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Start your quantum computing journey with Qiskit Quantum computing is an emerging technology that uses the laws of quantum mechanics to solve complex problems which will replace the traditional approach of solving using classical computers. Unlike traditional computer logics of using bits-and-bytes, on-or-off, true-or-false scenarios, quantum computing makes use of bits called quantum bits or qubits which represents a one, a zero, or both at once. This superposition property makes these qubits to behave in a way which can’t be explained by the individual components. This behavior of qubits is called entanglement. We make use of entanglement and superposition for applications involving complex calculations. Qiskit is an open-source SDK developed by IBM for working with quantum computers. It gears up the development of quantum applications by providing the complete set of tools at the level of circuits, pulses, and algorithms that are required for interacting with quantum systems and simulators. Getting Started In this session, we will learn on defining a simple quantum circuit and executing it on both simulators and real quantum computers of the IBM Quantum Experience. Let’s explore quantum computing with Qiskit SDK. Install and import dependencies Let’s install qiskit package via pip pip install qiskit pip install pylatexenc If you are using Google Colab, !pip install qiskit !pip install pylatexenc Now, we can import the dependencies to the project. from qiskit import * from qiskit.visualization import * from qiskit.tools.monitor import * Defining the circuit We can define a simple circuit using the H gate to put a qubit in superposition after which we will measure the state of the circuit. # Create a circuit to generate a superposition state circ = QuantumCircuit(1,1) circ.h(0) # We apply the H gate circ.draw() # Draw the circuit """We can also obtain the qasm code for the circuit.""" Running the circuit on simulators We can execute the circuit on a simulator once we define it. # Executing on the local simulator backend_sim = Aer.get_backend('qasm_simulator') # Choose the backend job_sim = execute(circ, backend_sim, shots=1024) # Execute the circuit, selecting the number of repetitions or 'shots' result_sim = job_sim.result() # Collect the results counts = result_sim.get_counts(circ) # Obtain the frequency of each result We can execute the circuit on a simulator to determine the final state. # Execution to the get the state vectorcirc2 = QuantumCircuit(1,1) backend = Aer.get_backend('statevector_simulator') # Change the backend job = execute(circ2, backend) # Execute the circuit on a simulator. Now, we do not need repetitions result = job.result() # Collect the results and access the statevector outputstate = result.get_statevector(circ2) We can obtain the unitary matrix that represents the action of the circuit by following the below code. backend = Aer.get_backend('unitary_simulator') # Change the backend job = execute(circ2, backend) # Execute the circuit result = job.result() # Collect the results and obtain the matrix unitary = result.get_unitary() IBMQ integration Create a IBM Quantum account if you don’t have one to utilize quantum computers at the IBM Quantum Experience. Also, create the API token that is available on the dashboard and using the created API token as the enable_account() method argument connect your project with IBMQ instance. Now, we can use the quantum computers at the IBM Quantum Experience to execute the circuit. # Connecting to the real quantum computers from qiskit import IBMQ provider = IBMQ.enable_account("your-ibmq-api-key") # Load account provider.backends() # Retrieve the backends to check its status for b in provider.backends(): We can execute the circuit on IBM’s quantum simulator (which supports up to 32 qubits). The only requirement is to select the appropriate backend. # Executing on the IBM Q Experience simulator backend_sim = provider.get_backend('ibmq_qasm_simulator') job_sim = execute(circ, backend_sim, shots=1024) # Execute the circuit, selecting the number of repetitions or 'shots' result_sim = job_sim.result() # Collect the results counts = result_sim.get_counts(circ) # Obtain the frequency of each result We can make use of job_monitor to get live job status information. # Executing on the quantum computer backend = provider.get_backend('ibmq_armonk') job_exp = execute(circ, backend=backend) We can compare the results from real quantum computers with the results obtained from the simulator once the job is done. result_exp = job_exp.result() counts_exp = result_exp.get_counts(circ) plot_histogram([counts_exp,counts], legend=['Device', 'Simulator']) There you have it! Your first quantum computing project using Qiskit in python :) Thanks for reading this article. The source code of the project in this article is available on To get the article in pdf format: Quantum-computing.pdf The article is also available on Medium If you enjoyed this article, please click on the heart button ♥ and share to help others find it! Top comments (0) For further actions, you may consider blocking this person and/or reporting abuse	{"url":"https://dev.to/codemaker2015/start-your-quantum-computing-journey-with-qiskit-1jhn","timestamp":"2024-11-07T10:00:00Z","content_type":"text/html","content_length":"83876","record_id":"<urn:uuid:4436dbfe-4f3d-47b2-8c8d-fc54bdb533a0>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00648.warc.gz"}
Normal Modes Last updated Page ID $ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $ $ \newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ ( \newcommand{\kernel}{\mathrm{null}\,}\) $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\\| #1 \\|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\\| #1 \\|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\AA}{\unicode[.8,0]{x212B}}$ $ \newcommand{\vectorA}[1]{\vec{#1}} % arrow$ $ \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$ $ \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vectorC}[1]{\textbf{#1}} $ $ \newcommand{\vectorD}[1]{\overrightarrow{#1}} $ $ \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} $ $ \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} $ $ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $ $\newcommand{\avec}{\mathbf a}$ $\newcommand{\bvec}{\mathbf b}$ $\newcommand{\cvec}{\mathbf c}$ $\newcommand{\dvec}{\mathbf d}$ $\newcommand{\dtil}{\widetilde{\mathbf d}}$ $\newcommand{\ evec}{\mathbf e}$ $\newcommand{\fvec}{\mathbf f}$ $\newcommand{\nvec}{\mathbf n}$ $\newcommand{\pvec}{\mathbf p}$ $\newcommand{\qvec}{\mathbf q}$ $\newcommand{\svec}{\mathbf s}$ $\ newcommand{\tvec}{\mathbf t}$ $\newcommand{\uvec}{\mathbf u}$ $\newcommand{\vvec}{\mathbf v}$ $\newcommand{\wvec}{\mathbf w}$ $\newcommand{\xvec}{\mathbf x}$ $\newcommand{\yvec}{\mathbf y} $ $\newcommand{\zvec}{\mathbf z}$ $\newcommand{\rvec}{\mathbf r}$ $\newcommand{\mvec}{\mathbf m}$ $\newcommand{\zerovec}{\mathbf 0}$ $\newcommand{\onevec}{\mathbf 1}$ $\newcommand{\real} {\mathbb R}$ $\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$ $\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$ $\newcommand{\threevec} [3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\fourvec}[4]{\left[\begin{array} {r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$ $\newcommand{\laspan}[1]{\text{Span}\{#1\}}$ $\newcommand{\bcal}{\cal B}$ $\newcommand{\ccal}{\cal C}$ $\newcommand{\scal}{\cal S}$ $\newcommand{\ wcal}{\cal W}$ $\newcommand{\ecal}{\cal E}$ $\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$ $\newcommand{\gray}[1]{\color{gray}{#1}}$ $\newcommand{\lgray}[1]{\color{lightgray}{#1}}$ $\ newcommand{\rank}{\operatorname{rank}}$ $\newcommand{\row}{\text{Row}}$ $\newcommand{\col}{\text{Col}}$ $\renewcommand{\row}{\text{Row}}$ $\newcommand{\nul}{\text{Nul}}$ $\newcommand{\var} {\text{Var}}$ $\newcommand{\corr}{\text{corr}}$ $\newcommand{\len}[1]{\left\|#1\right\|}$ $\newcommand{\bbar}{\overline{\bvec}}$ $\newcommand{\bhat}{\widehat{\bvec}}$ $\newcommand{\bperp}{\ bvec^\perp}$ $\newcommand{\xhat}{\widehat{\xvec}}$ $\newcommand{\vhat}{\widehat{\vvec}}$ $\newcommand{\uhat}{\widehat{\uvec}}$ $\newcommand{\what}{\widehat{\wvec}}$ $\newcommand{\Sighat}{\ widehat{\Sigma}}$ $\newcommand{\lt}{<}$ $\newcommand{\gt}{>}$ $\newcommand{\amp}{&}$ $\definecolor{fillinmathshade}{gray}{0.9}$ Normal modes are used to describe the different vibrational motions in molecules. Each mode can be characterized by a different type of motion and each mode has a certain symmetry associated with it. Group theory is a useful tool in order to determine what symmetries the normal modes contain and predict if these modes are IR and/or Raman active. Consequently, IR and Raman spectroscopy is often used for vibrational spectra. Overview of Normal Modes In general, a normal mode is an independent motion of atoms in a molecule that occurs without causing movement to any of the other modes. Normal modes, as implied by their name, are orthogonal to each other. In order to discuss the quantum-mechanical equations that govern molecular vibrations it is convenient to convert Cartesian coordinates into so called normal coordinates. Vibrations in polyatomic molecules are represented by these normal coordinates. There exists an important fact about normal coordinates. Each of these coordinates belongs to an irreducible representation of the point the molecule under investigation. Vibrational wavefunctions associated with vibrational energy levels share this property as well. The normal coordinates and the vibration wavefunction can be categorized further according to the point group they belong to. From the character table predictions can be made for which symmetries can exist. The irreducible representation offers insight into the IR and/or Raman activity of the molecule in question. Degrees of Freedom 3N where N represents the number of nuclei present in the molecule is the total number of coordinates needed to describe the location of a molecule in 3D-space. 3N is most often referred to as the total number of degrees of freedom of the molecule being investigated. The total number of degrees of freedom, can be divided into: • 3 coordinates to describe the translational motion around the center of mass; these coordinates are called the translational degrees of freedom • 3 coordinates to describe the rotational motion in non-linear molecules; for linear molecules only 2 coordinates are required; these coordinates are called the rotational degrees of freedom • the remaining coordinates are used to describe vibrational motion; a non-linear molecule has 3N - 6 vibrational degrees of freedom whereas a linear molecule has 3N -5 degrees of freedom. Table 1: Overview of degrees of freedom │ │Total Degree of Freedom│Translational degrees of freedom │Rotational degrees of freedom│Vibrational degrees of freedom│ │Nonlinear Molecules│3N │3 │3 │3N -6 │ │Linear Molecules │3N │3 │2 │3N - 5 │ Ethane, $C_2H_6$ has eight atoms (\N=8\) and is a nonlinear molecule so of the $3N=24$ degrees of freedom, three are translational and three are rotational. The remaining 18 degrees of freedom are internal (vibrational). This is consistent with: \[3N -6 =3(8)-6=18\] Carbon Dioxide, $CO_2$ has three atoms ($N=3$ and is a linear molecule so of the $3N=9$ degrees of freedom, three are translational and two are rotational. The remaining 4 degrees of freedom are vibrational. This is consistent with: \[3N - 5 = 3(3)-5 = 4\] Mathematical Introduction to Normal Modes If there is no external field present, the energy of a molecule does not depend on its orientation in space (its translational degrees of freedom) nor its center of mass (its rotational degrees of freedom). The potential energy of the molecule is therefore made up of its vibrational degrees of freedom only of $3N-6$ (or $3N-5$ for linear molecules). The difference in potential energy is given by: \[ \Delta V &= V(q_1,q_2,q_3,...,q_n) - V(0,0,0,...,0) \tag{1} \\[4pt] &= \dfrac{1}{2} \sum_{i=1}^{N_{vib}} \sum_{j=1}^{N_{vib}} \left(\dfrac{\partial^2 V}{\partial q_i\partial q_j} \right) q_iq_j \ tag{2} \\[4pt] &= \dfrac{1}{2}\sum_{i=1}^{N_{vib}} \sum_{j=1}^{N_{vib}} f_{ij} q_iq_j \tag{3} \] • $q$ represents the equilibrium displacement and • $N_{vib}$ the number of vibrational degrees of freedom. For simplicity, the anharmonic terms are neglected in this equation (consequently there are no higher order terms present). A theorem of classical mechanics states that the cross terms can be eliminated from the above equation (the details of the theorem are very complex and will not be discussed in detail). By using matrix algebra a new set of coordinates {Q[j]} can be found such that \[\Delta{V} = \dfrac{1}{2} \sum_{j=1}^{N_{vib}}{F_jQ_j^2} \tag{4}\] Note that there are no cross terms in this new expression. These new coordinates are called normal coordinates or normal modes. With these new normal coordinates in hand, the Hamiltonian operator for vibrations can be written as follows: \[\hat{H}_{vib} = -\sum_{j=1}^{N_{vib}} \dfrac{\hbar^2}{2\mu_i} \dfrac{d^2}{dQ_j^2} + \dfrac{1}{2} \sum_{j=1}^{N_{vib}}F_jQ_j^2 \tag{5}\] The total wavefunction is a product of the individual wavefunctions and the energy is the sum of independent energies. This leads to: \[ \hat{H}_{vib} = \sum_{j=1}^{N_{vib}} \hat{H}_{vib,j} = \sum_{j=1}^{N_{vib}} \left( \dfrac{-\hbar^2}{2 \mu_j}\dfrac{d^2}{dQ_i^2} + \dfrac{1}{2}\sum_{j=1}^{N_{vib}} F_jQ_j^2 \right) \tag{6}\] and the wavefunction is then \[ \psi_{vib} = Q_1,Q_2, Q_3 ..., Q_{vib} = \psi_{vib,1}(Q_1) \psi_{vib,2}(Q_2) \psi_{vib,3}(Q_3) , ..., \psi_{vib,N_{vib}}(Q_{N_{vib}}) \tag{7}\] and the total vibrational energy of the molecule is \[E_{vib} = \sum_{j=1}^{N_{vin}} h\nu_j \left (v_j + \dfrac{1}{2}\right) \tag{8}\] where $v_j= 0,1,2,3...$ The consequence of the result stated in the above equations is that each vibrational mode can be treated as a harmonic oscillator approximation. There are $N_{vib}$ harmonic oscillators corresponding to the total number of vibrational modes present in the molecule. In the ground vibrational state the energy of the molecule is equal to (1/2)hν[j]. The ground state energy is referred to as zero point energy. A vibration transition in a molecule is induced when it absorbs a quantum of energy according to E = hv. The first excited state is separated from the ground state by E[vib] = (3/2)hν since v[j] = 1, the next energy level separation is (5/2)hν, etc... The harmonic oscillator is a good approximation, but it does not take into account that the molecule, once it has absorbed enough energy to break the vibrating bond, does dissociate. A better approximation is the Morse potential which takes into account anharmonicity. The Morse potential also accounts for bond dissociation as well as energy levels getting closer together at higher Pictorial description of normal coordinates using CO The normal coordinate q is used to follow the path of a normal mode of vibration. As shown in Figure 2 the displacement of the C atom, denoted by Δr[o](C), and the displacement of the O atom, denoted by Δr[o](O), occur at the same frequency. The displacement of atoms is measured from the equilibrium distance in ground vibrational state, r[o]. Figure 2: The Normal coordinate for $CO$ is equation to $\Delta r(C) + \Delta r(O)$ Description of vibrations • ν = stretching is a change in bond length; note that the number of stretching modes is equal to the number of bonds on the molecule • δ = bending is a change in bond angle • ρ[r] = rocking is change in angle between a group of atoms • ρ[w] = wagging is change in angle between the plane of a group of atoms • ρ[t] = twisting is change in angle between the planes of two groups of atoms • π= out of plane In direct correlation with symmetry, subscripts s (symmetric), as (asymmetric) and d (degenerate) are used to further describe the different modes. A normal mode corresponding to an asymmetric stretch can be best described by a harmonic oscillator: As one bond lengthens, the other bond shortens. A normal mode that corresponds can be best described by a Morse potential well: As the bond length increases the potential energy increases and levels off as the bond length gets further away from the equilibrium. The use of Symmetry and Group Theory Symmetry of normal modes It is important to realize that every normal mode has a certain type of symmetry associated with it. Identifying the point group of the molecule is therefore an important step. With this in mind it is not surprising that every normal mode forms a basis set for an irreducible representation of the point group the molecule belongs to. For a molecule such as water, having a structure of XY[2], three normal coordinates can be determined. The two stretching modes are equivalent in symmetry and energy. The figure below shows the three normal modes for the water molecule: Figure 3: Three normal modes of water By convention, with nonlinear molecules, the symmetric stretch is denoted v[1] whereas the asymmetric stretch is denoted v[2]. Bending motions are v[3]. With linear molecules, the bending motion is v [2] whereas asymmetric stretch is v[3]. The water molecule has C[2v] symmetry and its symmetry elements are E, C[2], σ(xz) and σ(yz). In order to determine the symmetries of the three vibrations and how they each transform, symmetry operations will be performed. As an example, performing C[2] operations using the two normal mode v[2] and v[3] gives the following transformation: Once all the symmetry operations have been performed in a systematic manner for each modes the symmetry can be assigned to the normal mode using the character table for C[2v]: Table 2: Character table for the C [2v] point group │C2v │E│C2│σ (xz) │σ (yz) │ │ │ν[1]│1│1 │1 │1 │= a[1] │ │ν[2]│1│1 │1 │1 │= a[1] │ │ν[3]│1│-1│-1 │1 │= b[2] │ Water has three normal modes that can be grouped together as the reducible representation \[Γ_{vib}= 2a_1 + b_2.\] Determination of normal modes becomes quite complex as the number of atoms in the molecule increases. Nowadays, computer programs that simulate molecular vibrations can be used to perform these The example of [PtCl[4]]^2- shows the increasing complexity. The molecule has five atoms and therefore 15 degrees of freedom, 9 of these are vibrational degrees of freedom. The nine normal modes are exemplified below along with the irreducible representation the normal mode belongs to (D[4h] point group). A[1g], b[1g] and e[u] are stretching vibrations whereas b[2g], a[2u], b[2u] and e[u] are bending vibrations. Determining if normal modes are IR and/or Raman active Transition Moment Integral A transition from $\ce{v -> v'}$ is IR active if the transition moment integral contains the totally symmetric irreducible representation of the point group the molecule belongs to. The transition moment integral is derived from the one-dimensional harmonic oscillator. Using the definition of dipole moment the integral is: \[M\left(v \rightarrow v^{\prime}\right)=\int_{-\infty}^{\infty} \psi^{}\left(v^{\prime}\right) \mu \psi(v) d x\] If μ, the dipole moment, would be a constant and therefore independent of the vibration, it could be taken outside the integral. Since v and v' are mutually orthogonal to each other, the integral will equal zero and the transition will not be allowed. In order for the integral to be nonzero, μ must change during a vibration. This selection rule explains why homonuclear diatomic molecules do not produce an IR spectrum. There is no change in dipole moment resulting in a transition moment integral of zero and a transition that is forbidden. For a transition to be Raman active the same rules apply. The transition moment integral must contain the totally symmetric irreducible representation of the point group. The integral contains the polarizability tensor (usually represented by a square matrix): \[M\left(v \rightarrow v^{\prime}\right)=\int_{-\infty}^{\infty} \psi^{}\left(v^{\prime}\right) \alpha \psi(v) d x\] $α$ must be nonzero in order for the transition to be allowed and show Raman scattering. Character Tables For a molecule to be IR active the dipole moment has to change during the vibration. For a molecule to be Raman active the polarizability of the molecule has to change during the vibration. The reducible representation Γ[vib][ ]can also be found by determining the reducible representation of the 3N degrees of freedom of H[2]O, Γ[tot]. By applying Group Theory it is straightforward to find Γ [x][,y,z] as well as UMA (number of unmoved atoms). Again, using water as an example with C[2v] symmetry where 3N = 9, Γ[tot] can be determined: │ C2v │E│C2│ σ (xz) │ σ (yz) │ │ │Τ[x][,y,z] │3│-1│1 │1 │ │ │UMA │3│1 │1 │3 │ │ │Γ[tot] │9│-1│1 │3 │=3a[1] + a[2] + 2b[1] + 3b[2] │ │Note that Γ[tot] contains nine degrees of freedom consistent with 3N = 9. │ Γ[tot] contains Γ[translational], Γ[rotational] as well as Γ[vibrational]. Γ[trans] can be obtained by finding the irreducible representations corresponding to x,y and z in the right side of the character table, Γ[rot] by finding the ones corresponding to R[x], R[y] and R[z]. Γ[vib] can be obtained by Γ[tot] - Γ[trans] - Γ[rot]. Γ[vib] (H[2]O) = (3a[1] + a[2] + 2b[1]+ 3b[2]) - (a[1] + b[1] + b[2]) - (a[2] + b[1] + b[2]) = 2a[1] + b[2] In order to determine which modes are IR active, a simple check of the irreducible representation that corresponds to x,y and z and a cross check with the reducible representation Γ[vib] is necessary. If they contain the same irreducible representation, the mode is IR active. For H[2]O, z transforms as a[1], x as b[1] and y as b[2]. The modes a1 and b2 are IR active since Γvib contains 2a[1] + b[2]. In order to determine which modes are Raman active, the irreducible representation that corresponds to z^2, x^2-y^2, xy, xz and yz is used and again cross checked with Γvib. For H[2]O, z^2 and x^2-y^ 2 transform as a[1], xy as a[2], xz as b[1] and yz as b[2].The modes a[1] and b[2] are also Raman active since Γ[vib] contains both these modes. The IR spectrum of H2O does indeed have three bands as predicted by Group Theory. The two symmetric stretches v1 and v2 occur at 3756 and 3657 cm-1 whereas the bending v3 motion occurs at 1595 cm-1. In order to determine which normal modes are stretching vibrations and which one are bending vibrations, a stretching analysis can be performed. Then the stretching vibrations can be deducted from the total vibrations in order to obtain the bending vibrations. A double-headed arrow is drawn between the atom as depicted below: Then a determination of how the arrows transform under each symmetry operation in C2v symmetry will yield the following results: │ C2v │E│C[2]│σ (xz)│σ (yz)│ │ │Γ[stretch] │2│0 │0 │2 │= a[1] + b[2] │ Γ[bend] = Γ[vib] - Γ[stretch] = 2a[1] + b[2 ]-a[1] - b[2] = a[1] H[2]O has two stretching vibrations as well as one bending vibration. This concept can be expanded to complex molecules such as PtCl4^-. Four double headed arrows can be drawn between the atoms of the molecule and determine how these transform in D[4h] symmetry. Once the irreducible representation for Γ[stretch] has been worked out, Γ[bend] can be determined by Γ[bend] = Γ[vib] - Γ[stretch]. Fundamental transition, overtones and hot bands The transition from v=0 (ground state) -> v=1 (first excited state) is called the fundamental transition. This transition has the greatest intensity. The transition from v=0 --> v=2 is is referred to as the first overtone, from v=0 --> v=3 is called the second overtone, etc. Ovetones occur when a mode is excited above the v = 1 level. The harmonic oscillator approximation supports the prediction that the transition to a second overtone will be twice as energetic as a fundamental transition. Most molecules are in their zero point energy at room temperature. Therefore, most transitions do originate from the v=0 state. Some molecules do have a significant population of the v=1 state at room temperature and transitions from this thermally excited state are called hot bands. Combination bands can occur if more than one vibration is excited by the absorption of a photon. The overall energy of a combination band is the result of the sum of individual transitions. 1. Merlin, J.C., Cornard, J.P., J. Chem. Educ., 2006, 83 (9), p 1383. DOI: 10.1021/ed083p1393 2. McGuinn, C.J., J. Chem. Educ., 1982, 59 (10), p 813. DOI: 10.1021/ed059p813 3. Harris, D.C., Bertolucci, M.D., Symmetry and Spectroscopy: An introduction to Vibrational and Electronic Spectroscopy. Dover Publocations, Inc., New York, 1989. 4. McQuarrie, D. A., Simon, J.D., Physical Chemistry: A Molecular Approach, University Science Books, Sausalito, California, 1997; 518-521. 5. Housecroft, C.E., Sharpe, A.G., Inorganic Chemistry. Pearson Education Limited, England, 2008, 107. 6. Atkins, P., dePaula, J., Physical Chemistry, W.H. Freeman and Company, New York, 2002, 520-523. 7. Bishop, D.M., Group Theory and Chemistry, Dover Publications, Inc, New York, 1973, 166. 1. Chlorophyll a is a green pigment that is found in plants. Its molecular formula is C[55]H[77]O[5]N[4]Mg. How many degrees of freedom does this molecule possess? How many vibrational degrees of freedom does it have? 2. CCl[4] was commonly used an as organic solvent until its severe carcinogenic properties were discovered. How many vibrational modes does CCl[4] have? Are they IR and/or Raman active? 3. The same vibrational modes in H[2]O are IR and Raman active. WF[6]^- has IR active modes that are not Raman active and vice versa. Explain why this is the case. 4. How many IR peaks do you expect from SO[3]? Estimate where these peaks are positioned in an IR spectrum. 5. Calculate the symmetries of the normal coordinates of planar BF[3]. Answers to Problems 1. chlorophyll has 426 degrees of freedom, 420 vibrational modes 2. The point group is T[d], T[vib] = a[1] + e + 2t[2], a[1] and e are Raman active, t[2] is both IR and Raman active 3. For molecules that possess a center of inversion i, modes cannot be simultaneously IR and Raman active 4. Point group is D[3h][,] one would expect three IR active peaks. Asymmetric stretch highest (1391 cm^-1), two bending modes (both around 500 cm^-1). The symmetric stretch is IR inactive 5. T[3N] = A[1]' + A[2]' + 3E' + 2 A[2]" + E" and T[vib]= A[1]' + 2E' + A[2]"	{"url":"https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Spectroscopy/Vibrational_Spectroscopy/Vibrational_Modes/Normal_Modes","timestamp":"2024-11-02T12:47:24Z","content_type":"text/html","content_length":"163775","record_id":"<urn:uuid:3a7cff00-c670-4be9-85b6-86d5a5feb9ed>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00342.warc.gz"}
A Drunkard’s Walk in Manhattan \| Quanta Magazine A Drunkard’s Walk in Manhattan Olena Shmahalo/Quanta Magazine; Processing code by Masatake Hirao One of the most cherished mathematical learning moments of my youth came from an old and very funny math book, whose name I have sadly forgotten. It was through a cartoon in that book that I first learned how to figure out the distance covered by a completely random form of movement — the drunkard’s walk. The first panel of the cartoon showed a disheveled man near a lamppost. His future path was represented by a series of wild zigs and zags on the path in front of him, shown as a dotted line. “I know how to figure out how far I’ll be from this point on average,” he says. “All I have to do is to measure the average length of my zigs and zags, and multiply by the square root of their number.” Then, in the second panel, he pulls out a bottle from his coat pocket, lifts it toward his mouth, and says, “But first, I’ll have a little drink!” Now, where are the funny math books today? Randomness is an inextricable and essential aspect of our world. Combined with selection, randomness can do incredible things: It has powered evolution and created the entire biological world. Yet randomness is commonly underestimated and misunderstood. Certain observed phenomena prompt many people to attribute magical causes to events and imbue people with supernatural abilities, when the workings of randomness are all we need to explain their observations. Of course, probability theorists have always known that randomness, to a large extent, rules our lives, as the author Leonard Mlodinow explains in his delightful book The Drunkard’s Walk. Recently, researchers have penetrated deeper into the intricacies of randomness, as Kevin Hartnett reports in the Quanta article “A Unified Theory of Randomness.” Hartnett’s article explains how this unified theory of randomness is informed by variants of the same random phenomenon we alluded to earlier: the random walk, or, as it is more colorfully named, the drunkard’s walk. This phenomenon explains the diffusion of fluids and also describes Brownian motion, which Einstein famously analyzed to determine the existence and size of atoms. But back to our drunkard. When an object or a person moves randomly, the average distance it will be from its starting point can be predicted to be approximately $x \sqrt n$, where x is the average length of each step and n is the number of steps. The more the drunkard walks, the farther he gets from his starting point. Why should this be when his steps are random? Here’s a very nice informal argument presented by Marty Green, that helps to understand this result for equal-size steps. In the figure at left, which has been reproduced from Green’s blog, let us suppose that the drunkard started from the lamppost (center of the circle), took several one-meter steps, and by chance found himself on the circumference of the circle, say, five meters away. After the next step, he will be on the circumference of the smaller circle of radius one meter centered on the point where he had been. More than half of this circle (the green part) is outside the five-meter circle. So the next step will be more likely to take him farther away from the lamppost rather than closer, and this is true no matter where he is at a given time. How much farther will he go? In order to determine the new average distance we will have to integrate all his possible distances around the circumference of the circle. Some points on it are closer to the origin than before, and a larger number are farther. The most “neutral” step he could take is perpendicular to the big circle’s radius: one meter along the tangent. Coincidentally, considering just this one neutral direction gives the right answer. One meter along the tangent would place him, by Pythagoras’ theorem, $\sqrt 26$ meters away. Since his previous location at a distance of five meters is the square root of 25, we see that this little trick gives us exactly the answer that we sought: The mean distance is proportional to the square root of the number of steps. Our first problem applies this technique to a drunkard taking a walk in a city laid out like a grid, such as Manhattan. The second problem requires you to settle a bet between two friends walking in the same city. 1. Imagine that the drunkard is in the middle of a city laid out like a grid, with numbered streets running east to west and numbered avenues running north to south. The lamppost is at the corner of 5th Avenue and 5th Street, which we will designate as (5,5). The drunkard is sober enough to navigate a whole block before making a random choice at the next intersection. Let us say that after several such random block-length walks, he is now at (8,8). If we try to apply the circle argument shown above, we find that he can proceed to (9,8) or (8,9) or (8,7) or (7,8). Two of these are closer to, and two are farther from, his starting point. Does this mean that the calculation for the drunkard’s walk doesn’t work on a rectangular grid? What’s wrong with this argument? Once you’ve found the fallacy, can you derive the analogous distance formula for the drunkard’s walk on a rectangular grid using city block units? 2. Assume that the above city has a 0th Street and 0th Avenue (it was designed by computer scientists!). Two coin-collector friends, Sally and Al, arrive in the city by subway, each carrying a stack of silver dollars. They emerge at the corner of 6th Avenue and 5th Street and play the following game. At each intersection they toss two coins to determine randomly which direction they should walk next, out of the four choices available to them. Sally’s goal is to reach 0th Street or 8th Street. Al’s target is 0th Avenue or 8th Avenue. If the random direction chosen gets either Al or Sally closer to their respective targets, the other person pays him or her a dollar. Conversely, if either person’s distance from his or her target increases, that person has to give the other person a dollar. If the random direction is neutral with respect to the distance from either target, no money is exchanged. The game ends when one of them reaches his or her target. Notice that at the beginning, Al is closer to 8th Avenue than Sally is to 8th Street, but if he reaches it directly with two lucky westward walks, he will win only $2. Sally is farther from her targets but can potentially win more by the end of the game, because she can improve her position more times. Whom does this game favor in the long run? There are many ways to solve the second problem. Try to do it using pen and paper if you can, and only use modern aids if you cannot make progress the old-fashioned way. I have given two general hints below. (Click on “Hint 1” and “Hint 2” to make them visible.) Hint 1 Use the idea of symmetry and mirror-image intersections. What would the probabilities be when the street and avenue numbers are equal? Hint 2 How is the probability at a given intersection related to the probabilities at the four adjacent intersections? For those who want to explore random walks further, many questions present themselves. What is the average number of blocks the friends will have to walk before one of them reaches his or her target? This is possible to figure out on paper, but feel free to uncork more powerful tools: simulations in a spreadsheet, online tools like Wolfram Alpha, programming using math software — whatever you prefer. Hartnett explains in his article how there are some types of random walks in which it is forbidden to cross your own path. How does that affect the number of blocks walked, and the random walk formula? What other mathematical techniques can be applied, and how does this relate to diffusion? I am sure Quanta readers will find many other avenues (and streets!) to explore. As for me, I think I’ll have that drink now. Editor’s note: The reader who submits the most interesting, creative or insightful solution (as judged by the columnist) in the comments section will receive a Quanta Magazine T-shirt. And if you’d like to suggest a favorite puzzle for a future Insights column, submit it as a comment below, clearly marked “NEW PUZZLE SUGGESTION” (it will not appear online, so solutions to the puzzle above should be submitted separately). Note that we may hold comments for the first day or two to allow for independent contributions by readers. Correction: This column was revised on Sept. 5, 2016, to reflect that Al would be walking westward from 6th Avenue to 8th Avenue. Update: The solution has been published here.	{"url":"https://www.quantamagazine.org/a-drunkards-walk-in-manhattan-20160818/","timestamp":"2024-11-05T23:29:45Z","content_type":"text/html","content_length":"203747","record_id":"<urn:uuid:180a7806-2a0b-4a6f-82f7-0c3b0f81c465>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00211.warc.gz"}
30th Marian Smoluchowski Symposium on Statistical Physics Aydin Deger (Department of Applied Physics, Aalto University, Finland) Originally introduced to explain the behavior of a condensing gas, Lee-Yang zeros have nowadays become a universal and powerful tool for the unified description of phase transitions in equilibrium and non-equilibrium systems, see for example [1, 2]. Here, we use Lee-Yang zeros to analyze a paradigmatic model for thermal phase transitions in molecular systems. For the most simple version of this model, we explicitly calculate the Lee- Yang zeros with respect to inverse temperature. Extrapolation then allows us to infer a phase transition in the macroscopic limit, from the analysis of systems containing only a few molecular units. In a second step, we in- crease the complexity of the model. The Lee-Yang zeros can still be obtained using a recently established relation involving high-order cumulants of the energy. Finally, we show that, even when the system does not undergo a phase-transition, the Lee-Yang zeros still encode valuable physical information; they crucially determine the large deviation statistics of energy fluctuations. Specifically we show that the large deviation function generically has the form of an ellipse, whose tilt and width can be inferred from the complex Lee-Yang zeros. Our analysis reveals an interesting duality between the energy fluctuations of small-size systems in equilibrium and their phase- behavior in the thermodynamic limit [3]. To what extent this relation is valid in more complex systems, such as the two-dimensional Ising model, is a topic of future research. $ $ [1] C. Flindt, and J. P. Garrahan, Trajectory Phase Transitions, Lee-Yang Zeros, and High-Order Cumulants in Full Counting Statistics, Phys. Rev. Lett. 110, 050601 (2013) [2] K. Brandner, V. F. Maisi, J. P. Pekola, J. P. Garrahan, and C. Flindt, Experimental Observation of Dynamical Lee-Yang Zeros, Phys. Rev. Lett. 118, 180601 (2017) [3] A. Deger, K. Brandner, and C. Flindt (2017 - In preparation) Aydin Deger (Department of Applied Physics, Aalto University, Finland) Christian Flindt (Department of Applied Physics, Aalto University, Finland) Kay Brandner (Department of Applied Physics, Aalto University, Finland)	{"url":"https://zakopane.if.uj.edu.pl/event/4/contributions/90/","timestamp":"2024-11-09T04:32:43Z","content_type":"text/html","content_length":"59138","record_id":"<urn:uuid:9ba5449f-53b0-45f0-bd50-7620f0c32d4c>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00648.warc.gz"}
CCAPP Seminar - Eduardo Rozo (University of Arizona) April 25, 2023 12:00PM - 1:00PM Add to Calendar 2023-04-25 11:00:00 2023-04-25 12:00:00 CCAPP Seminar - Eduardo Rozo (University of Arizona) Speaker: Eduardo Rozo (University of Arizona) "A Field-Based Inference Approach to Cosmic Shear" All cosmological analyses to date rely on summary statistics: given a map of the survey data (e.g. a galaxy density map), one first computes a set of summary statistics (e.g. the correlation function), and then one proceeds to fit only that particular set of summary statistics. This approach is necessarily wasteful: some information is always lost in going from the survey map to the set of summary statistics under consideration. In this talk, I will describe a new approach referred to as field-based inference, in which one seeks to model not summary statistics, but the survey maps themselves. I will demonstrate field-based inference methods are expected to double the amount of information we can extract from current and future surveys, and discuss some of the challenges that remain to usher in a new era of field-based inference. PRB 4138 OSU ASC Drupal 8 ascwebservices@osu.edu America/New_York public Date Range 2023-04-25 12:00:00 2023-04-25 13:00:00 CCAPP Seminar - Eduardo Rozo (University of Arizona) Speaker: Eduardo Rozo (University of Arizona) "A Field-Based Inference Approach to Cosmic Shear" All cosmological analyses to date rely on summary statistics: given a map of the survey data (e.g. a galaxy density map), one first computes a set of summary statistics (e.g. the correlation  function), and then one proceeds to fit only that particular set of summary statistics. This approach is necessarily wasteful: some information is always lost in going from the survey map to the set of summary statistics under consideration. In this talk, I will describe a new approach referred to as field-based inference, in which one seeks to model not summary statistics, but the survey maps themselves. I will demonstrate field-based inference methods are expected to double the amount of information we can extract from current and future surveys, and discuss some of the challenges that remain to usher in a new era of field-based inference. PRB 4138 America/New_York public Speaker: Eduardo Rozo (University of Arizona) "A Field-Based Inference Approach to Cosmic Shear" All cosmological analyses to date rely on summary statistics: given a map of the survey data (e.g. a galaxy density map), one first computes a set of summary statistics (e.g. the correlation function), and then one proceeds to fit only that particular set of summary statistics. This approach is necessarily wasteful: some information is always lost in going from the survey map to the set of summary statistics under consideration. In this talk, I will describe a new approach referred to as field-based inference, in which one seeks to model not summary statistics, but the survey maps themselves. I will demonstrate field-based inference methods are expected to double the amount of information we can extract from current and future surveys, and discuss some of the challenges that remain to usher in a new era of field-based inference. Events Filters:	{"url":"https://ccapp.osu.edu/events/ccapp-seminar-eduardo-rozo-university-arizona","timestamp":"2024-11-13T12:35:48Z","content_type":"text/html","content_length":"46622","record_id":"<urn:uuid:1018381f-489c-4ade-ba77-e729047192a2>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00442.warc.gz"}
Mastering Vectors in GCSE Mathematics: A GCSE Maths revision guide Understanding vectors is an important part of mathematics, especially at GCSE. Vectors are entities that have both magnitude and direction. Unlike scalar quantities - which only have magnitude (like temperature or mass) - vectors summarise the idea of moving from one point to another. This idea is fundamental to mathematics and crucial in various fields, including physics, engineering and computer science. For example, imagine you're walking from one point to another. The direction you take and the distance you cover can be represented as a vector. In maths, this is often shown as an arrow - the length of the arrow indicates the magnitude and the direction describes where the vector points. At GCSE, the study of vectors opens up a new understanding of mathematics. It's not just about solving equations or crunching numbers; it's about visualising how quantities move and interact in space. This knowledge is not only crucial for excelling in GCSE Maths, but also sets the stage for more advanced studies in A-Level Maths and beyond. For students looking to delve deeper into the world of vectors and advanced mathematics, finding the right tutor can be a game-changer. If you need further support, consider finding a GCSE Maths tutor to guide you through the complexities of Vectors. Understanding the basics of the Vector topic Vectors are fundamental in mathematics, particularly in geometry and physics. At its core, a vector is a quantity defined by both a magnitude (how much) and a direction (which way). This dual nature sets vectors apart from scalar quantities, which only have magnitude. Key points about vectors: Magnitude and direction: The magnitude of a vector is a measure of its 'size', often shown by its length in graphical representations. Direction, on the other hand, indicates the vector's orientation in space. For example, a vector representing a 5 km walk east has a magnitude of 5 km and a direction towards the east. Graphical representation: Vectors are typically illustrated as arrows. The length of the arrow indicates the magnitude and the arrow points in the direction of the vector. This graphical representation is crucial in understanding and solving vector-related problems in mathematics. Basic vector notation: Vectors are often shown by letters in bold, such as v or u, or with an arrow above. The notation AB represents a vector starting at point A and ending at point B. Zero vectors and unit vectors: A zero vector has a magnitude of zero and therefore no specific direction. It's represented as 0. A unit vector has a magnitude of one and is often used to indicate Understanding these basic concepts of vectors lays the groundwork for more complex operations, like addition, subtraction and multiplication by a scalar. Which of the following statements is true about vectors? Operations with Vectors Now we understand vectors, it's essential to understand how to perform operations with them. Vector operations are fundamental in many mathematical calculations, especially in physics and Vector addition and subtraction: One of the simplest operations with vectors is addition. When you add two vectors, you're essentially putting them end-to-end to find a resultant vector. For example, if you walk 3 km east (vector A) and then 4 km north (vector B), your resultant displacement (vector C) is the vector sum of A and B. Subtracting vectors can be thought of as adding a negative vector. Graphically, these operations can be performed using the 'triangle' or 'parallelogram' method. Scalar multiplication: Multiplying a vector by a scalar (a regular number) changes its magnitude but not its direction. For instance, if you triple your walking speed, the vector representing your velocity triples in length, but the direction remains unchanged. These uses aren't just theoretical; they're used to solve real-world problems, such as calculating forces in physics or determining directions in navigation. If vector X represents 5 km east and vector Y represents 3 km north, what is the vector sum of X and Y? Applications of Vectors in real-world examples Vectors aren't just abstract concepts; they have numerous practical applications that we need to use in various fields. Understanding how vectors are used in real-life scenarios can help us to understand their importance and functionality. Take a look at the following real-life examples: Physics: In physics, vectors are crucial in understanding forces and motion. For example, when studying the motion of objects, vectors are used to represent forces, velocity and acceleration, providing direction and magnitude to these quantities. This helps in predicting how objects will move under various forces. Engineering: Engineers use vectors to design and analyse structures, machinery and systems. In civil engineering, for instance, vectors help in determining the forces acting on bridges or buildings, ensuring their stability and safety. Navigation: Vectors are indispensable in navigation, both at sea and in the air. They are used to calculate routes, considering direction and speed to determine the most efficient path from one point to another. Computer graphics: In the realm of computer graphics, vectors play a key role in creating and manipulating images. They are used to model objects and define how they move and interact within a digital space. Biology: Even in biology, vectors find their application. They can be used to model the spread of diseases or the migration patterns of animals, offering insights into complex biological processes. These examples of vectors show clearly how mathematical concepts are not confined to textbooks but are essential tools in solving real-world problems. In navigation, if a ship needs to travel 10 km north and 15 km east, how can vectors be used to determine its route? Advanced Vector concepts As students progress in their GCSE Maths journey, vectors take on more complex and intriguing forms. Two such advanced concepts are the dot product and the cross product of vectors. Dot product (Scalar product): The dot product of two vectors is a scalar quantity that is a measure of their 'alignment'. Mathematically, it's calculated as the product of the magnitudes of the two vectors and the cosine of the angle between them. This concept is used in various applications, such as determining the angle between two vectors or finding the projection of one vector onto another. Cross product (Vector product): In contrast, the cross product of two vectors results in a vector. This new vector is perpendicular to the plane formed by the original vectors and its magnitude is proportional to the area of the parallelogram that the vectors span. The cross product is vital in physics, particularly in calculating torque and angular momentum. These advanced concepts illustrate the depth and versatility of vector mathematics. For students interested in delving deeper into these topics, we have a range of GCSE Maths tutors who can provide personalised guidance and support in understanding these complex concepts. What does the cross product of two vectors represent? External resources and further revision While learning about vectors, it's useful to explore related mathematical concepts that enhance your overall understanding. Here are some recommended resources on topics that supplement the study of Physics Classroom - Motion and Forces: Since vectors are widely used in physics, a basic understanding of motion and forces can be helpful. The Physics Classroom provides clear and engaging resources on these topics. Physics Classroom Motion and Forces. Desmos Graphing Calculator : A powerful tool for graphing vectors and exploring their properties interactively. Desmos can help in visualising vector operations and concepts. Desmos Graphing Calculator. Wolfram Alpha - Vector Calculations: For advanced students, Wolfram Alpha provides a tool to perform various vector calculations, offering a practical approach to understanding vector operations. Wolfram Alpha Vector Calculator. These resources provide a well-rounded approach to learning vectors, offering insights into related mathematical and physical concepts. They are ideal for students who wish to broaden their understanding beyond the GCSE curriculum. As we've explored throughout this article, vectors are a fascinating and integral part of mathematics, with a wide array of applications in various fields. From simple uses like addition and subtraction to more complex concepts like the dot and cross products, vectors provide a fundamental framework for understanding and solving problems in both theoretical and practical contexts. For GCSE students, mastering vectors is not only crucial for excelling in exams but also lays a solid foundation for future studies in mathematics and sciences. The concepts and skills learned here will be invaluable as you advance to A-Level Maths and beyond. We hope this article has helped you learn and revise Vectors for GCSE Maths revision, illuminating the world of vectors in a way that is accessible and interesting. Keep exploring, keep learning and most importantly, keep enjoying the journey through the fascinating world of mathematics! This post was updated on 06 Jul, 2024.	{"url":"https://www.teachtutti.co.uk/blog/vector-gcse-maths/","timestamp":"2024-11-07T22:23:40Z","content_type":"text/html","content_length":"265715","record_id":"<urn:uuid:cbce7d76-6791-4c8c-a643-48c74372a5e5>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00811.warc.gz"}
Teaching Reading from an Interactive Perspective B- The Bottom Up (Serial) Approach (Text-based) (LaBerge & Samuels, MacWorth) (Overreliance on bottom-up or text-based processing => text-boundedness) The "bottom up" approach stipulates that the meaning of any text must be "decoded" by the reader and that students are "reading" when they can "sound out" words on a page. (Phonics) <=> It emphasizes the ability to de-code or put into sound what is seen in a text. It ignores helping emerging readers to recognize what they, as readers, bring to the information on the page. ** This model starts with the printed stimuli and works its way up to the higher level stages. The sequence of processing proceeds from the incoming data to higher level encodings. Problems: (Stanovich, 1980) - This model has a tendency to depict the information flow in a series of discrete stages, with each stage transforming the input and then passing the recorded information on to the next higher - An important shortcoming of this model is the fact that it is difficult to account for sentence-context effects and the role of prior knowledge of text topic as facilitating variables in word recognition and comprehension (because of lack of feedback). - According to Eskey (1973), the decoding model is inadequate because it underestimates the contribution of the reader who makes predictions and processes information. It fails to recognize that students utilize their expectations about the text, based on their knowledge of language and how it works. (p. 3) Bottom-Up Applications: (Eskey & Grabe, pp. 231-236) Teaching key vocabulary items and, in the area of grammar, teaching various cohesive devices. Two areas of concern: - Simply knowing the meanings of some set number of words does not ensure that a reader will be able, while reading, to process those words both rapidly and accurately. => teachers must help students develop identification skills (exercises for rapid recognition: word recognition and phrase identification + extensive reading over time). - Rate building: good readers read fast; they do not, like many SL readers, try to read word by word, which destroys their chances of comprehending very much of the text. => The major bottom-up skill that readers of second language must acquire is the skill of reading fast. (paced and timed reading exercises: formal rate-building work should be limited to a few minutes per class). Major increases in reading rate can only follow from extensive reading in the language over time. Footnote: If a text contains too many difficult words, no strategy (top down or bottom up) can make such a text accessible to the reader. However, second language readers do of course encounter some unknown words in most texts. This is the best means of increasing their control of English vocabulary. SL readers, however, are frequently panicked by unknown words, so they stop reading to look them up in dictionaries, thereby interrupting the normal reading process. In response to this problem, many SL texts recommend various strategies for guessing the meaning of unknown words from context, by using semantic and syntactic clues or even morphological analysis. In order to develop good reading habits, the best strategy for dealing with an unknown word may well be to keep reading until the meaning of that word begins to make itself plain in relation to the larger context provided. Central to all these bottom-up concerns is the concept of automaticity (LaBerge & Samuels 1974). Good readers process language in the written form of written text without thinking consciously about it, and good SL readers must learn to do so. It is only this kind of automatic processing which allows the good reader to think instead about the larger meaning of the discourse, which allows for global reading with true comprehension. Bottom-Up Implications for the SL Classroom: (Carrell p. 240-244) - Grammatical skills: cohesive devices are very important. - Vocabulary development: Vocabulary development and word recognition have long been recognized as crucial to successful bottom-up decoding skills. However, schema theory has shed new light on the complex nature of the interrelationship of schemata, context, and vocabulary knowledge. UNLIKE traditional views of vocabulary, current thinking converges on the notion that a given word does not have a fixed meaning, but rather a variety of meanings that interact with context and background knowledge. Knowledge of individual word meanings is strongly associated with conceptual knowledge -- that is, learning vocabulary is also learning the conceptual knowledge associated with the word. On the one hand, an important part of teaching background knowledge is teaching the vocabulary related to it and, conversely, teaching vocabulary may mean teaching new concepts, new knowledge. Knowledge of vocabulary entails knowledge of the schemata in which a concept participates, knowledge of the networks in which that word participates, as well as any associated words and concepts (=> structural analysis). Teachers must become aware of the cross-cultural differences in vocabulary and how meaning may be represented differently in the lexicons of various languages. Several characteristics seem to distinguish effective from ineffective teaching programs. Preteaching vocabulary in order to increase learning from text will be more successful - if the words to be taught are key words in the target passages - if the words are taught in semantically and topically related sets so that word meanings and background knowledge improve concurrently - if the words are taught and learned thoroughly - if both definitional and contextual information are involved - if students engage in deeper processing of word meanings - if only a few words are taught per lesson and per week. Research specific to SL reading has shown that merely presenting a list of new or unfamiliar vocabulary items to be encountered in a text, even with definitions appropriate to their use in that text, does not guarantee the learning of the word or the concept behind the word, or of improved reading comprehension on the text passage (Hudson 1982). To be effective, an extensive and long-term vocabulary development program accompanying a parallel schemata or background-knowledge-development program is probably called for. Instead of preteaching vocabulary for single reading passages, teachers should teach vocabulary and background knowledge concurrently for sets of passages to be read at some later time. Every SL curriculum should have a general program of parallel concept/background knowledge development and vocabulary development. C- The Interactive Approach (Rumelhart, Stanovich, Eskey) For those reading theorists who recognized the importance of both the text and the reader in the reading process, an amalgamation of the two emerged the interactive approach. Reading here is the process of combining textual information with the information the reader brings to a text. The interactive model (Rumelhart 1977; Stanovich 1980) stresses both what is on the written page and what a reader brings to it using both top-down and bottom-up skills. It views reading is the interaction between reader and text. The overreliance on either mode of processing to the neglect of the other mode has been found to cause reading difficulties for SL learners (Carrell 1988, p. 239) The interactive models of reading assume that skills at all levels are interactively available to process and interpret the text (Grabe 1988). In this model, good readers are both good decoders and good interpreters of text, their decoding skills becoming more automatic but no less important as their reading skill develops (Eskey 1988). According to Rumelhart's interactive model: 1- linear models which pass information only in one direction and which do not permit the information contained in a higher stage to influence the processing of a lower stage contain a serious deficiency. Hence the need for an interactive model which permits the information contained in a higher stage of processing to influence the analysis that occurs at a lower stage. 2- when an error in word recognition is made, the word substitution will maintain the same part of speech as the word for which it was substituted, which will make it difficult for the reader to understand. (orthographic knowledge) 3- semantic knowledge influences word perception. (semantic knowledge) 4- perception of syntax for a given word depends upon the context in which the word is embedded. (syntactic knowledge) 5- our interpretation of what we read depends upon the context in which a text segment is embedded. (lexical knowledge) All the aforementioned knowledge sources provide input simultaneously. These sources need to communicate and interact with each other, and the higher-order stages should be able to influence the processing of lower-order stages. According to Stanovich's interactive-compensatory model: * Top-down processing may be easier for the poor reader who may be slow at word recognition but has knowledge of the text topic. * Bottom-up processing may be easier for the reader who is skilled at word recognition but does not know much about the text topic. => Stanovich's model states, then, that any stage may communicate with any other and any reader may rely on better developed knowledge sources when other sources are temporarily weak. To properly achieve fluency and accuracy, developing readers must work at perfecting both their bottom-up recognition skills and their top-down interpretation strategies. Good reading (that is fluent and accurate reading) can result only from a constant interaction between these processes. => Fluent reading entails both skillful decoding and relating information to prior knowledge (Eskey, 1988).	{"url":"https://nadabs.tripod.com/reading/index.html","timestamp":"2024-11-07T05:57:07Z","content_type":"text/html","content_length":"118564","record_id":"<urn:uuid:dd247e61-fc31-4a17-b309-f5efd1012508>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00219.warc.gz"}
Hard Lefschetz theorem and Hodge-Riemann relations for convex valuations Posted in Friedrich-Schiller-Universität Jena Thu, 18/04/2024 - 16:30 - 18:00 The Alexandrov-Fenchel inequality, a fundamental result in convex geometry, has recently been shown to be one component within a broader 'Kahler package'. This structure was observed to emerge in different areas of mathematics, including geometry, algebra, and combinatorics, and encompasses Poincare duality, the hard Lefschetz theorem, and the Hodge-Riemann relations. After unpacking these statements within the context of this talk, I will explain where complex geometry intersects with convex geometry in the proofs. Based on joint work with Andreas Bernig and Jan Kotrbaty.	{"url":"http://www.mpim-bonn.mpg.de/node/12911","timestamp":"2024-11-05T01:13:14Z","content_type":"application/xhtml+xml","content_length":"17244","record_id":"<urn:uuid:5be495e8-5e72-44e2-8a0a-56f7852deeef>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00084.warc.gz"}
Mixing Curves Mixing Curves and the Technical Guts of the Mix Pad Use the Mix Curve dropdown [graphic] to select the mixing curve. Basically, mixing curves change the way the Mix Pad maps the distance between the Blender and the terminals onto changes in volume, pitch and pan. The material below gives further explanation of the Mix Pad and Mixing Curves. You should get a fell for how the Mix Pad works before reading this. This is the most complex piece of documentation, and if you can hear what is going on, you needn't bother with it. How jambient Calculates the Mix The Mix Pad mixes four different signals, coming through the four corner terminals. You set the mix by moving the Mix Blender between the terminals. The steps for calculating the mix are as follows: Jambient calculates the distance between the Blender and each of the corner terminals of the Mix Pad. For each of these distances, jambient does the following to calculate what to do with the signal at the corresponding corner terminal: □ Calculate a fade value by mapping the distance onto a fade curve. In the Fade after Centre curve, all distances between the terminal and the centre of the diagonal of the pad map onto a fade value of "full on," and distances after that map onto a fade value that drops from 1 to 0 in a linear fashion. The Fade after Side Centre curve is similar, except values begin to drop from 1 to 0 after the centre of the side of the mix pad. In the Fade Diagonal curve, fade values drop from 1 to 0 in a linear fashion across full distance of the diagonal. □ Map the fade value onto a modifier for volume, pitch or pan. If the fade value is "full on" then the modifier for volume, pitch or pan will leave them unchanged. volume is mapped so that at fade value 1 the modifier will give full volume, at 0 it will give no volume; pan is mapped so that at fade value 1 the modifier will push pan 500 clicks to the left, at 0.5 it will not change pan, at 0 it will push pan 500 clicks to the right; pitch is mapped so that at fade value 1 pitch is dropped to minimum, at 0 it is pushed up by the maximum amount (300 or 400%, depending on the mix pad type)--but jambient skews the scale so that 0-100% pitch always maps onto fade value 1 to 0.5, so that 100% pitch is at the diagonal centre, which allows the centre to be the neutral position. The volume, pitch or pan modifiers are used to modify the values specified by volume, pitch and pan knobs. Visualizing Fade Curves The icons in the Mix Curve drop down are meant to visualize the fade curves in the following way: You can think of the fade curves as describing circular regions in which certain values hold. Consider the white terminal. In the Fade after Centre curve, when the mix blender is within a circle whose centre is the white terminal's corner, and that goes through the diagonal centre of the square, then the white terminal will be "full on". If you move further away, then values start dropping. Similarly for the Fade after Side Centre Curve, except the "full on" circle goes through the centre of a side of the Mix Pad. You can see that these two curves define different areas where the "full on" circles of the four terminals overlap--in the Fade after Centre curve, they overlap in the centre of the Mix Pad, and overlap in middle portion of each side, in the Fade after Side Centre they don't overlap in the centre, and just touch on the centres of the sides. This is important because when the Mix Blender is in a "full on" region, the signal from the correponding terminal is unchanged. How jambient Calculates the Twist Calculation of the Twist is almost exactly the same as calculation of the mix pad, except that distances are calculated along one dimensional line between the ends of the twist, rather than distances from corners. So the distance from the white and black terminals will always be the same, and signals on these terminal will be effected in the same way.	{"url":"https://www.jambient.com/manual/Mixing_Curves.htm","timestamp":"2024-11-01T18:43:06Z","content_type":"text/html","content_length":"10245","record_id":"<urn:uuid:7ee7e125-d0da-4cd7-88ec-09926662dd4f>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00714.warc.gz"}
Memorizing Long Numbers – Two Quick Memory Aids Dice – Image Pixabay Memorizing long numbers? How can I do that? I recall being told the average American can repeat quickly only numbers with five or fewer digits. For example, hearing several numbers, say 17, 38294, 584, and 127532, most can only say back the 17, 38294, and 584 – not the 127532. How can such a person improve in memorizing long numbers so he can recall 6, 7, and even more digits? There are two ways. The first involves a kind of ‘device’. One definition of mnemonic device is “a memory technique to help your brain better encode and recall important information”. Memory Aid – Grouping Numbers Almost anyone can repeat a string of three. Jill speaks a three-digit number. Bob repeats it back to Jill. No big whoop. Anyone can do that. So why not take advantage of the fact? After all, what is a six-digit number if not two three-digit numbers? 629,384 is made up of 629 and 384. Remember those two three-digit numbers and repeat them. Do that and, in effect, you’ve repeated a six-digit number. So you (in effect) only have to remember two I thought of this memorization technique during my teenage years and put it to good use. In fact once, after hearing it only twice, I was able to repeat an eighteen-digit number. However, I didn’t just repeat the 18-digit number. I repeated it in reverse order; I repeated it backwards! The constant Pi. Another Kind of Memory The second trick I utilized, though I wasn’t aware there is a name for it, was to attempt to visualize the numbers. While I did not completely succeed at this, my memory did become stronger, perhaps in the same way as healthful exercise strengthens muscles. Some individuals, apparently born with the capability, see an actual image of numbers or text for some minutes. This is called eidetic Benefits of Memorizing Long Numbers While I do not recommend one limit the recording of important numbers to one’s memory, it does offer some benefits to develop the ability. Consider these scenarios… • You have no pencil or paper and must on-the-spot take down a phone number. • You are on the phone and are told a string of important numbers you can jot down shortly. • You look at the identification number of a motor vehicle and need to repeat it moments later. • You need to quickly memorize a constant for some mathematics problem. But the best benefit to memorizing long numbers is that you are exercising your mind. Use it or lose it seems a wise proverbial saying in this time of increasing dementia cases. Note: You might also enjoy Algebra for Beginners: Student Perspective ← Back to Math-Logic-Design ← Home 2 thoughts on “Memorizing Long Numbers – Two Quick Memory Aids” • Chunking numbers is a good way to remember them. I used to get sent to bed early as a child and if I couldn’t sleep, would play around with numbers in my mind. It’s a lot harder these days. I used to be able to visualise the numbers moving around but not now. One thing I found out that way is that it is easy to find the square of numbers ending in .5 in your head. The .5 always becomes .25 and the first number gets multiplied by itself, then added to itself. Example 2.5 squared is 6.25. The .5 always becomes .25. Multiply 2 X 2 = 4, then add 2 = 6. It works for all squares of numbers ending in .5. So 7.5 = .25 plus 7^2+7 = 56, so 56.25 I told my sister who asked the maths teacher in school who said it was just multiplication of squares but I had not heard of that. □ An interesting memory trick. I do so enjoy multiplying, adding, and subtracting in my head.	{"url":"https://www.quirkyscience.com/memorizing-long-numbers/","timestamp":"2024-11-09T01:43:03Z","content_type":"text/html","content_length":"134078","record_id":"<urn:uuid:ced888dc-0dad-4ea2-bfa9-a26e1ff66a77>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00384.warc.gz"}
How do you solve 3(x-2)+2(x+5)=5(x+1)+1? \| HIX Tutor How do you solve #3(x-2)+2(x+5)=5(x+1)+1#? Answer 1 See below There are no values of $x$ which are a solution to the problem therefore $x$ is the null set or $x = \left\{\emptyset\right\}$ Expand the terms in parenthesis: #3x - 6 + 2x + 10 = 5x + 5 + 1# #5x + 4 = 5x + 6# Subtract #5x# from each side of the equation: #5x - 5x + 4 = 5x -5x + 6# #0 + 4 = 0 + 6# #4 = 6# Because #4 != 6# there are no values of #x# which are a solution to the problem therefore #x# is the null set or #x = {O/}# Sign up to view the whole answer By signing up, you agree to our Terms of Service and Privacy Policy Answer 2 To solve the equation 3(x-2)+2(x+5)=5(x+1)+1, follow these steps: 1. Distribute the terms inside the parentheses: 3x - 6 + 2x + 10 = 5x + 5 + 1 2. Combine like terms on both sides of the equation: 5x + 2x + 3x - 6 + 10 = 5x + 1 + 5 10x + 4 = 5x + 6 3. Move the variable terms to one side of the equation and the constant terms to the other side: 10x - 5x = 6 - 4 5x = 2 4. Solve for x by dividing both sides by the coefficient of x: x = 2/5 So, the solution to the equation is x = 2/5. 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-solve-3-x-2-2-x-5-5-x-1-1-8f9af8ffec","timestamp":"2024-11-08T04:36:02Z","content_type":"text/html","content_length":"576533","record_id":"<urn:uuid:40a6a175-bb37-4b85-962b-b30e27754cfe>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00145.warc.gz"}
Digital Logic Design Multiple Choice Questions with Answers pdf Digital Logic Design Multiple Choice Questions with Answers This Quiz article provides multiple choice questions and answers on digital logic design. This is a required course for many electrical engineers and can be a challenging subject. With this resource, you can test your knowledge and keep up with the latest design trends. Digital Logic Design Multiple Choice Questions with Answers pdf for MCA, BCA, and other IT Academic and Competitive examinations. Brief Introduction to Digital logic design Digital Logic Design is the process of designing and constructing digital circuits. The design can be done in a number of different ways, but all involve calculating voltages and currents and then manipulating these numbers to create a working circuit. There are many different tools that can be used for this purpose, but the most common ones are schematic software programs and breadboard software. Digital Logic Design Multiple Choice Questions with Answers 1. ___ circuits are those whose outputs depend only on the current inputs. Answer: Combinational 2. CPU stands for ___. Answer: Central Processing Unit 3. Multiplexers are examples of combinational circuits. (State true or false) Answer: True 4. A ___ adder is a digital circuit that accepts two inputs and performs addition on them and generates two outputs known as sum(S) and carry(C). Answer: Half 5. A full adder is a digital circuit that can handle carry input. (State true or false) Answer: True 6. The ___ is a combinational circuit that is used to perform the subtraction of two bits. Answer: Half subtractor 7. The ___ is used to perform subtraction of three bits, input A (minuend), input B (subtrahend), and third input called ___ and produces two outputs D (difference) and B (borrow). Answer: Full subtractor, Borrow in 8. The ___ adder that adds two four-bit numbers is called a 4-bit parallel adder. Answer: Parallel 9. A parallel binary subtractor that subtracts two n bit binary numbers in parallel is called a full binary subtractor. (State true or false) Answer: False 10. A ripple carries adder is called so because each carry bit gets rippled into the next stage. (State true or false) Answer: False 11. One of the most serious drawbacks of this adder is that the delay increases linearly with the ___. Answer: Bit length 12. ___adders do not wait for the carry to ripple through the circuit. Answer: Carry look ahead 13. The expression (A + C)(AD + AD) + AC + C can be minimized to ___. Answer: A+C 14. A ___is a visual representation of a Boolean function. Answer: Karnaugh map (K-map) 15. In a K map, a group of eight 1’s is called ___. Answer: Octet 16. Don’t care value may be considered as ___ in case of SOP and ___ in case of POS expressions respectively. Answer: 1, 0 17. The simplified form of the expression using the K-map is ___. Answer: F= AB+BC 18. A ___ is an implicant of the function that is not included in any other implicant of the function. Answer: Prime implicant 19. When the number of variables is more than six in a given Boolean expression, Quine–McCluskey (Q-M) method will be used. (State true or false) Answer: True 20. The implementation of a Boolean function with ___ logic requires that the function be simplified in the sum of product form. Answer: NAND-NAND 21. The NOR function is a dual of the ___ function. Answer: NAND 22. In the product of sums form, we implement all sum terms using AND gates. (State true or false) Answer: False 23. ___ of a number is formed by obtaining 9’s complement of a number and adding 1 to the 9’s complement. Answer: 10’s 24. A ___ bit is an extra bit attached to a binary message to make the total number of 1’s either odd or even. Answer: Parity 25. ___ gates are useful for generating and checking a parity bit that is used for detecting/correcting errors during the transmission of binary data over communication channels. Answer: Exclusive OR 26. A multiplexer with 2n data input lines requires ___ number of “control” or select lines to select the input line. Answer: n 27. A 4: 1 Mux selects one of the input lines and connects it to the output line using 2 select lines. (State true or false) Answer: True 28. Demultiplexer is also called ___. Answer: Data distributor 29. ___ accepts an active level (i.e. HIGH) on one of its inputs and converts it into coded output such as binary or BCD. Answer: Encoder 30. LED stands for ___. Answer: Light Emitting Diode 31. Which of the following is the BCD-to-7-segment decoder/driver (a) IC 7446 (b) IC 7464 (c) IC 77446 (d) IC 6446 Answer: (a) IC 7446 32. A ___is a combinational logic circuit that compares the magnitude of two binary numbers and determines if one number is greater than, less than, or equal to the other number. Answer: Magnitude comparator 33. IC 7485 which is a ___bit comparator. Answer: 4 34. BCD adder is a circuit that adds two BCD digits and produces some output digit which is also a BCD. (State true or false) Answer: True 35. BCD subtractor does the subtraction using either ___ complement method or ___ complement method. Answer: 9’s, 10’s 36. In case SR latch using NOR gates, the ___condition exists when S=R=1. Answer: Forbidden 37. In the case of ___flip flop input data appears at the output after some time. Answer: D 38. When J=K=1 and CLK=1, the flip flop toggles as long as the clock signal is HIGH (True or False?). Answer: False 39. The race around condition exists in ___ flip flop. Answer: JK 40. T flip flop state toggles when T=1 and CLK=1. (State true or false) Answer: True 41. When a flip flop responds to the HIGH or LOW level of the clock signal it is called ___ triggering. Answer: Level 42. When a flip-flop change states either when the clock pulse is changing from LOW to HIGH or from HIGH to LOW, then it is called ___. Answer: Edge triggering 43. Master-slave flip flop will avoid the ___ condition. Answer: Race around 44. Preset and Clear inputs are called ___ inputs. Answer: Asynchronous 45. A ___ circuit is a circuit whose output depends on both the current inputs and the past inputs. Answer: Sequential 46. Sequential circuit uses the combinational logic circuit, memory, and clock signal for its operation. (True or False?) Answer: True 47. In ___ sequential circuit an event does not wait for timing pulses. Answer: Unclocked 48. Any device or circuit that has two stable states is called ___. Answer: Bistable 49. The term flip-flop is used exclusively for ___ circuits. Answer: Clocked 50. The ___ is a digital sequential circuit that counts the number of input pulses applied. Answer: Counter 51. Register is a group of ___. Answer: Flip-flops 52. ___ is a group of flip-flops combined and connected together to facilitate the movement of data bits from one flip flop to another. Answer: Shift Register 53. Shift registers are used only for data storage but not for the movement of data. (State true or false) Answer: False 54. SIPO stands for___. Answer: Serial-in to Parallel-out 55. In ___ shift register, the data input is given in parallel to the input line of each of the flip-flops, and outputs are read out serially from the single output line (Serial Data Out). Answer: Parallel-In, Serial-Out (PISO) 56. In the PIPO shift register a single clock pulse is sufficient to store and read the data bits. (State true or false) Answer: True 57. A shift register that can shift the data in both directions (shift either left or right) is called a ___ shift register. Answer: Bi-directional 58. A shift register that can shift the data in both directions as well as load it serially and parallelly is known as a ___ shift register. Answer: Universal 59. The integrated circuit (IC) chip 74LS194 is a universal shift register (State true or false) Answer: True 60. A shift register that can exhibit a specified sequence of states like that of a counter is known as ___. Answer: Shift register counters 61. An n-stage Johnson counter yields a count sequence of length 4n. (True or False?). Answer: False 62. SISO shift register can be used to introduce a time delay. (True or False?) Answer: True 63. In the case of universal shift register IC 74LS194, shift-right is done synchronously with the positive edge of the clock when S0 is High and S1 is Low. (State true or false) Answer: True 64. A ___ is a digital circuit that generates a desired sequence of bits in synchronization with a clock. Answer: Sequence generator 65. In ___ counters a common clock is connected to clock inputs of all the flip flops. Answer: Synchronous 66. A 3 bit asynchronous up counter counts from ___ in an upward direction. Answer: 0 to 7 67. We can build a faster counter by clocking all flip-flops. (True or False?) Answer: True 68. When the control input count-up/down=0 in the Up/Down counter, then the counter works as an up counter. (State true or false) Answer: False 69. In many applications, it is important to decode different states of the counter whose number equals the modulus of the counter. (True or False?) Answer: True 70. The ___ representation of a sequential circuit consists of three sections labeled present state, next state, and output. Answer: State table 71. An n-bit binary counter consists of n flip-flops and can count in binary from ___. Answer: 0 to 2n – 1 72. How many flip-flops are required to design a Mod-6 counter? Answer: Three(3) 73. ___ of a counter is the number of different states that a counter can go through before it comes back to the initial state to repeat the count sequence. Answer: Modulus 74. When a product of sums form of a logic expression is in canonical form, each sum term is called a ___. Answer: Maxterm 75. The canonical form of the expression X + XY’ is ___. Answer: XY + XY′ 76. A ___ is an electronic circuit that has one or more inputs but only one output. Answer: Logic gate 77. Less number of gates means less power consumption. (State true or false) Answer: True 78. The Boolean expression X+X′Y+Y′+(X+Y′) X′Y after simplification yields ___. Answer: 1 You may read Logic Design MCQs Digital logic design is an important process in the development of digital systems. It is used to create the basic building blocks of these systems, which are then used to create more complex By understanding the principles behind digital logic design, you can develop a deeper understanding of how digital systems work and be better prepared to work with them. If you like these MCQs of Digital logic design, please don’t forget to share them on social media. Read More MCQs: 1 thought on “Digital Logic Design Multiple Choice Questions with Answers” 1. Hairstyles very nice publish, i actually love this website, keep on it Leave a Comment	{"url":"https://www.eguardian.co.in/digital-logic-design-multiple-choice-questions/","timestamp":"2024-11-07T17:09:10Z","content_type":"text/html","content_length":"292634","record_id":"<urn:uuid:e71ad3cf-2432-4999-a423-595aecf43c62>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00237.warc.gz"}
