-PBEPW91, and DSD-PBEPBE. In addn., DSD-PBEP86 and DSD-PBEPBE can be used without source code modifications in a wider variety of electronic structure codes. Sample job decks for several commonly used such codes are supplied as electronic Supporting Information. Copyright © 2013 Wiley Periodicals, Inc. 25. 25 Brémond, É.; Ciofini, I.; Sancho-García, J. C.; Adamo, C. Nonempirical Double-Hybrid Functionals: An Effective Tool for Chemists. Acc. Chem. Res. 2016, 49, 1503, DOI: 10.1021/ Nonempirical Double-Hybrid Functionals: An Effective Tool for Chemists Bremond, Eric; Ciofini, Ilaria; Sancho-Garcia, Juan Carlos; Adamo, Carlo Accounts of Chemical Research (2016), 49 (8), 1503-1513CODEN: ACHRE4; ISSN:0001-4842. (American Chemical Society) A review. D. functional theory (DFT) emerged in the last two decades as the most reliable tool for the description and prediction of properties of mol. systems and extended materials, coupling in an unprecedented way high accuracy and reasonable computational cost. This success rests also on the development of more and more performing d. functional approxns. (DFAs). Indeed, the Achilles' heel of DFT is represented by the exchange-correlation contribution to the total energy, which, being unknown, must be approximated. Since the beginning of the 1990s, global hybrids (GH) functionals, where an explicit dependence of the exchange-correlation energy on occupied Kohn-Sham orbitals is introduced thanks to a fraction of Hartree-Fock-like exchange, imposed themselves as the most reliable DFAs for chem. applications. However, if these functionals normally provide results of sufficient accuracy for most of the cases analyzed, some properties, such as thermochem. or dispersive interactions, can still be significantly improved. A possible way out is represented by the inclusion, into the exchange-correlation functional, of an explicit dependence on virtual Kohn-Sham orbitals via perturbation theory. This leads to a new class of functionals, called double-hybrids (DHs). In this Account, we describe our nonempirical approach to DHs, which, following the line traced by the Perdew-Burke-Ernzerhof approach, allows for the definition of a GH (PBE0) and a DH (QIDH) model. In such a way, a whole family of nonempirical functionals, spanning on the highest rungs of the Perdew's quality scale, is now available and competitive with other-more empirical-DFAs. Discussion of selected cases, ranging from thermochem. and reactions to weak interactions and excitation energies, not only show the large range of applicability of nonempirical DFAs, but also underline how increasing the no. of theor. constraints parallels with an improvement of the DFA's numerical performances. This fact further consolidates the strong theor. framework of nonempirical DFAs.Finally, even if nonempirical DH approaches are still computationally expensive, relying on the fact that they can benefit of all tech. enhancements developed for speeding up post-Hartree-Fock methods, there is substantial hope for their near future routine application to the description and prediction of complex chem. systems and reactions. 26. 26 Su, N. Q.; Xu, X. The XYG3 Type of Doubly Hybrid Density Functionals. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2016, 6, 721, DOI: 10.1002/wcms.1274 The XYG3 type of doubly hybrid density functionals Su, Neil Qiang; Xu, Xin Wiley Interdisciplinary Reviews: Computational Molecular Science (2016), 6 (6), 721-747CODEN: WIRCAH; ISSN:1759-0884. (Wiley-Blackwell) Doubly hybrid (DH) functionals have emerged as a new class of d. functional approxns. (DFAs), which not only have a nonlocal orbital-dependent component in the exchange part, but also incorporate the information of unoccupied orbitals in the correlation part, being at the top rung of Perdew's view of Jacob's ladder in DFAs. This review article focuses on the XYG3 type of DH (xDH) functionals, which use a low rung functional to perform the self-consistent-field calcn. to generate orbitals and densities, with which a top rung DH functional is used for final energy evaluation. We will discuss the theor. background of the xDH functionals, briefly reviewing the adiabatic connection formalism, coordinate scaling relations, and Goerling-Levy perturbation theory. General performance of the xDH functionals will be presented for both energies and structures. In particular, we will present the fractional charge behaviors of the xDH functionals, examg. the self-interaction errors, the delocalization errors and the deviation from the linearity condition, as well as their effects on the predicted ionization potentials, electron affinities and fundamental gaps. This provides a theor. rationale for the obsd. good performance of the xDH functionals. WIREs Comput Mol Sci 2016, 6:721-747. doi: 10.1002/wcms.1274 For further resources related to this article, please visit the . 27. 27 Feyereisen, M.; Fitzgerald, G.; Komornicki, A. Use of approximate integrals in ab initio theory. An application in MP2 energy calculations. Chem. Phys. Lett. 1993, 208, 359, DOI: 10.1016/ Use of approximate integrals in ab initio theory. An application in MP2 energy calculations Feyereisen, Martin; Fitzgerald, George; Komornicki, Andrew Chemical Physics Letters (1993), 208 (5-6), 359-63CODEN: CHPLBC; ISSN:0009-2614. Authors use the resoln. of the identity (RI) as a convenient way to replace the use of four-index two-electron integrals with linear combinations of three-index integrals. The method is broadly applicable to a wide range of problems in quantum chem. Authors demonstrate the effectiveness of RI for the calcn. of MP2 energies. For the water dimer, agreement within 0.1 kcal/mol is obtained with respect to exact MP2 calcns. The RI-MP2 energies require only about 10% of the time required by conventional MP2. 28. 28 Gyevi-Nagy, L.; Kállay, M.; Nagy, P. R. Integral-direct and parallel implementation of the CCSD(T) method: Algorithmic developments and large-scale applications. J. Chem. Theory Comput. 2020, 16, 366, DOI: 10.1021/acs.jctc.9b00957 Integral-Direct and Parallel Implementation of the CCSD(T) Method: Algorithmic Developments and Large-Scale Applications Gyevi-Nagy Laszlo; Kallay Mihaly; Nagy Peter R Journal of chemical theory and computation (2020), 16 (1), 366-384 ISSN:. A completely integral-direct, disk I/O, and network traffic economic coupled-cluster singles, doubles, and perturbative triples [CCSD(T)] implementation has been developed relying on the density-fitting approximation. By fully exploiting the permutational symmetry, the presented algorithm is highly operation count and memory-efficient. Our measurements demonstrate excellent strong scaling achieved via hybrid MPI/OpenMP parallelization and a highly competitive, 60-70% utilization of the theoretical peak performance on up to hundreds of cores. The terms whose evaluation time becomes significant only for small- to medium-sized examples have also been extensively optimized. Consequently, high performance is also expected for systems appearing in extensive data sets used, e.g., for density functional or machine learning parametrizations, and in calculations required for certain reduced-cost or local approximations of CCSD(T), such as in our local natural orbital scheme [LNO-CCSD(T)]. The efficiency of this implementation allowed us to perform some of the largest CCSD(T) calculations ever presented for systems of 31-43 atoms and 1037-1569 orbitals using only four to eight many-core CPUs and 1-3 days of wall time. The resulting 13 correlation energies and the 12 corresponding reaction energies and barrier heights are added to our previous benchmark set collecting reference CCSD(T) results of molecules at the applicability limit of current implementations. 29. 29 Almlöf, J. Elimination of energy denominators in Møller-Plesset perturbation theory by a Laplace transform approach. Chem. Phys. Lett. 1991, 181, 319, DOI: 10.1016/0009-2614(91)80078-C Elimination of energy denominators in Moeller-Plesset perturbation theory by a Laplace transform approach Almlof, Jan Chemical Physics Letters (1991), 181 (4), 319-20CODEN: CHPLBC; ISSN:0009-2614. It is shown how the energy denominators encountered in various schemes for electronic structure calcn. can be removed by a Laplace transform technique. The method is applicable to a wide variety of electronic structure calcns. 30. 30 Häser, M.; Almlöf, J. Laplace transform techniques in Møller-Plesset perturbation theory. J. Chem. Phys. 1992, 96, 489, DOI: 10.1063/1.462485 There is no corresponding record for this reference. 31. 31 Ayala, P. Y.; Scuseria, G. E. Linear scaling second-order Møller-Plesset theory in the atomic orbital basis for large molecular systems. J. Chem. Phys. 1999, 110, 3660, DOI: 10.1063/1.478256 Linear scaling second-order Moeller-Plesset theory in the atomic orbital basis for large molecular systems Ayala, Philippe Y.; Scuseria, Gustavo E. Journal of Chemical Physics (1999), 110 (8), 3660-3671CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) We have used Almlof and Haser's Laplace transform idea to eliminate the energy denominator in second-order perturbation theory (MP2) and obtain an energy expression in the AO basis. We show that the asymptotic computational cost of this method scales quadratically with mol. size. We then define AO domains such that selective pairwise interactions can be neglected using well-defined thresholding criteria based on the power law decay properties of the long-range contributions. For large mols., our scheme yields linear scaling computational cost as a function of mol. size. The errors can be controlled in a precise manner and our method reproduces canonical MP2 energies. We present benchmark calcns. of polyglycine chains and water clusters contg. up to 3040 basis functions. 32. 32 Surján, P. R. The MP2 energy as a functional of the Hartree-Fock density matrix. Chem. Phys. Lett. 2005, 406, 318– 320, DOI: 10.1016/j.cplett.2005.03.024 The MP2 energy as a functional of the Hartree-Fock density matrix Surjan, Peter R. Chemical Physics Letters (2005), 406 (4-6), 318-320CODEN: CHPLBC; ISSN:0009-2614. (Elsevier B.V.) The explicit E[2][P] functional is presented, where E [2] is the second order Moller-Plesset correlation energy and P is the std. Hartree-Fock d. matrix. The ideas leading to this functional are implicit in previous studies, but the significance of its existence has not yet been sufficiently emphasized and its simple explicit form has not been presented. With the proposed functional one may obtain the correlation energy in the absence of MOs, knowing merely the d. matrix. This may further facilitate linear scaling computation of the correlation energy. 33. 33 Kobayashi, M.; Nakai, H. Implementation of Surján’s density matrix formulae for calculating second-order Møller-Plesset energy. Chem. Phys. Lett. 2006, 420, 250– 255, DOI: 10.1016/ Implementation of Surjan's density matrix formulae for calculating second-order Moller-Plesset energy Kobayashi, Masato; Nakai, Hiromi Chemical Physics Letters (2006), 420 (1-3), 250-255CODEN: CHPLBC; ISSN:0009-2614. (Elsevier B.V.) We numerically assess the method for obtaining second-order Moller-Plesset (MP2) energy from the Hartree-Fock d. matrix (DM) recently proposed by Surjan [Surjan, Chem. Phys. Lett. 406 (2005) 318]. It is confirmed that Surjan's method, referred to as DM-Laplace MP2, can obtain MP2 energy accurately by means of appropriate integral quadrature and a matrix exponential evaluation scheme. Numerical tests reveal that the Euler-Maclaurin and the Romberg numerical integration schemes can achieve milli-hartree accuracy with small quadrature points. This Letter also indicates the possibility of the application of DM-Laplace MP2 to linear-scaling SCF techniques, which give approx. DM. 34. 34 Doser, B.; Lambrecht, D. S.; Kussmann, J.; Ochsenfeld, C. Linear-scaling atomic orbital-based second-order Møller-Plesset perturbation theory by rigorous integral screening criteria. J. Chem. Phys. 2009, 130, 064107 DOI: 10.1063/1.3072903 Linear-scaling atomic orbital-based second-order Moller-Plesset perturbation theory by rigorous integral screening criteria Doser, Bernd; Lambrecht, Daniel S.; Kussmann, Joerg; Ochsenfeld, Christian Journal of Chemical Physics (2009), 130 (6), 064107/1-064107/14CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) A Laplace-transformed second-order Moller-Plesset perturbation theory (MP2) method is presented, which allows to achieve linear scaling of the computational effort with mol. size for electronically local structures. Also for systems with a delocalized electronic structure, a cubic or even quadratic scaling behavior is achieved. Numerically significant contributions to the AO (AO)-MP2 energy are preselected using the so-called multipole-based integral ests. (MBIE) introduced earlier by us. Since MBIE provides rigorous upper bounds, numerical accuracy is fully controlled and the exact MP2 result is attained. While the choice of thresholds for a specific accuracy is only weakly dependent upon the mol. system, our AO-MP2 scheme offers the possibility for incremental thresholding: for only little addnl. computational expense, the numerical accuracy can be systematically converged. We illustrate this dependence upon numerical thresholds for the calcn. of intermol. interaction energies for the S22 test set. The efficiency and accuracy of our AO-MP2 method is demonstrated for linear alkanes, stacked DNA base pairs, and carbon nanotubes: e.g., for DNA systems the crossover toward conventional MP2 schemes occurs between one and two base pairs. In this way, it is for the first time possible to compute wave function-based correlation energies for systems contg. more than 1000 atoms with 10 000 basis functions as illustrated for a 16 base pair DNA system on a single-core computer, where no empirical restrictions are introduced and numerical accuracy is fully preserved. (c) 2009 American Institute of Physics. 35. 35 Schäfer, T.; Ramberger, B.; Kresse, G. Quartic scaling MP2 for solids: A highly parallelized algorithm in the plane wave basis. J. Chem. Phys. 2017, 146, 104101 DOI: 10.1063/1.4976937 Quartic scaling MP2 for solids: A highly parallelized algorithm in the plane wave basis Schafer Tobias; Ramberger Benjamin; Kresse Georg The Journal of chemical physics (2017), 146 (10), 104101 ISSN:. We present a low-complexity algorithm to calculate the correlation energy of periodic systems in second-order Moller-Plesset (MP2) perturbation theory. In contrast to previous approximation-free MP2 codes, our implementation possesses a quartic scaling, O(N(4)), with respect to the system size N and offers an almost ideal parallelization efficiency. The general issue that the correlation energy converges slowly with the number of basis functions is eased by an internal basis set extrapolation. The key concept to reduce the scaling is to eliminate all summations over virtual orbitals which can be elegantly achieved in the Laplace transformed MP2 formulation using plane wave basis sets and fast Fourier transforms. Analogously, this approach could allow us to calculate second order screened exchange as well as particle-hole ladder diagrams with a similar low complexity. Hence, the presented method can be considered as a step towards systematically improved correlation energies. 36. 36 Pulay, P.; Saebø, S. Orbital-invariant formulation and second-order gradient evaluation in Møller-Plesset perturbation theory. Theor. Chem. Acc. 1986, 69, 357, DOI: 10.1007/BF00526697 Orbital-invariant formulation and second-order gradient evaluation in Moeller-Plesset perturbation theory Pulay, Peter; Saeboe, Svein Theoretica Chimica Acta (1986), 69 (5-6), 357-68CODEN: TCHAAM; ISSN:0040-5744. Based on the Hylleraas functional form, the second and third orders of the Moeller-Plesset (MP) perturbation theory were reformulated in terms of arbitrary (e.g., localized) internal orbitals, and AOs in the virtual space. The results are strictly equiv. to the canonical formulation if no further approxns. are introduced. The new formalism permits the extension of the local correlation method to MP theory. It also facilitates the treatment of weak pairs at a lower (e.g., second-order) level of theory in CI and coupled-cluster methods. Based on the formalism, an MP2 gradient algorithm is outlined, which does not require the storage of deriv. integrals, integrals with three external MO indexes, and, using the method of N. C. Handy and H. F. Schaefer III (1984), the repeated soln. of the coupled-perturbed SCF equations. 37. 37 Kats, D.; Usvyat, D.; Schütz, M. On the use of the Laplace transform in local correlation methods. Phys. Chem. Chem. Phys. 2008, 10, 3430, DOI: 10.1039/b802993h On the use of the Laplace transform in local correlation methods Kats, Danylo; Usvyat, Denis; Schuetz, Martin Physical Chemistry Chemical Physics (2008), 10 (23), 3430-3439CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry) The applicability of the Laplace transform ansatz of Almlof in the context of local correlation methods with a priori restricted sets of wavefunction parameters is explored. A new local MP2 method based on the Laplace transform ansatz is described, its relation to the local MP2 method based on the Pulay ansatz is elucidated, and its accuracy and efficiency are compared to the 38. 38 Nagy, P. R.; Samu, G.; Kállay, M. An integral-direct linear-scaling second-order Møller-Plesset approach. J. Chem. Theory Comput. 2016, 12, 4897, DOI: 10.1021/acs.jctc.6b00732 An Integral-Direct Linear-Scaling Second-Order Moller-Plesset Approach Nagy, Peter R.; Samu, Gyula; Kallay, Mihaly Journal of Chemical Theory and Computation (2016), 12 (10), 4897-4914CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society) An integral-direct, iteration-free, linear-scaling, local second-order Moller-Plesset (MP2) approach is presented, which is also useful for spin-scaled MP2 calcns. as well as for the efficient evaluation of the perturbative terms of double-hybrid d. functionals. The method is based on a fragmentation approxn.: the correlation contributions of the individual electron pairs are evaluated in domains constructed for the corresponding localized orbitals, and the correlation energies of distant electron pairs are computed with multipole expansions. The required electron repulsion integrals are calcd. directly invoking the d. fitting approxn.; the storage of integrals and intermediates is avoided. The approach also utilizes natural auxiliary functions to reduce the size of the auxiliary basis of the domains and thereby the operation count and memory requirement. Our test calcns. show that the approach recovers 99.9% of the canonical MP2 correlation energy and reproduces reaction energies with an av. (max.) error below 1 kJ/mol (4 kJ/mol). Our benchmark calcns. demonstrate that the new method enables MP2 calcns. for mols. with more than 2300 atoms and 26000 basis functions on a single processor. 39. 39 Saebø, S. Linear-Scaling Techniques in Computational Chemistry and Physics: Methods and Applications; Springer: Netherlands, 2011; pp 65– 82. There is no corresponding record for this reference. 40. 40 Zienau, J.; Clin, L.; Doser, B.; Ochsenfeld, C. Cholesky-decomposed densities in Laplace-based second-order Møller-Plesset perturbation theory. J. Chem. Phys. 2009, 130, 204112 DOI: 10.1063/ Cholesky-decomposed densities in Laplace-based second-order Moller-Plesset perturbation theory Zienau, Jan; Clin, Lucien; Doser, Bernd; Ochsenfeld, Christian Journal of Chemical Physics (2009), 130 (20), 204112/1-204112/4CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) Based on our linear-scaling AO second-order Moller-Plesset perturbation theory (AO-MP2) method , we explore the use of Cholesky-decompd. pseudodensity (CDD) matrixes within the Laplace formulation. Numerically significant contributions are preselected using our multipole-based integral ests. as upper bounds to two-electron integrals so that the 1/R6 decay behavior of transformed Coulomb-type products is exploited. In addn., we combine our new CDD-MP2 method with the resoln. of the identity (RI) approach. Even though the use of RI results in a method that shows a quadratic scaling behavior in the dominant steps, gains of up to one or two orders of magnitude vs. our original AO-MP2 method are obsd. in particular for larger basis sets. (c) 2009 American Institute of Physics. 41. 41 Maurer, S. A.; Clin, L.; Ochsenfeld, C. Cholesky-decomposed density MP2 with density fitting: Accurate MP2 and double-hybrid DFT energies for large systems. J. Chem. Phys. 2014, 140, 224112 DOI: 10.1063/1.4881144 Cholesky-decomposed density MP2 with density fitting: Accurate MP2 and double-hybrid DFT energies for large systems Maurer, Simon A.; Clin, Lucien; Ochsenfeld, Christian Journal of Chemical Physics (2014), 140 (22), 224112/1-224112/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) Our recently developed QQR-type integral screening is introduced in our Cholesky-decompd. pseudo-densities Moller-Plesset perturbation theory of second order (CDD-MP2) method. We use the resoln.-of-the-identity (RI) approxn. in combination with efficient integral transformations employing sparse matrix multiplications. The RI-CDD-MP2 method shows an asymptotic cubic scaling behavior with system size and a small prefactor that results in an early crossover to conventional methods for both small and large basis sets. We also explore the use of local fitting approxns. which allow to further reduce the scaling behavior for very large systems. The reliability of our method is demonstrated on test sets for interaction and reaction energies of medium sized systems and on a diverse selection from our own benchmark set for total energies of larger systems. Timings on DNA systems show that fast calcns. for systems with more than 500 atoms are feasible using a single processor core. Parallelization extends the range of accessible system sizes on one computing node with multiple cores to more than 1000 atoms in a double-zeta basis and more than 500 atoms in a triple-zeta basis. (c) 2014 American Institute of Physics. 42. 42 Helmich-Paris, B.; Repisky, M.; Visscher, L. Relativistic Cholesky-decomposed density matrix MP2. Chem. Phys. 2019, 518, 38, DOI: 10.1016/j.chemphys.2018.11.009 Relativistic Cholesky-decomposed density matrix MP2 Helmich-Paris, Benjamin; Repisky, Michal; Visscher, Lucas Chemical Physics (2019), 518 (), 38-46CODEN: CMPHC2; ISSN:0301-0104. (Elsevier B.V.) We introduce the relativistic Cholesky-decompd. d. (CDD) matrix second-order Moller-Plesset perturbation theory (MP2) energies. The working equations are formulated in terms of the usual intermediates of MP2 when employing the resoln.