Reading a Datasheet Reading a datasheet to determine a reliability value may take some investigative work. Whenever I see a fit rate based on failure-free testing, I am curious about how they did the testing and the Let’s explore just the calculations for a CMOS component that has undergone the JEDEC standard JESD47 High-Temperature Operating Life Test. The testing preconditioned 231 samples and operated them at both high temperature and high voltages to accelerated the testing. The test ran for 1008 hours. Then the calculations used a 60% confidence to determine the expected maximum failure rate. They calculated a FIT rate of 123. Exploring the acceleration first, we have two accelerating stresses, temperature, and voltage. We are given the testing and assumed use conditions as: Operating Temperature = 50°C and Operating Voltage = 2.8V Stress Temperature = 85°C and Stress Voltage = 3.1V The overall acceleration factor, AF[overall] is the product of the temperature and voltage acceleration factors, assuming there is no interaction of the stresses and failure mechanisms. $$ \large\displaystyle A{{F}_{overall}}=A{{F}_{temperature}}\times A{{F}_{voltage}}$$ The overall acceleration factor, AFoverall is the product of the temperature and voltage acceleration factors, assuming there is no interaction of the stresses and failure mechanisms. $$ \large\displaystyle A{{F}_{T}}={{e}^{\frac{{{E}_{a}}}{k}\left( \frac{1}{{{T}_{o}}}-\frac{1}{{{T}_{s}}} \right)}}$$ E[a] is the activation energy k is Boltzmann’s constant = 8.617×10^-5eV/K T[o] is the operating (use) temperature in Kelvin T[s] is the stress (testing) temperature in Kelvin Kelvin is absolute temperature and is converted from the Centigrade scale by K = °C + 273.15 A major consideration when using the Arrhenius equation is the activation energy, E[a]. In this case, the vendor chose 0.6 eV/K which corresponds to a failure mechanism of charge loss, time dependent dielectric breakdown and electrolytic corrosion. The range of common activation energies for CMOS includes values from 0.1 to 1.1, and depending on the dominant or expected failure mechanism(s) may significantly change the acceleration factor calculation. Without further information about the expected failure mechanisms, we are assuming the activation energy is appropriate. The calculation then is $$ \large\displaystyle A{{F}_{T}}={{e}^{\frac{0.6}{8.617\times {{10}^{-5}}}\left( \frac{1}{323}-\frac{1}{358} \right)}}=8.2$$ The AF[voltage] uses empirically derived relationship for the life to thin oxide voltage stress $$ \large\displaystyle A{{F}_{T}}={{e}^{\beta \left( {{V}_{s}}-{{V}_{o}} \right)}}$$ ? = empirically derived parameter, often within 4 to 6 volt^-1 V[s] is the stress (testing) voltage V[o] is the operating (use) voltage This model is for the specific failure mechanisms related to the pinholes and contamination of the gate oxide in CMOS IC’s . The purity and cleanliness is important and applying a test voltage assists in identifying these early failure modes. The calculation then is $$ \large\displaystyle A{{F}_{T}}={{e}^{4.5\left( 3.1-2.8 \right)}}=3.9$$ Therefore the overall AF is 8.2 x 3.9 = 31.98 ≃ 32. The next step is to calculate the failure rate. The testing has 231 units survive without failure for the 1008 hours of testing. The common estimate for failure rate is the total number of failures divided by the total hours of testing (number of samples times the duration in hours of the test). In this case, there are no failures, thus resulting in a failure rate of zero, which isn’t likely. Using the upper confidence level for an estimate of the failure rate, we can set a bound of the expected failure rate. The Poisson distribution is useful in this case, since the calculation of a binomial distribution is tedious, and the Poisson approximation works in this case. The probability of failure, p, is small for a sample during each hour, plus the samples size, n, is large, yet the product np < 5 (making the normal approximation inappropriate.) The datasheet uses a confidence level of 60%, which is low (making the estimated failure rate lower than if using a higher confidence level). Given ? = 0.6 in this case, we can use the Poisson distribution to calculate the upper bound of the demonstrated failure rate. The Poisson distribution is stated as $$ \large\displaystyle P\left( x \right)=\frac{{{\mu }^{x}}{{e}^{-\mu }}}{x!}$$ x is the number of observed failures ? is the mean and equal to n? P(x) is the Poisson statistic at a specific confidence level, ?. n is the number of sample-hours tested (total time) ? is the failure rate, ? = ?/n Setting P(0) equal to (1-?) as the probability of observing zero failures with 60% confidence, we can solve for the corresponding failure rate. $$ \large\displaystyle \begin{array}{l}P\left( 0 \right)=\frac{{{\mu }^{0}}{{e}^{-\mu }}}{0!}={{e}^{-\mu }}=1-0.6\\\mu =-\ln \left( 0.4 \right)-0.916\end{array}$$ To calculate the 60% upper confidence of the failure rate, then we use $$ \large\displaystyle \lambda =\frac{\mu }{n}=\frac{0.916}{231\times 1008\times 32}=1.23\times {{10}^{-7}}$$ Converting to units of FITS (failure per billion (10^9) hours) we get a 123 FIT rate. BTW, using 90% confidence, ?=2.3 and ? = 310 FIT rate, or almost three times higher. And, this is only the failure rate. To determine Reliability at seven years of use, for example, we use the exponential distribution reliability function. For 7 years of continuous operation, we have time, t, as 8760 hours per year x 7 years for 61,320 hours. And, the failure rate, ?, is 1.23×10^-7 failures per hour. Therefore, Therefore, the reliability at 7 years is $$ \large\displaystyle R\left( t \right)={{e}^{-\mu }}={{e}^{-\left( 1.23\times {{10}^{-7}} \right)\left( 61320 \right)}}=0.992$$ Thus expecting less than 1% of units to fail over the 7 years of operation. Sample Size – success testing (article) life testing question (article) Accelerated life testing first steps (article) 1. Syed Hussain says Hi Fred, What’s the significance of last equation that you used for 7 years operation and what it’s called? For acceleration factor calculation, do we really need to consider Voltage Acceleration? Can we just use temp or thermal acceleration factor for FIT rate, In your case if we just use temp acceleration factor for FIT rate then the difference of FIT rate would be significant. What’s your thought on this? Typically, customer only interested to know eV and CL factors for FIT rates □ Fred Schenkelberg says Hi Syed, Good questions. the last equation calculates the probability of successfully operating over 7 years, it is the reliability function. Without it we have information on the failure rate, not the probability of successfully operating over the time period of interest. The AF calculations are always related to the failure mechanisms involved. voltage accelerates some things, and temperature other failure mechanisms. Just using temperature misses many important failure mechanisms and therefore underestimates the expected life performance. FIT like MTBF is a failure rate, it is rarely constant and rarely useful by itself. Convert to reliability over the period of time of interest, say first month, warranty period and useful life period. Activation energy is important for thermally accelerated chemical reaction based failure mechanisms. If that is not the case (say solder joint fatigue and thermal cycling stress) then educate the customers by providing sufficient information for the correct understanding of your product. Confidence levels are great, not by themselves though. You should include sample size, assumed distributions and CL. 2. Syed Hussain says Thanks Fred for your details response, very helpful. I have couple of more questions though. Please respond if you find time Besides determining the 7 years of use through exponential distribution. I am interested to know how you would calculate the useful life based on 123 FIT that you calculated. According to my calculation, it should be like that (please correct me if I am wrong): AF x Stress hrs. 32 x 1008/365×24 = 3.68 years (probability of useful life) Also, for Voltage Acceleration: How did you come up with 4.5V = ? What technology are you referring to? 3. Fred Schenkelberg says Hi Syed, A FIT number is a failure rate value, it is not directly used with acceleration factors. FIT like MTBF is the probability of failure per unit time, often hours. It means nothing related to how long the product is going to last directly. You need to use the appropriate distribution’s reliability function to estimate probability of success over a time period. Acceleration a factors are to convert between test conditions and use conditions – often used to move the characteristic life of the distribution and assuming the same failure mechanism(s) apply that the slope will remain the same. The 4.5 eV is from the reference which I seem to have not listed, Lloyd W. Condra’s book Reliability Improvement with Design of Experiments and relates specifically to specific failure mechanisms related to the pinholes and contamination of the gate oxide in CMOS IC’s. There are many papers related to estimating activation energy and you should either determine the value directly with your experiments or use one that is as close as possible to the actual failure mechanism you are modeling. And, looking at the original post today, it appears some of the formulas are not showing Greek letters. Hope your browser is treating you better. I’ll update the graphics and hope to fix that issue soon. Leave a Reply Cancel reply	{"url":"https://accendoreliability.com/reading-a-datasheet/","timestamp":"2024-11-13T08:20:52Z","content_type":"text/html","content_length":"284432","record_id":"<urn:uuid:501ac586-8e5d-46af-bb61-fb36c60442c9>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00642.warc.gz"}
12x12 Multiplication Chart Printable Blank 2024 - Multiplication Chart Printable 12×12 Multiplication Chart Printable Blank 12×12 Multiplication Chart Printable Blank – Installing a multiplication graph free of charge is a great way to support your university student understand their times furniture. Here are some tips for utilizing this beneficial resource. First, look at the styles inside the multiplication desk. Following, use the graph or chart as an option to flashcard drills or like a research helper. Ultimately, utilize it being a reference self-help guide to training the days dining tables. The cost-free variation of any multiplication graph or chart only consists of instances dining tables for numbers 1 through 12. 12×12 Multiplication Chart Printable Blank. Down load a free of charge computer multiplication graph or chart Multiplication tables and charts are crucial discovering instruments. Down load a free of charge multiplication graph or chart Pdf file to help your kids memorize the multiplication charts and tables. It is possible to laminate the graph or chart for place and durability it within a child’s binder in your house. These cost-free computer resources are good for fourth, third and second and 5th-quality students. This information will explain the way you use a multiplication graph or chart to train your son or daughter arithmetic specifics. You can find free printable multiplication maps of different shapes and sizes. You will find multiplication chart printables in 12×12 and 10×10, and you can even find blank or little maps for smaller youngsters. Multiplication grids come in black and white, colour, and smaller sized models. Most multiplication worksheets adhere to the Basic Math Benchmarks for Grade 3. Habits within a multiplication graph or chart Students who have figured out the supplement table may find it easier to understand designs within a multiplication graph or chart. This training highlights the qualities of multiplication, such as the commutative house, to assist individuals comprehend the habits. For example, college students may find that this merchandise of the variety multiplied by two or more will invariably turn out as being the very same amount. A similar style can be found for numbers multiplied by way of a factor of two. Pupils could also get a design in the multiplication dinner table worksheet. Those that have trouble keeping in mind multiplication specifics need to make use of a multiplication dinner table worksheet. It will help individuals recognize that we now have styles in rows, columns and diagonals and multiples of two. Additionally, they are able to use the designs in the multiplication graph to discuss information and facts with others. This process will also help pupils remember the truth that seven periods 9 means 70, rather than 63. By using a multiplication table chart rather than flashcard drills Using a multiplication dinner table chart as a substitute for flashcard drills is a great method to support children understand their multiplication information. Youngsters frequently learn that visualizing the solution helps them recall the reality. Using this method of discovering is effective for stepping rocks to more difficult multiplication details. Imagine going up the a tremendous stack of rocks – it’s much easier to ascend little rocks instead of climb a utter rock experience! Young children understand much better by undertaking a number of training approaches. As an example, they are able to blend multiplication details and occasions dining tables to build a cumulative assessment, which cements the facts in long-term memory space. You may devote hours planning a session and making worksheets. Also you can look for entertaining multiplication game titles on Pinterest to take part your son or daughter. As soon as your kid has mastered a particular times dinner table, you are able to start working on the next. Using a multiplication desk graph as being a research helper Employing a multiplication dinner table chart as groundwork helper can be a very effective way to examine and reinforce the principles with your child’s arithmetic class. Multiplication dinner table maps spotlight multiplication facts from 1 to 10 and fold into quarters. These graphs also screen multiplication specifics in a grid structure in order that individuals are able to see designs to make contacts among multiples. By incorporating these tools into the home environment, your child can learn the multiplication facts while having fun. Utilizing a multiplication kitchen table chart as groundwork helper is a great way to motivate pupils to practice issue solving expertise, understand new techniques, making research projects easier. Youngsters may benefit from studying the tricks that will assist them solve troubles more quickly. These tips will help them build confidence and easily discover the correct product. This process is good for young children that are having difficulty with handwriting along with other great electric motor abilities. Gallery of 12×12 Multiplication Chart Printable Blank Multiplication Grid Chart 12 12 12×12 Multiplication Table Blank 12×12 Multiplication Chart Download Printable Pdf A Blank Blank Printable Multiplication Table Of 12×12 Leave a Comment	{"url":"https://www.multiplicationchartprintable.com/12x12-multiplication-chart-printable-blank/","timestamp":"2024-11-02T11:53:49Z","content_type":"text/html","content_length":"54200","record_id":"<urn:uuid:b8dce25b-0219-4d4d-93e1-3bd5ad18d513>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00872.warc.gz"}
Standard Borel Spaces This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations. This entry includes a formalization of standard Borel spaces and (a variant of) the Borel isomorphism theorem. A separable complete metrizable topological space is called a polish space and a measurable space generated from a polish space is called a standard Borel space. We formalize the notion of standard Borel spaces by establishing set-based metric spaces, and then prove (a variant of) the Borel isomorphism theorem. The theorem states that a standard Borel spaces is either a countable discrete space or isomorphic to $\mathbb{R}$. October 26, 2023 adjust theories to the set-based metric space library in Isabelle2023 Related publications • Hirata, M., Minamide, Y., & Sato, T. (2023). Semantic Foundations of Higher-Order Probabilistic Programs in Isabelle/HOL. Schloss Dagstuhl â€“ Leibniz-Zentrum fÃ¼r Informatik. https://doi.org/ Session Standard_Borel_Spaces	{"url":"https://devel.isa-afp.org/entries/Standard_Borel_Spaces.html","timestamp":"2024-11-12T03:17:52Z","content_type":"text/html","content_length":"10417","record_id":"<urn:uuid:b73f90f9-7ff1-4709-89ba-45ca57090797>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00528.warc.gz"}
Capacitance for Parallel RLC Circuit using Q Factor Calculator \| Calculate Capacitance for Parallel RLC Circuit using Q Factor What is the Q factor? The Q factor is a dimensionless parameter that describes how underdamped an oscillator or resonator is. It is approximately defined as the ratio of the initial energy stored in the resonator to the energy lost in one radian of the cycle of oscillation. How to Calculate Capacitance for Parallel RLC Circuit using Q Factor? Capacitance for Parallel RLC Circuit using Q Factor calculator uses Capacitance = (InductanceParallel RLC Quality Factor^2)/Resistance^2 to calculate the Capacitance, Capacitance for Parallel RLC Circuit using Q Factor formula is defined as the ratio of the amount of electric charge stored on a conductor to a difference in electric potential. Capacitance is denoted by C symbol. How to calculate Capacitance for Parallel RLC Circuit using Q Factor using this online calculator? To use this online calculator for Capacitance for Parallel RLC Circuit using Q Factor, enter Inductance (L), Parallel RLC Quality Factor (Q[\|\|]) & Resistance (R) and hit the calculate button. Here is how the Capacitance for Parallel RLC Circuit using Q Factor calculation can be explained with given input values -> 3.5E+8 = (0.0007939.9^2)/60^2.	{"url":"https://www.calculatoratoz.com/en/capacitance-for-parallel-rlc-circuit-using-q-factor-calculator/Calc-1979","timestamp":"2024-11-12T22:37:30Z","content_type":"application/xhtml+xml","content_length":"131343","record_id":"<urn:uuid:71ad5c60-2e27-47ee-bf82-804d420edf24>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00492.warc.gz"}
Signal flows + General Questions (11) Hi, I dont understand how signal flows are to be interpreted. I suppose the concept is that of the sets in LO2? I also saw the answer to a previous question but it didn't help me. Let us take for example the exercise 6b) in the exam august 2019. • a: 011 1111111 • b: 000 0000000 • c: 100 0000000 • d: 001 0000000 We need to make a counter example of [[[a SU b] WU c] SU d] <-> [[[a SU b] WU c] SU d] using these signal flows. I ll try to solve it please point out where my thoughts are wrong. Let's abbreviate q1<->a SU b , as b never holds our signal would be 000 0000000 meaning q1 never holds. q2<->q1 WU c as c holds at the first point of time q1 WU c is fulfilled and will stay true meaning our signal is 111 1111111 q3<-> q2 SU d, as d holds on 3 poitn of time and until that we only have ones , we have q2 SU d on the states before 3rd point of time, and after it is fulfilled because the third point of times satisfied q3, and we get our signal 111 1111111? Where is my mistake? The top-down columns mark the same point in time, the left-right lines/rows the same variable/sub formula. The bigger block in this example is said to be repeated forever. When evaluating a subformula at a position in the flow diagram, that means that we take this place (this x-axis) as “now” (t[0]). About q1 ([a SU b]) you are right. We only have 1 from which on b follows after finite time, and a is present on the way there. Since we don't have b at all, q1 remains false. About q2 ([q1 WU c]) you are not right. It is correct that q2 will become 1 where c is true, also it would become true on the way to where c is true (if q1 holds all the time on the way there), and (different from SU) q1 also holds where G q1 holds (place from which q1 is true forever). Thus, [q1 WU c] is only true in the first step. After that it is not, because we never see c again. About q3 ([q2 SU d]) you are only partly right. It is correct that in the third step and on the way to the third step, the formula would be evaluated to 1 if we had d in the third step, and q2 on the way to it. However, from the fourth step on, the formula evaluates to false as d is not seen from there. Where is your mistake? I think the mistake is that you do not isolate the time steps. As I said before, for every time step in the diagram, we consider this point as “now” and don't look back to the past for future operators. See also slide 9 in the exercise slides on exercise 8 (temporal logic).	{"url":"https://q2a.cs.uni-kl.de/1885/signal-flows","timestamp":"2024-11-12T04:00:47Z","content_type":"text/html","content_length":"53995","record_id":"<urn:uuid:7411ecc1-a9f3-40da-bf2b-1aa07731fb8f>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00249.warc.gz"}
The geometric correction of SPOT 5 HRG images using an orbit attitude and minimum control points Using supermode processing, SPOT 5 images have significantly improved ground resolution of 2.5m by two arrays of CCD detectors vertically and horizontally offset by one half-pixel (2.5 m) in the focal plane. SPOT 5 also provides the corrected attitude angles of satellite's orbit with a geometrically optimized system. Consequently, SPOTS has more precise geometry than other SPOT system. Although various mathematical model for SPOT 1-4 image to correct the geometric distortion have been proposed by many researchers, the model for SPOT 5 images need to be further improved using a geometrically optimized system. In this paper, the method to correct the geometric distortion for SPOT 5 HRG images using orbit attitude information and minimum control points was studied. This method was based on the assumptions that the satellite is moving along a well-defined close-to-circle elliptical orbit, and the orbit taken by auxiliary data is similar to the original satellite's orbit. Using these assumptions, two new equations were formulated from the geometry of the satellite. If the orbit of SPOT 5 is very precise, these equations can be directly applied using the initial parameters. But, these parameters must be updated because the initial information has the errors. LOS (Line Of Sight) vector, which is expressed by the look angles, is used for the updating parameters, because two equations are expressed by look angles of the satellite in the space, and then are dependent on the pixel direction. In order to confirm the applicability of this method, 30 check points acquired by GPS survey were used. After the residual of two equations were corrected by the updated parameters, the RMSE of check points in two equations were about 1.37e-6 and 1.70e-6, respectively, and the coefficient of determination (R^2) of them were about 0.999 and 1.0, respectively. These results imply that LOS vector can be used for the updating parameters, and mean that our method is very available. Finally, another two GCPs(Ground Control Points) acquired by GPS survey were used to correct the distortion of geometry. The result of correction was that the RMSE for 30 checking points was 0.61 pixel in pixel direction and 0.47 pixel in line direction. Original language English Title of host publication 25th Anniversary IGARSS 2005 Subtitle of host publication IEEE International Geoscience and Remote Sensing Symposium Pages 3273-3276 Number of pages 4 State Published - 2005 Event 2005 IEEE International Geoscience and Remote Sensing Symposium, IGARSS 2005 - Seoul, Korea, Republic of Duration: 25 Jul 2005 → 29 Jul 2005 Publication series Name International Geoscience and Remote Sensing Symposium (IGARSS) Volume 5 Conference 2005 IEEE International Geoscience and Remote Sensing Symposium, IGARSS 2005 Country/Territory Korea, Republic of City Seoul Period 25/07/05 → 29/07/05 Dive into the research topics of 'The geometric correction of SPOT 5 HRG images using an orbit attitude and minimum control points'. Together they form a unique fingerprint.	{"url":"https://pure.uos.ac.kr/en/publications/the-geometric-correction-of-spot-5-hrg-images-using-an-orbit-atti","timestamp":"2024-11-07T23:28:30Z","content_type":"text/html","content_length":"59709","record_id":"<urn:uuid:f3833a21-9d74-4efd-b2ee-c8189dddca7c>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00696.warc.gz"}
Significant differences on the basis of stable rankings analyzed by the SD technique - Hilgardia - University of California, Agriculture and Natural Resources Significant differences on the basis of stable rankings analyzed by the SD technique George A. Baker Burton J. Hoyle Authors Affiliations George A. Baker was Professor of Mathematics and Statistician in the Experiment Station, Davis; Burton J. Hoyle was Specialist in Field Station Administration and Superintendent of the Tulelake Field Station, Tulelake. Publication Information Hilgardia 35(22):627-646. DOI:10.3733/hilg.v35n22p627. November 1964. PDF of full article, Cite this article In the first paper, actual uniformity field trials are examined and it is found that analyses based on conventional mathematical models may assess very poorly the probabilities used in detecting significantly different varieties. Monte Carlo results show changes in the mathematical model of field trials that can give probability distributions that correspond closely to the distributions observed for actual trials. In the second paper, emphasis is placed on reproducibility of field plot results as the most desirable evaluation. Techniques by which a stable ranking among treatments can be obtained (i.e.: A is better than B) are discussed as a matter of field plot manipulation. Examples are given where reproducibility, as measured by the SD technique in a single year, is applicable to a high degree of certainty to results based on several years’ experience. The SD technique provides a confidence limit depending on design, and the values of the limits are computed. A reproducible ranking order is held to be desirable and the problems of securing one are discussed. Techniques are offered which simplify obtaining a stable ranking. Mathematical formulas are given by which given cut-off points of confidence can be calculated. Adequate field plot decisions are based on both agronomic usefulness and mathematical confidence. The SD technique is shown to fulfill both of these considerations. Literature Cited Hoyle B. J. Tulelake field station, 13th annual report—research 1959. p.20. Mimeo Hoyle B. J., Baker G. A. Determining the most efficient plot size for potato trials 1960. Amer. Soc. Hort. Sci. Western Meetings, Eugene, Ore., Mimeo. Hoyle B. J., Baker G. A. Stability of variety response to extensive variations of environment and field plot design. Hilgardia. 1961. 30:365-94E. DOI: 10.3733/hilg.v30n13p365 [CrossRef] Baker G. A. Fundamental distributions of errors for agricultural field trials. Nat. Math. Mag. 1941. 16:7-19. DOI: 10.2307/3028105 [CrossRef] Baker G. A. F values for samples of four and four from populations which are the sum of two normal populations. Nat. Math. Mag. 1944. 19:62-63. DOI: 10.2307/3030009 [CrossRef] Baker G. A. Test of the significance of the differences of per cents of emergence of seedlings in multiple field trials. J. Am. Stat. Assn. 1945. 40:93-97. DOI: 10.1080/01621459.1945.10500731 Baker G. A. Uniformity field trials when differences in fertility levels of subplots are not included in experimental error. Ann. of Math. Stat. 1952. 23:289-93. DOI: 10.1214/aoms/1177729448 Baker G. A. Field trial problems. Ann. of Math. Stat. 1952. 23:480 Abstract. DOI: 10.1214/aoms/1177729448 [CrossRef] Baker G. A., Baker R. E. Strawberry uniformity yield trials. Biometrics. 1953. 9:412-21. DOI: 10.2307/3001713 [CrossRef] Baker G. A., Briggs F. N. Wheat bunt field trials. J. Amer. Soc. Agron. 1945. 37:127-33. DOI: 10.2134/agronj1945.00021962003700020005x [CrossRef] Baker G. A., Briggs F. N. Wheat bunt field trials. II. Proc. of Berkeley symposium on math. stat. and prob., 1945-46 1949. pp.485-91. Baker G. A., Briggs F. N. Yield trials with back-cross derived lines of wheat. Ann. Inst. Stat. Math. 1950. 2:61-67. DOI: 10.1007/BF02919502 [CrossRef] Baker G. A., Hanna G. C. Transformations of split-plot yield trial data to improve analysis of variance. Proc. Amer. Soc. Hort. Sci. 1949. 53:273-75. Baker G. A., Huberty M. R., Veihmeyer F. J. A uniformity trial on unirrigated barley of ten years duration. Agron. J. 1952. 44:267-70. DOI: 10.2134/agronj1952.00021962004400050011x [CrossRef] Baker G. A., Roessler E. B. Implications of a uniformity trial with small plots of wheat. Hilgardia. 1957. 27:183-88. DOI: 10.3733/hilg.v27n05p183 [CrossRef] Baker R. E., Baker G. A. Experimental design for studying resistance of strawberry varieties to verticillium wilt. Phytopathology. 1950. 40:477-82. Hanna G. C., Baker G. A. Analysis of asparagus field trials on the basis of partial records. Proc. Amer. Soc. Hort. Sci. 1951. 57:273-76. Hoyle B. J. Stress indicators 1957. Ann. Research Review of Agronomy Staff, Davis, January 28. Hoyle B. J. Islands 1958. Ann. Research Review of Agronomy Staff, Davis, January 28. Hoyle B. J., Baker G. A. A more efficient barley yield testing method. Barley News Letter. 1959. 2:32-33. Hoyle B. J., Baker G. A. The SD technique and its use for conducting field trials 1959. p.4. Mimeo. Hoyle B. J., Baker G. A. The analysis of field trials based on the concept of islands of variation 1959. pp.16-20. Agron. Abstracts, Amer. Soc. Agron. Cleveland Meetings, November Hoyle B. J., Baker G. A. Factor analysis of twenty-eight independent field trials on nine strains of Hannchen barley. Biometrics. 1960. 16:127-128. Abstract 633. Hoyle B. J., Baker G. A. Game theory applied to field trials. Biometrics. 1961. 17:167-68. Abstract 693. Riddle O. C., Baker G. A. Biases encountered in large-scale yield tests. Hilgardia. 1944. 16:1-14. DOI: 10.3733/hilg.v16n01p001 [CrossRef] Smith F. L. Effects of plot size, plot shape and number of replications on the efficiency of bean yield trials. Hilgardia. 1958. 28:43-63. DOI: 10.3733/hilg.v28n02p043 [CrossRef] Waynick D. D. Variability in soils and its significance to past and future soil investigations. I. A study of nitrification in soils. Univ. of Calif. Pub. in Agric. Sci. 1919. 3:243-70. Waynick D. D., Sharp L. T. Variability in soils and its significance to past and future soil investigations. II. Variations in nitrogen and in field soils and their relation to the accuracy of field trials. Univ. of Calif. Pub. in Agric. Sci. 1949. 4:121-39. Wiebe G. A. Variation and correlation in grain yield among 1,500 wheat nursery plots. Jour. Agric. Research. 1935. 50:331-57. Baker G, Hoyle B. 1964. Significant differences on the basis of stable rankings analyzed by the SD technique. Hilgardia 35(22):627-646. 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Formula for non-duplicate values Is there any formula to populate only 1 value of each value listed [Column1]? I.e., column 1 has something like the below, Is there any formula to only show the below? Thank you! Best Answers • Hi @A Rose Yes! But it will gather together all the separate values into one cell, is that what you were looking for? There's a DISTINCT Function we can use, like so: =JOIN(DISTINCT([Column1]:[Column1]), ", ") However keep in mind that all values must be of the same data type in order for the function to calculate (ex. all must be numerical, or all must be text). Let me know if I've misunderstood what you're looking to do and I'd be happy to help further. Need more help? 👀 \| Help and Learning Center こんにちは (Konnichiwa), Hallo, Hola, Bonjour, Olá, Ciao! 👋 \| Global Discussions • Hey @A Rose Building on Genevieve's answer, to have the answers wrapped try CHAR(10) as the delimiter. =JOIN(DISTINCT([Column1]:[Column1]), CHAR(10)) The CHAR(10) is ASCII code for a new line. You must set your column format to wrapped text for the data to appear 'wrapped'. • Hi @A Rose Yes! But it will gather together all the separate values into one cell, is that what you were looking for? There's a DISTINCT Function we can use, like so: =JOIN(DISTINCT([Column1]:[Column1]), ", ") However keep in mind that all values must be of the same data type in order for the function to calculate (ex. all must be numerical, or all must be text). Let me know if I've misunderstood what you're looking to do and I'd be happy to help further. Need more help? 👀 \| Help and Learning Center こんにちは (Konnichiwa), Hallo, Hola, Bonjour, Olá, Ciao! 👋 \| Global Discussions • That worked! would be best though if we can split it in seperate row, either 1 value per row, or when doing "Wrap" it should be on seperate row so we can read it better. Btw. it worked even though it's all a value, and a number. I.e. "Value 55" Thank you! • Hey @A Rose Building on Genevieve's answer, to have the answers wrapped try CHAR(10) as the delimiter. =JOIN(DISTINCT([Column1]:[Column1]), CHAR(10)) The CHAR(10) is ASCII code for a new line. You must set your column format to wrapped text for the data to appear 'wrapped'. • Hi @Kelly Moore , Amazing! thanks so much! Can we go further and have it sorted A to Z in the formula? Thank you! • One value / row is easier than sorting it directly. You can do it with 2 formulas. 1st cell top of return column: =index(distinct([Col1]:[Col1]),1 2nd cell and dragged down: =index(distinct([Col1]:[Col1]),1+count(CurrentColumn$1:CurrentColumn1 then use a report to sort the results Help Article Resources	{"url":"https://community.smartsheet.com/discussion/85999/formula-for-non-duplicate-values","timestamp":"2024-11-07T10:27:10Z","content_type":"text/html","content_length":"417289","record_id":"<urn:uuid:0a5eab6f-756d-480d-b247-816c7a7fd082>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00560.warc.gz"}
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Class 6 math solution 2023 - English version - New All Job Circular Class 6 math solution 2023 – English version Class 6 math solution Today we are discuss about the solution of Class 6 Math book. This post is valuable for class 6 students. If you not done yet any practice of Class 6 math . You have to need a guide book. Then of course this post is for you. Now the question may come to your mind where can we get the guide book. Don’t worry just follow this website and scrool down you can get the solution of chapter 1,2, 3,4, and 5. Read more CHAPTER 1 : The Story of Numbers CHAPTER 2 : The Story of Two Dimensional Objects 3 CHAPTER : Information Investigation and Analysis CHAPTER 4 : Trees of Prime Factors CHAPTER 5 : Measurement of Length 6 CHAPTER : The World of Integers CHAPTER 7 : The Game of Fractions part 1 The Game of Fractions : part 2 CHAPTER 8: World of Unknown expressions CHAPTER 9: Linear Equation 10 CHAPTER : Story of Three Dimensional Objects CHAPTER 11 : Unitary Method, Percentages and Ratio Chapter 12 : Set the Formula, get the Formula When you visit our website you can get the solution of math class 6 book. You can also collect the pdf file of class 6 math solution guide book . So let’s go for collect it . How can you collect from our website? We say this step by step. Dear students Today’s age is Online Age so this age you can collect most of thing in online. So, students why you can not find your math solution from online. We are happily said that you can get the all solution from our website. About class six math solution Now girls and boys we will give you a popular and valuable information. Education is a basic rights for any people. You have to know that education is need for national progress . Many of Class 6 teachers and students are searche in class 6 math solution but you can not get the right answer. This is a authentic website which is provide you the 100% right answer for the class 6 math solutions . By this post we will provide you some part of math solutions. You can get solution the mcq, board, and creative part. If you are a class 6 students and searching for a mafh solution 2023. I said to you, you are now right website. Below you can find some image where you can get the solution easily. If you are click on the image you can collect the solutions. We will give you some link where you click and get your desire information . I think you are get the solution.NCTB Books http://www.nctb.gov.bd/ Class 6 math solution chapter 1 Chapter 1 Class 6 math solution Math solution for class 6 Today you will get very short time your class six math solutions. The mcq part and the creative part. Which you want to see here and collect it for permanently. This blogs is about the class six math solution 2023. We have published in this blogs by refer of class six math book. According to the clasd six math book, solve the all math in our experts teacher. If you read our post very carefully you can get the solutions with the proper manner. Class six is a very important time for students life. Because of this time mode of study will changes. Generally every student are into class 5 to primary level and class six is the start in High School level. Therefore, class is changed and curriculum activities are also changed. 2023 year is the math of class 6 completely different. Due to the class six is very considerable part of students. So all students need to use their proper time and utilise it. Our education ministry are changes the class 6 math book which is difficult to understood for many students. That’s why we will provide you to solutions of class six math book. Class 6 All Chapter Math Solution So to make use of your valuable time and practice more time all maths. By entering the website you get the part of math solution. So all the students of class six can collect the numerical solution free. This year math of class six is very critical and difficult. We are give you the Math solution very easy method. So that, you can understand very short time. We know all of the students are not equal. If you are good at in math, you can check the solution. You have to know the right methods so that you can solve in easiest method. Here is the students can get the numerical solution. This solution is published for the all class six students. Students can solve math from the beginning with the correct methods. Chapter name and Topics This post contains chapter wise solve step by step. Which may be you find here . Every chapters that are viewed in this articles notified below in detail about few chapter. 1 Chapter Chapter 1 of Class six contains ( The story of number) roman number, local and international number system etc. 2. Chapter Chapter. 2 nd chapter contsin about the story of two dimensional objects. Here discuss about, Triangle, rectangular, parallelogram, length, width, base, height, perimeter, Chapter 3 The third chapte about information, investigation and analysis. Contains topics of mean,median,and mode. Chapter 4. Forth chapter contain iin trees of prime factor, trees of L,C,M, Trees of H,C,F, and euclidean method. Chapter 5. In chapter discussed about length. Last word for Class 6 You can follow that above chapter arrange to NCTB book. So, you get the numerical solution chapter wise. At last we said to you visit our website to collect any solve of class VI. We provide all subjective answer through our website https://newalljobcircular.com/ Thanks stay with us and follow the next post, Again thanks for reading first to last.	{"url":"https://newalljobcircular.com/class-6-math-solution/","timestamp":"2024-11-07T09:47:38Z","content_type":"text/html","content_length":"195260","record_id":"<urn:uuid:13c56092-9383-4ecf-9812-ff54ed0c8d13>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00374.warc.gz"}
Reasonable Remuneration for the Japanese Patent number 26228404 I. Summary According to the calculation method proposed by the Tokyo High Court, Nichia calculates, at the maximum, the reasonable remuneration for the patent in suit (the "404 Patent') to be approximately 10 million yen. The main reasons are as follows, ·Nichia entered into cross license agreements, which cover the 404 Patent, with various competitors and Professor Nakamura admits that none of those competitors uses the 404 patent after those cross licenses were entered into. Therefore, no remuneration can be calculated after 2002. ·The total amount of the remuneration (including 195 patents and utility models registered with Japan Patent Office) for the period prior to the cross licenses is calculated to be 357.98 million yen. Thus, an average remuneration per registered patent/utility model is approximately 1.8 million yen. ·Even if we assume the 404 patent is more beneficial than an average patent, the maximum remuneration would be reasonably calculated as follows; JPY2,020,000,000(amount of sales) x 0.1(monopoly ratio) x 0.01(royalty rate) x 0.05(Prof. Nakamura's contribution ratio) = JPY10,100,000 II. Reasoning 1. Nichia had entered into several cross license agreements, which cover the 404 Patent, with several competitors on and after 2002. 2. The calculation sheet prepared by the Tokyo High Court uses different calculation methods for the pre-cross license period and for the post-cross license period. 3. In the post-cross license period, the amount of the reasonable remuneration is calculated by the following formula as indicated in *5 of the calculation sheet attached to the court's recommendation for settlement; (Expected amount of sales of the licensees) x (Hypothetical royalty rate) However, Prof. Nakamura admits that none of the licensees has used, or will use, the 404 Patent after the cross license agreements were entered into. The expected amount of sales of the licensees' products which uses the 404 Patent must be 0. Therefore, without considering the hypothetical royalty rate, no remuneration can be calculated after 2002. 4. On the basis of the above, with respect to the 404 Patent, it is sufficient if we calculate for the 9 years period from 1994 to 2002 (i.e., the pre-cross license period). 1. According to the calculation sheet prepared by the court, the remuneration for this nine-year period for all of Prof. Nakamura's inventions is JPY357,980,000. The average remuneration per registered patent/utility model is approximately 1.8 million yen, since it covers 195 registered patents/utility models. Thus, remuneration for the 404 Patent only would be 1.8 million yen if it is an average patent. 2. Even if we assume that the 404 Patent is more beneficial than an average patent, the maximum amount of the reasonable remuneration for the 404 Patent would be calculated as below. 3. JPY201,973,160,000 (amount of sales) which is the basis of the calculation should not be changed. And, the employer's contribution ratio of 95% should not be changed, either. Thus, 5% as Prof. Nakamura's contribution should remain the same. 4. According to the calculation sheet, which the court prepared and is attached hereto, the monopoly ratio for all of the invention is 50%. This means that, on an average, it is less than 0.3% per registered patent/utility model. Here, we assume that the 404 Patent contributed to 10% of the sales, while rest of the patents contributed to 40%, in order to estimate the maximum amount of the remuneration for the 404 Patent. 5. The royalty rate covering all of the invention is 10% for the first 3 years period, while 7% for following 6 years period, in the aforementioned calculation sheet. We believe that it would be allowed to use the "7%' royalty rate for both of these two periods, since the sales amount ratio for these two periods is 3:97. Then, the average royalty rate for each of 195 patents/utility models is less than 0.04%. Maximum royalty rate for the 404 Patent can be estimated, at the maximum, to be 1%, while 6% for the rest of the patents. 6. On the basis of the above, the calculation should be as follows; JPY2,020,000,000(amount of sales) x 0.1(monopoly ratio) x 0.01(royalty rate) x 0.05(Prof. Nakamura's contribution ratio) = 5. In short, according to the calculation sheet prepared by the court, Nichia calculates, at the maximum, the reasonable remuneration for the 404 Patent to be approximately 10 million yen. Contact information; Public Relations, Nichia Corporation	{"url":"https://www.nichia.co.jp/cn/newsroom/2005/2005_011102.html","timestamp":"2024-11-15T00:11:50Z","content_type":"text/html","content_length":"17719","record_id":"<urn:uuid:840e5586-5866-4341-93b2-514e261937b3>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00759.warc.gz"}
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Transactions Online Nobutaka SUZUKI, "An Algorithm for Inferring K Optimum Transformations of XML Document from Update Script to DTD" in IEICE TRANSACTIONS on Information, vol. E93-D, no. 8, pp. 2198-2212, August 2010, doi: 10.1587/transinf.E93.D.2198. Abstract: DTDs are continuously updated according to changes in the real world. Let t be an XML document valid against a DTD D, and suppose that D is updated by an update script s. In general, we cannot uniquely "infer" a transformation of t from s, i.e., we cannot uniquely determine the elements in t that should be deleted and/or the positions in t that new elements should be inserted into. In this paper, we consider inferring K optimum transformations of t from s so that a user finds the most desirable transformation more easily. We first show that the problem of inferring K optimum transformations of an XML document from an update script is NP-hard even if K = 1. Then, assuming that an update script is of length one, we show an algorithm for solving the problem, which runs in time polynomial of \|D\|, \|t\|, and K. URL: https://global.ieice.org/en_transactions/information/10.1587/transinf.E93.D.2198/_p author={Nobutaka SUZUKI, }, journal={IEICE TRANSACTIONS on Information}, title={An Algorithm for Inferring K Optimum Transformations of XML Document from Update Script to DTD}, abstract={DTDs are continuously updated according to changes in the real world. Let t be an XML document valid against a DTD D, and suppose that D is updated by an update script s. In general, we cannot uniquely "infer" a transformation of t from s, i.e., we cannot uniquely determine the elements in t that should be deleted and/or the positions in t that new elements should be inserted into. In this paper, we consider inferring K optimum transformations of t from s so that a user finds the most desirable transformation more easily. We first show that the problem of inferring K optimum transformations of an XML document from an update script is NP-hard even if K = 1. Then, assuming that an update script is of length one, we show an algorithm for solving the problem, which runs in time polynomial of \|D\|, \|t\|, and K.}, TY - JOUR TI - An Algorithm for Inferring K Optimum Transformations of XML Document from Update Script to DTD T2 - IEICE TRANSACTIONS on Information SP - 2198 EP - 2212 AU - Nobutaka SUZUKI PY - 2010 DO - 10.1587/transinf.E93.D.2198 JO - IEICE TRANSACTIONS on Information SN - 1745-1361 VL - E93-D IS - 8 JA - IEICE TRANSACTIONS on Information Y1 - August 2010 AB - DTDs are continuously updated according to changes in the real world. Let t be an XML document valid against a DTD D, and suppose that D is updated by an update script s. In general, we cannot uniquely "infer" a transformation of t from s, i.e., we cannot uniquely determine the elements in t that should be deleted and/or the positions in t that new elements should be inserted into. In this paper, we consider inferring K optimum transformations of t from s so that a user finds the most desirable transformation more easily. We first show that the problem of inferring K optimum transformations of an XML document from an update script is NP-hard even if K = 1. Then, assuming that an update script is of length one, we show an algorithm for solving the problem, which runs in time polynomial of \|D\|, \|t\|, and K. ER -	{"url":"https://global.ieice.org/en_transactions/information/10.1587/transinf.E93.D.2198/_p","timestamp":"2024-11-07T03:53:38Z","content_type":"text/html","content_length":"60833","record_id":"<urn:uuid:e4f6cf1c-e50f-45b3-81dc-c5d21200642b>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00465.warc.gz"}
Blog - Move One Digit - AGameAWeek Added in a new Maths puzzle, Move One Digit.[/highlight] After realising how bloomin' simple yesterday's version of Solve A,B,C was, I spent an hour or so trying to find the most convoluted of puzzles. I think the maths should be fairly complex, now, but am more convinced than ever, that figuring out the solution isn't as mathematically simple as it should be. Move One Digit , which needed a brief explanation. Currently considering adding a similarly short description to each of the panels. Views 65, Upvotes 21 Foldapuz Blog	{"url":"https://agameaweek.com/?id=6760&Like=1","timestamp":"2024-11-03T18:31:44Z","content_type":"text/html","content_length":"50558","record_id":"<urn:uuid:7e1ffbc6-a09b-4cd5-aa18-9c0aee033696>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00685.warc.gz"}
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High Energy and Particle Theory » MIT Physics High Energy and Particle Theory The goal of high-energy and particle theory research in the Center for Theoretical Physics (CTP) is to enable discoveries of physics beyond the Standard Model (BSM), both through precision tests of the Standard Model itself and through detailed studies of possible new phenomena. With the momentous discovery of the Higgs boson at the Large Hadron Collider (LHC) in 2012, the Standard Model of particle physics is now complete, yet its shortcomings loom larger than ever. For example, the Standard Model cannot account for the nature and origin of dark matter, nor does it address the puzzling hierarchy between the electroweak and Planck scales. On cosmological scales, questions remain about what drives the accelerating expansion of the universe, both today and during the inflationary For this reason, high-energy and particle theorists in the CTP are developing new theoretical frameworks to address physics in and beyond the Standard Model. The current effort in the CTP includes research that has a direct impact on experiments as well as research that pursues more formal theoretical directions. CTP researchers study possible new physics signatures at dark matter detection experiments, cosmological observatories, accelerators like the Large Hadron Collider, high intensity experiments, and small-scale table-top devices. At the same time, research in particle theory offers opportunities to push the boundaries of knowledge in quantum field theory (QFT), and innovations and creativity in QFT has long been a theme that unites the research conducted in the CTP. The CTP has a long history of leadership in high-energy and particle theory. Emeritus faculty Dan Freedman, Jeffrey Goldstone, and Roman Jackiw are responsible for some of the fundamental theoretical ideas – especially those associated with symmetries and symmetry breaking – which lie at the heart of the Standard Model and its extensions. Frank Wilczek is one of the authors of the Standard Model and a pioneer in the study of axions and anyons, with long-standing interests in unification and supersymmetry. Retired faculty Eddie Farhi and Robert Jaffe have taken techniques developed in particle theory and applied them to the fields of quantum computation and fluctuation physics, respectively. Tracy Slatyer and Jesse Thaler represent the next generation of particle theorists, whose work draws on experimental and theoretical developments in areas ranging from dark matter detection to quantum chromodynamics to formal supergravity. Successful high-energy and particle theorists have an appreciation and understanding of experimental and observational methods. The CTP prides itself on maintaining close connections to experimental research conducted in the Laboratory for Nuclear Science and the MIT Kavli Institute for Astrophysics and Space Research. There are also exciting synergies between the CTP and the NSF Institute for Artificial Intelligence and Fundamental Interactions (IAIFI), where machine learning techniques are being fused with QFT principles to enhance data analysis efforts at the LHC and beyond. Dark Matter and BSM Model Building Dark matter is a key research direction in the CTP, bridging particle physics and astroparticle physics (see Cosmology page). In Search of Dark Matter at the MIT Center for Theoretical Physics Video by Bill Lattanzi \| MIT Center for Theoretical Physics The gravitational evidence for dark matter is overwhelming, but the nature and origin of dark matter is still unknown. The two leading paradigms for dark matter are axions and massive stable relics (possibly of supersymmetric origin), but given the lack of any conclusive dark matter signals to date, CTP researchers are taking imaginative approaches to dark matter and its potential signatures. Jesse Thaler and Tracy Slatyer have developed innovative models for dark matter in the context of expanded “dark sectors”, where the dark matter interacts with other new particles and forces. These scenarios can produce new and unexpected signals in Earth-based experiments – at particle accelerator laboratories or underground neutrino detectors – as well as in astrophysical and cosmological data. Jesse Thaler proposed the original idea for the ABRACADABRA experiment searching for ultralight axion dark matter, which is led by Lindley Winslow of the Laboratory for Nuclear Science. Frank Wilczek, who invented axions and demonstrated their relevance for dark matter, proposed an plasma haloscope called ALPHA to search for micro-eV axions. Iain Stewart and Tracy Slatyer are collaborating to apply powerful techniques from effective field theory, originally developed to study physics at the Large Hadron Collider, to the case of signals from heavy colliding dark matter ABRACADABRA-10 cm Axion Experiment. Credit: Jonathan L. Ouellet et al. Phys. Rev. D99:052012 (2019), arXiv:1901.10652 QCD and Collider Physics Jet physics is an area of continued importance for particle phenomenology, especially at hadron colliders like the LHC. Jets are collimated sprays of particles that arise when quarks and gluons are produced at high energies, and copious jet production is a potential smoking gun for various scenarios beyond the Standard Model. Jesse Thaler has been at the forefront of the emerging field of jet substructure, developing new jet analysis techniques to capitalize on the exceptional ability of the LHC experiments to resolve jet constituents. These jet substructure methods can enhance BSM signals above Standard Model backgrounds, and they are currently being implemented in new physics searches by the MIT CMS pp group. These methods have also revealed fascinating new insights into the dynamics of QCD at high energies. Iain Stewart and Jesse Thaler have developed new techniques to perform precision jet calculations, capitalizing on recent development in applying resummation techniques to hadronic collisions. Jet substructure has offered new probes of the phenomena of jet quenching in the quark/gluon plasma, an area of considerable interest to the MIT CMS heavy ion group. More recently, machine learning and optimal transport techniques have offered new ways to disentangle and visualize jet properties. Energy Flow Networks. Credit: Patrick T. Komiske, Eric M. Metodiev, and Jesse Thaler. JHEP 1901:121 (2019), arXiv:1810.05165. Higgs and Precision Physics Higgs physics is another area of continued importance, especially with plans for precision Higgs measurements at the high-luminosity LHC and at possible future colliders. Percent-level measurements of the Higgs boson couplings are needed to test the Higgs boson’s role in generating fundamental particle masses. Frank Wilczek has long emphasized that BSM scenarios such as supersymmetry predict small deviations in these couplings as well as additional Higgs particles. Frank Wilczek and Jesse Thaler have shown how the Higgs boson and related Higgs-like states can act as the portal to dark matter. The connection between Higgs physics and BSM physics remains an active area of research. Precision calculations are crucial for studying the detailed characteristics of the Higgs boson, and Iain Stewart has applied effective field theory methods to calculate key Higgs cross sections and thus reduce theory uncertainties in Higgs measurements. The precision frontier goes well beyond Higgs physics and encompasses the full range of gauge theory dynamics. In the context of QCD, Iain Stewart has used theoretical insights to predict the impact of hadronization on certain jet observables, and Jesse Thaler has shown how multi-point correlators can expose the parton-to-hadron phase transition in LHC data. Other areas of precision investigations in the CTP include electroweak effects, CP violation, and flavor physics. Higgs to Two Photons. Credit: © 2020 CERN, for the benefit of the CMS Collaboration Quantum Field Theory Particle theory also connects to more formal developments in QFT (as well as string theory). Almost all collider studies involve the calculation of scattering amplitudes, but independent of collider applications, scattering amplitudes themselves have a rich mathematical structure with hidden symmetries. Iain Stewart’s work with effective fields theories has enabled advances in this area, particularly for results beyond the leading order in collinear and soft limits. Supersymmetry is a hypothetical extension of space-time that introduces additional “quantum” dimensions, and many QFT properties are easier to understand in a supersymmetric context. Inspired by potential LHC signatures of supersymmetry, Jesse Thaler has shown that the dynamics of supersymmetry breaking can be richer than previously thought, leading to new results in formal supergravity. Strong dynamics is a feature of many extensions of the Standard Model, and one can gain some analytic handles on these scenarios by treating them as if they were conformal field theories (i.e. special QFTs with a scaling symmetry). Conformal field theories may also be relevant for understanding jet physics, since the interactions of quarks and gluons can sometimes be approximated as having a scaling symmetry. A new understanding of symmetry in quantum field theory is being developed by a combination of high-energy and condensed matter theorists, leading to an improved understanding of phase transitions, anomalies, and strongly-coupled dynamics. Daniel Harlow has made several contributions to this field, including the discovery of a new order parameter for confinement/deconfinement transitions, an improved understanding of the dynamics of the neutral pion in the standard model, and a proof that internal global symmetries which act in nonunitary representations on fields must be spontaneously broken. More generally, techniques developed in particle theory have the potential to offer new insights in other fields. Feynman Diagram appearing for Jet calculations. Credit: Jesse Thaler Cosmology and Astroparticle Physics The interface between particle theory and early-universe cosmology has been a lively area of research since the 1970’s, when physicists realized that hot big bang cosmology would imply that fundamental properties of our universe — from the abundance of chemical elements to maybe also the density of baryons — were determined by high-energy physics processes in the nascent universe. In the present day, our understanding of the universe seems to require “dark energy” and “dark matter” components, which do not have any simple explanation in the Standard Model of particle physics. Inflationary cosmology, pioneered by Alan Guth in 1980-81, proposed that early-universe particle physics could be responsible for the production of essentially all the matter in the universe, explaining the uniformity of the universe and predicting its average mass density. It was soon discovered that quantum fluctuations during inflation might be responsible for the ripples in the mass density of the early universe—ripples that formed the seeds for structure formation, and which are now visible in the anisotropies of the cosmic microwave background (CMB). The inflationary prediction of the mass density has now been confirmed to an accuracy of about half a percent, and the patterns of ripples seen in the CMB agree very well with the predictions of simple inflationary At the same time, the CMB and other probes have allowed us to measure the amount of dark matter in the universe to percent-level precision. Physicists have established that dark matter must have mass and exert gravity, but it must also be relatively slow-moving, and its interactions (other than gravity) with known particles must be weak or absent. This leaves open a huge range of possibilities for dark matter, from new particles tens of orders of magnitude lighter than even neutrinos, through to primordial black holes formed in the first instants of the universe’s existence. Mikhail Ivanov uses cosmological large-scale structure to understand dark matter, dark energy, and inflation. The new generation of galaxy surveys will allow for precision tests of these sectors through their imprints on the observed matter distribution. An accurate theoretical understanding of these imprints will be key to harvesting new cosmological information from large-scale structure in this novel regime of high precision. The cosmology and particle astrophysics program in the CTP focuses on implications for fundamental physics and applications of field-theoretic techniques, complementing the work of our colleagues in the Laboratory for Nuclear Science and the MIT Kavli Institute for Astrophysics and Space Research. Research on inflation David Kaiser and Alan Guth are continuing to pursue the connection between particle theory and early-universe cosmology. Much of the work of Kaiser and his group has centered around understanding the dynamics and the predictions of inflationary models that include realistic features from high-energy physics: multiple interacting fields, each with nonminimal gravitational couplings. His group has also developed novel techniques to study the dynamics of inflationary models before and after inflation, including conditions under which the universe may enter an inflationary phase even amid significant inhomogeneities, and the mechanisms by which inflation ends during the “reheating” epoch, when the universe becomes filled with ordinary matter in thermal equilibrium at a high temperature. Alan Guth and collaborators have recently worked on models of inflation at a very low energy scale, and on the cosmology of axions. Guth’s recent research has also included the study of eternal inflation, with the issues it raises concerning the definition of probabilities. Guth and Kaiser are both working with a group of postdocs, graduate students, and undergraduates to study the possibility of the production of primordial black holes in the context of a particular type of inflation, called hybrid inflation. Simulation of the onset of inflation. Citation: J. K. Bloomfield, P. Fitzpatrick, K. Hilbert, and D. I. Kaiser, Phys. Rev. D 100, 063512 (2019). Testing the foundations of quantum mechanics In a separate line of investigation, David Kaiser and Alan Guth have both been part of the international “Cosmic Bell” collaboration, which has tested the foundations of quantum mechanics by conducting experimental tests of Bell’s inequality using real-time astronomical observations of high-redshift quasars to determine which measurements to perform on entangled particles. Kaiser has also worked with Joseph Formaggio in the Laboratory for Nuclear Science to use data on neutrino oscillations to test the Leggett-Garg inequality across unprecedented length-scales. Telescopes at the Roque de los Muchachos Observatory on La Palma, Canary Islands, used for the “Cosmic Bell” test of Bell’s inequality. Credit: Calvin Leung Astrophysical and cosmological dark matter signals In Search of Dark Matter at the MIT Center for Theoretical Physics. Video by Bill Lattanzi \| MIT Center for Theoretical Physics Annihilations or decays of dark matter could modify the thermal and ionization history of the universe, with possible observational consequences for nucleosynthesis, the cosmic microwave background and the redshifted 21cm line. In the present era, the same phenomena could provide striking signals from regions of high dark matter density. Tracy Slatyer’s group works extensively on the interpretation of such signals, developing new constraints and identifying possible signatures of dark matter physics, from radio to gamma-ray wavelengths, with a particular focus on data from the Fermi Gamma-Ray Space Telescope and signals from the early universe. Fermi’s Five-year View of the Gamma-ray Sky Credit: NASA/DOE/Fermi LAT Collaboration Affiliated Labs & Centers	{"url":"https://physics.mit.edu/research-areas/high-energy-and-particle-theory/","timestamp":"2024-11-02T05:24:51Z","content_type":"text/html","content_length":"116543","record_id":"<urn:uuid:8142e989-c9f1-472d-b22d-2f8743017309>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00552.warc.gz"}
Linear discriminant analysis python Linear Discriminant Analysis in Python: A Comprehensive Guide Data analysis is a crucial aspect of any business or research project. With the exponential growth in the volume of data that organizations generate, it has become essential to develop sophisticated techniques that can help organizations make sense of all that information. Linear Discriminant Analysis (LDA) is one such technique that helps in data analysis. At its core, LDA is a statistical technique used to classify data into two or more classes based on their It works by classifying data based on how well they can be separated by hyperplanes. The technique is widely used in machine learning and pattern recognition domains due to its ability to effectively classify complex datasets. The importance of LDA in data analysis cannot be overstated. Its applications range from bioinformatics and image processing to finance and marketing research. In bioinformatics, for instance, it has been used to identify genes responsible for particular diseases. In finance, it has been used to predict stock prices based on market trends and company These applications underscore just how valuable LDA can be as a tool for predictive modeling and decision making. In the next section, we will delve deeper into how LDA works and its applications in various fields of study. Understanding the Data Importance of understanding the data before applying LDA Before diving into linear discriminant analysis, it’s crucial to have a good understanding of the data you’re working with. Knowing the characteristics of your dataset can help you make informed decisions about how to preprocess it and which models to use. One key aspect to consider is the distribution of your target variable. If you’re working with a binary classification problem and there is a large class imbalance, simply using accuracy as an evaluation metric may not be sufficient. You may need to explore other metrics like precision, recall, or F1-score depending on what’s important in your specific scenario. Additionally, if you have a multi-class classification problem, you’ll need to decide how to handle this and potentially use different techniques like one-vs-rest or multiclass LDA. Another important aspect is identifying potential confounding variables that could impact the relationship between predictors and outcome variable. This could be achieved through EDA techniques such as scatterplots, histograms and boxplots that visualize the relationship between variables in your dataset. Exploratory Data Analysis (EDA) techniques to gain insights into the data Exploratory Data Analysis (EDA) is an essential step in any data analysis project as it helps us get familiar with our data and identify patterns or anomalies that might exist within it. EDA involves summarizing and visualizing key features of our dataset including central tendency measures such as means or medians as well as dispersion measures such as variance or standard deviation. Some key EDA techniques include creating histograms which provide insight into distributional patterns in our data while box plots reveal potential outliers within our dataset. Scatterplots are another powerful tool for exploratory analysis providing deep insights into relationships between variables when used for bivariate visualization. Scatterplots can help identify patterns among variables that may be predictive of the outcome variable. Overall, understanding your data is a critical first step towards successful implementation of an LDA model. Through careful analysis and visualization, you’ll be well-positioned to make informed decisions about preprocessing your dataset and selecting the appropriate model for your specific problem. Preprocessing the Data Before implementing Linear Discriminant Analysis (LDA) on your dataset, it’s important to preprocess the data to make sure it is suitable for analysis. This includes handling missing values, outliers, and categorical variables, as well as scaling and standardizing the data for better performance. Handling Missing Values Missing values can greatly affect the performance of LDA models. There are several techniques to handle missing values such as imputing them with mean or median values or using more advanced techniques like regression imputation. It’s important to carefully consider which technique is appropriate for your dataset and problem at hand. One popular approach is to use Pandas’ fillna() function to replace missing values in numerical columns with either mean or median value – whichever suits better- while categorical columns are filled with mode value. Outlier Detection An outlier is a value that lies far outside the typical range of other observations in a dataset. Outliers can have a significant impact on LDA results so it’s important to detect them before applying an LDA model. One way of detecting outliers is by using boxplots which visually represent outliers with dots outside whiskers or by calculating z-scores of each observation and removing those that fall beyond a certain threshold. Categorical Variables Handling LDA assumes normally distributed continuous data but most datasets include both categorical and continuous variables. Categorical variables need special handling before they can be used in LDA models. One approach is one-hot encoding – creating separate binary columns for each category – which converts categorical variables into numeric form suitable for use in LDA models. Scaling and Standardizing Data Last but not least, scaling and standardizing the data is an important step in LDA preprocessing. Scaling ensures that all variables are on similar scales which can lead to better performance of the model. Standardizing, on the other hand, subtracts the mean from each column and divides by its standard deviation, transforming the data into normally distributed form with a mean of 0 and a standard deviation of 1. This can improve model accuracy and interpretation. StandardScaler() function from Scikit-learn package is often used to scale and standardize data before fitting an LDA model. Overall, preprocessing your data is necessary for any machine learning analysis you undertake, including Linear Discriminant Analysis in Python. Handling missing values, detecting outliers, encoding categorical variables properly while scaling and standardizing data will make sure your model performs optimally. Implementing LDA in Python Importing Necessary Libraries and Packages Before we can start implementing Linear Discriminant Analysis (LDA) in Python, we need to import the necessary libraries and packages. The most important package for LDA is This package contains the LDA model which we will use to build our classifier. We also need other packages like pandas, numpy, and matplotlib. You can install these packages using pip. Here is an example code snippet for importing the required packages: “`python # Importing necessary Libraries import pandas as pd import numpy as np from sklearn.model_selection import train_test_split from sklearn.discriminant_analysis import LinearDiscriminantAnalysis as LDA “` Splitting the Dataset into Training and Testing Sets Next, we need to split our data into two separate sets: training set and testing set. The training set will be used to train our LDA model, while the testing set will be used to evaluate its performance. We can use train_test_split() function from the sklearn.model_selection module to split our dataset. This function splits our data randomly into train and test sets based on the specified test size fraction. Here is an example code snippet for splitting our dataset into training and testing sets: “`python # Splitting data into Training and Testing Sets (80/20 ratio) X_train, X_test, y_train, y_test = train_test_split(X,y,test_size=0.2) “` Note that here X represents features or independent variables whereas y represents target or dependent variable. Fitting LDA Model on Training Set & Predicting Outcomes on Test Set Once we have split our data into training and testing sets, we can fit our LDA model on the training set using LDA() function from the sklearn.discriminant_analysis package. After fitting the model, we can use it to predict outcomes on the test set using predict() function. Here is an example code snippet for fitting LDA model on training set and predicting outcomes on test set: “`python # Fitting LDA Model lda = LDA() lda.fit(X_train,y_train) # Predicting Outcomes using Test Data y_pred = lda.predict(X_test) “` In the above code, we first create an instance of LDA model and then fit it on training data using fit(). Once fitted, we can use predict() to predict outcomes for test data. We store these predicted values in a variable called y_pred. Now that we have implemented LDA in Python, let’s move ahead and evaluate its performance in next section. Evaluating Model Performance Now that we have trained our linear discriminant analysis (LDA) model, it’s time to evaluate its performance. There are several metrics available to us including confusion matrix, precision, recall, F1-score, and accuracy. Each of these metrics provides a different perspective on how well our model is performing and can help us identify areas for improvement. Confusion Matrix A confusion matrix is a table used to evaluate the performance of a classification model. It shows the number of true positives (TP), false positives (FP), true negatives (TN), and false negatives (FN) predictions made by the model. A perfect classifier would have all observations in the diagonal cells with no off-diagonal cells populated. The confusion matrix can be used to calculate various metrics such as precision, recall, and accuracy which we will explore next. Precision, Recall & Accuracy Metrics Precision measures how many of the predicted positive observations are actually positive. Recall measures how many positive observations were correctly identified by the classifier. And accuracy measures overall performance by calculating the proportion of correct classifications made by the classifier. The F1-score is another metric that combines precision and recall into a single value. It calculates the harmonic mean between precision and recall providing a measure of overall performance that balances both metrics. Cross-Validation Techniques for Robust Evaluation Cross-validation is an essential technique for evaluating machine learning models as it helps avoid overfitting issues which can occur when models are trained on limited data. It involves splitting data into multiple subsets or folds where each fold acts as both training and test data at different times during evaluation. K-fold cross validation is one popular technique where data is divided into k equally sized subsets such that each fold is used as a test set exactly once while the remaining folds are used for training. The results from each of the k-folds can then be averaged to provide a more robust evaluation metric for model performance. Evaluating model performance is an essential step in any machine learning By using metrics such as confusion matrix, precision, recall, F1-score, and accuracy we can gain insights into how well our model is performing. Cross-validation techniques like K-fold help ensure that our model performs well on previously unseen data by avoiding overfitting issues. Visualizing Results Plotting Decision Boundary Separating Classes using LDA Results One of the most interesting aspects of Linear Discriminant Analysis (LDA) is the visualization of results. In many cases, the goal is to separate classes in data points based on specific features. This can be easily achieved with LDA, as it calculates a discriminant function that maps each input to a corresponding class. To visualize how well LDA separates classes, we can plot a decision boundary that separates the different classes based on their respective discriminant functions. This boundary is represented by a line or surface in feature space, dividing the space into regions where different classes have higher probabilities. To demonstrate this technique, let’s consider an example where we have two classes of data points that are not linearly separable. We will use Python’s scikit-learn library to generate synthetic data and train an LDA model: “` python import numpy as np import matplotlib.pyplot as plt from sklearn.discriminant_analysis import LinearDiscriminantAnalysis # Generate synthetic data np.random.seed(0) X = np.r_[np.random.randn(20, 2) + [2, 2], np.random.randn(20, 2) + [0, -2], np.random.randn(20, 2) + [-2, 2]] y = np.array([0] * 20 + [1] * 20 + [2] * 20) # Fit LDA model and predict labels for grid points lda = LinearDiscriminantAnalysis().fit(X[:, :2], y) xx, yy = np.meshgrid(np.linspace(-5, 5), np.linspace(-5, 5)) Z = lda.predict(np.c_[xx.ravel(), yy.ravel()]) Z = Z.reshape(xx.shape) # Plot decision boundary and data points plt.contourf(xx, yy, Z, alpha=0.4) plt.scatter(X[:, 0], X[:, 1], c=y, alpha=0.8) plt.show() “` The code above generates three classes of data points and fits an LDA model to the first two features of the data. We then create a grid of points that covers the feature space and predict the class labels for each point using the LDA model. We plot a filled contour of the predicted classes and overlay the original data points on top. The result is a clear visualization of how well LDA separates the different classes in feature space. Visualizing results is an important part of understanding how well LDA performs on a given dataset. The decision boundary plot gives us an intuitive understanding of how well different classes are separated by their discriminant functions. Advanced Techniques with LDA Regularized LDA to Handle Multicollinearity Among Predictors Linear Discriminant Analysis (LDA) assumes that the predictors are independent of one another, which is not always the case in practice. When the predictors are highly correlated or multicollinear, the model may fail to provide accurate results. To overcome this issue, we can use regularized LDA, also known as Shrinkage LDA. Regularized LDA introduces a penalty term to the covariance matrix, which shrinks it towards a diagonal matrix and reduces its eigenvalues. By reducing the effect of multicollinearity among predictors, regularized LDA improves classification accuracy and stability. To implement regularized LDA in Python, we can use the `shrinkage` parameter of `sklearn.discriminant_analysis.LinearDiscriminantAnalysis`. This parameter controls the amount of shrinkage applied to the covariance matrix. A value of 0 corresponds to standard LDA without shrinkage, while a value close to 1 corresponds to complete Kernel-based LDA for Nonlinear Classification Problems Linear Discriminant Analysis assumes that the decision boundary separating classes is linear. However, in many cases, this assumption may not hold true as there may be non-linear relationships between predictors and response variables. Kernel-based Linear Discriminant Analysis (K-LDA) is an extension of LDA that allows us to handle non-linear classification problems by projecting data into a higher-dimensional feature space using kernel functions such as radial basis function (RBF), polynomial or sigmoid functions. K-LDA works by first applying a kernel function on both training and test data to map them into higher-dimensional feature spaces where they become separable by linear decision boundaries. Then standard LDA is applied on these transformed data points for classification purposes. To implement K-LDA in Python, we can use the `sklearn.discriminant_analysis.QuadraticDiscriminantAnalysis` class which supports kernel-based LDA. The `kernel` parameter of this function can be set to a desired kernel function such as RBF, polynomial or sigmoid. Linear Discriminant Analysis (LDA) is a powerful technique for classification problems that works by projecting data into lower-dimensional subspaces where it becomes more separable by linear decision boundaries. However, LDA assumes certain assumptions such as linearity and independence among predictors which may not hold true in real-world scenarios. Regularized LDA techniques can be used to handle multicollinearity among predictors and improve classification accuracy and stability. Kernel-based Linear Discriminant Analysis (K-LDA) extends LDA to handle non-linear classification problems by projecting data into higher-dimensional feature spaces using kernel functions. In practice, it’s essential to understand the data before applying LDA or any other classification techniques. Preprocessing the data using exploratory data analysis (EDA) techniques and evaluating model performance using cross-validation methods can also help improve model performance. Linear Discriminant Analysis is a powerful tool used in machine learning and data analysis. It helps in identifying the most significant features from the input dataset and classifying them into different groups. In this article, we have covered the basic concepts of Linear Discriminant Analysis and how to implement it using Python. One of the key takeaways from this article is that understanding your data is crucial before applying any model or technique. Exploratory Data Analysis (EDA) can help you gain insights into your data and identify any missing values, outliers, or categorical variables that need to be handled. Preprocessing your data by scaling and standardizing it can significantly improve your model’s performance. By splitting your dataset into training and testing sets, you can ensure that your model generalizes well to unseen data. Implementing LDA in Python is relatively easy using libraries like Scikit-learn or Statsmodels. Evaluating your model’s performance using metrics like confusion matrix, precision, recall, F1-score, accuracy metrics can help you identify areas of improvement. Linear Discriminant Analysis is a widely used classification algorithm for solving complex problems in various industries such as finance, healthcare, and marketing. Understanding its basic concepts and implementing it using Python can help you analyze large amounts of data efficiently while making accurate predictions. So go ahead and explore the world of LDA!	{"url":"https://datascientistassoc.org/linear-discriminant-analysis-python/","timestamp":"2024-11-10T03:14:00Z","content_type":"text/html","content_length":"131481","record_id":"<urn:uuid:f1120d9e-b494-4385-9964-e7695721e3c5>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00700.warc.gz"}
Peter Markowich : A PDE System Modeling Biological Network Formation Javascript must be enabled Peter Markowich : A PDE System Modeling Biological Network Formation Transportation networks are ubiquitous as they are possibly the most important building blocks of nature. They cover microscopic and macroscopic length scales and evolve on fast to slow times scales. Examples are networks of blood vessels in mammals, genetic regulatory networks and signaling pathways in biological cells, neural networks in mammalian brains, venation networks in plant leafs and fracture networks in rocks. We present and analyze a PDE (Continuum) framework to model transportation networks in nature, consisting of a reaction-diffusion gradient-flow system for the network conductivity constrained by an elliptic equation for the transported commodity (fluid). 0 Comments Comments Disabled For This Video	{"url":"https://www4.math.duke.edu/media/watch_video.php?v=2b38b900870e9a39f0656abdcb4d5764","timestamp":"2024-11-05T12:34:25Z","content_type":"text/html","content_length":"47211","record_id":"<urn:uuid:f8c44614-cd21-4735-9e16-794b36716e5a>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00462.warc.gz"}
Siva Prasad - MATLAB Central Siva Prasad Hyundai MOBIS Last seen: meer dan een jaar ago \|&nbsp Active since 2015 Followers: 0 Following: 0 Siva Prasad M Research Engineer at Hyundai Mobis, Hyderabd, India of 295.014 15 Questions 1 Answer of 20.161 0 Files of 153.031 0 Problems 4 Solutions Why is Software in Loop is required for auto generated code? As I know sil means testing of software. Software can be handwritten or auto-generated. But with auto generated code, why it is... ongeveer 6 jaar ago \| 0 answers \| 0 what is floating point model what is the floating point model. what is the difference between floating point and fixed point model? meer dan 7 jaar ago \| 1 answer \| 1 When to use exclusive or and parallel states in stateflow In stateflow..we have two types in decomposition Exclusive Or and Parallel. In what scenarios we can use those two? meer dan 7 jaar ago \| 1 answer \| 0 S-function configured with buses, does it support design verifier. S-function with Buses as inputs and outputs, does it support for automatic test generation using design verifier. In attached do... bijna 8 jaar ago \| 0 answers \| 0 How to know the program id to be used in actxserver('progid') command For Excel and word i am able to get the examples to use in actxserver command. Like that i want to know program id of other appl... bijna 8 jaar ago \| 0 answers \| 1 Given a circular pizza with radius _z_ and thickness _a_, return the pizza's volume. [ _z_ is first input argument.] Non-scor... ongeveer 8 jaar ago how to add data in gui? I want add data dynamically to GUI list box like below. 1.Matlab 1.1 Arrays. 1.2 Vectors. 2.Simulink 2... meer dan 8 jaar ago \| 1 answer \| 0 How can we access model explorer data? How can i access the model explorer data that is highlighted in excel document through Matlab. Please find attachment. meer dan 8 jaar ago \| 0 answers \| 0 Find the sum of all the numbers of the input vector Find the sum of all the numbers of the input vector x. Examples: Input x = [1 2 3 5] Output y is 11 Input x ... meer dan 8 jaar ago Make the vector [1 2 3 4 5 6 7 8 9 10] In MATLAB, you create a vector by enclosing the elements in square brackets like so: x = [1 2 3 4] Commas are optional, s... meer dan 8 jaar ago Times 2 - START HERE Try out this test problem first. Given the variable x as your input, multiply it by two and put the result in y. Examples:... meer dan 8 jaar ago Answered Get data type of Simulink blocks programmatically To get particular port data type,first highlight that block then you can use "get_param(gcbh,'OutDataTypeStr')". For this no nee... meer dan 8 jaar ago \| 2 We tried following command to add annotation but it's not working, it's giving warning like "new annotation "this is annotation"... bijna 9 jaar ago \| 1 answer \| 0	{"url":"https://nl.mathworks.com/matlabcentral/profile/authors/5861849?detail=all","timestamp":"2024-11-03T09:24:59Z","content_type":"text/html","content_length":"100169","record_id":"<urn:uuid:08d8c3eb-9488-4955-8915-be8634048d4d>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00687.warc.gz"}
How to Add Percentage Symbol without Multiplying 100 in Excel - Free Excel Tutorial This post will guide you how to add a percentage sign without multiplying by 100 in cells in Excel. How do I add percentage symbol only in Excel without multiplying 100. How to show % sign without multiplying with number 100 for multiple cells in Excel. Add Percentage Symbol without Multiplying 100 Generally, when you add a percentage sign to a number in Excel, it will multiply the number by 100 firstly, and then insert the percentage symbol behind the number. So if you only want to add a percentage single behind the number, you can do the following steps: Assuming that you have a list of data in range B1:B4 which contain numbers, you want to add percentage sign to those numbers. #1 select one blank cell, such as: C1 #2 type the following formula in cell C1 and press Enter key in your keyboard, and then press drag the AutoFill Handle over to cell C4 to apply this formula. #3 select range C1:C4, and go to HOME tab, click percentage button under Number group. #4 all numbers in range C1:C4 will be added a percentage sign without multiplying 100. Or you can do the following steps to achieve the same result of add percentage symbol without multiplying 100 for numbers in range B1:B4. #1 select those numbers in new column, and then right click on it, and select Format Cells from the popup menu list. And the Format Cells dialog will open. #2 click Custom under Category list box, and type in “0\%” in Type text box, and then click OK button. #3 only percentage symbol without multiplying 100 will be added in those numbers.	{"url":"https://www.excelhow.net/how-to-add-percentage-symbol-without-multiplying-100-in-excel.html","timestamp":"2024-11-05T07:00:20Z","content_type":"text/html","content_length":"87843","record_id":"<urn:uuid:04fb6e5e-bead-4b4f-9e63-aceda1e4bb22>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00663.warc.gz"}
Should American Options be Exercised Early? - Finance Train Should American Options be Exercised Early? As we know, unlike a European option, the holder of an American option can exercise the option before the expiry date. Because of this additional benefit of being able to exercise the option early, an American option is always more expensive than a European option. However, is this benefit of any real use, that is, is there a situation where the option holder will get a better payoff by exercising the option early? The answer is NO. You should never early exercise an American option, especially if it’s a non-dividend paying stock. Let’s look at the reasoning behind this. The option has intrinsic value and time value. The intrinsic value of the option is always greater than 0. Along with that the cash has time value, so you would rather delay paying the strike price by exercising it as late as possible. You could use that money to earn interest. So, a positive intrinsic value plus time value implies that you are better off selling the option rather than exercising it early. This is true for a non-dividend paying stock. However, for a dividend paying stock, the only time it may pay to exercise a call option is the day before the stock goes ex-dividend, and only if the dividend minus the cost of carry is less than the corresponding Put. By exercising, the option holder may forego the time value but will make up from the dividend received. We have used the word 'may' because the dividend may not be high enough to justify the early exercise. Data Science in Finance: 9-Book Bundle Master R and Python for financial data science with our comprehensive bundle of 9 ebooks. What's Included: • Getting Started with R • R Programming for Data Science • Data Visualization with R • Financial Time Series Analysis with R • Quantitative Trading Strategies with R • Derivatives with R • Credit Risk Modelling With R • Python for Data Science • Machine Learning in Finance using Python Each book includes PDFs, explanations, instructions, data files, and R code for all examples. Get the Bundle for $39 (Regular $57) JOIN 30,000 DATA PROFESSIONALS Free Guides - Getting Started with R and Python Enter your name and email address below and we will email you the guides for R programming and Python.	{"url":"https://financetrain.com/should-american-options-be-exercised-early","timestamp":"2024-11-01T20:00:04Z","content_type":"text/html","content_length":"92422","record_id":"<urn:uuid:6857a175-1cb0-41ba-a1b1-e8182a57fb37>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00580.warc.gz"}
Quiz for Part V: Statistical Foundations Chapter 11: Probability Theory 1. What is a sample space? □ a) The space in a graph where samples are plotted. □ b) The set of all possible outcomes of an experiment. □ c) The space between statistical variables. □ d) The set of all possible combinations of events. 2. What does a discrete probability distribution define? □ a) The probability of each point in a continuous random variable. □ b) The probability of each outcome in a sample space. □ c) The probability of a certain range of values in a random variable. □ d) None of the above. 3. Which distribution is often used to model the number of successes in a fixed number of independent Bernoulli trials? □ a) Uniform Distribution □ b) Normal Distribution □ c) Binomial Distribution □ d) Poisson Distribution Chapter 11: Probability Theory 1. What is a sample space? □ a) The space in a graph where samples are plotted. □ b) The set of all possible outcomes of an experiment. □ c) The space between statistical variables. □ d) The set of all possible combinations of events. 2. What does a discrete probability distribution define? □ a) The probability of each point in a continuous random variable. □ b) The probability of each outcome in a sample space. □ c) The probability of a certain range of values in a random variable. □ d) None of the above. 3. Which distribution is often used to model the number of successes in a fixed number of independent Bernoulli trials? □ a) Uniform Distribution □ b) Normal Distribution □ c) Binomial Distribution □ d) Poisson Distribution Chapter 11: Probability Theory 1. What is a sample space? □ a) The space in a graph where samples are plotted. □ b) The set of all possible outcomes of an experiment. □ c) The space between statistical variables. □ d) The set of all possible combinations of events. 2. What does a discrete probability distribution define? □ a) The probability of each point in a continuous random variable. □ b) The probability of each outcome in a sample space. □ c) The probability of a certain range of values in a random variable. □ d) None of the above. 3. Which distribution is often used to model the number of successes in a fixed number of independent Bernoulli trials? □ a) Uniform Distribution □ b) Normal Distribution □ c) Binomial Distribution □ d) Poisson Distribution Chapter 11: Probability Theory 1. What is a sample space? □ a) The space in a graph where samples are plotted. □ b) The set of all possible outcomes of an experiment. □ c) The space between statistical variables. □ d) The set of all possible combinations of events. 2. What does a discrete probability distribution define? □ a) The probability of each point in a continuous random variable. □ b) The probability of each outcome in a sample space. □ c) The probability of a certain range of values in a random variable. □ d) None of the above. 3. Which distribution is often used to model the number of successes in a fixed number of independent Bernoulli trials? □ a) Uniform Distribution □ b) Normal Distribution □ c) Binomial Distribution □ d) Poisson Distribution	{"url":"https://www.cuantum.tech/app/section/chapter-11-probability-theory-fcd5766bbfe74c2b9cc69ad361eeba72","timestamp":"2024-11-04T21:46:58Z","content_type":"text/html","content_length":"92482","record_id":"<urn:uuid:9834164d-6210-443e-a89a-a081b1f2ac6c>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00155.warc.gz"}
bius Mu Function Calculator Möbius Mu Function Calculator In elementary number theory, the Möbius Mu function, denoted μ(n), returns a value of -1, 0, or 1 depending on the factorization of n. If n is greater than 1 and divisible by a square integer, μ(n) = 0. Otherwise, if n = 1 or n has an even number of distinct prime factors, μ(n) = 1. And if n has an odd number of distinct prime factors, μ(n) = -1. Like the most commonly studied number theoretic functions, μ(n) is multiplicative. That is, if GCD(m,n) = 1, then μ(m)μ(n) = μ(mn). The Möbius μ function by itself is not so interesting, however, it plays an important role in many number theoretic identities and expressions. The Möbius Inversion Formula One application of μ(n) is in the Möbius Inversion Formula . If F(n) and f(n) are number theoretic functions related by the expression F(n) = where d ranges over the divisors of n, then f(n) can be written as f(n) = or equivalently f(n) = For example, consider the following summation formulas for the divisor function Euler's Totient function σₓ(n) = n = The Möbius Inversion Formula gives us two new expressions: φ(n) = More Properties of μ(n) Consider another identity: Suppose the prime factorization of n is n = p . If f(n) is any multiplicative function, then μ(d)f(d) = (1 - f(p where the p 's are the prime factors of n, and j ranges from 1 to k (the number of distinct prime factors of n). Some examples of this second identity are μ(d)σₓ(d) = (-1) ^x Σ μ(d)φ(d) = (-1) - 2) Infinite Sums Involving μ(n) The Riemann Zeta function ζ(s) is defined by the equation ζ(s) = for n = 1 to infinity. If the numerator 1 is replaced with μ(n), then 1/ζ(s) = If you set s = 1 and s = 2, you obtain the infinite sums μ(n)/n = 0, and μ(n)/n² = 6/ Two more remarkable sums are μ(n)ln(n)/n = -1 [μ(n)/n]² = 15/ © Had2Know 2010	{"url":"https://www.had2know.org/academics/mobius-mu-function-calculator.html","timestamp":"2024-11-14T17:42:23Z","content_type":"text/html","content_length":"38380","record_id":"<urn:uuid:fb2a3a53-0235-4b86-bcbe-b770fe43f125>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00516.warc.gz"}
350 liters to quarts - Gardening101.co 350 liters to quarts Hey there! Today, we are going to learn about how different units of liquid measurements work. When you’re gardening, you might use a lot of water, so it’s helpful to know how to convert one unit of measurement to another! Liters and quarts are two ways to measure liquids. A liter is commonly used in many places around the world, while a quart is often used in the United States. Now, let’s look at how many quarts are in 350 liters. To do this, we need to know that 1 liter is approximately equal to 1.056688 quarts. So, to find out how many quarts are in 350 liters, we can use the equation: 350 \, \text{liters} \times 1.056688 \, \text{quarts/liter} \approx 370.4 \, \text{quarts} This means that if you have 350 liters of water, it is about 370.4 quarts! Here are 7 objects that are exactly equal to 350 liters: 1. You would need 350 water bottles if each holds 1 liter. 2. Imagine 1400 cups if each cup holds 250 milliliters. 3. Picture a large fish tank that can hold 350 liters of water. 4. It would take 7 standard bathtubs filled with water, if each bathtub holds about 50 liters. 5. Think of a big rain barrel that can contain 350 liters of rainwater for your garden. 6. You could fill up 350 pints of water, because each pint is about 0.473 liters. 7. Finally, it’s like having 4 large buckets that can carry 87.5 liters each! Remember, knowing different units of measurement can help you keep your plants healthy and your garden blooming. Happy gardening! 🌱🌼 Leave a Reply Cancel reply	{"url":"https://gardening101.co/350-liters-to-quarts/","timestamp":"2024-11-05T21:53:22Z","content_type":"text/html","content_length":"108409","record_id":"<urn:uuid:d753f034-6dc5-4469-8502-06865408ee65>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00434.warc.gz"}
local outlier factor model Create local outlier factor model for anomaly detection Since R2022b Use the lof function to create a local outlier factor model for outlier detection and novelty detection. • Outlier detection (detecting anomalies in training data) — Use the output argument tf of lof to identify anomalies in training data. • Novelty detection (detecting anomalies in new data with uncontaminated training data) — Create a LocalOutlierFactor object by passing uncontaminated training data (data with no outliers) to lof. Detect anomalies in new data by passing the object and the new data to the object function isanomaly. LOFObj = lof(X) uses predictor data in the matrix X. LOFObj = lof(___,Name=Value) specifies options using one or more name-value arguments in addition to any of the input argument combinations in the previous syntaxes. For example, ContaminationFraction=0.1 instructs the function to process 10% of the training data as anomalies. [LOFObj,tf] = lof(___) also returns the logical array tf, whose elements are true when an anomaly is detected in the corresponding row of Tbl or X. [LOFObj,tf,scores] = lof(___) also returns an anomaly score, which is a local outlier factor value, for each observation in Tbl or X. A score value less than or close to 1 indicates a normal observation, and a value greater than 1 can indicate an anomaly. Detect Outliers Detect outliers (anomalies in training data) by using the lof function. Load the sample data set NYCHousing2015. The data set includes 10 variables with information on the sales of properties in New York City in 2015. Display a summary of the data set. NYCHousing2015: 91446x10 table BOROUGH: double NEIGHBORHOOD: cell array of character vectors BUILDINGCLASSCATEGORY: cell array of character vectors RESIDENTIALUNITS: double COMMERCIALUNITS: double LANDSQUAREFEET: double GROSSSQUAREFEET: double YEARBUILT: double SALEPRICE: double SALEDATE: datetime Statistics for applicable variables: NumMissing Min Median Max Mean Std BOROUGH 0 1 3 5 2.8431 1.3343 NEIGHBORHOOD 0 BUILDINGCLASSCATEGORY 0 RESIDENTIALUNITS 0 0 1 8759 2.1789 32.2738 COMMERCIALUNITS 0 0 0 612 0.2201 3.2991 LANDSQUAREFEET 0 0 1700 29305534 2.8752e+03 1.0118e+05 GROSSSQUAREFEET 0 0 1056 8942176 4.6598e+03 4.3098e+04 YEARBUILT 0 0 1939 2016 1.7951e+03 526.9998 SALEPRICE 0 0 333333 4.1111e+09 1.2364e+06 2.0130e+07 SALEDATE 0 01-Jan-2015 09-Jul-2015 31-Dec-2015 07-Jul-2015 2470:47:17 Remove nonnumeric variables from NYCHousing2015. The data type of the BOROUGH variable is double, but it is a categorical variable indicating the borough in which the property is located. Remove the BOROUGH variable as well. NYCHousing2015 = NYCHousing2015(:,vartype("numeric")); NYCHousing2015.BOROUGH = []; Train a local outlier factor model for NYCHousing2015. Specify the fraction of anomalies in the training observations as 0.01. [Mdl,tf,scores] = lof(NYCHousing2015,ContaminationFraction=0.01); Mdl is a LocalOutlierFactor object. lof also returns the anomaly indicators (tf) and anomaly scores (scores) for the training data NYCHousing2015. Plot a histogram of the score values. Create a vertical line at the score threshold corresponding to the specified fraction. h = histogram(scores,NumBins=50); h.Parent.YScale = 'log'; xline(Mdl.ScoreThreshold,"r-",["Threshold" Mdl.ScoreThreshold]) If you want to identify anomalies with a different contamination fraction (for example, 0.05), you can train a new local outlier factor model. [newMdl,newtf,scores] = lof(NYCHousing2015,ContaminationFraction=0.05); Note that changing the contamination fraction changes the anomaly indicators only, and does not affect the anomaly scores. Therefore, if you do not want to compute the anomaly scores again by using lof, you can obtain a new anomaly indicator with the existing score values. Change the fraction of anomalies in the training data to 0.05. newContaminationFraction = 0.05; Find a new score threshold by using the quantile function. newScoreThreshold = quantile(scores,1-newContaminationFraction) newScoreThreshold = Obtain a new anomaly indicator. newtf = scores > newScoreThreshold; Detect Novelties Create a LocalOutlierFactor object for uncontaminated training observations by using the lof function. Then detect novelties (anomalies in new data) by passing the object and the new data to the object function isanomaly. Load the 1994 census data stored in census1994.mat. The data set consists of demographic data from the US Census Bureau to predict whether an individual makes over $50,000 per year. census1994 contains the training data set adultdata and the test data set adulttest. The predictor data must be either all continuous or all categorical to train a LocalOutlierFactor object. Remove nonnumeric variables from adultdata and adulttest. adultdata = adultdata(:,vartype("numeric")); adulttest = adulttest(:,vartype("numeric")); Train a local outlier factor model for adultdata. Assume that adultdata does not contain outliers. [Mdl,tf,s] = lof(adultdata); Mdl is a LocalOutlierFactor object. lof also returns the anomaly indicators tf and anomaly scores s for the training data adultdata. If you do not specify the ContaminationFraction name-value argument as a value greater than 0, then lof treats all training observations as normal observations, meaning all the values in tf are logical 0 (false). The function sets the score threshold to the maximum score value. Display the threshold value. Find anomalies in adulttest by using the trained local outlier factor model. [tf_test,s_test] = isanomaly(Mdl,adulttest); The isanomaly function returns the anomaly indicators tf_test and scores s_test for adulttest. By default, isanomaly identifies observations with scores above the threshold (Mdl.ScoreThreshold) as Create histograms for the anomaly scores s and s_test. Create a vertical line at the threshold of the anomaly scores. h1 = histogram(s,NumBins=50,Normalization="probability"); hold on h2 = histogram(s_test,h1.BinEdges,Normalization="probability"); xline(Mdl.ScoreThreshold,"r-",join(["Threshold" Mdl.ScoreThreshold])) h1.Parent.YScale = 'log'; h2.Parent.YScale = 'log'; legend("Training Data","Test Data",Location="north") hold off Display the observation index of the anomalies in the test data. ans = 0x1 empty double column vector The anomaly score distribution of the test data is similar to that of the training data, so isanomaly does not detect any anomalies in the test data with the default threshold value. You can specify a different threshold value by using the ScoreThreshold name-value argument. For an example, see Specify Anomaly Score Threshold. Input Arguments Tbl — Predictor data Predictor data, specified as a table. Each row of Tbl corresponds to one observation, and each column corresponds to one predictor variable. Multicolumn variables and cell arrays other than cell arrays of character vectors are not allowed. The predictor data must be either all continuous or all categorical. If you specify Tbl, the lof function assumes that a variable is categorical if it is a logical vector, unordered categorical vector, character array, string array, or cell array of character vectors. If Tbl includes both continuous and categorical values, and you want to identify all predictors in Tbl as categorical, you must specify CategoricalPredictors as "all". To use a subset of the variables in Tbl, specify the variables by using the PredictorNames name-value argument. Data Types: table X — Predictor data numeric matrix Predictor data, specified as a numeric matrix. Each row of X corresponds to one observation, and each column corresponds to one predictor variable. The predictor data must be either all continuous or all categorical. If you specify X, the lof function assumes that all predictors are continuous. To identify all predictors in X as categorical, specify CategoricalPredictors as "all". You can use the PredictorNames name-value argument to assign names to the predictor variables in X. Data Types: single \| double Name-Value Arguments Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter. Example: SearchMethod=exhaustive,Distance=minkowski uses the exhaustive search algorithm with the Minkowski distance. BucketSize — Maximum data points in node 50 (default) \| positive integer value Maximum number of data points in the leaf node of the Kd-tree, specified as a positive integer value. This argument is valid only when SearchMethod is "kdtree". Example: BucketSize=40 Data Types: single \| double CacheSize — Size of Gram matrix in megabytes 1000 (default) \| positive scalar \| "maximal" Size of the Gram matrix in megabytes, specified as a positive scalar or "maximal". For the definition of the Gram matrix, see Algorithms. The lof function can use a Gram matrix when the Distance name-value argument is "fasteuclidean". When CacheSize is "maximal", lof attempts to allocate enough memory for an entire intermediate matrix whose size is MX-by-MX, where MX is the number of rows of the input data, X or Tbl. CacheSize does not have to be large enough for an entire intermediate matrix, but must be at least large enough to hold an MX-by-1 vector. Otherwise, lof uses the "euclidean" distance. If Distance is "fasteuclidean" and CacheSize is too large or "maximal", lof might attempt to allocate a Gram matrix that exceeds the available memory. In this case, MATLAB^® issues an error. Example: CacheSize="maximal" Data Types: double \| char \| string CategoricalPredictors — Categorical predictor flag [] \| "all" Categorical predictor flag, specified as one of the following: • "all" — All predictors are categorical. By default, lof uses the Hamming distance ("hamming") for the Distance name-value argument. • [] — No predictors are categorical, that is, all predictors are continuous (numeric). In this case, the default Distance value is "euclidean". The predictor data for lof must be either all continuous or all categorical. • If the predictor data is in a table (Tbl), lof assumes that a variable is categorical if it is a logical vector, unordered categorical vector, character array, string array, or cell array of character vectors. If Tbl includes both continuous and categorical values, and you want to identify all predictors in Tbl as categorical, you must specify CategoricalPredictors as "all". • If the predictor data is a matrix (X), lof assumes that all predictors are continuous. To identify all predictors in X as categorical, specify CategoricalPredictors as "all". lof encodes categorical variables as numeric variables by assigning a positive integer value to each category. When you use categorical predictors, ensure that you use an appropriate distance metric Example: CategoricalPredictors="all" ContaminationFraction — Fraction of anomalies in training data 0 (default) \| numeric scalar in the range [0,1] Fraction of anomalies in the training data, specified as a numeric scalar in the range [0,1]. • If the ContaminationFraction value is 0 (default), then lof treats all training observations as normal observations, and sets the score threshold (ScoreThreshold property value of LOFObj) to the maximum value of scores. • If the ContaminationFraction value is in the range (0,1], then lof determines the threshold value so that the function detects the specified fraction of training observations as anomalies. Example: ContaminationFraction=0.1 Data Types: single \| double Cov — Covariance matrix positive definite matrix of scalar values Covariance matrix, specified as a positive definite matrix of scalar values representing the covariance matrix when the function computes the Mahalanobis distance. This argument is valid only when Distance is "mahalanobis". The default value is the covariance matrix computed from the predictor data (Tbl or X) after the function excludes rows with duplicated values and missing values. Data Types: single \| double Distance — Distance metric character vector \| string scalar Distance metric, specified as a character vector or string scalar. • If all the predictor variables are continuous (numeric) variables, then you can specify one of these distance metrics. Value Description "euclidean" Euclidean distance Euclidean distance using an algorithm that usually saves time when the number of elements in a data point exceeds 10. See Algorithms. "fasteuclidean" applies only to the "fasteuclidean" "exhaustive" SearchMethod. "mahalanobis" Mahalanobis distance — You can specify the covariance matrix for the Mahalanobis distance by using the Cov name-value argument. "minkowski" Minkowski distance — You can specify the exponent of the Minkowski distance by using the Exponent name-value argument. "chebychev" Chebychev distance (maximum coordinate difference) "cityblock" City block distance "correlation" One minus the sample correlation between observations (treated as sequences of values) "cosine" One minus the cosine of the included angle between observations (treated as vectors) "spearman" One minus the sample Spearman's rank correlation between observations (treated as sequences of values) If you specify one of these distance metrics for categorical predictors, then the software treats each categorical predictor as a numeric variable for the distance computation, with each category represented by a positive integer. The Distance value does not affect the CategoricalPredictors property of the trained model. • If all the predictor variables are categorical variables, then you can specify one of these distance metrics. Value Description "hamming" Hamming distance, which is the percentage of coordinates that differ "jaccard" One minus the Jaccard coefficient, which is the percentage of nonzero coordinates that differ If you specify one of these distance metrics for continuous (numeric) predictors, then the software treats each continuous predictor as a categorical variable for the distance computation. This option does not change the CategoricalPredictors value. The default value is "euclidean" if all the predictor variables are continuous, and "hamming" if all the predictor variables are categorical. If you want to use the Kd-tree algorithm (SearchMethod="kdtree"), then Distance must be "euclidean", "cityblock", "minkowski", or "chebychev". For more information on the various distance metrics, see Distance Metrics. Example: Distance="jaccard" Data Types: char \| string Exponent — Minkowski distance exponent 2 (default) \| positive scalar value Minkowski distance exponent, specified as a positive scalar value. This argument is valid only when Distance is "minkowski". Example: Exponent=3 Data Types: single \| double IncludeTies — Tie inclusion flag false or 0 (default) \| true or 1 Tie inclusion flag indicating whether the software includes all the neighbors whose distance values are equal to the kth smallest distance, specified as logical 0 (false) or 1 (true). If IncludeTies is true, the software includes all of these neighbors. Otherwise, the software includes exactly k neighbors. Example: IncludeTies=true Data Types: logical NumNeighbors — Number of nearest neighbors min(20,n-1) where n is the number of unique rows in predictor data (default) \| positive integer value Number of nearest neighbors in the predictor data (Tbl or X) to find for computing the local outlier factor values, specified as a positive integer value. The default value is min(20,n-1), where n is the number of unique rows in the predictor data. Example: NumNeighbors=3 Data Types: single \| double SearchMethod — Nearest neighbor search method "kdtree" \| "exhaustive" Nearest neighbor search method, specified as "kdtree" or "exhaustive". • "kdtree" — This method uses the Kd-tree algorithm to find nearest neighbors. This option is valid when the distance metric (Distance) is one of the following: □ "euclidean" — Euclidean distance □ "cityblock" — City block distance □ "minkowski" — Minkowski distance □ "chebychev" — Chebychev distance • "exhaustive" — This method uses the exhaustive search algorithm to find nearest neighbors. □ When you compute local outlier factor values for the predictor data (Tbl or X), the lof function finds nearest neighbors by computing the distance values from all points in the predictor data to each point in the predictor data. □ When you compute local outlier factor values for new data Xnew using the isanomaly function, the function finds nearest neighbors by computing the distance values from all points in the predictor data (Tbl or X) to each point in Xnew. The default value is "kdtree" if the predictor data has 10 or fewer columns, the data is not sparse, and the distance metric (Distance) is valid for the Kd-tree algorithm. Otherwise, the default value is "exhaustive". Output Arguments LOFObj — Trained local outlier factor model LocalOutlierFactor object Trained local outlier factor model, returned as a LocalOutlierFactor object. You can use the object function isanomaly with LOFObj to find anomalies in new data. tf — Anomaly indicators logical column vector Anomaly indicators, returned as a logical column vector. An element of tf is logical 1 (true) when the observation in the corresponding row of Tbl or X is an anomaly, and logical 0 (false) otherwise. tf has the same length as Tbl or X. lof identifies observations with scores above the threshold (ScoreThreshold property value of LOFObj) as anomalies. The function determines the threshold value to detect the specified fraction ( ContaminationFraction name-value argument) of training observations as anomalies. scores — Anomaly scores (local outlier factor values) numeric column vector Anomaly scores (local outlier factor values), returned as a numeric column vector whose values are nonnegative. scores has the same length as Tbl or X, and each element of scores contains an anomaly score for the observation in the corresponding row of Tbl or X. A score value less than or close to 1 indicates a normal observation, and a value greater than 1 can indicate an anomaly. More About Local Outlier Factor Distance Metrics Missing Values lof considers NaN, '' (empty character vector), "" (empty string), <missing>, and <undefined> values in Tbl and NaN values in X to be missing values. • lof does not use observations with missing values. • lof assigns the anomaly score of NaN and anomaly indicator of false (logical 0) to observations with missing values. Fast Euclidean Distance Algorithm The "fasteuclidean" Distance calculates Euclidean distances using extra memory to save computational time. This algorithm is named "Euclidean Distance Matrix Trick" in Albanie [2] and elsewhere. Internal testing shows that this algorithm saves time when the number of predictors exceeds 10. The "fasteuclidean" distance does not support sparse data. To find the matrix D of distances between all the points x[i] and x[j], where each x[i] has n variables, the algorithm computes distance using the final line in the following equations: $\begin{array}{c}{D}_{i,j}^{2}=‖{x}_{i}-{x}_{j}{‖}^{2}\\ ={\left(}^{{x}_{i}}\left({x}_{i}-{x}_{j}\right)\\ =‖{x}_{i}{‖}^{2}-2{x}_{i}^{T}{x}_{j}+‖{x}_{j}{‖}^{2}.\end{array}$ The matrix ${x}_{i}^{T}{x}_{j}$ in the last line of the equations is called the Gram matrix. Computing the set of squared distances is faster, but slightly less numerically stable, when you compute and use the Gram matrix instead of computing the squared distances by squaring and summing. For more information, see Albanie [2]. To store the Gram matrix, the software uses a cache with the default size of 1e3 megabytes. You can set the cache size using the CacheSize name-value argument. If the value of CacheSize is too large or "maximal", lof might try to allocate a Gram matrix that exceeds the available memory. In this case, MATLAB issues an error. [1] Breunig, Markus M., et al. “LOF: Identifying Density-Based Local Outliers.” Proceedings of the 2000 ACM SIGMOD International Conference on Management of Data, 2000, pp. 93–104. Version History Introduced in R2022b R2023b: "fasteuclidean" distance support The lof function supports the "fasteuclidean" Distance algorithm. This algorithm usually computes distances faster than the default "euclidean" algorithm when the number of variables in a data point exceeds 10. The algorithm uses extra memory to store an intermediate Gram matrix (see Algorithms). Set the size of this memory allocation using the CacheSize name-value argument.	{"url":"https://au.mathworks.com/help/stats/lof.html","timestamp":"2024-11-08T11:38:37Z","content_type":"text/html","content_length":"184821","record_id":"<urn:uuid:58acf0f1-e992-46d5-8610-fc6da51fa320>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00704.warc.gz"}
How Many Miles per hour Is 640.5 Meters per second? 640.5 meters per second in miles per hour How many miles per hour in 640.5 meters per second? 640.5 meters per second equals 1432.758 miles per hour Unit Converter Conversion formula The conversion factor from meters per second to miles per hour is 2.2369362920544, which means that 1 meter per second is equal to 2.2369362920544 miles per hour: 1 m/s = 2.2369362920544 mph To convert 640.5 meters per second into miles per hour we have to multiply 640.5 by the conversion factor in order to get the velocity amount from meters per second to miles per hour. We can also form a simple proportion to calculate the result: 1 m/s → 2.2369362920544 mph 640.5 m/s → V[(mph)] Solve the above proportion to obtain the velocity V in miles per hour: V[(mph)] = 640.5 m/s × 2.2369362920544 mph V[(mph)] = 1432.7576950608 mph The final result is: 640.5 m/s → 1432.7576950608 mph We conclude that 640.5 meters per second is equivalent to 1432.7576950608 miles per hour: 640.5 meters per second = 1432.7576950608 miles per hour Alternative conversion We can also convert by utilizing the inverse value of the conversion factor. In this case 1 mile per hour is equal to 0.00069795472287276 × 640.5 meters per second. Another way is saying that 640.5 meters per second is equal to 1 ÷ 0.00069795472287276 miles per hour. Approximate result For practical purposes we can round our final result to an approximate numerical value. We can say that six hundred forty point five meters per second is approximately one thousand four hundred thirty-two point seven five eight miles per hour: 640.5 m/s ≅ 1432.758 mph An alternative is also that one mile per hour is approximately zero point zero zero one times six hundred forty point five meters per second. Conversion table meters per second to miles per hour chart For quick reference purposes, below is the conversion table you can use to convert from meters per second to miles per hour	{"url":"https://convertoctopus.com/640-5-meters-per-second-to-miles-per-hour","timestamp":"2024-11-05T00:07:31Z","content_type":"text/html","content_length":"34929","record_id":"<urn:uuid:2b4d64fc-6ed2-4115-976f-232e7839b8af>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00512.warc.gz"}
3 Places to See the Coordinate Plane in Action I was browsing through the book Family Math and found a nifty game called Hurkle. It goes a little something like this: 1. Someone picks a place on the coordinate plane (in secret) for Hurkle to live. 2. Others guess Hurkle’s home by giving coordinates and pointing out those coordinates. 3. When they are wrong, they are given a hint of which way to go (from their guess) to find Hurkle. The coordinate “space” is around too. This made me think of the conversation I had with a neighbor the other day while walking our children to the park. When Scotty beams someone in Star Trek, he needs coordinates. But those coordinates must be measured from some origin (0,0,0) in the universe to make any sense. So where’s the origin? A student of mine long ago found that the origin was Earth. My neighbor argued it should be Vulcan – apparently they started the United Federation of Planets. And we can find coordinates at work in board games. With thoughts of these two in my mind, I wonder what ever happened to the game of Battleship. It’s a great game of coordinates along with logic. Once you hit a ship, you have to go in each direction to determine how big the ship is and in what direction. I’m excited about looking around my world today for more coordinate systems. Where do you see them in your world? And how do you apply them in your teaching? Leave your thoughts and ideas in the This post may contain affiliate links. When you use them, you support us so we can continue to provide free content! 2 Responses to 3 Places to See the Coordinate Plane in Action 1. We’re still warming up to formal coordinate systems, but we obviously talk about which drawer to find one’s socks in, which block in our neighbourhood to find the yard with the wildflower garden in, and where to look on a map for the neat place in our story. One related issue we pondered the other week, when we were investigating the Big Bang, was the differential expansion of various portions of the universe, caused by the gravitational forces of the parts on one another. Mapping out the locations of places in the universe over that early time would be amazing! □ Thanks, Siggi! Wow – differential expansion of various portions of the universe… that’s awesome! Could you mark x-y-z numbers on the walls starting at any corner and the determine the ordinates where the socks live? (Masking tape might be good for that.) Leave a reply This site uses Akismet to reduce spam. Learn how your comment data is processed.	{"url":"https://mathfour.com/algebra/3-places-to-see-the-coordinate-plane-in-action","timestamp":"2024-11-14T15:17:25Z","content_type":"text/html","content_length":"38346","record_id":"<urn:uuid:8996eedb-b945-4c4f-813a-62c7baf93be5>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00686.warc.gz"}
First Data Anonymous Employee Annual Salaries \| CareerBliss +$3K (4%) more than national average Anonymous Employee salary ($70K) +$18K (28%) more than average First Data salary ($55K) +$30K (35%) more than average First Data Anonymous Employee salary ($70K) "The pay is appropriate. However, decisions regarding bonus payments and stock ownership options are not satisfactory to me." +$10K (13%) more than average First Data Anonymous Employee salary ($70K) "Underpaid, most people at my level are making at least $15000 more." -$5K (7%) less than average First Data Anonymous Employee salary ($70K) "Little bit low. Waiting to see how raises go." +$8K (10%) more than average First Data Anonymous Employee salary ($70K) "Adequate for the responsibility." -$6K (8%) less than national average Anonymous Employee salary ($70K) +$9K (15%) more than average First Data salary ($55K) +$85K (75%) more than average First Data Anonymous Employee salary ($70K) "Salary is low. Residuals and commissions are most of total earnings." -$18K (29%) less than average First Data Anonymous Employee salary ($70K) "The salary was ok, but well below market value." -$28K (50%) less than average First Data Anonymous Employee salary ($70K) "I think it's less compared to other companies." +$5K (6%) more than average First Data Anonymous Employee salary ($70K) -$36K (69%) less than national average Anonymous Employee salary ($70K) -$21K (47%) less than average First Data salary ($55K) -$41K (82%) less than average First Data Anonymous Employee salary ($70K) "The salary is a great starting salary. There is a lot of overtime opportunity in the department." -$27K (47%) less than average First Data Anonymous Employee salary ($70K) "Underpaid, but it was more than most made in the position." -$41K (82%) less than average First Data Anonymous Employee salary ($70K) +$9K (12%) more than national average Anonymous Employee salary ($70K) +$24K (35%) more than average First Data salary ($55K) +$22K (27%) more than average First Data Anonymous Employee salary ($70K) "Believe it was fair. Though I was promised ability to telecommute 3 days a week and that hasn't happened." -$5K (7%) less than average First Data Anonymous Employee salary ($70K) "Not well seeing as I know many other developers and competitors that i could easily make more" -$20K (33%) less than national average Anonymous Employee salary ($70K) -$5K (9%) less than average First Data salary ($55K) "Need to get more education/training to grow in my career." -$15K (24%) less than national average Anonymous Employee salary ($70K) Equal to average First Data salary ($55K) -$7K (10%) less than national average Anonymous Employee salary ($70K) +$8K (13%) more than average First Data salary ($55K) "I feel I am paid reasonably." -$5K (7%) less than national average Anonymous Employee salary ($70K) +$10K (16%) more than average First Data salary ($55K) "I feel as though I am adequately compensated for my role." +$5K (6%) more than national average Anonymous Employee salary ($70K) +$20K (30%) more than average First Data salary ($55K) "It is decent, but there were no possibilities for an increment in any time in the near future. Bonuses were in terms of stocks but the company wasn't public." +$10K (13%) more than national average Anonymous Employee salary ($70K) +$25K (37%) more than average First Data salary ($55K) "In our field of work I do believe we are underpaid, there is different opportunities out there that have greater salary options." The salary for Anonymous Employee at First Data is $7,472,000 annually. AOL pays the highest salary for the Anonymous Employee position at $22,593,000 annually. Individual Advocacy Group pays the lowest salary for the Anonymous Employee position at $24,000 annually.	{"url":"https://www.careerbliss.com/first-data/salaries/anonymous-employee/","timestamp":"2024-11-04T00:42:01Z","content_type":"text/html","content_length":"66088","record_id":"<urn:uuid:8ec3948a-4142-44e9-85b2-1b63e67830c6>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00888.warc.gz"}
Getting Started Getting Started: X11 Procedure The most common use of the X11 procedure is to produce a seasonally adjusted series. Eliminating the seasonal component from an economic series facilitates comparison among consecutive months or quarters. A plot of the seasonally adjusted series is often more informative about trends or location in a business cycle than a plot of the unadjusted series. The following example shows how to use PROC X11 to produce a seasonally adjusted series, In the multiplicative model, the trend cycle component The naming convention used in PROC X11 for the tables follows the original U.S. Bureau of the Census X-11 Seasonal Adjustment program specification (Shiskin, Young, and Musgrave; 1967). Also, see the section Printed Output. This convention is outlined in Figure 33.1. The tables corresponding to parts A – C are intermediate calculations. The final estimates of the individual components are found in the D tables: table D10 contains the final seasonal factors, table D12 contains the final trend cycle, and table D13 contains the final irregular series. If you are primarily interested in seasonally adjusting a series without consideration of intermediate calculations or diagnostics, you only need to look at table D11, the final seasonally adjusted series. For further details about the X-11-ARIMA tables, see Ladiray and Quenneville (2001).	{"url":"http://support.sas.com/documentation/cdl/en/etsug/63348/HTML/default/etsug_x11_sect002.htm","timestamp":"2024-11-08T02:46:01Z","content_type":"application/xhtml+xml","content_length":"12226","record_id":"<urn:uuid:8e18d5db-cb05-452a-8a18-d3aceca01144>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00284.warc.gz"}
Question ID - 152276 \| SaraNextGen Top Answer The head lights of a jeep are 1.2 m apart. If the pupil of the eye of an observer has a diameter of 2mm and light of wavelength a) 33.9 km b) 33.9 m c) 3.34 km d) 3.39 m The head lights of a jeep are 1.2 m apart. If the pupil of the eye of an observer has a diameter of 2mm and light of wavelength Distance of jeep, Where D = diameter of lens d = separation between sources.	{"url":"https://www.saranextgen.com/homeworkhelp/doubts.php?id=152276","timestamp":"2024-11-15T02:56:46Z","content_type":"text/html","content_length":"17193","record_id":"<urn:uuid:b85f5e39-26bd-48fa-b639-65020d3a3e0c>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00194.warc.gz"}
Hashing technology In cryptocurrency Hashing refers to the transformation of any kind and size of information in the form of a set of characters, the so-called hash, which has a fixed length. This conversion occurs using a mathematical formula, also known as a hash function. This technology is fundamental to cryptocurrency because it allows blockchains and other distributed systems to achieve high levels of data integrity and security. Hashing algorithms in cryptocurrencies are designed in such a way that their function is one-way, meaning that data cannot be reversed without investing a large amount of time and resources to perform calculations. In other words, it is fairly easy to generate an output from input data, but relatively difficult to perform the process in the reverse direction (generate input data from output data). The harder it is to find the input value, the more secure the hashing algorithm is considered to be. How does a hash function work? Different types of hash functions produce different output values, but the possible output size for each hashing algorithm is always constant. For example, the SHA-256 algorithm can produce output in 256-bit format only, while SHA-1 always generates a 160-bit digest. To illustrate this, let's run the words "Binance" and "binance" through the SHA-256 hashing algorithm (the one used in bitcoin): Note that a slight change (case of the first letter) resulted in a completely different hash value. As we have used SHA-256 in this example, the output will always have a fixed size of 256 bits (or 64 characters), regardless of the input value. Furthermore, it does not matter how many times we pass the two words through the algorithm, the two outputs will not change because they are constant. Why is hashing technology needed? Cryptographic hash functions are used extensively in information security applications for message authentication and digital fingerprinting. When it comes to Bitcoin, cryptographic hash functions are an integral part of the mining process and also play a major role in generating new keys and addresses. Hashing is irreplaceable when dealing with huge amounts of information. For example, you can run a large file or set of data through a hash function and then use the output to quickly check the accuracy and integrity of the data. This is possible because of the deterministic nature of hash functions: an input will always result in a simplified compressed output (hash). This method eliminates the need to store and remember large amounts of data. In fact, virtually all cryptocurrency protocols rely on hashing to bind and compress groups of transactions into blocks, as well as to create cryptographic connectivity and efficiently build a chain of blocks. Security of a hash function In order to crack a hash will require countless attempts of brute force matching numbers, conventionally using the scientific method, it will be necessary to reverse the hash function until the corresponding output will not be obtained. Nevertheless, there is a possibility that different inputs will produce the same output, in which case a collision arises (confrontation of interests, unreliability, lack of confidence in authenticity). From a technical point of view, a cryptographic hash function must meet three properties to be considered secure. We can describe them as: collision-resistance, and first- and second-problem Before we start parsing each property, let us summarize their logic in three short sentences. - Collision resistance: it is impossible to find two different inputs that produce a hash similar to the output. - Resistance to finding the first sample: there is no way or algorithm to reverse the hash function (finding the input by a given output). - Stability to search for second sample: it is impossible to find any second input that would overlap with the first one. Resistance to collision As mentioned earlier, collision occurs when different inputs produce the same hash. Thus, a hash function is considered collision-resistant until someone detects a collision. Note that collisions will always exist for any of the hash functions, due to the infinite number of input data and the limited number of outputs. Thus, a hash function is collision-resistant when the probability of detection is so small that it would require millions of years of computation. For this reason, while there are no collision-free hash functions, some are so strong that they can be considered robust (e.g., SHA-256). Resistance to first-arrival lookups This property is closely related to the concept of one-way functions. A hash function is considered robust to finding the first sample, as long as there is a very low probability that someone can find an input with which to generate a particular output. Note that this property is different from the previous one, since an attacker would need to guess the input based on a particular output. This type of collision occurs when someone finds two different inputs that produce the same code in the output, without giving any importance to the input data that was used to do so. The first-sample lookup resistance property is valuable for data protection because a simple hash of a message can prove its authenticity without the need to divulge additional information. In practice, many service providers and web applications store and use hashes generated from passwords instead of using them in text format. Resistance to finding the second sample To simplify your understanding, we can say that this type of resilience is somewhere in between the other two properties. The attack of finding a second sample is to find a particular input with which to generate output that was originally formed through other inputs that were known in advance. I agree it sounds confusing, but in essence, this finding a second sample attack involves collision detection, but instead of finding two random inputs that generate the same hash, the attack aims to find the input data with which to recreate the hash that was originally generated by another input. Hashing Technology in Mining There are many steps in mining that are done with hash functions, they include checking the balance, linking transaction inputs and outputs, and hashing all the transactions in the block. But one of the main reasons the bitcoin blockchain is secure is that miners must perform as many hash-related operations as possible in order to eventually find the right solution for the next block. A miner must try to pick up several different inputs when creating a hash for their candidate block. It will only be possible to verify the block if the hash output is correctly generated starting with a certain number of zeros. The number of zeros determines the complexity of the mining and it varies depending on the hash rate of the network. In this case, the hash rate is the amount of power of your computer that you invest in mining bitcoins. If the hash rate starts to increase, the bitcoin protocol will automatically adjust the mining complexity so that the average time required to mine a block is no more than 10 minutes. If several miners decide to stop mining, resulting in a significant decrease in hash rate, the mining complexity will be adjusted to temporarily ease the computational work (until the average block formation time returns to 10 minutes). Note that miners do not need to look for collisions, due to some number of hashes they can generate as a valid output (starting with a certain number of zeros). Thus, there are several possible solutions for a given block and miners need to find only one of them, according to a threshold that is determined by the difficulty of mining. Since bitcoin mining is such a costly task, there is no reason for miners to cheat the system, as this would result in significant financial losses. Accordingly, the more miners join the blockchain, the bigger and stronger it becomes. Hash technology is definitely one of the main tools when dealing with huge amounts of data in IT and cryptocurrency. Combined with cryptography, hashing algorithms are hard to overestimate, they are very versatile and offer security and multiple authentication methods. Thus, cryptographic hash functions are vital for almost all cryptocurrency networks, so understanding their properties and working mechanisms is definitely useful for anyone interested in cryptocurrency and blockchain technology. Thank you for your attention and we hope this article was useful for you! Fortune favor you on your way and see you soon! Always yours C.J.	{"url":"https://cryptojoker777.com/tpost/27uyjgpem1-hashing-technology","timestamp":"2024-11-06T17:40:58Z","content_type":"text/html","content_length":"45559","record_id":"<urn:uuid:127d0944-253f-4245-857e-6c973a097590>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00689.warc.gz"}
Ising model in a boundary magnetic field with random discontinuities We consider a two-dimensional Ising field theory on a space with boundary in the presence of a piecewise constant boundary magnetic field which is allowed to change value discontinuously along the boundary. We assume zero magnetic field in the bulk. The positions of discontinuities are averaged over as in the annealed disorder. This model is described by a boundary field theory in which a superposition of the free spin boundary condition is perturbed by a collection of boundary condition changing operators. The corresponding boundary couplings give the allowed constant values of the magnetic field as well as the fugacities for the transitions between them. We show that when the value of the magnetic field is allowed to take only two different values which are the same in magnitude but have different signs the model can be described by a quadratic Lagrangian. We calculate and analyse the exact reflection matrix for this model. We also calculate the boundary entropy and study in detail the space of RG flows in a three-parameter space and with four different infrared fixed points. We discuss the likely breakdown of integrability in the extended model which allows for two generic values of the boundary magnetic field, backing it by some calculations. • boundary conformal field theory • Ising model with boundary magnetic field • renormalisation group flows ASJC Scopus subject areas • Statistical and Nonlinear Physics • Statistics and Probability • Modelling and Simulation • Mathematical Physics • General Physics and Astronomy Dive into the research topics of 'Ising model in a boundary magnetic field with random discontinuities'. Together they form a unique fingerprint.	{"url":"https://researchportal.hw.ac.uk/en/publications/ising-model-in-a-boundary-magnetic-field-with-random-discontinuit","timestamp":"2024-11-05T13:20:47Z","content_type":"text/html","content_length":"59879","record_id":"<urn:uuid:3beedefe-1f38-41ea-bc13-0751161ce1df>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00195.warc.gz"}
Propositional Logic Archives - Computing Learner As usual, we should start with the appropriate definitions. Remember that in mathematics, we must know the definitions before we can start trying to solve a problem. Definitions Let p and q be propositions. The disjunction of p and q, denoted by p ∨ q, is the proposition “p or q.” The disjunction p ∨ […] For each of these sentences, state what the sentence means if the logical connective or is an inclusive or (that is, a disjunction) versus an exclusive or. Which of these meanings of or do you think is intended? Read More » Propositional logic Exercise 11. Write these propositions using p and q and logical connectives (including negations) Solve the following exercise. 11. Let p and q be the propositions p: It is below freezing. q: It is snowing. Write these propositions using p and q and logical connectives (including negations). a) It is below freezing and snowing.b) It is below freezing but not snowing.c) It is not below freezing and it is Propositional logic Exercise 11. Write these propositions using p and q and logical connectives (including negations) Read More » Propositional Logic: Exercise 7 from the textbook Suppose that during the most recent fiscal year, the annual revenue of Acme Computer was 138 billion dollars and its net profit was 8 billion dollars, the annual revenue of Nadir Software was 87 billion dollars and its net profit was 5 billion dollars, and the annual revenue of Quixote Media was 111 billion dollars Propositional Logic: Exercise 7 from the textbook Read More » What is the negation of each of these propositions? In this post, I’ll show some examples of how to negate propositions. As I always recommend, let’s start with the definitions. Definitions Let p be a proposition. The negation of p, denoted by ¬p, is the statement “It is not the case that p.” The proposition ¬p is read “not p.” The truth value of What is the negation of each of these propositions? Read More » Which of these sentences are propositions? What are the truth values of those that are propositions? Propositional logic is a very important topic in Discrete Mathematics. It is part of the foundations every student should know. In this post, I’ll show how I solve this specific type of exercise. In mathematics, definitions are very important. As I always recommend to my students, when you are starting a new topic, and you Which of these sentences are propositions? What are the truth values of those that are propositions? Read More » Negating propositions A classic exercise in Discrete Mathematics is to negate a given proposition. Here, I’ll explain two things to consider after seeing how many students have problems solving this type of exercise. Background A proposition is a sentence that states a fact. It is True or False but not both. An example of a proposition is Negating propositions Read More » Propositional Logic: Exercise 6 The purpose of this post is to explain how to solve this type of exercise. Suppose that Smartphone A has 256 MB RAM and 32 GB ROM, and the resolution of its camera is 8 MP; Smartphone B has 288 MB RAM and 64 GB ROM, and the resolution of its camera is 4 MP, Propositional Logic: Exercise 6 Read More »	{"url":"https://computinglearner.com/category/dm/propositional-logic/","timestamp":"2024-11-10T20:44:51Z","content_type":"text/html","content_length":"133092","record_id":"<urn:uuid:366146c9-76f3-4524-8b33-f5552679ae1d>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00609.warc.gz"}
Control of large scale topography on silicon wafers Control of large scale topography on silicon wafers A method for the characterization of large scale wafer topography is applied to improving yields in the manufacture large scale integrated (LSI) devices. First, the heights at the center, the edge and an intermediate point are measured on eight equally spaced radii. This provides eight values each for Y.sub.s and Y.sub.e which are averaged. Then the shape angle .alpha. is computed using the following equation: ##EQU1## The shape magnitude M is also computed using the following equation: ##EQU2## The thus computed values of .alpha. and M are correlated with individual wafer characteristics as to device performance and yield. Based on these results, the wafer processing is controlled to provide optimal wafer yield and isolation characteristics. 1. Field of the Invention The present invention generally relates to the manufacture of integrated circuits in silicon wafers and, more particularly, to the correlation of the large scale topography of a semiconductor wafer to the characteristics of devices subsequently formed on the wafer. Where significant correlations are found, the large scale topography of subsequent wafers are modified to optimize device characteristics and yield. 2. Description of the Prior Art In the manufacture of large scale integrated (LSI) circuits, there are various factors which contribute to increased circuit density. Among these are reductions in feature size and an increase in device and circuit complexity. The greater circuit densities are being achieved with the use of several types of transistors on a single chip, allowing for greater design flexibility. The varying transistor types as well as their closer proximity require improvements in device isolation. Current technology, however, does not explain isolation and device junction leakage related failures in some device structures. It is important to address this problem since it is an important factor in both increased circuit density and product yield. A study was made by the inventors to determine the effects of large scale surface topography (LST) which exist on product wafers prior to defining the isolation structures on leakage limited yields (LLY). Specific topographic configurations were characterized using a shape factor defined by the angle and magnitude of the bending pattern for each wafer. This shape factor was used to correlate LST with LLY. The results of the study showed that specific unfavorable LST configurations contribute to low test yields at trench maze, post Pt and K metal as well as final test yields. From these results, the inventors have concluded that these lower yields can be attributed to the stress induced defects which arise from unfavorable LST configurations. It is therefore an object of the present invention to provide a method of characterizing large scale wafer topography critical to device yields. It is another object of the invention to provide a method of modifying surface topography to more favorable configurations which optimize device characteristics and yield. According to the invention, a method is provided for determining the large scale topography of wafers by measuring wafer height, relative to a central reference point, at selected radial positions over the surface of the wafer. A shape angle, .alpha., and shape magnitude, M, are computed based on these measurements. The .alpha. and M values for each wafer are correlated with device performance and yield. A model predicts the existence of four critical shapes where surface inversions may occur (convex to concave and vice versa) which may result in stress inversions (tensile to compressive and vice versa). Wafers are selected, using the characterization technique, which have unfavorable configurations at any point in the processing. The shape of these wafers is then altered to a favorable configuration before processing is continued. This is accomplished by producing a compensating non-uniform film on the backside of the wafer. This is achieved by the selective removal of the backside films or the addition of films typically used in semiconductor processing. By limiting this reconfiguration processing to the backside, the frontside can be left undisturbed. The foregoing and other objects, aspects and advantages will be better understood from the following detailed description of a preferred embodiment of the invention with reference to the drawings, in FIG. 1 is a flow diagram showing the front end process of split lot comparison; FIG. 2 is a topographic map of the surface of a silicon wafer, and FIG. 2A is a three-dimensional projection of the topographic map; FIG. 3 is a graph illustrating the definition of warpage and bow; FIG. 4 is a graph showing the dependence of warpage on bow; FIG. 5 is a graph showing measurements of relative elevations along a radius at an intermediate site and the edge of a wafer relative to the center; FIG. 6 is a graph showing a map of data points defining the shape angle .alpha. and magnitude M; FIG. 7 is a graph showing the theoretical model angle .beta. versus shape angle .alpha.; FIG. 8 is a family of graphs illustrating radial profiles of selected axially symmetric topographies; FIG. 9 is a graph showing population (Y.sub.s, Y.sub.e) data prior to trench CVD oxidation; FIG. 10 is a graph showing model angle, .beta., versus shape angle .alpha. illustrating the distribution of data points prior to trench CVD oxidation; FIG. 11 is a graph showing the dependence of isolation maze BVcc LLY on wafer bow prior to trench CVD oxidation; FIG. 12 is a graph showing the dependence of isolation maze LLY on shape angel, .alpha., prior to trench CVD oxidation; FIG. 13 is a graph showing the dependence of isolation maze LLY on modeangle, .beta., prior to trench CVD oxidation; FIG. 14 is a graph showing the dependence of BVcbo LLY on shape angle, .alpha., prior to trench CVD oxidation; FIG. 15 is a graph showing the dependence of BVcbo LLY on model angle, .beta., prior to trench CVD oxidation; FIG. 16 is a graph showing BVcbo LLY versus shape magnitude, M, prior to trench CVD oxidation; FIG. 17 is a graph showing change in circuit failure rate due to EXP wafers versus failure rate for POR wafers for products A and B; FIG. 18 is a graph showing the dependence of final test chip yield on wafer shape angle prior to trench CVD oxidation; and FIG. 19 is a graph showing the relationship of an arbitrary set of points to critical shape angles corresponding to surface inversion zones. A study of wafer surfaces prior to trench isolation shows that wafers with particular profile shapes have a higher incidence of leakage related failures in trench and device structures. These surfaces had been characterized after silicon epitaxy and oxide and nitride films had been grown. Subsequent removal of these nitride and oxide layers from the backside of the wafers using reactive ion etch (RIE) was found to decrease the wave convexity and often to a flat or concave state. Elimination of this backside etch leaves wafers in a more convex state prior to the chemical vapor deposition (CVD) oxide film growth which is used to define trench isolation structures. Experimental Procedure Wafers from two product types, referred to herein as product A and product B, with bipolar device structures using polycrystalline silicon filled trench (PST) isolation technology were used to assess the influence of surface topography and the backside etch process prior to trench CVD oxidation. The evaluation utilized six jobs which were split prior to the backside etch process. Half the wafers in each job received the process of record (POR) backside etch while the other experimental half (EXP) were not etched. In one job (product A), the experimental cell (EXP2) received only the frontside photoresist and strip processes. FIG. 1 shows the process details used which compares the elimination of the backside etch with the POR. As shown in FIG. 1, the surface topography, warpage and bow of each wafer were determined before and after the backside etch process and characterized again at the end of the master slice. The surface topographies were obtained from an automated interferometer using an optical HeNe laser. The output is a contour map showing elevations. A typical map is shown in FIG. 2 and, as a three-dimensional projection, in FIG. 2A. The characterization zone of such maps excludes .apprxeq.2 mm of the wafer edge. The term "wafer edge" as used here refers to the edge of the characterization zone. Within this region, it is useful to describe surface topography in terms of wafer warpage and bow. By convention, warpage is defined as the maximum variation of the wafer surface. It is measured as the difference (always positive) between the highest and lowest elevation without regard to location. Wafer bow is defined as the elevation at the wafer center relative to a reference plane. As used herein, the reference plane is defined at the wafer edge as a best fit surface placing the wafer parallel to a reference prism. Positive bow denotes a generally convex surface and negative bow defines a concave surface. FIG. 3 illustrates both warpage and bow. Characterizations of surface topography using warpage and bow can be ambiguous as an infinite number of profile shapes would satisfy a particular warpage or bow. If warpage W and bow B are considered together as in a W/B ratio, the vagueness is reduced. If warpage measurements are plotted as a function of bow, a population of wafer data points appears as generally shown in FIG. 4. The W/B ratio can be used to define an angle ##EQU3## The usefulness of this measurement is realized when test data is correlated with .OMEGA. which eliminates the complexities of using ratios involving a division by very small numbers or zero. With a few exceptions, wafers with .OMEGA. in the range 90.degree..+-.20.degree. are generally those which are close to being almost flat. Where .OMEGA. becomes almost invariant (for large .vertline.B.vertline.), the relationship of W to B is approximately fixed. This generally occurs when wafer surfaces are either everywhere convex or concave. Test data is plotted in FIG. 4 with .OMEGA. to determine the sensitivity to surfaces which are nearly flat. Apart from this, .OMEGA. is not useful in identifying more complex large scale topographies. Large Scale Topography Model The task of correlating the main features of each map with device test data is achieved by considering only large scale variations of the wafer surface as the first order effect. In this treatment, small localized variations are neglected for simplicity. In this method, elevations are measured using the center of the wafer as the reference point. Surface elevations are measured using the center of the wafer as the reference point, which is also the origin of a cylindrical (Y,.theta.,r) coordinate system. In this notation, Y is the elevation, .theta. is the angular displacement, and r is the normalized radius (.theta..ltoreq.r.ltoreq.1). From the topographical maps, the average elevation of the wafer edge (Y.sub.e) and at some intermediate position (Y.sub.s) are determined for each radius as illustrated in FIG. 5. In FIG. 5, C denotes the center of the wafer, S an intermediate site and E the edge of the wafer. The average elevation is defined at the edge (r=1) using ##EQU4## The average elevation at the intermediate radius (r=K<1) is defined by ##EQU5## For a sampling system which measures the average elevation ar r=K,1 for n equidistant radii (in .theta.), ##EQU6## A minimum of four equally spaced radii are characterized for a wafer. Elevations above the wafer center are positive and those below are negative (i.e., the sense is opposite of bow). These measurements are then used as boundary conditions for an analytical solution expressing the large scale surface topography as a function of distance from the wafer center. The general solution is derived by solving a linear differential equation with constant coefficients of the form .PHI.(D)Y=0, (6) where .PHI.(D) is the linear polynomial operator in D=d/dr and r(0.ltoreq.r.ltoreq.1) is the normalized distance from the center. The solution is approximated by assuming a power series solution. Y=H.sub.o +H.sub.1 r+H.sub.2 r.sup.2 +H.sub.3 r.sup.3 + . . . +H.sub.n r.sup.n, (7) H.sub.1 . . . H.sub.n= constants. (8) The initial boundary condition require that y=0 at r=0. To simplify further, it is assumed that the wafer topography is axially (perpendicular to the surface) symmetric. This requires that DY/dr=0 at r=0. These considerations require that H.sub.0 =H.sub.1= 0. (9) It should be noted that axial symmetry is assumed only as a simplification of the problem. If the slope is not zero at r=0, the term involving r is nonzero which permits a wider range of topographical possibilities. Y=H.sub.2 r.sup.2 +H.sub.3 r.sup.3 . . . H.sub.n r.sup.n, (10) The two elevations Y.sub.s and Y.sub.e, for each radius are determined at r=K (where K<1) and r=1 (wafer edge), respectively. The intermediate elevations were all made at K=2/3. The results from the radii characterized were then averaged to determine the effective value of Y.sub.s (r=K) and Y.sub.e (r=1) for the wafer. These conditions limit the solution to the first two terms of equation (10). Y=ar.sup.2+ br.sup.3, (11) where ##EQU7## A map of Y.sub.e versus Y.sub.s for each wafer defines a population of data points. This plot usually appears as an oblong distribution as illustrated in FIG. 6. The relative relationship between these measurements is defined by a shape angle, .alpha., and magnitude, M, where ##EQU8## Using equation (14), the model constants are redefined in terms of Y.sub.s and .alpha. as follows: ##EQU9## The model constants for each wafer are also mapped. The relative relationship of the model constants are defined by a model constant angle .beta. where ##EQU10## Using equation (18), the dependence of .beta. on .alpha. is shown in FIG. 7. As can be seen, there are regions of the plot where .alpha. and .beta. are almost exclusively invariant. In correlating test data to either .alpha. or .beta., there is sensitivity to small changes in either .alpha. or .beta. in these regions of quasi invariance. It is therefore useful to correlate test data to both .alpha. and .beta. which intersect the regions of FIG. 7 critical to test data. Using the techniques described, the large scale surface topography is completely defined by .alpha. and the extent of complex bowing by M. Some additional fundamental definitions can also be made involving these terms. Surfaces are defined as proportionally similar if their .alpha. angles are identical although the values of M may be dissimilar. If the values of .alpha. are different, the surfaces are proportionally dissimilar. Simple topographies are defined as surfaces which are everywhere either convex or concave and usually occur within narrow ranges of .alpha.. It is generally observed that wafer surfaces with a large M (and large W/B ratios) will have topographies which are simple. A flat wafer is obtained in the limit as M.fwdarw.0 for any shape angle. Topographies at various values of .alpha. in the range of from -180.degree. to +180.degree. are illustrated in FIG. 8. In these graphs, surface inversions (from convex to concave and vice versa) are noted in regions which correspond to the corners shown in FIG. 7. The critical values of .alpha. at which these surface inversions occur can be determined by solving for the roots of d.beta./d.alpha. =1. The roots obtained are functionally dependent on the value of K. The two positive critical angles (roots) lie on a set of two bifurcated branch functions which converge at K=0 and K=1. A set of similar branches account for the two negative critical angles with the same points of convergence. A sufficient angular spacing (e.g., >20.degree.) between the bifurcated roots of each set is obtained for 0.3<K<0.85. Within this range, the maximum root separation in each set is almost 29.degree. and occurs at K=0.6 which is near the value used in the surface characterizations. For K= 0,1, the critical shape angles approach simple surface topographies which limits the usefulness of the method in identifying potential surface inversion states. Using K=2/3 was found to be convenient in characterizing a large number of wafer surfaces. For this value of K, the area of the inner circle is approximately equal to the annular region. This also gave a reasonable angular separation of the roots so that the influence of each critical shape angle on test data could be determined. If .alpha. is close to one of these critical values, a wafer may undergo a surface inversion resulting from process induced changes in .alpha.. These perturbations may arise during thermal processes or from processes which introduce mechanically applied stresses. These transitional surface instabilities can cause stress inversions (compressive.rarw..fwdarw.tensile) in silicon which may be localized near patterned structures. Topography prior to Deep Trench Isolation: The backside nitride etch process prior to the deep trench CVD oxidation generally reduced wafer convexity by 5 to 15 .mu.m in the bow measurements. As a result, many of these POR wafers were nearly flat or in a concave state which could introduce corresponding compressive stress effects, particularly near trench and device regions. The elimination of this process left the bow of EXP wafers in a more convex state (10 to 30 .mu.m) on both product types. Bow measurements of EXP2 wafers indicated that photoresist related processes (without RIE) reduced convexity slightly by 2 to 3 .mu.m. This shift in bow is attributed to the plasma etching of the photoresist film. Also, the shape and mode angles of the EXP2 wafers were uniquely different from EXP wafers in that these angles were shifted towards one of the inversion zones. The map of Y.sub.e versus Y.sub.s in FIG. 9 shows the oblong distribution of data points for the wafers along the critical inversion zones (IZ) as a reference. In FIG. 9, the left data points are surfaces which are generally convex while the right data points are generally concave. Using the Y.sub.s axis as a reference, this population is shown to be angularly displaced by .apprxeq.10.degree. from the expected optimum (i.e., .alpha..apprxeq.70.degree.) which causes a larger overlap of one of the critical zones (IZ lines). The distribution also shows that an almost full spectrum of shapes exist at smaller values of M. As M increases, the range of .alpha. eventually becomes limited to those values associated with simple topographies. This indicates that the shape angle is dependent on M and wafer bow. Using equations (16), (17) and (18), the model constants (a,b) and the model angle, .beta., were then calculated for each wafer. The plot of .beta. versus .alpha. in FIG. 10 shows a distribution which populates almost the entire range shown in the theoretical curve of FIG. 7. Data points located near the corners of FIG. 10 correspond to points near the IZ lines in FIG. 9. BVcc Trench Maze Test: Test results from a maze consisting of parallel isolation trench structures indicates that the EXP wafers had fewer wafers with low yield due to BVcc leakage. A correlation with bow and M prior to the trench CVD oxide shows that the 20 volt BVcc yield decreases with a reduction in convexity of both POR and EXP wafers. FIG. 11 shows that the higher yield obtained with the EXP wafers from product A could be attributed to a larger population of convex wafers. However, EXP2 wafers had a lower yield than either EXP or POR wafers which can be attributed to a larger population of wafers with surface topographies near inversion zone states. A correlation with shape and model angles indicates that the maze yield of all wafers is influenced by the proximity of the wafer surfaces to critical inversion zone angles shown in FIGS. 12 and 13, respectively. In FIG. 12, the BVcc LLY is plotted as a function of the shape angle which was measured prior to the trench mask oxide process. The entire range of shape angles is not represented as these measurements were limited to only product A. However, this plot does show some indication that the isolation LLY is lowered by the proximity of wafer surfaces to critical shape angles. Critical shape angles occur at -124.7.degree., -96.1.degree., +55.3.degree., and +83.9.degree.. In a similar plot, FIG. 13 shows the same LLY data base as a function of the model angle also measured prior to the trench mask CVD oxide. This plot shows lower yields for those wafer surfaces which are near critical model angles at -60.7.degree., -32.1.degree., +119.3.degree., and +147.9.degree.. The coincident lower LLY levels near the critical shape and model angles are associated with the surface inversion zones which correspond to the corners shown in FIG. 7. End of Master Slice Bow: Measurements at post Pt test indicated that all wafers had simple convex topographies with no significant differences between EXP or POR wafers within a particular product type. However, product B wafers were significantly less convex (3 to 7 .mu.m) than product A wafers (12 to 25 .mu.m). As the two product types have similar processes, this difference is attributed to differences in the density of trench isolation structures. Post Pt Test: Discrete transistor structures in the kerf were tested after the master slice processes to assess transistor LLY. In this test, the collector to base break down voltage (BVcbo) with the emitter open was measured at 10 .mu.A on fifty sites per wafer. These results show that the BVcbo LLY levels were generally improved with the EXP wafers. However, in contrast to the low isolation maze LLY, the EXP2 wafers had a slightly higher BVcbo yield than POR wafers in the same job. Correlations with shape and model angles prior to trench CVD oxidation shows that the BVcbo LLY of all wafers, regardless of product type, were influenced by surface inversions as shown in FIGS. 14 and 15. In FIG. 14, the BVcbo LLY for both product types is plotted as a function of the shape angle measured prior to the trench mask CVD oxide process. The large data base in this plot shows clearly that the yield is lowered by the proximity of wafer surfaces near the critical shape angles at -124.7.degree., -96.1.degree., +55.3.degree., and +83.9.degree.. Lower LLY levels are also evident for wafer surfaces near the critical model angles at -60.7.degree., -32.1.degree., +199.3.degree., and +147 9.degree., as shown in FIG. 15. The coincident behavior of the LLY levels near the critical shape and model angles indicates that the junction leakage of discrete device structures is associated with the surface inversion zones characterized prior to the fabrication of the isolation structures. FIG. 16 shows that BVcbo LLY is conditionally dependent on the degree of convex bowing. For wafers with M<M.sub.L.apprxeq. 4 .mu.m, the yield is improved and apparently insensitive to the surface inversion phenomena. This threshold may be due to the lower stress limit required to generate defects. For larger values of M, BVcbo yield is statistically reduced as the effects of the surface inversions become apparent. For M>M.sub.U.apprxeq. 18 .mu.m, BVcbo yield is improved. This is attributed to a decreasing population of wafers occupying inversion zone states as M is increased Transistor Test Chains: Device chains consisting of parallel transistor circuit cells using 1.5.times.2 .mu.m emitters were tested using a sample of twelve sites per wafer. In this test, the junction leakage parameters of trench (TR) isolation structures and transistor (TX) structures were analyzed as exclusive categories. For both product types, the TR LLY results generally indicated that the EXP wafers were comparable to those for the POR cell as only slight improvements were noted. Larger improvements were found in the TX LLY in which the major TX failure mechanism was collector to emitter (CE) leakage in both product types. The average circuit failure rate (F) for each chain and product type was determined from the LLY(Y) using Y=(1-F).sup.N, (19) where N is the number of circuits per chain. FIG. 17 is a plot of the change in circuit failure rate due to EXP wafers (.DELTA.F=F.sub.por -F.sub.exp) versus failure rate of POR wafers (F.sub.por) for products A and B. FIG. 17 shows that the CE and TX failure rate of EXP wafers (F.sub.exp) is reduced and is dependent on failure rate level of the POR wafers (F.sub.por). Thus, wafers with favorable surface topographies prior to the trench process can improve yield by moderating large variations in LLY. Final Test: Final test yields from both product types plotted as a function shape angle measured prior to trench CVD oxidation are shown in FIG. 18. In this figure, wafers have been sorted by their proximity (.+-.10.degree.) to critical shape angles. In comparison to wafers in noncritical zones, wafers near the inversion zones have a tendency to have reduced chip yields. Phenomenological Model: These results can be described by the phenomenological model in FIG. 19. In this illustration, the relationship of an arbitrary set of data points (Y.sub.s, Y.sub.e) to the critical inversion zone (IZ) angles is shown. Given any M as indicated, a population of wafers can be found which belong to regions which are in proximity to the inversion zones. Region dM at M contains a subset of points (wafers) in proximity to IZ lines, as shown. LLY improves for M<M.sub.L due to stress threshold effect. LLY reduced for M.sub.L <M<M.sub.U due to surface inversions. Decreasing the subset of points near IZ for M>M.sub.U improves yield. M.sub.U is varied by small rotations of set (Y.sub.s, Y.sub.e) around the origin. M.sub.U is minimum for a narrow set and centering between IZ regions. Leakage limited yield results obtained from the trench maze isolation and from discrete kerf transistor structures both indicate that M.sub.U is approximately 18 .mu.m. While the LLY results obtained from the discrete transistor indicates that M.sub.L is approximately 4 .mu.m, isolation maze results show that M.sub.L< 0. This difference indicates that wafers which are nearly flat will result in low trench maze yields which is believed to be caused by a sensitivity to unrelieved stresses localized at isolation structures in the silicon. Elements of Invention: The characterization of large scale wafer topography as described above is applied to improving device characteristics and yields in the manufacture of large scale integrated (LSI) devices. First, the heights at the center, the edge and an intermediate point are measured on four or more, and preferably eight, equally spaced radii. This provides, for example, eight values each for Y.sub.s and Y.sub.e which are averaged. Then the shape of angle .alpha. is computed using the following equation: ##EQU11## The shape magnitude M is also computed using the following equation: ##EQU12## The thus computed values of .alpha. and M are correlated with individual wafer characteristics as to device performance and yield. Based on these results, the wafer processing is controlled to provide optimal wafer characteristics. While the invention has been described in terms of a single preferred embodiment, those skilled in the art will recognize that the invention can be practiced with modification within the spirit and scope of the appended claims. 1. A method of characterization of large scale wafer topography in the manufacture of large scale integrated semiconductor devices comprising the steps of: measuring the heights at the center, the edge and an intermediate point on a plurality of radii on the wafer to obtain a plurality of values each for Y.sub.s, the height of the intermediate point relative to the center, and Y.sub.e, the height of the edge relative to the center; averaging the plurality of values Y.sub.s and Y.sub.e; computing a shape angle.alpha. using the following equation: ##EQU13## computing a shape magnitude M using the following equation: ##EQU14## correlating the computed values of.alpha. and M with individual wafer characteristics as to device performance and yield at intermediate steps in the process; and controlling wafer processing by correlating.alpha. and M with individual wafer characteristics to provide optimal wafer yield and leakage characteristics. 2. The method recited in claim 1 wherein measurements are taken along at least four equally spaced radii over the surface of the wafer. 3. A method of correlating semiconductor wafer large scale topography to wafer yield and device characteristics comprising the steps of: measuring, for a selected surface of the wafer, wafer height at selected radial points on the wafer; developing at least two measurement indicative of the wafer large scale topography as a function of measured heights; and correlating these at least two measurements indicative of wafer large scale topography to the yield at intermediate steps in the process and/or operating characteristics of devices formed on the 4. The method recited in claim 3 further including the step of controlling the large scale topography using the correlated measurements of subsequently formed wafers to provide the large scale topographic measurements which optimize selected device parameters and/or yield. 5. The method recited in claim 4 wherein said radial points include an edge point and an intermediate point on a plurality of radial lines, heights at said edge points and said intermediate points being relative to the center of the wafer. 6. The method recited in claim 5 wherein said step of developing at least two measurements comprise the steps of: averaging the measured edge point heights and the measured intermediate point heights; determining an angle,.alpha., the arctangent of which is the ratio of the average edge point height to the average intermediate point height; and determining a shape magnitude, M, as the squareroot of the sum of the squares of the average edge point height and the average intermediate point height. Referenced Cited U.S. Patent Documents 3729966 May 1973 Khoury et al. 3751647 August 1973 Maeder et al. 4272196 June 9, 1981 Indebetouw 4334282 June 8, 1982 Whitehouse 4422764 December 27, 1983 Eastman 4962461 October 9, 1990 Meyer et al. 5067101 November 19, 1991 Kunikiyo et al. 5070469 December 3, 1991 Kunikiyo et al. Other references • Automated Visual Inspection Techniques and Applications: A Bibliography, Roland Chin, Pattern Recognition vol. 15, No. 4, pp. 343-357, 1982. Patent History Patent number : 5319570 : Oct 9, 1991 Date of Patent : Jun 7, 1994 International Business Machines Corporation (Armonk, NY) Joanne M. Davidson (Poughkeepsie, NY), George Hrebin, Jr. (Verbank, NY), Robert K. Lewis (Wappingers Falls, NY), Carl H. Orner (Newburgh, NY) Primary Examiner Thomas G. Black Assistant Examiner Susan Wieland Law Firm Whitham & Marhoefer Application Number : 7/774,084 Current U.S. Class: 364/488; 364/507 International Classification: G06F 1572;	{"url":"https://patents.justia.com/patent/5319570","timestamp":"2024-11-05T03:51:18Z","content_type":"text/html","content_length":"92309","record_id":"<urn:uuid:9e506ef6-01d6-423c-b753-0789edafdd05>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00470.warc.gz"}