-of-the-identity approxn. (RI) for two-electron integrals. Those intermediates are obtained by substituting the occupied and virtual quaternion pseudo-d. matrixes of our previously proposed two-component (2C) AO-based MP2 (Helmich-Paris et al., 2016) by the corresponding pivoted quaternion Cholesky factors. While working within the Kramers-restricted formalism, we obtain a formal spin-orbit overhead of 16 and 28 for the Coulomb and exchange contribution to the 2C MP2 correlation energy, resp., compared to a non-relativistic (NR) spin-free CDD-MP2 implementation. This compact quaternion formulation could also be easily explored in any other algorithm to compute the 2C MP2 energy. The quaternion Cholesky factors become sparse for large mols. and, with a block-wise screening, block sparse-matrix multiplication algorithm, we obsd. an effective quadratic scaling of the total wall time for heavy-element contg. linear mols. with increasing system size. The total run time for both NR and 2C calcns. was dominated by the contraction to the exchange energy. We have also investigated a bulky Te-contg. supramol. complex. For such bulky, three-dimensionally extended mols. the present screening scheme has a much larger prefactor and is less effective. 43. 43 Glasbrenner, M.; Graf, D.; Ochsenfeld, C. Efficient Reduced-Scaling Second-Order Møller-Plesset Perturbation Theory with Cholesky-Decomposed Densities and an Attenuated Coulomb Metric. J. Chem. Theory Comput. 2020, 16, 6856, DOI: 10.1021/acs.jctc.0c00600 Efficient Reduced-Scaling Second-Order Moller-Plesset Perturbation Theory with Cholesky-Decomposed Densities and an Attenuated Coulomb Metric Glasbrenner, Michael; Graf, Daniel; Ochsenfeld, Christian Journal of Chemical Theory and Computation (2020), 16 (11), 6856-6868CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society) We present a novel, highly efficient method for the computation of second-order Moller-Plesset perturbation theory (MP2) correlation energies, which uses the resoln. of the identity (RI) approxn. and local MOs obtained from a Cholesky decompn. of pseudodensity matrixes (CDD), as in the RI-CDD-MP2 method developed previously in our group [Maurer, S.A. et al., J. Chem. Phys., 2014, 140, 224112]. In addn., we introduce an attenuated Coulomb metric and subsequently redesign the RI-CDD-MP2 method in order to exploit the resulting sparsity in the three-center integrals. Coulomb and exchange energy contributions are computed sep. using specialized algorithms. A simple, yet effective integral screening protocol based on Schwarz ests. is used for the MP2 exchange energy. The Coulomb energy computation and the preceding transformations of the three-center integrals are accelerated using a modified version of the natural blocking approach [Jung, Y., Head-Gordon, M., Phys. Chem. Chem. Phys., 2006, 8, 2831]. Effective subquadratic scaling for a wide range of mol. sizes is demonstrated in test calcns. in conjunction with a low prefactor. The method is shown to enable cost-efficient MP2 calcns. on large mol. systems with several thousand basis functions. 44. 44 Neuhauser, D.; Rabani, E.; Baer, R. Expeditious Stochastic Approach for MP2 Energies in Large Electronic Systems. J. Chem. Theory Comput. 2013, 9, 24, DOI: 10.1021/ct300946j Expeditious Stochastic Approach for MP2 Energies in Large Electronic Systems Neuhauser, Daniel; Rabani, Eran; Baer, Roi Journal of Chemical Theory and Computation (2013), 9 (1), 24-27CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society) A fast stochastic method for calcg. the second order Moller-Plesset (MP2) correction to the correlation energy of large systems of electrons is presented. The approach is based on reducing the exact summation over occupied and unoccupied states to a time-dependent trace formula amenable to stochastic sampling. We demonstrate the abilities of the method to treat systems with thousands of electrons using hydrogen passivated silicon spherical nanocrystals represented on a real space grid, much beyond the capabilities of present day MP2 implementations. 45. 45 Willow, S. Y.; Kim, K. S.; Hirata, S. Stochastic evaluation of second-order many-body perturbation energies. J. Chem. Phys. 2012, 137, 204122 DOI: 10.1063/1.4768697 Stochastic evaluation of second-order many-body perturbation energies Willow, Soohaeng Yoo; Kim, Kwang S.; Hirata, So Journal of Chemical Physics (2012), 137 (20), 204122/1-204122/5CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) With the aid of the Laplace transform, the canonical expression of the second-order many-body perturbation correction to electronic energy is converted into a sum of two 13-dimensional integrals, the 12-dimensional parts of which are evaluated by Monte Carlo integration. Wt. functions are identified that are anal. normalizable, finite, non-neg. everywhere, and share the same singularities as the integrands. They generate appropriate distributions of four-electron walkers via the Metropolis algorithm, yielding correlation energies of small mols. within a few mEh of the correct values after 108 Monte Carlo steps. This algorithm does away with the integral transformation as the hotspot of the usual algorithms, has a far superior size dependence of cost, does not suffer from the sign problem of some quantum Monte Carlo methods, and potentially easily parallelizable and extensible to other more complex electron-correlation theories. (c) 2012 American Institute of Physics. 46. 46 Barca, G. M. J.; McKenzie, S. C.; Bloomfield, N. J.; Gilbert, A. T. B.; Gill, P. M. W. Q-MP2-OS: Møller-Plesset Correlation Energy by Quadrature. J. Chem. Theory Comput. 2020, 16, 1568, DOI: Q-MP2-OS: Moller-Plesset Correlation Energy by Quadrature Barca, Giuseppe M. J.; McKenzie, Simon C.; Bloomfield, Nathaniel J.; Gilbert, Andrew T. B.; Gill, Peter M. W. Journal of Chemical Theory and Computation (2020), 16 (3), 1568-1577CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society) We present a quadrature-based algorithm for computing the opposite-spin component of the MP2 correlation energy which scales quadratically with basis set size and is well-suited to large-scale parallelization. The key ideas, which are rooted in the earlier work of Hirata and co-workers, are to abandon all two-electron integrals, recast the energy as a seven-dimensional integral, approx. that integral by quadrature, and employ a cutoff strategy to minimize the no. of intermediate quantities. We discuss our implementation in detail and show that it parallelizes almost perfectly on 840 cores for cyclosporine (a mol. with roughly 200 atoms), exhibits scaling for a sequence of polyglycines, and is principally limited by the accuracy of its 47. 47 Martínez, T. J.; Carter, E. A. Pseudospectral Møller-Plesset perturbation theory through third order. J. Chem. Phys. 1994, 100, 3631, DOI: 10.1063/1.466350 Pseudospectral Moeller-Plesset perturbation theory through third order Martinez, Todd J.; Carter, Emily A. Journal of Chemical Physics (1994), 100 (5), 3631-8CODEN: JCPSA6; ISSN:0021-9606. The authors present a formulation and implementation of Moeller-Plesset perturbation theory in a pseudospectral framework. At the second-order level, the pseudospectral formulation is a formally a factor of N/n faster than conventional approaches, while the third order is formally faster by a factor of n, where N is the no. of AOs and n is the no. of occupied orbitals. The accuracy of the resulting energies is probed for a no. of test cases. Practical timings are presented and show conclusively that the pseudospectral formulation is faster than conventional 48. 48 Kossmann, S.; Neese, F. Efficient Structure Optimization with Second-Order Many-Body Perturbation Theory: The RIJCOSX-MP2 Method. J. Chem. Theory Comput. 2010, 6, 2325, DOI: 10.1021/ct100199k Efficient Structure Optimization with Second-Order Many-Body Perturbation Theory: The RIJCOSX-MP2 Method Kossmann, Simone; Neese, Frank Journal of Chemical Theory and Computation (2010), 6 (8), 2325-2338CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society) Efficient energy calcns. and structure optimizations employing second-order Moller-Plesset perturbation theory (MP2) are presented. The application of the RIJCOSX approxn., which involves different approxns. for the formation of the Coulomb- and exchange-type matrixes, to MP2 theory is demonstrated. The RIJCOSX approxn. incorporates the resoln. of the identity' approxn. in terms of a Split-RI-J variant for the evaluation of the Coulomb matrixes and a seminumeric exchange treatment via the chain-of-spheres' algorithm for the formation of the exchange-type matrixes. Beside the derivation of the working equations, the RIJCOSX-MP2 method is benchmarked against the original MP2 and the already highly efficient RI-MP2 method. Energies as well as gradients are computed employing various basis sets and are compared to the conventional MP2 results concerning accuracy and total wall clock times. Speedups of typically a factor of 5-7 in comparison to MP2 can be obsd. for the largest basis set employed in our study. Total energies are reproduced with an av. error of ≤0.8 kcal/mol and min. energy geometries differ by ∼0.1 pm in bond lengths and typically ∼0.2 degrees in bond angles. The RIJCOSX-MP2 gradient parallelizes with a speedup of 8.2 on 10 processors. The algorithms are implemented into the ORCA electronic structure package. 49. 49 Maslen, P. E.; Head-Gordon, M. Non-iterative local second order Møller-Plesset theory. Chem. Phys. Lett. 1998, 283, 102, DOI: 10.1016/S0009-2614(97)01333-X Non-iterative local second order Moller-Plesset theory Maslen, P. E.; Head-Gordon, M. Chemical Physics Letters (1998), 283 (1,2), 102-108CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.) Second order Moller-Plesset perturbation theory (MP2) is formulated in terms of atom-centered occupied and virtual orbitals. Both the occupied and the virtual orbitals are non-orthogonal. A new parameter-free atoms-in-mols. local approxn. is employed to reduce the cost of the calcn. to cubic scaling, and a quasi-canonical two-particle basis is introduced to enable the soln. of the local MP2 equations via explicit matrix diagonalization rather than iteration. 50. 50 Jung, Y.; Shao, Y.; Head-Gordon, M. Fast evaluation of scaled opposite-spin second-order Møller-Plesset correlation energies using auxiliary basis expansions and exploiting sparsity. J. Comput. Chem. 2007, 28, 1953, DOI: 10.1002/jcc.20590 Fast evaluation of scaled opposite spin second-order Moller-Plesset correlation energies using auxiliary basis expansions and exploiting sparsity Jung, Yousung; Shao, Yihan; Head-Gordon, Martin Journal of Computational Chemistry (2007), 28 (12), 1953-1964CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.) The scaled opposite spin Moller-Plesset method (SOS-MP2) is an economical way of obtaining correlation energies that are computationally cheaper, and yet, in a statistical sense, of higher quality than std. MP2 theory, by introducing one empirical parameter. But SOS-MP2 still has a fourth-order scaling step that makes the method inapplicable to very large mol. systems. We reduce the scaling of SOS-MP2 by exploiting the sparsity of expansion coeffs. and local integral matrixes, by performing local auxiliary basis expansions for the occupied-virtual product distributions. To exploit sparsity of 3-index local quantities, we use a blocking scheme in which entire zero-rows and columns, for a given third global index, are deleted by comparison against a numerical threshold. This approach minimizes sparse matrix book-keeping overhead, and also provides sufficiently large submatrixes after blocking, to allow efficient matrix-matrix multiplies. The resulting algorithm is formally cubic scaling, and requires only moderate computational resources (quadratic memory and disk space) and, in favorable cases, is shown to yield effective quadratic scaling behavior in the size regime we can apply it to. Errors assocd. with local fitting using the attenuated Coulomb metric and numerical thresholds in the blocking procedure are found to be insignificant in terms of the predicted relative energies. A diverse set of test calcns. shows that the size of system where significant computational savings can be achieved depends strongly on the dimensionality of the system, and the extent of localizability of the MOs. 51. 51 Förster, A.; Franchini, M.; van Lenthe, E.; Visscher, L. A Quadratic Pair Atomic Resolution of the Identity Based SOS-AO-MP2 Algorithm Using Slater Type Orbitals. J. Chem. Theory Comput. 2020 , 16, 875, DOI: 10.1021/acs.jctc.9b00854 A Quadratic Pair Atomic Resolution of the Identity Based SOS-AO-MP2 Algorithm Using Slater Type Orbitals Forster Arno; Franchini Mirko; Visscher Lucas; Franchini Mirko; van Lenthe Erik Journal of chemical theory and computation (2020), 16 (2), 875-891 ISSN:. We report a production level implementation of pair atomic resolution of the identity (PARI) based second-order Moller-Plesset perturbation theory (MP2) in the Slater type orbital (STO) based Amsterdam Density Functional (ADF) code. As demonstrated by systematic benchmarks, dimerization and isomerization energies obtained with our code using STO basis sets of triple-ζ-quality show mean absolute deviations from Gaussian type orbital, canonical, basis set limit extrapolated, global density fitting (DF)-MP2 results of less than 1 kcal/mol. Furthermore, we introduce a quadratic scaling atomic orbital based spin-opposite-scaled (SOS)-MP2 approach with a very small prefactor. Due to a worst-case scaling of [Formula: see text], our implementation is very fast already for small systems and shows an exceptionally early crossover to canonical SOS-PARI-MP2. We report computational wall time results for linear as well as for realistic three-dimensional molecules and show that triple-ζ quality calculations on molecules of several hundreds of atoms are only a matter of a few hours on a single compute node, the bottleneck of the computations being the SCF rather than the post-SCF energy correction. 52. 52 Förster, A.; Visscher, L. Double hybrid DFT calculations with Slater type orbitals. J. Comput. Chem. 2020, 41, 1660, DOI: 10.1002/jcc.26209 Double hybrid DFT calculations with Slater type orbitals Forster Arno; Visscher Lucas Journal of computational chemistry (2020), 41 (18), 1660-1684 ISSN:. On a comprehensive database with 1,644 datapoints, covering several aspects of main-group as well as of transition metal chemistry, we assess the performance of 60 density functional approximations (DFA), among them 36 double hybrids (DH). All calculations are performed using a Slater type orbital (STO) basis set of triple-ζ (TZ) quality and the highly efficient pair atomic resolution of the identity approach for the exchange- and Coulomb-term of the KS matrix (PARI-K and PARI-J, respectively) and for the evaluation of the MP2 energy correction (PARI-MP2). Employing the quadratic scaling SOS-AO-PARI-MP2 algorithm, DHs based on the spin-opposite-scaled (SOS) MP2 approximation are benchmarked against a database of large molecules. We evaluate the accuracy of STO/PARI calculations for B3LYP as well as for the DH B2GP-PLYP and show that the combined basis set and PARI-error is comparable to the one obtained using the well-known def2-TZVPP Gaussian-type basis set in conjunction with global density fitting. While quadruple-ζ (QZ) calculations are currently not feasible for PARI-MP2 due to numerical issues, we show that, on the TZ level, Jacob's ladder for classifying DFAs is reproduced. However, while the best DHs are more accurate than the best hybrids, the improvements are less pronounced than the ones commonly found on the QZ level. For conformers of organic molecules and noncovalent interactions where very high accuracy is required for qualitatively correct results, DHs provide only small improvements over hybrids, while they still excel in thermochemistry, kinetics, transition metal chemistry and the description of strained organic systems. 53. 53 Hohenstein, E. G.; Parrish, R. M.; Martínez, T. J. Tensor hypercontraction density fitting. I. Quartic scaling second- and third-order Møller-Plesset perturbation theory. J. Chem. Phys. 2012, 137, 044103 DOI: 10.1063/1.4732310 Tensor hypercontraction density fitting. I. Quartic scaling second- and third-order Moller-Plesset perturbation theory Hohenstein, Edward G.; Parrish, Robert M.; Martinez, Todd J. Journal of Chemical Physics (2012), 137 (4), 044103/1-044103/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics) Many approxns. have been developed to help deal with the O(N4) growth of the electron repulsion integral (ERI) tensor, where N is the no. of one-electron basis functions used to represent the electronic wavefunction. Of these, the d. fitting (DF) approxn. is currently the most widely used despite the fact that it is often incapable of altering the underlying scaling of computational effort with respect to mol. size. We present a method for exploiting sparsity in three-center overlap integrals through tensor decompn. to obtain a low-rank approxn. to d. fitting (tensor hypercontraction d. fitting or THC-DF). This new approxn. reduces the 4th-order ERI tensor to a product of five matrixes, simultaneously reducing the storage requirement as well as increasing the flexibility to regroup terms and reduce scaling behavior. As an example, we demonstrate such a scaling redn. for second- and third-order perturbation theory (MP2 and MP3), showing that both can be carried out in O(N4) operations. This should be compared to the usual scaling behavior of O(N5) and O(N6) for MP2 and MP3, resp. The THC-DF technique can also be applied to other methods in electronic structure theory, such as coupled-cluster and CI, promising significant gains in computational efficiency and storage redn. (c) 2012 American Institute of Physics. 54. 54 Bangerter, F. H.; Glasbrenner, M.; Ochsenfeld, C. Low-Scaling Tensor Hypercontraction in the Cholesky Molecular Orbital Basis Applied to Second-Order Møller-Plesset Perturbation Theory. J. Chem. Theory Comput. 2021, 17, 211, DOI: 10.1021/acs.jctc.0c00934 Low-Scaling Tensor Hypercontraction in the Cholesky Molecular Orbital Basis Applied to Second-Order Moller-Plesset Perturbation Theory Bangerter, Felix H.; Glasbrenner, Michael; Ochsenfeld, Christian Journal of Chemical Theory and Computation (2021), 17 (1), 211-221CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society) We employ various reduced scaling techniques to accelerate the recently developed least-squares tensor hypercontraction (LS-THC) approxn. [Parrish, R.M. et al., J. Chem. Phys. 2012, 137, 224106] for electron repulsion integrals (ERIs) and apply it to second-order Moller-Plesset perturbation theory (MP2). The grid-projected ERI tensors are efficiently constructed using a localized Cholesky MO basis from d.-fitted integrals with an attenuated Coulomb metric. Addnl., rigorous integral screening and the natural blocking matrix format are applied to reduce the complexity of this step. By recasting the equations to form the quantized representation of the 1/r operator Z into the form of a system of linear equations, the bottleneck of inverting the grid metric via pseudoinversion is removed. This leads to a reduced scaling THC algorithm and application to MP2 yields the (sub-)quadratically scaling THC-ω-RI-CDD-SOS-MP2 method. The efficiency of this method is assessed for various systems including DNA fragments with over 8000 basis functions and the subquadratic scaling is illustrated. 55. 55 Del Ben, M.; Hutter, J.; VandeVondele, J. Second-Order Møller-Plesset Perturbation Theory in the Condensed Phase: An Efficient and Massively Parallel Gaussian and Plane Waves Approach. J. Chem. Theory Comput. 2012, 8, 4177, DOI: 10.1021/ct300531w Second-Order Moller-Plesset Perturbation Theory in the Condensed Phase: An Efficient and Massively Parallel Gaussian and Plane Waves Approach Del Ben, Mauro; Hutter, Jurg; VandeVondele, Joost Journal of Chemical Theory and Computation (2012), 8 (11), 4177-4188CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society) A novel algorithm, based on a hybrid Gaussian and plane waves (GPW) approach, is developed for the canonical second-order Moller-Plesset perturbation energy (MP2) of finite and extended systems. The key aspect of the method is that the electron repulsion integrals (ia\|λσ) are computed by direct integration between the products of Gaussian basis functions λσ and the electrostatic potential arising from a given occupied-virtual pair d. ia. The electrostatic potential is obtained in a plane waves basis set after solving the Poisson equation in Fourier space. In particular, for condensed phase systems, this scheme is highly efficient. Furthermore, our implementation has low memory requirements and displays excellent parallel scalability up to 100 000 processes. In this way, canonical MP2 calcns. for condensed phase systems contg. hundreds of atoms or more than 5000 basis functions can be performed within minutes, while systems up to 1000 atoms and 10 000 basis functions remain feasible. Solid LiH has been employed as a benchmark to study basis set and system size convergence. Lattice consts. and cohesive energies of various mol. crystals have been studied with MP2 and double-hybrid functionals. 56. 56 Katouda, M.; Naruse, A.; Hirano, Y.; Nakajima, T. Massively parallel algorithm and implementation of RI-MP2 energy calculation for peta-scale many-core supercomputers. J. Comput. Chem. 2016, 37, 2623, DOI: 10.1002/jcc.24491 Massively parallel algorithm and implementation of RI-MP2 energy calculation for peta-scale many-core supercomputers Katouda, Michio; Naruse, Akira; Hirano, Yukihiko; Nakajima, Takahito Journal of Computational Chemistry (2016), 37 (30), 2623-2633CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.) A new parallel algorithm and its implementation for the RI-MP2 energy calcn. utilizing peta-flop-class many-core supercomputers are presented. Some improvements from the previous algorithm have been performed: (1) a dual-level hierarchical parallelization scheme that enables the use of more than 10,000 Message Passing Interface (MPI) processes and (2) a new data communication scheme that reduces network communication overhead. A multi-node and multi-GPU implementation of the present algorithm is presented for calcns. on a central processing unit (CPU)/graphics processing unit (GPU) hybrid supercomputer. Benchmark results of the new algorithm and its implementation using the K computer (CPU clustering system) and TSUBAME 2.5 (CPU/GPU hybrid system) demonstrate high efficiency. The peak performance of 3.1 PFLOPS is attained using 80,199 nodes of the K computer. The peak performance of the multi-node and multi-GPU implementation is 514 TFLOPS using 1349	{"url":"https://pubs.acs.org/doi/10.1021/acs.jctc.1c00093","timestamp":"2024-11-02T02:57:39Z","content_type":"text/html","content_length":"1050574","record_id":"<urn:uuid:bd41b4d8-32a0-4384-8f6a-cd6eac16a50f>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00674.warc.gz"}