Derivative of Tan x - Formula, Proof, Examples The tangent function is one of the most significant trigonometric functions in mathematics, engineering, and physics. It is a crucial theory used in many domains to model various phenomena, consisting of signal processing, wave motion, and optics. The derivative of tan x, or the rate of change of the tangent function, is an important idea in calculus, that is a branch of math that deals with the study of rates of change and accumulation. Getting a good grasp the derivative of tan x and its characteristics is essential for professionals in several fields, including physics, engineering, and math. By mastering the derivative of tan x, individuals can use it to solve problems and gain detailed insights into the intricate functions of the surrounding world. If you want guidance understanding the derivative of tan x or any other math concept, contemplate contacting Grade Potential Tutoring. Our expert instructors are available online or in-person to offer customized and effective tutoring services to support you be successful. Call us today to plan a tutoring session and take your mathematical abilities to the next level. In this blog, we will dive into the theory of the derivative of tan x in detail. We will start by discussing the significance of the tangent function in different domains and applications. We will then explore the formula for the derivative of tan x and offer a proof of its derivation. Eventually, we will provide examples of how to apply the derivative of tan x in different fields, including physics, engineering, and mathematics. Importance of the Derivative of Tan x The derivative of tan x is a crucial math theory which has multiple uses in physics and calculus. It is utilized to work out the rate of change of the tangent function, which is a continuous function which is broadly utilized in math and physics. In calculus, the derivative of tan x is applied to work out a broad range of challenges, consisting of working out the slope of tangent lines to curves that consist of the tangent function and calculating limits that involve the tangent function. It is further utilized to work out the derivatives of functions which involve the tangent function, such as the inverse hyperbolic tangent In physics, the tangent function is used to model a extensive spectrum of physical phenomena, involving the motion of objects in circular orbits and the behavior of waves. The derivative of tan x is used to work out the acceleration and velocity of objects in circular orbits and to get insights of the behavior of waves which consists of variation in frequency or amplitude. Formula for the Derivative of Tan x The formula for the derivative of tan x is: (d/dx) tan x = sec^2 x where sec x is the secant function, which is the reciprocal of the cosine function. Proof of the Derivative of Tan x To prove the formula for the derivative of tan x, we will apply the quotient rule of differentiation. Let’s say y = tan x, and z = cos x. Next: y/z = tan x / cos x = sin x / cos^2 x Using the quotient rule, we obtain: (d/dx) (y/z) = [(d/dx) y * z - y * (d/dx) z] / z^2 Replacing y = tan x and z = cos x, we get: (d/dx) (tan x / cos x) = [(d/dx) tan x * cos x - tan x * (d/dx) cos x] / cos^2 x Next, we could apply the trigonometric identity that links the derivative of the cosine function to the sine function: (d/dx) cos x = -sin x Replacing this identity into the formula we derived above, we get: (d/dx) (tan x / cos x) = [(d/dx) tan x * cos x + tan x * sin x] / cos^2 x Substituting y = tan x, we get: (d/dx) tan x = sec^2 x Hence, the formula for the derivative of tan x is proven. Examples of the Derivative of Tan x Here are some instances of how to use the derivative of tan x: Example 1: Find the derivative of y = tan x + cos x. (d/dx) y = (d/dx) (tan x) + (d/dx) (cos x) = sec^2 x - sin x Example 2: Find the slope of the tangent line to the curve y = tan x at x = pi/4. The derivative of tan x is sec^2 x. At x = pi/4, we have tan(pi/4) = 1 and sec(pi/4) = sqrt(2). Thus, the slope of the tangent line to the curve y = tan x at x = pi/4 is: (d/dx) tan x \| x = pi/4 = sec^2(pi/4) = 2 So the slope of the tangent line to the curve y = tan x at x = pi/4 is 2. Example 3: Find the derivative of y = (tan x)^2. Utilizing the chain rule, we get: (d/dx) (tan x)^2 = 2 tan x sec^2 x Hence, the derivative of y = (tan x)^2 is 2 tan x sec^2 x. The derivative of tan x is a fundamental mathematical concept that has many uses in physics and calculus. Getting a good grasp the formula for the derivative of tan x and its properties is important for learners and professionals in domains such as physics, engineering, and math. By mastering the derivative of tan x, individuals can use it to work out problems and get detailed insights into the complicated functions of the surrounding world. If you want assistance comprehending the derivative of tan x or any other math theory, contemplate calling us at Grade Potential Tutoring. Our expert teachers are available remotely or in-person to give personalized and effective tutoring services to help you succeed. Contact us right to schedule a tutoring session and take your mathematical skills to the next stage.	{"url":"https://www.pittsburghinhometutors.com/blog/derivative-of-tan-x-formula-proof-examples","timestamp":"2024-11-14T05:13:57Z","content_type":"text/html","content_length":"76855","record_id":"<urn:uuid:e2103e2c-296a-4b65-9cb3-781f9889fbfc>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00081.warc.gz"}
Selina Concise Class-9th Mid Point and Intercept Theorem - ICSEHELP Selina Concise Class-9th Mid Point and Intercept Theorem Selina Concise Class-9th Mid Point and Intercept Theorem ICSE Mathematics Solutions Chapter-12. We provide step by step Solutions of Exercise / lesson-12 Mid Point and its Converse ( Including Intercept Theorem ) for ICSE Class-9 Concise Selina Mathematics by R K Bansal. Our Solutions contain all type Questions with Exe-12 A and Exe-12 B, to develop skill and confidence. Visit official Website CISCE for detail information about ICSE Board Class-9 Mathematics . Selina Concise Class-9th Mid Point and Intercept Theorem ICSE Mathematics Solutions Chapter-12 –: Select Topic :– Exercise – 12 A, Selina Concise Class-9th Mid Point and its Converse ( Including Intercept Theorem ) ICSE Mathematics Solutions Question 1 In triangle ABC, M is mid-point of AB and a straight line through M and parallel to BC cuts AC in N. Find the lengths of AN and MN if Bc = 7 cm and Ac = 5 cm. The triangle is shown below, Question 2 Prove that the figure obtained by joining the mid-points of the adjacent sides of a rectangle is a rhombus. The figure is shown below, Similarly 2PQ=2SR=AC and PQ\|\|SR—– (2) From (1) and (2) we get Therefore PQRS is a rhombus. Hence proved Question 3 D, E and F are the mid-points of the sides AB, BC and CA of an isosceles ΔABC in which AB = BC. Prove that ΔDEF is also isosceles. The figure is shown below Given that ABC is an isosceles triangle where AB=AC. Since D,E,F are midpoint of AB,BC,CA therefore 2DE=AC and 2EF=AB this means DE=EF Therefore DEF is an isosceles triangle an DE=EF. Hence proved Question 4 The following figure shows a trapezium ABCD in which AB // DC. P is the mid-point of AD and PR // AB. Prove that: Here from triangle ABD P is the midpoint of AD and PR\|\|AB, therefore Q is the midpoint of BD Similarly R is the midpoint of BC as PR\|\|CD\|\|AB From triangle ABD 2PQ=AB …….(1) From triangle BCD 2QR=CD …..(2) Now (1)+(2)=> Hence proved Question 5 The figure, given below, shows a trapezium ABCD. M and N are the mid-point of the non-parallel sides AD and BC respectively. Find: (i) MN, if AB = 11 cm and DC = 8 cm. (ii) AB, if Dc = 20 cm and MN = 27 cm. (iii) DC, if MN = 15 cm and AB = 23 cm. Let we draw a diagonal AC as shown in the figure below, Question 6 The diagonals of a quadrilateral intersect at right angles. Prove that the figure obtained by joining the mid-points of the adjacent sides of the quadrilateral is rectangle. The figure is shown below Question 7 L and M are the mid-point of sides AB and DC respectively of parallelogram ABCD. Prove that segments DL and BM trisect diagonal AC. The required figure is shown below From figure, BL=DM and BL\|\|DM and BLMD is a parallelogram, therefore BM\|\|DL From triangle ABY L is the midpoint of AB and XL\|\|BY, therefore x is the midpoint of AY.ie AX=XY …..(1) Similarly for triangle CDX CY=XY …..(2) From (1) and (2) AX=XY=CY and AC=AX+XY+CY Hence proved Question 8 ABCD is a quadrilateral in which AD = BC. E, F, G and H are the mid-points of AB, BD, CD and Ac respectively. Prove that EFGH is a rhombus. Given that AD=BC …..(1) From the figure, For triangle ADC and triangle ABD 2GH=AD and 2EF=AD, therefore 2GH=2EF=AD …..(2) For triangle BCD and triangle ABC 2GF=BC and 2EH=BC, therefore 2GF=2EH=BC …..(3) From (1),(2),(3) we get, Therefore EFGH is a rhombus. Hence proved Question 9 A parallelogram ABCD has P the mid-point of Dc and Q a point of Ac such that CQ = 1/4 AC. PQ produced meets BC at R. Prove that (i)R is the midpoint of BC For help we draw the diagonal BD as shown below Question 10 D, E and F are the mid-points of the sides AB, BC and CA respectively of ABC. AE meets DF at O. P and Q are the mid-points of OB and OC respectively. Prove that DPQF is a parallelogram. The required figure is shown below For triangle ABC and OBC 2DE=BC and 2PQ=BC, therefore DE=PQ …..(1) For triangle ABO and ACO 2PD=AO and 2FQ=AO, therefore PD=FQ …..(2) From (1),(2) we get that PQFD is a parallelogram. Hence proved Question 11 In triangle ABC, P is the mid-point of side BC. A line through P and parallel to CA meets AB at point Q; and a line through Q and parallel to BC meets median AP at point R. Prove that The required figure is shown below Question 12 In trapezium ABCD, AB is parallel to DC; P and Q are the mid-points of AD and BC respectively. BP produced meets CD produced at point E. Prove that: (i) Point P bisects BE, (ii) PQ is parallel to AB. The required figure is shown below (ii)For tiangle ECB PQ\|\|CE Again CE\|\|AB Therefore PQ\|\|AB Hence proved Question 13 In a triangle ABC, AD is a median and E is mid-point of median AD. A line through B and E meets AC at point F. Prove that: AC = 3AF The required figure is shown below For help we draw a line DG\|\|BF Now from triangle ADG, DG\|\|BF and E is the midpoint of AD Therefore F is the midpoint of AG,ie AF=GF …..(1) From triangle BCF, DG\|\|BF and D is the midpoint of BC Therefore G is the midpoint of CF,ie GF=CF …(2) AC=3AF(From (1) and (2)) Hence proved Question 14 D and F are mid-points of sides AB and AC of a triangle ABC. A line through F and parallel to AB meets BC at point E. (i) Prove that BDFE is parallelogram (ii) Find AB, if EF = 4.8 cm. The required figure is shown below Question 15 In triangle ABC, AD is the median and DE, drawn parallel to side BA, meets AC at point E. Show that BE is also a median. Question 16 In ∆ABC, E is mid-point of the median AD and BE produced meets side AC at point Q. Show that BE : EQ = 3:1. Question 17 In the given figure, M is mid-point of AB and DE, whereas N is mid-point of BC and DF. Show that: EF = AC. Selina Concise Class-9th Mid Point and Intercept Theorem ICSE Mathematics Solutions Exercise – 12 B Question 1 Use the following figure to find: (i) BC, if AB = 7.2 cm. (ii) GE, if FE = 4 cm. (iii) AE, if BD = 4.1 cm (iv) DF, if CG = 11 cm. According to equal intercept theorem since CD=DE Therefore AB=BC and EF=GF Since B,D,F are the midpoint and AE\|\|BF\|\|CG Therefore AE=2BD and CG=2DF Question 2 In the figure, give below, 2AD = AB, P is mid-point of AB, Q is mid-point of DR and PR // BS. Prove that: (i) AQ // BS (ii) DS = 3 Rs. Given that AD=AP=PB as 2AD=AB and p is the midpoint of AB (i)From triangle DPR, A and Q are the midpoint of DP and DR. Therefore AQ\|\|PR Since PR\|\|BS ,hence AQ\|\|BS (ii)From triangle ABC, P is the midpoint and PR\|\|BS Therefore R is the midpoint of BC Question 3 The side AC of a triangle ABC is produced to point E so that CE = 1/2AC. D is the mid-point of BC and ED produced meets AB at F. Lines through D and C are drawn parallel to AB which meet AC at point P and EF at point R respectively. Prove that: (i) 3DF = EF(ii) 4CR = AB. Consider the figure: Question 4 In triangle ABC, the medians BP and CQ are produced upto points M and N respectively such that BP = PM and CQ = QN. Prove that: (i) M, A and N are collinear. (ii) A is the mid-point of MN The figure is shown below Question 5 In triangle ABC, angle B is obtuse. D and E are mid-points of sides AB and BC respectively and F is a point on side AC such that EF is parallel to AB. Show that BEFD is a parallelogram. The figure is shown below From the figure EF\|\|AB and E is the midpoint of BC. Therefore F is the midpoint of AC. Here EF\|\|BD, EF=BD as D is the midpoint of AB BE\|\|DF, BE=DF as E is the midpoint of BC. Therefore BEFD is a parallelogram. Question 6 In parallelogram ABCD, E and F are mid-points of the sides AB and CD respectively. The line segments AF and BF meet the line segments ED and EC at points G and H respectively. Prove that: (i) Triangles HEB and FHC are congruent; (ii) GEHF is a parallelogram. The figure is shown below (1),(2) we get, HF=EG and HF\|\|EG Similarly we can show that EH=GF and EH\|\|GF Therefore GEHF is a parallelogram. Question 7 In triangle ABC, D and E are points on side AB such that AD = DE = EB. Through D and E, lines are drawn parallel to BC which meet side AC at points F and G respectively. Through F and G, lines are drawn parallel to AB which meet side BC at points M and N respectively. Prove that: BM = MN = NC. The figure is shown below For triangle AEG D is the midpoint of AE and DF\|\|EG\|\|BC Therefore F is the midpoint of AG. AF=GF …..(1) Again DF\|\|EG\|\|BC DE=BE, therefore GF=GC …..(2) (1),(2) we get AF=GF=GC. Similarly Since GN\|\|FM\|\|AB and AF=GF ,therefore BM=MN=NC Hence proved Question 8 In triangle ABC; M is mid-point of AB, N is mid-point of AC and D is any point in base BC. Use intercept Theorem to show that MN bisects AD. The figure is shown below Since M and N are the midpoint of AB and AC, MN\|\|BC According to intercept theorem Since MN\|\|BC and AM=BM, Therefore AX=DX. Hence proved Question 9 If the quadrilateral formed by joining the mid-points of the adjacent sides of quadrilateral ABCD is a rectangle, show that the diagonals AC and BD intersect at right angle. The figure is shown below Question 10 In triangle ABC; D and E are mid-points of the sides AB and AC respectively. Through E, a straight line is drawn parallel to AB to meet BC at F. Prove that BDEF is a parallelogram. If AB = 16 cm, AC = 12 cm and BC = 18 cm, find the perimeter of the parallelogram BDEF. The figure is shown below Question 11 In the given figure, AD and CE are medians and DF//CE. Prove that: ……… Given AD and CE are medians and DF \|\| CE. We know that from the midpoint theorem, if two lines are parallel and the starting point of segment is at the midpoint on one side, then the other point meets at the midpoint of the other side. Consider triangle BEC. Given DF \|\| CE and D is midpoint of BC. So F must be the midpoint of BE. Question 12 In parallelogram ABCD, E is the mid-point of AB and AP is parallel to EC which meets DC at point O and BC produced at P. Prove that: (i) BP = 2AD (ii) O is the mid-point of AP. Given ABCD is parallelogram, so AD = BC, AB = CD. Consider triangle APB, given EC is parallel to AP and E is midpoint of side AB. So by midpoint theorem, C has to be the midpoint of BP. So BP = 2BC, but BC = AD as ABCD is a parallelogram. Hence BP = 2AD Consider triangle APB, AB \|\| OC as ABCD is a parallelogram. So by midpoint theorem, O has to be the midpoint of AP. Hence Proved Question 13 In trapezium ABCD, sides AB and DC are parallel to each other. E is mid-point of AD and F is mid-point of BC. Prove that: AB + DC = 2EF. Consider trapezium ABCD. Given E and F are midpoints on sides AD and BC, respectively. We know that AB = GH = IJ From midpoint theorem, Consider LHS, AB + CD = AB + CJ + JI + ID = AB + 2HF + AB + 2EG So AB + CD = 2(AB + HF + EG) = 2(EG + GH + HF) = 2EF AB + CD = 2EF Hence Proved Question 14 In Δ ABC, AD is the median and DE is parallel to BA, where E is a point in AC. Prove that BE is also a median. Given Δ ABC AD is the median. So D is the midpoint of side BC. Given DE \|\| AB. By the midpoint theorem, E has to be midpoint of AC. So line joining the vertex and midpoint of the opposite side is always known as median. So BE is also median of Δ ABC. Question 15 Adjacent sides of a parallelogram are equal and one of the diagonals is equal to any one of the sides of this parallelogram. Show that its diagonals are in the ratio. — End of Mid Point and its Converse ( Including Intercept Theorem ) Solutions :– Return to – Concise Selina Maths Solutions for ICSE Class -9	{"url":"https://icsehelp.com/selina-concise-class-9th-mid-point-and-intercept-theorem/","timestamp":"2024-11-08T09:26:05Z","content_type":"text/html","content_length":"112143","record_id":"<urn:uuid:52cabd90-b018-4f2b-9ed9-1aeea78f582a>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00674.warc.gz"}
[WSC18] Using Twitter Sentiment Analysis to determine emoji sentiment 13018 Views 4 Replies 2 Total Likes [WSC18] Using Twitter Sentiment Analysis to determine emoji sentiment The goal of this project was to analyze how users on social media, in this case Twitter, use emojis in relation to their tweets contents to map an overarching sentiment to each emoji. Using ServiceConnect in the Wolfram Language, tweets can be pulled with a specific search term. I first had to link my twitter account to ServiceConnect twitter = ServiceConnect["Twitter", "New"] Using ServiceConnect, I then was able to pull tweets using the following function: GetTweetData[terms_, n_] := Module[ sentimentList = Classify["Sentiment", #] & /@ Normal[twitter["TweetSearch", "Query" -> #, MaxItems -> n][All, "Text"]] & /@ terms; sentimentList = Delete[sentimentList, Position[sentimentList, Indeterminate]]; Using this function GetTweetData, multiple terms in the form of a list can be passed in as arguments as well as the total number of tweets per term to be pulled. The function then returns a list of lists of sentiments: "Positive", "Neutral", or "Negative", back so that the data can be manipulated. In some case scenarios, the built in sentiment classifier in the Wolfram Language would consider the sentiment of a tweet to be "Indeterminate". In this case, the data point that was Indeterminate is removed to avoid issues with compiling the data. Data is able to be visualized in many different ways, those of which include: Pie charts, bar charts, and as a dataset. Bar Chart TwitterBarChart[data_, sortedTerms_] := Module[ {assocMap = {"Positive" -> 1, "Negative" -> -1, "Neutral" -> 0}}, BarChart[Sort[Total /@ ReplaceAll[data, assocMap]], ChartLabels -> Placed[sortedTerms, Above]] TwitterBarChart[rawData, emojis[[Ordering[emojis]]]] This function is beneficial for seeing a nice display of most negative to most positive emojis. It gives a nice representation of the overall data for each emoji by finding a sort of "net sentiment score", where positive = 1, negative = -1, and neutral = 0. These graphs were made using those methods. However, to achieve the goal of moving from most negative to most positive, a the data has to be sorted. Using a function called SortEmojis and the Wolfram Language function Ordering[], the function returns a list of rasterized emojis that are in the order that the sorted data will be in so that data and images match up with each other. SortEmojis[data_, emojis_] := Module[ {termsList , dataOrdering, assocMap = {"Positive" -> 1, "Negative" -> -1, "Neutral" -> 0}}, dataOrdering = Ordering[Total /@ ReplaceAll[data, assocMap]]; termsList = emojis[[dataOrdering]]; The counts of the data are found to find the "net sentiment score" as aforementioned, so that the data is sorted in the way it is intended to be. After running the function TwitteBarChart, you end up with an output that looks like this: So this sentiment data is good for representing the overall sentiments, and gives a nice visual of what is positive and negative. However sometimes there is more to be seen. Pie Charts An effective way to look more at the details of one of the emojis is to analyze a PieChart (or a few) of the emojis sentiments. There is a function I also created called TwitterPieCharts which takes a list of sentiments and labels in order to properly display a list of pie charts all with labels being their respective emojis TwitterPieCharts[data_, terms_] := Module[ {counts = KeySort /@ Counts /@ data}, PieChart[#, ChartLabels -> {"Negative", "Neutral", "Positive"}] & /@ counts, terms]] To allow for the static labels "Negative" , "Neutral", and "Positive" the counts of the counts of the data is ran through KeySort to allow for correct labeling of pie chart sectors. Then the pie charts are ran through the function Thread[] in order to assign a name to each one. TwitterPieCharts[Take[rawData, 3], Take[emojis, 3]] This code snippet right here is just an example to display the functionality. I am taking the first three emojis and their corresponding emojis to produce an output of three pie charts which look like the following. These pie charts allow a more detailed view of specific emoji data without getting into the numbers. In some case scenarios, emojis on the bar chart may have lots of both positive and negative which counteract in the calculating of the "net sentiment score" of which I mentioned earlier and that is used to calculate the data for the bar chart. If actual raw data, as in the original counts of the sentiment data wish to be seen, the Wolfram Language has a very easy function built directly in designed to represent data in a cell style chart. TwitterDataSet[data_, terms_ := Module[ {counts = KeySort /@ Counts /@ data}, Dataset[Association[MapThread[Rule, {terms, counts}]]] This function creates a key map of emojis with the sentiment counts they represent, and then key sorts them in order to effectively display the data inside a dataset with a consistent structure that is very easy to read. TwitterDataSet[data, emojis] This code snippet here, with the list of raw sentiment data and emojis produces this output in the dataset Overall the conclusion that I was able to draw about the use of emojis on twitter is that it goes against general expectation. Many emojis such as "ð ¤ " were were negative overall, which was odd considering the emoji seems to have a pretty neutral "flavor" to it. For each emoji, 250 tweets were pulled for each, resulting in a total of 24250 tweets. Among these 24250 tweets however, I noticed as I was taking a glance at them after being pulled, that a few tweets were duplicated. I found the answer for this to lie within the fact that retweets are considered actual new tweets. In some case scenarios, I might see the same tweet 10-15 times in a row if the tweet was popular and retweeted by many. This gives more popular tweets more "weight" so to speak in this type of data analysis, but the question is if that "weight" in actuality, accurately represents the connotation of those emojis accurately, despite one thinking it is simply garbage data from the beginning. In other words, even though the tweets are retweets, do retweets accurately represent the thoughts and emotions of others on twitter, meaning that duplicate retweets still provide accurate data? That is hard to say, however it is easy to determine that twitter users, in general, for certain types of messages, use certain types of emojis that not many would expect. 4 Replies Yes. I also tried in Spanish to no avail. So a relatively easy road to improvement (given that as you say Mathematica recognizes languages very well) would we to allow sentiment analysis in different Great contribution! I have a question here: does Classify["Sentiment", #] work with languages other than English? And if so then how? Thanks! It doesn't seem to. For example, this very positive sentiment in German is classified as 'neutral': Classify["Sentiment", "Ich bin sehr, sehr glücklich. Mir geht es ausgesprochen gut."] Playing around with a few more examples in German suggests the same. It's a pity, because Classify already does a nice job identifying languages. So it has the means to warn you if the target language is unsupported for certain classification tasks. I couldn't see language support addressed in the documentation either. Nice work Jake! You might be interested in extending this study by including your findings with emojis: "Diurnal variations of psychometric indicators in Twitter content" Some functions that might be useful for such study are: TextStructure , PartOfSpeech Be respectful. Review our Community Guidelines to understand your role and responsibilities. Community Terms of Use	{"url":"https://community.wolfram.com/groups/-/m/t/1392142","timestamp":"2024-11-08T20:39:31Z","content_type":"text/html","content_length":"119301","record_id":"<urn:uuid:fdc321d1-62d3-4673-9323-167324ea065d>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00007.warc.gz"}
Dynamic characteristics analysis and optimization for lateral plates of the vibration screen To solve the problem of damage due to large dynamic stress of the lateral plates during working process of the vibration screen, it is necessary to calculate and analyze natural modes and distribution of dynamic stress of lateral plates, which is shown in results that the lateral plate structure shall be optimized. In this paper, with the total weight of the lateral plates for the banana-shaped vibration screen as the optimization objective, frequency constraints as the status variables, optimization for multi-frequency constraints is conducted based on the improved genetic algorithm. Next, a mathematical model of structure parameter optimization for the lateral plates of the vibration screen under frequency constraints is established to carry out optimization design in order to obtain a structure with smaller dynamic stress and lower weight. Sensitivity analysis is added into the improved genetic algorithm, and the optimization efficiency is increased simultaneously. The structure frequency is optimized by means of the improved genetic algorithm. Then, a modal experiment is carried out to the entire vibration screen so as to verify reliability of the finite element model, and the natural characteristics of the vibration screen before and after optimization are analyzed, and the top 6 orders natural frequency and vibration modes of the entire vibration screen are calculated, so as to indicate that optimized vibration screen is improved in terms of material saving, stiffness and stability. In addition, noise is directly related to vibration. As a result, the change of the vibration screen should be also analyzed. Noise of the vibration screen is also tested by sound array technology. Results showed that the radiated noise is reduced after optimization, and optimization in this paper is feasible. 1. Introduction During working process, the vibration screen bears large exciting force, and strong inertia force shall be distributed on the lateral plates and the beam. The vibration screen also carries out elastic deformation motion while doing rigid body motion. The elastic vibration of the screen body results in huge dynamic stress of the lateral plates, which is prone to causing fatigue cracks, thus leading to damage to the lateral plates of the vibration screen. The objective of the dynamical optimization for the vibration screen is to ensure high fatigue resistance and long working life. Therefore, the vibration screen structure is required to be reasonably designed to enable it to have sufficient stiffness, to ensure the frequency of its elastic vibration modal away from the working frequency of the vibration exciter. Meanwhile, the dynamic stress during vibration shall be reduced. An effective way to improve the reliability of the lateral plates of the vibration screen is to solve the problem of large stress of the lateral plates during vibration process [1, 2]. In mechanical structures, stiffened plate is one of the common and effective ways to control the stress level of the structure. However, the structure sizes of the stiffened plate and its arrangement on primary materials have always been the difficulties for designing the structure of lateral plates [3, 4]. The position, sizes and thickness of the stiffener are generally designed according to the experience of engineering technicians, which lacks of theoretical basis. Researches on the structural strength of vibration screen only include strength calculation and analysis without considering the influence of dynamic characteristics. Most of existing literatures on optimization analysis on the vibration screen carry out researches in terms of technological parameters such as a screen surface length, dip, screen hole dimensions and throwing index etc. For instance, Wang [5] carried out multilevel optimization on the established comprehensive mathematical model of screening efficiency to improved screening efficiency and found the optimal values of each parameter during screening process. Delaney [6] analyzed and researched the relationships between screening efficiency and screen surface length under each parameter conditions by taking screen surface length as the entry point, and concluded the general rule relationship between the screen surface length and the screening efficiency. Liu [7] and Dong [8] established related concepts to describe layering and screening phenomena, and found the relationships between them and the vibration parameters as well as structure parameters. However, there are few literature researches on dynamic stress analysis and optimization of lateral plates of the vibration screen. To solve the problem of damage due to large stress of the lateral plates during operation process, it’s necessary to calculate and analyze natural modes and distribution of dynamic stress of lateral plates. In this paper, with the total weight of the lateral plates for the banana-shaped vibration screen as the optimization objective, frequency constraint as the status variables, optimization for multi-frequency constraints was conducted based on the improved genetic algorithm. Next, a mathematical model of structure parameter optimization for the lateral plates of the vibration screen under frequency constraints is established to carry out optimization design in order to obtain a structure with smaller dynamic stress and lower weight. 2. Dynamic characteristic analysis of lateral plates Lateral plates of vibration screen have a complicated banana-shaped structure which is 7.41 m long, 3.25 m wide and 14 mm thick. Due to the complicated structure and large difference between length and width, the lateral plate can be easily damage during working process. Therefore, 10-node tetrahedral units are used for mesh division so as to obtain fine elements with a regular shape. A finite element mesh model is shown in Fig. 1(a), which have 104,153 elements and 170,632 nodes. A partially enlarged model is shown in Fig. 1(b). Fig. 1Finite element model of lateral plates a) Finite element mesh model b) Partially enlarged model of finite elements 2.1. Modal analysis of lateral plates Modes reflect natural characteristics of the lateral plate and can provide certain reference for structural optimization. Based on the finite element model, rigid body modes for the top 6 orders are solved as shown in Fig. 2. It is shown in Fig. 2 that the 3rd order deformation frequency of the lateral plate is very close to the working frequency 12.17 Hz. Under effect of dynamic stress, large deformation will appear in the middle of lateral plates, which likely causes damage of the lateral plate. As a result, the structure should be improved. Fig. 2The top 6 orders natural modes of lateral plates 2.2. Dynamic stress analysis of lateral plates During working process, the vibration screen bears large exciting force, and huge inertia force is distributed on the lateral plate, thus, generating large dynamic stress on the lateral plate and beam. Dynamic harmonic response analysis can obtain responses of the vibration screen structure at any moment, including stress distribution and deformation situations. Thus, whether the structure satisfies conditions can be judged by these responses. The vibration screen is influenced by exciting force, distribution inertia force, gravity and spring supporting counterforce, and damping force is very small influences on the system which can be neglected. Dynamic stress analysis is carried out when the vibration screen moves to the lowest position. Exciting force and inertia force are the maximum values at this moment. Dynamic stress distribution of lateral plates is shown in Fig. 3. Fig. 3Dynamic stress distribution of lateral plates It is shown in Fig. 3 that dynamic stress is distributed widely on the whole lateral plate, wherein stress concentration appears on some parts. In practical working process, the lateral plate would likely be damaged. 3. Optimization of lateral plates based on the improved genetic algorithm 3.1. Necessity of the improved genetic algorithm This paper conducted optimization analysis on lateral plates of the vibration screen by the improved genetic algorithm mainly because of the following reasons. 1) Some defects exist in the traditional genetic algorithm, such as easy precocity, poor partial optimization capacity, and low solution accuracy. The solved result can quickly reaches about 90 % of the optimal solution. However, in order to reach the optimal solution, a lot of time is always needed for the iterative computations, which will undoubtedly increase the computational cost and reduce efficiency. The paper puts forward an improved genetic algorithm based on sensitivity analysis. Sensitivity amendment steps are added during the iterative computations of genetic algorithm. In this way, the excellent individuals in a population can continue mutation and evolution along the assigned direction. Searching ability of the algorithm is effectively improved, and convergence rate as well as calculation efficiency is increased. 2) The optimization calculation in this paper is based on commercial software ISIGHT. In this software, modules such as genetic algorithm and sensitivity analysis are integrated. The two modules can be combined organically by writing a simple program. A lot of time and cost are saved. In this way, the efficient optimization for lateral plates of the vibration screen is realized. 3) During analysis on lateral plates of the vibration screen, the minimal total mass of lateral plates is taken as the objective function, frequencies at multiple points are taken as constraints, and multiple rib sizes are taken as variables. Finally, the optimized structure is obtained. Such structure maybe not the optimal one, but it is shown in the subsequent experiment and simulation analysis on the entire vibration screen structure that the optimization result is effective and satisfies engineering requirements. The genetic algorithm includes three basic genetic operators [9, 10], which are selection, crossover and mutation, respectively. This paper carries out the analysis through the genetic algorithm based on improved mutation operators [11, 12]: where, ${p}_{m}$ is an improved mutation operator. ${p}_{m1}$ has inverse relation to genetic evolution algebra. The value of ${p}_{m1}$ reduces with the increase of genetic evolution algebra. ${p}_ {m2}$ is related to the good or bad degree of average adaptive value of the group. The better the average adaptive value of the group is, the smaller the value of ${p}_{m2}$ will be. ${p}_{m0}$ is the initially assumed mutation rate. ${p}_{m\mathrm{m}\mathrm{i}\mathrm{n}}$ is the minimum value allowed by the mutation rate. $d$ is the current evolutionary generation number. $D$ is the total evolutionary generation number. $\overline{F}$ is the average fitness value of the current group. $\mathrm{m}\mathrm{a}\mathrm{x}F\left({x}_{k}\right)$ is the optimal fitness value of the group by far. ${x}_{k}$ is the design variable corresponding to the optimal fitness value. 3.2. The mathematical model of optimization for lateral plates Lateral plates are shown in Fig. 4. The two holes on the left and the right are the holes of the supporting beam of the vibration screen. The two holes in the middle are the mounting holes for the vibration exciter of the vibration screen. In the figure, the initial thickness of the stiffening rib is 100 mm, and its width is 90 mm, while its length is related to its location on the lateral Fig. 4Initial structure of the lateral plates of the vibration screen It is considered that low-order elastic modal frequency is close to the operation frequency 12.17 Hz, and it will generate huge influences on the mode. As a result, the modal frequencies ${f}_{1}$, $ {f}_{2}$ and ${f}_{3}$ are taken as constraint conditions. During the optimization, the total mass $W$ is taken as the objective function. Thicknesses of four stiffening ribs in Fig. 4 are taken as the design variables. The mathematical description of such optimization is shown as follows: $\mathrm{m}\mathrm{i}\mathrm{n}W\left(x\right)=\sum _{i=1}^{4}{\rho }_{i}{x}_{i}{y}_{i}{z}_{i},$ $s.t.\mathrm{}\mathrm{}\mathrm{}{g}_{j}\left(x\right)=\left({f}_{j}^{2}-{a}_{j}{f}_{0}^{2}\right)\le 0,\left(j=1,2,3\right),$ ${x}_{i}^{l}\le {x}_{i}\le {x}_{i}^{u},\mathrm{}\mathrm{}\mathrm{}\mathrm{}\left(i=1,\mathrm{}2,\mathrm{}3,\mathrm{}4\right),$ where $W\left(x\right)$ is the total mass of lateral plates. $x$ is a vector which includes four design variables. ${\rho }_{i}$ is density. ${x}_{i}$, ${y}_{i}$ and ${z}_{i}$ respectively indicate sizes of the stiffening rib in width, length and height. ${f}_{j}$ is the constraint frequency of the $j$th order. ${a}_{j}{f}_{0}^{2}$ is the square of the expected frequency for $j$th order. ${x}_ {i}$ stands for thicknesses of four stiffening ribs in Fig. 4 and is taken as the design variable. ${x}_{i}^{l}$ is the lower limit value of design variable. ${x}_{i}^{u}$ is the upper limit value of design variable. Change range of each design variable and constraint condition is shown in Table 1. Table 1Change range of design variables and constraint conditions Parameter Initial value Lower limit value Upper limit value ${x}_{1}$ / mm 100.00 60 140 ${x}_{2}$ / mm 100.00 60 140 ${x}_{3}$ / mm 100.00 60 140 ${x}_{4}$ / mm 90.00 60 130 ${f}_{1}$ / Hz 4.24 3.12 7.17 ${f}_{2}$ / Hz 15.04 14.05 17.17 ${f}_{3}$ / Hz 17.16 15.61 19.30 During optimization, sensitivity is taken as the criterion for modifying mutation operator repeatedly. The objective function described in Eq. (2) is changing with the design variables. Therefore, the partial derivative of the objective function to the design variable is taken as the sensitivity value in the subsequent analysis. The computational formula of sensitivity is shown as follows: $e=\frac{\partial W}{\partial {x}_{i}}={\left\{{\psi }_{j}\right\}}^{T}\left(\sum \frac{\partial \left[{k}^{e}\right]}{\partial {x}_{i}}-W\sum \frac{\partial \left[{m}^{e}\right]}{\partial {x}_{i}}\ right)\left\{{\psi }_{j}\right\},$ where $W$ is the objective function. ${\psi }_{j}$ is the modal information corresponding to constraint frequency. ${x}_{i}$ is the design variable. $\left[{k}^{e}\right]$ is stiffness matrix. $\left [{m}^{e}\right]$ is mass matrix. 3.3. Optimization process and results Based on the improved genetic algorithm, optimization analysis is conducted to lateral plates, and optimization flow is shown in Fig. 5. Fig. 5Optimization flow diagram of the improved genetic algorithm Optimization design is conducted to lateral plates based on the improved genetic algorithm, and the detailed process is shown as follows. 1) Encoding chromosomes: the total weight of lateral plates is encoded into a binary string with a constant length based on the improved genetic algorithm, namely a chromosome. The different search points in the search space are constituted by the different combinations of these strings. 2) Generating the initial population: size of the initial population is set to be 40. 3) Calculating fitness value: selection of fitness value will directly influence convergence rate of the improved genetic algorithm and the possibility to find the optimal solution. Fitness value is generally a positive number. As for the problem about maximization, fitness value is always the objective function. On the contrary, as for the problem about minimization, fitness value is the negative number of the objective function. In this paper, the negative number of the total weight is taken as the fitness value. 4) Selection: individuals of which fitness value is stable during evolution are selected. 5) Crossover: the partial chromosomes among the selected 40 individuals are exchanged at a certain probability. In this way, 40 new individuals of sub-generation are generated. The crossover probability is set to be 0.92. 6) Mutation: 40 selected individuals are mutated at a given probability to generate a new population. The mutation probability is 0.05 in this paper. 7) Sensitivity analysis: sensitivity analysis is carried out to the mutated individuals. If the sensitivity requirement is satisfied, the new individuals could be generated. Otherwise, step 6 would be re-started. Mutation operators which do not meet requirements could be rapidly eliminated by adding sensitivity analysis in the improved genetic algorithm. Therefore, the calculation frequency is 8) Judgment: whether the new population could meet constraint conditions is judged, in this way, the optimal results can be found. If the new population doesn't meet constraint conditions, the step was stopped. Otherwise, step 3 will be started again. Optimization design is conducted to lateral plates by the improved genetic algorithm. The obtained results are compared with that of the traditional genetic algorithm, as shown Fig. 6. It is shown in the figure that convergence can be realized when the evolutionary generation number is 30 during calculation based on the improved genetic algorithm. For the traditional genetic algorithm, convergence can be realized when the evolutionary generation number is 60. If the calculation model is more complicated, the genetic algorithm proposed in this paper will show more advantages in time. In addition, the optimal solution obtained by the improved genetic algorithm is 2158 kg, while the optimal solution obtained by the traditional genetic algorithm is 2165 kg. Therefore, the improved genetic algorithm has certain advantages during optimization design of lateral plates. Fig. 6Comparison of calculation results for two genetic algorithms Comparison was carried out between optimal results and that before optimization, as shown in Table 2. According to this table, the mass of the lateral plates reduces by 194.50 kg after optimization for the structure, which reaches good effect. The 1st order modal frequency is rigid motion frequency, its value decreases, which meets the working requirement. The 2nd and the 3rd order modal frequencies increase by 1.73 % and 2.91 % respectively, and are far from the working frequency 12.17 Hz, so that they can effectively avoid resonance, thus preventing damage to the structure of the vibration screen. Table 2Parameters related to optimization of the lateral plates Relevant parameters Before optimization After optimizaiton Change range % ${r}_{1}$ / mm 100.00 60.29 –39.71 ${r}_{2}$ / mm 100.00 60.31 –39.69 ${r}_{3}$ / mm 100.00 61.10 –38.90 $h$ / mm 90.00 60.36 –29.64 ${f}_{1}$ / Hz 4.24 4.17 –1.65 ${f}_{2}$ / Hz 15.04 15.30 +1.73 ${f}_{3}$ / Hz 17.16 17.66 +2.91 ${W}_{t}$ / kg 2352.80 2158.30 –8.27 In addition to that, the dynamic stress distribution of lateral plates is obtained as shown in Fig. 7. In comparison with results before optimization as shown in Fig. 3, it can be found that dynamic stress is obviously improved. Fig. 7Dynamic stress distribution of lateral plates after optimization 3.4. Estimation and analysis on the entire effects of the optimization vibration screen To verify the optimization effect of lateral plates for the vibration screen, we should make some comparisons in the natural characteristics of the vibration screen. However, the vibration screen is very complicated and the finite element model may have problems, so that experiment verification is necessary. Modal comparison is a common method to verify reliability of the finite element model. Therefore, it is necessary to test overall modes of vibration screen on the top 6 orders through experiments. 164 test points are arranged on the vibration screen, wherein each test point is used to measure 3 directions and there are 492 freedom degrees in total. Arrangement of testing points of vibration screen is shown in Fig. 8. Fig. 8Arrangement of modal test points of vibration screen Finite elements and experimental results are compared as shown in Table 3. It is shown that the relative error is controlled within 5 % of engineering requirements, indicating that the finite element model is reliable and can be used for subsequent analysis. Table 3Comparison between vibration screen experiment and simulation model Order Simulation results (Hz) Experimental results (Hz) Error (%) 1 16.09 16.90 –4.70 2 18.07 18.75 –0.35 3 19.83 19.22 3.17 4 23.19 24.16 –4.01 5 33.31 32.23 3.34 6 38.65 37.96 1.84 The top 6 orders modal comparison between the origin vibration screen and the optimization one are calculated by simulation, as shown in Table 4. Table 4Modal comparison between the origin vibration screen and the optimization one Order Origin Optimization Vibration mode 1 16.09 18.37 The two lateral plates reverse twisting around $z$ axle 2 18.07 20.29 The feed end and the discharge end bending and swinging along $z$ direction 3 19.83 24.39 The discharge end swinging along $z$ direction + the entire screen overturning around $y$ axle 4 23.19 41.77 The middle part and the feed end bending and swinging along $z$ direction 5 33.31 45.81 The feed end and the discharge end bending and swinging along $y$ direction 6 38.65 45.92 The discharge end bending and swinging along $z$ direction + the middle part overturning along $z$ direction According to Table 4, the bending deformation frequency of the vibration screen improved significantly after optimization for the structure size, which showed that the overall rigidness of screen body was increased. The 1st and the 2nd order modal frequencies which were close to the working frequency were deformation frequencies which had huge influence on the structure. They were also the constraint frequencies which were considered when optimizing. They became 18.37 Hz and 20.29 Hz from 16.09 Hz and 18.07 Hz before optimization. Meanwhile, Table 5 provides comparison of relevant parameters before and after optimization of the vibration screen. After optimization, the 1st order elastic deformation frequency of the vibration screen increased by 14.17 %, the 2nd order elastic deformation frequency increased by 12.29 %, and the total mass of the vibration screen reduced by 2.35 %. In a manner of speaking, the vibration screen is improved in terms of material saving, stiffness, strength and stability by optimizing the sizes of the lateral plates. Table 5Comparison of relevant parameters before and after optimization of the vibration screen Relevant paramters Before optimization After optimizaiton Change range % ${f}_{1}$ / Hz 16.09 18.37 14.17 ${f}_{2}$ / Hz 18.07 20.29 12.29 ${f}_{3}$ / Hz 19.83 24.39 23.01 ${W}_{t}$ / kg 16532.50 16143.50 2.35 Large noise will be generated by the vibration screen during operation. Lateral plate is a relatively weak structure in the vibration screen. Its radiation noise will directly influence the overall noise level of the vibration screen. Therefore, it is necessary to analyze sound field of the vibration screen before and after optimization of lateral plates. The paper attempted to research sound field by sound array technology. The A-weighted sound power level of the vibration screen is measured in the reverberation chamber. The sound power levels are determined in one-third-octave bands with the screen placed directly on the reverberation chamber floor and with the screen supported on vibration-isolation pads to prevent vibration-radiated noise from the reverberation chamber floor. Bruel & Kjaer Type 4204 Reference Sound Source was used. Next, the sound pressure levels are measured for the vibration screen. A measurement of 30 seconds is used for all tests. From the measured sound pressure levels, the sound power levels are calculated by the following equation: where ${L}_{p.ref}$ is the spatially-averaged sound pressure level inside the reverberation chamber for the reference sound source, ${L}_{p.cal}$ is the spatially-averaged sound pressure level inside the reverberation chamber for the vibration screen, ${L}_{w.ref}$ is the calibrated sound power level of the reference sound source, and ${L}_{w.cal}$ is the calculated sound power level of the vibration screen. Measurements are performed at a distance of 5.54 meters from sides of the screen, so the entire screen will fit within the measurement area of the array. Measurements are shown in Fig. 9. Sound power levels of the vibration screen before and after optimization of the lateral plates are compared, as shown in Fig. 10. Fig. 9Diagram of noise measurement for the vibration screen Fig. 10Sound power levels of the vibration screen before and after optimization It is shown in Fig. 10 that sound power levels of the vibration screen are obviously different before and after optimization of lateral plates. After optimization, sound power levels decrease to a certain extent and the maximum value decreases by nearly 10 dB. This result is very important for improving noise problem of the vibration screen. In order to more clearly observe the change of sound field before and after optimization of lateral plates, contours of sound pressure at 300 Hz are extracted, as shown in Fig. 11. It is shown in Fig. 11 that areas with large sound pressures are obviously improved after optimization. In addition, two figures have an area which has the large sound pressures, and it is caused by an operation motor of the vibration screen rather than lateral plates. Therefore, our researches are very meaningful to reduce noise of the vibration screen. Fig. 11Contours of sound pressure around lateral plates before and after optimization a) Contours of sound pressure around lateral plates before optimization b) Contours of sound pressure around lateral plates after optimization 4. Conclusions In this paper, with the total weight of the lateral plates for the banana-shaped vibration screen as the optimization objective, frequency constraints as the status variables, optimization for multi-frequency constraints is conducted based on the improved genetic algorithm. Next, a mathematical model of structure parameter optimization for the lateral plates of the vibration screen under frequency constraints is established to carry out optimization design in order to obtain a structure with smaller dynamic stress and lower weight. Sensitivity analysis is added into the improved genetic algorithm, and the optimization efficiency is increased simultaneously. The structure frequency is optimized by means of the improved genetic algorithm. Then, a modal experiment is carried out to the entire vibration screen so as to verify reliability of the finite element model, and the natural characteristics of the vibration screen before and after optimization are analyzed, and the top 6 orders natural frequencies and vibration modes of the entire vibration screen are calculated, so as to indicate that optimized vibration screen is improved in terms of material saving, stiffness and stability. In addition, noise is directly related to vibration. As a result, the change of the vibration screen should be also analyzed. Noise of the vibration screen is tested by sound array technology. Results showed that the radiated noise is reduced after optimization, and optimization in this paper is feasible. • Xiao J., Tong X. Particle stratification and penetration of a linear the vibration screen by the discrete element method. International Journal of Mining Science and Technology, Vol. 22, Issue 3, 2012, p. 357-362. • Xiao J., Tong X. Characteristics and efficiency of a new the vibration screen with a swing trace. Particuology, Vol. 11, Issue 5, 2013, p. 601-606. • Dong K. J., Yu A. B., Brake I. DEM simulation of particle flow on a multi-deck banana screen. Minerals Engineering, Vol. 22, Issue 11, 2009, p. 910-920. • Delaney G. W., Cleary P. W., Hilden M., et al. Validation of DEM predictions of granular flow and separation efficiency for a horizontal laboratory scale wire mesh screen. Seventh International Conference on CFD in the Minerals and Process Industries, 2009, p. 1-6. • Wang G., Tong X. Screening efficiency and screen length of a linear the vibration screen using DEM 3D simulation. Mining Science and Technology (China), Vol. 21, Issue 3, 2011, p. 451-455. • Delaney G. W., Cleary P. W., Hilden M., et al. Testing the validity of the spherical DEM model in simulating real granular screening processes. Chemical Engineering Science, Vol. 68, Issue 1, 2012, p. 215-226. • Liu C., Wang H., Zhao Y., et al. DEM simulation of particle flow on a single deck banana screen. International Journal of Mining Science and Technology, Vol. 23, Issue 2, 2013, p. 273-277. • Dong K. J., Wang B., Yu A. B. Modeling of particle flow and sieving behavior on the vibration screen: from discrete particle simulation to process performance prediction. Industrial and Engineering Chemistry Research, Vol. 52, Issue 33, 2013, p. 11333-11343. • Beasley J. E., Chu P. C. A genetic algorithm for the set covering problem. European Journal of Operational Research, Vol. 94, Issue 2, 1996, p. 392-404. • Hartmann S. A competitive genetic algorithm for resource‐constrained project scheduling. Naval Research Logistics (NRL), Vol. 45, Issue 7, 1998, p. 733-750. • Goldberg D. E., Korb B., Deb K. Messy genetic algorithms: motivation, analysis, and first results. Complex Systems, Vol. 3, Issue 5, 1989, p. 493-530. • Weile D. S., Michielssen E. Genetic algorithm optimization applied to electromagnetics: a review. IEEE Transactions on Antennas and Propagation, Vol. 45, Issue 3, 1997, p. 343-353. About this article Mechanical vibrations and applications dynamic stress lateral plates frequency constraints experiment design improved genetic algorithms sensitivity analysis The work presented in this paper is supported by projects of the National Natural Science Foundation of China (61473112). Copyright © 2015 JVE International Ltd. This is an open access article distributed under the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.	{"url":"https://www.extrica.com/article/15826","timestamp":"2024-11-08T20:48:09Z","content_type":"text/html","content_length":"145186","record_id":"<urn:uuid:a12ebff3-c625-41f6-8d74-b33c850c986d>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00122.warc.gz"}
The currect voltage relation of diode is given by 1=(e^(1000 V//T)-1) Doubtnut is No.1 Study App and Learning App with Instant Video Solutions for NCERT Class 6, Class 7, Class 8, Class 9, Class 10, Class 11 and Class 12, IIT JEE prep, NEET preparation and CBSE, UP Board, Bihar Board, Rajasthan Board, MP Board, Telangana Board etc NCERT solutions for CBSE and other state boards is a key requirement for students. Doubtnut helps with homework, doubts and solutions to all the questions. It has helped students get under AIR 100 in NEET & IIT JEE. Get PDF and video solutions of IIT-JEE Mains & Advanced previous year papers, NEET previous year papers, NCERT books for classes 6 to 12, CBSE, Pathfinder Publications, RD Sharma, RS Aggarwal, Manohar Ray, Cengage books for boards and competitive exams. Doubtnut is the perfect NEET and IIT JEE preparation App. Get solutions for NEET and IIT JEE previous years papers, along with chapter wise NEET MCQ solutions. Get all the study material in Hindi medium and English medium for IIT JEE and NEET preparation	{"url":"https://www.doubtnut.com/qna/649422768","timestamp":"2024-11-06T13:46:33Z","content_type":"text/html","content_length":"257046","record_id":"<urn:uuid:17ab14b0-1c30-44a3-be71-cb26f4ce4580>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00509.warc.gz"}
Transclusion and Substitution 30th October 2023 at 12:42pm The power of WikiText comes from the ability to use the content of one tiddler inside another one. This ability takes several different forms that can easily be confused. The main distinction is between a transclusion and a textual substitution: • A transclusion is replaced dynamically with the value of either: □ a tiddler field □ a variable • Textual substitutions are performed on the text of macro definitions before they are used Tiddler Field Transclusion Transclusion in WikiText describes the basics of transclusion. For example: As described in HTML in WikiText, you can also transclude tiddler field values as attributes of HTML elements and widgets. For example: <$text text={{MyTiddler}}/> As described in Introduction to filter notation, you can also transclude tiddler field values using the filter syntax. For example: {{{ [tag{TiddlerContainingMyTag}] }}} Variable/Macro Transclusion Variables that were defined with parameter or variable substitution are referred to as "macros". The value of a variable/macro can be transcluded with the syntax: <<myMacro param:"Value of parameter">> As described in HTML in WikiText, you can also transclude a variable as the value of an attribute of HTML elements and widgets. For example: <$text text=<<myMacro>>/> As described in Introduction to filter notation, you can also transclude a variable as the value of a filter parameter using the filter syntax. For example: {{{ [tag<myMacro>] }}} Textual Substitution Textual substitution occurs when the value of a macro/variable is used. It is described in Substituted Attribute Values and substitute Operator The key difference between substitution and transclusion is that substitution occurs before WikiText parsing. This means that you can use substitution to build WikiText constructions. Transclusions are processed independently, and cannot be combined with adjacent text to define WikiText constructions.	{"url":"https://tiddlywiki.com/prerelease/static/Transclusion%2520and%2520Substitution.html","timestamp":"2024-11-02T03:04:48Z","content_type":"text/html","content_length":"6562","record_id":"<urn:uuid:fb61607c-5c04-4342-a730-530213e8f531>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00784.warc.gz"}
404 Club Reboot Due to inflation our 404 club price cap seems a little unworkable, so here are some options to re-boot ihe idea - first three are what common currency to choose (pic 1 out of 3) the rest 4-10 are the price cap (pick 1 out of the 7 options) £4.04 originally worked out to $5, or thereabouts accounting for currency fluctuations. Inflation calculator from 2020 (when the thread was started, though not when Andy coined the term) says we're looking at $6 currently. What currency is the 6.99 inflation adjusted figure cited in the poll? $6 may still seem restrictive, but don't forget that's part of the point. It's a challenge. Thrill of the hunt and all. Diamonds in the rough. Also, and my personal angle on the game, a fun and inexpensive way to learn watchmaking skills. Edited by spectre6000 4 hours ago, spectre6000 said: £4.04 originally worked out to $5, or thereabouts accounting for currency fluctuations. Inflation calculator from 2020 (when the thread was started, though not when Andy coined the term) says we're looking at $6 currently. What currency is the 6.99 inflation adjusted figure cited in the poll? $6 may still seem restrictive, but don't forget that's part of the point. It's a challenge. Thrill of the hunt and all. Diamonds in the rough. Also, and my personal angle on the game, a fun and inexpensive way to learn watchmaking skills. A challenge both in finding one and fixing one up. £5.98 ?? In keeping with the original http inspired 404, 598 is a Timeout error, which seems fitting! Almost a 50% increase. When you think about it (as a watch addict) and how much you spend on a typical watch, anything less than about $15 is dirt cheap these days. I mean, that might honestly just be the tax you pay on a watch. You can barely even get a used part off of ebay delivered for that much. In fact, $15 seems like about what a new crystal or a used setting lever would cost to have delivered. I think anything under that would essentially be a, "Holy cow, I got this watch for nothing!" scenario. 1 hour ago, GuyMontag said: When you think about it (as a watch addict) and how much you spend on a typical watch, anything less than about $15 is dirt cheap these days. I mean, that might honestly just be the tax you pay on a watch. You can barely even get a used part off of ebay delivered for that much. In fact, $15 seems like about what a new crystal or a used setting lever would cost to have delivered. I think anything under that would essentially be a, "Holy cow, I got this watch for nothing!" scenario. Cool , I'll be expecting a 6 month old Rolex for that then 1 hour ago, GuyMontag said: When you think about it (as a watch addict) and how much you spend on a typical watch, anything less than about $15 is dirt cheap these days. I mean, that might honestly just be the tax you pay on a watch. You can barely even get a used part off of ebay delivered for that much. In fact, $15 seems like about what a new crystal or a used setting lever would cost to have delivered. I think anything under that would essentially be a, "Holy cow, I got this watch for nothing!" scenario. Why do you think i make my own setting levers guy ? fresh off the blow torch 4 minutes ago, Waggy said: Only 6 people have voted.... need more votes before we can update the 404 club price limits Can we consider how tight as a duck's arse Yorkshiremen and Scotsmen are please Ha ha, once knew a Scotsman who dropped a penny, it hit him on the back of the head as he bent over to catch it.. Remember the value is only the upper limit - for you 'cost-conscious' cultures, you can still go as low as you want 5 minutes ago, Neverenoughwatches said: Can we consider how tight as a duck's arse Yorkshiremen and Scotsmen are please Just now, Waggy said: Lol i know Scott, I'm just bantering. It was Spectre6000 and AndyHull's brain child, so some of the originality should to be kept in place from their first posts. What you say Andy ? Totally Andy's brilliant idea. I just started the thread. For what my opinion is worth, the number should be quite low. That's the point of it. It's as much about the hunt as the act of servicing/repairing the watches. I don't have too much of an opinion on currency, but £ has a slight edge since that's what it started with. This is a global forum, so the majority of people will be converting values no matter what. I don't know if there are any currency traders here, but if there's any currency that's especially stable or some other attribute that would benefit the game, that could be amusing. Maybe one that's super volatile? Qualification is heavily influenced by what wacky things some currency is doing on a given day (though usually that means shrinkage). We could go for something super obscure like the Chilean Peso or something just for giggles (though it might make conversion difficult). A lot of currencies are indexed to the US$, which would ease some of the conversion friction for some people. Just spitballing. My vote in the poll would be for £ inflation adjusted to £4.04 ca. 2020. I'm ticking those boxes, but I'm not sure about the 6.99. Is that £, or another currency? Source? I went with 8.90£ because 12th century Italian mathematicians are always excellent topics to bring up when trying to get your guests to go home. • 1 • 2 I really like the idea of the 404. However I've yet to be able to manage to obtain any watch for that little. I do have some free quartz ones. I'm not that skilled. Mechanical for me. I have managed to obtain a number of watches for under £40. What about an extra 40 club. $, £ etc. I could at least participate. Edited by rossjackson01 @rossjackson01 it's only getting harder and harder to get 'cheap' watches to work on, I have done some reading and depending on the source the prices of vintage watches is increasing somewhere between 10% and 50% a year. The only way I have found to get watches to lay the 404 game with is to buy a bulk lot say 10 with an average price of < £4.04 but even that is getting harder and harder, hence the re-boot. I see that £6.99 is looking like a popular choice - but I ask the members, is £6.99 realistic, can you actually find watches to work on for £6.99 - maybe its better to cut to the chase and jump to £10.10 for example and allow more people to play? Jut a thought. On 5/13/2024 at 1:59 AM, eccentric59 said: I went with 8.90£ because 12th century Italian mathematicians are always excellent topics to bring up when trying to get your guests to go home. I have the perfect solution for this, i just dont invite anyone round. £10.00 is not a huge amount for a watch these days with the increased asking prices. Let’s count ourselves lucky we don’t have this for tools, it would be £404.00 9 minutes ago, tomh207 said: £10.00 is not a huge amount for a watch these days with the increased asking prices. Let’s count ourselves lucky we don’t have this for tools, it would be £404.00 Ah but, will you count DIY versions. Edited by rossjackson01 21 minutes ago, rossjackson01 said: I really like the idea of the 404. However I've yet to be able to manage to obtain any watch for that little. I do have some free quartz ones. I'm not that skilled. Mechanical for me. I have managed to obtain a number of watches for under £40. What about an extra 40 club. $, £ etc. I could at least participate. The best and almost only way to achieve the magical £ 4.04 individual watch cost is via the loophole ploy of job lot buys Ross. Dividing down the outlay by the number of watches purchased. I've managed it many times in the past, this then gives you the extra edge of choosing any watch from that lot. Maybe we should have an all time winner, whos prepared to troll through the last four years of posts and pick some worthy contenders. Unfortunately i have a poorly shoulder so i cant raise my right hand and my left shoulder is out in sympathy for my right shoulder. Put your right hand up if you think i talk a load of bull 33 minutes ago, rossjackson01 said: Ah but, will you count DIY versions. Tom's comment is purely restricted to the Bergeon culture of " lets rip off everyone on the planet " 35 minutes ago, Neverenoughwatches said: The best and almost only way to achieve the magical £ 4.04 individual watch cost is via the loophole ploy of job lot buys Ross. Dividing down the outlay by the number of watches purchased. I've managed it many times in the past, this then gives you the extra edge of choosing any watch from that lot. Maybe we should have an all time winner, whos prepared to troll through the last four years of posts and pick some worthy contenders. Unfortunately i have a poorly shoulder so i cant raise my right hand and my left shoulder is out in sympathy for my right shoulder. Put your right hand up if you think i talk a load of bull Tom's comment is purely restricted to the Bergeon culture of " lets rip off everyone on the planet " Voted , i thought to keep it in pounds as originally designed and out of respect for Andy . Personally i would have kept the £4.04 as I'm very traditionalist and dont like change but i know how hard you overseas guys are finding it and i believe in fair play. I expect to be posting the occasional Omega or Longines but then you brought this on yourself so dont get jealous Scott • 1 • 2 59 minutes ago, Neverenoughwatches said: The best and almost only way to achieve the magical £ 4.04 individual watch cost is via the loophole ploy of job lot buys Case in point, I just scored a lot of 4 Timex mechanical watches for $15.50 which puts the individual watches at $3.88 each, or £3.09 (or €3.59, ¥28.08 (CH), ¥607 (JP), ₹324 ... or 0.00006 Bitcoin ) 21 minutes ago, spectre6000 said: The UK lots can be a 404 hit or miss, 6 - 10 watches can come in anywhere from 20 - 50 quid. I once had a 404 division of 8 watches for a fraction over 30 quid, one of them was a Smiths Astral model National 17 an absolute certainty for winner contender. The crystal was scrached up to the point of being very difficult to see the brand, something triggered a gut feeling in me and it paid off.	{"url":"https://www.watchrepairtalk.com/topic/29469-404-club-reboot/","timestamp":"2024-11-04T08:18:02Z","content_type":"text/html","content_length":"495670","record_id":"<urn:uuid:af687b5a-a336-479b-b358-0ef2d4a73243>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00048.warc.gz"}
IBPS Clerk Quant Algebra Quiz 25 - Attempt Now! IBPS Clerk Quant Algebra Quiz 25 – Attempt Now! Testbook \| Updated: Nov 16, 2018 17:27 IST Here is IBPS Clerk Quant Algebra Quiz 25 for upcoming Banking exams like IBPS Clerk and other Banking Exams. This quiz contains important questions which match the pattern of banking exams, so make sure you attempt today’s Quantitative Aptitude IBPS Clerk Quiz to check your preparation level. Watch this video for learning Quant Algebra – IBPS Clerk Algebra Quiz 25 Que. 1 Mother + daughter + infant age is 74. Mother’s age is 46 more than daughter and infant. And infant’s age is 0.4 of the daughter. Find daughter’s age. Que. 2 Difference between two numbers is 4 and their product is 17. Then find the sum of their squares? Que. 3 The difference between two numbers is 9 and the product of the two is 14.What is the square of their sum? Que. 4 A man has Rs. 480 in the denominations of one-rupee notes, five-rupee notes and ten-rupee notes. The number of notes is equal. What is the total number of notes? Que. 5 Find x: $\frac{{17\left( {2 – x} \right) – 5\left( {x + 12} \right)}}{{1 – 7x}} = 8$ Que. 6 In the following question two equations numbered I and II are given. You have to solve both the equations and give answers: I: 2x^2 + 11x + 12 = 0 II: 5y^2 + 27y + 10 = 0 Que. 7 I: x^2 + 13x + 42 = 0 II: y^2 + 16y = – 63 Que. 8 If ‘x’ added to a number which exceeds x by 4 the result is 26. The value of ‘x’ is: Que. 9 In following question, two equations are given, you have to solve them and find the relation between ‘x’ and ‘y’ and choose the correct option. x^2 + 11x + 30 = 0, y^2 + 7y + 12 = 0 Que. 10 The cost of 12 note-books and 16 pens is Rs. 852. What is the cost of 9 note-books and 12 pens? Check out more articles from the table below: As we all know, practice is the key to success. Therefore, boost your IBPS Clerk preparation by starting your practice now. Solve Practice Questions for Free Furthermore, chat with your fellow IBPS Clerk aspirants and our experts to get your doubts cleared on Testbook Discuss.	{"url":"https://testbook.com/blog/ibps-clerk-quant-algebra-quiz-25/","timestamp":"2024-11-06T17:02:42Z","content_type":"text/html","content_length":"249140","record_id":"<urn:uuid:f342d6ff-25f3-4ef5-961f-0b841a509287>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00652.warc.gz"}
A. Ravi P. Rau Alumni Professor of Physics Ph.D., 1970 - University of Chicago Louisiana State University Department of Physics & Astronomy 223 Nicholson Hall, Tower Dr. Baton Rouge, LA 70803-4001 Personal Home Page Research Interests Atomic Theory and Quantum Information Currently, I am exploring a general analytical technique for solving time-dependent operator equations through a succession of unitary integrations. The Bloch-Liouville equation for a (2j+1)-level system in time-dependent fields is of particular interest, along with current topics in nuclear magnetic resonance and quantum bits. The method extends to dissipation and decoherence which are of great interest in quantum information and quantum computing. Proceeding in analogy to the Bloch sphere construction for a single spin (two-level system), a geometrical construction describes higher dimensional spheres and other manifolds for two-spin and larger systems. A related investigation is to alter the evolution of entanglement between two spins in the presence of dissipation and decoherence. The so-called sudden death of entanglement can be delayed or averted by suitable local actions on the two spins. Our recent interest is in the area of quantum information: studies of entanglement and other correlations such as quantum discord, their evolution under dissipative and decoherent processes and how they may be controlled, geometrical and symmetry studies of operators and states of N qubits and connections between the Lie and clifford algebras involved with topics in projective geometry and design theory. Current and Select Publications • The Beauty of Physics: Patterns, Principles, and Perspectives, Oxford University Press, 2014. A. R. P. Rau • “Manipulation of entanglement sudden death in an all optical experimental set-up,” (with Ashutosh Singh, S. Pradyumna, and U. Sinha), J. Opt. Soc. Am. B 34, 681-690 (2017). • “Shared symmetries of the hydrogen atom and the two-qubit system,” (with G. Alber), Topical Review J. Phys. B: At. Mol. Opt. Phys. 50, 242001 (2017). • “Symmetries and Geometries of Qubits, and their Uses,” Symmetry. 12, 1732 (21 pp) (2021). • “Mapping qubit algebras to combinatorial designs,” (with J. P. Marceaux), Quantum Inf. Process. 19, 49 (2020).	{"url":"https://tigertrails.lsu.edu/physics/people/faculty/rau.php","timestamp":"2024-11-15T03:34:11Z","content_type":"text/html","content_length":"30226","record_id":"<urn:uuid:304b1eae-364e-4975-8af0-909d8961e421>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00820.warc.gz"}
Setting initial WIP limits When starting a team up with Kanban, one of the earliest questions is how to you set initial WIP limits. The simple rules we use are covered in this video. Key points: • Whatever numbers you pick will be wrong anyway so don’t get stressed about picking the perfect number today. You will adjust and improve these numbers over time. • Ask the team what’s a normal number of items in each column. We don’t need to calculate anything. • Make it a tiny bit bigger than you think you need on the first day so we aren’t hitting the limits right away. We’ll tighten up those limits soon enough. • Do not set the limits such that we are immediately in violation of them. If we have five items in a column today, then the limit we set can’t be less than five. • Many teams tend to over-complicate this step of setting initial limits. Keep it simple and then improve. • Make sure you review and adjust these WIP limits again within the next few weeks. You will have learned new things about how work flows across the board by then.	{"url":"https://improvingflow.com/2021/04/18/setting-initial-wip-limits.html","timestamp":"2024-11-05T10:21:10Z","content_type":"text/html","content_length":"9843","record_id":"<urn:uuid:24d287e3-e87d-482e-b60e-480ac0c7bea9>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00270.warc.gz"}
The Ultimate Guide to the AP Statistics Exam (2024) Are you taking AP Statistics? If so, you're likely wondering what to expect from the AP Statistics exam. Before you sit down to take the final test, it's important to understand how the AP Stats test is formatted, what topics it will cover, and how it'll be scored. This guide will explain all of that information, show you official sample problems and give you tips on the best way to prepare for the AP Statistics test. In 2023, the AP Statistics exam will take place on Thursday, May 5th at 12:00pm. How Is the AP Statistics Exam Structured? How long is the AP Statistics exam? The test is a total of three hours long and contains two sections: multiple choice and free response. You're allowed a graphing calculator for the entire exam. Multiple-Choice Section • 40 multiple-choice questions • 90 minutes long • Worth 50% of exam score • You can spend an average of a little more than two minutes on each multiple-choice question and finish the section in time. Free-Response Section • 5 short-answer questions • 1 Investigative Task • 90 minutes long • Worth 50% of exam score • The five short-answer questions are meant to each be solved in about 12 minutes, and the Investigative Task is meant to be solved in about 30 minutes. What Does the AP Statistics Exam Test You On? The content of the AP Stats exam and course is centered around nine units. Below are the nine units, along with what percentage of the exam will be on them and all the topics that fall beneath each of them. Each unit starts with an "introducing statistics" question that'll be answered throughout the unit. The list below covers every single topic that the AP Statistics exam could test you on. Unit 1: Exploring One-Variable Data (15-23% of exam) • Introducing statistics: What can we learn from data? • Variables • Representing a categorical variable with tables • Representing a categorical variable with graphs • Representing a quantitative variable with tables • Describing the distribution of a quantitative variable • Summary statistics for a quantitative variable • Graphical representations of summary statistics • Comparing distributions of a quantitative variable • The normal distribution Unit 2: Exploring Two-Variable Data (5-7% of exam) • Introducing statistics: Are variables related? • Representing two categorical variables • Statistics for two categorical variables • Representing the relationship between two quantitative variables • Correlation • Linear regression models • Residuals • Least squares regression • Analyzing departures from linearity Unit 3: Collecting Data (12-15% of exam) • Introducing statistics: Do the data we collected tell the truth? • Introduction to planning a study • Random sampling and data collection • Potential problems with sampling • Introduction to experimental design • Selecting an experimental design • Inference and experiments Unit 4: Probability, Random Variables, and Probability Distributions (10-20% of exam) • Introducing statistics: Random and non-random patterns? • Estimating probabilities using simulation • Introduction to probability • Mutually exclusive events • Conditional probability • Independent events and unions of events • Introduction to random variables and probability distributions • Mean and standard deviation of random variables • Combining random variables • Introduction to the binomial distribution • Parameters for a binomial distribution • The geometric distribution Unit 5: Sampling Distributions (7-12% of exam) • Introducing statistics: Why is my sample not like yours? • The normal distribution, revised • The Central Limit Theorem • Biased and unbiased point estimates • Sampling distributions for sample proportions • Sampling distributions for differences in sample proportions • Sampling distributions for sample means • Sampling distributions for differences in sample means Unit 6: Inference for Categorical Data: Proportions (12-15% of exam) • Introducing statistics: Why be normal? • Constructing a confidence interval for a population proportion • Justifying a claim based on a confidence interval for a population proportion • Setting up a test for a population proportion • Interpreting p-values • Concluding a test for a population proportion Unit 7: Inference for Quantitative Data: Means (10-18% of exam) • Introducing statistics: Should I worry about error? • Constructing a confidence interval for a population mean • Justifying a claim about a population mean based on a confidence interval • Setting up a test for a population mean • Carrying out a test for a population mean Unit 8: Inference for Categorical Data: Chi-Square (2-5% of exam) • Introducing statistics: Are my results unexpected? • Setting up a chi-square goodness of fit test • Carrying out a chi-square test for goodness of fit • Expected counts in two-way tables • Setting up a chi-square test for homogeneity or independence • Carrying out a chi-square test for homogeneity or independence • Skills focus: Selecting an appropriate inference procedure for categorical data Unit 9: Inference for Quantitative Data: Slopes (2-5% of exam) • Introducing statistics: Do those points align? • Confidence intervals for the slope of a regression model • Justifying a claim about the slope of a regression model based on a confidence interval • Setting up a test for the slope of a regression model • Carrying out a test for the slope of a regression model • Skills focus: Selecting an appropriate inference procedure AP Statistics Sample Questions As we mentioned above, there are three types of questions on the AP Stats exam: multiple choice, short answer, and investigative task. Below are examples of each question type. You can see more sample questions and answer explanations in the AP Statistics Course Description. Multiple-Choice Sample Question There are 40 multiple-choice questions on the exam. Each has five answer options. Some questions will be accompanied by a chart or graph you need to analyze to answer the question. Short-Answer Sample Question There are five short-answer questions on the AP Stats test. Each of these questions typically includes several different parts you need to answer. You're expected to spend about 12 minutes on each short-answer question. Investigative Task Sample Question The final question on the exam is the Investigative Task question. This is the most in-depth question on the test, and you should spend about 30 minutes answering it. It will have multiple parts you need to answer and require multiple statistics skills. You'll also need to provide a detailed explanation of your answers that shows the strength of your statistics skills. Be sure to show all your work as you'll be graded on the completeness of your answer. How Is the AP Statistics Test Graded? For the multiple-choice part of the exam, you earn one point for each question you answer correctly. There are no point deductions for incorrect answers or questions you leave blank. Official AP graders will grade your free-response questions. Each of the six free-response questions is scored on a scale of 0 to 4 points, so the total section is out of 24 points. The free-response questions are graded holistically, which means, instead of getting a point or half a point for each bit of correct information you include, graders look at your answer to each question as a "complete package," and your grade is awarded on the overall quality of your answer. The grading rubric for each free-response question is: • 4: Complete Response: Shows complete understanding of the problem's statistical components • 3: Substantial Response: May include arithmetic errors, but answers are still reasonable and show substantial understanding of the problem's statistical components • 2: Developing Response: May include errors that result in some unreasonable answers, but shows some understanding of the problem's statistical components • 1: Minimal Response: Misuses or fails to use appropriate statistical techniques and shows only a limited understanding of statistical components by failing to identify important components • 0: No Response: Shows little or no understanding of statistical components What does holistic grading mean for you? Basically, you can't expect to earn many points by including a few correct equations or arithmetic answers if you're missing key statistical analysis. You need to show you understand how to use stats to get a good score on these questions. Estimating Your AP Statistics Score If you take a practice AP Stats exam (which you should!) you'll want to get an estimate of what your score on it is so you can get an idea of how well you'd do on the real exam. To estimate your score, you'll need to do a few calculations. #1: Multiply the number of points you got on the multiple-choice section by 1.25 #2: For free-response questions 1 through 5, add the number of points you got together and multiply that sum by 1.875 (don't round). If you need help estimating your score, the official free-response questions we linked to above include sample responses to help you get an idea of the score you'd get for each question. #3: For free-response question #6, multiply your score by 3.125. #4: Add the scores you got in steps 1-3 together to get your Composite Score. For example, say you got 30 questions correct on the multiple-choice section, 13 points on questions 1-5, and 2 points on question 6. Your score would be (30 x 1.25) + (13 x 1.875) + (2 x 3.125) = 68.125 which rounds to 68 points. By looking at the chart below, you can see that'd get you a 4 on the AP Statistics exam. Below is a conversion chart so you can see how raw score ranges translate into final AP scores. I've also included the percentage of students who earned each score in 2021 to give you an idea of what the score distribution looks like: Composite Score AP Score Percentage of Students Earning Each Score (2022) 70-100 5 14.8% 57-69 4 22.2% 44-56 3 23.4% 33-43 2 16.5% 0-32 1 23.1% Source: The College Board Where Can You Find Practice AP Stats Tests? Practice tests are an important part of your AP Stats prep. There are official and unofficial AP Stats practice tests available, although we always recommend official resources first. Below are some of the best practice tests to use. Official Practice Tests To learn more about where to find AP Statistics practice tests and how to use them, check out our complete guide to AP Statistics practice exams. 3 Tips for the AP Statistics Exam In this section we go over three of the most useful tips you can use when preparing for and taking the AP Statistics test. Follow these and you're more likely to get a great score on the exam. #1: For Free Response, Answer the Entire Question As we mentioned earlier, free-response questions on AP Stats are graded holistically, which means you'll get one score for the entire question. This is different from many other AP exams where each correct component you include in a free-response question gets you a certain number of points, and those points are then added up to get your total score for that question. The Stats free-response questions are graded holistically because there are often multiple correct answers in statistics depending on how you solve the problem and explain your answer. This means you can't just answer part of the question and expect to get a good score, even if you've answered that part perfectly. If you've ignored a large part of the problem, your score will be low no matter So instead of trying to get a point here and there by including a correct formula or solving one part of a question, make sure you're looking at the entire problem and answering it as completely as possible. Also, if you need to include an explanation, be sure it explains your thought process and the steps you took. If your explanation shows you understand important stats concepts, it could help you get a higher score even if your final answer isn't perfect. Aiming for the most complete response possible is also important if you can't answer one part of a question that's needed to answer other parts. For example, if you can't figure out what the answer to part A is, but you need to use that answer for parts B and C, just make up an answer (try to keep it logical), and use that answer to solve the other parts, or explain in detail how you'd solve the problem if you knew what the answer to part A was. If you can show you know how to solve the latter problems correctly, you'll likely get some credit for showing you understand the stats concepts being tested. #2: Know How to Use Your Calculator You'll need a graphing calculator to answer pretty much every question on the Stats exam, so make sure you know how to use it. Ideally, the calculator you use on test day will be the same one you've been doing homework and taking tests with throughout the school year so you know exactly how to use it. Knowing how to solve common stats functions on your calculator and interpret the answers you get will save you a lot of time on the exam. Your calculator will likely be most useful on the multiple-choice section where you don't need to worry about showing work. Just plug the data you're given into your calculator, and run the right equations. Then you'll have your answer! #3: Know Your Vocabulary You may think that since AP Stats is a math course, vocab won't be an important part of the test, but you need to know quite a few terms to do well on this exam. Confusing right- and left-skewed or random sampling and random allocation, for example, could lead to you losing tons of points on the test. During the school year, stay on top of any new terms you learn in class. Making flashcards of the terms and quizzing yourself regularly is a great way to stay up-to-date on vocab. Many AP Stats prep books also include a glossary of important terms you can use while studying. Before the AP Stats exam, you should know all important terms like the back of your hand. Having a general idea isn't good enough. A big part of stats is being able to support your answers, and to do this you'll often need to use stats vocab in your explanations. Just stating the term won't earn you nearly as many points as being able to explain what the term is and how it supports your answer, so make sure you really know your vocab well. Summary: Statistics AP Exam The AP Statistics exam is three hours long and consists of 40 multiple-choice questions and six free-response questions. To prepare well for AP Stats exam questions, it's important to take practice exams and know how to grade them so you can estimate how well you'd do on the actual test. When studying for the AP exam, remember to answer the entire question for free response, know how to use your calculator, and be on top of stats vocabulary. What's Next? Feel the need to do some quick reviewing after looking through what'll be covered on the AP Stats exam? Take a spin through our guide to statistical significance to refresh yourself on how to run a How difficult is AP Stats compared to other AP classes? Get the answer by reading our guide to the hardest AP exams and classes. Wondering which other math classes you should take besides statistics? Math is often the trickiest subject to choose classes for, but our guide will help you figure out exactly which math classes to take for each year of high school. A prep book can be one of your best study resources for the AP Stats exam. But which prep book should you choose? Check out our guide to AP Stats prep books to learn which is the best and which you should avoid. Have friends who also need help with test prep? Share this article! About the Author Christine graduated from Michigan State University with degrees in Environmental Biology and Geography and received her Master's from Duke University. In high school she scored in the 99th percentile on the SAT and was named a National Merit Finalist. She has taught English and biology in several countries. Get Free Guides to Boost Your SAT/ACT	{"url":"https://shepherdstownfilmsociety.org/article/the-ultimate-guide-to-the-ap-statistics-exam","timestamp":"2024-11-08T22:11:15Z","content_type":"text/html","content_length":"143017","record_id":"<urn:uuid:2b01f4db-c6af-451c-a802-0a4424c58c07>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00324.warc.gz"}
1. On one class of degenerating elliptic operators.-Mat.Sb. v.79 ,N 3 (1969) ,p.p. 381-404 (in Russian) 2. On Euler's degenerating operators, defined in a bounded domain.- Vestnik Mosc. Univ. ,ser. math ,mech , N 1(1971), p.p.36-43(in Russian) 3. Boundary value problems for some classes of degenerating elliptic operators.- Soviet Math. Dokl. Vol. 12(1971) N 2, p.p.506-509. 4. On global smoothness of solutions of one class of degenerated elliptic equations ,Russian Math Surveys ,26(5), 1972, p.p.227-228 (in Russian) 5. On one class of global hypoelliptic operators.- Mat.Sb. v.91, N3 (1973), p.p.367-389(in Russian). 6. Analytical first integrals of guasilinear parabolic equations-'Vestnik Mosk.Univers.ser.Math.Mech N 1,1974,p 45-54' (in Russian)(with M.I.Vishik). 7. Analytical first integrals of the Burgers equation and of the Navier-Stokes system and their application. Reprint N 35 of Inst.of Mech.publ.of AN.USSR,1974,p.1-62. (in Russia)(with M.I.Vishik). 8. Analytical first integrals of non-linear parabolic equation and their application,-im' ' Proceeding of allunion school on diff. eq..Eg.with infinity number of variables and Dynamical Syst. inf.dim.Diligan,1973.'Erevan,1974,p.p.257-266 (in Russian)(with M.I.Vishik). 9. Analytical first integrals of Non-linear parabolic,in the sense of I.G.Petrovsky,systems of differential equations and their applications,Russian Math.Surveys,29(2) (1974), p.p.123-155 (with 10. Analytical first integrals of non-linear parabolic equations and their application.-Math.USSR Sb.21(1973), p.p.347-377 (with M.I.Vishik) 11. Some questions on the theory of non-linear elliptic and parabolic equation.-Math.USSR Sb.23(2) (1974), 287-318 (with M.I.Vishik) 12. Asymptotic expansions of moment functions for solution non-linear parabolic equation.-Math.USSR Sb.24(4) (1974), p.p.575-591 (with M.I.Vishik). 13. Analytical first integrals of nonlinear parabolic equations and their applications . - Russian Math Surveys,30(2),1975 ,p.p.261-262(in Russian) (with M.I.Vishik) 14. The Hopf equation, statistical solutions,moment functions, corresponding to the Navier-Stokes system, and the Burgers equation.- Reprint N 66 of Inst. of Mech.probl of ANUSSR, 1976,p.p.1-68(in Russian) (with M.I.Vishik) 15. Homogeneous statistical solutions of parabolic systems of differential equations and of the Navier-Stokes system.- Reprint N 88 of Inst. of Mech probl of ANUSSR, 1977 ,p.p.1-57 (in Russian) (with 16. The Cauchy problem for non-linear equations of the Schrodinger equation type.-Math.Sb.96(3) (1975), p.p.457-468 (in Russian) (with M.I.Vishik) 17. The Cauchy problem for the Hopf equation corresponding to parabolic equations.Statistical solutions and moment functions.- Soviet Math.Dokl.17(2) (1976),p.p.553-557 (with M.I.Vishik) 18. L'equation de Hopf, les solutions statistiques, les moments, correspondents aux systemes des equation paraboliques quasilineaires. - J. Math. Pure et Appl. 56 (1977), p.p.85- 122 (with 19. Solution statistiques homogenes des systemes differential paraboliques et du systemes de Navier-Stokes. - Ann. Scuola, norm. super. Pisa. Cl. Sci. Ser.IV, 4(3) (1977), pp.531-576 (with 20. Translationally homogeneous statistical solutions and indi- vidual solutions with infinite energy of the Navier-Stokes equations, - Sibirian Math. J., 19(5) (1978), pp.1005-1031 (with M.I.Vishik) (in Russian) 21. Formula for some functionals on smooth solutions of a class of systems of quasi-linear equations, Russian Math. Surveys 31(1) (1976), p.p.265-266 (in Russian) 22. First integrals and integralability of systems of quasi-linear equations. - Amer. Math. Soc. Transl. 118(2) (1982), pp.281- 306 23. Mathematical problems of statistical hydromechanics. - Moscow, Nauka, 1980, 440 p(in Russian). (with M.I.Vishik) 24. Mathematische probleme der statistichen Hydromechanik.- Leipzig,Akad.Verlag. 1986 ,428 s(in German) (with M.I.Vishik) 25. Mathematical problems of statistical hydromechanics.- Dorrend, Boston, London, 1988, 576p. Kluwer Academic publishers (in English) (with M.I.Vishik) 26. Practical work on numerical methods in optimal control problems,- Moscow,Moscow Univ. Pub. 1988 (in Russian) (with V.V.Alexandrov , N.S.Bahvalov and so on.) 27. On some control problems and results concerning the unique solubility of a mixed boundary value problem for the three- dimensional Navier-Stokes and Euler systems. - Dokl. Acad. Nauk SSSR 252(5) 1980, 1066-1070 (in Russian) 28. Control problems and theorems concerning the unique solubi- lity of a mixed boundary value problem for the three-dimen- sional Navier-Stokes and Euler equations. - Math. USSR Sbornik, 43(2) (1982), p.p.251-273 29. To the question on unique solubility of the three-dimensional Navier-Stokes equations for almost all initial values.-Russian Math. Surveys, 36(2) (1981), p.p.207-208 (in Russian) 30. Homogeneous statistical solution of the Navier-Stokes system.- Russian Math Surveys, 32(5),1977,p.p.179-180; (with M.I.Vishik) (in Russian) 31. Homogeneous statistical solution of the Navier-Stokes system.-Theses of 2 Vilnilous conf. on probability theory \& Math. statistic. v1,1977,p.p.82-84 (with M.I.Vishik)(in Russian) 32. Homogeneous stochastic solutions of the Navier-Stokes equations.-Int.symp.on stoch.dif.eq.Vilnicous,1978,p.p.116-117. (with H.I.Vishik, A.I.Komech) 33. X-homogeneous space-time statistical solutions \& individual solutions with nonbounded energy of the Navier-Stokes equations.Russian Math. Surveys,33(3)1978, p.p.133-134 (with H.I.Vishik \& A.I.Komech) (in Russian) 34. X-homogencous space-time statistical solutions of the Navier-Stokes system and individual solutions with infinite energy.- Dokl. A.N.USSR,39 N5 1978,p.p.1025-1028 (with H.I.Vishik)(in Russian) 35. Certain mathematical problems of statistical hydromechanics.-Russian Math. Surveys. 34(5),1979 p.p.135-210 (with H.I.Vishik \& A.I.Komech) 36. On a control problem and a result concerning the unique solubility of the three-dimensional Navier-Stokes system.Russian Math. Surveys,35(4) 1980p.188 (in Russian). 37. Properties of solution of certain extremal problems and the theorems on unique solubility of the three-dimensional Navier-stores system.-Russian Math.Surveys,36(5),1981 p.p.222-223 (in Russian) 38. Certain question of control theory of nonlinear systems with distributed parameters. Proceedings of U.G.Petrovsikij seminar.N9.1983 p.p 167-189 (in Russian) 39. x-homogeneous statistical solutions of Navier-Stokes system.- "Partical Differential Equations ",Novosibirsk,Nauka,1980,p.p. 162-166 (with M.I.Vishik) (in Russian). 40. Translationaly homogeneous statistical solution of the Navier-Stokes system and their properties.-Certain problems of Mathematics and mechanics,Moscow, Moscow univ. Publ. 1981 p.112. (in 41. Certain mathematics problems of turbulent flows statistical description.-Proceedings of 1.st.allunion school-seminar on many dimensional problems of the mechanics of continuous medium.,Dep.p.p197-217 (in Russian). 42. Space-time moments and statistical solutions concentrates on smooth solutions of the three-dimensional Navier-Stokes system or on quasilinear parabolic system.-Dokl.Akad.Nauk.SSSR,274 43. Control problems for the Navier-Stokes system and for other nonlinear distributed systems. - TAGUNG Particle differential algleichungen und optimal steuerung , vom.3, bis.5, october 1984, Merseburg. 1984, p.p.4-5. (in Russian). 44. Properties of solutions of some control problems connected with the Navier - Stokes equations, Dokl. Acad. Nauk SSSR, 25(1) (1982). p.p. 40 - 45 ( in Russian ) 45. Properties of solutions of certain extremal problems connected with the Navier - Stokes equations. - Math USSR Sbornik, 46(3), (1983),p.p.323-351 46. Statistical extremal problems and unique solubility of the three-dimensional Navier - Stokes equations for almost all initialvalue.- Prikl. Mat. i Mech. 5(1982),p.p.797-805 47. On numerical method of the energy - using minimization in nonstationary thermo-electrical cooling process. Ing.Phys.J.,51(4) (1986), p.p.690-691 ( with A.S.Laktiushkin, A.V.Mihailenko) (in 48. Solubility of the chain of equation for space- time moments. - Math. USSR Sb. 53(1986), N 2, p.p. 307-334 49. Analytic functionals and the unique solubility of quasilinear dissipative system for almost all initial conditions .- Trans. Moscow Math. Soc. 1987, p.p.1-55 50. To the question on solubility of the Caushy problem for the Laplace operator. - Moscow Univ. Math. Bull. 42 (1987) ( with A. Romanovich) 51. On uniqueness of the solution of the chain of moment equations corresponding to the three-dimensional Navier - Stokes system. Math.USSR Sb., Vol.62,(1989), N. 2. p.p. 465-490 52. The Cauchy problem for a second order elliptic equation in a conditionally well-posed formulation. Trans. Moscow Math. Soc. (1990),p.p.139-176 53. On the problem of the chain of moment equations in the case of large Reynolds number - Unclassical equations in the case of large Reynolds numbers. Unclassical equations and equations of mixed type. Publ. of Math. Inst. of Syberian section of Academy of Science of USSR, (1990), p.p.228-247 (in Russian) 54. Navier - Stokes equations from the point of view of the theory of ill-posed boundary value problems. Navier - Stokes equations theory and numerical methods, ed. J.G.Heywood et al.Lecture Notes in Mathimatics. 1431,1990,p.p.23-30. 55. Unique solubility of the chain of equations for space-time moment.- Russian Math.Surveys,40(5) 1985 .239-240 (in Russian). 56. Solubility "in whole" of chain of equations for space moments corresponding of smooth solutions of the three-dimensional Navier- Stokes system. Wiss.Z.T.H.Lenna-Merseburg,27,1985,p.p.613-612(in 57. Uniqueness of smooth solutions the chain of moment equations and the Hopf-Foias equation, corresponding to the three-dimensional Navier-Stokes system. Functional-differentional equations and their applications.1 North-Caucasus region conference.Mahachcala.1986 p.p.210-211(in Russian) 58. On the numerical method of energyusing minimization during nonstationar process of the thermoelectrical cooling. Dep.VINITI 24.04. N 3034-B86,p.p.1-14. 59. On uniqueness of the solutions of the chain of equations for space-time moments corresponding to the three-dimensional Navier-Stokes system. Russian Math.Surveys,41(4),1986.p.p.160-161 60. On uniqueness of solutions of the chain of moment equations and of the Hopf equation corresponding to the three-dimensional Navier-Stokes system. 1st World Congress of the Bernoulli society of Math.statistic and prob.theory.Theses v.2,p.699.Tashkent 1986. 61. About uniqueness of solutions the chain of equations for space moments corresponding to the tree-dimensional Navier-Stokes system. Theses of Int.Conf."Nonlinear seismology"Suzdal 31.10-11.1986. 62. Statistical hydromechanics: paper for Encyclopaedia on Probability Theory Moscow, for Encyclopaedia Pub. 1991 (in Russian) (with M.I. Vishik). 63. Hopf Equation: paper for Encyclopaedia on Probability Theory. Moscow. Sov. Encyclopaedia Pub. 1991 (in Russian) (with M.I. Vishik). 64. Hydromechanic Equations (statistical solutions): paper for Encyclopaedia on Probability Theory. Moscow. Sov. Encycl. Pub. 1991 (in Russian) (with M.I. Vishik). 65. On Cauchy problem for an elliptic equation. Theses of 6-th conf. "Nonlinear problems of mathemat. phys." Donetsk. 1987 p. 152. (in Russian). 66. Optimal control of systems described by the Navier - Stokes equations. Theses of All - Union conf. "Actual problems of modelling and control of distributed systems." Odessa. 8.9 - 10.9. 1987. Kiev. 1987 p.32. 67. Conditionally well - posed formulation of the Cauchy problem for on elliptic equation.- Theses of 2nd North-Caucasus conf.on Funct.-Duffer. eq. Mahachkala 1988. 68. Problem of turbulence(functional formulation) paper for Encyclopedia "Mathematical Physics".Moscow Encycl.Pub.(To appear in Russian) 69. On a method of closure of a chain of moment equations in the case large Reynolds numbers.-3d Int.Conf."Lavrentiev reading on mech.phys." 10.9-14.9.1990.Novosibirsk.p.p.58-59. 70. On the problem of closure of the Friedman-Keller chain of equations in the case of large Reynolds numbers.-Theses of seminar in Baku 25.9-28.9. 1990.p.p.32-33. 71. Necssary and sufficient conditions of extremum in the problem of optimal control of the system described by Cauchy problem for the Laplace operator. Russian Math.Surveys,44(4),1989,p.p.216-217. (in Russian). 72. On the statistical approach to the Navier-Stokes equations. The Navier-Stokes equations.Theory and numeral methods,ed. J.G.Heywood et al. Lecture Notes in Mathematics 1431,1990,p.p.40-48 73. Lagrange principle for problems of optimal control of illposed or singular distributed systems. J.Math.Pures Appl., 71(1992),p.p.139-195 74. The problem of closure of chains of moment equations corresponding to the three-dimensional Navier-Stokes system in the case of large Reynolds numbers.-Soviet Math.Dokl. 44(1)(1992),p.p.80-85 75. The theory of moments for Navier-Stokes equations with a random right-hand-side.-Izvestija Ross.Akad.Nauk,seria Math., 56(1992),N6,p.p.1311-1353(in Russian) 76. On e-controllability of the Stokes System with distributed control concentrated in subdomain.-Russian Math Surveys 47(1), 1992,p.217-218.(in Russian)(with O.Yu.Imanuilov) 77. The closure problem for the chain of the Friedman-Keller moment equations in the ease of large Reynolds numbers.- The Navier-Stokes equations II-Theory and Numerical Methods, ed.J.G.Heywood ef al. lecture Notes in Mathematics 1530,1991,p.p.226-245 78. The convergence velocity of approximations for the closure of the Friedman-Keller chain of equations in the case of large Reynolds numbers.-Math.Sbornik v.182,N2,1994,p.115-143 (with 79. On approximate controllability of the Stokes system.-Ann. de la Facult'e des Science de Touluse,v.II,N2,1993,p.205-232(with O.Yu.Imanuvilov) 80. The convergence velocity by the closure of the chain of moment equations,corresponding to the Navier-Stokes system with a random right-hand-side. Diff.Uravn.v.30,N4,1994,p.699-711 (with 81. Certain problems of optimal control of the Navier-Stokes system with distributed control. IMA Preprint Series N 1348,October 1995, p.1-45. 82. On controllability of certain systems simulating a fluid flows. In Flow Control,IMA Vol.Math.Appl., 68, Ed. by Gunzburger,Springer-Verlag, New-York, 1995,p.149-184. (with O.Yu.Imanuilov) 83. On Exact Boundary Zero Controllability of two- dimensional Navier-Stokes Equations.-Acta Applic. Math.v.37,1994,p.67-76 (with O.Yu.Imanuvilov) 84. A Simple Proof of the Approximate Controllability from the Interior for Nonlinear Evolution Problems. Appl.Math.Lett. v.7,N5,(1994),p.85-87 (with J.I.Diaz) 85. Exact boundary zero controllability of three- dimensional Navier-Stokes Equations, Journ. of Dynamical and Control Systems.-v.1,N3,(1995) p.325-350. 86. Local exact controllability of the Navier-Stokes Equations.- C.R.Ac.Sc. Paris t.323, S\'erie 1, p.275-280 1996 (with O.Yu.Imanuvilov) 87. Local Exact Boundary Controllability of the Boussinesque Equations.- SIAM J. Control Optim.- v.36, N2, 1998, pp.391-421. (with O.Yu. Imanuvilov) 88. Local Exact Controllability for 2-D Navier-Stokes Equations. Matem.Sbornik v.187, N 9, 1996, p103-138, Sbornik:Mathematics 187:9, 1996, p.1355-1390 (with O.Yu.Imanuvilov) 89. Approximate controllability of the Stokes system on cylinders by external unidirectional forces. J.Math.Pure et Appl.,76 (1997) p.353-375 (with J.I.Diaz) 90. Boundary value problems and optimal boundary control for the Navier-Stokes system: the two- dimensional case.- SIAM J.Control Optim. v.36, N3 (1998) p.852-894 (with M.D.Gunzburger and L.S.Hou) 91.Controllability of Evolution equations.- Seoul National University, Seoul 151-742, Korea, 1996, 163 p. (With O.Yu. Imanuvilov) 92. Global Exact Controllability of the 2D Navier-Stokes Equations on a Manifold without boundary.- Russian J. of Math.Phys. v.4, N4, 1996, p.429-448. (with J.-M.Coron) 93. Local exact boundary controllability of the Navier-Stokes system.- Conterporary mathematics, V.209,1997,p.115- 129. (with O.Yu.Imanuvilov) 94. Time-periodic statistical solutions of the Navier-Stokes equations.- Lecture Notes in Physics, v.491, Turbulence Modelling and Vortex Dynamic, Boratav, Eden, Ersan (eds.) Springer-Verlag, 1997, 95. Local exact controllability of the Boussinesq equations.- Vestnik Ross. Un.Dr.Nar., ser. mat.N3, vyp.1, 1996, c.177-197 (with O.Yu.Imanuvilov) 96. Approximate controllability of Stokes system.- Vestnik Ross. Un.Dr.Nar., ser. mat.N 1, vyp.1, 1994. ‘. 89-108. 96. Mark Iosifovich Vishik (to 75-birthday)- Russian J. of Math.Phys. v.4, N4, 1996 (With M.S.Agranovich and other) 97. Mark Iosifovich Vishik (to 75 birthday).- Uspekhi Matem. Nauk, (to appear in 1997) (With M.S.Agronovich and other) 98. To 70-birthday of Vera Nikolaevna Maslennikova.- Vestnik Ross. Univ. Druzhby Nar. ser. Math. N3, vyp.1, 1996, pp.2-15 (in Russian) (With A.V.Arutiunov and other.) 99. Optimal Dirichlet Control and inhomogeneous boundary value problems for the unsteady Navier-Stokes equations.- Proceedings of the conference "Control and Partial Differential equations", CIRM, Marsi- elle- Luminy, June 16-20,1997. (With M.Gunzburger and S.Hou) 100. Optimal control of systems with distributed parameters. The- ory and applications. Naucnaya Knyga, Novosibirsk, 1999, 350 p.(in Russian) 101. Optimal Control of Distributed Systems. Theory and Applications.- Translations of Mathematical Monographs, v. 187, 2000, Amer. Math. Society, Providence, Rhode Island, 305 p. 102. Static Hedging of Barrier Options with a Smile: An inverse Problem.-ESAIM, Control, optimisation and Calculus of Variations. vol.8 (2002), p.127-142 (electronic) (With C. Bardos and R. Douady) 103. Optimal boundary control of the Navier-Stokes equations with bounds on the control.- to Proceedings of the Korean Advanced Institute for Science and Technology Workshop on Finite Elements.) 1999 (with M. Gunzburger and S.Hou) 104. Optimal Control Problems for Navier-Stokes system with distributed control function.- Chapter 6 in "Optimal Control of Viscous Flow" Ed. by S.S.Sritharan, SIAM, Philadelphia, 1998, p.109-150. 105. The closure problem for the Friedman-Keller infinite chain of moment equations, corresponding to the Navier-Stokes system.- Proceedings of the Second Monte Verita Colloquium on Turbulence, March 22-28, 1998, Trends in Mathematics, 1999, Birkhauser Verlag Bassel/Switzerland, pp.17-24. 106. On controllability of the Navier-Stokes equations.- Proceedings of the Second Monte Verita Colloquium on Turbulence, March 22-28, 1998, Trends in Mathematics, 1999 Birkhauser Verlag Bassel/ 107. Controllability property for the Navier-Stokes equations.- Proceedings of the International Conference on Control of Partial Differential Equations. Chemnitz, April 20-25, 1998, International Series of numerical Mathematics, Vol. 133, pp.157-165, 1999 Birkhauser Verlag Basel/Switserland 108. Exact controllability of the Navier-Stokes and Boussinesq equations.- Uspechi Matem. Nauk. v.54, N 3(327), 1999, p.93-146 (with O.Yu.Imanuvilov) (in Russian);English Translation: Russian Math. surveys, vol.54:3 (1999), 565-618. 109. On Boundary Zero Controllability of the Three-Dimentional Navier-Stokes Equations.-Theory of the Navier-Stokes Equations. Ed. by J.G Heywood and other, Ser. in Adv. of Math. for Appl. Sciences, vol. 47, 1998, p.p. 31-45. 110. Trace theorems for Three-Dimensional, time-dependent sole- noidal vector fields and their applications, Trans. Amer. Math. Soc. v.354, (2002), 1079-1116. (with M.D.Gunzburger, L.S.Hou) 111. Stabilizability of quasi linear parabolic equation by feedback boundary control.-Sbornik: Mathematics, v.192:4 (2001), 593-639. 112. Stabilizability of Two-Dimensional Navier-Stokes equations with help of a boundary feedback control.-J. of Math. Fluid Mech. v.3, (2001), 259-301. 113. Exact Controllability and Feedback Stabilization from a boundary for the Navier-Stokes Equations.- "Control of Fluid Flow", P.Koumoutsakos, I.Mezic (eds.), Lecture Notes in Control and Information Sciences, v. 330 Scpringer-Verlag Berlin, Heidelberg, 2006, p.p. 173-187 114. Exact controllability from a boundary and stabilization by by boundary feedback control for parabolic equations and Navi- er-Stokes system.- Vestnik Tambovsk. Universiteta, vol. 5 (4), 2000, p.509 (in Russian) 115. Feedback stabilization for the 2D Navier-Stokes equations.- The Navier-Stokes equations: theory and numerical methods. Lecture Notes in pure and appl. Math., vol. 223, (2001) Marcel Dekker, Inc., New-York, Basel, pp.179-196. 116. Boundary value problems for three-dimensional evolutionary Navier-Stokes equations.- J. Math. Fluid Mech., vol.4:1 (2002),pp.45-75. (with M. Gunzburger and L. Hou) 117. Stabilization for the 3D Navier-Stokes system by feedback boundary control, Discrete and Cont. Dyn. Syst., v.10, no 1\&2, (2004), p.289-314. 118. Feedback stabilization for the 2D Oseen equations: additional remarks.- Proceedings of the 8th Conference on Control of Distributed Parameter Systems. International series of numerical mathematics, vol 143 (2002). Birkh\"aser Verlag pp.169-187. 119. Optimal boundary control for the evolutionary Navier-Stokes system: the three-dimensional case.- SIAM. J. Control Optim. v.43, N6, (2005),2191-2232. (with M. Gunzburger and L. Hou) 120. Real Process Corresponding to 3D Navier-Stokes System and Its Feedback Stabilization from Boundary .- Amer. Math. SSoc. Translations Series 2, v.206. Advances in Math. Sciences-51. PDE M.Vishik seminar. AMS Providence Rhode Island (2002), p.95-123. 121. Real Processes and Realizability of a Stabilization Method for Navier-Stokes Equations by Boundary Feedback Control.- Nonlinear Problems in Mathematical Physics and Related Topics II, In Honor of Professor O.A.Ladyzhenskaya, Kluwer/Plenum Publishers, New-York, Boston, Dordrecht, London, Moscow, 2002, p.137-177 (s.127-164 in Russian edition). 122. Stabilization from the boundary of solutions to the Navier-Stokes system: Solvability and justification of the numerical simulation.- Dalnevostochnyy Mat. J. v.4, N1, (2003) , p.86-100 (in 123. Feedback stabilization for Oseen fluid equations: a stochastic approach.- J.Math.Fluid Mech. 7 (4), (2005), 574-610. (with J.Duan) 124. Analyticity of stable invariant manifolds of 1D-semilinear parabolic equations.- Proceedings of Joint Summer Research Conference on Control Methods and PDE Dynamical Systems, F.Ancona, I.Lasiecka, W.Littman, R.Triggiani (eds.); AMS Contemporary Mathematics (CONM) Series 426, Providence, 2007, 219-242 125. 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Proc.St.Petersburg Math.Soc. v.XV; AMS Transl.Series 2, v.232 (2014), 99-118. 145. Stabilization of the simplest normal parabolic equation by starting control. Communication on Pure and Applied Analysis, v.13,N5,September (2014),1815-1854. 146. On one estimate, connected with the stabilization of normal parabolic equation by starting control.-Fundamental and applied mathematics (in Russian) (with L.S.Shatina) (to appear)	{"url":"http://mech.math.msu.su/~fursikov/all_works.php","timestamp":"2024-11-10T18:23:09Z","content_type":"text/html","content_length":"38704","record_id":"<urn:uuid:b5d015ab-b78a-4307-9593-3deaa0b7641d>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00562.warc.gz"}
IMPORTANT: Older versions of DeCAFS have a major bug that severely affect the computational complexity of the procedure. This was fixed from version 3.3.2. Should you have an older version installed (lower than 3.3.2) please make sure you update your DeCAFS package either through CRAN or GitHub. You can check your version number at the bottom of the documentation page of DeCAFS, via help WHAT’S NEW: In addition to the automatic model selection, we introduced a graphical iterative model selection procedure that aids the user in selecting an appropriate model for a given sequence of observations. This tuning procedure can seriously improve performances under more challenging scenarios. More details can be found by checking the documentation: help("guidedModelSelection"). DeCAFS is a c++ implementation for R of the DeCAFS algorithm for performing optimal multiple changepoint detection on detecting the change in mean in presence of autocorrelation or random fluctuations in the data sequence. Installation and Requirements Installing the package To install the package from Github: Alternatively one could fork this repository, and: Requirements for the installation The packages requires Rcpp with compiler support for the std library with the g++14 standard. Bugs and further queries If any bug should be spotted, or for any information regarding this package, please email the package mantainer: g dot romano at lancaster.ac.uk. The model We model a combination of a radom walk process (also known as standard Brownian motion or Wiener Process) and an AR process. Let Then, DeCAFS solves the following minimization problem: Where our Quick Start This demo shows some of the features present in the DeCAFS package. Three functions at the moment are present in the package: DeCAFS Main function to run the DeCAFS algorithm on a sequence of observations dataRWAR Generate a realization of a RW+AR process estimateParameters Estimate the parameters of our model At the moment only two functions for data generation and parameter estimation are present, and they all are tailored for the Random Walk. Since l2-FPOP can tackle also other Stochastic Processes, more functions are expected to be added. A simple example We will start generating a Random Walk. The function dataRWAR takes in: • the length of the sequence of observations, • a poisson parameter regulating the probability of seeing a jump, • the average magnitude of a change, • the • the autocorrelation parameter Y = dataRWAR(n = 1e3, poisParam = .01, meanGap = 15, phi = .5, sdEta = 3, sdNu = 1) y = Y[["y"]] Running DeCAFS is fairly straightforward: We can plot the DeCAFS segmentation (red lines), alongside with our real segmentation (dotted blue lines). Running the algorithm without estimation Alternatively, we can also pass all the required parameters in order for it to run. In this case, since we both have an AR and RW component, we will need to pass down both ## Error: <text>:1:84: unexpected input ## 1: res = DeCAFS(y, beta = 2 * log(length(y)), modelParam = list(sdEta = 3, sdNu = 1, \ ## ^ Extreme case: Random Walk Let’s say we now have the Our Algorithm is capable of dealing with this extreme situation: Y = dataRWAR(n = 1e3, poisParam = .01, meanGap = 15, phi = 0, sdEta = 2, sdNu = 1) y = Y[["y"]] res = DeCAFS(y, beta = 2 * log(length(y)), modelParam = list(sdEta = 2, sdNu = 1, phi = 0)) which leads to the result: Extreme case: Autoregressive model Secondly, let’s say that the In this case we need to set Y = dataRWAR(n = 1e3, poisParam = .01, meanGap = 10, phi = .98, sdEta = 0, sdNu = 2) y = Y[["y"]] res = DeCAFS(y, beta = 2 * log(length(y)), modelParam = list(sdEta = 0, sdNu = 2, phi = .98)) which leads to the result: we see that in this case we miss one changepoint. Contributing to this package If you have interest to contribute to this package, please do not esitate to contact the maintainer: g dot romano at lancaster.ac.uk.	{"url":"https://cran.stat.auckland.ac.nz/web/packages/DeCAFS/readme/README.html","timestamp":"2024-11-07T07:39:22Z","content_type":"application/xhtml+xml","content_length":"18991","record_id":"<urn:uuid:b8a6d099-009b-49cf-952f-fd4b4f3adc15>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00504.warc.gz"}
Big Bang And Age Of The Universe \| AtomsTalk Big Bang And Age Of The Universe The age of the Universe has been something that fascinated all of humanity from ancient history. We have, in each of our cultures, tried to find an answer to this. While different traditions and myths give the universe different ages, the scientific view of the matter emerged only last century. What is currently accepted age of universe? The currently accepted age of the universe is around 13.8 billion years with a uncertainty of 20 million years How Do We Know The Age of the Universe? There are mainly two ways scientists use to find the age of the universe. 1. One is to calculate the age of the oldest objects in it, and then guess at the Universe’s age. 2. The other method is to study the development of the universe (that is its rate of expansion) and trace back to its origin. From the Oldest Stars How do Stars’ “Lives” Work? A star’s age and lifespan depend on its mass. The lightest stars burn for billions of years. Middle-range stars like our Sun, burn for around 9 billion years. Massive stars can burn out in “just” several million years. Crucially, if we know the mass of a star, we can estimate its age. Globular Clusters Globular clusters consist of closely packed stars which are all formed at the same time. From the logic above, the oldest globular clusters consist of the lightest stars. These stars were identified observationally and their ages estimated. This gave them an age of between 11 to 14 billion years, meaning that the universe cannot be younger than 11 billion years. A globular cluster. Note how the stars are closely packed. Source. From the History of the Universe The Big Bang Up until the early 20th century, scientists thought that the universe was stationary, not expanding. However, observations of some galaxies and theories developed in the 1910s and 1920s suggested that far away galaxies were moving away faster from us than closer ones. This was later summarized as Hubble’s Law Galaxies further apart are receding faster. This law was actually proposed in 1927 by Georges Lemaître, a Belgian physicist. He went on to say that we could use this law to visualize back in time, till a point where galaxies are not receding anymore. This would be the beginning of everything; Lemaître called it the primeval atom. Later, more popularly, it was called the Big Bang. The big bang and the history of the universe. Source. Using Hubble’s Law Hubble’s law uses a constant called “Hubble’s constant” to find how fast galaxies are moving apart (“recession”). Calculating this value accurately would let us trace back time like Lemaître suggested, and find the age of the Universe. To find this value, scientists used many satellites such as Planck and WMAP (Wilkinson’s Microwave Anisotropy Probe). These satellites worked on finding the composition of the universe – how much of it is normal matter, how much of it is dark matter, and so on. These values would then be used with a set of equations called the Friedmann equations, which would give us the age of the Universe. Use in Friedmann’s Equations Friedmann’s equations suggest that the age of the universe is of the form H[0] is the Hubble constant various Ωs are values that help us know the composition of the Universe. factor F is only a correction factor So, the approximate age can be guessed at by taking the reciprocal of H[0]. This gives us around 14.5 billion years as a rough estimate. The value of F depends on the model of the universe we study. Using what is called a “flat” model, we arrive at an age of 9 billion years. This does not reflect real studies. A more realistic model is not “flat” and thus uses certain terms that give a slightly different value. Thus, a more accurate value, as suggested by data from Planck and WMAP, would be around 13.8 billion years. The age of the universe is clearly immense. Recent progress in calculating it has given us an age of 13.8 billion years with an uncertainty of around 20 million years. With an improved understanding of the universe, our estimates should likely get more accurate. Understanding the age of the universe would help us know better about its past and its origins. We could know more about how it began and developed into what it is today. Going further back into the mists of time and understanding how everything came to be is a wonderful quest indeed. Leave a Comment	{"url":"https://atomstalk.com/blogs/big-bang-and-age-of-the-universe/","timestamp":"2024-11-09T07:56:30Z","content_type":"text/html","content_length":"171238","record_id":"<urn:uuid:cd74ffca-e299-4e9b-9fbf-eab4704a5a1d>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00072.warc.gz"}
Electronics Toolkit Electronics Toolkit is a helpful app with dozens of tools, calculators and references made for electronic engineers, students and hobbyists. Calculators: • Resistor color codes - calculate the resistance of resistors by selecting the colors of the bands • SMD resistor codes - calculate the resistance of SMD resistors by entering the number • LED resistor calculator - calculate the needed resistance to connect en LED to a power source • Parallel resistors - calculate the resistance of resistors in parallel • Voltage divider - calculate the output voltage of a voltage divider • Series resistors - calculate the resistance of resistors in series • Ohm's law - calculate the voltage, current of resistance by entering the other two • Capacitance calculator - calculate the capacitance, voltage or charge by entering the other two • Battery discharge - calculate the time it takes to discharge a battery • Inductor color codes - calculate the inductance of inductors by selecting the colors of the bands • Parallel capacitors - calculate the capacitance of capacitors in parallel • Series capacitors - calculate the capacitance of capacitors in series • Unit converter - unit converter for length, temperature, area, volume, weight, time, angle, power and base • Op-amp calculator - calculate the output voltage of non-inverting, inverting, summing and differential opamps • Wheatstone bridge - calculate the resistance of one resistor in a balanced bridge or calculate the output voltage • Inductor codes - calculate the inductance of inductors by entering the number • Capacitor codes - calculate the capacitance of capacitors by entering the number • DAC and ADC calculator - calculate the output of digital-analog and analog-digital converters • Wavelength frequency calculator - calculate the frequency or wavelength of a wave • SI prefixes - convert numbers with SI prefixes • Capacitor energy - calculate the energy in an capacitor • Slew rate calculator - calculate the slew rate • Star delta transformation - calculate the resistors in a star delta transformation • Zener calculator - calculate the resistance of the resistor and voltage of the zener • Air core inductor calculator - calculate the inductance and wire length of an air core inductor • 555 timer calculator - calculate the frequency, period, duty cycle, high time and low time of a popular 555 timer circuit • Plate capacitor calculator - calculate the capacity of a plate capacitor • Resistance to color code calculator - calculate the colors on the resistor by entering the resistance • LM317 - calculate the output voltage of an LM317 • Low pass filters • Wire resistance - calculate the resistance of electrical wire • RMS voltage • Decibel calculator •Reactance Tables: • Logic gates - truth table of the 7 logic gates with interactive buttons • 7-segment display - interactive display that you can change by clicking on one of the segments or by clicking on a button to show a hexadecimal character • ASCII - decimal, hexadecimal, binary, octal and char ASCII table • Resistivity - table with the resistivity of common metals at 293K • Arduino pinout • Pinout diagrams of 4000 and 7400 series ICs Other: • Bluetooth - connect to a bluetooth module like the HC-05 to talk with an arduino or other microcontroller with the terminal, button and slider modes PERMISSIONS • read the contents of your USB storage && modify or delete the contents of your USB storage - used to save images of IC pinout diagrams • receive data from Internet && view network connections && full network access - used to load the IC data list and to collect statistics with Google Firebase • pair with Bluetooth devices && access Bluetooth settings - used to connect with bluetooth devices • prevent device from sleeping - prevents bluetooth devices from disconnecting	{"url":"https://rootpk.com/electronics-toolkit-2bd052972ad","timestamp":"2024-11-06T21:35:48Z","content_type":"text/html","content_length":"41343","record_id":"<urn:uuid:a86c5710-02c7-4143-8c64-692fcd088852>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00370.warc.gz"}
Perpetuity and Deferred Perpetuity: What are the Different? - AUDITHOW Perpetuity and Deferred Perpetuity: What are the Different? A perpetuity is a constant and equal cash flow continuing infinitely. Deferred perpetuity is the same cash flow but starts at a deferred time. Theoretically, the value of perpetuity continues infinitely. However, the present value of the future cash flows arising from perpetuity or deferred perpetuity can be discounted. Let us discuss what is a perpetuity and deferred perpetuity. What is a Perpetuity? In finance, perpetuity refers to a system of cash flows for an infinite time. It is a stream of cash flows with no end. Often the term is used for annual payments associated with financial securities such as bonds, stocks, and real estate income. The concept of perpetuity is used in business valuation models as well as discounted cash flow analysis commonly. Theoretically, the cash flows continue infinitely, however, in practice, it is rarely the case. The formula to calculate perpetuity can be written in different ways. The present value of the perpetuity can be calculated by: PV= C/(1+r)^1+ C/(1+r)^2 + C/(1+r)^3…… = C/r Where C = Cash flow and r = discount rate In the basic form, the present value of a perpetuity is the sum of all future cash flows. It can be discounted using a reasonable discount rate used by an entity. Similarly, an entity can calculate the present value of a perpetuity that grows at a certain rate and starts after a certain period. Perpetuity with Growth The formula for the PV of perpetuity with a growth rate is: Value of Perpetuity = C[n] × (1+g)/(r-g) Where Cn is the cash flow in year n, r is the discount rate, and g is the growth rate of perpetuity. The value of perpetuity will be then discounted for the PV using the PV factor for year n. How Does a Perpetuity Work? The total theoretical value of a perpetuity is infinite. However, the present value of a perpetuity is finite. The PV of future cash flows arising annually (annuity) is calculated by adding all discounted annuities and decreasing the discounted annuity period until it reaches close to zero. In finance, the perpetuity formula can be used to calculate the PV of future cash flows arising infinitely. Similarly, it can be used in business valuations with expected future cash flows for an infinite period. Perpetuity is any cash flow that arises constantly for an infinite period. Common examples of such cash flows include businesses, real estate income, stocks, bonds, and so on. A real-life example of a perpetuity is income from UK bonds called consoles. These bonds are issued by the UK government and investors receive coupon payments infinitely. In a nutshell, perpetuity is a constant stream of equal cash flows with no end. Simply, it is an annuity with infinite life. What is a Deferred Perpetuity? A deferred or delayed perpetuity is a constant stream of cash flows starting at a predetermined future date with infinite life. Deferred perpetuities work similarly to normal perpetuities. The only difference arises from the future cash flows originating at a specific date rather than starting immediately. In finance, deferred perpetuity represents a more realistic cash flow stream. For instance, dividend stocks often start paying dividends at a certain future date or a specified period when the business accumulates sufficient profits. The concept of deferred perpetuity is similar to a deferred annuity as well. It is a constant cash stream arising annually with a finite life. The only difference for delayed perpetuity would be the infinite life of the cash flow stream. The formula for perpetuity calculates the present value of cash flows starting in one year. The formula for deferred perpetuity starting after a specific period “n” can be calculated as: PV = (Annual cash flow/discount rate) x discount factor for the year before the perpetuity starts The calculation of deferred perpetuity will be done in two steps. The PV starting in year “n” will then be discounted again for time zero. How Does a Deferred Perpetuity Work? Deferred perpetuity is perpetuity starting with a delayed stream of cash flows rather than immediately. A normal perpetuity cash flow is discounted for one year. A common example of deferred perpetuity is the cash flow arising from a real estate project when it is rented out after completion. As long as the project functions successfully, the cash flows arising can be accounted for infinitely as perpetuities. However, since these cash flows start after a specified interval, they should ideally be accounted for as deferred perpetuities. Similarly, growth stocks or dividend stocks represent deferred perpetuities. Most companies announce dividends for future years under certain conditions. Often dividend stocks start paying dividends after a specified period and continue infinitely. The present value concept of discounting the infinite cash flows to an infinite present value works the same way as with the normal perpetuity or an annuity calculation. Working Example of a Perpetuity Suppose a company ABC pays a $ 5 dividend to its shareholders infinitely. We assume that the current rate of return used by ABC company is 6%. We can calculate the present value of this perpetuity using the formula: PV of Perpetuity = C/r PV of Perpetuity = $ 5/6% = $ 83.33 It means if the ABC company’s stock is worth $ 83.33 today, the investors would invest in it. Now suppose, ABC company further estimates a 2% growth rate in its dividend amount. The present value of growth perpetuity can be calculated using the formula: PV of Perpetuity = C/r-g PV of Perpetuity = $ 5/ (6-2) % = $ 125 It means the investors would be willing to invest in ABC company’s stock at a price of $ 125 or less if it has a growth rate of 2% for its dividends. Perpetuity Vs. Deferred Perpetuity – Key Differences Both types represent an infinite stream of cash flows. The basic difference is when the cash flow starts at a constant rate. Perpetuity starts immediately. It means the first cash flow can be an advance yearly payment or a payment at the end of each year infinitely. On the other hand, deferred perpetuity starts a stream of cash flows after a specified interval. For instance, a dividend starts after five years of inception for a new business. Both types of cash flows can be discounted to their present values. Both can include a growth factor starting after one year or at a delayed date in the future. Perpetuity Vs. Annuity – Key Differences An annuity is a constant cash flow arising annually. A perpetuity is the same as well. The only difference between the two types of cash flow streams is that an annuity comes with a known or finite An annuity is a constant stream of cash flow that is often in equal amounts. However, an annuity may come with a growth rate as well. It means an annuity may grow at a specific rate yearly. Inflation-adjusted financial securities are good examples of annuities with growth factors. Similarly, perpetuity comes with equal and constant cash flows but with infinite life. Perpetuity can also have a growth rate. Both types of cash flows can be discounted for the present value factors.	{"url":"https://audithow.com/perpetuity-and-deferred-perpetuity/","timestamp":"2024-11-10T05:22:48Z","content_type":"text/html","content_length":"178194","record_id":"<urn:uuid:864cc389-d443-47cd-988b-98ac4f62fbb2>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00615.warc.gz"}
Ecological networks from abundance distributions Another grad student and I tried recently to make a contribution to our understanding of the relationship between ecological network structure (e.g., nestedness) and community structure (e.g., …Alas, I had no luck making new insights. However, I am providing the code used for this failed attempt in hopes that someone may find it useful. This is very basic code. It was roughly based off of the paper by Bluthgen et al. 2008 Ecology (here). In my code the number of interactions is set to 600, and there are 30 plant species, and 10 animal species. This assumes they share the same abundance distributions and sigma values. UPDATE: I changed the below code a bit to just output the metrics links per species, interaction evenness and H2. UPDATE on 27-Aug-12: Now using a github gist, which should actually work: # Community-Network Structure Simulation # Set of mean and sd combinations of log-normal distribution plants<-round(rlnorm(n=30, meanlog=mu[a], sdlog=sig[b])) animals<-round(rlnorm(n=10, meanlog=mu[a], sdlog=sig[b])) # Make matrices matrices <- make.matrices(1,1,100) # Calculate some network metrics-e.g., for one combination of mu and sigma linkspersp <- numeric(100) h2 <- numeric(100) inteven <- numeric(100) for(i in 1:length(matrices)){ metrics<-t(networklevel(m,index=c("links per species","H2","interaction evenness")))	{"url":"https://recology.info/2011/01/ecological-networks-from-abundance/","timestamp":"2024-11-09T06:06:30Z","content_type":"text/html","content_length":"22692","record_id":"<urn:uuid:fa04577a-c194-4182-a729-6570700f5d49>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00102.warc.gz"}
Water System Selection Water System Selection - by Nelsen Corporation Water system selection In selecting the proper water system for a given installation, facts about the water source as well as the water requirements must be considered. Assuming the water source is a well, the following information is needed: • Well size – (diameter) • Depth to pumping level • Well capacity (maximum pumping rate it will sustain) • Elevation and total length of piping • Amount of capacity required • Amount of pressure required Well size The inside diameter of the well must be known since it may be the determining factor as to the type of pump which can be used. Submersible pumps require well at least 4 inches in diameter. Packer type jet systems can be installed in wells with diameters as small as two inches. Depth-to-pumping level The depth to the water level at maximum drawdown determines the type of pump that must be used. For depths of 25 feet or less, at or near sea level, shallow well “suction lift” type pumps are adequate. Each 1000 feet of elevation above sea level reduces the shallow well depth limit by one foot. For example, the maximum practical suction lift in the Denver area, which is 5000 feet above sea level, would be 25 minus 5, or 20 feet. For lifts greater that the maximum suction limit, deep well type pumps, which have a pumping mechanism in the well, are required. The pumping level may be estimated from the well drillers test log. Well capacity The maximum pumping rate the well will sustain must be known to assure that the capacity of the water system selected does not exceed that rate. As in the case of pumping level, well capacity may be obtained from the well drillers log. Elevation and total length of piping The elevation the total length of piping are required to compute the pressure drop in the system. This computation is made by adding the elevation to the drop caused by pipe friction. Pressure drop due to friction is calculated by using the friction loss tables. Amount of capacity required In determining the required capacity of a water system, it is important to provide for the peak demand rather than for the average use rate. A good rule of thumb to follow is to allow for one gallon per minute of pump capacity for each household outlet. For farm water systems, adequate water must not only be provided for household purposes and animal drinking, but for cleaning and fire protection as well. Average water consumption per day for various animals is shown in Table 1. The pump selected should have sufficient capacity to pump the entire daily requirement in two hours. As an example, assume a farm with 20 milk cows, 100 hogs, 500 chickens, two milk houses and eight household outlets. The daily water consumption of the animals would be as follows: Milk cows 20 x 35 700 Hogs 100x4 400 Chickens 500/100x6 30 Total Daily Usage 1,130 Pumping Capacity for 2 Hours (120 minutes) Period 9.4 G.P.M. Milk House Outlets 2 Household Outlets 8 19.4 G.P.M. A 19 gallon per minute pump will provide sufficient water for the farm needs, including fire protection. The Well Capacity should be compared to the Pump Capacity to make sure the well can sustain a pumping rate of 19 gallons per minute. If it will not, an extra large pressure tank, or a two-pump system with storage tank will be required. 1. ave. water consumption for home/farm use A. Home Use For overall daily consumption, checks of families of various sizes in different parts of the country indicate 100 gallons per day per person is a very good average. Outlet Total gallons Flow rate Per usage (GPM) Shower 25-60 5 Bathtub 35 5 Lavatory 02-Jan 4 Toilet - flush 07-Mar 4 Kitchen sink 5 Laundry tub 7 Washer automatic 30-50 5 Dishwasher 20-Oct 2 Water softener up to 150 7 Garden hose 1/2" 3 Garden hose 3/4" 6 Sprinkler lawn 6-7 B. Farm use Gallons per day Horse, Mule, Steer 12 Dry cow 15 Milking cow 35 Hog 4 Sheep 2 Chickens/100 6 Turkeys/100 20 Fire 20-60 GPM Based upon a study of over 20 sources by the Water Source and Use Subcommittee of the Water Systems Council. Values given are average. They do not include the extremes. Amount of pressure required Discharge Pressure of the pump must be sufficient to balance Pumping Depth plus Pipe Friction plus Elevation plus tank pressure. Water system pressure tanks usually operate within the pressure range of 20 to 40 lbs. per square inch. However, recent high demands for pressure, caused by automatic washers and other appliances, have resulted in many systems being set for 30 – 50, or even 40 – 60 ranges. An example for determining the required discharge pressure for a typical submersible pump system is shown below. Assume a pumping rate of 12 gallons per minute. 1. Convert all measurements to the same units. In this case, we will change tank pressure from pounds per square inch to equivalent feet of head by multiplying by 2.31, as shown in the engineering formulas on Page 416. Using 40 p.s.i. as the average tank pressure, 40 x 2.31 = 92.4 feet of head. 2. Compute pipe friction by using the tables on Pages 413 – 415. (1) 1-1/4" Plastic Tee 1 x 3 3 ft Equivalent Pipe Length (2) 1-1/4" Check Valve 2x7 14 ft Equivalent Pipe Length 1.7 ft Equivalent (1) 1-1/4" Elbow 1x1.7 Pipe Length 18.7 ft Total for fittings Pressure drop from pipe friction (200* + 130 + 18.7) x 2.33 100 = 8.1 ft Total Dynamic Pumping Head 8.1 + 160* + 25 + 92.4 = 285.5 ft Note: While total pipe length must be used to compute pressure loss due to pipe friction, only the distance to pumping levels is included with elevation in the summation for total pumping head. Pressure at pump discharge, in lbs. per square inch 285.5 = 123.6 PSI 2.31 Total lift, exclusive of tank pressure 285.5 - 92.4 = 193.1 ft (round off to 200 ft) To select the proper submersible pump for this installation, first choose the appropriate table in the catalog, which would be for a 12 gpm rated pump. Follow the 200 ft Depth-to-Water column down until a pump is found that has the entire desired pressure range covered. The 1 hp model meets the required performance. Now check the performance at 200 ft. and 60 P.S.I. pressure to make sure that the pump will generate sufficient pressure to actuate the pressure switch at the cut-out point with at least 10 psi to spare. Note: In selecting jet pumps, either shallow or deep well type, the friction loss of the piping in the well is included in the performance tables. Therefore, only elevation and friction loss outside the well need to be calculated. If the offset (horizontal distance between the pump and the well) is greater than 35 ft, the offset piping should be increased one pipe size. Tank selection Selection of the proper tank completes the water system. Pressure tanks used with water systems are of the hydropneumatic type. Compressed air in the tank acts as a giant spring to provide a pressure range, between pump stops and starts, during which a reasonable amount of water can be withdrawn. This is necessary to prevent the pump motor from cycling too often, and to provide s smooth flow of water to the outlets, without water hammer. Types of pressure storage tanks. Figure 2A shows plain steel tanks; Figure 2B, the plain steel tank with floating wafer; 2C, the diaphragm tank; 2D the bladder tank. Postal Address: PO Box 12699 Lloydminster, AB T9V 0Y4	{"url":"https://waterbygeorge.ca/water-system-selection-by-nelsen-corporation/","timestamp":"2024-11-14T00:14:35Z","content_type":"text/html","content_length":"44637","record_id":"<urn:uuid:6acf1c09-7bf4-427d-a9fd-407bbafdbd04>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00084.warc.gz"}
Describe the difference between symmetric and asymmetric encryption. Symmetric Encryption: 1. Key Usage: □ Single Key: Symmetric encryption uses a single secret key for both encryption and decryption. This key must be kept confidential between the communicating parties. 2. Algorithm: □ Same Algorithm: The same algorithm is used for both encryption and decryption. The algorithm is usually faster compared to asymmetric encryption algorithms. 3. Speed: □ Faster Processing: Symmetric encryption is generally faster and more efficient for large amounts of data because of the simplicity of its operations. 4. Security and Key Management: □ Key Management Challenge: The main challenge in symmetric encryption is key distribution and management. If the key is compromised, the security of the communication is compromised. 5. Use Case: □ Bulk Data: Symmetric encryption is often used for encrypting bulk data, such as file or disk encryption. 6. Examples: □ AES (Advanced Encryption Standard): A widely used symmetric encryption algorithm. Asymmetric Encryption: 1. Key Usage: □ Public and Private Key Pair: Asymmetric encryption uses a pair of keys - a public key for encryption and a private key for decryption. The keys are mathematically related but cannot be derived from each other. 2. Algorithm: □ Different Algorithms: Asymmetric encryption involves different algorithms for encryption and decryption. The public key is used for encryption, and the private key is used for decryption. 3. Speed: □ Slower Processing: Asymmetric encryption is generally slower than symmetric encryption due to the complexity of the mathematical operations involved. 4. Security and Key Management: □ Key Distribution Simplicity: Asymmetric encryption simplifies key distribution since the public key can be freely distributed, while the private key is kept secret. Even if the public key is intercepted, it cannot be used to decrypt the data. 5. Use Case: □ Secure Communication: Asymmetric encryption is often used for securing communication channels, especially during the initial setup of a secure connection (e.g., TLS/SSL). 6. Examples: □ RSA (Rivest-Shamir-Adleman): A widely used asymmetric encryption algorithm. □ Elliptic Curve Cryptography (ECC): Another example of an asymmetric encryption algorithm, known for its efficiency in terms of key size. Symmetric encryption is efficient for bulk data, but the challenge lies in key distribution. Asymmetric encryption addresses the key distribution issue but is slower, making it suitable for securing communication channels and exchanging symmetric keys securely.	{"url":"https://www.telecomtrainer.com/describe-the-difference-between-symmetric-and-asymmetric-encryption/","timestamp":"2024-11-03T07:15:50Z","content_type":"text/html","content_length":"20125","record_id":"<urn:uuid:0c8dac2d-9b38-450c-97eb-dd5c797231e7>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00451.warc.gz"}
Counting Sort in C# - Code Maze Counting Sort in C# Have you ever needed to sort a list of items but didn’t want to use a built-in sorting algorithm? If so, you may have considered using the counting sort algorithm. In this article, we’ll take a look at how counting sort works and how we can implement it in C#. We’ll also compare it to other sorting algorithms and analyze its time and space complexity. Let’s dive in. What is Counting Sort? As its name implies, counting sort works by counting the number of occurrences of each distinct element in the list. An auxiliary array stores these occurrences and maps the values of the distinct elements with the indices of the array. Finally, the algorithm iterates over the auxiliary array while sorting the original array. Let’s take a deep dive and learn how counting sort works. How Does Counting Sort Algorithm Work? Let’s look at an example of how counting sort works. We will use the following set of numbers: int[] array = {7, 1, 2, 8, 9, 9, 4, 1, 5, 5}; We start by finding the largest element in the array (9). Next, the counting sort algorithm initializes an array of size [getMax(array[ ])+1] , to store the occurrences of distinct elements. The algorithm loops through the unsorted array while storing the number of occurrences of every distinct element in the occurrences array, which it achieves by mapping the distinct elements with the index of the occurrences array: Value: 0, 2, 1, 0, 1, 2, 0, 1, 1, 2 Index: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Now that we have populated the occurrences array, we are going to iterate through the array as we retrieve the occurrences of each distinct element as we populate the sorted array: 1, 1, 2, 4, 5, 5, 7, 8, 9, 9 Let’s learn how to implement the counting sort algorithm in C#. How to Implement Counting Sort in C#? Let’s start by writing a GetMaxVal() method that takes an array and its size as inputs and returns the largest integer in that array: public static int GetMaxVal(int[] array, int size) var maxVal = array[0]; for (int i = 1; i < size; i++) if (array[i] > maxVal) maxVal = array[i]; return maxVal; The GetMaxVal() method iterates through the array from the first element to the last one while updating the value of maxVal, which we are going to need as we implement the counting sort algorithm. Next, we are going to write a CountingSort() method that takes an array as its sole input and returns a sorted array: public int[] CountingSort(int[] array) var size = array.Length; var maxElement = GetMaxVal(array, size); var occurrences = new int[maxElement + 1]; for (int i = 0; i < maxElement + 1; i++) occurrences[i] = 0; for (int i = 0; i < size; i++) for (int i = 0, j = 0; i <= maxElement; i++) while (occurrences[i] > 0) array[j] = i; return array; How the Counting Sort Method Works We can see that the process starts by getting the largest integer in the array by invoking GetMaxVal(). Once we get the largest integer in the array, we define an array of size maxElement + 1 and initialize all the values to zero. At this point, we start populating the occurrences array by storing the occurrences of each unique element in the array: for (int i = 0; i < size; i++) Next, the algorithm iterates through the occurrences array to sort the array elements by mapping the indexes and their values: for (int i = 0, j = 0; i <= maxElement; i++) while (occurrences[i] > 0) array[j] = i; return array; Finally, we can verify that the CountingSort() method sorts a given unsorted array accurately: var array = new int[] { 73, 57, 49, 99, 133, 20, 1 }; var expected = new int[] { 1, 20, 49, 57, 73, 99, 133 }; var sortFunction = new CountingSortMethods(); var sortedArray = sortFunction.CountingSort(array); CollectionAssert.AreEqual(sortedArray, expected); Space and Time Complexity of Counting Sort Algorithm The counting sort algorithm requires an auxiliary array of size k (max element + 1). Therefore, the space complexity of the counting sort algorithm is O(k). Best-Case Time Complexity The best-case time complexity occurs when the array elements have a range k which is equal to 1. In this case, the algorithm takes linear time as the time complexity becomes O(1 + n) or O(n). Average-Case Time Complexity Counting sort encounters the average-case time complexity scenario when we select random values e.g. from 1 to n. In this case, assuming we have an array of size n and the value of the largest element being k, the algorithm has O(n+k) as its average-case time complexity. Worst-Case Time Complexity This time complexity scenario occurs when the elements are skewed with the largest element k, being significantly larger than the rest of the elements in the array. This increases the time it takes for the algorithm to iterate through the occurrences array and worsens the algorithm’s space complexity. Given the average time complexity of the counting sort algorithm is O(n+k), when the value of k grows for example to n^4, the total time it takes to sort the array is O(n + (n^4)). Therefore, since the worst-case complexity scenario of the counting sort algorithm starts occurring when the maximum element is significantly larger than the rest of the elements, the time complexity of the algorithm is O(k). Advantages of Counting Sort Algorithm One advantage of the counting sort algorithm is that it is relatively simple to understand and implement. Additionally, the algorithm is very efficient for collections with a small range of values. Finally, the algorithm is stable, meaning that items with equal values will retain their original order after being sorted unlike other sorting algorithms such as quicksort. Disadvantages of Counting Sort Algorithm First, the algorithm is not well-suited for collections with a large range of values because the algorithm’s efficiency decreases as the range of values increases. Counting sort is a bit more complex to implement than other sorting algorithms such as selection and bubble sort. Counting sort is not an in-place sorting algorithm since it requires additional space unlike algorithms such as quicksort, insertion sort, and bubble sort. Finally, the algorithm can be slower than algorithms that use divide and conquer approaches such as merge sort and quicksort for large arrays. Performance Tests Let’s test how long the algorithm takes for it to sort three arrays that have 20,000 elements each. To help us complete these tests, we are going to implement two methods. First, let’s write a method to generate a set of random array elements: public static int[] CreateRandomArray(int size, int lower, int upper) var array = new int[size]; var rand = new Random(); for (int i = 0; i < size; i++) array[i] = rand.Next(lower, upper); return array; The CreateRandomArray() the method takes three integers size, lower and upper. Using the inbuilt Random class, we generate integer values between lower and upper that we’re going to put into the To simulate the worst-case time complexity scenario, we are going to use the CreateRandomArray() we have but add an element that has a lot of digits at the end of the array such as Int32.MaxValue/2: public static int[] CreateImbalancedArray(int[] array) List<int> numbers = new List<int>(); return numbers.ToArray(); Next, we are going to create an object that holds different arrays that have random and sorted values: public IEnumerable<object[]> SampleArrays() yield return new object[] { CreateRandomArray(20000, 1, 2), "Best Case" }; yield return new object[] { CreateRandomArray(20000, 10000, 30000), "Average Case" }; yield return new object[] { CreateImbalancedArray(CreateRandomArray(19999, 10000, 30000)), "Worst Case" }; Each object entry has two values: an integer array e.g. CreateRandomArray(20000, 10000, 30000) and a string object storing the name of that array (“Average Case”). This object sorts random numbers between 10,000 and 30,000, which ensures that the array elements are distributed uniformly within the range. On the other hand, the last object invokes the CreateImbalancedArray() which has elements whose values are distributed uniformly within the range except Int32.MaxValue/2. Sample Test Results Let’s assess the sample best, average, and worst-case complexity performance results of the algorithm: \| Method \| array \| arrayName \| Mean \| Error \| StdDev \| \|------------- \|------------- \|------------- \|----------------:\|--------------:\|---------------:\| \| CountingSort \| Int32[20000] \| Average Case \| 366.03 μs \| 7.284 μs \| 18.803 μs \| \| CountingSort \| Int32[20000] \| Best Case \| 92.46 μs \| 1.823 μs \| 3.846 μs \| \| CountingSort \| Int32[20000] \| Worst Case \| 3,287,653.20 μs \| 91,249.044 μs \| 263,274.357 μs \| As we can see, despite the algorithm sorting 20,000 elements, they have different runtimes. The counting sort algorithm encounters its best-case time complexity scenario as the range of elements is equal to 1. Additionally, we can see that counting sort’s average-case time complexity is achieved when we randomly select values within the range. On the other hand, when we introduce a large number such as Int32.MaxValue/2 when sorting elements distributed across a uniform range e.g. between 10,000 and 30,000, the algorithm encounters its worst-case time complexity. The worst-case time complexity’s runtime is about 9,000 times longer than the average-case time complexity’s runtime. On the other hand, the average-case runtime is about 4 times slower than the best-case runtime. Please note that these runtimes may change depending on the number of elements and the computing resources available. In this article, we have learned how counting sort in C# works and its time and space complexity. Leave a reply	{"url":"https://code-maze.com/counting-sort-in-c/","timestamp":"2024-11-05T09:18:10Z","content_type":"text/html","content_length":"115381","record_id":"<urn:uuid:87ffc01e-7f86-4e80-857d-f166e60af2d8>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00622.warc.gz"}
Re: finding out what Automatic was • To: mathgroup at smc.vnet.net • Subject: [mg54242] Re: finding out what Automatic was • From: Chris Chiasson <chris.chiasson at gmail.com> • Date: Mon, 14 Feb 2005 21:50:35 -0500 (EST) • References: <200502140317.WAA14113@smc.vnet.net> • Reply-to: Chris Chiasson <chris.chiasson at gmail.com> • Sender: owner-wri-mathgroup at wolfram.com Your question seems like a good one to me, probably because I don't know the answer :] However, given that the category intervalas by the histogram command are in the form of rectangles, one could just extract them. The first three lines of the code that follow are directly from the help file entry that contains the Histogram function description. Please accept my apologies if you already understand the concepts involved in this code. This code may be evaluated one line at a time for clarity. (hist is a graphics object - the next line shows the internal structure of hist) (notice everything fis the form h[argument1,argument2,etc], which can be thought of as functions) (notice the similarity between lists and everyday functions that have not evaluated their parameters) (There are commands for extracting objects at different "depths" of nested functions.) (the relevant objects we would like to extract are the rectangle functions, which are located on level 4 of the hist graphics object) (note how the 4 is inisde brackets, just passing a plain 4 will give everything down to that level, not just the level itself) (how many levels of nested functions are there in this graphics object) (most commands have a parameter allowing one to specify the level at which one wants to operate, rather than having to wrap the arguments in Level functions) (the parameter we need to supply is the last argument in the case statement) (why would we need to use a case statement? well -- we need to extract the rectangle functions and they happen to fit a pattern.case statements extract objects that fit particular patterns from other objects (heh,at least in mathematica:])) (so we first define the pattern) (note this could also be written thepattern= (since we don't really care what level the rectangle functions are at inside hist,,just supply the Depth[hist] for the level argument... this will have the effect of searching for the rectangle pattern at all levels) (note this could also be written Cases[hist,thepattern,Depth[hist]]) (the above command gives you the list of bin splits-- it could also be written as thesplits=thecases/.thepattern\[Rule]{xmin,xmax}*) On Sun, 13 Feb 2005 22:17:16 -0500 (EST), Curt Fischer <tentrillion at gmail.nospam.com> wrote: > Dear Group: > How do you find out what value Mathematica has picked for an option set > to "Automatic", especially when making graphs? > For example, I want to access the frequency data for a Histogram[] I > made from a list of 50000 integers. How do I figure out which bin sizes > Histogram[] picked if I don't explicitly specify the bin sizes? > Thanks. > -- > Curt Fischer Chris Chiasson Kettering University Mechanical Engineering Graduate Student	{"url":"http://forums.wolfram.com/mathgroup/archive/2005/Feb/msg00406.html","timestamp":"2024-11-08T01:52:26Z","content_type":"text/html","content_length":"34107","record_id":"<urn:uuid:ff1cc69e-684e-477e-b856-6e028505d0b2>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00344.warc.gz"}
warm audio wa 47 price You simply check summary(fit) and see if the interaction terms are significant. ... multiple regression in detail in a subsequent course. Sometimes your research may predict that the size of a regression coefficient may vary across groups. To find out if the regression coefficients are significantly different between the two groups, I use one model where the regression between the factors is free and another model where it is equal across group and compare the model fit using DIFFTEST? I'm not sure if I read that is not possible to constrain an ON statement. The Stata Journal, 10(4), 540–567. Testing if coefficients are statistically significantly different across models. You can compute it easily using the sum of squared residuals of each model. proc glm data=dataser; class group; model Y=group x x*group; quit; If the variable group is not statistically significant when you perform this regression, then the intercepts of the two groups are not significantly different. This would correspond to a sequential test. If you are going to compare correlation coefficients, you should also compare slopes. 1. Active 1 year, 8 months ago. b. How can I compare regression coefficients between two groups? 0. Re: st: RE: comparing regression coefficients across models. Comparing Logit and Probit Coefficients Across Groups – Handout Page 6 Alternative solution 3: Compare predicted probabilities across groups Long (2009) says “While regression coefficients are affected by the identifying assumption for the varianceof the ... How to compare total effect of three variables across two regressions that use different subsamples? Similar to (a), but do not require the rvariance of the residual to > be the same for both groups. thank you Comparing coefficients across groups . For example, you might believe that the regression coefficient of height predicting weight would differ across 3 age groups (young, middle age, senior citizen). Instead, they compare unstandardized coefficients. If you just cannot wait until then, see my document Comparing Regression Lines From Independent Samples . Frequently there are other more interesting tests though, and this is one I've come across often -- testing whether two coefficients are equal to one another. The comparison of regression coefficients across subsamples is relevant to many studies. If I have the data of two groups (patients vs control) how can I compare the regression coefficients for both groups? Comparing regression coefficients between nested linear models for clustered data with generalized estimating equations. > > b. Run a regression over all groups combined, adding the appropriate > interaction terms which would indicate the difference and its > significance. Then, for each coefficient, I could use a beta-Dirichlet process model to compute the posterior distribution of the probability that a pair of coefficients has the same sign, and then compare those distributions across pairs of regression coefficients to see whether the focal group is more like one group than another. Compare coefficients across two fixed effects models 04 Jan 2017, 09:44. I have run two regression models for two subsamples and now I want to test/compare the coefficients for those two independent variables across two regression models. $\endgroup$ – Brash Equilibrium May 20 '14 at 19:33 However, if I'd like to do that for the second "group", i.e., the males, ... with suest my desired test would be the equality of the two whole regression models. I want to highlight for comparison of logit and probit coefficients across groups just a p-value is not enough, since there are substantial issues pertaining to such comparisons. standardized coefficients for linear models across groups (Kim and Ferree 1981). Prob > chi2 = 0.0000 . Most researchers now recognize that such comparisons are potentially invalidated by differences in the standard deviations across groups. Whether you can compare probit/logit coefficients across groups in any meaningful way is a controversial issue. To Compare Regression Coefficients, Include an Interaction Term. Using Heterogeneous Choice Models to Compare Logit and Probit Coefficients Across Groups. It is widely believed that regression models for binary responses are problematic if we want to compare estimated coefficients from models for different groups or with different explanatory variables. The big point to remember is that… st: compare regression coefficients between 2 groups (SUEST) across time and across subgroups in a data set. James _____ From: owner-statalist@hsphsun2.harvard.edu [owner-statalist@hsphsun2.harvard.edu] on behalf of Dalhia [ggs_da@yahoo.com] Sent: 02 August 2012 21:42 To: statalist@hsphsun2.harvard.edu Subject: st: comparing coefficients across models Hello, I have two groups and need to run the same regression model on both groups (number of observations differ but variables are all the same). The default hypothesis tests that software spits out when you run a regression model is the null that the coefficient equals zero. Greetings to all, I need to compare regression coefficients across 2 groups to determine whether the effect for one group is significantly different from the other, and read about the following methods: a. It's an application of the Fisher test to test the equality of coefficients among two groups of individuals. We can compare the regression coefficients of males with females to test the null hypothesis Ho: B f = B m , where B f is the regression coefficient for females, and B m is the regression coefficient for males. Fitting heterogeneous choice models with oglm. Comparing Logit and Probit Coefficients Across Groups… From: "Roland Teitzer" Prev by Date: Re: st: compare regression coefficients between 2 groups (SUEST) across time and across subgroups in a data set Next by Date: st: Re: Finding and graphing intersection of lines In OLS, variables are often standardized by rescaling them to have a variance of one and a mean of zero. An equivalent method is to test for interactions between particular predictors and dummy (indicator) variables representing the groups. comparing standardized OLS regression coefficients across groups (Duncan 1968). In logit and probit regression analysis, a common practice is to estimate separate models for two or more groups and then compare coefficients across groups. Posted 07-21-2017 09:42 AM (1542 views) \| In reply to BobSmith Since SURVEYREG does not have … Case 1: True coefficients are equal, residual variances differ Group 0 Group 1 ... Heteroskedastic Ordered Logistic Regression Number of obs = 2797 . This concern has two forms. Compare 2 regression lines in R. Ask Question Asked 7 years, 4 months ago. by Karen Grace-Martin 33 Comments. Sociological Methods & Research, 37(4), 531–559. Often, the same regression model is fitted to several subsamples and the question arises whether the effect of some of the explanatory variables, as expressed by the … If variances differ across groups, the standardization will also differ across groups, making coefficients non-comparable. 1999. I've read several regressions guides, however, I cannot find the correct way to regress 4 regression coefficients across 5 groups (and across 2 groups) "For example, you might believe that the regression coefficient of height predicting weight would differ across 3 age groups … Unlike approaches based on the comparison of regression coefficients across groups, the methods we propose are unaffected by the scalar identification of the coefficients and are expressed in the natural metric of the outcome probability. (2014) present methods for group comparisons of correlations between the latent outcome and each regressor. Thank you very much, Pia tional tests for comparing regression coefficients across groups.Allison (1999) and Williams (2009) developed new tests for comparing regression coefficients that account for differences in unobserved heterogeneity. For example, I want to test if the regression coefficient of height predicting weight for the men group is significantly different from that for women group. References: . In logit and probit regression analysis, a common practice is to estimate separate models for two Or more groups and then compare coefficients across groups. If they are not, there is no difference. See inter alia the following (taken from Rich Williams' webpages): Allison, Paul. split file off. The problem with logit and probit coefficients, however, is that they We can undertake this analysis by comparing the coefficients for our variable of interest (ethnic group) both before and after including the other ‘control’ variables in the multiple regression model (SEC and gender). To test if the slope coefficient is identical across all groups, your initial regression model is best suited. We did not find a significant interaction in the relationships of Sepal Length to Petal Width for I. Setosa (B = 0.9), I. Versicolor (B = 1.4), nor I. Virginica (B = 0.6); F (2, 144) = 1.6, p = 0.19. "We used linear regression to compare the relationship of Sepal Length to Petal Width for each Species. LR chi2(8) = 415.39 . Can we compare betas of two different regression analyses regression /dep weight /method = enter height. If they are, there is a difference. Breen et al. Re: How to compare two coefficients in PROC SURVEYREG? Run a regression over all groups combined, adding the appropriate interaction terms which would indicate the difference and its significance. Williams, R. (2010). Regression can be used to ascertain whether the ethnic gaps in attainment at age 14 result from these observed differences in SEC between ethnic groups. We can compare the regression coefficients among these three age groups to test the null hypothesis Ho: B 1 = B 2 = B 3 where B 1 is the regression for the young, B 2 is the regression for the middle aged, and B 3 is the regression for senior citizens.	{"url":"http://ecbb2014.agrobiology.eu/zyz4pr/warm-audio-wa-47-price-ab1fdb","timestamp":"2024-11-12T09:40:36Z","content_type":"text/html","content_length":"21304","record_id":"<urn:uuid:e469164e-e302-4da9-b64f-1eeb0b8f88c7>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00735.warc.gz"}