Single-Layer Perceptrons Single-Layer Perceptrons A single-layer perceptron is like a building block in a neural network. It's made up of one layer of artificial neurons, called perceptrons, placed one after the other. Each perceptron does some math with the inputs it gets, and then decides whether to give an output or not. This decision is made using something called an "activation function," which helps the network make choices. One cool thing about single-layer perceptrons is that they can learn from data. It's like they're getting smarter over time. This learning process is about changing the weights connected to each input, so the output becomes closer to what we want it to be. The process follows a set of rules called the "perceptron learning rule." It keeps adjusting the weights until things look good. Think of a perceptron as a tiny decision-maker. It takes a bunch of inputs, plays around with their weights, adds them up, and then passes the result through the activation function. This helps it decide if it should say "yes" or "no" to a question. Here's how it looks in math: • $x_1, x_2, \ldots, x_n$ be the input values. • $w_1, w_2, \ldots, w_n$ be the corresponding weights. • $b$ be the bias term. • $z$ be the weighted sum of inputs and bias: $z = \sum_{i=1}^{n} (w_i \cdot x_i) + b$. • $a$ be the output after applying an activation function: $a = \text{activation}(z)$. Mathematically, the output $a$ of the perceptron can be written as: \[ a = \text{activation}\left(\sum_{i=1}^{n} (w_i \cdot x_i) + b\right) \] The activation function is usually a step function, sigmoid function, or ReLU (Rectified Linear Unit), depending on the specific design of the perceptron and the neural network it's a part of. Single-layer perceptrons serve as the building blocks of more complex neural networks. Their simplicity and effectiveness make them an essential topic in the study of artificial intelligence. By understanding the structure, learning algorithms, and limitations of single-layer perceptrons, we can gain valuable insights into the foundations of neural networks and their applications in solving real-world problems. You can see a simple representation of a perceptron in action in single-layer-perceptrons. Perceptron Learning Algorithm The perceptron learning algorithm is used to adjust the weights and bias of a perceptron in response to labeled training data, allowing it to learn patterns and make accurate predictions. The algorithm follows these steps: Here's a simplified description of how the perceptron learning algorithm works: 1. Initialization: • Initialize the weights $w$ and bias $b$ to small random values or zeros. 2. Forward Pass: • For each training example, compute the weighted sum of inputs: $z = \sum_{i=1}^{n} w_ix_i + b$, where $x_i$ is the input feature, $w_i$ is the weight associated with that feature, $b$ is the bias term, and $n$ is the number of input features. • Apply a thresholding function, such as the step function or a sigmoid function, to $z$. If the result is greater than or equal to a certain threshold, the perceptron outputs one class (e.g., 1); otherwise, it outputs the other class (e.g., 0). 3. Error Calculation and Weight Updates: • If the perceptron's output for a training example is incorrect, an error is computed as the difference between the predicted output and the true target value (0 or 1). • The weights are updated using a learning rate ($\alpha$) and the error. The general weight update rule is: \[ w_i = w_i + \alpha \times \text{error} \times x_i \] • The bias term $b$ is updated in a similar manner, where $\alpha$ is the learning rate. 4. Repeat: • Steps 2 and 3 are repeated for each training example in the dataset. • The process may go through multiple iterations (epochs) over the entire dataset until the perceptron's performance improves, and the classification error decreases. The perceptron training algorithm aims to adjust the weights and bias so that the perceptron correctly classifies as many training examples as possible. However, the perceptron is limited to linearly separable problems; it can only learn tasks where a linear decision boundary exists. For more complex tasks, multiple perceptrons or more advanced neural network architectures are required. Activation Functions in Perceptrons Activation functions play an important role in the behavior and learning capabilities of artificial neural networks, including perceptrons. These functions introduce non-linearity to the network, enabling it to capture and represent complex relationships in data. Several commonly used activation functions, including the step function, sigmoid function, and rectified linear unit (ReLU), have been pivotal in shaping the field of neural networks. One of the simplest activation functions is the Step Function. Sometimes called the Heaviside step function, it's a basic tool used in neural networks. This function takes an input and gives a result of either 0 or 1, kind of like an on-off switch. It decides this based on a specific point we call the "threshold." Here's how it works: Imagine you have a number, let's call it $x$. If $x$ is smaller than the threshold, the output will be 0. But if $x$ is equal to or bigger than the threshold, the output becomes 1. In a mathematical way, it looks like this: \[ f(x) = \begin{cases} 0 & \text{if } x < \text{threshold} \\ 1 & \text{if } x \geq \text{threshold} \end{cases} \] Even though the step function is quite simple, it has a limitation. It can't really show gradual changes or smooth transitions between values. This makes it not very good for lots of real-world problems. In the past, people used it in early versions of neural networks, like perceptrons. But nowadays, we have better options – more flexible tools called activation functions. We'll learn about some of these in the next section, especially the ones used in multi-layer perceptrons. Limitations of Single-Layer Perceptrons Single-layer perceptrons, while valuable in specific contexts, exhibit certain limitations that must be considered when applying them in machine learning and pattern recognition tasks. These constraints include: • Linear Separability Constraint: Single-layer perceptrons are restricted to handling linearly separable data, which limits their ability to model complex, non-linear relationships between input and output. • Inability to Handle Complex Patterns: They struggle with capturing intricate patterns or recognizing non-linear associations in data, making them unsuitable for many real-world applications. • Inefficient Function Approximation: Single-layer perceptrons struggle to efficiently approximate certain functions, especially those that require more intricate transformations or complex decision boundaries. • Limited Representation Power: Compared to multi-layer architectures like neural networks, single-layer perceptrons have restricted representation capabilities, which can hinder their performance on intricate tasks. • Convergence Issues: Some problems may pose convergence challenges for single-layer perceptrons, making it difficult for them to find optimal weight values during training. • Lack of Hierarchical Learning: Single-layer perceptrons lack the capacity to learn hierarchical features, limiting their ability to model high-level abstractions in data. Applications of Single-Layer Perceptrons Conversely, single-layer perceptrons find their niche in various practical applications where their simplicity aligns with the task requirements. Some of the key applications include: • Basic Linear Classification: Single-layer perceptrons find utility in straightforward linear classification tasks where data can be separated by a single hyperplane. • Logical Operations: They excel in performing elementary logical operations like AND, OR, and NOT, which are inherently linearly separable. • Simple Pattern Recognition: For tasks involving simple and linear patterns, single-layer perceptrons can be effective. • Binary Decision Problems: They are well-suited for binary decision problems where the goal is to categorize data into one of two classes based on linear criteria. • Linear Regression: Single-layer perceptrons are useful for linear regression problems, where the objective is to approximate linear functions and predict continuous output values. While single-layer perceptrons have limitations in handling complex data relationships, they remain valuable tools in scenarios where linear separability and simplicity are paramount, such as in basic classification tasks and logical operations. However, for more intricate and non-linear problems, multi-layer neural networks offer enhanced modeling capabilities.	{"url":"https://cstopics.com/books/artificial-intelligence/05-neural-networks/02-single-layer-perceptrons/","timestamp":"2024-11-04T02:06:33Z","content_type":"text/html","content_length":"112851","record_id":"<urn:uuid:60191968-c5a8-4577-afd6-43e412ea1d0c>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00482.warc.gz"}
12 linear_equation_solvers (release) The linear_equation_solvers objects define parameters for linear equation solution strategies. Name Type Description Default label STRING The user defined identifier for this linear solver object. direct (release) SUB OBJECT See linked documentation Example Usage: linear_equation_solvers: [ { label: "multi_frontal", direct: { multi_frontal: { use_diagonal_scaling: true } } } ] 12.1 direct (release) Specifies that a direct solver will be used in the linear solve. Name Type Description Default lu (release) \| multi_frontal (release) SUB OBJECT \| SUB OBJECT User must define one of the given keywords that specify the type of direct solver to be used. Example Usage: direct: { multi_frontal: { use_diagonal_scaling: true } } 12.1.1 lu (release) Specifies a direct linear solver that uses a Lower-Upper (LU) factorization. Name Type Description Default use_diagonal_scaling true\| If true, symmetric diagonal scaling is performed on the system before solving. For a linear system $K d = f$, this transforms the system into $(D^{-\frac{1}{2}} K [optional] false D^{-\frac{1}{2}})D^{\frac{1}{2}} d = D^{-\frac{1}{2}} f$, where $D_{ii}$ is $\frac{1}{\|{K_{ii}\|}}$ unless $K_{ii}$ is zero and then it is 1. This should NOT be false used if the linear solver is being used by a nonlinear solver that utilizes a line-search. Example Usage: lu: { use_diagonal_scaling: false } 12.1.2 multi_frontal (release) Specifies a direct linear solver that uses multifrontal factorization. This method performs LU or LDL factorizations (depending on symmetry of the linear system) on multiple subsets of the linear system. Each subset is a “front” and is essentially the transition region between the part of the system already finished and the part not touched yet. Because of this approach, the multifrontal solver typically uses considerably less memory than the LU method and, as a result, is typically faster than the LU approach. Internal testing at Coreform has found the multifrontal method to be slightly less robust than the LU method, sometimes resulting in inexplicable failures to converge. Name Type Description Default use_diagonal_scaling true\| If true, symmetric diagonal scaling is performed on the system before solving. For a linear system $K d = f$, this transforms the system into $(D^{-\frac{1}{2}} K [optional] false D^{-\frac{1}{2}})D^{\frac{1}{2}} d = D^{-\frac{1}{2}} f$, where $D_{ii}$ is $\frac{1}{\|{K_{ii}\|}}$ unless $K_{ii}$ is zero and then it is 1. This should NOT be false used if the linear solver is being used by a nonlinear solver that utilizes a line-search. Example Usage: multi_frontal: { use_diagonal_scaling: true }	{"url":"https://docs.coreform.com/content/iga-reference/linear-equation-solvers.html","timestamp":"2024-11-03T04:45:24Z","content_type":"text/html","content_length":"22578","record_id":"<urn:uuid:00669ea8-f3fc-4acb-a874-d2fa555d26a4>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00672.warc.gz"}
Spheroid - Wikiwand A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has circular symmetry. If the ellipse is rotated about its major axis, the result is a prolate spheroid, elongated like a rugby ball. The American football is similar but has a pointier end than a spheroid could. If the ellipse is rotated about its minor axis, the result is an oblate spheroid, flattened like a lentil or a plain M&M. If the generating ellipse is a circle, the result is a sphere. Due to the combined effects of gravity and rotation, the figure of the Earth (and of all planets) is not quite a sphere, but instead is slightly flattened in the direction of its axis of rotation. For that reason, in cartography and geodesy the Earth is often approximated by an oblate spheroid, known as the reference ellipsoid, instead of a sphere. The current World Geodetic System model uses a spheroid whose radius is 6,378.137 km (3,963.191 mi) at the Equator and 6,356.752 km (3,949.903 mi) at the poles. The word spheroid originally meant "an approximately spherical body", admitting irregularities even beyond the bi- or tri-axial ellipsoidal shape; that is how the term is used in some older papers on geodesy (for example, referring to truncated spherical harmonic expansions of the Earth's gravity geopotential model).^[1] The assignment of semi-axes on a spheroid. It is oblate if (left) and prolate if (right). The equation of a tri-axial ellipsoid centred at the origin with semi-axes a, b and c aligned along the coordinate axes is ${\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1.}$ The equation of a spheroid with z as the symmetry axis is given by setting a = b: ${\displaystyle {\frac {x^{2}+y^{2}}{a^{2}}}+{\frac {z^{2}}{c^{2}}}=1.}$ The semi-axis a is the equatorial radius of the spheroid, and c is the distance from centre to pole along the symmetry axis. There are two possible cases: • c < a: oblate spheroid • c > a: prolate spheroid The case of a = c reduces to a sphere. An oblate spheroid with c < a has surface area ${\displaystyle S_{\text{oblate}}=2\pi a^{2}\left(1+{\frac {1-e^{2}}{e}}\operatorname {arctanh} e\right)=2\pi a^{2}+\pi {\frac {c^{2}}{e}}\ln \left({\frac {1+e}{1-e}}\right)\qquad {\mbox{where}}\ quad e^{2}=1-{\frac {c^{2}}{a^{2}}}.}$ The oblate spheroid is generated by rotation about the z-axis of an ellipse with semi-major axis a and semi-minor axis c, therefore e may be identified as the eccentricity. (See ellipse.)^[2] A prolate spheroid with c > a has surface area ${\displaystyle S_{\text{prolate}}=2\pi a^{2}\left(1+{\frac {c}{ae}}\arcsin \,e\right)\qquad {\mbox{where}}\quad e^{2}=1-{\frac {a^{2}}{c^{2}}}.}$ The prolate spheroid is generated by rotation about the z-axis of an ellipse with semi-major axis c and semi-minor axis a; therefore, e may again be identified as the eccentricity. (See ellipse.) ^ These formulas are identical in the sense that the formula for S[oblate] can be used to calculate the surface area of a prolate spheroid and vice versa. However, e then becomes imaginary and can no longer directly be identified with the eccentricity. Both of these results may be cast into many other forms using standard mathematical identities and relations between parameters of the ellipse. The volume inside a spheroid (of any kind) is ${\displaystyle {\tfrac {4}{3}}\pi a^{2}c\approx 4.19a^{2}c.}$ If A = 2a is the equatorial diameter, and C = 2c is the polar diameter, the volume is ${\displaystyle {\tfrac {\pi }{6}}A^{2}C\approx 0.523A^{2}C.}$ Let a spheroid be parameterized as ${\displaystyle {\boldsymbol {\sigma }}(\beta ,\lambda )=(a\cos \beta \cos \lambda ,a\cos \beta \sin \lambda ,c\sin \beta ),}$ where β is the reduced latitude or parametric latitude, λ is the longitude, and −π/2 < β < +π/2 and −π < λ < +π. Then, the spheroid's Gaussian curvature is ${\displaystyle K(\beta ,\lambda )={\frac {c^{2}}{\left(a^{2}+\left(c^{2}-a^{2}\right)\cos ^{2}\beta \right)^{2}}},}$ and its mean curvature is ${\displaystyle H(\beta ,\lambda )={\frac {c\left(2a^{2}+\left(c^{2}-a^{2}\right)\cos ^{2}\beta \right)}{2a\left(a^{2}+\left(c^{2}-a^{2}\right)\cos ^{2}\beta \right)^{\frac {3}{2}}}}.}$ Both of these curvatures are always positive, so that every point on a spheroid is elliptic. Aspect ratio The aspect ratio of an oblate spheroid/ellipse, c : a, is the ratio of the polar to equatorial lengths, while the flattening (also called oblateness) f, is the ratio of the equatorial-polar length difference to the equatorial length: ${\displaystyle f={\frac {a-c}{a}}=1-{\frac {c}{a}}.}$ The first eccentricity (usually simply eccentricity, as above) is often used instead of flattening.^[4] It is defined by: ${\displaystyle e={\sqrt {1-{\frac {c^{2}}{a^{2}}}}}}$ The relations between eccentricity and flattening are: {\displaystyle {\begin{aligned}e&={\sqrt {2f-f^{2}}}\\f&=1-{\sqrt {1-e^{2}}}\end{aligned}}} All modern geodetic ellipsoids are defined by the semi-major axis plus either the semi-minor axis (giving the aspect ratio), the flattening, or the first eccentricity. While these definitions are mathematically interchangeable, real-world calculations must lose some precision. To avoid confusion, an ellipsoidal definition considers its own values to be exact in the form it gives. The most common shapes for the density distribution of protons and neutrons in an atomic nucleus are spherical, prolate, and oblate spheroidal, where the polar axis is assumed to be the spin axis (or direction of the spin angular momentum vector). Deformed nuclear shapes occur as a result of the competition between electromagnetic repulsion between protons, surface tension and quantum shell Spheroids are common in 3D cell cultures. Rotating equilibrium spheroids include the Maclaurin spheroid and the Jacobi ellipsoid. Spheroid is also a shape of archaeological artifacts. Prolate spheroids A rugby ball. The prolate spheroid is the approximate shape of the ball in several sports, such as in the rugby ball. Several moons of the Solar System approximate prolate spheroids in shape, though they are actually triaxial ellipsoids. Examples are Saturn's satellites Mimas, Enceladus, and Tethys and Uranus' satellite Miranda. In contrast to being distorted into oblate spheroids via rapid rotation, celestial objects distort slightly into prolate spheroids via tidal forces when they orbit a massive body in a close orbit. The most extreme example is Jupiter's moon Io, which becomes slightly more or less prolate in its orbit due to a slight eccentricity, causing intense volcanism. The major axis of the prolate spheroid does not run through the satellite's poles in this case, but through the two points on its equator directly facing toward and away from the primary. This combines with the smaller oblate distortion from the synchronous rotation to cause the body to become triaxial. The term is also used to describe the shape of some nebulae such as the Crab Nebula.^[7] Fresnel zones, used to analyze wave propagation and interference in space, are a series of concentric prolate spheroids with principal axes aligned along the direct line-of-sight between a transmitter and a receiver. The atomic nuclei of the actinide and lanthanide elements are shaped like prolate spheroids.^[8] In anatomy, near-spheroid organs such as testis may be measured by their long and short axes.^[9] Many submarines have a shape which can be described as prolate spheroid.^[10] Dynamical properties For a spheroid having uniform density, the moment of inertia is that of an ellipsoid with an additional axis of symmetry. Given a description of a spheroid as having a major axis c, and minor axes a = b, the moments of inertia along these principal axes are C, A, and B. However, in a spheroid the minor axes are symmetrical. Therefore, our inertial terms along the major axes are:^[11] {\displaystyle {\begin{aligned}A=B&={\tfrac {1}{5}}M\left(a^{2}+c^{2}\right),\\C&={\tfrac {1}{5}}M\left(a^{2}+b^{2}\right)={\tfrac {2}{5}}M\left(a^{2}\right),\end{aligned}}} where M is the mass of the body defined as ${\displaystyle M={\tfrac {4}{3}}\pi a^{2}c\rho .}$	{"url":"https://www.wikiwand.com/en/articles/Spheroid","timestamp":"2024-11-14T23:27:55Z","content_type":"text/html","content_length":"354583","record_id":"<urn:uuid:06ae8ba5-65dd-47fe-985f-3feb71714832>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00047.warc.gz"}