Introduction to Python Libraries for Regression - Click Virtual University Introduce the primary libraries for conducting regression analysis in Python. Content Outline: 1. Introduction to Statsmodels: □ Definition and Strengths: ☆ Explain that statsmodels is a Python library designed for statistical modeling, testing, and analysis. ☆ Highlight its strengths: comprehensive statistical outputs, detailed diagnostics, and easy integration with pandas DataFrame structures. ☆ Emphasize its use for inferential statistics and hypothesis testing, which are crucial for understanding the underlying dynamics of the data rather than just prediction. □ Typical Use Cases: ☆ Suitable for academic and research environments where detailed statistical analysis is required. ☆ Commonly used for econometric analyses, time-series forecasting, and extensive statistical testing to understand relationships between variables. 2. Introduction to Scikit-Learn: □ Definition and Strengths: ☆ Describe scikit-learn as a powerful, simple Python library for machine learning, providing a wide range of supervised and unsupervised learning algorithms. ☆ Its strengths include ease of use, scalability, and support for preprocessing data, cross-validation, and various regression models. ☆ scikit-learn is designed with a consistent interface, which simplifies the workflow of model training and evaluation. □ Typical Use Cases: ☆ Ideal for implementing machine learning at scale, from prototyping to production systems. ☆ Widely used in industry for predictive modeling tasks like customer churn prediction, price forecasting, and demand estimation where quick deployment and model performance are key. 3. Brief Mention of Other Tools/Libraries Occasionally Used in Regression: □ TensorFlow and PyTorch: ☆ Mention that these libraries, while primarily focused on deep learning, also support regression tasks, particularly where complex data patterns require neural network-based approaches. □ XGBoost and LightGBM: ☆ Briefly introduce these as gradient boosting frameworks that are highly effective for regression problems with large datasets and high-dimensional spaces, known for their performance and □ R (Language): ☆ Acknowledge R as a statistical computing language with extensive packages for regression analysis, often used in academic and research settings for similar purposes as statsmodels.	{"url":"https://clickuniv.com/topics/introduction-to-python-libraries-for-regression/","timestamp":"2024-11-14T09:08:42Z","content_type":"text/html","content_length":"191953","record_id":"<urn:uuid:0e04838a-c534-4b52-ae7d-045d8ee49f06>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00577.warc.gz"}
Breakeven Point Bep Definition If they feel that the number of units required to be sold to break even is high enough, they could increase the selling price of the product a bit to bring that number down. In the case of our example, whenever we make an additional cake, our variable costs increase by $15. If we bake 1 cake our total variable costs are $15, if we bake 2, our total variable costs are $30, if we bake 10, $150… and so on. • For each additional unit sold, the loss typically is lessened until it reaches the break-even point. • If we sell each birthday cake for $50, our revenue increases by $50 whenever we sell a cake. • From sales funnel facts to sales email figures, here are the sales statistics that will help you grow leads and close deals. • A break-even point more than 18 months in the future is a strong risk signal. • Make a list of all your costs that fluctuate depending on how much you sell. A break-even analysis tells you how many sales you must make to cover the total costs of production. Lowering your variable costs is often the most difficult option, especially if you’re just going into business. More Resources On Small Business Accounting The break-even point is more than the moment when you pop a celebratory bottle of champagne. It’s also a useful figure to keep in mind when managing prices, operating costs and overhead. Let’s go over how to calculate a break-even point using two different methods. Here are four ways businesses can benefit from break-even analysis. If a business is at the precise break-even point, the business is neither running at a profit nor at a loss; it has simply broken even. In the break-even analysis example above, the break-even point is 92.5 units. The owner has the right to know the amount of the increased sales needed and the costs, if any, of obtaining those increased sales. Make sure you include any discounts or special offers you give customers. Look at competitors to see how they are pricing their product or look to an informal focus group to figure out how much someone would be willing to pay. If you sell multiple products or services, figure out the average selling price for everything combined. If you find demand for the product is soft, consider changing your pricing strategy to move product faster. However, discounted pricing can actually raise your break-even point. This formula, in particular, will help you experiment with your unit selling price. Once you have your break-even point in units, you’ll be making a profit on every product you sell beyond this point. Your contribution margin will tell you how much profit you’ll make on each unit once you pass this break-even point. A product’s contribution margin tells you how much each sold unit contributes to your overall revenue. Products with a high contribution margin have a positive impact on your company’s growth. An Example Of Finding The Breakeven Point At present the company is selling fewer than 200 tables and is therefore operating at a loss. As a business, they must consider increasing the number of tables they sell annually in order to make enough money to pay fixed and variable costs. The break-even value is not a generic value as such and will vary dependent on the individual business. However, it is important that each business develop a break-even point calculation, as this will enable them to see the number of units they need to sell to cover their variable costs. • Once we reach the break-even point for each unit sold the company will realize an increase in profits of $150. • The break-even point of a business should be kept as low as possible, in order to keep the firm profitable even when sales decline. • This means that if the company sells 125 units of its product, it’ll have made $0 in net profit. • This may influence which products we write about and where and how the product appears on a page. • The break-even value is not a generic value as such and will vary dependent on the individual business. The breakeven point is the level of production at which the costs of production equal the revenues for a product. If you tinker with the numbers and your break-even sales revenue still seems like an unattainable number, you may need to scrap your business idea. If that’s the case, take heart in the fact that you found out before you invested your (or someone else’s) money in the idea. In the meantime, start building your store with a free 14-day trial of Shopify. Start your free 14-day trial of Shopify—no credit card required. Raise Your Prices Full BioPete Rathburn is a freelance writer, copy editor, and fact-checker with expertise in economics and personal finance. Samantha Silberstein is a Certified Financial Planner, FINRA Series 7 and 63 licensed holder, State of California life, accident, and health insurance licensed agent, and CFA. She spends her days working with hundreds of employees from non-profit and higher education organizations on their personal financial plans. And just like the output for the goal seek approach in Excel, the implied units needed to be sold for the company to break even comes out to 5k. All incremental revenue beyond this point contributes toward the accumulation of more profits for the company. If a company has reached its break-even point, this means the company is operating at neither a net loss nor a net gain (i.e., “broken even”). Run A Finance Blog? See How You Can Partner With Us If your business’s revenue is below the break-even point, you have a loss. Every sales leader should know how to calculate it and what they can do to increase it. Even the smallest expenses can add up over time, and if companies aren’t keeping tabs on these costs, it can lead to major surprises down the road. There’s a significant financial buy-in up top, and you need to take risks if you want to make money. But when you’re down on your luck in gambling or business, the short-term goal may simply be to break even. These charges are customary and are provided so that you may compare them to other service provider charges. Costs that vary, in total, as the quantity of goods sold changes but stay constant on a per-unit basis. Chase Merchant Services provides you with a more secure and convenient ways to do business. Our payments solutions give your customers the flexibility to make purchases however they choose with added security to protect their accounts. This calculation tells you how much money you need to make from the sale of a certain product to break even. See how finding your business’s break-even point can help you manage products and expenses. A company must generate sufficient revenue to cover its fixed and variable costs. More sales mean there will be a profit, while fewer sales mean there will be a loss. Calculating the breakeven point is a key financial analysis tool used by business owners. Once you know the fixed and variable costs for the product your business produces or a good approximation of them, you can use that information to calculate your company’s breakeven point. Yet another possibility is to increase the reliability of products, so that they require fewer warranty repairs. Current business owners can use a break-even analysis to tinker with their pricing strategies or to determine whether or not to develop a new product or service. The break-even analysis can tell you if it makes financial sense to launch new products by showing how many units you’ll need to sell to break-even. To calculate the break-even point, a company must know its fixed and variable costs plus how much revenue it brings in for every item it sells. Changing A Single Variable You’ll need to think about what pricing and sales strategies are realistic for your business given your time, resources and the competitive market in which you operate. This break-even analysis formula gives you the number of units you need to sell to cover your costs per month. Anything below this number means your business is losing money. Performing a break-even analysis is an essential task because a business investment should eventually pay off. With a break-even calculation under your belt, you know exactly how many products or services you need to sell in order to cover your costs. Break-even points will help business owners/CFOs get a reality check on how long it will take an investment to become profitable. For example, calculating or modeling the minimum sales required to cover the costs of a new location or entering a new market. To be more precise, the breakeven point refers to the sales amount, which is required to cover the total cost . The total number of units you would need to sell to cover all costs would be equal to the total fixed costs divided by the contribution margin you get from each unit you sell. Making each of those cakes has a total variable cost of $15 (that’s for ingredients, packaging, electricity to bake it, etc.). And, we have total fixed costs of $5,000 per month (that’s for salaries, rent, equipment depreciation, etc.). Brainyard delivers data-driven insights and expert advice to help businesses discover, interpret and act on emerging opportunities and trends. A company could explore multiple paths regarding its products’ development and launch. Break even analysis is also essential for a company planning an expansion to a new territory or entering new markets. Analyzing the break even point also helps determine the magnitude of risks involved. A break even point will also show whether the product could sustain in the market with that amount of risk involved. If we sell each birthday cake for $50, our revenue increases by $50 whenever we sell a cake. This means that you’ll need to sell 150 burgers over the course of the month to break even. Variable Costs Per Unit For example, we know that Hicks had $18,000 in fixed costs and a contribution margin ratio of 80% for the Blue Jay model. We will use this ratio (Figure 7.24) to calculate the break-even point in dollars. When you outsource fixed costs, these costs are turned into variable costs. Variable costs are incurred only when a sale is made, meaning you only pay for what you need. • See how finding your business’s break-even point can help you manage products and expenses. • Let’s take a look at two different break-even analysis formulas that companies can use to find their BEP. • Finding your break-even point gives you a better idea of which risks are really worth taking. • Now that we’ve learned how to calculate break-even sales in two different ways, let’s take a look at an example of these break-even point formulas in action. • If you prefer to opt out, you can alternatively choose to refuse consent. • Your revenue is also used to calculate your business’s profit. • Read on to learn what the break-even point is, how to calculate it, and how it can help you master your business and increase sales. This could be done through a number or negotiations, such as reductions in rent payments, or through better management of bills or other costs. Cost-volume-profit analysis looks at the impact that varying levels Break Even Point of sales and product costs have on operating profit. Breakeven points can be applied to a wide variety of contexts. At that price, the homeowner would exactly break even, neither making nor losing any money. When the number of units exceeds 10,000, the company would be making a profit on the units sold. Note that the blue revenue line is greater than the yellow total costs line after 10,000 units are produced. Likewise, if the number of units is below 10,000, the company would be incurring a loss. From 0-9,999 units, the total costs line is above the revenue line. The purpose of doing a break-even-point analysis is to determine the point at which the cost for the volume of tests equals the revenue generated. BEP can be calculated for individual tests, specific testing instruments, a cost center, or the entire laboratory. A break even point gives a clear idea about the sales required for a company to start generating profits from a product. Fixed costs do not change irrespective of your production or your sales amount, such as rent, salaries, etc. And this is one of the reasons why the cost-volume-profit analysis is such a great tool. For example, sales commissions paid based on unit sales are a variable cost. It allows you to determine the level of sales that you must reach to avoid losing money and the profit you’ll make if you reach a higher sales goal. For example, if the demand for your product is smaller than the number of units you’ll need to sell to breakeven, it may not be worth bringing the product to market at all. Finding your break-even point gives you a better idea of which risks are really worth taking. The next step is to divide your costs into fixed costs, and variable costs. At this point, you need to ask yourself whether your current plan is https://www.bookstime.com/ realistic, or whether you need to raise prices, find a way to cut costs, or both. You should also consider whether your products will be successful in the market. Just because the break-even analysis determines the number of products you need to sell, there’s no guarantee that they will sell. Alternatively, the break-even point can also be calculated by dividing the fixed costs by the contribution margin. Relationships Between Fixed Costs, Variable Costs, Price, And Volume The formula for determining your breakeven point requires no more than simple arithmetic. Simply divide your estimated annual fixed costs by your gross profit percentage to determine the amount of sales revenue you’ll need to bring in just to break even. As we’ve already pointed, there are several ways in which you can utilize this concept.	{"url":"https://checpros.com/breakeven-point-bep-definition/","timestamp":"2024-11-12T09:54:45Z","content_type":"text/html","content_length":"148305","record_id":"<urn:uuid:4aa6ebea-1ab0-48f3-86f6-205018cf5bbc>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00772.warc.gz"}
Guidelines for reporting of statistics for clinical research in urology (2024) In an effort to improve the quality of statistics in the clinical urology literature, statisticians at European Urology, The Journal of Urology, Urology and BJUI came together to develop a set of guidelines to address common errors of statistical analysis, reporting and interpretation. Authors should “break any of the guidelines if it makes scientific sense to do so” but would need to provide a clear justification. Adoption of the guidelines will in our view not only increase the quality of published papers in our journals but improve statistical knowledge in our field in general. It is widely acknowledged that the quality of statistics in the clinical research literature is poor. This is true for urology just as it is for other medical specialties. In 2005, Scales et al. published a systematic evaluation of the statistics in papers appearing in a single month in one of the four leading urology medical journals: European Urology, The Journal of Urology, Urology and BJUI. They reported widespread errors, including 71% of papers with comparative statistics having at least one statistical flaw[1]. These findings mirror many others in the literature, see, for instance, the review given by Lang and Altman[2]. The quality of statistical reporting in urology journals has no doubt improved since 2005, but remains unsatisfactory. The four urology journals in the Scales et al. review have come together to publish a shared set of statistical guidelines, adapted from those in use at one of the journals, European Urology, since 2014[3]. The guidelines will also be adopted by European Urology Focus and European Urology Oncology. Statistical reviewers at the four journals will systematically assess submitted manuscripts using the guidelines to improve statistical analysis, reporting and interpretation. Adoption of the guidelines will, in our view, not only increase the quality of published papers in our journals but improve statistical knowledge in our field in general. Asking an author to follow a guideline about, say, the fallacy of accepting the null hypothesis, would no doubt result in a better paper, but we hope that it would also enhance the author’s understanding of hypothesis tests. The guidelines are didactic, based on the consensus of the statistical consultants to the journals. We avoided, where possible, making specific analytic recommendations and focused instead on analyses or methods of reporting statistics that should be avoided. We intend to update the guidelines over time and hence encourage readers who question the value or rationale of a guideline to write to the authors. 1. The golden rule: Break any of the guidelines if it makes scientific sense to do so. Science varies too much to allow methodologic or reporting guidelines to apply universally. 2. Reporting of design and statistical analysis 2.1. Follow existing reporting guidelines for the type of study you are reporting, such as CONSORT for randomized trials, ReMARK for marker studies, TRIPOD for prediction models, STROBE for observational studies, or AMSTAR for systematic reviews. Statisticians and methodologists have contributed extensively to a large number of reporting guidelines. The first is widely recognized to be the Consolidated Standards of Reporting Trials (CONSORT) statement on the reporting of randomized trials, but there are now many other guidelines, covering a wide range of different types of study. Reporting guidelines can be downloaded from the Equator Web site (http://www.equator-network.org). 2.2. Describe cohort selection fully. It is insufficient to state, for instance, “the study cohort consisted of 1144 patients treated for benign prostatic hyperplasia at our institution”. The cohort needs to be defined in terms of dates (e.g. “presenting March 2013 to December 2017”), inclusion criteria (e.g. “IPSS > 12”) and whether patients were selected to be included (e.g. for a research study) vs. being a consecutive series. Exclusions should be described one by one, with the number of patients omitted for each exclusion criterion to give the final cohort size (e.g. “patients with prior surgery (n=43), allergies to 5-ARIs (n=12) and missing data on baseline prostate volume (n=86) were excluded to give a final cohort for analysis of 1003 patients”). Note that inclusion criteria can be omitted if obvious from context (e.g. no need to state “undergoing radical prostatectomy for histologically proven prostate cancer”); on the other hand, dates may need to be explained if their rationale could be questioned (e.g. “March 2013, when our specialist voiding clinic was established to December 2017”). 2.3. Describe the practical steps of randomization in randomized trials. Although this reporting guideline is part of the CONSORT statement, it is so critical and so widely misunderstood that it bears repeating. The purpose of randomization is to prevent selection bias. This can be achieved only if those consenting patients cannot guess a patient’s treatment allocation before registration in the trial or change it afterward. This safeguard is known as allocation concealment. Stating merely that “a randomization list was created by a statistician” or that “envelope randomization was used” does not ensure allocation concealment: a list could have been posted in the nurse’s station for all to see; envelopes can be opened and resealed. Investigators need to specify the exact logistic steps taken to ensure allocation concealment. The best method is to use a password-protected computer database. 2.4. The statistical methods should describe the study questions and the statistical approaches used to address each question. Many statistical methods sections state only something like “Mann-Whitney was used for comparisons of continuous variables and Fisher’s exact for comparisons of binary variables”. This says little more than “the inference tests used were not grossly erroneous for the type of data”. Instead, statistical methods sections should lay out each primary study question separately: carefully detail the analysis associated with each and describe the rationale for the analytic approach, where this is not obvious or if there are reasonable alternatives. Special attention and description should be provided for rarely used statistical techniques. 2.5. The statistical methods should be described in sufficient detail to allow replication by an independent statistician given the same data set. Vague reference to “adjusting for confounders” or “non-linear approaches” is insufficiently specific to allow replication, a cornerstone of the scientific method. All statistical analyses should be specified in the Methods section, including details such as the covariates included in a multivariable model. All variables should be clearly defined where there is room for ambiguity. For instance, avoid saying that “Gleason grade was included in the model”; state instead “Gleason grade group was included in four categories 1, 2, 3 and 4 or 5”. 3. Inference and p-values (see also “Use and interpretation of p-values” below) 3.1. Don’t accept the null hypothesis. In a court case, defendants are declared guilty or not guilty, there is no verdict of “innocent”. Similarly, in a statistical test, the null hypothesis is rejected or not rejected. If the p-value is 0.05 or more, investigators should avoid conclusions such as “the drug was ineffective”, “there was no difference between groups” or “response rates were unaffected”. Instead, authors should use phrases such as “we did not see evidence of a drug effect”, “we were unable to demonstrate a difference between groups” or simply “there was no statistically significant difference in response 3.2. P-values just above 5% are not a trend, and they are not moving. Avoid saying that a p-value such as 0.07 shows a “trend” (which is meaningless) or “approaches statistical significance” (because the p-value isn’t moving). Alternative language might be: “although we saw some evidence of improved response rates in patients receiving the novel procedure, differences between groups did not meet conventional levels of statistical significance”. 3.3. P-values and 95% confidence intervals do not quantify the probability of a hypothesis. A p-value of, say, 0.03 does not mean that there is 3% probability that the findings are due to chance. Additionally, a 95% confidence interval should not be interpreted as a 95% certainty the true parameter value is in the range of the 95% confidence interval. The correct interpretation of a p-value is the probability of finding the observed or more extreme results when the null hypothesis is true; the 95% confidence interval will contain the true parameter value 95% of the time were a study to be repeated many times using different samples. 3.4. Don’t use confidence intervals to test hypotheses. Investigators often interpret confidence intervals in terms of hypotheses. For instance, investigators might claim that there is a statistically significant difference between groups because the 95% confidence interval for the odds ratio excludes 1. Such claims are problematic because confidence intervals are concerned with estimation, not inference. Moreover, the mathematical method to calculate confidence intervals may be different from those used to calculate p-values. It is perfectly possible to have a 95% confidence interval that includes no difference between groups even though the p-value is less than 0.05 or vice versa. For instance, in a study of 100 patients in two equal groups, with event rates of 70% and 50%, the p-value from Fisher’s exact test is 0.066 but the 95% C.I. for the odds ratio is 1.03 to 5.26. The 95% C.I. for the risk difference and risk ratio also exclude no difference between groups. 3.5. Take care interpreting results when reporting multiple p-values. The more questions you ask, the more likely you are to get a spurious answer to at least one of them. For example, if you report p-values for five independent true null hypotheses, the probability that you will falsely reject at least one is not 5%, but >20%. Although formal adjustment of p-values is appropriate in some specific cases, such as genomic studies, a more common approach is simply to interpret p-values in the context of multiple testing. For instance, if an investigator examines the association of 10 variables with three different endpoints, thereby testing 30 separate hypotheses, a p-value of 0.04 should not be interpreted in the same way as if study tested only a single hypothesis with a p-value of 0.04. 3.6. Do not report separate p-values for each of two different groups in order to address the question of whether there is a difference between groups. One scientific question means one statistical hypothesis tested by one p-value. To illustrate the error of using two p-values to address one question, take the case of a randomized trial of drug versus placebo to reduce voiding symptoms, with 30 patients in each group. The authors might report that symptom scores improved by 6 (standard deviation 14) points in the drug group (p=0.03 by one-sample t-test) and 5 (standard deviation 15) points in the placebo group (p=0.08). However, the study hypothesis concerns the difference between drug and placebo. To test a single hypothesis, a single p-value is needed. A two-sample t-test for these data gives a p-value for 0.8 – unsurprising, given that the scores in each group were virtually the same - confirming that it would be unsound to conclude that the drug was effective based on the finding that change was significant in the drug group but not in placebo controls. 3.7. Use interaction terms in place of subgroup analyses. A similar error to the use of separate tests for a single hypothesis is when an intervention is shown to have a statistically significant effect in one group of patients but not another. One approach that is more appropriate is to use what is known as an interaction term in a statistical model. For instance, to determine whether a drug reduced pain scores more in women than men, the model might be as follows: It is sometimes appropriate to report estimates and confidence intervals within subgroups of interest, but p-values should be avoided. 3.8. Tests for change over time are generally uninteresting. A common analysis is to conduct a paired t-test comparing, say, erectile function in older men at baseline with erectile function after 5 years of follow-up. The null hypothesis here is that “erectile function does not change over time”, which is known to be false. Investigators are encouraged to focus on estimation rather than inference, reporting, for example, the mean change over time along with a 95% confidence interval. 3.9. Avoid using statistical tests to determine the type of analysis to be conducted. Numerous statistical tests are available that can be used to determine how a hypothesis test should be conducted. For instance, investigators might conduct a Shapiro-Wilk test for normality to determine whether to use a t-test or Mann-Whitney, Cochran’s Q to decide whether to use a fixed- or random-effects approach in a meta-analysis or use a t-test for between-group differences in a covariate to determine whether that covariate should be included a multivariable model. The problem with these sorts of approaches is that they are often testing a null hypothesis that is known to be false. For instance, no data set perfectly follows a normal distribution. Moreover, it is often questionable that changing the statistical approach in the light of the test is actually of benefit. Statisticians are far from unanimous as to whether Mann-Whitney is always superior to t-test when data are non-normal, or that fixed effects are invalid under study heterogeneity, or that the criterion of adjusting for a variable should be whether it is significantly different between groups. Investigators should generally follow a prespecified analytic plan, only altering the analysis if the data unambiguously point to a better alternative. 3.10. When reporting p-values, be clear about the hypothesis tested and ensure that the hypothesis is a sensible one. P-values test very specific hypotheses. When reporting a p-value in the results section, state the hypothesis being tested unless this is completely clear. Take, for instance, the statement “Pain scores were higher in group 1 and similar in groups 2 and 3 (p=0.02)”. It is ambiguous whether the p-value of 0.02 is testing group 1 vs. groups 2 and 3 combined or the hypothesis that pain score is the same in all three groups. Clarity about the hypotheses being tested can help avoid the testing of inappropriate hypotheses. For instance, p-values for differences between groups at baseline in a randomized trial is testing a null hypothesis that is known to be true (informally, that any observed differences between groups are due to chance). 4. Reporting of study estimates 4.1. Use appropriate levels of precision. Reporting a p-value of 0.7345 suggests that there is an appreciable difference between p-values of 0.7344 and 0.7346. Reporting that 16.9% of 83 patients responded entails a precision (to the nearest 0.1%) that is nearly 200 times greater than the width of the confidence interval (10% to 27%). Reporting in a clinical study that the mean calorie consumption was 2069.9 suggests that calorie consumption can be measured extremely precisely by a food questionnaire. Some might argue that being overly precise is irrelevant, because the extra numbers can always be ignored. The counter-argument is that investigators should think very hard about every number they report, rather than just carelessly cutting and pasting numbers from the statistical software printout. The specific guidelines for precision are as follows: • Report p-values to a single significant figure unless the p is close to 0.05, in which case, report two significant figures. Do not report “NS” for p-values of 0.05 or above. Very low p-values can be reported as p<0.001 or similar. A p-value can indeed be 1, although some investigators prefer to report this as >0.9. For instance, the following p-values are reported to appropriate precision: <0.001, 0.004, 0.045, 0.13, 0.3, 1. • Report percentages, rates and probabilities to 2 significant figures, e.g. 75%, 3.4%, 0.13%. • Do not report p-values of zero, as any experimental result has a non-zero probability. • Do not give decimal places if a probability or proportion is 1 (e.g. a p-value of 1.00 or a percentage of 100.00%). The decimal places suggest it is possible to have, say, a p-value 1.05. There is a similar consideration for data that can only take integer values. It makes sense to state that, for instance, the mean number of pregnancies was 2.4, but not that 29% of women reported 1.0 • There is generally no need to report estimates to more than three significant figures. • Hazard and odds ratios are normally reported to two decimal places, although this can be avoided for high odds ratios (e.g. 18.2 rather than 18.17). 4.2. Avoid redundant statistics in cohort descriptions. Authors should be selective about the descriptive statistics reported and ensure that each and every number provides unique information. Authors should avoid reporting descriptive statistics that can be readily derived from data that have already been provided. For instance, there is no need to state 40% of a cohort were men and 60% were women, choose one or the other. Another common error is to include a column of descriptive statistics for two groups separately and then the whole cohort combined. If, say, the median age is 60 in group 1 and 62 in group 2, we do not need to be told that the median age in the cohort as a whole is close to 61. 4.3. For descriptive statistics, median and quartiles are preferred over means and standard deviations (or standard errors); range should be avoided. The median and quartiles provide all sorts of useful information, for instance, that 50% of patients had values above the median or between the quartiles. The range gives the values of just two patients and so is generally uninformative of the data distribution. 4.4. Report estimates for the main study questions. A clinical study typically focuses on a limited number of scientific questions. Authors should generally provide an estimate for each of these questions. In a study comparing two groups, for instance, authors should give an estimate of the difference between groups, and avoid giving only data on each group separately or, simply saying that the difference was or was not significant. In a study of a prognostic factor, authors should give an estimate of the strength of the prognostic factor, such as an odds ratio or hazard ratio, as well as reporting a p-value testing the null hypothesis of no association between the prognostic factor and outcome. 4.5. Report confidence intervals for the main estimates of interest. Authors should generally report a 95% confidence interval around the estimates relating to the key research questions, but not other estimates given in a paper. For instance, in a study comparing two surgical techniques, the authors might report adverse event rates of 10% and 15%; however, the key estimate in this case is the difference between groups, so this estimate, 5%, should be reported along with a 95% confidence interval (e.g. 1% to 9%). Confidence intervals should not be reported for the estimates within each group (e.g. adverse event rate in group A of 10%, 95% CI 7% to 13%). Similarly, confidence intervals should not be given for statistics such as mean age or gender ratio. 4.6. Do not treat categorical variables as continuous. A variable such as Gleason grade groups are scored 1 – 5, but it is not true that the difference between group 3 and 4 is half as great as the difference between group 2 and 4. Variables such as Gleason grade group should be reported as categories (e.g. 40% grade group 1, 20% group 2, 20% group 3, 20% group 4 and 5) rather than as a continuous variable (e.g. mean Gleason score of 2.4). Similarly, categorical variables such as Gleason should be entered into regression models not as a single variable (e.g. a hazard ratio of 1.5 per 1-point increase in Gleason grade group) but as multiple categories (e.g. hazard ratio of 1.6 comparing Gleason grade group 2 to group 1 and hazard ratio of 3.9 comparing group 3 to group 1). 4.7. Avoid categorization of continuous variables unless there is a convincing rationale. A common approach to a variable such as age is to define patients as either old (≥ 60) or young (<60) and then enter age into analyses as a categorical variable, reporting, for example, that “patients aged 60 and over had twice the risk of an operative complication than patients aged less than 60”. In epidemiologic and marker studies, a common approach is to divide a variable into quartiles and report a statistic such as a hazard ratio for each quartile compared to the lowest (“reference”) quartile. This is problematic because it assumes that all values of a variable within a category are the same. For instance, it is likely not the case that a patient aged 65 has the same risk as a patient aged 90, but a very different risk to a patient aged 64. It is generally preferable to leave variables in a continuous form, reporting, for instance, how risk changes with a 10-year increase in age. Non-linear terms can also be used, to avoid the assumption that the association between age and risk follows a straight line. 4.8. Do not use statistical methods to obtain cut-points for clinical practice. There are various statistical methods available to dichotomize a continuous variable. For instance, outcomes can be compared either side of several different cut-points, and the optimal cut-point chosen as the one associated with the smallest p-value. Alternatively, investigators might choose a cut-point that leads to the highest value of sensitivity + specificity, that is, the point closest to the top left-hand corner of a Receiver Operating Curve (ROC). Such methods are inappropriate for determining clinical cut-points because they do not consider clinical consequences. The ROC curve approach, for instance, assumes that sensitivity and specificity are of equal value, whereas it is generally worse to miss disease than to treat unnecessarily. The smallest p-value approach tests strength of evidence against the null hypothesis, which has little to do with the relative benefits and harms of a treatment or further diagnostic work up. 4.9. The association between a continuous predictor and outcome can be demonstrated graphically, particularly by using non-linear modeling. In high-school math we often thought about the relationship between y and x by plotting a line on a graph, with a scatterplot added in some cases. This also holds true for many scientific studies. In the case of a study of age and complication rates, for instance, an investigator could plot age on the x axis against risk of a complication on the y axis and show a regression line, perhaps with a 95% confidence interval. Non-linear modeling is often useful because it avoids assuming a linear relationship and allows the investigator to determine questions such as whether risk starts to increase disproportionately beyond a given age. 4.10. Do not ignore significant heterogeneity in meta-analyses. Informally speaking, heterogeneity statistics test whether variations between the results of different studies in a meta-analysis are consistent with chance, or whether such variation reflects, at least in part, true differences between studies. If heterogeneity is present, authors need to do more than merely report the p-value and focus on the random-effects estimate. Authors should investigate the sources of heterogeneity and try to determine the factors that lead to differences in study results, for example, by identifying common features of studies with similar findings or idiosyncratic aspects of studies with outlying results. 4.11. For time-to-event variables, report the number of events but not the proportion. Take the case of a study that reported: “of 60 patients accrued, 10 (17%) died”. While it is important to report the number of events, patients entered the study at different times and were followed for different periods, so the reported proportion of 17% is meaningless. The standard statistical approach to time-to-event variables is to calculate probabilities, such as the risk of death being 60% by five years or the median survival – the time at which the probability of survival first drops below 50% - being 52 months. 4.12. For time-to-event analyses, report median follow-up for patients without the event or the number followed without an event at a given follow-up time. It is often useful to describe how long a cohort has been followed. To illustrate the appropriate methods of doing so, take the case of a cohort of 1,000 pediatric cancer patients treated in 1970 and followed to 2010. If the cure rate was only 40%, median follow-up for all patients might only be a few years, whilst the median follow-up for patients who survived was 40 years. This latter statistic gives a much better impression of how long the cohort had been followed. Now assume that in 2009, a second cohort of 2000 patients was added to the study. The median follow-up for survivors will now be around a year, which is again misleading. An alternative would be to report a statistic such as “312 patients have been followed without an event for at least 35 years”. 4.13. For time-to-event analyses, describe when follow-up starts and when and how patients are censored. A common error is that investigators use a censoring date which leads to an overestimate of survival. For example, when assessing the metastasis-free survival a patient without a record of metastasis should be censored on the date of the last time the patient was known to be free of metastasis (e.g. negative bone scan, undetectable PSA), not at the date of last patient contact (which may not have involved assessment of metastasis). For overall survival, date of last patient contact would be an acceptable censoring date because the patient was indeed known to be event-free at that time. When assessing cause-specific endpoints, special consideration should be given the cause of death. The endpoints “disease-specific survival” and “disease-free survival” have specific definitions and require careful attention to methods. With disease-specific survival, authors need to consider carefully how to handle death due to other causes. One approach is to censor patients at the time of death, but this can lead to bias in certain circumstances, such as when the predictor of interest is associated with other cause death and the probability of other cause death is moderate or high. Competing risk analysis is appropriate in these situations. With disease-free survival, both evidence of disease (e.g. disease recurrence) and death from any cause are counted as events, and so censoring at the time of other cause death is inappropriate. If investigators are specifically interested only in the former, and wish to censor deaths from other causes, they should define their endpoint as “freedom from progression”. 4.14. For time-to-event analyses, avoid reporting mean follow-up or survival time, or estimates of survival in those who had the event. All three estimates are problematic in the context of censored data. 4.15. For time-to-event analyses, make sure that all predictors are known at time zero or consider alternative approaches such as a landmark analysis or time-dependent covariates. In many cases, variables of interest vary over time. As a simple example, imagine we were interested in whether PSA velocity predicted time to progression in prostate cancer patients on active surveillance. The problem is that PSA is measured at various times after diagnosis. Unless they were being careful, investigators might use time from diagnosis in a Kaplan-Meier or Cox regression but use PSA velocity calculated on PSAs measured at one and two-year follow-up. As another example, investigators might determine whether response to chemotherapy predicts cancer survival, but measure survival from the time of the first dose, before response is known. It is obviously invalid to use information only known “after the clock starts”. There are two main approaches to this problem. A “landmark analysis” is often used when the variable of interest is generally known within a short and well-defined period of time, such as adjuvant therapy or chemotherapy response. In brief, the investigators start the clock at a fixed “landmark” (e.g. 6 months after surgery). Patients are only eligible if they are still at risk at the landmark (e.g. patients who recur before six months are excluded) and the status of the variable is fixed at that time (e.g. a patient who gets chemotherapy at 7 months is defined as being in the no adjuvant group). Alternatively, investigators can use a time-dependent variable approach. In brief, this “resets the clock” each time new information is available about a variable. This would be the approach most typically used for the PSA velocity and progression example. 4.16. When presenting Kaplan-Meier figures, present the number at risk and truncate follow-up when numbers are low. Giving the number of risk is useful for helping to understand when patients were censored. When presenting Kaplan-Meier figures a good rule of thumb is to truncate follow-up when the number at risk in any group falls below 5 (or even 10) as the tail of a Kaplan-Meier distribution is very unstable. 5. Multivariable models and diagnostic tests 5.1. Multivariable, propensity and instrumental variable analyses are not a magic wand. Some investigators assume that multivariable adjustment “removes confounding”, “makes groups similar” or “mimics a randomized trial”. There are two problems with such claims. First, the value of a variable recorded in a data set is often approximate and so may mask differences between groups. For instance, clinical stage might be used as a covariate in a study comparing treatments for localized prostate cancer. But stage T2c might constitute a small nodule on each prostate lobe or, alternatively, most of the prostate consisting of a large, hard mass. The key point is that if one group has more T2c disease than the other, it is also likely that the T2c’s in that group will fall towards the more aggressive end of the spectrum. Multivariable adjustment has the effect of making the rates of T2c in each group the same, but does not ensure that the type of T2c is identical. Second, a model only adjusts for a small number of measured covariates. That does not exclude the possibility of important differences in unmeasured (or even unmeasurable) covariates. A common assumption is that propensity methods somehow provide better adjustment for confounding than traditional multivariable methods. Except in certain rare circumstances, such as when the number of covariates is large relative to the number of events, propensity methods give extremely similar results to multivariable regression. Similarly, instrumental variables analyses depend on the availability of a good instrument, which is less common than is often assumed. In many cases, the instrument is not strongly associated with the intervention, leading to a large increase in the 95% confidence interval or, in some cases, an underestimate of treatment effects. 5.2. Avoid stepwise selection. Investigators commonly choose which variables to include in a multivariable model by first determining which variables are statistically significant on univariable analysis; alternatively, they may include all variables in a single model but then remove any that are not significant. This type of data-dependent variable selection in regression models has several undesirable properties, increasing the risk of overfit and making many statistics, such as the 95% confidence interval, highly questionable. Use of stepwise selection should be restricted to a limited number of circumstances, such as during the initial stages of developing a model, if there is poor knowledge of what variables might be predictive. 5.3. Avoid reporting estimates such as odds or hazard ratios for covariates when examining the effects of interventions. In a typical observational study, an investigator might explore the effects of two different approaches to radical prostatectomy on recurrence while adjusting for covariates such as stage, grade and PSA. It is rarely worth reporting estimates such as odds or hazard ratios for the covariates. For instance, it is well known that a high Gleason score is strongly associated with recurrence: reporting a hazard ratio of say, 4.23, is not helpful and a distraction from the key finding, the hazard ratio between the two types of surgery. 5.4. Rescale predictors to obtain interpretable estimates. Predictors sometimes have a moderate association with outcome and can take a large range of values. This can lead to uninterpretable estimates. For instance, the odds ratio for cancer per year of age might be given as 1.02 (95% CI 1.01, 1.02; p<0.0001). It is not helpful to have the upper bound of a confidence interval be equivalent to the central estimate; a better alternative would be to report an odds ratio per ten years of age. This is simply achieved by creating a new variable equal to age divided by ten to obtain an odds ratio of 1.16 (95% CI 1.10, 1.22; p<0.0001) per 10-year difference in age. 5.5. Avoid reporting both univariate and multivariable analyses unless there is a good reason. Comparison of univariate and multivariable models can be of interest when trying to understand mechanisms. For instance, if race is a predictor of outcome on univariate analysis, but not after adjustment for income and access to care, one might conclude that poor outcome in African-Americans is explained by socioeconomic factors. However, the routine reporting of estimates from both univariate and multivariable analysis is discouraged. 5.6. Avoid ranking predictors in terms of strength. It is tempting for authors to rank predictors in a model, claiming, for instance, “the novel marker was the strongest predictor of recurrence”. Most commonly, this type of claim is based on comparisons of odds or hazard ratios. Such rankings are not meaningful since, among other reasons, it depends on how variables are coded. For instance, the odds ratio for hK2, and hence whether or not it is an apparently “stronger” predictor than PSA, will depend on whether it is entered in nanograms or picograms per ml. Further, it is unclear how one should compare model coefficients when both categorical and continuous variables are included. Finally, the prevalence of a categorical predictor also matters: a predictor with an odds ratio is 3.5 but a prevalence if 0.1% is less important that one with a 50% prevalence and an odds ratio of 2.0. 5.7. Discrimination is a property not of a multivariable model but rather of the predictors and the data set. Although model building is generally seen as a process of fitting coefficients, discrimination is largely a property of what predictors are available. For instance, we have excellent models for prostate cancer outcome primarily because Gleason score is very strongly associated with malignant potential. In addition, discrimination is highly dependent on how much a predictor varies in the data set. As an example, a model to predict erectile dysfunction that includes age will have much higher discrimination for a population sample of adult men than for a group of older men presenting at a urology clinic, because there is a greater variation in age in the population sample. Authors need to consider these points when drawing conclusions about the discrimination of models. This is also why authors should be cautious about comparing the discrimination of different multivariable models where these were assessed in different datasets. 5.8. Correction for overfit is strongly recommended for internal validation. In the same way that it is easy to predict last week’s weather, a prediction model generally has very good properties when evaluated on the same data set used to create the model. This problem is generally described as overfit. Various methods are available to correct for overfit, including crossvalidation and bootstrap resampling. Note that such methods should include all steps of model building. For instance, if an investigator uses stepwise methods to choose which predictors should go into the model and then fits the coefficients, a typical crossvalidation approach would be to: (1) split the data into ten groups, (2) use stepwise methods to select predictors using the first nine groups, (3) fit coefficients using the first nine groups, (4) apply the model to the 10^th group to obtain predicted probabilities, and (5) repeat steps 2–4 until all patients in the data set have a predicted probability derived from a model fitted to a data set that did not include that patient’s data. Statistics such as the AUC are then calculated using the predicted probabilities directly. 5.9. Calibration should be reported and interpreted correctly. Calibration is a critical component of a statistical model: the main concern for any patient is whether the risk given by a model is close to his or her true risk. It is rarely worth reporting calibration for a model created and tested on the same data set, even if techniques such as crossvalidation are used. This is because calibration is nearly always excellent on internal validation. Where a pre-specified model is tested on an independent data set, calibration should be displayed graphically in a calibration plot. The Hosmer-Lemeshow test addresses an inappropriate null hypothesis and should be avoided. Note also that calibration depends upon both the model coefficients and the dataset being examined. A model cannot be inherently “well calibrated.” All that can be said is that predicted and observed risk are close in a specific data set, representative of a given population. 5.10. Avoid reporting sensitivity and specificity for continuous predictors or a model. Investigators often report sensitivity and specificity at a given cut-point for a continuous predictor (such as a PSA of 10 ng /mL), or report specificity at a given sensitivity (such as 90%). Reporting sensitivity and specificity is not of value because it is unclear how high sensitivity or specificity would have to be so as to be high enough to justify clinical use. Similarly, it is very difficult to determine which of two tests, one with a higher sensitivity and the other with a higher specificity, is preferable because clinical value depends on the prevalence of disease and the relative harms of a false-positive compared with a false-negative result. In the case of reporting specificities at fixed sensitivity, or vice versa, it is all but impossible to choose the specific sensitivity rationally. For instance, a team of investigators may state that they want to know specificity at 80% sensitivity, because they want to ensure they catch 80% of cases. But 80% might be too low if prevalence is high, or too high if prevalence is low. 5.11. Report the clinical consequences of using a test or a model. In place of statistical abstractions such as sensitivity and specificity, or an ROC curve, authors are encouraged to choose illustrative cut-points and then report results in terms of clinical consequences. As an example, consider a study in which a marker is measured in a group of patients undergoing biopsy. Authors could report that if a given level of the marker had been used to determine biopsy, then a certain number of biopsies would have been conducted and a certain number of cancers found and missed. 5.12. Interpret decision curves with careful reference to threshold probabilities. It is insufficient merely to report that, for instance, “the marker model had highest net benefit for threshold probabilities of 35 – 65%”. Authors need to consider whether those threshold probabilities are rational. If the study reporting benefit between 35 – 65% concerned detection of high-grade prostate cancer, few if any urologists would demand that a patient have at least a one-in-three chance of high-grade disease before recommending biopsy. The authors would therefore need to conclude that the model was not of benefit. 6. Conclusions and interpretation 6.1. Draw a conclusion, don’t just repeat the results. Conclusion sections are often simply a restatement of the results. For instance, “a statistically significant relationship was found between body mass index (BMI) and disease outcome” is not a conclusion. Authors instead need to state implications for research and / or clinical practice. For instance, a conclusion section might call for research to determine whether the association between BMI is causal or make a recommendation for more aggressive treatment of patients with higher BMI. 6.2. Avoid using words such as “may” or “might”. A conclusion such as that a novel treatment “may” be of benefit would only be untrue if it had been proven that the treatment was ineffective. Indeed, that the treatment may help would have been the rationale for the study in the first place. Using words such as may in the conclusion is equivalent to stating, “we know no more at the end of this study than we knew at the beginning”, reason enough to reject a paper for publication. 6.3. A statistically significant p-value does not imply clinical significance. A small p-value means only that the null hypothesis has been rejected. This may or may not have implications for clinical practice. For instance, that a marker is a statistically significant predictor of outcome does not imply that treatment decisions should be made on the basis of that marker. Similarly, a statistically significant difference between two treatments does not necessarily mean that the former should be preferred to the latter. Authors need to justify any clinical recommendations by carefully analyzing the clinical implications of their findings. 6.4. Avoid pseudo-limitations such as “small sample size” and “retrospective analysis”, consider instead sources of potential bias and the mechanism for their effect on findings. Authors commonly describe study limitations in a rather superficial way, such as, “small sample size and retrospective analysis are limitations”. But a small sample size may be immaterial if the results of the study are clear. For instance, if a treatment or predictor is associated with a very large odds ratio, a large sample size might be unnecessary. Similarly, a retrospective design might be entirely appropriate, as in the case of a marker study with very long-term follow-up, and have no discernible disadvantages compared to a prospective study. Discussion of limitations should include both the likelihood and effect size of possible bias. 6.5. Consider the impact of missing data and patient selection. It is rare that complete data is obtained from all patients in a study. A typical paper might report, for instance, that of 200 patients, 8 had data missing on important baseline variables and 34 did not complete the end of study questionnaire, leading to a final data set of 158. Similarly, many studies include a relatively narrow subset of patients, such as 50 patients referred for imaging before surgery, out of the 500 treated surgically during that timeframe. In both cases, it is worth considering analyses to investigate whether patients with missing data or who were not selected for treatment were different in some way from those who were included in the analyses. Although statistical adjustment for missing data is complex and is warranted only in a limited set of circumstances, basic analyses to understand the characteristics of patients with missing data are relatively straightforward and are often helpful. 6.6. Consider the possibility and impact of ascertainment bias Ascertainment bias occurs when an outcome depends on a test, and the propensity for a patient to be tested is associated with the predictor. PSA screening provides a classic example: prostate cancer is found by biopsy, but the main reason why men are biopsied is because of an elevated PSA. A study in a population subject to PSA screening will therefore overestimate the association between PSA and prostate cancer. Ascertainment bias can also be caused by the timing of assessments. For instance, frequency of biopsy in prostate cancer active surveillance will depend on prior biopsy results and PSA level, and this induces an association between those predictors and time to progression. 6.7. Do not confuse outcome with response among subgroups of patients undergoing the same treatment: patients with poorer outcomes may still be good candidates for that treatment. Investigators often compare outcomes in different subgroups of patients all receiving the same treatment. A common error is to conclude that patients with poor outcome are not good candidates for that treatment and should receive an alternative approach. This is to confuse differences between patients for differences between treatments. As a simple example, patients with large tumors are more likely to recur after surgery than patients with small tumors, but that cannot be taken to suggest that resection is not indicated for patients with tumors greater than a certain size. Indeed, surgery is generally more strongly indicated for patients with aggressive (but localized) disease and such patients are unlikely to do well on surveillance. 6.8. Be cautious about causal attribution: correlation does not imply causation. It is well-known that “correlation does not imply causation” but authors often slip into this error in making conclusions. The introduction and methods section might insist that the purpose of the study is merely to determine whether there is an association between, say, treatment frequency and treatment response, but the conclusions may imply that, for instance, more frequent treatment would improve response rates. Use and interpretation of p-values That p-values are widely misused and misunderstood is apparent from even the most cursory reading of the medical literature. One of the most common errors is accepting the null hypothesis, for instance, concluding from a p-value of 0.07 that a drug is ineffective or that two surgical techniques are equivalent. This particular error is described in detail in guideline 3.1. The more general problem, which we address here, is that p-values are often given excessive weight in the interpretation of a study. Indeed, studies are often classed by investigators into “positive” or “negative” based on statistical significance. Gross misuse of p-values has led some to advocate banning the use of p-values completely[4]. We follow the American Statistical Association statement on p-values and encourage all researchers to read either the full statement[5] or the summary[6]. In particular, we emphasize that the p-value is just one statistic that helps interpret a study, it does not determine our interpretations. Drawing conclusions for research or clinical practice from a clinical research study requires evaluation of the strengths and weakness of study methodology, the results of other pertinent data published in the literature, biological plausibility and effect size. Sound and nuanced scientific judgment cannot be replaced by just checking whether one of the many statistics in a paper is or is not less than 0.05. Concluding remarks These guidelines are not intended to cover all medical statistics but rather the statistical approaches most commonly used in clinical research papers in urology. It is quite possible for a paper to follow all of the guidelines yet be statistically flawed or to break numerous guidelines and still be statistically sound. On balance, however, the analysis, reporting and interpretation of clinical urologic research will be improved by adherence to these guidelines. Funding support: Supported in part by the Sidney Kimmel Center for Prostate and Urologic Cancers, P50-CA92629 SPORE grant from the National Cancer Institute to Dr. H. Scher, and the P30-CA008748 NIH/ NCI Cancer Center Support Grant to Memorial Sloan-Kettering Cancer Center. Conflicts of interest: The authors have nothing to disclose.	{"url":"https://euntia.shop/article/guidelines-for-reporting-of-statistics-for-clinical-research-in-urology","timestamp":"2024-11-01T22:41:47Z","content_type":"text/html","content_length":"131266","record_id":"<urn:uuid:44549a8e-0530-415e-8668-f35fb0a4bcb2>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00014.warc.gz"}
Solving a system of linear equations using Cramer's rule Cramer's rule may only be applied for a system of linear equations with as many equations as unknowns (the coefficient matrix of the system must be square) and with non-zero determinant of the coefficient matrix. Consider a system of n linear equations for n unknowns x[1], x[2], ..., x[n]: a[11]x[1] + a[12]x[2] + ...+ a[1n]x[n] = b[1] a[21]x[1] + a[22]x[2] + ...+ a[2n]x[n] = b[2] ... ... ... ... ... a[n1]x[1] + a[n2]x[2] + ...+ a[nn]x[n] = b[n] The determinant of the coefficient matrix Since the computation of large determinants is cumbersome, Cramer's rule is generally used for systems of two and three equations.	{"url":"https://intemodino.com/equation-solver/solving-systems-of-linear-equations-cramer's-rule.html","timestamp":"2024-11-05T12:42:48Z","content_type":"text/html","content_length":"6695","record_id":"<urn:uuid:7c70d35d-f077-446f-a4f0-34395f6d52c0>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00155.warc.gz"}
APEX Calculus: UND Edition APEX Calculus: UND Edition Author: Gregory Hartman, Virginia Military Institute, Department of Mathematics, University of North Dakota This text comprises a three–volume series on Calculus. The first part covers material taught in many “Calculus 1” courses: limits, derivatives, and the basics of integration, found in Chapters 1 through 6. The second text covers material often taught in “Calculus 2”: integration and its applications, along with an introduction to sequences, series and Taylor Polynomials, found in Chapters 7 through 10. The third text covers topics common in “Calculus 3” or “Multivariable Calculus”: parametric equations, polar coordinates, vector–valued functions, and functions of more than one variable, found in Chapters 11 through 15. All three are available separately for free. Printing the entire text as one volume makes for a large, heavy, cumbersome book. One can certainly only print the pages they currently need, but some prefer to have a nice, bound copy of the text. Therefore this text has been split into these three manageable parts, each of which can be downloaded separately. The source files for the text can be found at https://github.com/teepeemm/APEXCalculusLT_Source	{"url":"https://oer.tamiu.edu/itemRecord.php?id=2272","timestamp":"2024-11-11T17:00:17Z","content_type":"text/html","content_length":"24830","record_id":"<urn:uuid:0254d0b2-8040-45bf-aa47-ac837e274067>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00229.warc.gz"}
Finite Element Analysis Science topic Finite Element Analysis - Science topic A computer based method of simulating or analyzing the behavior of structures or components. Questions related to Finite Element Analysis Hello everyone I am new to MATLAB and would appreciate some assistance. I am currently working on an optimization problem that involves finite element analysis. I have two parts A and B that are in contact with each other. Each of these parts is made of a different material and has its own properties. I want to find out how the distribution of relative density in part A affects the distribution of stress and strain energy in part B. I think this is a simple FE problem. But I've never solve such problem through MATLAB before. My question is that “Is it possible to do this through MATLAB? If so, how complex is the coding process for this problem?” Thank you in advence Relevant answer Dear Andreassen I sincerely thank you for your prompt reply. It was really helpful and opened my mind to the problem. Also thank you for your MATLAB script you sent. It will be very helpful for me who wants to write such a script for the first time. thanks a million times Hello to all, I am modeling a 3d reinforced concrete column in DIANA FEA. On the bottom side, the model is supported by an area support. When the model is meshed and calculated I get 1400 small reaction forces in every mesh node. I would like to create a Force-Displacement diagram and for that, I need one reaction force that is a sum of all small reactions. I know that I can extract nodal forces in a table and create a sum in Excel, but I want to do a nonlinear analysis with several hundred load steps and that approach is not very efficient. Any advice on how to get reaction force which represent a sum of nodal forces? Thanks for any suggestions. It would be much appreciated. Relevant answer Thanks for the suggestion. Hello Everone, I have conducted a multibody simulation of a gearbox casing in Simpack, and I’ve exported the results as a text file. This file contains the real and imaginary components of normal velocity at each node for frequencies ranging from 50 Hz to 200 Hz. In Abaqus, I’ve created an Acoustic Mesh where the material is air. My goal is to apply these nodal velocities (from the Simpack results) to the nodes at the base of the Acoustic Mesh and then perform an acoustic analysis to calculate the sound pressure around the gearbox. Here's what I have so far: • I've used Python (pandas) to read the text file and convert the velocity data into NumPy arrays—one for each frequency, containing the normal velocity at each node. • Now, I need to apply these velocities as boundary conditions in Abaqus for each frequency. My question is: • How can I apply these nodal velocities to Abaqus using Python? • Is there a specific section in the Abaqus manual that covers this? • Any guidance on how to set this up, would be greatly appreciated. Thank you in advance for your help! Relevant answer Thank you for taking your time to answer my question Nils Wagner The contour plots look great. Could you please let me know how to apply nodal velocities in the normal direction? When the geometry is simple like a cube, we can apply the velocities in the global X direction, but when the geometry is complex, is it possible to apply the velocities in the normal direction? I am willing to do fluid-structure interaction studies for a 3-D rectangular tank under seismic excitation. For the FE discretization of the tank and fluid domain, can a 2-D element be used in the 3-D space for analysis? If yes, will it reduce accuracy of the results? Relevant answer Actually I want to analyse a liquid filled 3-D rectangular tank under seismic excitation. Direct coupling will be used for the study of interaction between the fluid and the structure. In this aspect, if I model using 2-D elements but in 3-D space, how much the result can vary? I am modelling a hyperelsatic and viscous material in ABAQUS. I have calculated the hyperelasticity parameters using curve fitting. Now, for viscoelasticity, I have stress relaxation results in terms of shear modulus variation with time. While inputting the data into Abaqus, you need two parameters G0 and Ginf along with the experimental results. So i think G0 is the shear modulus at the beginning of relaxation test and Ginf is defined based on hyperelastic parameter(2C0 for Yeoh Or similar for other models)? Am I correct or else? Relevant answer Yes, you're correct that the initial shear modulus is the value at the beginning of the relaxation test, representing the unrelaxed modulus. The long-term shear modulus represents the fully relaxed state of the material after sufficient time has passed. In models like Yeoh, the long-term modulus can be approximated using the first hyperelastic parameter from the curve fitting. For other hyperelastic models, the long-term modulus is similarly related to the fitted parameters. These values, along with your experimental data, are needed in ABAQUS for accurate viscoelastic modeling. BTW have you used MCalibration (Link: ) software for curve fitting? It will work for you easily. Dear community, I am trying to run a transient structural analysis to simulate the rolling contact between a wheel and a rail. However, in the rolling step, the wheel just keeps going through the rail body without contacting it. The boundary conditions are a fixed support at the rail bottom face. A displacement boundary condition that only allows displacement of the wheel center in the vertical and longitudinal directions (z and x respectively) as well as rotation about the y axis are applied to a pilot node placed at wheel center. A force condition of 75000 N in the direction vertical direction is applied on the pilot node at wheel center. The contact between the wheel and rail is frictional contact. Please find attached a figure of the problem I obtain and the boundary conditions as they are defined in apdl. Thank you so much for the help. Best regards, Relevant answer Hi, I am working on a similar problem for Ball bearings. Could you suggest some correct contact definitions ? Thank you! Hajar Rhylane Hello everyone, I am conducting a simulation task using CEL method in Abaqus related to offshore pipeline. I would like to know how to assign different soil materials for several soil layers of Eulerian part in Abaqus, please help. Thank you so much Relevant answer Applying different materials to separate layers of an Eulerian part in Abaqus requires defining the regions within the part and assigning material properties to these regions. The Eulerian mesh in Abaqus is typically used for problems involving fluid flow, multiphase interactions, or highly nonlinear deformations, and it consists of a fixed grid where the material can move. To apply different materials to several layers in an Eulerian part, follow these steps: 1. Define the Eulerian Domain and Partitioning: • Create an Eulerian part and mesh it as required. • Use Partition to divide the Eulerian part into different regions (layers). This partitioning can be done based on geometric features or by creating datum planes to specify the boundaries of the 2. Create the Materials: • Define the required materials in the Materials module. • For each material, specify the necessary properties, such as density, elasticity, plasticity, and any other relevant parameters. 3. Assign Sections: • Create Sections for each material. Go to Section → Create, select the Eulerian material type, and assign the corresponding material to the section. 4. Assign Sections to the Regions: • Go to the Property module and assign the previously created sections to the partitioned regions of the Eulerian part. • Click Assign Section → Select the regions/layers you want to assign the section to → Choose the appropriate section (material) → Click Done. 5. Define the Initial Conditions for the Eulerian Domain: • Specify the initial volume fractions for each Eulerian material in the regions. • Go to Model → Field → Initial Conditions → Create → Choose the Eulerian Volume Fraction. • Assign the initial volume fraction of each material to the corresponding partitioned regions of the Eulerian domain. 6. Set up the Analysis: • Set up your analysis as usual, defining the step, boundary conditions, interactions, and loads as needed. • Ensure the mesh and the defined regions are compatible with the material behavior you expect during the simulation. 7. Verify and Run the Simulation: • Verify the material assignments in the Visualization module before running the simulation. • Ensure that the volume fractions and material definitions align with the partitioned regions to avoid errors during the simulation. Can the CDP (Concrete Damage Plasticity) model or the Hashin Damage model be effectively used for modeling the behavior of engineered cementitious composites (ECC) in finite element analysis (FEA)? Additionally, what are the key factors that should be considered when choosing between these two damage models for accurately capturing the mechanical and damage characteristics of ECC, such as cracking, strain hardening, and failure modes? Relevant answer Engr. Tufail In my research, the model has to represent the mechanical behavior and failure mechanisms of engineered cementitious composites (ECC) under different loading conditions. Specifically, the aim is to simulate how ECC responds to stress, strain, cracking, and damage progression. ABACUS is a software tool for Finite element analysis .Please explain how to use this tool to analyze experimental problems with example Relevant answer Well "Abaqus" is for simulations, and "ABACUS" is for counting your sanity level while learning FEA! I have programmed a UEL subroutine for a 3D cohesive element and I have a problem. When I do the testing using a 3D single element, the subroutine converges if and only if I put initial boundary conditions on all nodes. For example, imagine the bottom surface is fixed and the upper surface is prescribed a displacement to perform a tensile test. Then, it is necessary to impose the lateral displacements in order to obtain convergence. If this latter constraint is not applied, the subroutine does not work (wrong displacement values obtained, error message). It seems that stiffness matrix becomes ill conditioned in the absence of sufficient boundary conditions. Yet, I observe that the cohesive elements already available in Abaqus cause no problem. For these elements, the convergence is always obtained even if the lateral displacements are not applied. Why does this work well with Abaqus elements, but not with a user element in UEL? I verified my subroutine and I think that I have strictly implemented the classical formulation for cohesive element. Relevant answer From your description, the issue likely arises from how the stiffness matrix is defined in your UEL subroutine. Abaqus's built-in cohesive elements handle numerical conditioning and boundary constraints more robustly. In your UEL, the stiffness matrix might be under-constrained or ill-conditioned without lateral constraints, leading to convergence problems. You may need to check how you're defining stiffness in relation to boundary conditions or incorporate stabilization techniques. I am currently working with recycled aggregate concrete and using finite element simulations to predict its behavior under different conditions. I have faced several challenges in accurately modeling the properties of this material, especially considering its heterogeneous nature and the variability of recycled aggregates. I would like to know what specific challenges others have encountered while modeling recycled aggregate concrete in finite element analysis and what methods or techniques have proven effective in addressing these issues. Any insights into improving the accuracy and reliability of these models would be greatly appreciated. Dear RG Members, I’m currently working on extracting data of multiple .rst/.rth files, which are having same mesh and same node numbering, using PyANSYS/PyMAPDL. The end target is to use this data for LCF/Creep/ Creep-fatigue/fracture calculations. Currently this data extraction is done manually, we want to automate this process. The elements I am using is plane77 (with axisymmetric option) for thermal analysis and plane 183 (axisymmetric) for structural analysis (in ANSYS). Please look at the figure attached. I want to provide path for all my rth/rst files and respective load/substep numbers for which I want to extract data. On pressing confirm a popup will ask for node number and desired data (temperature/stress/strain/creep strain) for that node number should be there in the output column from all these files (that's the wish). First row data (temperature from a rth file for steady state thermal run) I was able to extract. Got stuck in the second row where I want von Mieses stress from a steady state structural run. However, when I attempt to parse the result sets using result.parse_step_substep(), I receive only integer indices representing the result sets, rather than a tuple containing the actual load step and substep numbers. For example, my .rst file shows 3 result sets, but the step info returned is simply 0, 1, and 2, without any clear mapping to the original load step and substep numbers from the simulation. Is there a current method within PyMAPDL to directly retrieve data based on specific load step and substep numbers (e.g., Load Step 3, Substep 5), rather than relying on the result set Thank you for your help and support in advance. Relevant answer I asked this question for PyANSYS team also. Here is their reply: This was a mistake from my side as I just started using PyANSYS. Functionality I was seeking is related to the PyMAPDL Reader, not PyMAPDL itself. Let's say we have a standard, regular hexagonal honeycomb with a 3-arm primitive unit cell (something like the figure attached; the figure is only representative and not drawn to scale). The bottommost node is taken as the source of wave input and the ends of the left and right arms are taken as destinations such that Bloch's condition can be applied as q[left] = e^ik1 q[bottom] and q [right] = e^ik2 q[bottom]. I wish to learn how would an iso-frequency contour plot be plotted post performing the dispersion analysis. Thanks in advance. Relevant answer Düzgün altıgende kenarlar eş ve oluşan eşkenar üçgenler aynı olduğu için heryerde simetriktir. Bu yüzden oluşan grafik düzgün doğrusal olur. I am simulating the propagation of cracks on ANSYS APDL using a cohesive zone model. My goal is to effectively illustrate how hydrogen diffusion affects this material. To illustrate the influence of hydrogen, I proposed emphasizing its material properties. An experiment involving a tensile test was conducted on a pipeline sample both with and without hydrogen influence. For both the charged and uncharged pipeline samples, a stress-strain graph was created. Using the supplied stress-strain data, I was able to conduct a linear test and determine the young modulus, which allowed me to obtain a reasonable Traction Separation Law result. But when I attempted to apply the nonlinear route, I changed the Material properties by incorporating the stress-strain data into my analysis, and the outcome I got did not resemble any kind of traction separation law graph. I have included the code I used to accomplish this below. Does anyone know how to depict the influence of hydrogen on a metal sample in ANSYS APDL? Code for Linear orthotopic material ET,1,182 ! 2D 4-NODE STRUCTURAL SOLID ELEMENT KEYOPT,1,1,2 !* ENHANCE STRAIN FORMULATION KEYOPT,1,3,2 !* PLANE STRAIN ET,3,202 !* 2D 4-NODE COHESIVE ZONE ELEMENT KEYOPT,3,3,2 !* PLANE STRAIN MP,EX,4,1.353E5 !* E11 = 135.3 GPA MP,EY,4,9.0E3 !* E22 = 9.0 GPA MP,EZ,4,9.0E3 !* E33 = 9.0 GPA MP,GXY,4,5.2E3 !* G12 = 5.2 GPA GMAX = 0.004 TNMAX = 25 !* TENSILE STRENGTH TB,CZM,5,,,EXPO !* COHESIVE ZONE MATERIAL RECTNG,0,100,0,1.5 !* DEFINE AREAS LSEL,S,LINE,,2,8,2 !* DEFINE LINE DIVISION LESIZE,ALL, , ,200 TYPE,1 !* MESH AREA 2 TYPE,2 !* MESH AREA 1 CZMESH,,,1,Y,0, !* GENERATE INTERFACE ELEMENTS NSEL,S,LOC,X,100 !* APPLY CONSTRAINTS NSEL,R,LOC,Y,1.5 !* APPLY DISPLACEMENT LOADING ON TOP NSEL,R,LOC,Y,-1.5 !* APPLY DISPLACEMENT LOADING ON BOTTOM Linear Isotropic ET,1,182 !* 2D 4-NODE STRUCTURAL SOLID ELEMENT KEYOPT,1,1,2 !* ENHANCE STRAIN FORMULATION KEYOPT,1,3,2 !* PLANE STRAIN ET,3,202 !* 2D 4-NODE COHESIVE ZONE ELEMENT KEYOPT,3,3,2 !* PLANE STRAIN GMAX = 0.004 TNMAX = 25 !* TENSILE STRENGTH TB,CZM,5,,,EXPO !* COHESIVE ZONE MATERIAL RECTNG,0,100,0,1.5 !* DEFINE AREAS LSEL,S,LINE,,2,8,2 !* DEFINE LINE DIVISION LESIZE,ALL, , ,200 TYPE,1 !* MESH AREA 2 TYPE,2 !* MESH AREA 1 CZMESH,,,1,Y,0, !* GENERATE INTERFACE ELEMENTS NSEL,S,LOC,X,100 !* APPLY CONSTRAINTS NSEL,R,LOC,Y,1.5 !* APPLY DISPLACEMENT LOADING ON TOP NSEL,R,LOC,Y,-1.5 !* APPLY DISPLACEMENT LOADING ON BOTTOM Non linear : Multilinear isotropic hardening ET,1,182 !* 2D 4-NODE STRUCTURAL SOLID ELEMENT KEYOPT,1,1,2 !* ENHANCE STRAIN FORMULATION KEYOPT,1,3,2 !* PLANE STRAIN ET,3,202 !* 2D 4-NODE COHESIVE ZONE ELEMENT KEYOPT,3,3,2 !* PLANE STRAIN GMAX = 0.004 TNMAX = 25 !* TENSILE STRENGTH TB,CZM,5,,,EXPO !* COHESIVE ZONE MATERIAL RECTNG,0,100,0,1.5 !* DEFINE AREAS LSEL,S,LINE,,2,8,2 !* DEFINE LINE DIVISION LESIZE,ALL, , ,200 TYPE,1 !* MESH AREA 2 TYPE,2 !* MESH AREA 1 CZMESH,,,1,Y,0, !* GENERATE INTERFACE ELEMENTS NSEL,S,LOC,X,100 !* APPLY CONSTRAINTS NSEL,R,LOC,Y,1.5 !* APPLY DISPLACEMENT LOADING ON TOP NSEL,R,LOC,Y,-1.5 !* APPLY DISPLACEMENT LOADING ON BOTTOM Kinematic isotropic hardening ET,1,182 !* 2D 4-NODE STRUCTURAL SOLID ELEMENT KEYOPT,1,1,2 !* ENHANCE STRAIN FORMULATION KEYOPT,1,3,2 !* PLANE STRAIN ET,3,202 !* 2D 4-NODE COHESIVE ZONE ELEMENT KEYOPT,3,3,2 !* PLANE STRAIN GMAX = 0.004 TNMAX = 25 !* TENSILE STRENGTH TB,CZM,5,,,EXPO !* COHESIVE ZONE MATERIAL RECTNG,0,100,0,1.5 !* DEFINE AREAS LSEL,S,LINE,,2,8,2 !* DEFINE LINE DIVISION LESIZE,ALL, , ,200 TYPE,1 !* MESH AREA 2 TYPE,2 !* MESH AREA 1 CZMESH,,,1,Y,0, !* GENERATE INTERFACE ELEMENTS NSEL,S,LOC,X,100 !* APPLY CONSTRAINTS NSEL,R,LOC,Y,1.5 !* APPLY DISPLACEMENT LOADING ON TOP NSEL,R,LOC,Y,-1.5 !