Shivam Parikh The main goal of this project was to implement a mesh editor to manipulate half-edge meshes and implement loop subdivisions to improve the resolution of the mesh. As part of the project, we also implemented de Casteljau's algorithm to display/render Bezier curves and surfaces. The final product was an editor/viewer that allowed us to load meshes and curves, flip edges, split edges, and subdivide the mesh. Section I: Bezier Curves and Surfaces Part 1: Bezier curves with 1D de Casteljau subdivision de Casteljau's algorithm is a recursive technique to evaluate the polynomials we use to represent Bezier curves. An n degree Bezier curve is defined by (n+1) control points, which we recursively evaluate in levels to narrow from (n+1) points to a single point based on a parameter t between [0,1]. The linear interpolation step is conducted as shown by the math below: We repeat these steps of linear interpolation until we narrow down to a single point for each parameter in the continuous range of t. The steps of evaluation are demonstrated by the screenshots of our editor below. In white are the original control points, and in blue are the next evaluated level of recursive points. The red point is the final point for the given parameter t. The points used for the screenshots. The first level of evaluation. The second level of evaluation. The third level of evaluation. The fourth level of evaluation. The fifth level of evaluation. The fifth level of evaluation with the actual curve evaluated at all points of t drawn. Because our viewer is also an editor, I moved some of the points around to create a new curve, and also varied the value of the parameter t to show the evaluation at different points in time. The moved points. The moved points at one point on the parameter t. The moved points at another point on the parameter t. Part 2: Bezier surfaces with separable 1D de Casteljau subdivision While the last part was about two dimensional curves based on a parameter t, this part on Bezier surfaces was working with three dimensional curves to form planar surfaces. The process of evaluation was actually very similar to the first part, but just repeated. In this part, we were given a 4x4 matrix of control points defining the cubic Bezier surface. In total, there were 16 control points. We were also given two parameters, u and v. I simply replicated the two dimensional evaluation using de Casteljau that took in four control points and one parameter, and evaluated this on each row of the matrix with the parameter u. Then with the four resulting points, I evaluated using de Casteljau again, this time with the v parameter to evaluate our resultant cubic plane over the range of [0,1] very similar to the single row evaluation. The de Casteljau algorithm extends pretty simply between the 1D and 2D control points cases, as is shown by the way we evaluate over the two parameters using different rows of the matrix. When rendering the bez/teapot.bez file, we end up with this beautiful rendering of a teapot. And we also have another bez file, the wavy cube! Section II: Sampling After finishing the rendering of Bezier curves and surfaces, it was time to delve deeper into the Halfedge mesh datastructure. Part 3: Average normals for half-edge meshes To start working on this part, I began by reading through the documentation for the halfedge mesh class and understanding the various class variables each attribute had. Because the Vector3D supports addition and subtraction and a unit operation, getting the area weighted average normals was not particularly difficult. After assigning the correct pointers and end condition for the while loop, I looped through all the neighboring vertices (using the halfedges and twins to find them), found the vector between the adjacent vertices and the position of the vertex we were operating from, and then summed the cross product of the two vectors (the area between the two vectors) into an accumulating Vector3D. Then normalizng the accumulating Vector3D provides us with the area-weighted average normal vector at a vertex. The results of the normal vectors are displayed below. With face normals. With the average vertex normal vectors. This is with face normals while toggling the GLSL Shaders. This is with the average vertex normals while toggling the GLSL Shaders. Part 4: Half-edge flip Part 4 started the more difficult parts of the project. Flipping edges meant keeping track of a lot of pointers and ensuring the correct reassignment of each pointer that was changed. To approach this, I started by loading literally every variable for a particular pair of triangles into memory. For instance, I loaded all 9 halfedges, all 4 vertices, all 2 faces, and all 4 edges. Then, I drew out a diagram of where each new pointer should point after the flip. I used the CMU document for reference. Before using the setNeighbors function, I first reassigned each indivdual pointer for every object so that I could keep track of the number of reassignments and not worry about reassigning things like the faces for external half-edges that should not be changed anyways. Then, I went back and after confirming it worked, I used the setNeighbors function. The original edges. The flipped edges. Part 5: Half-edge split My approach to solving half-edge split was almost identical to the process for flipping edges. The only caveat is that in splitting, we have to build one new vertex, two new faces, three new edges, and six new halfedges. So after loading all the variables, I create new variables for these new pointers, and then procede to perform the reassignments to all the variables I loaded in the memory. I also drew out the paths and reassignments of all the variables on a whiteboard before reassigning them. The before picture of our teapot. Performing split operations on a bunch of edges in a variety of different orders and directions. A closeup of flips and splits and its result. A zoom out of flips and splits and its result. Part 6: Loop subdivision for mesh upsampling Part 6 was probably the toughest of all the parts, and debugging was rather difficult as well because upsampling took advantage of the flips and split operators I wrote earlier for our meshes. Which meant that my errors could have been anywhere in the three largest functions I wrote. Much of the beginning was formulaic, following the weighted position of new vertices based on old vertices was as simple as finding the degree, calculating the neighbor_position_sum (a sum of the positions of all the neighboring vertices) and applying a formula: I did run into a bug where I was calculating the edge position wrong, but after re-reading the spec a good number of times, I found my mistake in the midpoint calculation and the images turned out ok. At the very bottom are some failures :P. Another interesting issue I ran into was how to avoid using the isNew check in my while loop. First, I tackled this by simply counting the number of edges in the original mesh in an earlier loop, and only allowing my edge splitting loop to run over the pointers in the range of the original edges. However, after a trek to Office Hours, I worked out a solution to this, to avoid checking the !isNew on an edge, we instead check to see if an edge is connecting two old points. If it's connecting two old vertices, then it is an edge that still needs to be split. All other cases would not need to be split, so we adjust the condition to check for that. The results for cube. The original cube Level 2 subsample. Level 3 subsample. Level 4 subsample. The results for cube with the pre-splitting. I pre-process by splitting all the edges that exist already so that I increase the resolution along the sharp edges, leaving more information before the subsample occurs. As we can see in the torus and isocahedron examples before, the sharp edges and faces end up shaping into curves due to the interpolation, but as is seen in the simple example of the cube, adding more information along the edges shows an improvement. The original cube My pre-processing technique to make sure the result is symmetrical. When using the original cube, the asymmetric result. Another perspective of the asymmetric result. When using the pre-processed cube, the symmetric result. Another perspective of the symmetric result. Some other results of my subsampling The Isocahedron The Torus	{"url":"http://www.sparikh.me/uploads/blog/html/meshedit/index.html","timestamp":"2024-11-13T05:52:35Z","content_type":"application/xhtml+xml","content_length":"16759","record_id":"<urn:uuid:6ac3b0fc-ecfc-4085-8e84-6f5f34597687>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00430.warc.gz"}
Worksheets For Distributive Property Of Multiplication Math, particularly multiplication, develops the foundation of many scholastic techniques and real-world applications. Yet, for many learners, grasping multiplication can posture a challenge. To address this difficulty, instructors and moms and dads have actually accepted a powerful device: Worksheets For Distributive Property Of Multiplication. 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Advantages of Using Worksheets For Distributive Property Of Multiplication Distributive Property Of Multiplication Worksheets 4th Grade Times Tables Worksheets Distributive Property Of Multiplication Worksheets 4th Grade Times Tables Worksheets According to the distributive property we can break this down into two smaller multiplication problems 4 3 4 2 Now we can solve each part 12 8 And finally we can add the results 20 So 4 3 2 20 In the Distributive Property of Multiplication Worksheets you ll find lots of problems like this to practice Distributive Property of Multiplication We use the distributive property to break apart problems with larger numbers to make them easier to solve In this worksheet third graders learn what the distributive property is and how to use it to solve multiplication problems They then solve 10 one and two digit multiplication problems using the Boosted Mathematical Skills Constant practice hones multiplication effectiveness, improving general math capabilities. 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Urging consistent practice, offering aid, and creating a positive understanding setting are valuable steps.	{"url":"https://crown-darts.com/en/worksheets-for-distributive-property-of-multiplication.html","timestamp":"2024-11-12T23:39:17Z","content_type":"text/html","content_length":"29870","record_id":"<urn:uuid:19ed53b5-f4f0-46b0-a6d2-a8400b26dc40>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00572.warc.gz"}
How To Write Quadratic Equations In Vertex Form Converting an equation to vertex form can be tedious and require an extensive degree of algebraic background knowledge, including weighty topics such as factoring. The vertex form of a quadratic equation is y = a(x – h)^2 + k, where "x" and "y" are variables and "a," "h" and k are numbers. In this form, the vertex is denoted by (h, k). The vertex of a quadratic equation is the highest or lowest point on its graph, which is known as a parabola. Step 1 Ensure that your equation is written in standard form. The standard form of a quadratic equation is y = ax^2 + bx + c, where "x" and "y" are variables and "a," "b" and "c" are integers. For instance, y = 2x^2 + 8x – 10 is in standard form, whereas y – 8x = 2x^2 – 10 is not. In the latter equation, add 8x to both sides to put it in standard form, rendering y = 2x^2 + 8x – 10. Step 2 Move the constant to the left side of the equals sign by adding or subtracting it. A constant is a number lacking an attached variable. In y = 2x^2 + 8x – 10, the constant is -10. Since it is negative, add it, rendering y + 10 = 2x^2 + 8x. Step 3 Factor out "a," which is the coefficient of the squared term. A coefficient is a number written on the variable's left-hand side. In y + 10 = 2x^2 + 8x, the coefficient of the squared term is 2. Factoring it out yields y + 10 = 2(x^2 + 4x). Step 4 Rewrite the equation, leaving an empty space on the right side of the equation after the "x" term but before the end parenthesis. Divide the coefficient of the "x" term by 2. In y + 10 = 2(x^2 + 4x), divide 4 by 2 to get 2. Square this result. In the example, square 2, producing 4. Place this number, preceded by its sign, in the empty space. The example becomes y + 10 = 2(x^2 + 4x + 4). Step 5 Multiply "a," the number you factored out in Step 3, by the result of Step 4. In the example, multiply 2*4 to get 8. Add this to the constant on the left side of the equation. In y + 10 = 2(x^2 + 4x + 4), add 8 + 10, rendering y + 18 = 2(x^2 + 4x + 4). Step 6 Factor the quadratic inside the parentheses, which is a perfect square. In y + 18 = 2(x^2 + 4x + 4), factoring x^2 + 4x + 4 yields (x + 2)^2, so the example becomes y + 18 = 2(x + 2)^2. Step 7 Move the constant on the left-hand side of the equation back over to the right by adding or subtracting it. In the example, subtract 18 from both sides, producing y = 2(x + 2)^2 – 18. The equation is now in vertex form. In y = 2(x + 2)^2 – 18, h = -2 and k = -18, so the vertex is (-2, -18). Cite This Article Harris, Amy. "How To Write Quadratic Equations In Vertex Form" sciencing.com, https://www.sciencing.com/write-quadratic-equations-vertex-form-8529869/. 24 April 2017. Harris, Amy. (2017, April 24). How To Write Quadratic Equations In Vertex Form. sciencing.com. Retrieved from https://www.sciencing.com/write-quadratic-equations-vertex-form-8529869/ Harris, Amy. How To Write Quadratic Equations In Vertex Form last modified August 30, 2022. https://www.sciencing.com/write-quadratic-equations-vertex-form-8529869/	{"url":"https://www.sciencing.com:443/write-quadratic-equations-vertex-form-8529869/","timestamp":"2024-11-09T15:25:44Z","content_type":"application/xhtml+xml","content_length":"72024","record_id":"<urn:uuid:833042ac-2eb1-432f-acd0-7f4e13dfc30a>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00042.warc.gz"}
Converting Between Fractions And Mixed Numbers Worksheet Converting Between Fractions And Mixed Numbers Worksheet serve as foundational tools in the realm of mathematics, supplying an organized yet flexible platform for students to check out and grasp numerical principles. These worksheets supply a structured technique to understanding numbers, nurturing a solid foundation upon which mathematical effectiveness prospers. From the easiest counting exercises to the details of innovative calculations, Converting Between Fractions And Mixed Numbers Worksheet accommodate students of varied ages and ability degrees. 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Supporting Number Sense and Reasoning Mixed Numbers And Fractions Worksheets Mixed Numbers And Fractions Worksheets Mixed Numbers Improper Fractions In each problem below an improper fraction is represented by blocks beneath a number line Use the number line to determine what the equivalent mixed number form would be Notice that some number lines have diferent sub divisions thirds fourths ifths 0 0 0 0 0 0 3 4 2 2 2 2 2 2 3 3 3 3 3 To convert from an improper fraction to a mixed number follow the steps below Step 1 Rewrite the fraction as a division problem between the numerator and denominator Step 2 Perform long division Step 3 Rewrite the long division problem as a mixed number by setting the quotient equal to the whole number The heart of Converting Between Fractions And Mixed Numbers Worksheet hinges on cultivating number sense-- a deep comprehension of numbers' definitions and affiliations. 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From Theory to Real-World Application Grade 5 Fractions Worksheets Convert Decimals To Mixed Numbers K5 Learning Converting Grade 5 Fractions Worksheets Convert Decimals To Mixed Numbers K5 Learning Converting Improper Fraction Worksheets Welcome to our Improper Fraction Worksheets page Here you will find a wide range of free printable Fraction Worksheets which will help your child understand and practice how to convert improper fractions to mixed numbers Question 1 Match up the improper fractions and mixed numbers Question 2 Arrange these improper fractions in order starting with the smallest Question 3 Write down a mixed number between and Question 4 Gregory feeds his cat of a can of cat food each day Work out how many cans of cat food are eaten each fortnight Converting Between Fractions And Mixed Numbers Worksheet work as avenues connecting academic abstractions with the palpable realities of day-to-day life. 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Improper Fractions To Mixed Numbers Worksheet Converting Between Fractions Decimals And Mixed Numbers Worksheets Check more of Converting Between Fractions And Mixed Numbers Worksheet below Converting Improper Fractions To Mixed Numbers Worksheet Improper Fractions To Mixed Numbers Worksheets Converting Fractions To Mixed Numbers Worksheets Mixed Number Worksheet 3Rd Grade Improper Fractions And Mixed Numbers Worksheet Grade 4 Math Worksheets Convert Decimals To Mixed Numbers K5 Learning Converting Fractions And Convert Between Mixed Fraction And Improper Fraction Printable Math Worksheets www mathworksheets4kids Write as mixed fraction 12 7 25 3 39 8 49 11 23 4 35 6 44 9 22 5 16 3 31 2 Write as improper fraction 4 2 9 1 5 6 6 4 7 7 2 3 8 4 5 5 1 8 2 3 4 7 3 7 9 1 2 6 4 5 Convert between Mixed Fraction and Improper Fraction Sheet 1 Convert Improper Fractions And Mixed Numbers Worksheets Convert between Improper Fractions and Mixed Numbers Worksheets Explore this pack of printable converting between improper fractions and mixed numbers worksheets and excel in the step by step conversion of both an improper fraction to a mixed number and a mixed number to an improper fraction Printable Math Worksheets www mathworksheets4kids Write as mixed fraction 12 7 25 3 39 8 49 11 23 4 35 6 44 9 22 5 16 3 31 2 Write as improper fraction 4 2 9 1 5 6 6 4 7 7 2 3 8 4 5 5 1 8 2 3 4 7 3 7 9 1 2 6 4 5 Convert between Mixed Fraction and Improper Fraction Sheet 1 Convert between Improper Fractions and Mixed Numbers Worksheets Explore this pack of printable converting between improper fractions and mixed numbers worksheets and excel in the step by step conversion of both an improper fraction to a mixed number and a mixed number to an improper fraction Mixed Number Worksheet 3Rd Grade Improper Fractions To Mixed Numbers Worksheets Improper Fractions And Mixed Numbers Worksheet Grade 4 Math Worksheets Convert Decimals To Mixed Numbers K5 Learning Converting Fractions And Brimwood Boulevard Junior Public School Classrooms Mme Law s Gr 4 5 Mixed Numbers And Improper Fractions Worksheet Mixed Numbers And Improper Fractions Worksheet 41 How To Convert Improper Fractions To Mixed Numbers Worksheet Worksheet Information	{"url":"https://szukarka.net/converting-between-fractions-and-mixed-numbers-worksheet","timestamp":"2024-11-09T01:30:49Z","content_type":"text/html","content_length":"27003","record_id":"<urn:uuid:2ad463d4-1544-4a3a-a0b9-2392ddde318b>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00156.warc.gz"}
Bias and Variance in Machine Learning - A Fantastic Guide for Beginners! • Learn to interpret Bias and Variance in a given model. • What is the difference between Bias and Variance? • How to achieve Bias and Variance Tradeoff using Machine Learning workflow Let us talk about the weather. It rains only if it’s a little humid and does not rain if it’s windy, hot or freezing. In this case, how would you train a predictive model and ensure that there are no errors in forecasting the weather? You may say that there are many learning algorithms to choose from. They are distinct in many ways but there is a major difference in what we expect and what the model predicts. That’s the concept of Bias and Variance Tradeoff. In this article, you will get to know about the bias variance tradeoff, with bias and variance in machine learning, also you will get to know about What is bias and variance in machine learning. Usually, Bias and Variance Tradeoff is taught through dense mathematical formulas. But in this article, I have attempted to explain Bias and Variance as simply as possible! My focus will be to spin you through the process of understanding the problem statement and ensuring that you choose the best model where the Bias and Variance errors are minimal. For this, I have taken up the popular Pima Indians Diabetes dataset. The dataset consists of diagnostic measurements of adult female patients of Native Indian Pima Heritage. For this dataset, we are going to focus on the “Outcome” variable – which indicates whether the patient has diabetes or not. Evidently, this is a binary classification problem and we are going to dive right in and learn how to go about it. In this article, you will explore the bias-variance tradeoff in machine learning, learning how bias and variance affect model performance and the importance of balancing these two factors for optimal If you are interested in this and data science concepts and want to learn practically refer to our course- Introduction to Data Science Evaluating your Machine Learning Model The primary aim of the Machine Learning models is to learn from the given data and generate predictions based on the pattern observed during the learning process. However, our task doesn’t end there. We need to continuously make improvements to the models, based on the kind of results it generates. We also quantify the model’s performance using metrics like Accuracy, Mean Squared Error(MSE), F1-Score, etc and try to improve these metrics. This can often get tricky when we have to maintain the flexibility of the model without compromising on its correctness. A supervised Machine Learning model aims to train itself on the input variables(X) in such a way that the predicted values(Y) are as close to the actual values as possible (Modafinil). This difference between the actual values and predicted values is the error and it is used to evaluate the model. The error for any supervised Machine Learning algorithm comprises of 3 parts: 1. Bias error 2. Variance error 3. The noise While the noise is the irreducible error that we cannot eliminate, the other two i.e. Bias and Variance are reducible errors that we can attempt to minimize as much as possible. In the following sections, we will cover the Bias error, Variance error, and the Bias-Variance tradeoff which will aid us in the best model selection. And what’s exciting is that we will cover some techniques to deal with these errors by using an example dataset. Problem Statement and Primary Steps As explained earlier, we have taken up the Pima Indians Diabetes dataset and formed a classification problem on it. Let’s start by gauging the dataset and observe the kind of data we are dealing with. We will do this by importing the necessary libraries: Now, we will load the data into a data frame and observe some rows to get insights into the data. Python Code: import numpy as np # linear algebra import pandas as pd # data processing, CSV file I/O (e.g. pd.read_csv) from sklearn.model_selection import train_test_split from sklearn.preprocessing import StandardScaler from sklearn.neighbors import KNeighborsClassifier from sklearn.neighbors import KNeighborsRegressor from sklearn.metrics import confusion_matrix from sklearn import metrics import matplotlib.pyplot as plt #%matplotlib inline data_file_path = 'diabetes.csv' data_df = pd.read_csv(data_file_path) We need to predict the ‘Outcome’ column. Let us separate it and assign it to a target variable ‘y’. The rest of the data frame will be the set of input variables X. Now let’s scale the predictor variables and then separate the training and the testing data. Since the outcomes are classified in a binary form, we will use the simplest K-nearest neighbor classifier(Knn) to classify whether the patient has diabetes or not. However, how do we decide the value of ‘k’? • Maybe we should use k = 1 so that we will get very good results on our training data? That might work, but we cannot guarantee that the model will perform just as well on our testing data since it can get too specific • How about using a high value of k, say like k = 100 so that we can consider a large number of nearest points to account for the distant points as well? However, this kind of model will be too generic and we cannot be sure if it has considered all the possible contributing features correctly. Let us take a few possible values of k and fit the model on the training data for all those values. We will also compute the training score and testing score for all those values. To derive more insights from this, let us plot the training data(in red) and the testing data(in blue). To calculate the scores for a particular value of k, We can make the following conclusions from the above plot: • For low values of k, the training score is high, while the testing score is low • As the value of k increases, the testing score starts to increase and the training score starts to decrease. • However, at some value of k, both the training score and the testing score are close to each other. This is where Bias and Variance come into the picture. What is Bias? In the simplest terms, Bias is the difference between the Predicted Value and the Expected Value. To explain further, the model makes certain assumptions when it trains on the data provided. When it is introduced to the testing/validation data, these assumptions may not always be correct. In our model, if we use a large number of nearest neighbors, the model can totally decide that some parameters are not important at all. For example, it can just consider that the Glusoce level and the Blood Pressure decide if the patient has diabetes. This model would make very strong assumptions about the other parameters not affecting the outcome. You can also think of it as a model predicting a simple relationship when the datapoints clearly indicate a more complex relationship: Mathematically, let the input variables be X and a target variable Y. We map the relationship between the two using a function f. Y = f(X) + e Here ‘e’ is the error that is normally distributed. The aim of our model f'(x) is to predict values as close to f(x) as possible. Here, the Bias of the model is: Bias[f'(X)] = E[f'(X) – f(X)] As I explained above, when the model makes the generalizations i.e. when there is a high bias error, it results in a very simplistic model that does not consider the variations very well. Since it does not learn the training data very well, it is called Underfitting. What is a Variance? Contrary to bias, the Variance is when the model takes into account the fluctuations in the data i.e. the noise as well. So, what happens when our model has a high variance? The model will still consider the variance as something to learn from. That is, the model learns too much from the training data, so much so, that when confronted with new (testing) data, it is unable to predict accurately based on it. Mathematically, the variance error in the model is: Since in the case of high variance, the model learns too much from the training data, it is called overfitting. In the context of our data, if we use very few nearest neighbors, it is like saying that if the number of pregnancies is more than 3, the glucose level is more than 78, Diastolic BP is less than 98, Skin thickness is less than 23 mm and so on for every feature….. decide that the patient has diabetes. All the other patients who don’t meet the above criteria are not diabetic. While this may be true for one particular patient in the training set, what if these parameters are the outliers or were even recorded incorrectly? Clearly, such a model could prove to be very costly! Additionally, this model would have a high variance error because the predictions of the patient being diabetic or not vary greatly with the kind of training data we are providing it. So even changing the Glucose Level to 75 would result in the model predicting that the patient does not have diabetes. To make it simpler, the model predicts very complex relationships between the outcome and the input features when a quadratic equation would have sufficed. This is how a classification model would look like when there is a high variance error/when there is overfitting: To summarise, • A model with a high bias error underfits data and makes very simplistic assumptions on it • A model with a high variance error overfits the data and learns too much from it • A good model is where both Bias and Variance errors are balanced Bias-Variance Tradeoff The bias-variance tradeoff is a fundamental concept in machine learning and statistics. It refers to the delicate balance between two sources of error in a predictive model: bias and variance. Bias represents the error due to overly simplistic assumptions in the learning algorithm. High bias can cause the model to underfit the data, leading to poor performance on both training and unseen Variance, on the other hand, reflects the model’s sensitivity to small fluctuations in the training data. High variance can lead to overfitting, where the model captures noise in the training data and performs poorly on new, unseen data. The goal is to find the right level of complexity in a model to minimize both bias and variance, achieving good generalization to new data. Balancing these factors is essential for building models that perform well on a variety of datasets. Understand Bias-Variance Tradeoff with the help of an example How do we relate the above concepts to our Knn model from earlier? Let’s find out! In our model, say, for, k = 1, the point closest to the datapoint in question will be considered. Here, the prediction might be accurate for that particular data point so the bias error will be less. However, the variance error will be high since only the one nearest point is considered and this doesn’t take into account the other possible points. What scenario do you think this corresponds to? Yes, you are thinking right, this means that our model is overfitting. On the other hand, for higher values of k, many more points closer to the datapoint in question will be considered. This would result in higher bias error and underfitting since many points closer to the datapoint are considered and thus it can’t learn the specifics from the training set. However, we can account for a lower variance error for the testing set which has unknown values. To achieve a balance between the Bias error and the Variance error, we need a value of k such that the model neither learns from the noise (overfit on data) nor makes sweeping assumptions on the data (underfit on data). To keep it simpler, a balanced model would look like this: Though some points are classified incorrectly, the model generally fits most of the datapoints accurately. The balance between the Bias error and the Variance error is the Bias-Variance Tradeoff. The following bulls-eye diagram explains the tradeoff better: The center i.e. the bull’s eye is the model result we want to achieve that perfectly predicts all the values correctly. As we move away from the bull’s eye, our model starts to make more and more wrong predictions. A model with low bias and high variance predicts points that are around the center generally, but pretty far away from each other. A model with high bias and low variance is pretty far away from the bull’s eye, but since the variance is low, the predicted points are closer to each other. In terms of model complexity, we can use the following diagram to decide on the optimal complexity of our model. So, what do you think is the optimum value for k? From the above explanation, we can conclude that the k for which • the testing score is the highest, and • both the test score and the training score are close to each other is the optimal value of k. So, even though we are compromising on a lower training score, we still get a high score for our testing data which is more crucial – the test data is after all unknown Let us make a table for different values of k to further prove this: To summarize, in this article, we learned that an ideal model would be one where both the bias error and the variance error are low. However, we should always aim for a model where the model score for the training data is as close as possible to the model score for the testing data. That’s where we figured out how to choose a model that is not too complex (High variance and low bias) which would lead to overfitting and nor too simple(High Bias and low variance) which would lead to underfitting. Bias and Variance plays an important role in deciding which predictive model to use. I hope this article explained the concept well. Hope you like the article! The bias-variance tradeoff in machine learning is an important idea. It helps us understand the balance between bias and variance in machine learning models for better Q1. What is the bias and variance tradeoff? A. The bias-variance tradeoff in machine learning involves managing two types of errors. Bias arises from overly simplistic models, leading to underfitting, while variance results from complex models capturing noise, causing overfitting. Balancing these errors is crucial for creating models that generalize well to new data, optimizing performance and robustness. Q2. What is the bias variance method? A. The bias-variance method is an approach in machine learning that analyzes the tradeoff between bias and variance to optimize model performance. By adjusting a model’s complexity, it aims to strike a balance between underfitting (high bias) and overfitting (high variance). This method guides the selection of appropriate models, helping to create accurate and robust predictions on new data. Q3. What is the purpose of bias and variance? In machine learning, bias and variance are two essential concepts that influence a model’s ability to generalize to unseen data. Bias represents the inherent error due to the model’s assumptions, while variance measures the model’s sensitivity to training data. A balance between these two error types is crucial for optimal performance. Q4. What is bias-variance tradeoff for dummies? The bias-variance tradeoff is about finding the right balance between simplicity and complexity in a machine learning model. High bias means the model is too simple and consistently misses the target, while high variance means the model is too complex and shoots all over the place. You want to aim for a model that’s just right – not too simple, not too complex – to make accurate predictions on new data. Responses From Readers Hi AWESOME POST!! One clarity is needed : From the bulls-eye diagram High Bias & Low Variance case , the points are away from target(Ground truth both in Training & Testing) then how by the defintion of variance ( high if model is unable to predict new unseen data) its low? Kindly help me improve myself on this please. Great explanation. I needed this. I think data normalization should be done after splitting the data and not before it because it adds is a potential bias in the evaluation of the performance. Let me know your opinion. Hi Padma, This is a great post. I was wondering though, how the "model can totally decide that some parameters are not important at all"? I can't picture this graphically. Each plot is a record of 9 dimensions, correct? All the features are computed to make a plot for each record. So I don't see how only the "Glucose level and the Blood Pressure decide if the patient has diabetes". I can see that these two features have relatively high values, but the plots to not represent sum totals of the records, but rather a plot. So how can they influence the model so much for a high k value? For any k value, for that matter. Please help me understand this. AWESOME explanation... I saved the post for the bookmark. One have one question. Why for the high variance the data points are catered (not near to each other). but for the low variance they do A very clear explanantion !!! The example under the section with title "Understand Bias-Variance Tradeoff with the help of an example" seems to be incorrect. When K = 1, the model learns from nearest datapoints and ignores the farther datapoints so it means it makes certain assumptions and learns very little from the data, then shouldn't be it High bias and low variance. Here the model is becoming underfitting. Similarly when K is very high value then the model will try to learn from as much as datapoints possible causing the High Variance. Here the model is becoming overfitting.	{"url":"https://www.analyticsvidhya.com/blog/2020/08/bias-and-variance-tradeoff-machine-learning/?utm_source=reading_list&utm_medium=https://www.analyticsvidhya.com/blog/2021/03/introduction-to-adaboost-algorithm-with-python/","timestamp":"2024-11-08T08:26:52Z","content_type":"text/html","content_length":"431880","record_id":"<urn:uuid:fdf9fc00-2c9b-417d-a8e6-c88a66f42c95>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00335.warc.gz"}
Know all about Pugh Matrix (Decision Matrix) - 6sigma In today’s business world, be it any department or of any hierarchy, individuals and teams need to make decisions that would best suit their requirements. But with a plethora of resources available in the market today, sometimes it becomes difficult to gauge and narrow out a specific solution. Sometimes the decision taken can turn out to be of good value, while sometimes it won’t. It then becomes obvious for individuals and teams to not rely on a hunch by making decisions and waiting to go their way. Rather they require a process in which they can track down the best viable option using statistics, reasoning, and feedback. The process we are talking about here is — Pugh Matrix (also known as Decision Matrix) — developed by a British product designer — Stuart Pugh. What is Pugh Matrix? Pugh Matrix is a qualitative technique used to rank the multi-dimensional options of option-set. It is majorly used in engineering processes to make design decisions. The technique is extensively used in other segments as well, like, investment options, vendor options, product options, or any other multi-dimensional entities. To ease up the understanding, Decision Matrix refines a list of ideas using a process to weigh and compare the conceptual ideas. It’s similar to the Pros/Cons list (with numbers). You basically use Pugh Matrix when- • A current process requires redesign or improvement • A current solution doesn’t meet the customer’s requirement • Choosing best of combinations/characteristics/concepts Types of Pugh Matrix: • Weighted Pugh Matrix – Assigns an order of importance to each criterion. • Unweighted Pugh Matrix – Considers all criteria of equal importance. Let’s go practical with Pugh Matrix To give you a short example, let’s say you bought a new phone last week. It’s obvious that you had already checked multiple options before deciding. The decision process is exactly what the Decision Matrix does. It takes every solution on paper and it charts down the best quality in each of them by giving out points — -1, 0, +1. Example 1 – Unweighted Pugh Matrix An unweighted decision matrix consists of 4 components: • Criteria – A qualitative component to help create meaning on how options compare relative to each other. • Options – These are the possible choices one can make. • Values – There are 3 possible values — -1 (below average), 0 (average), 1 (above average). • Total – The final score derived from calculating the values. Let’s take the phone example and use it in the Unweighted Pugh Matrix Criteria Option 1 – Apple Option 2 – Samsung Option 3 – Nokia Option 4 – One Plus Cost-Effectiveness -1 0 +1 +1 Quality Assurance +1 0 -1 +1 Accessibility +1 +1 -1 0 Customer Support 0 0 +1 -1 Customer Usage +1 +1 -1 0 Total 2 2 -1 1 Determining the value of Unweighted Pugh Matrix Here, you can see Apple & Samsung are of equivalent values. While for the rest, you can clearly decide on the values you got. This is where the Unweighted Pugh Matrix falls off. The value of the Unweighted Pugh Matrix lies at the intersection of simplicity and clarity. But at the same time, the reasoning is lost in the execution process. While all the criteria are important, some are more and some are not. Some criteria might be in high demand, some might not be! In the above process, we cannot decide what value each criterion has of its own. This is where the Weighted Pugh Matrix takes the show away. Example 2 – Weighted Pugh Matrix In the weighted pugh matrix we add a new component next to the criteria. The column will hold the value of each criterion, giving us an in-depth understanding of which parameters are important and which are not. Criteria Criteria Weight Option 1 – Apple Option 2 – Samsung Option 3 – Nokia Option 4 – One Plus Cost-Effectiveness 3 -1 0 +1 +1 Quality Assurance 2 +1 0 -1 +1 Accessibility 1 +1 +1 -1 0 Customer Support 1 0 0 +1 -1 Customer Usage 3 +1 +1 -1 0 Total 3 4 -2 4 Determining the value of Weighted Pugh Matrix Here, you can see how the tables have turned. Apparently, by giving a specific criteria weight, Samsung turns out to do as well as One Plus. The further process is to determine which factor you want more from the category and make the decision accordingly. By using the 10-base system, we show what percent of weight each criterion holds. For example, Customer usage has a weightage of 30% while Customer support has 10%. Pugh Matrix and Six Sigma Pugh Matrix is often associated as part of a technique in Six Sigma (a methodology used to eliminate defects in solutions/services). Both of these techniques are not inherently linked, but support each other in a way or another. The Six Sigma methodology addresses building on processes by defining, measuring, analyzing, improving, and controlling (DMAIC process). Pugh matrix is heavily used in the defining stage and finds its way waning throughout the process. After all, when you are asking to define a problem, what better way other than the Pugh Matrix? For more information on our Lean Six Sigma training classes and services, please visit 6sigma.com. 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Help with a formula Welcome to the Smartsheet Forum Archives The posts in this forum are no longer monitored for accuracy and their content may no longer be current. If there's a discussion here that interests you and you'd like to find (or create) a more current version, please Visit the Current Forums. Help with a formula This is probably a basic question... I am wondering how to input a formula that would allow me to calculate the value/price of something based on a drop-down selection. For instance, column A would have a drop-down options listing different tree species. For example, Redwood, Elm, and Walnut. Column B would have the board feet of a slab/board we have in stock. I want column C to calculate the price of each board based off of the known value of the species. It would know that if the first cell has 'redwood' selected than the price is $7 per board foot, 'elm' has a price of $8 per board foot, 'walnut' has a price of $17 per board foot, etc. That would be multiplied with the board feet in column B to give me the value of the board. I'm a spreadsheet rookie. Is this possible? • =IF([Column C Name]1 = "Walnut", [Column B Name]1 * 17, IF([Column C Name]1 = "elm", [Column B Name]1 * 8, IF([Column C Name]1 = "redwood", [Column B Name]1 * 7))) Should do the trick. □ Put your own column name in the brackets. If your column name has no white space and doesn't end in a number, then you don't need the brackets. □ This formula assumes row 1 (hence the 1 after the column name) □ Format the column you put that formula in with the $ in the tool panel, to get a monetary value. Hope that helps. • Thanks so much, Mike. This got me on track. I just had to swap out [Column C Name] with [Column A Name] because I was ending with a circular reference. Exactly what I needed. Thank you! This discussion has been closed.	{"url":"https://community.smartsheet.com/discussion/13501/help-with-a-formula","timestamp":"2024-11-07T06:57:33Z","content_type":"text/html","content_length":"406183","record_id":"<urn:uuid:0935fb6f-b9e6-4767-9e58-67d2daab7165>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00390.warc.gz"}
Triangle calculator Triangle calculator SAS To calculate the missing information of a triangle when given the SAS theorem, you can use the known side lengths and angles to find the remaining side length and angles using trigonometry or If you know the lengths of two sides (a and b) and the angle (C) between them, you can use the Law of Cosines to find the length of the third side (c) as: = a + b - 2ab * cos(C) Once you have the length of the third side, you can use the Law of Sines to find the remaining angles (A and B) as: a/sin(A) = b/sin(B) = c/sin(C) = 2R Where R is the circumradius of the triangle You can also use the given side lengths and angles to find the area of the triangle using Heron's formula or using trigonometric functions like Sin or Cos. It's important to note that you need to have the measures of two sides and the angle between them to use this theorem. If you have only two sides or one side and one angle, it would not be possible to determine the triangle completely. If you know two sides and one adjacent angle, use the SSA calculator Triangle SAS theorem math problems: more triangle problems » Look also at our friend's collection of math problems and questions: triangle basics on Wikipedia or more details on solving triangles	{"url":"https://www.triangle-calculator.com/?what=sas","timestamp":"2024-11-06T04:10:01Z","content_type":"text/html","content_length":"17475","record_id":"<urn:uuid:3b17f8e5-019e-40e0-a238-cad831913a54>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00771.warc.gz"}