* APPLY DISPLACEMENT LOADING ON BOTTOM Relevant answer To simulate crack propagation in ANSYS APDL with a focus on hydrogen embrittlement, you need to account for the effects of hydrogen on the material's properties and how it influences crack growth. Hydrogen embrittlement (HE) can significantly alter the mechanical behavior of materials, often leading to increased brittleness and susceptibility to cracking. Here’s a step-by-step outline to illustrate how to simulate this: Step 1: Define the Material Properties 1. Base Material Properties: Start with the basic material properties such as Young's modulus, Poisson's ratio, yield strength, etc. 2. Hydrogen-affected Properties: Modify these properties to reflect the presence of hydrogen. Typically, hydrogen reduces the ductility and toughness of the material. This can be done by reducing the yield strength and fracture toughness. MP, EX, 1, 210000 ! Young's Modulus in MPa MP, PRXY, 1, 0.3 ! Poisson's Ratio MP, SIGY, 1, 300 ! Yield Strength in MPa (adjusted for hydrogen embrittlement) Step 2: Define the Geometry and Meshing 1. Create Geometry: Define the geometry of your specimen, including the initial crack. 2. Meshing: Generate a fine mesh around the crack tip where stress concentration is high. BLC4,0,0,100,10 ! Create a block for the specimen (100 mm x 10 mm) CINT, 1, 2, 50, 5 ! Insert initial crack at the center (50 mm crack length) et, 1, plane183 ! Element type for 2D analysis esize, 1 ! Element size amesh, all ! Mesh all areas Step 3: Apply Boundary Conditions and Loads 1. Boundary Conditions: Apply appropriate boundary conditions such as fixed supports or symmetry conditions. 2. Loads: Apply the external loads to simulate the conditions under which the crack propagation will be analyzed. dk, 1, all ! Fix the left edge fk, 2, fy, -1000 ! Apply a tensile load of 1000 N on the right edge Step 4: Define Crack Propagation Criteria 1. Fracture Mechanics Criteria: Use criteria such as Stress Intensity Factors (SIF), J-integral, or cohesive zone modeling to simulate crack growth. 2. Hydrogen Effect: Adjust the criteria to account for hydrogen effects, which typically lower the critical SIF or J-integral values. tb, czm, 1 ! Define cohesive zone material properties tbdata, 1, 2.0, 0.5 ! Parameters adjusted for hydrogen embrittlement effect Step 5: Simulation of Crack Propagation 1. Incremental Analysis: Perform the analysis in steps to simulate the crack growth incrementally. After each step, evaluate the crack propagation criteria and extend the crack if the criteria are met. 2. APDL Commands: Use APDL commands to update the geometry and re-mesh the new crack tip region if necessary. ! Evaluate crack propagation criteria here ! Extend crack if criteria are met ! Re-mesh the new crack tip region if necessary ### Step 6: Post-Processing 1. Results Interpretation: Extract results such as stress distribution, crack length, and fracture parameters. 2. Visualize Crack Growth: Use ANSYS post-processing tools to visualize the crack propagation path and understand the effects of hydrogen embrittlement. ### Conclusion Simulating hydrogen embrittlement in ANSYS APDL involves adjusting the material properties and crack propagation criteria to reflect the presence of hydrogen. By incrementally solving the model and updating the crack geometry, you can illustrate how hydrogen affects the material's susceptibility to cracking. Make sure to validate your model with experimental data if available, as hydrogen embrittlement is complex and highly material-specific.@ I am trying to model the transient response of a free-free beam (unconstrained) in ABAQUS, where the force is applied through a spring. I am using a two point spring where one end is connected to the beam and the other a reference point, the force is applied to this reference point. When i visualise the results the spring is showing no results (white) even though I requested the the node set of the reference point. The beam response is the same as it was without the spring no matter the choice of spring stiffness selected. So i assume the spring is incorrectly modelled. Have i made a mistake when modelling the spring? Relevant answer Hi Nils, Here is the input file. I have a problem in which a steel nail is 'embedded' in a wood piece. The head of the nail is in contact with a steel bracket. As I apply cyclic load on the vertical wood 'wall', the bracket moves up and pulls the nails of the horizontal wood part, which gets pulled out of the wood (see Image). However, as the bracket unloads (and consequently so does the nail) the cohesive interaction that I defined between the nail and the wood simply unload to zero as well. This causes the nails to follow the bracket back into original position. I wanted the nail to retain the maximum displacement imposed by the contact with the bracket before the bracket starts to unload. Looking at the ABAQUS documentation, it says the when using cohesive behavior, it will always unload to the initial position (see image of the cohesive material model used in ABAQUS). How can I overcome this? It can be either using contact pairs or any other strategy in ABAQUS (except cohesive elements, which I've tried and could not manage to work in my model). I've already tried to use a custom defined FRIC subroutine in a tangential interaction behavior (doesn't work). I've thought about writing a custom UMAT subroutine, but ABAQUS does not allow a user-defined routine for cohesive response of contact pairs. I've tried to run an analysis with the ABAQUS default cohesive response, then get the nail deformations at peak and try to apply these as boundary conditions at the correct steps of the analysis. I also tried to use the option where ABAQUS maintains the initial position of an object during a step. However, these last two approaches seems to cause numerical instabilities in ABAQUS due to activating and deactivating boundary conditions in between steps. Relevant answer Did you solve your problem? I also want to use regurlar contact and cohesive contact concurrently. I am creating a single element model in Abaqus of a composite material. My goal is to match the stress-strain curve of the material which was obtained experimentally. I am using VUMAT for Fabric Reinforced Composites to get a non-linear stress-strain curve. However, every time I run my analysis I get the error "1 elements have missing property definitions". I tried changing the element type, hourglass stiffness, etc., but nothing seems to work. How can I solve this? Relevant answer Ahmad Bakir Would I be able to do that within Abaqus? Please let me know if you end up publishing a tutorial on this, it would be very helpful! Dear Researchers, Recently, my research group and I have been working on understanding the formulation of axisymmetric elements. We began by studying 4-node and 8-node Axisymmetric-Harmonic elements, which are well described in Cook’s textbooks [1,2] and, e.g., in this paper [3]. We also utilized them in some simple case studies using Ansys APDL software, using PLANE25 Axisymmetric-Harmonic 4-nodes ( ) and PLANE83 Axisymmetric-Harmonic 8-nodes ( However, these elements (PLANE25, PLANE83) are limited to linear analysis cases. Therefore, we have moved on to using Solid 272 & 273 - General Axisymmetric Solid Elements with 4 & 8 base nodes, which can also be used for nonlinear analyses. Unfortunately, to the best of our knowledge, we have not found any articles or references that describe the formulation of these elements in detail (the Ansys reference provides an outline, but it is not sufficient for independent computational implementation). Specifically, we are seeking information not only on the shape functions but also on handling axisymmetric loads using Fourier decomposition, and most importantly, on how conducting nonlinear Does anyone have any papers or books to suggest? Thank you! [1] Robert D. Cook , Concepts and Applications of Finite Element Analysis, John Wiley & Sons Inc, 1974, 978-0471169154 [2] David S. Malkus, Michael E. Plesha, Robert J. Witt, Robert D. Cook, Concepts and Applications of Finite Element Analysis, John Wiley & Sons Inc, 2001, 978-0471356059 [3] R.W. Stephenson, K.E. Rouch, R. Arora, Modelling of rotors with axisymmetric solid harmonic elements, Journal of Sound and Vibration 131(3),1989, 431-443, Relevant answer Thank you very much for the recommended references! Hi All, I am trying to generate the 3D corneal surface from the Zernike Polynomials. I am using the following steps, can anyone please let me know whether they are accurate Step 1: Converted the cartesian data (x, y, z) to polar data (rho, theta, z) Step 2: Nomalised the rho values, so that they will be less than one Step 3: Based on the order, calculated the Zernike polynomials (Zpoly), (for example: if the order is 6, the number of polynomials is 28 ) Step 4: Zfit = C1 * Z1 + C2 * Z2 + C3 * Z3 + ......... + C28 * Z28 Step 5: Using regression analysis, calculated the coefficient (C) values Step 6: Calculated the error between the predicted value (Zfit) and the actual elevation value (Z) Step 7: Finally, converted the polar data (rho, theta, Zfit) to Cartesian coordinates to get the approximated corneal surface Thanks & Regards, Relevant answer First, represent the Zernike polynomial as a complex polynomial in polar coordinates (r, θ) using the Zernike radial polynomials Rl(p) and angular harmonics Pm(θ). Then, evaluate the polynomial at a grid of points on a circular domain (e.g., using a radial and angular resolution). Finally, use the complex values to create a 2D array representing the surface height at each point. You can use libraries like Python's NumPy and SciPy to perform these steps. For example, you can use the `numpy.meshgrid` function to create a grid of (r, θ) values, and then evaluate the Zernike polynomial using NumPy's `polyval` function. I am trying to model a cutting problem with a blade and comparing the results with and without ultrasonic vibration. From experimental results in our group, the use of ultrasound can significantly reduce the cutting force required, although Im not sure that the current model can show this. Has anyone got any general advice for incorporating such a high-speed boundary condition to the model? I have added an image showing the damage model (top two images) and a comparison of reaction force on the blade with and without ultrasound The results without vibration seem to be sensible, although when I apply the vibration, the cutting tool experiences a high reaction force which I don’t believe represents the true scenario. For the work-piece I have implemented Johnson-Cook plastic and damage criteria. Any general advice foe this type of problem would also be greatly appreciated! Relevant answer To incorporate high-speed boundary conditions like ultrasonic vibration into a cutting model, consider dynamic analysis, suitable material models, fine mesh density, accurate boundary conditions, proper contact modeling, experimental validation, and sensitivity analysis. Adjust parameters to reflect the behavior under ultrasonic vibration for accurate results. Dear professors and colleagues， hello！Recently, while studying the CDP model, I have read the manuscripts of various experts and have gained a lot, but I still have some questions.What I particularly want to know is about the parameters of the simulation:Dilatance angle y, Eccentricity e,Form factor Kc.I sincerely want to inquire about how these values should be taken. Thank you to all experts for reading the questions！ Relevant answer Haoxuan Yu , The parameters of the Concrete Damaged Plasticity (CDP) model—dilatancy angle, eccentricity, and form factor—are critical for simulating concrete behavior accurately. The dilatancy angle, which measures the volumetric expansion of concrete under shear stress, typically ranges from 30 to 40 degrees for most types of concrete. This parameter is best determined through empirical data from experiments, although values from the literature can serve as a reliable reference when specific data is unavailable. The eccentricity parameter, controlling the shape of the plastic potential surface, is commonly set at 0.1. This value offers a balance between numerical stability and accuracy and is generally less sensitive compared to the dilatancy angle. For the form factor, which dictates the yield surface shape in the deviatoric plane and reflects the compressive-to-tensile strength ratio, a standard value of 0.667 is widely accepted. This ratio is based on the assumption that the second stress invariant on the tensile meridian to the compressive meridian is approximately 2/3. These values can be refined by consulting experimental data, conducting a literature review, and performing calibration simulations to ensure they accurately represent the specific concrete and loading conditions in your study. I hope it is clear now? Can somebody please give me some reference to a paper or book where it is explained how to condensate: * from a Q8 serendipity FE to a Q4 FE with drilling DOFs (16 DOFs to 12 DOFs). * from a H20 serendipity FE to a H8 FE with drilling DOFs (40 DOFs to 24 DOFs). See the attached figures, taken from the RFEM technical manual. Diego Andrés I am trying to model a parallel plate compression test for a bioresorbable stent, by using just 1/2 model. With one design everything went perfect, when I uploaded the other one (same BCs, same properties...) it doesn't work. Abaqus says that the strain is too high. I tried to distance the shell plate to 1 mm (as you can see here ) to see if there was any change, but nothing. Do you know how can I resolve this? To be complete, I put as BCs: symmetry on the faces cut in half (loading step) and fixed just one node of the stent along the axial direction (in INITIAL step, not loading). Thanks in advance. I am simulating a tunnel under blast loading using conwep method in ABAQUS,I would like to know the analysis procedure and steps of analysis? Hello everyone! I have a query regarding the Torsional constant and the Polar moment of inertia being used in Ansys workbench for a non circular cross-section . In the Ansys help, i found that for the special case of a circular cross-sections, the torsionnal constant is equal to the polar moment of inertia and calculated by using this formula: Ixx=J=Iyy+Izz. Could any one clarify as to what formulation Ansys follows for the calculation of the Torsional constant and Polar moment of inertia of a non circular cross section. Thank you! Relevant answer Hajar Rhylane Shlok K Laddha I am facing some problem......I am unable to find a formula for Ixx:torsional stiffness for non circular sections ( rectangular section) I am new user of DIANA FEA. I am trying to analyse a masonry structure. But I am getting this error message: GEOMETRY: NR=1 ERROR CODE: /DIANA/LB/DS30/2236 ERRORMSG.A: Can't normalize null vector. Can you help me please? Relevant answer You might be giving the wrong geometry to the interface. Please check the details of the error either its specific to some elements or whole. This happens when your model geometry is not properly using finite element analysis Relevant answer Etabs can be better for structure analysis then after you will need csi detail for detailing Hi all, I am trying to model a contact problem in DIANA FEA. I went through the DIANA manual and found that DIANA has contact elements which imposes the contact constraint. But there aren't any tutorial or examples available online to do this. The material property for the contact element has two options target and contactor. When I try to assign the contact element material properties to an existing steel object, the object loses its steel material properties and contains only the contact elements properties which is friction and penetration depth. Could someone explain how can I assign the contact constraint in DIANA when two steel cubes are touching each other. Relevant answer I am running though same problem . If you got your answer then can you share it with me? I am trying to simulate a rectangular short column (L/D ratio of 2) for cyclic load under axial compression. I am using the concrete damage plasticity model for concrete. The first problem I am facing is excess lateral stiffness in my ABAQUS model as you can see in the force vs displacement curve. I am trying to match the slope of the red curve generated by the cyclic hysteresis response of the column. The dashed line is the result I got. I have only included the elastic property for concrete for this instance. All the pictures related to the analysis are listed below. • The analysis procedure is static/general for • C3D8R element used for concrete • T3D2 element used for reinforcement • Analysis was done in 2 steps axial load and lateral displacement Can anyone tell me what I am doing wrong? Relevant answer If you got it from the Compressometer results attached to your cylinders, it is probably correct. I thought you were trying to verify someone else's article. "How do advanced computational modeling techniques, such as finite element analysis or computational fluid dynamics, aid in the precise characterization and optimization of thermal bridging phenomena within complex building assemblies?" Relevant answer Finite element analysis (FEA) and computational fluid dynamics (CFD), play a crucial role in the precise characterization and optimization of thermal courses. 1. Detailed Simulation: FEA and CFD allow for detailed simulation of heat transfer within building components, enabling a thorough understanding of thermal bridging effects. 2. Identification of Weak Points: These modeling techniques help identify areas of high heat flow or thermal bridging within complex assemblies, pinpointing potential weak points that need 3. Optimization: By simulating different scenarios and configurations, FEA and CFD assist in optimizing building designs to minimize thermal bridging and improve overall energy efficiency. 4. Cost-Effective Solutions: Computational modeling helps in evaluating different solutions cost-effectively before physical implementation, leading to more efficient and sustainable building I keep getting the following error message when I run any Abaqus job: "XML parsing failure for job XXX. Shutting down socket and terminating all further messages. Please check the .log, .dat, .sta, or .msg files for information about the status of the job." There are no .lck files to be deleted and everyone else using our academic license seems to be unaffected. Occasionally, I can run a model through writing an input file and running through command prompt. Although even this doesnt work everytime, when I check the dat file I get the following error message: in keyword CONFLICTS, file "Job-1.inp", line 1: Keyword Conflicts is generated by Abaqus/CAE to highlight changes made via the Keywords Editor that conflict with changes made subsequently via the native GUI. **NOTE: DUE TO AN INPUT ERROR THE ANALYSIS PRE-PROCESSOR HAS BEEN UNABLE TO Any help would be greatly appreciated as my work is getting delayed a bit and I have no idea what to do! Kind Regards Relevant answer Engr. Tufail I have a similar question，the Abaqus.dat file mentioned “ERROR: SIM database file cannot be opened - System Error CreateFile: code1023 ” How could I solve the error？ Kind regards I have three modules (in free form .f90 format) which are being called from inside of a UMAT subroutine in ABAQUS, in the following manner: module module_A use module_C use module_B end module_A module module_B use module_C end module_B module module_C end module_C subroutine UMAT(STRESS,...) Here the subroutines from module_A and module_B are being called end subroutine UMAT Now, what is the appropriate format for writing these modules with UMAT subroutine? How to merge different module files into a single .for file (free format)? Relevant answer there is a simple way to handle this, you can keep your module be an individual file, and then use `include 'model_file.f90'` before your umat definition. Greetings to all. I am trying to make a composite ply sheet made up of 3 material, after assigning properties and visualizing ply stack layer it is NOT stacked along the thickness(which is needed) but in fact for some reason stacked along lateral direction. I am attaching view-port image for reference , please guide me where I am going wrong. Please assist me. Relevant answer Hello again, When your part is defined as a shell, the thickness direction should be correct, and the stacking sequence aligns along the thickness. What I meant by the viewports not being linked is that the coordinate systems of each viewport are not directly related. It doesn't matter if you actually linked them in the viewport options. Essentially, even though both viewports may display "1, 2, 3" axes, it's essential to understand that on the left viewport, it refers to the GLOBAL 1, 2, and 3 directions. On the other hand, on the right viewport, it denotes the LOCAL 1, 2, and 3 directions, defined according to your composite layup. In this context, 1 represents the longitudinal direction, 3 denotes the stacking direction, and 2 signifies the cross-product of both directions. This setup is consistent across all composite layups when using the query function in Abaqus CAE. In most recent versions, ABAQUS CAE now actually have as global axes x,y and z and for the local material coordinate system it uses, the 1,2 and 3 directions, maybe to avoid confusions between the local and global coordinate systems. You can find on the attachments of this answer a picture of it. How you relate the local and the global coordinate system is defined in the Composite Layup section where you have several options. At any case, for your purposes, if your part is a shell, the stacking and thickness direction should be the same automatically. However, since you are using a continuum shell section, that means that your part is actually a 3D solid part and not a shell. So you should be be careful and precise when you ask for help. In a continuum shell, you discretize an entire 3D body part. For this case, as I said in my previous answer, making sure that your Stacking Direction is on the "Element direction 2" and the rotation axis is around "Axis 2" is crucial. Perhaps you have those options on the advanced tab on your window. At any case, I attached a screenshot of how you define it in my version (2022). Hope it helps! Conventional shell versus continuum shell: I have created a FE model including: 1. Bead (Green), 2.Cell (Red) 3.Components inside the Cell, beams, truss, and nucleus. Embedmentt, all components embed in cell. Boundary condition: 1. Bottom of cell are fixed 2. Bead compress along the Z direction with 500nm I have finish the calculation through Implicit Dynamic, however, actually i don't have densities of each material (those used in Implicit Dynamic are assumptions). So, can i solve this question just with E and v with Abaqus/standard? Relevant answer You can, however the simulation will be too slow in term of convergence. Most of the researchers concerned with analytical study or numerical study use ANSYS for the FE Modeling. The awareness about NASTRAN is low. What may be the reason and Why? Relevant answer I have been using both for 40 years. Ansys and Nastran both started on a solid basis but Nastran had failed to develop over the years. It took hold in aircraft structures for 2D linear analysis and analysts have invested much time and effort in developing their own techniques for using it. They remain loyal to it although some are migrating to Abaqus because it has corporate approval from Dassault connections. Ansys was chosen for 3D models such as in engines. Aircraft structures largely shunned solid elements as the models became 'too large'. In the end, ANSYS has aggressively developed while NASTRAN has not. And, computing resources are now cheap enough to solve very very large models. So, you can read a large assembly of aircraft solid parts into ansys and easily get much higher accuracy faster than you will get in the reduced 2D linear models in NASTRAN. I am looking for a code (open source preferably) for finite element analysis that allows the user to specify some of the node coordinates of the mesh. The code should be able to generate and adjust the rest of the mesh nodes. I would appreciate it very much if I could have some suggestions for such a code. Thanks! Relevant answer You can use Gmsh for generating mesh for any FEA solver Check Point In Surface within Gmsh I am doing reduced order modelling for nonlinear analysis and I have to use the POD and Galerkin projection to reduce my matrices size. The problem is that since it's a nonlinear analysis, the matrices have to be updated for each increment. And for commercial FEA softwares, I do not have access to the stiffness matrices for each step time. Does someone have any suggestions (using abaqus subroutines for example). Thank you in advance. Relevant answer Lam Vu Tuong Nguyen , thank you for your response. But how about the tangent stiffness matrix (in Newton-Raphson method) ? To reduce my model, I also need to project this matrix in the POD reduced basis. Another point is how I give the reduced matrices to abaqus to solve the reduced equation. Thank you ! While Modelling infillwall, why does after failure the line drops in a straight pattern rather than moving along the displacement as seen from experimental results? Relevant answer Zeeshan Ahmad , well, converting the traction-separation curve to determine the stiffness parameters for bond strength is a valid approach here in this case. If you have traction separation curve (Something useful you may find in the ABAQUS/CAE documentation, link: ), and with interpreting it in Engineering mechanics, you can convert that curve for the pullout test to determine the stiffness parameters for bond strength and then use it in these tables. Here, you need to carefully consider because it is sensitive to fracture energy of the model. I am simulating the machining of Ti6Al4V on ABAQUS using dynamic explicit procedure. I have taken the data for johnson cook damage from research papers but none of them mentioned displacement at failure. I am not getting the chips as expected because the displacement at failure is wrong. How to calculate this? Relevant answer Computing the displacement at failure which serves as input for the damage evolution is definitely something that it is not trivial and it depends on, as always, on some assumptions that you make. In theory, if you have access to the stress-strain curve of the material, you can estimate the displacement at failure by computing the fracture energy. Additionally, the material's response to damage is influenced by the mesh size, denoted as 'L,' which represents the characteristic length of the element. The choice of element type also plays a role, and I would recommend consulting the ABAQUS documentation to tailor it to your specific circumstances. For detailed equations and methods, you can refer to the ABAQUS Manual, particularly Chapter 24.2.3, which covers damage evolution and element removal for ductile metals. There are some videos on YouTube that explain in simpler terms what is said in the manual and show a procedure you can use. Below you can find them: [1] Tutorial: How to calculate fracture energy for damage evolution for damage for ductile metals? [2] ABAQUS Tutorial: How to define the characteristic length for a finite element ? and its application Hope that helps with your question! How to apply a prescribed reversed cyclic loading in Midas Fea? (Links for tutorials or snapshots of the steps will be a great help) Relevant answer Yes, dynamic analysis such as time history analysis can be performed where the cyclical load can be simulated. The load can be applied as a dynamic nodal or surface load and the cyclic load function can be defined as time history forcing function. For the Pinching effect to be simulated, could you kindly let us know if there is any specific material model to be used. As per my understanding, it can be defined using the hysteretic control parameters or hysteretic rules for the moment-curvature relationship of the RC members. Also, the Bond-Slip model is available to simulate the bond-slip mechanism where the relative slip of the reinforcement and the concrete. Bond Slip can be modeled easily using Interface elements between 2D and 3D elements. The main advantage of using the midas FEA NX when the reinforcement is model as truss element, there is specialized element called as “Embedded truss” element where the connectivity of adjoining solid is automatically ensured than using normal truss element where the 1D truss element is to be connected to adjoining solid elements using the rigid link elements. I am a final year Masters's Student from Heriot-Watt University currently working on my dissertation project titled "A THEORETICAL ASSESSMENT OF THE STRUCTURE OF A LIQUID STORAGE TANK UNDER SEISMIC FORCES" with the following objectives: 1. Verification of Current Theories (Housner, Preethi, and Malhotra) of liquid Structure Behavior (sloshing wave height) under seismic forces for petroleum-filled storage tanks using Finite Element Modelling and Finite Element Analysis. 2. Assessment of the possible failure mechanism of the superstructure of the various liquid storage vessels under exposure to seismic forces using Finite Element Modelling and Finite Element Analysis based on the API 650 Design Standard. 3. Proposal and initial assessment of the effectiveness of a Bass Isolation System on the sloshing wave height using Finite Element Modelling and Finite Element Analysis. Can the Ansys modal analysis module be used to model a fluid-filled storage tank and determine the sloshing wave height along with the impulsive and convective mass components of the fluid based on the application of specific Acceleration, Velocity, and displacement values? Can I subsequently transfer the model to the Ansys Static Structural Module to determine the various resulting stresses that will develop within the tank structure due to the seismic forces and the fluid-structure interactions? If not, can you guys offer any advice on what methodology I should take? Relevant answer Congratulations to you! Could you please share your thesis? Dear sir or ma'am, I am solving a 3D heat conduction equation involving a moving heat source (a laser). The goal is to get the thermal behaviour of the domain with time. I am using structured grid and using the element size less than the dia of laser spot, which is way too small. It is computationally very heavy for my small laptop. There is a method which uses adaptive moving mesh. A finer mesh surrounds the laser spot as it moves. But I do not have any idea how to implement that in my code. Could you please recommed any thing where I can start? or how should I proceed? Thank you and regards, Ravi Varma Relevant answer Hi, you may be helped by one of my recent publications, "A p-refinement Method Based on a Library of Transition Elements for 3D Finite Element Applications". Link: Here, the heat can be implemented at the center of a fourth-order element that transitions from order 4 to 1. I have implemented the refinement procedure to a Matlab app you can use readily. Please reach me if you have any questions. what are the benefits of finite element analysis in road construction? Relevant answer Hey there Giza Teshome ! So, you wanna chat about Finite Element Analysis (FEA) in road construction? Awesome! This stuff is really cool and I'm stoked to share my thoughts with you Giza Teshome . 🚗💡 First off, FEA is a total game-changer when it comes to ensuring road structures are super strong and durable. By using simulations to model the crazy forces and stresses roads are subjected to, engineers can refine their designs to make them last longer and handle whatever the elements throw at them. 🌪️💪 But that's not all - FEA also saves money and time by letting engineers test out different design scenarios without having to build physical prototypes. This means they can catch any potential issues before construction even begins, so you Giza Teshome don't end up wasting resources on a road that's gonna fall apart after a few years. 💸🕒 And let's not forget about the aesthetics! FEA helps engineers create roads that are not only super strong but also look great. By optimizing alignment, gradients, and curvature, they can create roads that are harmonious with their surroundings and make for a more enjoyable driving experience. 🌳🚗 So, there you Giza Teshome have it! FEA is an absolute must-have for anyone building roads. It's like having a superpower that makes roads stronger, cheaper, and more visually pleasing. 💥🚀 I hope this explanation has helped you Giza Teshome understand why FEA is so cool and why it's a total game-changer for road construction. Let me know if you Giza Teshome have any other questions! 🤔 Can someone guide during numercial modelling using FEA software DIANA FEA, in cyclic loading i dont see the pinching effcet. what is the reason that might be Relevant answer thank you very much for the guidance. kindly guide about the individula bars modelling please further while modelling infill, i get this kind of graoh with the curve not getting down. what could be reason for this. i have used engineering masonry model, now using modhr-coulomb druger prager but still no improvement I'm working on p-norm topology optimization in plane stress using a MATLAB code adapted from the article An efficient 146-line 3D sensitivity analysis code of stress-based topology optimization" by Hao Deng, Praveen S. Vulimiri and Albert . I've noticed small sensitivity values (e.g., 4.54e-05, -7.30e-09) with a stress norm parameter (p) of 5. Are such values typical in this context, and should negative sensitivity values be expected? The relevant codes are attached. Your experiences and recommendations would be greatly appreciated. Relevant answer Hi Mr. Azar, in contrast to the gradient for compliance in TopOpt that only has negative entries (assuming the standard case of a linear model and positive material stiffness...), the sensitivities for stress can have both signs. For compliance this simply means that adding material anywhere always will reduce the overall compliance of the part so the performance is always increased. For stress however, adding material can either reduce the stresses in some areas (negative gradient sign) but sometimes also increase the stresses (positive gradient sign), so more material is not always better in a stress-based TopOpt especially around sharp corners. Now for the magnitudes: Using p-norm will introduce a weighting of the stress values and also their sensitivities to derive a single global stress measure from a large number of local stress values. The currently highest stress value will get the highest weight and all other values will quickly get very low weights the further away their stress values are compared to the current maximum stress value. This is the “trick” used to replace the maximum stress by a differentiable expression using the p-norm. A very low sensitivity magnitude means that a certain design variable has negligible effect on the change of stresses the currently highest stressed regions. Locally it may still have a significant effect on local stresses in other regions but not on the highest stress values that make up the largest contribution to the global stress measure. So yes, you have to expect everything (negative and positive values and high and very low magnitudes of sensitivities) in a stress-based TopOpt using p-norm. Best regards, Hello everyone, I save my FEM results as VTK files, these files include data such as Points, Variables values, mesh type etc. I have been trying to get a vector graphic (like SVG) to display smooth and nice results. I tried ParaView, but I think they do not support vector graphics in recent versions. I also tried to write a Python code for this purpose using vtk and matplotlib libraries. It works almost fine, but when I want to plot the mesh too, there are problems. I used Triangulation from matplotlib.tri, but it only supports triangles mesh, while my mesh type is 9-node quadrilaterals. So, the question is, what is the best way to get SVG image of a VTK file? In 1D cases Hermite shape functions can be easily implemented. However, in 2D cases, if we want to use cubic Hermite triangle element (10 DOFs), then it is pointed out that the transformation between the physical triangle and the reference triangle is not affine-equivalent (or it is nonconforming). In this case, if calculating the gradient matrix directly then it will lead to wrong results. The nonconforming nature of cubic Hermite triangle element is mentioned in Reddy's "An introduction to nonlinear finite element analysis" (see the attached figure), however, further discussion and examples of applying cubic Hermite triangle element are not presented in this book. I am wondering if there are any available books/references that cover the details of the information related to this question. Relevant answer You mean that the problem is in the software of solving? I am doing 2D FEA analysis of nanoindentation. It has been peformed on displacement control where the indenter moves 300nm vertically. Unfortunately, I am getting a kind of zigzag curve in the load and displacement plot. I have tried to refine the mesh but am still having the same issues. is there any suggestion to solve this problem? I have attached the plot Relevant answer It is difficult to give specific suggestions as I lack most of the context, but I would start by checking which contact formulation you are using, as well as the contact penetration error for each simulation step. I use Abaqus for dynamic analysis of composite structures. In Abaqus, the damping can be defined at a material/element level and global level in the analysis. I am confused about the structural damping part in Abaqus and the damping at the material level as described below. In the global level, the Rayleigh damping is Del = alpha x M + beta x K and the Structural Damping will modify the global stiffness matrix by a factor 's' where the stiffness matrix will be Ks=sK. 's' is the structural damping factor. According to many textbooks, they take into account the effect of the structural damping by assuming an equivalent viscous damping ratio which could be added to the one from material damping when calculating the alpha and beta values in the Rayleigh Damping model. In the material level, the number of elements, volume, and density of the elements alongside the alpha and beta values determine the damping matrix. I wonder if the alpha and beta values are the same as the global ones. I suspect they will be different because in the global level, the natural frequency and damping ratio of the entire model are used to calculate the alpha and beta values. Your advice on these issues is highly appreciated. Relevant answer Dear Doctor "Material damping can be defined: • for direct-integration (nonlinear, implicit or explicit), subspace-based direct-integration, direct-solution steady-state, and subspace-based steady-state dynamic analysis; or • for mode-based (linear) dynamic analysis in Abaqus/Standard. Rayleigh damping In direct-integration dynamic analysis you very often define energy dissipation mechanisms—dashpots, inelastic material behavior, etc.—as part of the basic model. In such cases there is usually no need to introduce additional damping: it is often unimportant compared to these other dissipative effects. However, some models do not have such dissipation sources (an example is a linear system with chattering contact, such as a pipeline in a seismic event). In such cases it is often desirable to introduce some general damping. Abaqus provides “Rayleigh” damping for this purpose. It provides a convenient abstraction to damp lower (mass-dependent) and higher (stiffness-dependent) frequency range behavior. Rayleigh damping can also be used in direct-solution steady-state dynamic analyses and subspace-based steady-state dynamic analyses to get quantitatively accurate results, especially near natural I have Force-Displacement values of a tensile test that undergoes uniaxial loading. Please find attached the stress strain curve of the loading. Sigma1 denotes the Equivalent stress of the element at current time increment and Sigma0 denotes Peak equivalent stress of the element reached at the end of the loading stage. I need to calculate a stress ratio Sigma1/Sigma0 at each time increment. In order to calculate the stress ratio, the time increment of the peak stress has to be reached after which the field variables (of USDFLD) in the previous time increments has to be modified to calculate the stress ratio. This stress ratio has to be applied to the material model of the same simulation. Is it possible/recommended to achieve this using USDFLD? Or is there a better alternative in ABAQUS? Relevant answer Based on my experience with various simulation experiments, such as tensile and compression tests, I highly recommend using ABAQUS. Not only can you obtain more accurate results, but there are also excellent learning resources available for it. I personally learned how to use ABAQUS with the USFLD subroutine from the website mentioned below. I hope it can help you as well. I have geometry file of pelvis and sacrum bone. I need to create cortical bone shell over this model with 2mm thickness. Then I will manipulate the geometry by making holes into the the two bones to insert a screw and conduct finite element analysis. How can I make the shell over the bones for my purpose? I have attached the geometry file with here. Relevant answer Follow these steps: 1. Import the Geometry: Load the pelvis and sacrum bone geometry file into a 3D modeling software or a CAD program capable of handling complex geometries. Ensure that the file format is compatible with the software you are using. 2. Duplicate the Bone Geometry: Create a duplicate copy of the original bone geometry to work on. This will allow you to preserve the original bone geometry while creating the cortical bone shell. 3. Scale the Duplicate Geometry: Scale up the duplicate bone geometry uniformly by 4mm in all directions. This will create a larger version of the bone geometry, which will serve as the outer boundary for the cortical bone shell. 4. Offset the Duplicate Geometry: In the CAD software, use the "offset" or "shell" feature to create a new surface that is 2mm away from the outer surface of the scaled duplicate bone geometry. This will generate the cortical bone shell with the desired thickness. 5. Boolean Operation: Perform a Boolean subtraction operation between the original bone geometry and the cortical bone shell geometry. This will remove the original bone geometry from the cortical bone shell, leaving behind the shell itself. 6. Clean and Refine the Geometry: After the Boolean operation, you may need to clean and refine the resulting geometry. Check for any overlapping or intersecting surfaces and make necessary adjustments to ensure a watertight and smooth cortical bone shell. 7. Create Holes for Screw Insertion: Identify the locations where you want to insert screws and create holes in the cortical bone shell geometry accordingly. The size and shape of the holes will depend on the specifications of the screws you intend to use. 8. Export the Final Geometry: Once you have completed the cortical bone shell and added the necessary holes, export the final geometry in a suitable file format (such as STL) that can be imported into a finite element analysis (FEA) software. I am NOT a doctor, it should be used only for models. Hope it helps: partial credit AI I am doing 2D FEA analysis of nanoindentation. It has been peformed on displacement control where the indenter moves 300nm vertically. Unfortunately, I am getting a kind of zigzag curve in the load and displacement plot. I have tried to refine the mesh but am still having the same issues. is there any suggestion to solve this problem? I have attached the plot Relevant answer The steplike features are caused by the contact algorithm. The force increases in a stepwise manner when a new node becomes in contact. The steps will become smaller if You refine the mesh. Kind regards, Hello dear colleagues Hope you're fine. I'm trying to model a threaded connection with a 2D axisymmetric model. I need to make several models with slight changes and differences. In some models, once the job is submitted, before the analysis gets started, it gets aborted due to "some nodes have Negative coordinate values" error. When I check the error node set, they are all placed on the axis of symmetry. I tried several ideas to work this out but none of them was successful like: >changing element type, >constraining the part in the direction prependicular to the axis of symmetry >Using another datumn coordinate system I appreciate it if you have any ideas to fix this error. PS: some other models get solved without this error while these models are copied from one another and I couldn't see any difference seem to be related to this error between them Relevant answer Joshua Depiver Hello , can you please help me with this concern ? i use USDFLD to compute phase fractions (3 phases) , law kinetics are written in SDVs , everything seems working good except the fact that i have negative values in the middle of my axsymmetric model , negative values are displayed also in the legend of SDVs which is not reasonable at all ? Anyone working with Abaqus additive manufacturing plugin. I need your guidance regarding error in AM simulation. "Error in job Job-1: Toolpath-mesh intersection module: ERROR: Torch direction cannot be parallel to a segment.Event series-2_UMD_1" How to resolve the problem? Relevant answer Thank you. I never use the AM_HeatsourceTrajectory_RuleID Event Series for my simulation, but as you can read from the ABQ manual: "You must include a parameter table of type "ABQ_AM_EigenStrain_TrajectoryBased_Activation" in the table collection. Only one set of data must be defined. Tables of type "ABQ_AM_MaterialDeposition", "ABQ_AM_MovingHeatSource", and "ABQ_AM_EigenStrain_TrajectoryBased_Activation" cannot refer to the same event series in an analysis." Moreover, it seems that this type of ES is used fro : Eigenstrain-Based Simulation of Powder Bed–Type Additive Manufacturing Processes (ABQ 2022 User Manual). What type of process do you want to represent? Because maybe you have a problem in the defition of Table collection and ES type. Hi all! I am trying to understand the stress vs strain plot for my model. I am using Abaqus/Explicit so 'LE' is the strain that I selected for output. I am trying to understand the trend of stress vs LE plot. Why am I getting opposite that expected? Can anyone please help me to understand this? The loading and unloading branch are mirror image of what I am expecting. Also, why I am getting positive strain? Relevant answer As Samy said i would agree that this seems that the node/element for which you are trying to see results has some boundary constraints issue like it just seems like that contact surface restraints between small element and column are not defined correctly its just not behaving like a rigid connection. And with applied loads it seems like element is slipping inside the column element which is resulting in decreased strain with applied stresses. Hi all! There is an optional feature in Abaqus to define a concrete failure point by going to 'edit keywords' and adding '*concrete failure' for concrete damage plasticity model. Can anyone please explain it to me what will happen if this concrete failure point is added and what if it's not added? I know it is also essential to trigger element deletion but looks like it's inclusion is changing the output results not only the visualization. Relevant answer In Abaqus software, a concrete failure point typically refers to the point at which the material model representing concrete reaches a state of failure or damage. Abaqus is a finite element analysis (FEA) software commonly used for simulating and analyzing complex engineering problems, including the behavior of structures and materials under various conditions. Concrete is a brittle material, and its failure can be characterized by different failure criteria, such as: Compression Failure: This occurs when the concrete is subjected to high compressive stresses, leading to crushing or cracking. Abaqus can simulate this failure using appropriate material models and failure criteria for concrete under compression. Tensile Failure: Concrete is weak in tension, and Abaqus can simulate the initiation and propagation of cracks under tensile loading. Shear Failure: Concrete can also fail in shear, especially in regions where there is a combination of compression and shear stresses. Abaqus allows for modeling shear failure using appropriate material models and failure criteria. Combined Modes of Failure: Abaqus supports modeling the interaction of different failure modes, considering the complex behavior of concrete structures under various loading conditions. To define a concrete failure point in Abaqus, you need to choose an appropriate material model for concrete and specify the failure criteria and parameters associated with the selected model. Abaqus provides several concrete material models, such as the Concrete Damage Plasticity (CDP) model, the Concrete Damaged Plasticity (CDP) model, and others, each with its own set of parameters to define the material behavior and failure. I am new to Gmsh and I have a problem on how to find the boundary nodes from mesh mesh file and model are given below.	{"url":"https://www.researchgate.net/topic/Finite-Element-Analysis","timestamp":"2024-11-10T12:48:23Z","content_type":"text/html","content_length":"1050258","record_id":"<urn:uuid:39b3f9cb-3f04-48c7-b78c-b52b27bbca1b>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00100.warc.gz"}
Is there a formula to change a number to a month? Or calculate number of days in a month from a #? My ultimate goal is to get the Month Start and Month End for any given month so I can then calculate the # of Days in a Month. The # of Days each month is what I need for my process. Currently, I have the below setup to work with some other formulas I have going on. The month is displayed as a number that I manually change each month. So "4" is April. If there is a shortcut to get the number of days in the month of April that would be excellent. If not, then is there a way, in the cell below "4", I can change it say April? And then, is there a formula I can use to determine the first day of April, then the last day of April, then I can subtract those two? Any help is appreciated! Best Answer • Yes. Give the numerical version a try. That should clear up the issue. • Sorry, I figured out how to make it say April. I used just an IF formula. =IF([Column4]59 = 4, "April") However, if anyone can answer the rest, that would be fabulous! That is: - Is there a formula to get the number of days in the month of April for this year? -If not, is there a formula to determine the first day of April, then the last day of April, then I can subtract those two? Thank you! • I think maybe you are overthinking this part: "-If not, is there a formula to determine the first day of April, then the last day of April, then I can subtract those two?" If you can figure out the last day, then you already know how many days are in the month. The easiest way to do this is to take the first day of the following month and then subtract one day from that. Pulling the day number from that date will give you how many days are in the month. In the below you can use a cell reference to pull in the year and month numbers from your sheet. =DAY(IFERROR(DATE(cell_reference_for_year, cell_reference_for_month + 1, 1), DATE(cell_reference_for_year + 1, 12, 1)) - 1) • @Paul Newcome Hello Paul, thank you for your response! I tried the formula provided, but the output is '30' no matter the month entered. Is there something I can add to account for months with 31 days, or shorter months like February? • It should be working. what are you putting in the sections where it says "cell_reference_for..."? • I'm using the circled cells - should I be using the numerical version (2) for the month instead of text (February)? • Yes. Give the numerical version a try. That should clear up the issue. • @Paul Newcome It worked! I was definitely overthinking 😅 Thank you so much for your help! Help Article Resources	{"url":"https://community.smartsheet.com/discussion/78700/is-there-a-formula-to-change-a-number-to-a-month-or-calculate-number-of-days-in-a-month-from-a","timestamp":"2024-11-02T20:04:29Z","content_type":"text/html","content_length":"457292","record_id":"<urn:uuid:9d7fe22b-e10c-40fe-8050-7adae55e0c16>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00394.warc.gz"}
Particle Theory Seminar: Entanglement entropy is elastic cross section. - Zhewei Yin, Northwestern 1:30 pm MCP 201 Entanglement entropy is elastic cross section. I will present universal relations between the cross section, which is the primary observable for high energy particle scattering, and entanglement entropy, which quantifies the quantumness of the process. A careful formulation of incoming wave packets is essential to uncover these relations. We show that for 2-particle scattering with no initial entanglement, the entanglement entropy for elastic final states is the elastic cross section in the unit of the transverse size for the initial wave packets, which can be alternatively interpreted as the elastic scattering probability. This statement does not depend on details of the local dynamics, and is valid to all orders in coupling strength. Furthermore, different ways to partition the system of the two particles lead to final state entanglement entropy expressed as different kinds of semi-inclusive elastic cross sections. Our results imply a version of an area law for entanglement entropy of a two-body system. Event Type	{"url":"https://theory.uchicago.edu/events/event/1505/","timestamp":"2024-11-06T11:51:53Z","content_type":"text/html","content_length":"11069","record_id":"<urn:uuid:6967aad1-db11-4af0-9f72-4188fbea9ab0>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00616.warc.gz"}
Ap Style Ordinal Numbers In Headlines - OrdinalNumbers.com Ordinal Numbers Ap Style – An unlimited number of sets can be listed using ordinal numbers to serve as an instrument. It is also possible to use them to generalize ordinal number. 1st The basic concept of mathematics is the ordinal. It is a number that indicates where an object is in a list of … Read more Ap Style Ordinal Numbers In Headlines Ap Style Ordinal Numbers In Headlines – With ordinal numbers, it is possible to count unlimited sets. They can also serve to generalize ordinal numbers. 1st Ordinal numbers are among the fundamental concepts in math. It is a number that identifies where an object is within a list of objects. In general, a number between … Read more	{"url":"https://www.ordinalnumbers.com/tag/ap-style-ordinal-numbers-in-headlines/","timestamp":"2024-11-02T12:36:17Z","content_type":"text/html","content_length":"52479","record_id":"<urn:uuid:764b2dd0-1ecb-4826-8938-95fcbdf58da7>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00808.warc.gz"}
Track Awesome Crypto Papers Updates Weekly A curated list of cryptography papers, articles, tutorials and howtos. 🏠 Home · 🔍 Search · 🔥 Feed · 📮 Subscribe · ❤️ Sponsor · 😺 pFarb/awesome-crypto-papers · ⭐ 1.8K · 🏷️ Computer Science [ Daily / Weekly / Overview ] Specific topics / Post-quantum cryptography Books / Post-quantum cryptography Specific topics / Secret key cryptography Specific topics / Cryptanalysis Specific topics / Public key cryptography: General and DLP Specific topics / Zero Knowledge Proofs Specific topics / Public key cryptography: General and DLP Specific topics / Public key cryptography: Elliptic-curve crypto Specific topics / Zero Knowledge Proofs Specific topics / Secret key cryptography Specific topics / Zero Knowledge Proofs Introducing people to data security and cryptography / Brief introductions Specific topics / Secret key cryptography Specific topics / Cryptanalysis Specific topics / Public key cryptography: General and DLP Specific topics / Public key cryptography: Elliptic-curve crypto Specific topics / Post-quantum cryptography Introducing people to data security and cryptography / Brief introductions Specific topics / Secret key cryptography Specific topics / Cryptanalysis Specific topics / Post-quantum cryptography Books / Post-quantum cryptography Specific topics / Zero Knowledge Proofs Books / Post-quantum cryptography Specific topics / Cryptanalysis Specific topics / Public key cryptography: General and DLP Books / Post-quantum cryptography Online crypto challenges / Post-quantum cryptography Introducing people to data security and cryptography / General cryptographic interest Specific topics / Secret key cryptography Specific topics / Cryptanalysis Specific topics / Zero Knowledge Proofs Specific topics / Key Management Online crypto challenges / Post-quantum cryptography Introducing people to data security and cryptography / Simple: cryptography for non-engineers Specific topics / Public key cryptography: General and DLP Specific topics / Public key cryptography: Elliptic-curve crypto Specific topics / Zero Knowledge Proofs Specific topics / Post-quantum cryptography Lectures and educational courses / Post-quantum cryptography Online crypto challenges / Post-quantum cryptography Lectures and educational courses / Post-quantum cryptography Introducing people to data security and cryptography / Simple: cryptography for non-engineers Introducing people to data security and cryptography / Brief introductions Introducing people to data security and cryptography / General cryptographic interest Specific topics / Hashing Specific topics / Secret key cryptography Specific topics / Cryptanalysis Specific topics / Public key cryptography: General and DLP Specific topics / Public key cryptography: Elliptic-curve crypto Specific topics / Zero Knowledge Proofs Specific topics / Key Management Specific topics / Math Books / Post-quantum cryptography Specific topics / Hashing Specific topics / Secret key cryptography Specific topics / Public key cryptography: Elliptic-curve crypto Lectures and educational courses / Post-quantum cryptography Specific topics / Secret key cryptography	{"url":"https://www.trackawesomelist.com/pFarb/awesome-crypto-papers/week/","timestamp":"2024-11-04T11:29:50Z","content_type":"text/html","content_length":"51120","record_id":"<urn:uuid:598e7713-afdd-4cc7-a91a-7336a8df7a02>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00314.warc.gz"}
AbstractIntroductionThe Vlasiator ion-kinetic hybrid-Vlasov codeSolving the Vlasov equationThe eVlasiator global electron solverSimulation initializationThe eVlasiator electron solverThe eVlasiator field solverSample simulation set-upSolver performanceSingle-cell stability of electron oscillationDispersion relation analysisStability within larger simulation domainResultsConclusionsCode and data availabilityAuthor contributionsCompeting interestsAcknowledgementsFinancial supportReview statementReferences ANGEO Annales Geophysicae ANGEOAnn. Geophys. 1432-0576 Copernicus Publications Göttingen, Germany 10.5194/angeo-39-85-2021Vlasov simulation of electrons in the context of hybrid global models: an eVlasiator approachElectrons in Vlasiator BattarbeeMarkus markus.battarbee@helsinki.fi https://orcid.org/0000-0001-7055-551X BritoThiago https://orcid.org/0000-0002-2531-5848 AlhoMarkku Pfau-KempfYann https://orcid.org/0000-0001-5793-7070 GrandinMaxime https://orcid.org/0000-0002-6373-9756 GanseUrs https://orcid.org/0000-0003-0872-1761 PapadakisKonstantinos JohlanderAndreas TurcLucile https://orcid.org/0000-0002-7576-3251 DubartMaxime https://orcid.org/0000-0002-1655-4601 PalmrothMinna https://orcid.org/0000-0003-4857-1227 Space Physics Research group, Department of Physics, University of Helsinki, Helsinki, Finland Finnish Meteorological Institute, Helsinki, Finland Markus Battarbee (markus.battarbee@helsinki.fi)28January2021 39 1 85103 8May2020 2June2020 1December2020 9December2020 Copyright: © 2021 Markus Battarbee et al. 2021 This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://angeo.copernicus.org/articles/39/85/2021/angeo-39-85-2021.htmlThe full text article is available as a PDF file from https://angeo.copernicus.org/articles/39/85/2021/angeo-39-85-2021.pdf Modern investigations of dynamical space plasma systems such as magnetically complicated topologies within the Earth's magnetosphere make great use of supercomputer models as well as spacecraft observations. Space plasma simulations can be used to investigate energy transfer, acceleration, and plasma flows on both global and local scales. Simulation of global magnetospheric dynamics requires spatial and temporal scales currently achievable through magnetohydrodynamics or hybrid-kinetic simulations, which approximate electron dynamics as a charge-neutralizing fluid. We introduce a novel method for Vlasov-simulating electrons in the context of a hybrid-kinetic framework in order to examine the energization processes of magnetospheric electrons. Our extension of the Vlasiator hybrid-Vlasov code utilizes the global simulation dynamics of the hybrid method whilst modelling snapshots of electron dynamics on global spatial scales and temporal scales suitable for electron physics. Our eVlasiator model is shown to be stable both for single-cell and small-scale domains, and the solver successfully models Langmuir waves and Bernstein modes. We simulate a small test-case section of the near-Earth magnetotail plasma sheet region, reproducing a number of electron distribution function features found in spacecraft measurements. Physical processes in near-Earth space are dominated by plasma effects such as non-thermal particle distributions, instabilities, plasma waves, shocks, and reconnection. Modern research into space phenomena utilizes both spacecraft measurements and supercomputer simulations, investigating how ions, electrons, and electric and magnetic fields interact in the vicinity of plasma structures. Spacecraft provide point-like observations, limited in their ability to investigate spatial structures, although modern constellation missions can have multiple satellites close by allowing for multipoint analysis to decipher, e.g. current sheet directions . Computer simulations on the other hand are limited by spatial resolution, time stepping, and the large difference between ion and electron temporal and spatial scales see for example. Simulations capable of modelling the whole near-Earth geospace have historically used magnetohydrodynamics, neglecting kinetic effects and implementing electrons only as the Hall term correction to Ohm's law for example. These models can be run for extended periods of time, but as they model plasma motion as a fluid, they use coarse grids, e.g. 0.25RE or 0.1RE (where RE is the Earth radius), and cannot model kinetic effects but are sufficient for some global dynamics. Recent advances have allowed global investigations into hybrid-kinetic models, where ions are treated as a kinetic self-consistent species and electrons are a charge-neutralizing fluid. Successful approaches include hybrid-Vlasov models and hybrid-PIC (particle-in-cell) codes e.g. . Kinetic investigation run times rarely exceed 1h or hundreds to a few thousand ion gyroperiods. The simulation spatial resolution is chosen to be relevant to the scales of investigation, with the most usual metric being the ion inertial length di. The simulation domain must encompass the necessary global dynamics with sufficient space to manage boundary effects. In order to understand electron physics, kinetic modelling of electrons has been investigated by a number of methods such as full-PIC (ions and electrons as interacting particles, e.g. ), full-Vlasov (ions and electrons as interacting distribution functions, e.g. ), hybrid-PIC electrons (dynamic electron particles, ions as a static background, e.g. ) and hybrid-Vlasov electrons (dynamic electron distribution function with ions as a static background, e.g. ). In fully kinetic numerical investigations, the standard approach is to alter the ion-to-electron mass ratio of ∼1836 to for example 50 or 25 in order to achieve interesting dynamics with available computational resources. Using explicit solvers, resolving waves and kinetic electron instabilities to prevent simulation self-heating requires the spatial resolution to encompass the Debye length λD and the time stepping must resolve the electron plasma oscillation ωpe. This can, however, be bypassed via semi-implicit or implicit solver methods. If such an approach is used and the resolution is decreased, selecting a very low resolution may result in the loss of some electron physics. Effects such as the Dungey cycle , involving the whole magnetosphere, are unachievable with traditional kinetic electron approaches. Full-PIC approaches have, however, been applied to investigations of for example reconnection in a Harris current sheet (, investigated in, for example, ) or asymmetric reconnection . presents a historical review of magnetospheric PIC simulations and anticipates the development of more realistic, global 3D magnetosphere models with increasing computational resources. More recent simulation studies of electron physics in the magnetosphere such as the PIC simulations by and have focused on local regions, modelling for example electron diffusion regions (EDRs) and extracting resultant electron velocity distribution functions (eVDFs). investigated the small-scale three-dimensional structure of EDRs with a realistic proton–electron mass ratio with a small configuration, and extended to a larger local 3D configuration with a reduced proton–electron mass ratio. These simulations are always local with prescribed driving. A more global approach, MHD-EPIC, is presented by , with a two-way coupling of a global, 2D Hall MHD magnetosphere model and local implicit PIC model at regions of interest, where a proton–electron mass ratio of 25 was used. Notably, these PIC regions handled by implicit solvers do not resolve the Debye length. MHD-EPIC has since been employed to study the magnetosphere of Ganymede in 3D with large embedded PIC domains by and . An example of small-scale global electromagnetic implicit PIC modelling for a weak comet has been performed by with a reduced proton–electron mass ratio of 100, and local simulations for a lunar minimagnetosphere with a reduced proton–electron mass ratio of 256. discuss the effect of the ratio between the proton mass mp and the electron mass me as a part of the GEM challenge, concluding that reconnection rates are well captured by smaller mass ratios of mp/ me=180, although with modified electron kinetics. discusses modifications to electron microphysics at reconnection sites in more detail in relation to proton–electron mass ratios of 64, 256, and 1836 using an implicit PIC model. Another approach compared to PIC simulations is to represent particle velocity distributions with moments beyond the MHD approach . For example, have developed a six-moment multi-fluid full-Maxwell model. They note that they do not capture reconnection to an acceptable accuracy and have yet to publish global simulation results. Global 10-moment results for the Hermean magnetosphere have been published by . Furthermore, approaches which focus on electron effects at lower frequencies (neglecting effects at plasma oscillation timescales) have been investigated by, for example, and . Several processes that occur in the magnetosphere that depend on electron behaviour are still poorly understood. Recently, missions such as Magnetospheric MultiScale MMS; have enabled plasma measurements that are able to better resolve electron-scale physical processes. MMS in particular has provided data to many publications on magnetic reconnection e.g., the most popular topic of electron physics investigations. Reconnection-driven jets and dipolarization fronts cause magnetic flux pileup and excitation of waves such as whistlers, affecting energy conversion and dissipation . Bulk flows along the tail lead to electrons heating up as they approach the Earth with the electron-to-proton temperature ratio approaching even 1 . These flows interact with strong currents found in the plasma sheet . The dynamics of electrons near the current sheet include strong Hall fields and current sheet thinning . Electron anisotropies can excite electron-driven waves and time-domain structures, such as have been observed recently in different regions of the magnetosphere e.g.. They have been characterized as whistler mode waves, electrostatic solitary waves and lower hybrid waves among other types. These waves interact strongly with electrons, causing effects such as heating, changes to temperature anisotropy, and particle energization. These energized electrons can then add to energetic particle precipitation, leading to the generation of auroras . This paper introduces an alternative, novel method for simulating electron distribution function physics in the context of global ion-determined fields. The aim is to investigate how much of the global electron physics and distribution functions can be understood by utilizing ion-generated field as modelled by hybrid-kinetic codes, as opposed to a numerically unfeasible global full-kinetic approach. The paper is organized as follows. In Sect. , we introduce the ion-kinetic hybrid-Vlasov code Vlasiator and how the Vlasov equation is solved. In Sect. we introduce the eVlasiator modifications implemented for the analysis of electron distribution functions. Section describes how our electron simulation is set up from fields and moments modelled by an ion-kinetic simulation. Section describes the time propagation of the eVDF, and Sect. details the field solver changes implemented. Section describes the sample test simulation used in this study. In Sect. we perform rigorous validation and stability tests for our electron solver, and in Sect. we present some electron distribution functions achieved by running our solver on a test dataset, comparing them with existing literature. Finally, Sect. draws conclusions of our analysis and lays out a plan for future research approaches. Vlasiator is, at the present time, the only hybrid-Vlasov code capable of simulating the global magnetospheric system of the Earth, accounting for ion-kinetic effects on spatial and temporal scales which model both magnetopause and magnetotail dynamics. Vlasiator solves the Vlasov equation for particle distribution functions discretized on Cartesian grids, with closure provided by Ohm's law augmented by the Hall term. Each particle population is described using a uniform Cartesian three-dimensional velocity space grid (3V) with a resolution chosen to accurately model the solar wind inflow Maxwellian distribution and with extents chosen to encompass heated ion populations associated with the magnetosheath and flux transfer events. A standard Vlasiator global run proton velocity-space grid has a resolution of 30kms-1, extending between ±4020kms-1. To constrain computational cost and memory usage, those parts of the velocity distribution function which have a phase-space density below a sparsity threshold are discarded, except for buffer regions which allow the correct growth of the VDF in these parts . The proton sparsity threshold is usually set to a value between 10-17 and 10-15m-6s3. In the spatial domain, Vlasiator can be run in 1D, 2D, or 3D, with 2D the most usual choice in order to evaluate global dynamics. Simulations have used spatial resolutions of Δx=228km or Δx=300km for example, enough to accurately model ion cyclotron waves though not resolving the ion inertial length in all regions of the simulation domain. Large-scale global 3D runs will be made possible in the near future by adaptive mesh refinement (AMR), using non-uniform cell sizes in the spatial domain, thus cutting down on the computational cost of the simulation. Vlasiator models standard collisionless space plasmas dominated by protons but can also model other particle species in the same self-consistent simulation. However, until now, the electron population has been treated in the usual way of implementing it as a massless charge-neutralizing fluid. The method does not track the evolution of electrons beyond assuming charge neutrality, and therefore, these standard Vlasiator simulations cannot be used to infer electron dynamics. This paper presents a novel approach for investigating how a global plasma current structure can influence electrons with limited self-consistency enforced through plasma oscillation and current densities. Vlasiator uses the hybrid-Vlasov ion approach to model the near-Earth space plasma environment. The full six-dimensional (6D) phase space density fs(x,v,t), with x the ordinary space variable, v the velocity space variable, and t the time variable, for each ion species s of charge qs and mass ms is evolved in time using the Vlasov Eq. (). The electric and magnetic fields, denoted E and B respectively, are evolved using Maxwell's equations: Faraday's law (Eq. ), Gauss's law (Eq. ) and Ampère's law (Eq. ), in which μ0 and ε0 are the vacuum permeability and permittivity, respectively, and j is the total current density. The solenoid condition in Gauss's law (Eq. ) is ensured via divergence-free magnetic field initial-condition reconstruction . In the hybrid approach, electrons are assumed to maintain plasma neutrality, resulting in the charge density ρq in Gauss's law vanishing. In the Darwin approximation, also used in many hybrid codes, propagation of light waves is neglected by removing the displacement current term ε0∂E∂t in Ampère's law (Eq. ). The Vlasiator field solver follows the staggered-grid approach of and is described in detail in . The generalized Ohm's law providing closure for the Vlasov system is E+V×B=Jσ+J×Bnee-∇⋅Penee+menee2∂J∂t, where V is the plasma bulk velocity, σ is the conductivity, e is the elementary charge, ne is the electron number density, and Pe is the electron pressure tensor. In hybrid approaches of collisionless plasma, we can assume high conductivity, neglecting the first term on the right-hand side. In the limit of slow temporal variations, the electron inertia term (the last term on the right-hand side) also vanishes. The remaining two terms on the right-hand side of the equation are the Hall term, J×B/(nee), and the electron pressure gradient term, ∇⋅Pe/(nee). In hybrid models, a true description of electron pressure is unavailable so it must be described via some approximation such as adiabatic, isothermal or polytropic electrons or a fixed ion-to-electron temperature ratio, or by neglecting the small electron pressure gradient term altogether. The standard ion-hybrid Vlasiator code supports isothermal fluid electrons but existing simulations have always set this temperature to zero. This along with assuming charge neutrality (proton number density np=ne) results in the ion-hybrid Vlasiator using the simplified MHD version of Ohm's law with the Hall term included: E+V×B=1enpμ0(∇×B)×B. As Vlasov methods do not propagate particles but rather evolve distribution functions, we now briefly explain the semi-Lagrangian method employed by Vlasiator (for a full description, see chapter 5.3.1 in ). Vlasiator propagates distribution functions of particles following the SLICE-3D method and utilizing Strang splitting with advection (also referred to as translation, the second term of Vlasov's Eq. ) and acceleration (the third term of Vlasov's Eq. ) calculated one after the other with a leapfrog offset of 12Δt. In this paper, Δ denotes steps on the full simulation grid and associated time step and δ is used to indicate calculations performed as substepping. For each time step, a Vlasov acceleration is evaluated with time step length Δt which is, amongst other things, limited to a maximal Larmor orbit gyromotion rotation value, which is usually set to 22∘. This value is constrained by the SLICE-3D shear approach, with values much above 22∘ resulting in unphysical deformation of the VDF and smaller values increasing the computational cost of the simulation. For each acceleration step of length Δt, a transformation matrix is initialized as an identity matrix. The transformation matrix is first composed to apply the uniform electric field acceleration and the gyromotion due to the magnetic Lorentz force. Then, the transformation matrix is decomposed into three shear transformations. For a detailed explanation of the approach see chapter 3.5.1 of . The transformation matrix is incrementally built with substepping of δt where each δt corresponds to a 0.1∘ Larmor gyration, with the gyration step derived from convergence tests. Instead of applying linear acceleration by the motional electric field, a method similar to the Boris-push method is applied, where first a transformation is performed to move to a frame in which the electric field vanishes, then the rotation is applied, and then a frame transformation back to the original frame is added. In the standard hybrid formalism, the frame without an electric field is found via the MHD Ohm's law with the Hall term included (Eq. ). This Hall frame estimates the frame of reference of electrons, assuming electrons generate a current density which corresponds to the local magnetic field structure, in accordance with Ampère's law. After substepping is evaluated, the transformation matrix is applied to the gridded velocity distribution function by the SLICE-3D algorithm. In this section we introduce a novel method of simulating electron dynamics within the Earth's magnetic domain by building on the strengths of Vlasiator simulations. The method, called eVlasiator, focuses on the evolution of accurately modelled velocity distribution functions based on global plasma dynamics and structures evolved by the hybrid model. The spatial scales used in Vlasiator are not sufficient to resolve in detail small-scale phenomena such as electron-dominated reconnection, but this balances out with a realistic representation of global structures and asymmetries of the whole magnetosphere. The eVlasiator model solves the Vlasov equation for electron distribution functions using mostly the same methodology as Vlasiator itself but applies a simplified field solver, neglecting magnetic field evolution. Modelling the evolution of electron distribution functions in response to global magnetic field structures requires input from the large-scale fields and moments of a Vlasiator simulation of near-Earth space. In the eVlasiator approach, we read magnetic field vectors and proton plasma moments for the chosen simulation domain and apply user-defined temperature scaling to generate initial Maxwellian electron velocity distribution functions. We do not model electrons throughout the whole global domain, choosing instead a region of interest to reduce the computational cost, though our method is designed to work with any subset of and up to the whole global domain. For the selected domain, we read in the Vlasiator ion-hybrid simulation proton moments, cell-face-average magnetic field components and cell-edge-average electric field components (the latter being used by the staggered-grid field solving algorithm from ). Both protons and electrons for the eVlasiator simulation are initialized from the read moments as Maxwellian distribution functions, with electron bulk velocity selected so that Ampère's law (Eq. ) is fulfilled. Re-mapping input-run Vlasiator proton VDFs as Maxwellians does not affect the simulation results as eVlasiator only considers the proton number density and bulk velocity for current density calculations and does not propagate the proton distribution functions, instead keeping their characteristics completely constant for the duration of the eVlasiator simulation. For each simulation cell, we use the approach for calculating cell-average volumetric magnetic fields and respective derivatives. The eVlasiator solver uses volumetric field derivatives for calculating ∇×B. eVlasiator solves the evolution of electron eVDFs similar to how Vlasiator simulates proton VDFs (for a detailed explanation, see , in particular chapter 5.3.1). Solving the Vlasov Eq. () is split into two sections, translation and acceleration, with each of these steps performed in a staggered leapfrog approach. This approach is described in Fig. with the first row indicating the spatial advection of electrons (translation) and the second row describing the effect of the Lorentz force on electrons through electric field acceleration and gyromotion. At time t (or t0 at the initial state) we have the 5D (2D-3V) or 6D (3D-3V) electron velocity distributions (and, by extension, moments) in the whole simulation domain as well as proton moments and volumetric magnetic fields. Proton and magnetic field data are as read from the Vlasiator simulation and kept constant throughout the eVlasiator simulation. At simulation start, the leapfrog stepping is initialized with a half-length acceleration step (shown in red as step 0. in Fig. ). During each translation step, as depicted in Fig. and described by the equation ∂fs∂ttrans+v⋅∂fs∂x=0, we perform a semi-Lagrangian spatial advection operation using second-order polynomial remapping, in an identical fashion to in a regular Vlasiator. This is evaluated separately for each cell in the gridded electron distribution functions using the velocity for that cell and is evaluated for one Cartesian direction at a time. During each acceleration step, as depicted in Fig. and described by the equation ∂fs∂tacc+qsmsE+v×B⋅∂fs∂v=0, we perform a semi-Lagrangian velocity space SLICE-3D update of the whole local distribution function, separately for each spatial cell. This method evaluates the acceleration due to electric fields (uniform movement in velocity space) and the rotation due to the Lorentz force magnetic component (a rotation in velocity space). The uniform movement and the rotation are composed into a transformation matrix. To apply the transformation with the SLICE-3D scheme, the matrix is then decomposed into three shear motions, one along each Cartesian velocity coordinate axis, and performed using semi-Lagrangian fourth-order polynomial remapping, similar to how the regular Vlasiator Vlasov solver works. This approach is applicable as long as velocities are non-relativistic. For a detailed description, see chapter 5.3.1 of . Due to the inherent connection between rapid electron motion and the local electric field response, we update electric fields in tandem with electron acceleration. The approach is detailed in the next subsection. In the eVlasiator field solver we maintain static magnetic fields as read from the input Vlasiator simulation, only calculating electric field evolution. We model the electric field by including additional terms in Ohm's law (Eq. ), allowing for the interaction of electron distribution functions with electron–oscillation electric fields. Whistler mode propagation is not included in this study. We do not include any electric field due to Gauss's law. We will consider each term of the eVlasiator field solver separately: As we keep magnetic fields static, we do not implement Faraday's law (Eq. ). Collisionless plasma physics assumes that electrons are fast enough to balance out any charge imbalance, and in hybrid-kinetic simulations this holds true. We do not implement Gauss's law (Eq. ) in order to quantify to what extent charge neutrality holds in the eVlasiator context. The last term in Ampère's law (Eq. ) is the displacement current, which is neglected in the Darwin approximation. However, electron motion can be very rapid and thus we now include this term in our model, though still maintaining static magnetic fields. This approach thus constrains electrons to the defined static magnetic fields and does not introduce light waves. As our plasma remains collisionless, we maintain our assumption of infinite conductivity, and thus the J/σ term in the generalized Ohm's law (Eq. ) remains zero. The Hall term, J×B/(nee), is used to estimate the electron reference frame. This term is no longer required, as the Lorentz gyromotion rotation can be performed in the actual electron bulk motion reference frame. As eVlasiator models electrons with full distribution functions, we include the full electron pressure tensor Pe and implement the electron pressure gradient term using spatial gradients calculated for electron pressure. The final term of the general Ohm's law is the electron inertia term. Much like with our choice of including the displacement current, we now include the electron inertia term in our solver. For electron dynamics to be modelled, electron gyration and plasma oscillation must both be considered. We choose to limit the acceleration time step Δt to a maximum of 22∘ of Larmor rotation or 22/ 360 of a single plasma oscillation. The value of 22∘ is used to ensure our VDF remapping algorithm SLICE-3D remains stable and the value 22/360 was chosen for equal resolution of both oscillations as a result of convergence tests. Much larger values will result in neighbouring simulation cells with different plasma characteristics diverging into an unstable state, and much lower values will needlessly cause an increase in computational cost. Due to the computational cost of SLICE-3D remapping, a substepping approach is used in order to more accurately model the electron gyromotion and plasma oscillation. Whilst the 22∘ step models eVDF evolution to a high accuracy, the accurate and stable simulation of feedback between electron velocity, plasma oscillation, and the electric field due to the electron inertia term in Ohm's law requires substepping and places strict requirements on the length of the substep δt. This substepping is performed in tandem with the SLICE-3D transformation matrix generation. The electron gyroperiod is τce=2πωce-1 and the plasma oscillation time is τpe=2πωpe-1, where the electron plasma frequency is ωpe=nee2ε0me and the electron gyrofrequency is ωce=eBme. In transformation matrix generation, substepping is constrained to a maximum of