Inquiry & Proof (MATH 3318) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Welcome to overleaf, (formerly writeLaTeX) --- just edit % your LaTeX on the left, and we'll compile it for you on % the right. If you give someone the link to this page, they % can edit at the same time. If you create a free overleaf % account, all your projects will be saved. See the help % menu above for more info. Enjoy! % ----------------------------------------------------------- % My thanks to Dana Ernst of Northern Arizona University % for sharing his template with me. This is largely his work. % ----------------------------------------------------------- % This is all preamble stuff that you don't have to worry about. % Head down to where it says "Start here" % ----------------------------------------------------------- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass[12pt]{article} \usepackage[margin=1in]{geometry} \usepackage {amsmath,amsthm,amssymb} \usepackage{graphicx} \usepackage{hyperref} \newenvironment{theorem}[2][Theorem]{\begin{trivlist} \item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end {trivlist}} \newenvironment{lemma}[2][Lemma]{\begin{trivlist} \item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}} \newenvironment{conjecture}[2][Conjecture]{\begin {trivlist} \item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}} \newenvironment{question}[2][Question]{\begin{trivlist} \item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}} \newenvironment{corollary}[2][Corollary]{\begin{trivlist} \item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}} \ newenvironment{definition}[2][Definition]{\begin{trivlist} \item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}} \newenvironment{problem}[2][Problem]{\begin {trivlist} \item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}} \begin{document} % -------------------------------------------------------------- % Start here % -------------------------------------------------------------- \title{Inquiry \& Proof} \author{Your Name} % replace with your name %\date{A fixed date} % for a fixed date, uncomment this line and replace date, otherwise, today's date will be used. \maketitle \noindent Information about collaboration should go here. If you feel it is appropriate to include any other comments as an introduction, put them here too. \begin{problem}{x.yz} %You can use theorem, exercise, problem, or question here. Modify x.yz to be whatever number you are explaining/proving. Delete this text and write your statement here. Replace x.yz with the class's numbering convention for that item. \end{problem} \begin{proof} Blah, blah, blah. Delete this and write your proof here. \end{proof} \bigskip \noindent You'll generally be turning in one item per page, but here is another example, just so we can showcase a few more features of \LaTeX. \bigskip \begin{theorem}{w.xy} This is a theorem about flying pigs. \end{theorem} \begin{proof} Blah, blah, blah. This is another proof for you to delete. It has some math symbols in it: Let $\alpha \in (A \cup B) \cap \{\text{flying pigs}\}$ and $\ mathbb{Z} \subset \mathbb{R}$. \begin{description} \item[Case 1:] Suppose $x=\sin\theta$ ... thus pigs fly. \item[Case 2:] Suppose $x \neq \sin\theta$ ... therefore, pigs fly. \end{description} Since in every case pigs fly, we have proven that pigs do fly. \end{proof} \noindent If you want to include a figure, you have to upload the image file to overleaf (or store it in the same folder as your .tex file, if you're not using overleaf). Look in the .tex source file for the syntax, which has been commented out so it doesn't appear here in the pdf. %\begin{figure}[ht] %\centering %\ includegraphics[width=.25\textwidth]{myimage.png} %\caption{This is a picture of math.} %\end{figure} It is possible to change the way your figure is placed and sized. Play around with the parameters to see what happens. By the way, when you make figures with GeoGebra, you can easily ``export" as a .png file. Then upload into overleaf.com using the ``Project'' button, then "Add files..." % -------------------------------------------------------------- % Leave this at the bottom and replace URL with your own % -------------------------------------------------------------- \vfill \small {Read-only document can be found online at \url{https://www.overleaf.com/lotasymbols}} %copy and paste the READ-ONLY URL of your document into this line. Then others with this pdf can also access the .tex file. Avoid editing or deleting the project once you've shared this URL. (create a copy of this project to start a revised version of the same.) I'll use this mostly when a result gets published to the online class journal. % -------------------------------------------------------------- % You don't have to mess with anything below this line. % -------------------------------------------------------------- \end{document}	{"url":"https://it.overleaf.com/latex/templates/inquiry-and-proof-math-3318/ggzjzhpvmrjh","timestamp":"2024-11-11T13:58:15Z","content_type":"text/html","content_length":"41043","record_id":"<urn:uuid:dff2c378-04ad-440b-9805-0b2764d80e17>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00601.warc.gz"}
Rolling Offset (run and travel) `"Rolling Offset" = "Run and Travel"[ "U" , "S" , fA ]` Enter a value for all fields The Rolling Offset calculator computes the rolling offset length of the run (R) and travel (T) based on the vertical (V) and horizontal (H) offsets and the fitting angles (fA) of the elbow fittings. This is also known as pipe rolling. INSTRUCTIONS: Choose units and enter the following: • (V) Vertical Offset • (H) Horizontal Offset • (fA) Elbow Fitting Angle. (e.g. 22.5°, 30°, 45^o, 60^o, 90^o) Rolling Offset (R & T): The calculator returns the run (R) and travel (T) in inches. However, these can be automatically converted to other length units (e.g. centimeters or feet) via the pull-down The Math / Science When running pipes or conduits, it is common to have to change the run of the line of pipes by using two equal angled elbow fittings (see diagram). The length of the run (R) and travel (T) created with the elbows and the length of pipe between them, can be calculated if one knows the vertical and horizontal offsets and the angle of the fitting. To compute the Rolling Offset Travel and Run (aka pipe rolling), the relationship between the travel length and the offsets is as follows: ` x = sqrt( V^2 + H^2)` ` T = x / cos (90 - fA)` `R = sqrt(T^2 - x^2)` • R - Run • T - Travel • V - Vertical offset • H - Horizontal offset • fA - Fitting angle Note: vCalc allows for multiple units for both length (SI and English) and for angles (degrees and radians). The result will be in inches. However, this can automatically be converted to other length units (e.g. centimeters) with the pull-down menu. • Rolling Offsets (Run and Travel) – The Rolling OffsetRolling Offset Lengths Pipe Grading function computes the run and travel length a rolling offset based on the offsets and fittings. (see • Diagonal of a Square - This is a simple calculation to assist in computing the diagonal of a square. • Diagonal of a Box - This computes the length of the diagonal of a box (T) based on sides of length R, S and U. • Flow Rate - This computes flow rate based on the total volume and the time it took to accumulate. • Pipe Flow Volume - This computes the total volume from a pipe based on the flow rated and the duration of flow. • Weight of Water in a Tank - Computes the weight of water in a cylindrical tank based on the radius and height (or length). • Weight of Water in a Pipe - Computes the weight of water or other substances in a pipe based on the dimension and material density. • Weight of Sea Water in pipe - Computes the weight of sea water in a cylinder based on the radius and height (or length) • Pressure Head - The Potential Gravity-Fed Water Pressure from a Tank (a.k.a. Pressure Head) based on the height of storage. • Pipe Volume - Computes the volume in a pipe • Pipe Surface Area - Computes the surface area of a pipe. • Pipe Coating Amount - Computes the volume of coating material for a pipe such as paint, polyethylene, polyurethane, zinc, bitumen, FBE or mortar. • Pipe Grading - Compute the drop needed over a run to maintain a grade (e.g., 4" over 12') • Volume of Water in a Tank (e.g. hot water tanks), • Volume of a Spherical Container • Weight of Water in a Spherical Container • Volume of Water in Rectangular Tank • Weight of Water in a Rectangular Tank • Capillary Rise - The height of water in a small tube due to capillary force. • Snow Water Equivalence - The volume of water created by an area and depth of snow. • Pore Water Pressure - Pressure of uplift from the water table. • Pipe Stress Budget - Computes the pressure that a pipe can withstand based on the allowable stress, wall thickness and outside diameter. Enhance your vCalc experience with a free account Sign Up Now! Sorry, JavaScript must be enabled. Change your browser options, then try again.	{"url":"https://www.vcalc.com/equation/?uuid=0957baff-1bae-11e5-a3bb-bc764e2038f2","timestamp":"2024-11-13T07:51:01Z","content_type":"text/html","content_length":"61182","record_id":"<urn:uuid:d30d4906-108c-4401-9d1e-9749263d7a09>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00001.warc.gz"}
principle of induction The principle of induction is a technique used to prove the relationship between a smaller subset The following three statements are equivalent. Suppose $S \subset \mathbb{N}$, which is non-empty. If $S$ is a non-empty subset such that $0 \in S$, and for all $n \in \mathbb{N}$, $n \in S \implies n+1 \in S$. Then, $S = \mathbb{N}$. Suppose $S \subset \mathbb{N}$, which is non-empty. If $S$ is a non-empty subset such that $0 \in S$, and for all $n \in \mathbb{N}$, $\{0, \dots, n\} \in S \implies n+1 \in S$. Then, $S = well-ordering principle If \(S \subset \mathbb{N}$ is non empty, then it has a smallest element Given $S \in \mathbb{N}$, such that $0 \in S$, whenever $n \in S$, then $n+1$ is also in $S$. We desire that that $S$ is the natural numbers. Assume for the sake of contradiction $S \neq \mathbb{N}$. Define $T = \mathbb{N} \setminus S$. Assume $T$ is non-empty. The WOP tells us that $T$ has a smallest element $t \in T$. We know that $t \neq 0$, because $0 \in S$. Therefore, $t-1 \in \mathbb{N}$. But, $t-1 < t$, which means that $t-1 \in S$. But by the statement of the givens, $(t-1) \in S \implies (t-1)+1 = t \in S$. Reaching contradiction. $\blacksquare$ Assume $S$ has no smallest element. Create some $T = \mathbb{N} \setminus S$. Now, $0 \in T$ because otherwise $0 \in S$ would be the smallest element. Now, consider $0, 1, &mldr; n \in T$, we notice that $n+1$ must be in $T$ as well. By strong induction, we have that $T = \mathbb{N}$ and $S$ is empty.	{"url":"https://www.jemoka.com/posts/kbhprinciple_of_induction/","timestamp":"2024-11-03T13:33:15Z","content_type":"text/html","content_length":"7800","record_id":"<urn:uuid:3f9044c2-4aea-41d8-8b36-77bde7cb320f>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00724.warc.gz"}
Rod Cutting Let's solve the Rod Cutting problem using Dynamic Programming. You are given a rod of length n meters. You can cut the rod into smaller pieces, and each piece has a price based on its length. Your task is to earn the maximum revenue that can be obtained by cutting up the rod into smaller pieces. Let’s say you have a rod of length $4$ meters and you have two lists: one that defines the lengths, and the other that defines the price of each length. lengths = [1, 3, 4] prices = [2, 7, 8] You can cut the rod into the pieces of varying lengths and earn the following revenues: • Four pieces of length $1$ meter $= 2 + 2 + 2 + 2 = 8$ • One piece of length $1$ meter and another piece of length $3$ meters $= 2 + 7 = 9$ • One single piece of length $4$ meters $= 8$ Therefore, the maximum revenue you can generate by cutting the rod and selling the pieces is $9$, by cutting the rod into two pieces, one of length $1$ and the other of length $3$. • $1\leq$ n $\leq40000$ • $1\leq$ prices[i] $\leq10^6$ • $1\leq$ lengths[i] $\leq n$ • prices.length $=$ lengths.length Level up your interview prep. Join Educative to access 80+ hands-on prep courses.	{"url":"https://www.educative.io/courses/grokking-dynamic-programming-interview-cpp/rod-cutting","timestamp":"2024-11-11T17:02:12Z","content_type":"text/html","content_length":"778467","record_id":"<urn:uuid:a1dc4cd0-dd86-47ef-a994-1dcfa784f88f>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00696.warc.gz"}
Math Tutor - Integrals In this part we look at the series. We start with series of real numbers, which is a topic that we cover rather well, in particular we cover (absolute) convergence and properties of series, examples, summability and convergence tests. As usual, in practical mastery we get help from Methods Survey, Solved Problems and Exercises. We then briefly look at sequences and series of functions, which is a vast and advanced topic and we only scrap the surface. It will serve as a springing board for two notions that can be used also on a basic level, the Taylor series and the Fourier series. Theoretically they are quite advanced, too; we will not go into it here in Math Tutor, rather we will provide some sort of a cookbook.	{"url":"https://math.fel.cvut.cz/en/mt/txte3e.htm","timestamp":"2024-11-04T17:19:40Z","content_type":"text/html","content_length":"1788","record_id":"<urn:uuid:5b29f0ba-4075-49de-b4f7-aef4655510b9>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00059.warc.gz"}
Networks and Epidemics A guide to using EpiModel for epidemic modeling This ebook contains all the material needed to teach yourself how to use the EpiModel package for epidemic modeling. The book begins with an overview of the foundations of epidemic modeling – from deterministic compartmental modeling to stochastic network models – works through how to implement all of these models using EpiModel’s built-in functions, and provides an introduction to using EpiModel’s extension API to build more complex research-level models for your own work. While EpiModel is capable of implementing deterministic compartmental models and “agent-based” or “individual-based” models, this ebook focuses primarily on the EpiModel’s unique ability to implement stochastic network models. These models are based on a principled statistical framework known as “Exponential-family Random Graph Models” (ERGMs) that allow researchers to represent everything from simple random graphs (aka “Erdos-Reyni” or “Bernoulli random graphs”) to very complex networks. • Where did this ebook come from? The materials in this book were originally developed to teach a weeklong intensive workshop on epidemic modeling called Network Modeling for Epidemics. We refined the materials as we taught the course over about a decade, initially in-person on the University of Washington campus, and then with the Covid pandemic, online. An interdisciplinary team of instructors developed the materials and taught the course, including an Epidemiologist, an Anthropologist and a Statistician/Sociologist. Over the years the lectures and labs were updated and modified with the input from our research assistants, postdocs, students and users, so this final product owes much to their patience, enthusiasm and insight. • Who is this ebook intended for? These materials are intentionally designed to be accessible to a wide range of users – you definitely don’t need to be a mathematical modeler or statistician to learn from them! Those trained in traditional epidemic modeling can also learn things here, in particular, how to model epidemics on stochastic networks using principled, data-driven statistical methods. Our goal in this ebook is to make these tools available to applied researchers and practitioners from many backgrounds: from public health departments and veterinary science, to social science, epidemiology and mathematics. Our students have come from all of these fields over the years, and from around the world, including the global south. Some participants take the course to provide a foundation for their own epidemic modeling projects, others take it so that they can understand and critique the research on epidemic modeling that they are reading. If your goal is to use EpiModel for your own research projects, you will need to know (or learn) how to program in the R computing environment. We have provided some suggestions for getting started with R if you don’t already have this background. The lectures and labs provide a thorough introduction to using EpiModel in R, with many code examples. A brief overview of EpiModel and its capabilities can be found in this article Each chapter contains a mixture of “lecture” type materials and labs for practicing the concepts with exercises and solutions. The first couple chapters are designed to make sure all of the foundations are in place, so depending on your background, some of the material there may be review. Chapter 1: “Epidemic and Networks 101” If you’re a newbie to epidemic modeling, we recommend going through this chapter carefully, working though each of the labs to get the basic principles down. All of the foundation you need for the rest of the course is provided here. If you have prior experience with epidemic modeling, you can probably skim (not skip) this chapter, to make sure you’re familiar with each of the basic modeling frameworks and their benefits and drawbacks. The labs in this chapter will also give you a feel for simple EpiModel coding and functionality. Chapter 2: “Statistical models for networks” Everyone should work through this chapter (if you’re already familiar with ERGMs, you can skim). It provides, using simple intuitive examples, an introduction to the key concepts that make network analysis different from the usual empirical research frameworks taught in most disciplines. It shows why these concepts matter for epidemics, what kind of network data would be needed to explore these effects, how these data can be modeled using flexible statistical methods, and how the fitted models can in turn be used to simulate epidemics on dynamic networks that “look like” the data you Chapters 3 thru 5: The remaining chapters get into the details of using EpiModel to model epidemics on networks. Chapters 3 and 4 introduce EpiModel’s built-in models for representing epidemics with and without feedback processes. Chapter 5 provides instructions on using the EpiModel API to move beyond these built-in models and build models for your own research-level projects. • Acknowledgements. EpiModel has been developed with the generous support of the U.S. National Institutes of Health. The publication of this ebook was supported by the NIH grant R01 AI138783. • Prerequisites A basic working knowlege of the R computing environment is needed in order to install and use the EpiModel software. • This ebook is based on R version 4.3.2 (2023-10-31) and EpiModel package version 2.3.2 This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.	{"url":"https://epimodel.github.io/epimodel-training/index.html","timestamp":"2024-11-08T04:37:05Z","content_type":"application/xhtml+xml","content_length":"48474","record_id":"<urn:uuid:cff9671f-12f3-429d-8888-40f231c29673>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00271.warc.gz"}
Ultimate behavior and strength of cable-stayed composite 2-I-girder bridges Masatugu Nagai This paper describes ultimate behavior and strength of cable-stayed bridges with a steel-concrete composite 2-I-girder and proposes stability design method of the main girder. Using 150,400 and 600-meter span bridge models, their ultimate behavior and strength are studied based on elasto-plastic finite displacement analysis, in which the depth of the girder is chosen as a Under the condition that min. depth of 1.0 meters is used for the bridge with a span length up to 400 meters, and max. span-to-depth ratio of 400 is used for the bridge with a span length from 400 to 600 meter, the followings are the results obtained from this study. 1）Geometrical non-linear effect is minor and can be neglected. 2）Global buckling behavior of the girder is not observed．Hence, stability check on global buckling is not necessary. 3）Ultimate strength is controlled by the yielding of the steel material．The load parameter（the ratio of the total load at ultimate state to the design load）can be calculated from the yield point divided by the max. working stress under design load．	{"url":"https://cee.nagaokaut.ac.jp/HTML/yoshisyu/2002/kensetu/e/ken03607_e.htm","timestamp":"2024-11-11T01:05:19Z","content_type":"text/html","content_length":"1844","record_id":"<urn:uuid:689d08e8-2760-4618-ba67-b80d3fe4b4af>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00095.warc.gz"}
Automatic feedback provided to students’ work has the potential to improve their mathematical thinking and skills. A commonly used type of online feedback is elaborated feedback. Engaging students with elaborated feedback is a well-known challenge, which we addressed by examining two mediating tools that are part of a dynamic diagram environment: self-assessment, and interactive feedback. We focused on the students’ process of constructing position-over-time graphs, while they were working on an online example-eliciting task on the Seeing the Entire Picture platform. We present and compare two case studies of students' use of the elaborated feedback, engaging both mediating tools, and analyze changes in the students' examples after processing the elaborated feedback. Original language English Title of host publication Proceedings of the 44th Conference of the International Group for the Psychology of Mathematics Education, 2021 Editors Maitree Inprasitha, Narumon Changsri, Nisakorn Boonsena Publisher Psychology of Mathematics Education (PME) Pages 1-8 Number of pages 8 ISBN (Print) 9786169383024 State Published - 2021 Event 44th Conference of the International Group for the Psychology of Mathematics Education, PME 2021 - Virtual, Online Duration: 19 Jul 2021 → 22 Jul 2021 Publication series Name Proceedings of the International Group for the Psychology of Mathematics Education Volume 3 ISSN (Print) 0771-100X ISSN (Electronic) 2790-3648 Conference 44th Conference of the International Group for the Psychology of Mathematics Education, PME 2021 City Virtual, Online Period 19/07/21 → 22/07/21 Bibliographical note Publisher Copyright: © 2021 left to authors. ASJC Scopus subject areas • Mathematics (miscellaneous) • Developmental and Educational Psychology • Experimental and Cognitive Psychology • Education Dive into the research topics of 'ENGAGING STUDENTS WITH ONLINE ELABORATED FEEDBACK: TWO MEDIATING TOOLS'. Together they form a unique fingerprint.	{"url":"https://cris.haifa.ac.il/en/publications/engaging-students-with-online-elaborated-feedback-two-mediating-t","timestamp":"2024-11-14T10:46:35Z","content_type":"text/html","content_length":"54434","record_id":"<urn:uuid:74041707-ece6-4b8b-8548-8d21c08b7eff>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00897.warc.gz"}
Unlocking the Mysteries of Hyperspheres: The Science of Higher-Dimensional Geometry - Eye Of Unity FoundationUnlocking the Mysteries of Hyperspheres: The Science of Higher-Dimensional Geometry - Eye Of Unity FoundationUnlocking the Mysteries of Hyperspheres: The Science of Higher-Dimensional Geometry Unlocking the Mysteries of Hyperspheres: The Science of Higher-Dimensional Geometry In the realm of geometry, we are often familiar with shapes like circles, squares, and cubes. However, there exists a fascinating branch of mathematics that delves into the study of higher-dimensional shapes, including a concept known as hyperspheres. Unlocking the mysteries of hyperspheres can provide us with a deeper understanding of our universe and the possibilities that lie beyond our usual three dimensions. What are Hyperspheres? A hypersphere, also known as an n-sphere, is a generalization of a sphere to higher dimensions. While we typically envision a sphere as a three-dimensional object, a hypersphere extends this concept to any number of dimensions. Essentially, a hypersphere is the set of all points in a Euclidean space that are equidistant from a fixed center To better understand hyperspheres, let’s consider their lower-dimensional counterparts. A circle, for example, is a two-dimensional hypersphere, where all the points on the circle’s circumference are equidistant from its center. In three dimensions, we have the familiar sphere, where all the points on its surface are equidistant from the center. The concept of hyperspheres allows us to extend this idea to higher-dimensional spaces. Higher-Dimensional Geometry Exploring higher-dimensional geometry is a mind-bending adventure that challenges our conventional understanding of space. While it is difficult to visualize shapes beyond our three dimensions, mathematics provides us with the tools to reason about these abstract concepts. In higher-dimensional geometry, we encounter intriguing properties of hyperspheres that differ from their lower-dimensional counterparts. For instance, in two dimensions, the surface area of a circle is given by the formula A = πr². However, in three dimensions, the surface area of a sphere is given by A = 4πr². This relationship continues to hold true as we move into higher dimensions, where the surface area of an n-sphere is determined by a similar formula involving the radius. Another fascinating aspect of higher-dimensional geometry is the concept of volumes. Just as we can calculate the volume of a three-dimensional object like a cube or a sphere, mathematicians have devised formulas to calculate the volume of higher-dimensional shapes. These formulas involve intricate mathematical concepts such as integrals and Applications of Hyperspheres The study of hyperspheres has numerous applications in fields such as physics, computer science, and even data analysis. In physics, hyperspheres can be used to model the behavior of particles in high-energy collisions or to describe the curvature of spacetime in Einstein’s theory of general relativity. Computer scientists often utilize hyperspheres in machine learning algorithms, particularly in clustering methods such as k-means. Hyperspheres can help classify data points based on their proximity to a central point, allowing for efficient grouping and analysis. Hyperspheres also find applications in data visualization techniques, where they can be used to represent high-dimensional data in a more comprehensible manner. By projecting data onto a lower-dimensional hypersphere, we can visualize complex relationships and patterns that may not be immediately apparent in the original high-dimensional Q: Can we visualize hyperspheres? A: Visualizing hyperspheres in dimensions higher than three can be challenging for our human perception. However, mathematicians and scientists use mathematical models and projections to reason about hyperspheres and their Q: Are hyperspheres the only higher-dimensional shapes? A: No, hyperspheres are just one example of higher-dimensional shapes. There are several other intriguing shapes and objects that exist in higher-dimensional spaces, such as hypercubes or tori. Q: How are hyperspheres relevant to our daily lives? A: While hyperspheres may seem abstract and distant from our everyday experiences, their applications in various fields can indirectly impact our lives. From understanding the universe’s curvature to improving data analysis techniques, the study of hyperspheres contributes to advancements that shape our modern world. Q: Can we mathematically prove properties of hyperspheres in higher dimensions? A: Yes, mathematicians have developed rigorous proofs and formulas to study the properties of hyperspheres in any number of dimensions. These proofs rely on advanced mathematical techniques and principles. Q: How can I explore higher-dimensional geometry further? A: To delve deeper into the captivating world of higher-dimensional geometry, consider studying advanced mathematics or exploring online resources that provide interactive visualizations and explanations of these concepts.	{"url":"https://eyeofunity.com/unlocking-the-mysteries-of-hyperspheres-the-science-of-higher-dimensional-geometry/","timestamp":"2024-11-02T17:09:45Z","content_type":"text/html","content_length":"85859","record_id":"<urn:uuid:d92aa88f-4437-488e-8cff-336c68ae2f40>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00617.warc.gz"}
Long period oscillations in sunspots Issue A&A Volume 513, April 2010 Article Number A27 Number of page(s) 7 Section The Sun DOI https://doi.org/10.1051/0004-6361/200913683 Published online 16 April 2010 A&A 513, A27 (2010) Long period oscillations in sunspots N. Chorley^1 - B. Hnat^1 - V. M. Nakariakov^1 - A. R. Inglis^1 - I. A. Bakunina^2,3 1 - Centre for Fusion, Space and Astrophysics, Physics Department, University of Warwick, Coventry CV4 7AL, UK 2 - Radiophysical Research Institute, 25 B. Pecherskaya Street, Nizhny Novgorod, 603950, Russia 3 - Central Astronomical Observatory at Pulkovo, Russian Academy of Sciences, Pulkovskoe chaussee., 65/1, St. Petersburg, 196140, Russia Received 17 November 2009 / Accepted 23 January 2010 Long period oscillations of the gyroresonant emission from sunspot atmospheres are studied. Time series data generated from the sequences of images obtained by the Nobeyama Radioheliograph operating at a frequency of 17 GHz for three sunspots have been analysed and are found to contain significant periods in the range of several tens of minutes. Wavelet analysis shows that these periods are persistent throughout the observation periods. The presence of the oscillations is confirmed by several methods (periodogram, wavelets, Fisher randomisation and empirical mode decomposition). Spatial analysis using the techniques of period, power, correlation and time lag mapping reveals regions of enhanced oscillatory power in the umbral regions. Also seen are two regions of coherent oscillation of about 25 pixels in size, that oscillate in anti-phase with each other. Possible interpretation of the observed periodicities is discussed, in terms of the shallow sunspot model and the leakage of the solar g-modes. Key words: Sun: oscillations - sunspots 1 Introduction Oscillatory processes observed in sunspots have attracted attention for several decades (for a comprehensive review, see, e.g. Bogdan 2000). Such oscillations are usually placed into one of four categories: (a) umbral chromospheric oscillations with a period of 3 minutes, thought to be slow magnetoacoustic waves (e.g. Centeno et al. 2006); (b) umbral photospheric oscillations with a period of 5 minutes, which may be a response to driving by the well known 5 minute photospheric acoustic oscillations (Thomas et al. 1984); (c) long period oscillations (of the order of hours, e.g. Goldvarg et al. 2005; Efremov et al. 2007); (d) ultra-long period (torque or torsional) oscillations of sunspot umbrae, with periods of several days (Gopasyuk 2004). So far, most of the work on sunspot oscillations has been focused on the first two, short period categories above. However, long period sunspot oscillations are interesting for a number of reasons, in particular because of their possible association with the eigenmodes of the sunspot magnetic flux tube, the dynamical processes in the solar interior, including the generation and transfer of the magnetic field, and the possible role these oscillations play in solar coronal dynamics (see, e.g. Sych et al. 2009). Observational evidence of the long period sunspot oscillations is abundant in both the optical and radio bands. For example, 30-60 min oscillations have been detected in the intensity difference of two solar radio signals recorded at two close frequencies, 9.67 GHz and 9.87 GHz (Kobrin & Korshunov 1972). Oscillation periods of tens and hundreds of minutes have also been seen as modulation of the microwave emission (Gelfreikh et al. 2006). Recently, oscillations of the line-of-sight velocity with periods from 60 to 80 min have been detected in sunspots (Efremov et al. 2009) in the Doppler shift of lines in the sunspot spectrum formed at different heights. The oscillations were found to be well pronounced at a level of 100-200 km and decreasing rapidly above it. Theoretical interpretation of long-period radial oscillations in sunspots is in an embryonic state. Possibly, the most advanced approach is the consideration of the oscillations in terms of the shallow'' sunspot model (e.g. Solov'ev & Kirichek 2008, and references therein). According to this model, a sunspot can oscillate, similar to a raft in water, with the frequency determined by the ratio of the sunspot depth to its Alfvén speed, taking the mass density averaged over the depth. For typical sunspot parameters (the surface mass density of ^-3, the magnetic field of 2000 G, and the depth of 3000 km), the oscillation period of the global mode was found to be 1.5 h. This mode is localised near the umbra. Lower subharmonics with periods of integer multiples of the period of the global mode can occur too. Figure 1: White light images of the active regions studied in this paper. Open with DEXTER In this paper, we study long period sunspot oscillations in the microwave emission, utilising both time and spatial information. The paper is organised as follows: in Sect. 2, we present the data used; Sect. 3 details our analysis (Sects. 3.1-3.4 describe the analysis of time signals from the sunspots, while Sects. 3.5 and 3.6 describe the analysis of time signals from the quiet Sun and spatial analysis, respectively) and in Sect. 4, we discuss and interpret our results. 2 Observations We analysed three sunspots that were observed for periods of about 8 h with the Nobeyama Radioheliograph (NoRH, Nakajima et al. 1994), at a frequency of 17 GHz. A cadence of 1 s and a spatial resolution of 10'' per pixel can be obtained with NoRH. The sunspots were the isolated sunspot of active region AR0330 (08 April 2003 22:45-09 April 2003 06:29), the leading sunspot of active region of AR108 (10 September 2002 22:45-11 September 2002 06:44) and the large leading sunspot of active region AR673 (21 September 2004 22:45-22 September 2004 06:44). Fig. 1 shows white light images of active regions AR108, AR0330 (obtained with SOHO/MDI) and AR673 (obtained with the Big Bear Solar Observatory). Table 1 summarises the observational details for these sunspots. The measured emission is likely to be generated by the gyroresonant mechanism (e.g. Vourlidas et al. 2006; Shibasaki et al. 1994) coming from a narrow layer of plasma over the sunspot, where a low harmonic of the gyrofrequency coincides with the observational frequency of the instrument. Table 1: Summary of data analysis. 2.1 Data generation Partial disk images, measuring the radio brightness temperature were obtained from NoRH for the 3 considered sunspots and time series were generated from these by integrating the signal over the field of view (FOV). Figure 2 shows the time series of the emission intensity from the analysed sunspots. Since the observation periods were quite long, it was necessary to track the sunspots through their passage across the solar disk. First, the sunspots were matched with their active region numbers by obtaining a full disk image for the start of the observing period and then adding the active region numbers to the image using SolarSoft's NOAA active region database and its associated routines (get_nar and oplot_nar). The centre of the radio sunspot was then found (see Table 1) at the start of the observing period, as the position of the maximum of the microwave emission. These coordinates were converted to heliographic longitude and latitude and the change in longitude in one time step (i.e. 10 s) due to differential rotation was found for the given latitude. The position at the next time step was given by Figure 2: Original time series of microwave intensity generated from partial disk radio images for AR108 (top), AR0330 (middle) and AR673 (bottom), as observed by the Nobeyama Radioheliograph, at a frequency of 17 GHz. Open with DEXTER 3 Analysis 3.1 Trend removal and filtering It is evident from Fig. 2 that there is a large scale trend in all the datasets due to the motion of the Sun across the sky during the observing period, long period (e.g. daily variation of the Earth's ionosphere) and other ultra-long period processes. To remove the trend, we fitted a 4th order polynomial to the data and then subtracted it from the original signal. Following this, we filtered out ultra-long period, slow varying dynamics from the Fourier spectrum of the signal. Also evident in the time series for AR108 is a large spike, which is perhaps artificial or a short flare. Since we were only interested in the long period oscillations in the data, the spike was removed. This was done by finding the mean of 5 points to the left of the region and then replacing the points in the region with this mean. 3.2 Period analysis In order to investigate the various frequencies present in the datasets, we employed both the periodogram technique and wavelet analysis^ (see, e.g. Torrence & Compo 1998). Figure 3 shows the Lomb-Scargle periodograms (Scargle 1982) and wavelet power spectra computed from the detrended data. We see that in all three periodograms, there is power above the 99% significance level (calculated according to Horne & Baliunas 1986) around frequencies of 5-6 mHz, corresponding to the well-known 3 min chromospheric umbral oscillations. It is also clear that there are several low frequency peaks in all the datasets. For AR108 and AR673, there is significantly more power in the low frequency oscillations than in the 3 min oscillations. Evidently, much longer periodicities, of about an hour are present too. For the wavelet analysis, the Morlet wavelet was chosen, because of its good performance in the study of oscillatory signals. The 99% significance level has been estimated using the subroutine provided by Torrence & Compo (1998). In Fig. 3, we have only shown power that is above this level. In addition to computing the wavelet power spectrum for each time signal, the global wavelet spectrum was also computed. In the global wavelet spectrum, the power for each frequency is simply the time averaged wavelet power for that frequency. The global wavelet spectra are shown in Fig. 3, overlaid on the periodograms. These spectra show that AR108 and AR0330 have two low frequency peaks and possibly that the same is true of AR673, but the two peaks blend together and are not resolved All three applied techniques (periodogram, wavelet and global wavelet) show the presence of coinciding spectral peaks, corresponding to the 3-min oscillations and longer period peaks (18-80 min). The 3-min peaks have the same spectral position. The long periods we find have different ranges for each sunspot and are as follows: AR108: P[1] = 88^+16[-21] min, P[2] = 37^+8[-11] min; AR673: Figure 3: The Scargle periodograms and Morlet wavelet power spectra for AR0330 (top row), AR108 (middle row) and AR673 (bottom row) time series. The left panels show the periodograms (thin lines) and global wavelet spectra (thick lines). The dashed lines indicate the 99% significance level for the periodograms. Open with DEXTER 3.3 Significance testing The Lomb-Scargle periodogram is a redefinition of the traditional power spectrum, which has at its heart simple statistical behaviour, namely that the power at a given frequency has an exponential distribution (for a pure Gaussian noise signal). The cumulative distribution function then has a simple form (see Scargle 1982, Eq. (13)) and using it, we can determine a power threshold, z[0], such that the probability, p[0], that the highest peak above this threshold is due to chance is small. The expression for z[0] given by Scargle (1982) is is the false alarm probability and is the number of frequencies over which to search for the highest peak. According to Horne & Baliunas (1986) has to be normalised to the variance of the signal. According to this estimation, the long period sunspot oscillations are statistically significant. Also, we make use of a significance test based on Fisher's method of randomisation (see, e.g. Linnell Nemec & Nemec 1985) in order to confirm our results from the periodogram analysis. The main reason for doing this is that the test is distribution independent, whereas other techniques implicitly assume a certain model for the statistical distribution. We would like to calculate the probability that the highest peak in our power spectrum is due to the presence of harmonic oscillations, i.e. not due to chance. Suppose first that the signal does not contain a periodic component. Then, the measurements t[i]). If that is the case, there are n! permutations of the signal, where n is the number of measurements. Typically, n is large and as such, it is impractical to perform all n! permutations. Due to this limitation, we choose to perform The probability, p, of the highest peak in the spectrum occuring by chance is then given by ). It should be noted that both and (1 - ) are only estimates of the true probabilities and this is due to the limitation on the number of permutations that can be performed. The Fisher randomisation method was applied to all three datasets with m = 200. We performed m random permutations 1000 times for each dataset (i.e. 1000 random sets of 200 permutations were performed) and for all 1000 experiments, the estimated value of p was found to be less than 0.1%, i.e. there was almost no chance that the detected periods corresponding to the highest spectral peaks (i.e. 88 min for AR0330, 57 min for AR108 and 27 min for AR673) occurred due to chance. 3.4 Empirical mode decomposition Fourier analysis, while a commonly used technique, is not without problems. For example, any large scale trend in the signal must be removed before performing Fourier analysis, otherwise the spectrum will be dominated by the trend. Also, it requires that signals contain strictly periodic components. In real signals, this is rarely the case and oscillatory components can be modulated by non-stationary and non-linear effects. Fourier analysis does not give the user any time information and using the windowed Fourier transform can be a solution. However, the size of the window needs to be chosen and in general, several different window sizes need to be tested before an appropriate one is chosen. Wavelets allow the window width to be chosen automatically and allow us to study non-stationary signals and analyse frequency modulation, but one disadvantage of that method is that time resolution is limited by the finite width of the wavelet function. Also, these techniques cannot perform well on periodic but significantly anharmonic signals. Empirical mode decomposition (EMD, Huang et al. 1998) was designed for use with non-linear and non-stationary signals and assumes that the signal is a sum of intrinsic oscillations (which are usually referred to as intrinsic mode functions, or IMFs) that may have varying amplitude and frequency. As such, it is an excellent method for studying frequency modulation. In addition, it is not necessary to remove a trend component from the data before applying the technique - being an adaptive filter, EMD effectively removes the trend. The intrinsic mode functions found do not need to be harmonic oscillations, which allows the technique to be used to investigate the anharmonicity of oscillations in a signal. Applications of EMD in solar physics can be found in Komm et al. (2001) and Terradas et al. (2004). Here, we use EMD to confirm our results derived from other methods. The IMFs are found for a signal, x(t), by a sifting process. First, the local maxima are identified and are connected with a cubic spline. This is repeated for the local minima and the mean of the envelope created by the connected maxima and minima, m(t), is found. This mean is subtracted from the signal to give ) will be an IMF, provided that it satisfies the following two conditions: 1. The number of extrema and number of zero crossings must differ by at most one. 2. The mean of the upper and lower envelopes must be zero at all times. The first condition means that the IMFs contain a narrow range of frequencies and the second means that the oscillations have zero mean. Once the mean has been subtracted from the original signal, the residual signal, h[1](t) is used as input to the EMD procedure and the process is repeated until all IMFs are found. Huang et al. (1998) proposed the Hilbert spectrum for the visualisation of these components, but here, we make use of the standard wavelet power spectrum, because it is easier to interpret than the Hilbert spectrum. Thus, in this study, EMD is used as an adaptive filtering technique only. Figure 4: Three components derived by the empirical mode decomposition method for the signal from AR108. The wavelet power spectra are shown for the first two of these components. The third component (bottom left) is the trend component. Bottom right: the global wavelet spectrum of the two oscillatory components displayed. Open with DEXTER The EMD technique was applied to all three datasets, after filtering the signals to contain only frequencies in the range 0-8 mHz, to include the 3 min oscillations and remove high frequency noise. For AR673, 12 components were found and 10 components were found for both AR108 and AR0330. Figure 4 shows the three longest period intrinsic modes found by the EMD for AR108, two of which are oscillatory and the third is the trend component. Also shown is the global wavelet spectrum for the two oscillatory components. The two oscillatory components are the most significant components (i.e. they have the largest amplitudes, besides the daily trend) and wavelet power spectra were computed for them. It is clear that the periods are persistent throughout the observations and remain stable. Also, EMD analysis shows that the intrinsic modes of the analysed signals are quasi-monochromatic and harmonic. These features coincide with those found in the significant components of the other signals, which is consistent with the previous results. The global wavelet spectrum in the bottom right of Fig. 4 shows the same periodicities as those shown in Fig. 3. 3.5 Quiet Sun signals As a final test to determine whether the long period oscillations seen in the sunspot time series were real, we investigated the same time variations of the signals of the quiet Sun. One region, with a starting location of (-170, -500)'', which did not contain any bright microwave features, was studied for the same observation period as for AR0330 and with the same time cadence. The signals for AR0330 and the quiet Sun region were filtered to contain only long period spectral components. A gaussian filter was used for both signals, with a bandpass of 0.35-0.60 mHz, keeping typical long periods of interest. The two filtered signals are shown in Fig. 5. The amplitude of the oscillations in the quiet Sun is seen to be two orders of magnitude smaller than those in the sunspot and we conclude that the oscillations of the quiet Sun in this narrowband frequency interval are not significant. Figure 5: Time series for AR0330 (top) and a quiet Sun region (bottom), after narrowband filtering to contain only long period oscillations ( Open with DEXTER 3.6 Spatial structure of the oscillations We apply the method of periodmapping, developed by Nakariakov & King (2007) to investigate the spatial distribution of the oscillations seen in the sunspots. This method has been applied to flaring data from NoRH by Inglis et al. (2008), who used the technique to deduce coherency of quasi-periodic pulsations in different segments of a flaring loop. Figure 6: The left two columns are for the two separate frequency ranges for AR108 (left: right: two right columns are for the two separate frequency ranges for AR0330 (left: right: First row: periodmaps for each sunspot for the two frequency ranges given above. Second row: power maps normalised to the maximum power in the map. Third row: correlation maps showing the maximum value of the correlation coefficient for the two frequency ranges above. The correlation coefficient has been computed over the range of lags P is the period of the maximum power in the global wavelet spectrum for the given frequency range. The values of P used were: AR108: P[1] = 31 min (left), P[2] = 57 min (right); AR0330: P[1] = 37 min (left), P[2] = 88 min (right). Only pixels for which the correlation coefficient, P. The contours show the position of the sunspot from the first image in the datacube. Open with DEXTER For each pixel in the derotated datacube, we generate the time series (by simply taking the values for that pixel along the time dimension of the datacube) and calculate the power spectrum. The frequency of the highest peak in the spectrum is then assigned to this pixel in the periodmap. The periodmap is then a 2D map of frequencies corresponding to the dominant frequencies for each pixel in the images. Nakariakov & King (2007) used the traditional Fourier spectrum to determine the frequency assigned to each pixel in the periodmap. Here, we use the global wavelet spectrum instead, since it is smoother than the Fourier spectrum. In addition, we also make use of power, correlation and lag maps (see Inglis et al. 2008). The first are 2D maps of the maximum global wavelet power in a prescribed frequency range for the corresponding pixel time series, the second show the correlation coefficient, Inglis et al. 2008). For active regions AR108 and AR0330, we first took images with a larger field of view than the ones used for generating the FOV-integrated time series studied in Sects. 3.1-3.4. The images for both datasets were The global wavelet spectra in Fig. 3 show that there are two distinct low frequency peaks present for AR108 and AR0330. It was thus decided to produce period, power, correlation and lag maps for each of these peaks independently, so the datacubes were filtered to contain the peaks separately. The frequency ranges used were: AR108: P is the period of the maximum power in the global wavelet spectrum for the given frequency range. The values of P used were: AR108: P[1] = 31 min, P[2] = 57 min; AR0330: P[1] = 37 min, P[2] = 88 min. In addition, we produced lag maps, showing the lag at which the correlation coefficient of the signal of the current pixel with the signal of the reference pixel was maximal over the same range of lags. The period, power, correlation and lag maps are shown in Fig. 6 and it is quite evident that besides frequency range For frequency range 6 (fourth row, first image). The master pixel is located in this region and as such, we expect (and indeed see) that this large region oscillates in phase (or coherently). A region of almost equal size that oscillates also coherently, but in anti-phase with the first region is seen towards the right of this image. This region is made up of pixels for which the maximum correlation was found at the time lag of plus or minus about half period - these lags are of course almost equivalent when dealing with the maximum correlation value. The lag map for frequency range The lag map for frequency range 6). These regions are each about half of the size of the sunspot and are aligned with the sunspot's ``tilt'' with respect to the vertical. The region oscillating in phase with the master pixel is seen in the northern part of the sunspot and the anti-phase region seen in the southern part. 4 Discussion In this paper we analysed three eight-hour datasets of the microwave emission generated over three different sunspots, recorded with the Nobeyama Radioheliograph at 17 GHz. The main findings of this study are as follows: 1. Significant long period ( 2. In each of the spectra of the sunspots, there are at least two such components. In general, these components are found to have higher power than the three minute oscillations. The periodicities are: AR108: P[1] = 88^+16[-21] min, P[2] = 37^+8[-11] min; AR673: 3. The periodicities stay constant during the observing intervals, without any significant drift. 4. The spatial distribution of the oscillations shows, in general, (for both frequency ranges of AR0330 ( 5. There are regions of the sunspots that coherently oscillate both in phase and in anti-phase with the chosen master pixel. The typical size of the coherently oscillating regions is about 25 pixels. For frequency ranges The nature of the detected long period oscillations in sunspots is still not revealed. In the following, we discuss possible options. The observed periodicities are close to the periods of candidate spectral peaks associated with g-modes: e.g. 22-26 min and about 75 min (García et al. 2008) and the sunspot magnetic flux tubes can operate as waveguides, channelling signals of g-modes from deeper regions. Thus, one possible interpretation of the observed periodicities is their association with the leakage of g-modes, but this scenario requires a solid theoretical foundation. Also, the observed anti-phase oscillations in different parts of the sunspots do not seem to support this interpretation. The observed patterns of fluctuations are also consistent with the shallow sunspot model of Solov'ev & Kirichek (2008). In this model a cylindrical magnetic flux tube has a finite depth of L, and becomes a mostly horizontal flow below the depth L (Zhao et al. 2001). For such a configuration, radially structured fluctuations (that is fluctuations with a certain azimuthal symmetry) could be excited at the bottom of the flux tube and propagate vertically to generate patterns, such as these observed in the last row of Figure 6. It has been shown, using the variational principle, that the periods of such oscillations vary from 40-200 min (Solov'ev & Kirichek 2008), in agreement with periods detected in this work. Further investigations using numerical modelling will be carried out in order to study the generation of such modes in sunspots. Further work is also needed to investigate the modulation of the 3 min N.C. is supported by an EPSRC studentship. N.C. would also like to thank Kiyoto Shibasaki for discussions about this work and financial support for a recent trip to the Nobeyama Solar Radio Observatory. Part of this work was supported by the Royal Society British-Russian Collaboration grant. The software used for wavelet analysis was provided by C. Torrence and G. P. Compo and can be found at http://paos.colorado.edu/research/wavelets. Copyright ESO 2010 All Tables Table 1: Summary of data analysis. All Figures Figure 1: White light images of the active regions studied in this paper. Open with DEXTER In the text Figure 2: Original time series of microwave intensity generated from partial disk radio images for AR108 (top), AR0330 (middle) and AR673 (bottom), as observed by the Nobeyama Radioheliograph, at a frequency of 17 GHz. Open with DEXTER In the text Figure 3: The Scargle periodograms and Morlet wavelet power spectra for AR0330 (top row), AR108 (middle row) and AR673 (bottom row) time series. The left panels show the periodograms (thin lines) and global wavelet spectra (thick lines). The dashed lines indicate the 99% significance level for the periodograms. Open with DEXTER In the text Figure 4: Three components derived by the empirical mode decomposition method for the signal from AR108. The wavelet power spectra are shown for the first two of these components. The third component (bottom left) is the trend component. Bottom right: the global wavelet spectrum of the two oscillatory components displayed. Open with DEXTER In the text Figure 5: Time series for AR0330 (top) and a quiet Sun region (bottom), after narrowband filtering to contain only long period oscillations ( Open with DEXTER In the text Figure 6: The left two columns are for the two separate frequency ranges for AR108 (left: right: two right columns are for the two separate frequency ranges for AR0330 (left: right: First row: periodmaps for each sunspot for the two frequency ranges given above. Second row: power maps normalised to the maximum power in the map. Third row: correlation maps showing the maximum value of the correlation coefficient for the two frequency ranges above. The correlation coefficient has been computed over the range of lags P is the period of the maximum power in the global wavelet spectrum for the given frequency range. The values of P used were: AR108: P[1] = 31 min (left), P[2] = 57 min (right); AR0330: P[1] = 37 min (left), P[2] = 88 min (right). Only pixels for which the correlation coefficient, P. The contours show the position of the sunspot from the first image in the datacube. Open with DEXTER In the text Copyright ESO 2010 Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform. Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days. Initial download of the metrics may take a while.	{"url":"https://www.aanda.org/articles/aa/full_html/2010/05/aa13683-09/aa13683-09.html","timestamp":"2024-11-03T02:27:52Z","content_type":"text/html","content_length":"143901","record_id":"<urn:uuid:befae22f-6fee-4b14-bddd-a91971fe6b2b>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00012.warc.gz"}
An optimization approach to drift detection and clustering in time-series: Application to air quality data in India # 12 Recent developments in low-cost sensors, wireless network communication, and computational tools have paved the way for applications like monitoring with the high spatial and temporal resolution for example in the context of air quality. However, the reduced quality of sensing units necessitates robust drift detection and calibration schemes. The few existing methods are variants of outlier detection algorithms. We presented an optimization-based clustering algorithm that first smooths the data and then performs clustering for drift detection. We present the detection efficiency of the algorithm with a simulated dataset where the proposed algorithm detects sensor failures like random walks, reduced sensitivity and changes in bias. The system we consider consists of pumps delivering water to different reservoirs in a network, with each reservoir catering to time varying demand. Pumps and ON/OFF valves are used as manipulated variables to control the flow and pressure. The decision variables are the number of pumps to be turned on and the state of the valves in the network over a given horizon and the objective is to minimize energy consumption while meeting the time varying demand. Given the nonlinear nature of the pump operating curve and the hydraulics, this results in a Mixed Integer NonLinear Program (MINLP). We propose to solve by decomposing it into series of sub-problems that can be solved efficiently. Application of these ideas to distribution networks reveals potential significant savings in energy or improvement in supply. Experimental results will be shared followed by our field implementations. Alexandre Reiffers Alexandre Reiffers is a post-doctoral fellow at Robert Bosch Centre for Cyber-Physical Systems. He received the B.Sc. degree in mathematics (2010) from the University of Marseille, the master degree in applied mathematics (2012) from the University of Pierre et Marie CURIE and the Ph.D. degree in computer science (January 2015) from the INRIA (National research institute in computer science and control) and the University of Avignon. His supervisors were Eitan Altman and Yezekael Hayel. From July 2016 to December 2017, Alexandre Reiffers was a researcher at SafranTech where he was working on comparison of maintenance strategies. Most of his research projects concern the application of mathematical tools (game theory, optimization, stochastic process and machine learning) for a better understanding of real-world problems. The different issues that he studies touch topics such as social networks, speech between human and computer, economy and manufacturing.	{"url":"https://cni.iisc.ac.in/seminars/2019-11-26/","timestamp":"2024-11-08T13:58:10Z","content_type":"text/html","content_length":"15015","record_id":"<urn:uuid:8866cd54-f799-4803-b8e8-4d8ac0540212>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00156.warc.gz"}
General Theory Of Relativity - Cosmos The General Theory of Relativity was published by Albert Einstein in 1915. The General Theory of Relativity is the accepted name published by Albert Einstein in 1915 for the theory of gravity. The force of gravity is a representation of the local geometry of space-time, according to the General Theory of Relativity. While the modern […] What is General theory of relativity? Equations & Examples Read More ยป	{"url":"https://cosmos.theinsightanalysis.com/tag/general-theory-of-relativity/","timestamp":"2024-11-14T05:14:54Z","content_type":"text/html","content_length":"133199","record_id":"<urn:uuid:cf57b463-a892-4974-a2dd-4dbf68046a7e>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00305.warc.gz"}
Interagency Modeling and Analysis Group One compartment with constant elimination rate of a drug, a first order process, and instantaneous injection of drug dose. A one compartment model with intravenous (i.v.) injection is the simplest description of a drug time course through the body. It assumes that the concentration of the drug within the blood stream is representative of that throughout the body and that drug concentration within the tissue is instantaneously in equilibrium with blood concentration. With an i.v. injection the concentration at time t.min is equal to the amount injected into the vein. A compartment model has a volume and a concentration of a substance. Here the volume is equal to the dose (total quantity of drug) injected divided by the measured concentration in the blood stream. The change in the quantity of drug in the compartment is described by mass balance equations. The apparent volume of distribution is designated as V, the concentration as C, and the amount of material or drug as Q. The change in concentration, dQ/dt is governed by sources (which add material to Q) and sinks which subtract material from Q. A source will be a positive quantity. A sink will be a negative quantity. The change in Q can be written as: dQ/dt = d(VC)/dt = CdV/dt+VdC/dt. Assuming V is constant, dQ/dt = VdC/dt. The ODE equation describing the first order decay process is given as: VdC/dt = -ClearanceC, where Clearance is the amount of drug removed from the compartment. This equation is usually rewritten as: dC/dt = -(Kelim)*C, after dividing both sides by the volume, where Kelim (Elimination rate const) = Clearance/V The term on the right hand side is a sink term. It is negative and removes material from the compartment. A note on V, the apparent volume of distribution: This is not a physiological volume but rather a ratio that measures the extent of drug distribution within a compartment. If the compartment represented blood volume and the drug distributed only in the plasma and not the red blood cells (RBCs), then the volume of distribution for the drug would be less than the volume of distribution of another drug that distributed into the RBCs as well as plasma. The volume of distribution is drug dependent. The equations for this model may be viewed by running the JSim model applet and clicking on the Source tab at the bottom left of JSim's Run Time graphical user interface. The equations are written in JSim's Mathematical Modeling Language (MML). See the Introduction to MML and the MML Reference Manual. Additional documentation for MML can be found by using the search option at the Physiome home Download JSim model project file • Download JSim model MML code (text): • Download translated SBML version of model (if available): Model Feedback We welcome comments and feedback for this model. Please use the button below to send comments: Niazi S., Textbook of Biopharmaceutics and Clinical Pharmacokinetics, Appleton-Century-Crofts, NewYork, 1979, ISBN 0-8385-8868-9 Rosenbaum S.E., Basic Pharmacokinetics and Pharmacodynamics: An Integrated Textbook and Computer Simulations, Wiley, Hoboken NJ, 2011, ISBN: 978-0-470-56906-1 Key terms first order process Please cite https://www.imagwiki.nibib.nih.gov/physiome in any publication for which this software is used and send one reprint to the address given below: The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061. Model development and archiving support at https://www.imagwiki.nibib.nih.gov/physiome provided by the following grants: NIH U01HL122199 Analyzing the Cardiac Power Grid, 09/15/2015 - 05/31/2020, NIH /NIBIB BE08407 Software Integration, JSim and SBW 6/1/09-5/31/13; NIH/NHLBI T15 HL88516-01 Modeling for Heart, Lung and Blood: From Cell to Organ, 4/1/07-3/31/11; NSF BES-0506477 Adaptive Multi-Scale Model Simulation, 8/15/05-7/31/08; NIH/NHLBI R01 HL073598 Core 3: 3D Imaging and Computer Modeling of the Respiratory Tract, 9/1/04-8/31/09; as well as prior support from NIH/NCRR P41 RR01243 Simulation Resource in Circulatory Mass Transport and Exchange, 12/1/1980-11/30/01 and NIH/NIBIB R01 EB001973 JSim: A Simulation Analysis Platform, 3/1/02-2/28/07.	{"url":"https://www.imagwiki.nibib.nih.gov/physiome/jsim/models/webmodel/NSR/pkcomp1decay","timestamp":"2024-11-09T09:58:02Z","content_type":"text/html","content_length":"61877","record_id":"<urn:uuid:1e5e4cbe-7992-483a-95cc-fbf139bc4708>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00890.warc.gz"}
Sensitivity and specificity: Video, Causes, & Meaning \| Osmosis Let’s say a new screening test is developed to figure out if people have diabetes before they start showing symptoms. Before using the test, we have to make sure that the test works - in other words, can the test correctly identify if a person has diabetes or not? This is the test’s validity, and it has two components - sensitivity and specificity. A test with high sensitivity will correctly identify most people who have the condition, and a test with high specificity will correctly identify most people who don’t have the disease. So let’s say that we recruit a 1000 people - 100 people who have diabetes and 900 people who don’t to put our diabetes test to the test! We can organize the results using a 2 by 2 table, where the true disease status, positive or negative, of the individual is on the top of the box and the results of the screening test, positive or negative, are on the side, and each of the cells is labeled a, b, c, or d. In this situation, a positive test indicates that a person has diabetes. So let’s look at this table closer, a person who gets a positive test result and has positive disease status, so has diabetes, is called a true positive. A person who gets a negative test result and a negative disease status, so doesn’t have diabetes, would be a true negative. A person who gets a positive test result even though they don’t have diabetes, would be a false positive. And lastly a person who gets a negative test result even though they have diabetes, would be a false negative. To calculate sensitivity, we divide the number of true positives by the total number of people who have diabetes - so cell a divided by the sum of cell a and cell c. A test with perfect sensitivity would have 100 true positives in cell a, because the test would correctly identify everyone who has diabetes, and zero false negatives in cell c. To calculate specificity, we divide the number of true negatives by the total number of people who do not have diabetes - so cell d divided by the sum of cell d and cell b. A test with perfect specificity would have 900 true negatives in cell d, because the test would correctly identify everyone who doesn’t have diabetes, and zero false positives, in cell b. But no test is 100% perfect, so let’s say that cell a contains 80 true positives, cell b contains 100 false positives, cell c contains 20 false negatives, and cell d contains 800 true negatives.	{"url":"https://www.osmosis.org/learn/Sensitivity_and_specificity?from=/md/clerkships/internal-medicine/back-to-the-basic-sciences/clinical-conditions/cancer-screening","timestamp":"2024-11-09T23:45:18Z","content_type":"text/html","content_length":"356278","record_id":"<urn:uuid:88a9c112-1e35-440f-b78a-24728af437d9>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00310.warc.gz"}
4 Digit By 2 Digit Multiplication Worksheets Math, specifically multiplication, develops the keystone of countless academic techniques and real-world applications. Yet, for several students, grasping multiplication can present an obstacle. To resolve this hurdle, instructors and parents have embraced an effective tool: 4 Digit By 2 Digit Multiplication Worksheets. 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Encouraging constant method, giving assistance, and producing a positive knowing setting are useful steps.	{"url":"https://crown-darts.com/en/4-digit-by-2-digit-multiplication-worksheets.html","timestamp":"2024-11-13T21:54:23Z","content_type":"text/html","content_length":"28510","record_id":"<urn:uuid:fb77754b-05d3-4f43-8920-d2baaea1ae17>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00781.warc.gz"}