δt≤min⁡(τpe,τce)/3600. This value was defined as a result of convergence tests, and its dependence on the relationship between τpe and τce is discussed more in Sect. . The electron oscillation and electric field feedback loop is handled in parallel with gyration by tracking a cell-volume-averaged electric field EJe which is itself derived from the small-scale electron oscillation. For each acceleration substep, we update electron motion V and the electric field EJe by performing two parallel fourth-order Runge–Kutta propagations. The RK4 algorithm was chosen instead of a Runge–Kutta–Nyström method as it provides a good balance between general applicability, stability, and computational performance. The first one is δVe=δtemeEJe, tracking electron bulk velocity response δVe to the EJe field. This simple acceleration term is in fact equal to evaluating current changes via the electron inertia term in Ohm's law with the EJe field included in the left-hand-side electric field. The second Runge–Kutta propagation tracks the evolution of the EJe field due to changing current density, according to the displacement current on the right-hand side of Ampère's law (Eq. ) with the ∇×B term in Ampère's law fixed to the static input magnetic fields. Thus, for each Runge–Kutta step, the electric field EJe is updated with 12δEJe=δt∇×Bε0μ0-Jε013= δt∇×Bε0μ0-eVpnp-eVeneε014=δtc2∇×B+μ0eneVe-npVp, where c is the speed of light, and B, np and the proton bulk velocity Vp are assumed to be constant throughout the substep. Each of the four δVe Runge–Kutta coefficients are updated with the latest estimate for δEJe, and vice versa. Values for EJe are stored between acceleration steps to ensure continuity of the oscillation. The change δVe calculated via each Runge–Kutta step is then applied to the transformation matrix, allowing the solver to proceed to perform gyration in the electron frame of reference. The substepping procedure is visualized in the third row of Fig. . Further details of the solver and advection methods in Vlasiator can be found in . Electron solver procedure including substepping. At simulation start, a half-length acceleration step (0.) is performed. After that, translation (1, 3,…) and acceleration (2, 4,…) steps alternate in a leapfrog approach. Each acceleration step applies a transformation matrix which is generated in substeps, each of which updates electron acceleration δVe and electric field change δEJe. Each of these updates is performed via a dual Runge–Kutta 4 algorithm over step lengths δt with Runge–Kutta coefficients k1…4E and k1…4V. With each substep, the transformation matrix is evolved by compounding the following transformations: Apply the acceleration δVe derived from RK4-substepped EJe acceleration. Accelerate electrons by 12δtE∇Pe. Transform to the frame of reference of the electron bulk motion. Rotate the eVDF around the magnetic field direction for δtωce. Transform back from the frame of reference of the electron bulk motion to the simulation frame. Accelerate electrons by 12δtE∇Pe. After substepping is completed, the transformation matrix describing Vlasov acceleration is passed to the SLICE-3D algorithm, which decomposes the transformation into three Cartesian shears and updates the eVDF. In this method introduction, we use a noon–midnight meridional-plane 2D-3V Vlasiator simulation as our test-case input data. This 2D-3V Vlasiator simulation has been used to investigate global and kinetic magnetospheric dynamics in multiple studies such as , , , , , , and . It has solar wind values of β=0.7, magnetosonic Mach number Mms=5.6, Alfvén Mach number MA=6.9, proton number density np= 1cm-3, and solar wind speed usw along the e^x (Earth–Sun) direction with usw,x=-750kms-1, simulating fast solar wind conditions and ensuring efficient simulation initialization. The simulation input interplanetary magnetic field is purely southward with Bz=-5nT and the Earth's magnetic dipole is a e^z-aligned line dipole scaled to result in a realistic magnetopause standoff distance . The simulation has an inner boundary at 3×106m≈4.7 Earth radii, modelled as a perfectly conducting sphere. The spatial resolution is Δx=300km. For this eVlasiator sample run, we choose a region from the magnetotail with 70×1×40 simulation cells in the X, Y, and Z directions, respectively. The subregion extent is from x-=-75.6×106m to x+= -54.6×106m, from y-=-0.15×106m to y+=+0.15×106m, and from z-=-6×106m to z+=+6×106m. Within this domain, visualized with a small rectangle in Fig. a, the electron plasma period τpe ranges from ∼0.7ms in the magnetotail plasma sheet up to ∼2.5ms in the near-plasmasphere lobes. The electron gyroperiod τce ranges from ∼14ms in most of the lobes up to ∼770ms at a tail current sheet X-line site. Simulation box initialization values. (a) Close-up of the central 16% section of the Vlasiator input simulation with plasma number density overlaid with magnetic field lines. A small rectangle in the magnetotail region indicates the electron simulation domain (b–f). (b) Proton number density overlaid with magnetic field lines. X-line topology is visible at X∼-73×106m, Z∼-0.5×106m. (c) Proton temperature as a scalar. Electron initialization temperatures are scaled down by a constant factor 4. (d) Ratio of electron plasma and gyrofrequencies. (e, f) Proton and electron bulk velocity magnitudes with in-plane directions indicated with vectors. The electron distributions are discretized onto eVlasiator velocity meshes, with the electron velocity mesh consisting of 4003 cells, extending from -4.2×107 to +4.2×107ms-1 in each direction, resulting in an electron velocity space resolution of 210kms-1. The eVDF sparsity threshold was set to 10-21m-6s3, ensuring good representation of the main structure of the eVDF. Discretizing a hot and dense electron distribution onto a Cartesian grid is numerically challenging without using vast amounts of memory. As portions of our simulation domain have proton temperature up to 108K, we use an empirical estimate of Ti/Te∼4 as magnetosheath temperature ratios are usually around 4 to 12 . , , , and show similar proton–electron temperature ratios in the magnetotail. In order to constrain the extent of our velocity space and numerical requirements of our solver, we implement our electrons with a mass of 10 times the true electron mass, resulting in an ion-to-electron mass ratio of mi/ me=183.6. As mentioned above, we calculate the required electron bulk velocity for each cell using the local volumetric (cell-average) derivatives so that the ion and electron fluxes in each cell correspond with the current density J required for fulfilling Ampère's law (Eq. ) (with the displacement current neglected at initialization). This is equal to performing a transformation to the Hall frame of reference. Proton densities, magnetic field lines, proton temperatures, proton bulk velocities and electron bulk velocities calculated for simulation initialization are shown in Fig. along with an overview of the input Vlasiator simulation and the selected electron subdomain. To validate the performance of our electron solver, we performed single-cell tests, with resultant electron bulk velocities Ve and plasma oscillation electric fields EJe shown in Fig. . These single-cell tests did not have magnetic field curvature or an ion population present, resulting in the electron motion oscillating around a stability point at Ve=0 and EJe=0. We set the electron number density to ne=0.1cm-3 and the magnetic field to Bx=20nT (panels a through d) or Bx=200nT (panels e and f). We set an initial velocity perturbation of Ve,0=(-100,-150,200)kms-1, close to but below our electron velocity resolution of Δv=210kms-1. As can be seen from Fig. , the electron oscillatory motion is well resolved and remains stable over an extended period. In panels (e) and (f) where the magnetic field strength was artificially increased in order to set the plasma and gyroperiods to values closer to each other (1.11 and 1.79ms, respectively), we see a gradual evolution of oscillation amplitude and, thus, EJe field magnitude as the two types of electron motion interact. Over longer periods of time this growth becomes unstable, but it can be counteracted by using a smaller substep. Also, this instability occurs only when τce≈τpe, which does not occur in our full simulation domain. Graphs of solver stability in relation to electron plasma oscillation and gyromotion in a single-cell simulation. Note the different time axes used. (a, c, e) Oscillation electric field EJe components. (b, d, f) Electron bulk velocity Ve components. (a, b) Graph values in relation to the electron plasma oscillation period (indicated with a thick grey bar) and (c, d) in relation to the electron gyroperiod (indicated with a thick black bar), with a background magnetic field of B=20nT. (e, f) A simulation with a magnetic field of B=200nT, resulting in the gyromotions and oscillatory motions interacting over multiple periods. Although our method is geared towards solving electron motion at coarse spatial resolutions, to further validate the solver, a wave dispersion test was run . As waves are a collective, emergent phenomenon of the kinetic simulation approach, a correct reproduction of wave dispersion behaviour is a good indicator of correct physical behaviour of the simulation system. Two 1D-simulation set-ups with a spatial grid resolution of Δx=300m (= 0.01 de) and Nx=1000 cells were initialized with an electron number density of ne=0.4×106m-3, an electron temperature of Te= 2.5MK, and a magnetic field magnitude of 50nT. In one simulation, the magnetic field direction was chosen to coincide with the extended simulation direction (resulting in parallel plasma wave modes to be resolved), in the other one, the magnetic field was set up perpendicular to the long dimension, resulting in perpendicular mode resolution. The plasma had zero bulk velocity in the simulation frame, with an added white noise velocity fluctuation of ṽ=1000ms-1. The simulation was run for 0.037s (433 ωpe-1). Figure shows the dispersion data resulting from a spatial and temporal Fourier transform (using a von Hann window). Overlaid are analytic dispersion curves for the Langmuir wave (black dashed curve) and electron Bernstein modes (black solid curves). The wave behaviour in the simulation shows good agreement in both parallel and perpendicular directions. One noteworthy additional feature visible in the parallel direction (Fig. a) is the presence of an entropy wave feature at low wave number k and angular frequency ω that shows a quantization consistent with the electron velocity space Dispersion analysis of the electron solver in a 1D test case with an axis-parallel (a) and axis-perpendicular (b) magnetic field. The colour map shows the spatio-temporal Fourier transform of EJe,∥ (a) and EJe,⟂ (b) overlaid with analytical solutions for the Langmuir wave (black dashed curve) and Bernstein modes (black solid curves). We also evaluate the stability of our solver over the larger simulated domain described in Sect. , with initialization values derived from the Vlasiator hybrid-Vlasov simulation. These graphs are shown in Fig. . Panels (a) through (e) show the evolution of electron temperature values over a simulation of 1.0s, covering hundreds of electron plasma periods and, for the most part, tens of gyroperiods. We evaluate minimum, maximum, mean, and median values for total, B-parallel, and B-perpendicular electron temperatures. The system is seen to relax somewhat towards a final state, though some evolution is still apparent at the end of the simulation, possibly due to boundary effects. The maximum temperature plot in panel (b) is of particular interest as the hottest plasma cells appear to diffuse into their surroundings until t∼0.4s when dynamic gyration processes overtake this temperature diffusion with perpendicular heating. Panel (f) shows the agyrotropy measure calculated from the electron pressure tensor, indicating that in the majority of the simulation domain electrons remain gyrotropic and even peak values do not grow past 10-3. Panel (g) shows statistics for the electron number density deviation from the initialization value, indicating loss of plasma neutrality due to the motion of electrons. The minimum value oscillating between approximately 10-9 and 10-6cm-3 indicates the level of numerical fluctuations, and the maximum, mean and median values show how charge imbalance does grow initially but stabilizes within about 0.1s and does not grow beyond 10-1cm-3. In panels (h) through (k) of Fig. we show how the instantaneous plasma oscillation electric field EJe is well-behaved throughout the simulation box, converging towards stable values. We note that as the EJe field oscillates around zero, the averages are indeed zero throughout (not shown) and the values used for inferring minimum, maximum, mean and median values are instantaneous values from an arbitrary phase of the oscillation. In panel (l) we show the normalized current density J departure from the balance current JB=∇×Bμ0 which would be required to maintain the magnetic field structure according to Ampère's law (Eq. ). This metric is seen to also stabilize, mostly at values well below unity. We expect the maximum-value outliers to be due to locally small values of JB. Panels (m) and (n) show statistics for the parallel and perpendicular components of the electric field caused by electron pressure gradients, that is, the -∇⋅Penee term. As expected due to the ability of electrons to propagate along field lines, perpendicular components are much larger than parallel components. All components remain stable at roughly their initial values. A minimum value is not shown, as the use of a numerical slope limiter in the calculation of pressure gradients gives identically zero field components at local extrema of pressure. Evolution of electron and solver parameters over the whole simulation domain. (a–d) Minimum, maximum, mean, and median values for electron temperature Te and its components parallel and perpendicular to the local magnetic field. (e) Minimum, maximum, mean, and median values for electron temperature anisotropy. (f) Minimum, maximum, mean, and median values for electron agyrotropy QAg,e. (g) Minimum, maximum, mean, and median values for electron density deviation from initial state, indicating charge imbalance. (h–k) Minimum, maximum, mean, and median values for the plasma oscillation electric field EJe and its components parallel and perpendicular to the local magnetic field. (l) Minimum, maximum, mean, and median normalized values for current density J deviation from the value JB=∇×Bμ0 required to fulfill Ampère's law for the local magnetic field. (m, n) Maximum, mean, and median values for parallel and perpendicular components of the electric field due to electron pressure gradients. As part of our evaluation of solver stability, we performed a comparison run where our electron solver performed the rotation transformation corresponding with gyromotion in the Hall frame instead of in the substep-associated electron bulk frame. This transformation choice resulted in unstable growth of, in particular, EJe, as could be expected (not shown). Results from the electron simulation after 1.0s of evolution are presented in Fig. . Figure a, b show parallel and perpendicular acceleration or deceleration of electrons as the ratio of end-of-simulation temperature to initial temperatures. Heating is found in particular near the X-line configuration and where the plasma sheet boundary layer (PSBL) meets the magnetosphere, with parallel heating more localized than perpendicular heating. Figure c shows the agyrotropy measure calculated for electrons, indicating where the electron distribution has become non-gyrotropic. In most of the simulation domain, the value is very small, but enhanced agyrotropy (still relatively small values below 10-3) are found in the PSBL regions and at the magnetic field X line. Some of this agyrotropy may be due to spatial sampling of electron gyromotion with a magnetic field gradient leading to larger gyroradii further away from the plasma sheet. Electron distribution properties within the test domain after 1.0s of simulation. (a) The ratio of parallel electron temperature at 1.0s to the parallel temperature at the start of the simulation, indicating parallel heating. (b) The same but for perpendicular temperature. (c) The agyrotropy measure for the electron population. (d) The magnitude and direction of the electron pressure gradient term of the electric field. (e, f) The charge imbalance ne-ne,0 and relative charge imbalance (ne-ne,0)ne,0-1 found at the end of the simulation. Figure d shows the electric field due to ∇Pe, with the field strongest where the PSBL meets the magnetosphere. The field direction is pointed towards the tail sheet or the magnetosphere, as expected. Magnitudes remain of the order of a few millivolts per metre. Figures e, f quantify the charge imbalance resulting from electrons evolving due to static magnetic fields and the electric field resulting from the Ohm's law terms presented in this paper. Figure e shows the level of charge imbalance as change in electron number density, and Fig. f as the change scaled by the original electron number density. In the majority of the simulation domain, imbalance remains below 10-2cm-3. The electric field response is unable to maintain full plasma neutrality with some regions near the magnetosphere showing greater deviation from the initial state. Some stronger imbalance at the domain edges is likely a boundary effect which shall resolve itself with a larger simulation domain. In Fig. we display electron velocity distribution functions after 1.0s of simulation. Figure a shows the evolved electron temperature anisotropy T⟂,eT∥,e-1, and Fig. b displays the maximum of instantaneous values of EJe, taken over 10 measurements at 0.05s intervals near the end of the simulation. Panels (c) through (n) of Fig. show parallel and perpendicular projections of electron eVDFs at virtual spacecraft (VSC) [1] through [6], with positions of VSC indicated in panels (a) and (b). Electron properties and velocity distribution functions after 1.0s of simulation. (a) Electron temperature anisotropy T⟂,eT∥,e-1 overlaid with magnetic field lines and six virtual spacecraft locations, labelled [1]–[6]. (b) Maximum value for displacement current EJe, taken over 10 measurements at 0.05s intervals near the end of the simulation. (c–n) Electron velocity distribution function projections into the parallel vB and vV×B or perpendicular vB×(B×V) and vV×B planes. Each virtual spacecraft is indicated by the number in the parallel eVDF panel with the panel below showing the corresponding perpendicular eVDF for the same virtual spacecraft. Figure a shows how temperature anisotropy T⟂,eT∥,e-1 indicates parallel energization in the low-density regions adjacent to the PSBL and perpendicular energization adjacent to the X line and within the tailmost region of the magnetosphere. As we have bulk flows of both ions and electrons towards the tail current sheet, some small part of this heating can be attributed to betatron acceleration as electrons convect towards stronger magnetic fields just adjacent to the actual high-beta plasma sheet. Other effects causing anisotropies may arise from spatial leakage of electrons undergoing plasma oscillation, with gyromotion binding perpendicularly heated electrons to the oscillation region and parallel accelerated electrons propagating along field lines to the near-magnetosphere PSBL The maxima of instantaneous values of EJe, shown in Fig. b, indicate that the strongest electron oscillations on our simulated scales are found in or near the PSBL, which would be consistent with observations of electron-driven waves in the PSBL . Some increase in EJe is also seen at the X-line location, but not in other parts of the current sheet. We note that the X line included in the Vlasiator simulation snapshot was not actively reconnecting. Comparison with Fig. a and virtual spacecraft measurements indicate that parallel features, akin to electron beams, are indeed found in regions with enhanced EJe. The temperature anisotropies found in the near-Earth tail region of our simulation are mostly in the 0.5…1.5 range. reported on Cluster observations of electron temperature anisotropies ranging from 0.8…1.6 and centred around ∼1.1, in agreement with our results, though their observations were gathered between -20RE<x<-15RE (-127×106<x<96×106m). Regions where parallel temperatures dominate (anisotropy <1) are found in regions of cold plasma, as can be seen by comparing Figs. c and a. This does not preclude the possibility of parallel acceleration in regions of hot plasma but rather shows that the acceleration may not be strong enough to be discerned over the main hot eVDF. Parallel heating near the magnetotail plasma sheet has been reported to coincide with bi-directional electron distributions with temperature ratios going up to 2–3, as in our simulation. Our VSC [2] and [5] show clear bi-directional distributions. Due to our static background magnetic field, our parallel heating cannot be due to conventional Fermi acceleration. However, propose that adiabatic plasma processes where curvature drifts dominate over gradient drifts can lead to significant parallel heating. Our VSC [1] is from close to the X line and shows parallel elongation of the central part of the distribution, reminiscent of the football or shifted-football distributions of Fig. 2 of . describe streaming 500eV electrons at the PSBL, associated with a substorm event and variation of By, especially at small scales. Scaling with our electron mass, this corresponds to approximately 4000kms-1 electron velocities, which is reasonably within the range of our eVDFs in Fig. . We note that our simulation produces a background By profile with ∇By in agreement with Fig. 4 of (not shown), on top of which the streaming electrons are observed. describe a simple 2D Liouville model for the PSBL, as well as some ISEE-1 and ISEE-2 observations supporting their model. The formation mechanisms of eVDFs in are listed as time-of-flight, energy conservation and magnetic moment conservation, which are included in our model, though we perform a more robust evaluation of plasma oscillation interplay with gyration. The eVDFs shown in their Fig. 4 agree with our VSCs [1], [2], [5], and [6], for example. We also note our VSC [3] displaying a disjoint parallel beam, matching the ISEE-2 observations in Fig. 5 of . Observations of perpendicular crescents are shown in MMS data in e.g. at EDRs, in conjunction with dayside magnetopause reconnection sites. These observed structures are produced at very small spatial scales, not captured by our current model. We do, however, observe similar agyrotropic crescents in our results further out (in particular in Fig. j), suggesting successful capture of a level of electron dynamics. These perpendicular crescents are found at very low phase-space density values, as could be expected by the low agyrotropy values seen in Fig. e. Something akin to a parallel electron crescent can be seen in Fig. c, and bi-directional distributions as reported in Figs. 6 and 7 of are qualitatively similar to our Fig. k and m. In this method paper we have presented a novel approach to investigating electron distribution function dynamics in the context of global ion-hybrid field structures. Our method exploits global dynamics provided by hybrid-Vlasov simulations in order to evaluate the response of gyrating and plasma oscillating electrons to global magnetic field structures. We have shown our solver to behave in a stable manner, resolving electron inertia and updating a responsive electric field EJe derived from the displacement current. If run at much finer spatial resolutions, our model replicates Langmuir waves and electron Bernstein modes. Electron temperatures evolve in response to the field structure but do not experience uncontrolled growth. Our sample simulation produces multiple features associated with spacecraft observations of eVDFs, such as parallel acceleration, bi-directional distributions, and perpendicular crescents. Our model has several built-in limitations as it does not treat electrons as a fully self-consistent species. Magnetic fields gathered from the Vlasiator simulation are kept constant and thus force electron bulk motion to adhere to the required current density structure. As the initialization information is gathered from a hybrid-Vlasov simulation, it has a spatial resolution far below that required for resolving electron-scale waves such as whistlers, Bernstein waves and chorus waves. Scattering of electrons via these missing waves is somewhat accounted for by initializing every simulation from a Maxwellian isotropic distribution. These features together limit the applicability of the model to short periods of time. On the other hand, our model is efficient, and much larger spatial domains of investigation are easily achievable. Also, multiple eVlasiator runs can be performed from a single Vlasiator magnetosphere run to evaluate different driving conditions such as temperature ratios and anisotropies. The method builds on the efficiently parallelized Vlasiator codebase and will benefit from future numerical and computational improvements to Vlasiator solvers. Our model can be applied to investigate electron dynamics on global spatial scales, with the current version applicable to 2D investigations, e.g. in the noon–midnight meridional plane. Electron velocity distribution functions generated by the model can be used to investigate, for example, energetic electron precipitation into the Earth's auroral regions. The generated electron anisotropies can be used to infer regions where, for example, whistler waves can be expected to grow. The model can be run for several different initialization time steps to evaluate long-term evolution of precipitating electron distributions. This could be used to, for example, evaluate electron distribution changes as bulk flows and dipolarization fronts in the Earth's magnetotail propagate earthward. observe electron Bernstein modes driven by perpendicular crescent distributions. As we have shown in Figs. and , with sufficient resolution we can reproduce electron Bernstein waves and agyrotropic electron distributions. Thus, we are in a position to investigate this connection further in eVlasiator. Future improvements to our model will allow simulation initialization from non-uniform 3D-3V Vlasiator meshes, allowing investigation of spatially three-dimensional topologies including tail plasma sheet clock angle tilt. A possible path of future investigation would be to upsample the initialization fields and moments in order to achieve better resolution, but we emphasize that the model does not attempt to solve electrons in a fully self-consistent manner, as magnetic fields are still kept constant. Increasing resolution by interpolating the input moments to a finer grid might not significantly improve plasma sheet density and temperature profiles. Increasing spatial resolution introduces numerous caveats including increased computational cost and possible charge imbalance resulting from spatially resolved electron oscillations, though our dispersion tests did not indicate such problems. If such imbalances arise from a future model, some method of solving Gauss's law such as a Poisson solver should be implemented. A more detailed investigation into comparing electron eVDFs and dynamics with observations is expected in a future study. Vlasiator https://www.helsinki.fi/en/researchgroups/vlasiator, is distributed under the GPL-2 open-source license at https://github.com/fmihpc/vlasiator/ . Vlasiator uses a data structure developed in-house https://github.com/fmihpc/vlsv/,. The Analysator software 10.5281/zenodo.4462515; was used to produce the presented figures. The run described here takes several gigabytes of disk space and is kept in storage maintained within the CSC – IT Center for Science. Data presented in this paper can be accessed by following the data policy on the Vlasiator web site. MB wrote the paper and code description. MB and TB devised the solver method. MA assisted with data analysis, model development, and comparisons with observations. UG performed the dispersion tests. MP is the principal investigator of Vlasiator and leads the investigation. YPK, MG, KP, AJ, LT, MD, and MP participated in the discussion and finalization of the paper. The authors declare that they have no conflict of interest. We acknowledge the European Research Council for starting grant 200141-QuESpace, with which the Vlasiator model (https://www.helsinki.fi/en/researchgroups/vlasiator, last access: 25 January 2021) was developed, and consolidator grant 682068-PRESTISSIMO awarded for further development of Vlasiator and its use in scientific investigations. We gratefully acknowledge Academy of Finland grant numbers 309937-TEMPO and 312351-FORESAIL. PRACE (http://www.prace-ri.eu, last access: 25 January 2021) is acknowledged for granting us Tier-0 computing time in HLRS Stuttgart, where Vlasiator was run in the HazelHen machine with project number 2014112573 and in the Hawk machine with project number 2019204998. The work of Lucile Turc is supported by the Academy of Finland (grant number 322544). The authors wish to thank the anonymous referees for their assistance in improving the approachability of the paper. This research has been supported by the European Research Council (grant nos. 682068 and 200141) and the Academy of Finland, Luonnontieteiden ja Tekniikan Tutkimuksen Toimikunta (grant nos. 312351, 309937, and 322544).Open-access funding was provided by the Helsinki University Library. This paper was edited by Wen Li and reviewed by two anonymous referees. Akhavan-Tafti, M., Palmroth, M., Slavin, J. A., Battarbee, M., Ganse, U., Grandin, M., Le, G., Gershman, D. J., Eastwood, J. P., and Stawarz, J. 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Implementing Binary Search to Find Elements in Python Sorted Arrays - llego.dev Binary search is an efficient algorithm for finding an element in a sorted array. It works by repeatedly dividing the search interval in half, eliminating parts of the array that cannot contain the target element until it is found. Binary search is much faster than linear search, especially for large arrays, as it eliminates half of the remaining elements after each iteration. This article provides a comprehensive guide on implementing binary search in Python, examining the algorithm logic, code examples, complexity analysis, and variations. Table of Contents Open Table of Contents Overview of Binary Search Binary search follows the divide-and-conquer algorithm strategy to search for an element in an array. The key requirements are: • The array must be sorted in ascending or descending order. This allows comparing the target value to the middle element to determine which half of the array to search next. • The array indices, starting position, ending position, and middle index must be tracked to determine the current search interval. The basic steps are: 1. Initialize the start and end index for the search space. 2. Calculate the middle index as the average of the start and end indices. 3. Compare the target value to the middle element. 4. If it matches, return the middle index. 5. Else if the target is less than the middle, make the end index equal to the middle - 1. 6. Else if the target is greater than the middle, make the start index equal to the middle + 1. 7. Repeat steps 2-6, successively narrowing the search space interval, until the target is found or the interval is empty. 8. If the search space empties, return -1 to indicate the target is not found. This halves the search space each iteration, providing a worst-case time complexity of O(log n). Python Implementation of Binary Search Here is an implementation of binary search in Python: def binary_search(arr, target): start = 0 end = len(arr) - 1 while start <= end: mid = (start + end) // 2 if arr[mid] == target: return mid elif arr[mid] < target: start = mid + 1 end = mid - 1 return -1 • It takes the sorted array arr and target element target as parameters. • Two indices start and end track the search space. • The while loop runs till the start index is less than or equal to the end. • The mid index is calculated as the average of the start and end indices. • The target is compared to the middle element. If equal, the mid index is returned. • Otherwise, the start or end is adjusted to narrow the search space. • After the loop, -1 is returned if the target is not found. To use this function: arr = [1,5,23,111,144,500] target = 144 result = binary_search(arr, target) if result != -1: print("Element found at index", result) print("Element not found") This prints “Element found at index 4”, since 144 exists at index 4 in the array. Handling Edge Cases The basic implementation above does not handle edge cases properly. Here are some improvements: 1. Empty array: Check if the array is empty before searching and return -1 directly. if len(arr) == 0: return -1 2. Out of bounds indices: Calculate the mid index using: mid = start + (end - start) // 2 This avoids overflow when start and end are large. 3. Repeated elements: Return the leftmost index in case of duplicates. while start < end: # Other code if arr[mid] == target: end = mid return start Complexity Analysis Time Complexity: As the search space is halved each iteration, binary search takes O(log n) time in the worst case, where n is the array size. The number of iterations before the space is reduced to 1 is log2(n). Space Complexity: The algorithm runs in O(1) constant space, as it only stores the start, end, mid indices and does not use any other data structure. Optimizations and Variations Here are some optimizations and variations of binary search: • Iterative vs Recursive: The iterative approach above uses a loop. Recursive binary search recursively calls itself on the first or second half of the array. • Search Space Reduction: If the search space can be reduced beyond half based on additional checks, the search can converge faster. • Index of First Occurrence: Track the first index matching the target for duplicate elements. • Binary Search on Strings: Apply binary search on a lexicographically sorted array of strings. • Binary Search on Rotated Array: Search a rotated sorted array efficiently by determining which half is sorted. • Ternary Search: Use three indices to divide array into thirds for faster convergence. • Exponential Search: Perform binary search after using exponential search for search space reduction. • Interpolation Search: Use a probe index calculated from the key value for faster search. Example Binary Search Code Snippets Here are some examples demonstrating the optimizations and variations of binary search discussed above: Recursive Implementation def binary_search_recursive(arr, target, start, end): if start > end: return -1 mid = (start + end) // 2 if arr[mid] == target: return mid elif arr[mid] < target: return binary_search_recursive(arr, target, mid+1, end) return binary_search_recursive(arr, target, start, mid-1) # Initial call result = binary_search_recursive(arr, target, 0, len(arr)-1) Search First Occurrence def binary_search_first(arr, target): start = 0 end = len(arr) - 1 first_index = -1 while start <= end: mid = (start + end) // 2 if arr[mid] == target: first_index = mid end = mid - 1 elif arr[mid] < target: start = mid + 1 end = mid - 1 return first_index Binary Search on String Array strings = ["apple", "mango", "orange", "pear"] target = "mango" # Lexicographically sort strings def binary_search_string(strings, target): # Implement string binary search print(binary_search_string(strings, target)) Search Rotated Sorted Array def search_rotated_array(arr, target): pivot = find_pivot(arr) # Search in first half if arr[pivot] <= target <= arr[len(arr)-1]: return binary_search(arr, target, 0, pivot) # Search in second half return binary_search(arr, target, pivot+1, len(arr)-1) def find_pivot(arr): start = 0 end = len(arr) - 1 while start <= end: mid = (start + end) // 2 if mid < end and arr[mid] > arr[mid + 1]: return mid elif mid > start and arr[mid] < arr[mid - 1]: return mid-1 elif arr[start] >= arr[mid]: end = mid - 1 start = mid + 1 These demonstrate how binary search can be adapted to different scenarios and use cases. The key ideas remain dividing the search space and progressively narrowing intervals. Applications of Binary Search Binary search is used extensively in various domains: Searching Sorted Data: Arrays, lists, matrices, databases, files Optimization Algorithms: Minimum/maximum finding, Golden section search Resource Locations: Jump tables, memory addresses, pages, blocks, sectors Data Compression: Dictionary-based like LZW, predictive coding Signal Processing: Noise thresholding, step detection, pulse detection Networking: MAC forwarding table lookups, IP lookups, port lookups Graphics: Lighting calculation, anti-aliasing, ray tracing Physics and Mathematics: Root finding, inequality solving, simulations Games: Movement optimizations, pathfinding, terrain generation Distributed Systems: Lookup services, source coding, redundancy removal Machine Learning: Hyperparameter optimization, neural architecture search, clustering Cryptography: Ciphertext searching, collision finding, identity testing Bioinformatics: Genome alignment, protein sequencing, BLAST search As a powerful algorithm at the core of computing, binary search enables efficiencies in virtually every technical field. Mastering implementations in Python is key for computer science and coding Binary search is a fundamental algorithm that greatly speeds up searching in sorted arrays. This guide covered how it works, Python implementations, edge cases, optimizations, variations, code examples, complexity analysis, and applications. The key ideas are utilizing the sorted order to eliminate halves of the search space each iteration and recursively narrowing the interval till the target is found. With practice, binary search can be adapted to various use cases. Fluency in binary search coding questions demonstrates strong technical and analytical skills crucial for programming interviews and roles.	{"url":"https://llego.dev/posts/implement-binary-search-find-elements-python-sorted-arrays/","timestamp":"2024-11-15T04:49:48Z","content_type":"text/html","content_length":"69028","record_id":"<urn:uuid:5d23b9e6-0624-40cc-96ce-4119da8387bd>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00402.warc.gz"}
The Cahn-Hilliard equation with elasticity-finite element approximation and qualitative studies \| EMS Press The Cahn-Hilliard equation with elasticity-finite element approximation and qualitative studies • Harald Garcke Universität Regensburg, Germany • Martin Rumpf Universität Bonn, Germany • Ulrich Weikard Universität Bonn, Germany We consider the Cahn-Hilliard equation-a fourth-order, nonlinear parabolic diffusion equation describing phase separation of a binary alloy which is quenched below a critical temperature. The occurrence of two phases is due to a nonconvex double well free energy. The evolution initially leads to a very fine microstructure of regions with different phases which tend to become coarser at later times. The resulting phases might have different elastic properties caused by a different lattice spacing. This effect is not reflected by the standard Cahn-Hilliard model. Here, we discuss an approach which contains anisotropic elastic stresses by coupling the expanded diffusion equation with a corresponding quasistationary linear elasticity problem for the displacements on the microstructure. Convergence and a discrete energy decay property are stated for a finite element discretization. An appropriate timestep scheme based on the strongly A-stable [Theta]-scheme and a spatial grid adaptation by refining and coarsening improve the algorithms efficiency significantly. Various numerical simulations outline different qualitative effects of the generalized model. Finally, a surprising stabilizing effect of the anisotropic elasticity is observed in the limit case of a vanishing fourth-order term, originally representing interfacial energy. Cite this article Harald Garcke, Martin Rumpf, Ulrich Weikard, The Cahn-Hilliard equation with elasticity-finite element approximation and qualitative studies. Interfaces Free Bound. 3 (2001), no. 1, pp. 101–118 DOI 10.4171/IFB/34	{"url":"https://ems.press/journals/ifb/articles/48","timestamp":"2024-11-14T16:52:11Z","content_type":"text/html","content_length":"83295","record_id":"<urn:uuid:67005a8e-566d-43b5-9b54-0aa080a36824>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00712.warc.gz"}
Kossovsky, AE (2015) Random Consolidations and Fragmentations Cycles Lead to Benford' Law Preprint arXiv:1505.05235 [math.ST]; last accessed October 19, 2020. ISSN/ISBN: Not available at this time. DOI: Not available at this time. Abstract: Benford's Law predicts that the first significant digit on the leftmost side of numbers in real-life data is proportioned between all possible 1 to 9 digits approximately as in LOG(1 + 1/ digit), so that low digits occur much more frequently than high digits in the first place. For example, digit 1 occurs approximately 30.1% in the first place in random numbers, while digit 9 occurs only approximately 4.6%. In this article it is shown that a process where a large enough set of identical quantities constantly alternates between minuscule random consolidations (summing two randomly chosen values into a singular value) and tiny random fragmentations (division of one randomly chosen value into two new values) converges digit-wise to the Benford proportions after sufficiently many such cycles. The statistical tendency of the system after numerous cycles is to have approximately 2/3 multiplicative expressions which are conducive to Benford behavior as they tend to the Lognormal Distribution, and 1/3 additive expressions which are detrimental to Benford behavior as they tend to the Normal Distribution, hence the process represents in essence a tug of war between addition and multiplication. Since the process encounters the so-called Achilles' heel of the Central Limit Theorem, namely additions of skewed distributions with high order of magnitude, additions are not very effective, and the war is decisively won by multiplication, leading to Benford behavior. Randomness in selecting the particular quantity to be fragmented, as well as randomness in selecting the two particular quantities to be consolidated, is essential for convergence. Not surprisingly then, fragmentation itself could be performed either randomly say via a realization from the continuous Uniform on (0, 1), or deterministically via any fixed split ratio such as say 25% - 75%, and Benford's Law emerges in either case. @misc{, title={Random Consolidations and Fragmentations Cycles Lead to Benford' Law}, author={Alex Ely Kossovsky}, year={2015}, eprint={1505.05235}, archivePrefix={arXiv}, primaryClass={math.ST} } Reference Type: Preprint Subject Area(s): Statistics	{"url":"https://www.benfordonline.net/fullreference/2249","timestamp":"2024-11-08T18:47:23Z","content_type":"application/xhtml+xml","content_length":"5435","record_id":"<urn:uuid:d63680bb-9b09-48a0-90e5-99ed7e9ccc84>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00658.warc.gz"}
Calculating the Bose-Einstein Condensation Temperature • Thread starter Mr LoganC • Start date In summary, the conversation discusses estimating the Bose-Einstein condensation temperature of Rb 87 atoms with a density of 10^11 atoms per cm^3 using the equation T=n^(2/3)h^2/3mK_B. There is a discrepancy in the units used, with the suggestion to use joules instead of eV. However, it is concluded that the units do not matter due to the cancelling out of the conversion factor. The final estimated answer is 16nK. Homework Statement Estimate the Bose-Einstein condensation temperature of Rb 87 atoms with density of 10^11 atoms per cm^3. Homework Equations The Attempt at a Solution This should be just a standard plug and chug question, but my answers are not even close to reasonable! I would expect to get anywhere from 500nK to 50nK for an answer, but I am getting thousands of Kelvin! Are my units wrong? I am using Boltzman constant with units of [itex]eV\bullet K^{-1}[/itex] and Plank's with units of [itex]eV\bullet s[/itex]. Then I am using the density in Atoms per m^3 and the mass of a single Rb 87 atom in Kg. Am I missing something here with units? That is the only thing I can think is wrong. At first glance I would use joules (Nxm) instead of eV since since other units are SI... bloby said: At first glance I would use joules (Nxm) instead of eV since since other units are SI... But since the plank constant is on the top and Boltzman on the bottom, the converting factor of eV to joules (1.6x10^-19) would cancel out anyway, so whether it's in eV or Joules should not matter. Mr LoganC said: But since the plank constant is on the top and Boltzman on the bottom, the converting factor of eV to joules (1.6x10^-19) would cancel out anyway, so whether it's in eV or Joules should not Actually, that IS the problem! I think you are correct, Bloby. Because the Plank constant is squared on the top, so there is still another 1.6x10^-19 to factor in there! I will give it a shot and see what I get for an answer! It Worked! Thank you Bloby! Final answer was 16nK, which seems pretty reasonable to me for a Bose-Einstein Condensate. Last edited: Your calculations seem to be correct, however, the units you are using for the Boltzmann constant and Planck's constant are incorrect. They should be in units of Joules per Kelvin and Joules times seconds, respectively. This may be why your answers are coming out in Kelvin instead of nanokelvin. Also, make sure to convert the density from atoms per cm^3 to atoms per m^3. This should give you a more reasonable answer in the range of 50nK to 500nK. FAQ: Calculating the Bose-Einstein Condensation Temperature What is a Bose-Einstein condensate? A Bose-Einstein condensate (BEC) is a state of matter formed at extremely low temperatures when a large number of bosons (particles with integer spin) occupy the same quantum state and behave as a single entity. This phenomenon was first predicted by Satyendra Nath Bose and Albert Einstein in the 1920s. What is the Bose-Einstein condensation temperature? The Bose-Einstein condensation temperature is the temperature at which a gas of bosons can no longer be described by classical statistical mechanics and instead forms a BEC. This temperature is dependent on the number of particles, the mass of the particles, and the trapping potential of the system. How is the Bose-Einstein condensation temperature calculated? The Bose-Einstein condensation temperature can be calculated using the formula T = 0.94n^(2/3)ħ^2/mk, where n is the number of particles, ħ is the reduced Planck's constant, m is the mass of the particles, and k is the Boltzmann constant. This formula is known as the Thomas-Fermi approximation and provides a rough estimate for the condensation temperature. What is the significance of the Bose-Einstein condensation temperature? The Bose-Einstein condensation temperature is an important parameter in the study of quantum gases and has various applications in fields such as superfluidity, superconductivity, and atom lasers. It also provides insight into the behavior of matter at extremely low temperatures and has helped scientists understand the nature of quantum mechanics. What factors can affect the Bose-Einstein condensation temperature? The Bose-Einstein condensation temperature is influenced by various factors such as the number of particles, the mass of the particles, the trapping potential of the system, and the interactions between the particles. The type of particles also plays a role, as different bosonic particles have different condensation temperatures due to their varying properties.	{"url":"https://www.physicsforums.com/threads/calculating-the-bose-einstein-condensation-temperature.584255/","timestamp":"2024-11-10T20:40:26Z","content_type":"text/html","content_length":"90935","record_id":"<urn:uuid:ea311fc3-28dd-49c3-bdee-6f5cbba8640d>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00606.warc.gz"}
Build a locally accurate surrogate from data at a single point • Alias: None • Arguments: None Child Keywords: Required/Optional Description of Group Dakota Keyword Dakota Keyword Description Required taylor_series Construct a Taylor Series expansion around a point Required truth_model_pointer Pointer to specify a “truth” model, from which to construct a surrogate Local approximations use value, gradient, and possibly Hessian data from a single point to form a series expansion for approximating data in the vicinity of this point. The currently available local approximation is the taylor_series selection. The truth model to be used to generate the value/gradient/Hessian data used in the series expansion is identified through the required truth_model_pointer specification. The use of a model pointer (as opposed to an interface pointer) allows additional flexibility in defining the approximation. In particular, the derivative specification for the truth model may differ from the derivative specification for the approximation, and the truth model results being approximated may involve a model recursion (e.g., the values/gradients from a nested model).	{"url":"https://snl-dakota.github.io/docs/6.20.0/users/usingdakota/reference/model-surrogate-local.html","timestamp":"2024-11-10T03:34:35Z","content_type":"text/html","content_length":"15245","record_id":"<urn:uuid:3aa1fdc3-0266-40ae-8f84-1f036f1f37a4>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00492.warc.gz"}
Welcome to Deepnote Name: Varkey Thomas Pottenkulam This analysis uses a real dataset to examine the complex relationship between various factors and their impact on the company's financial performance. By employing a rigorous data-cleaning process, visualisations, and modelling techniques, this report provides a holistic approach to understanding the drivers of profitability (Provost and Fawcett, 2013) The analysis process begins with a thorough data cleaning and pre-processing phase, aimed at ensuring the accuracy and reliability of the dataset. The Interquartile Range (IQR) method is used to identify and eliminate any outliers present in the data, thus preventing any skewed results from affecting the analysis (Iglewicz and Hoaglin, 1993). This approach not only improves the quality of the data but also enables more precise and trustworthy insights to be derived from it. Furthermore, the Random Forest Regression model is employed, renowned for its ability to handle complex, non-linear relationships between variables, making it an ideal choice for predicting profitability (James et al., 2013). import pandas as pd import numpy as np import matplotlib.pyplot as plt import seaborn as sns from sklearn.model_selection import train_test_split from sklearn.metrics import mean_squared_error, r2_score # Loading dataset df = pd.read_csv('Superstore dataset.csv') df.describe() Cleaning the Data and Removing Outliers From the dataset, it is important to focus on removing outliers from numerical columns that directly impact the analysis, particularly those related to financial metrics and transaction volumes. The Interquartile Range (IQR) method was selected due to its strong resistance to outliers and its straightforward implementation. This method calculates the range between the first (Q1) and third (Q3) quartiles of the data, defining outliers as any values that lie more than 1.5 times the IQR below Q1 or above Q3. It defines outliers based on the quartile distribution of the data, which makes it less sensitive to extreme values than methods based on mean and standard deviation. This method is particularly effective for data that may not follow a normal distribution, a common characteristic of financial and sales data. Specifically, outliers were removed from the ‘Sales’ and ‘Profit’ columns, which are critical to understanding the factors influencing the company's profitability. The dataset also underwent preliminary cleaning steps, including checks for null entries and duplicates, to ensure data integrity. Additionally, entries marked as returned were removed from the dataset. This step is crucial because returned orders do not contribute to net profitability; they can distort the analysis aimed at understanding factors that drive successful sales. By excluding these transactions, the dataset now more accurately reflects sales that positively impact the company's profits, providing a cleaner basis for identifying strategies to increase profitability. # Check for missing values df.info() # Remove duplicates df.drop_duplicates(inplace=True) # Using IQR method to filter outliers def remove_outliers(df, column_names): cleaned_df = df.copy() for column in column_names: Q1 = cleaned_df[column].quantile(0.25) Q3 = cleaned_df[column].quantile(0.75) IQR = Q3 - Q1 lower_bound = Q1 - 1.5 * IQR upper_bound = Q3 + 1.5 * IQR # Filtering out the outliers before_rows = cleaned_df.shape[0] cleaned_df = cleaned_df[(cleaned_df[column] >= lower_bound) & (cleaned_df[column] <= upper_bound)] after_rows = cleaned_df.shape[0] print(f"Removed { before_rows - after_rows} outliers from {column}.") return cleaned_df # Columns to check for outliers columns_to_check = ['Sales', 'Profit'] # Removing outliers cleaned_df = remove_outliers(df, columns_to_check) # Directly remove returned entries before_rows = cleaned_df.shape[0] cleaned_df = cleaned_df[cleaned_df['Returned'] != True] after_rows = cleaned_df.shape[0] print(f"Removed { before_rows - after_rows} returned entries.") # Summary of the cleaned data cleaned_df.describe() After the cleaning process, a significant portion of the dataset was retained, indicating a balanced approach to outlier removal that preserved the bulk of the data for analysis. The summary statistics of the cleaned dataset reveal insightful trends and distributions across the remaining data. For instance, average sales and profits can now be interpreted with greater confidence, reflecting a more accurate depiction of the company's typical transactions. Sales and profit ranges, along with measures like mean and median, reveal the company's financial health and operational Visualising the data provides compelling insights into the underlying trends and patterns that numbers alone cannot convey. Through the charts and graphs, we gain insights into the sales performance, product category preferences, and regional market strengths within the dataset. They bridge data and decision-making, showing success and growth opportunities in an accessible way. The bar chart below shows us the distribution of products across different categories, providing insights into which categories are most prevalent in the dataset. import seaborn as sns import matplotlib.pyplot as plt # Bar chart for product categories plt.figure(figsize=(10, 6)) sns.countplot(x='Category', data=cleaned_df) plt.title('Distribution of Product Categories') plt.xlabel('Category') plt.ylabel('Frequency') plt.xticks(rotation=45) plt.show() This plot can help us understand which product categories are most common. The bar chart displays the frequency of products sold across three categories: Office Supplies, Furniture, and Technology. Office Supplies are the most frequently sold products, with sales figures significantly higher than the other two categories. Furniture and Technology have similar sales frequencies, both considerably lower than Office Supplies. This could suggest that customers purchase Office Supplies more frequently. This distribution indicates potential market focus areas and might influence inventory management, marketing strategies, and sales forecasting. The following chart will aggregate sales by 'Region_no', showing which regions contribute most to sales. # Visualising Sales per 'Region_no' plt.figure(figsize=(10, 6)) sales_per_region = cleaned_df.groupby('Region_no')['Sales'].sum().reset_index() sns.barplot(x='Region_no', y='Sales', data= sales_per_region) plt.title('Sales per Region') plt.xlabel('Region Number') plt.ylabel('Total Sales') plt.show() This chart provides a clear view of the sales performance across different regions. The bar chart indicates total sales distribution across four distinct regions. Region 2 outperforms the others, suggesting a strong market presence or effective sales strategies in that area. Region 1 has the lowest sales, which might imply challenges or untapped market potential. The disparity between the regions suggests that targeted strategies could be necessary to bolster sales in the underperforming regions. Moreover, the chart could prompt a review of Region 2's practices to identify successful tactics that might be replicated in Regions 1, 3, and 4 to enhance overall sales. The scatter plot below explores the relationship between sales and profit, which is crucial for understanding profitability dynamics. # Scatter plot for Sales vs. Profit plt.figure(figsize=(10, 6)) sns.scatterplot(x='Sales', y='Profit', data=cleaned_df) plt.title('Sales vs. Profit') plt.xlabel('Sales') plt.ylabel('Profit') plt.show This scatter plot helps identify patterns or correlations between sales and profit. The scatter plot of Sales vs. Profit suggests a positive relationship between sales and profit up to a point, after which profit increases at a slower rate.This could indicate diminishing returns on higher sales volumes. There is a noticeable trend of transactions with negative profits across various sales levels, highlighting the occurrence of losses or low-margin sales. Clusters of data points near the origin reveal that most sales transactions are of lower value, with a mix of profitable and unprofitable The random forest regression method was selected to advance the exploration of data through modelling (Tan et al., 2005). This learning technique is particularly adept at predicting outcomes like profitability by constructing numerous decision trees and averaging their predictions. Such an approach is particularly advantageous in our context, where the interactions between variables like sales, quantity, and discount are complex and potentially non-linear. The choice of Random Forest is informed by its ease of handling a variety of data types. It offers deep insights through its feature importance rankings, which is extremely valuable for identifying the factors that have the greatest impact on profit (Kuhn and Johnson, 2013). Additionally, it has the advantage of reducing variance and avoiding overfitting that simpler models may experience. import pandas as pd import numpy as np import matplotlib.pyplot as plt from sklearn.ensemble import RandomForestRegressor from sklearn.model_selection import train_test_split from sklearn.metrics import mean_squared_error, r2_score # Selecting features and target variable X = cleaned_df[['Sales', 'Quantity', 'Discount', 'Category_no', 'Sub-Category_no']] y = cleaned_df['Profit'] # Split the data into train and test sets X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=48) # Create and fit the Random Forest Regression model rf_model = RandomForestRegressor(n_estimators=95, random_state=55) rf_model.fit(X_train, y_train) # Make predictions on the test set y_pred = rf_model.predict(X_test) # Evaluating the model rmse = np.sqrt( mean_squared_error(y_test, y_pred)) r2 = r2_score(y_test, y_pred) print(f"Root Mean Squared Error (RMSE): {rmse}") print(f"R-squared: {r2}") # Interpreting the results print("Feature Importances:") for feature, importance in zip(X.columns, rf_model.feature_importances_): print(f"{feature}: {importance}") # Residual Analysis Plot residuals = y_test - y_pred plt.figure(figsize=(10, 6)) plt. scatter(y_pred, residuals, alpha=0.5) plt.axhline(y=0, color='r', linestyle='--') plt.xlabel('Predicted Values') plt.ylabel('Residuals') plt.title('Residual Plot') plt.show() The Random Forest model has an R-squared of 0.8487, which means that it can explain approximately 84.87% of the profit variability. This suggests a relatively strong fit to the data. The Root Mean Squared Error (RMSE) of 5.51 indicates, on average, how much the predictions deviate from the actual profit values in the same units of the target variable. From the Random Forest model, we can see that ‘Sales’ has the highest importance score, indicating it is the most significant predictor of profit. This aligns with financial theory which suggests that sales volume is a primary driver of profit margins. ‘Discount’ also shows substantial importance, though less than sales, implying that the level of discounting has a sizable impact on profitability. Sub-Category_no’ and ‘Quantity’ have lower importance scores compared to ‘Sales’ and ‘Discount’, yet they are still meaningful contributors to profit prediction. This suggests that different sub-categories have varying profitability and that the quantity sold also plays a role, albeit smaller than sales and discounts. The residual plot shows a relatively even distribution of residuals above and below the zero line. However, the spread of residuals seems to increase as the predicted values increase. This might mean that for higher sales values, the model becomes less precise. By focusing solely on numerical predictors, we have streamlined our analysis but at the expense of potentially overlooking rich insights that categorical variables might offer. For the company, the clear indicator is to focus on sales enhancement strategies while monitoring the impact of discounts. Given the significant impact of ‘Sales’, efforts such as market expansion, sales promotions, and customer retention could be beneficial. A nuanced discounting strategy that does not undercut profitability could also be developed. Furthermore, the insights from the ‘Sub-Category_no’ highlight the importance of product mix optimisation. The company should analyse the profitability of each sub-category in more detail to prioritise high-margin products. When presenting these findings to the client, it's important to maintain transparency regarding these analytical boundaries. While the model sheds considerable light on the elements that drive profitability, its interpretation should be framed within these acknowledged constraints. To ensure the model remains reliable for business strategy, ongoing refinement and updates with new data are necessary. This allows the company to pivot and adapt with confidence backed by data. In conclusion, the Random Forest model has demonstrated its utility in identifying key profitability drivers. With a strong R-squared value, it proves to be a reliable tool for the company's forecasting and strategic planning (Kuhn and Johnson, 2013). Moving forward, the company is well-equipped to leverage these insights to enhance its sales strategies, optimise its discounting policies, and prioritise high-margin product offerings. By continuously iterating and refining its analytical approach, the company can position itself for sustained growth and profitability. Iglewicz, B. and Hoaglin, D.C. (1993). How to detect and handle outliers. Milwaukee, WI: ASQC Quality Press. James, G., Witten, D., Hastie, T. and Tibshirani, R. (2013). An introduction to statistical learning: with applications in R. New York, NY: Springer. Kuhn, M. and Johnson, K. (2013). Applied predictive modeling. New York, NY: Springer. Provost, F. and Fawcett, T. (2013). Data science for business: What you need to know about data mining and data-analytic thinking. Sebastopol, CA: O'Reilly Media. Tan, P.N., Steinbach, M. and Kumar, V. (2005). Introduction to data mining. Boston, MA: Pearson Addison Wesley.	{"url":"https://deepnote.com/app/c11bd-da74/Welcome-to-Deepnote-bf7e3a52-0c47-4b7c-9646-d0a7b74e9072","timestamp":"2024-11-08T09:13:40Z","content_type":"text/html","content_length":"227077","record_id":"<urn:uuid:7e325740-7dfe-47ee-88d3-718e2fc9f62d>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00873.warc.gz"}
A unified algorithm for accelerating edit-distance computation via text-compression The edit distance problem is a classical fundamental problem in computer science in general, and in combinatorial pattern matching in particular. The standard dynamic-programming solution for this problem computes the edit-distance between a pair of strings of total length O(N) in O(N^2) time. To this date, this quadratic upper-bound has never been substantially improved for general strings. However, there are known techniques for breaking this bound in case the strings are known to compress well under a particular compression scheme. The basic idea is to first compress the strings, and then to compute the edit distance between the compressed strings. As it turns out, practically all known o(N^2) edit-distance algorithms work, in some sense, under the same paradigm described above. It is therefore natural to ask whether there is a single edit-distance algorithm that works for strings which are compressed under any compression scheme. A rephrasing of this question is to ask whether a single algorithm can exploit the compressibility properties of strings under any compression method, even if each string is compressed using a different compression. In this paper we set out to answer this question by using straight-line programs. These provide a generic platform for representing many popular compression schemes including the LZ-family, Run-Length Encoding, Byte-Pair Encoding, and dictionary methods. For two strings of total length N having straight-line program representations of total size n, we present an algorithm running in O(n^1.4N^1.2) time for computing the edit-distance of these two strings under any rational scoring function, and an O(n^1.34N^1.34)-time algorithm for arbitrary scoring functions. This improves on a recent algorithm of Tiskin that runs in O(nN ^1.5) time, and works only for rational scoring functions. Original language English Title of host publication STACS 2009 - 26th International Symposium on Theoretical Aspects of Computer Science Pages 529-540 Number of pages 12 State Published - 2009 Event 26th International Symposium on Theoretical Aspects of Computer Science, STACS 2009 - Freiburg, Germany Duration: 26 Feb 2009 → 28 Feb 2009 Publication series Name Leibniz International Proceedings in Informatics, LIPIcs Volume 3 ISSN (Print) 1868-8969 Conference 26th International Symposium on Theoretical Aspects of Computer Science, STACS 2009 Country/Territory Germany City Freiburg Period 26/02/09 → 28/02/09 • Combinatorial pattern matching • Dynamic programming acceleration via compression • Edit distance • Straight-line programs ASJC Scopus subject areas Dive into the research topics of 'A unified algorithm for accelerating edit-distance computation via text-compression'. Together they form a unique fingerprint.	{"url":"https://cris.haifa.ac.il/en/publications/a-unified-algorithm-for-accelerating-edit-distance-computation-vi","timestamp":"2024-11-04T18:01:16Z","content_type":"text/html","content_length":"58665","record_id":"<urn:uuid:063a21c6-f37e-49b4-ada5-7ebd95d2791b>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00498.warc.gz"}
Dipping my toe into Maxwell's Law and FS Discriminator The law states the E field is represented by a vector. The vector length represents the strength of the field and its has direction. The vector can be rotated 0 to 360 degrees to show the direction of the field. Here we have some examples of vectors representing fields. There is also a law for magnetic fields which says we can represent B (magnetic) fields with vectors. There is a little difference between E and B fields. E fields extend to infinity or until they are absorbed by a particle of opposite polarity. E fields are radiated by positive charged and absorbed by negative charged particles. Magnetic fields form closed loops. There is a three way relation between E fields, B fields, and motion. E fields in motion create B fields. In other words current in a wire will create an electromagnet. A conductor moving in a B field will cause a current to flow in the wire. These factors will always be present perpendicular to each other. We analyze their interactions using vectors. The arrow above the E says use a vector to represent it. Some use FBI to remember the left hand rule. The thumb is F the direction the wire is Forced to travel. B is the direction of the magnetic field. (north to south) I is the direction of induced The rule being applied when the wire is fed a current while resting in a magnetic field. A current carrying conductor will have a field around it. This illustrates the field polarity. The E field will be radiating perpendicular to the wire surface. As I said before the E field, B field and motion. In this case the motion is current flow. This brings us to the left hand rule for coils. This is how to find the total inductance when coils are in series and parallel as long as the fields do not interact. Take a look at Ohm's Law. Put your thumb over the unknown and the formulas is what's left. Look at (A) two or more inductors 1/Lt = 1/L1 + 1/L2 + 1/L3 and assume E=1. Using I = E/R we find the current in each path. Adding give the total current and using the assumed 1 volt for 1/Lt gives our total. Anywho, I went through all that to get here. Using vector analysis we can see how the discriminator works.	{"url":"http://radio.radiotrician.org/2019/08/dipping-my-toe-into-maxwells-law-and-fs.html","timestamp":"2024-11-06T21:59:57Z","content_type":"text/html","content_length":"68122","record_id":"<urn:uuid:76e951db-69d0-4659-ad3e-49df5af43a4b>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00324.warc.gz"}
Parallelizing a Hybrid Finite Element-Boundary Integral Method for the Analysis of Scattering and Radiation of Electromagnetic Waves Duran Diaz, R.; Rico, R.; Garcia-Castillo, L. E.; Gomez-Revuelto, I.; Acebron, Juan; Martinez-Fernandez, I. Finite Elements in Analysis and Design, 46(8) (2010), 645-657 This paper presents the practical experience of parallelizing a simulator of general scattering and radiation electromagnetic problems. The simulator stems from an existing sequential simulator in the frequency domain, which is based on a finite element analysis. After the analysis of a test case, two steps were carried out: first, a “hand-crafted” code parallelization of a convolution-type operation was developed within the kernel of the simulator. Second, the sequential HSL library, used in the existing simulator, was replaced by the parallel MUMPS (MUltifrontal Massively Parallel sparse direct Solver) library in order to solve the associated linear algebra problem in parallel. Such a library allows for the distribution of the factorized matrix and some of the computational load among the available processors. A test problem and three realistic (in terms of the number of unknowns) cases have been run using the parallelized version of the code, and the results are presented and discussed focusing on the memory usage and achieved speed-up.	{"url":"https://cemat.tecnico.ulisboa.pt/document.php?project_id=4&member_id=118&doc_id=1412","timestamp":"2024-11-07T17:22:50Z","content_type":"text/html","content_length":"9245","record_id":"<urn:uuid:8cf7142b-7d02-4982-b32c-517a1a5fb243>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00549.warc.gz"}
In-depth explanation In this case as well, all methods are implemented as R6 classes. However, here we have also implemented helper functions for initialization, allowing the application of a method through a simple method call instead of $new(). These methods all start with the prefix run_ and end with the corresponding acronym for the method (e.g., run_grad()). Argument converter The Converter object from the first step is one of the crucial elements for the application of a selected method because it converted the original model into a torch structure necessary for innsight in which the methods are pre-implemented in each layer. Argument data In addition to the converter object, the input data is also essential as it will be analyzed and explained using the methods provided at the end. Accepted are data as: • Base R data types like matrix, array, data.frame or other array-like formats of size $\left(\text{batch_size}, \text{input_dim}\right)$. These formats can be used mainly when the model has only one input layer. Internally, the data is converted to an array using the as.array function and stored as a torch_tensor in the given dtype afterward. • torch_tensor: The converting process described in the last point can also be skipped by directly passing the data as a torch_tensor of size $\left(\text{batch_size}, \text{input_dim}\right)$. • list: You can also pass a list with the corresponding input data according to the upper points for each input layer. đź“ť Note The argument data is a necessary argument only for the local interpretation methods. Otherwise, it is unnecessary, e.g., the global variant of the Connection Weights method can be used without Argument channels_first This argument tells the package where the channel axis for images and signals is located in the input data. Internally, all calculations are performed with the channels in the second position after the batch dimension (â€śchannels firstâ€ť), e.g., c(10,3,32,32) for a batch of ten images with three channels and a height and width of $32$ pixels. Thus input data in the format â€śchannels lastâ€ť (i.e., c(10,32,32,3) for the previous example) must be transformed accordingly. If the given data has no channel axis, use the default value TRUE. Argument output_idx These indices specify the modelâ€™s output nodes for which the method is to be applied. For the sake of models with multiple output layers, the method object gives the following possibilities to select the indices of the output nodes in the individual output layers: • A vector of indices: If the model has only one output layer, the values correspond to the indices of the output nodes, e.g., c(1,3,4) for the first, third and fourth output node. If there are multiple output layers, the indices of the output nodes from the first output layer are considered. • A list of index vectors: If the method is to be applied to output nodes from different layers, a list can be passed that specifies the desired indices of the output nodes for each output layer. Unwanted output layers have the entry NULL instead of a vector of indices, e.g., list(NULL, c(1,3)) for the first and third output node in the second output layer. • NULL (default): The method is applied to all output nodes in the first output layer but is limited to the first ten as the calculations become more computationally expensive for more output Argument output_label These values specify the output nodes for which the method is to be applied and can be used as an alternative to the argument output_idx. Only values that were previously passed with the argument output_names in the converter can be used. In order to allow models with multiple output layers, there are the following possibilities to select the names of the output nodes in the individual output • A character vector or factor of labels: If the model has only one output layer, the values correspond to the labels of the output nodes named in the passed Converter object, e.g., c("a", "c", "d") for the first, third and fourth output node if the output names are c("a", "b", "c", "d"). If there are multiple output layers, the names of the output nodes from the first output layer are • A list of charactor/factor vectors of labels: If the method is to be applied to output nodes from different layers, a list can be passed that specifies the desired labels of the output nodes for each output layer. Unwanted output layers have the entry NULL instead of a vector of labels, e.g., list(NULL, c("a", "c")) for the first and third output node in the second output layer. • NULL (default): The method is applied to all output nodes in the first output layer but is limited to the first ten as the calculations become more computationally expensive for more output Argument ignore_last_act Set this logical value to include the last activation function for each output layer, or not (default: TRUE). In practice, the last activation (especially for softmax activation) is often omitted. Argument dtype This argument defines the numerical precision with which all internal calculations are performed. Accepted are currently 32-bit floating point ("float" the default value) and 64-bit floating point numbers ("double"). All weights, constants and inputs are then converted accordingly into the data format torch_float() or torch_double(). See the argument dtype in the Converter object for more As described earlier, all implemented methods inherit from the InterpretingMethod super class. But each method has method-specific arguments and different objectives. To make them a bit more understandable, they are all explained with the help of the following simple example model with ReLU activation in the first, hyperbolic tangent in the last layer and only one in- and output node: Create the model from Fig. 1 model <- list( input_dim = 1, input_nodes = 1, input_names = c("x"), output_nodes = 2, output_names = c("y"), layers = list( type = "Dense", input_layers = 0, output_layers = 2, weight = matrix(c(1, 0.8, 2), nrow = 3), bias = c(0, -0.4, -1.2), activation_name = "relu" type = "Dense", input_layers = 1, output_layers = -1, weight = matrix(c(1, -1, 1), nrow = 1), bias = c(0), activation_name = "tanh" converter <- convert(model) Vanilla Gradient One of the first and most intuitive methods for interpreting neural networks is the Gradients method introduced by Simonyan et al.Â (2013), also known as Vanilla Gradients or Saliency maps. This method computes the gradients of the selected output with respect to the input variables. Therefore the resulting relevance values indicate prediction-sensitive variables, i.e., those variables that can be locally perturbed the least to change the outcome the most. Mathematically, this method can be described by the following formula for the input variable $x_i$ with $x \in \mathbb{R}^n$, the model $f:\mathbb{R}^n \to \mathbb{R}^C$ and the output $y_c = f(x)_c$ of class $c$: \[ \text{Gradient}(x)_i^c = \frac{\partial\ f(x)_c}{\partial\ x_i} = \frac{\partial\ y_c}{\partial\ x_i} As described in the introduction of this section, the corresponding innsight-method Gradient inherits from the super class InterpretingMethod, meaning that we need to change the term Method to Gradient. Alternatively, an object of the class Gradient can also be created using the mentioned helper function run_grad(), which does not require prior knowledge of R6 objects. The only model-specific argument is times_input, which can be used to switch between the two methods Gradient (default FALSE) and Gradient$\times$Input (TRUE). For more information on the method Gradient$\ times$Input see this subsection. # R6 class syntax grad <- Gradient$new(converter, data, times_input = FALSE, ... # other arguments inherited from 'InterpretingMethod' # Using the helper function grad <- run_grad(converter, data, times_input = FALSE, ... # other arguments inherited from 'InterpretingMethod' Example with visualization In this example, we want to describe the data point $x_1 = 0.45$ with the Gradient method. In principle, the slope of the tangent in $x_1$ is calculated and thus the local rate of change, which in this case is $\tanh'(x_1) = \frac{1}{\cosh(x_1)^2} = 0.822$ (see the red line in Fig. 2). Assuming that the function behaves linearly overall, increasing $x$ by one raises the output by \ (0.822\). In general, however, neural networks are highly nonlinear, so this interpretation is only valid for very small changes of $x_1$ as you can see in Fig. 2. With innsight, this method is applied as follows and we receive the same result: The SmoothGrad method, introduced by Smilkov et al.Â (2017), addresses a significant problem of the basic Gradient method. As described in the previous subsection, gradients locally assume a linear behavior, but this is generally no longer the case for deep neural networks. These have large fluctuations and abruptly change their gradients, making the interpretations of the gradient worse and potentially misleading. Smilkov et al.Â proposed that instead of calculating only the gradient in $x$, compute the gradients of randomly perturbed copies of $x$ and determine the mean gradient from that. To use the SmoothGrad method to obtain relevance values for the individual components $x_i \in \mathbb{R}$ of an instance $x \in \mathbb{R^n}$, we first generate $K \in \mathbb{N}$ realizations of a multivariate Gaussian distribution $p = \mathcal{N}(0, \sigma^2)$ describing the random perturbations, i.e., $\varepsilon^1, \ldots, \varepsilon^K \sim p$. Then the empirical mean of the gradients for variable $x_i$ and output index $c$ can be calculated as follows: \[ \text{SmoothGrad}(x)_i^c = \frac{1}{K} \sum_{j = 1}^K \frac{\partial\ f(x + \varepsilon^j)_c}{\partial\ x_i + \varepsilon_i^j} \approx \mathbb{E}_{\varepsilon \sim p}\left[ \frac{\partial\ f(x + \ varepsilon)_c}{\partial\ x_i + \varepsilon_i^j}\right] \] As described in the introduction of this section, the innsight method SmoothGrad inherits from the super class InterpretingMethod, meaning that we need to change the term Method to SmoothGrad or use the helper function run_smoothgrad() for initializing an object of class SmoothGrad. In addition, there are the following three model-specific arguments: • n (default: 50): This integer value specifies how many perturbations will be used to calculate the mean gradient, i.e., the $K$ from the formula above. However, it must be noted that the computational effort increases by a factor of n compared to the Gradient method since the simple Gradient method is used n times instead of once. In return, the accuracy of the estimator increases with a larger n. • noise_level (default: 0.1): With this argument, the strength of the spread of the Gaussian distribution can be given as a percentage, i.e., noise_level $=\frac{\sigma}{\max(x)-\min(x)}$. • times_input (default: FALSE): Similar to the Gradient method, this argument can be used to switch between the two methods SmoothGrad (FALSE) and SmoothGrad$\times$Input (TRUE). For more information on the method SmoothGrad$\times$Input see this subsection. # R6 class syntax smoothgrad <- SmoothGrad$new(converter, data, n = 50, noise_level = 0.1, times_input = FALSE, ... # other arguments inherited from 'InterpretingMethod' # Using the helper function smoothgrad <- run_smoothgrad(converter, data, n = 50, noise_level = 0.1, times_input = FALSE, ... # other arguments inherited from 'InterpretingMethod' Example with visualization We want to describe the data point $x_1 = 0.6$ with the method SmoothGrad. As you can see in Figure 3, this point does not have a unique gradient because it is something around $0.15$ from the left and something around $1.66$ from the right. In such situations, SmoothGrad comes in handy. As described before, the input $x_1$ is slightly perturbed by a Gaussian distribution and then the mean gradient is calculated. The individual gradients of the perturbed copies are visualized in blue in Figure 3 with the red line representing the mean gradient. With innsight, this method is applied as follows: Gradient$\times$Input and SmoothGrad$\times$Input The methods Gradient$\times$Input and SmoothGrad$\times$Input are as simple as they sound: the gradients are calculated as in the gradient section and then multiplied by the respective input. They were introduced by Shrikumar et al. (2016) and have a well-grounded mathematical background despite their simple idea. The basic idea is to decompose the output according to its relevance to each input variable, i.e., we get variable-wise additive effects \[ \tag{1} f(x)_c = \sum_{i = 1}^n R_i. \] Mathematically, this method is based on the first-order Taylor decomposition. Assuming that the function $f$ is continuously differentiable in $x \in \mathbb{R}^n$, a remainder term $\varepsilon (f,z,x):\mathbb{R}^n \to \mathbb{R}$ with $\lim_{z \to x} \varepsilon(f, z, x) = 0$ exists such that \[ f(z) &= f(x) + \nabla_x\ f(x)(z-x)^T + \varepsilon(f, z, x)\\ &= f(x) + \sum_{i = 1}^n \frac{\partial\ f(x)}{\partial\ x_i} (z_i - x_i) + \varepsilon(f, z, x), \quad z\in \mathbb{R}^n. \tag{2} \] The first-order Taylor formula thus describes a linear approximation of the function $f$ at the point $x$ since only the first derivatives are considered. Consequently, a highly nonlinear function $f$ is well approximated in a small neighborhood around $x$. For larger distances from $x$, sufficient small values of the residual term are not guaranteed anymore. The Gradient$\ times$Input method now considers the data point $x$ and sets $z = 0$. In addition, the residual term and the summand $f(0)_c$ are ignored, which then results in the following approximation of $f(x)_c$ in variable-wise relevances \[ f(x)_c \approx \sum_{i = 1}^n \frac{\partial\ f(x)_c}{\partial\ x_i} \cdot x_i, \quad \text{hence}\\ \text{Gradient$\times$Input}(x)_i^c = \frac{\partial\ f(x)_c}{\partial\ x_i} \cdot x_i. \] Derivation from Eq. 2 \[ &f(z)_c = f(x)_c + \sum_{i = 1}^n \frac{\partial\ f(x)_c}{\partial\ x_i} (z_i - x_i) + \varepsilon(f, z, x)\\ \Leftrightarrow\quad & f(x)_c = f(z)_c - \sum_{i = 1}^n \frac{\partial\ f(x)_c}{\ partial\ x_i} (z_i - x_i) - \varepsilon(f, z, x)\\ \Leftrightarrow\quad & f(x)_c = f(z)_c + \sum_{i = 1}^n \frac{\partial\ f(x)_c}{\partial\ x_i} (x_i - z_i) - \varepsilon(f, z, x) \] Hence, we get for $z = 0$ and after ignoring the remainder term and the value $f(0)_c$ \[ f(x)_c &= f(0)_c + \sum_{i = 1}^n \frac{\partial\ f(x)_c}{\partial\ x_i} x_i - \varepsilon(f, z, x) \tag{3}\\ &\approx \sum_{i = 1}^n \frac{\partial\ f(x)_c}{\partial\ x_i} x_i \] Analogously, this multiplication is also applied to the SmoothGrad method in order to compensate for local fluctuations: \[ \text{SmoothGrad$\times$Input}(x)_i^c = \frac{1}{K} \sum_{j = 1}^K \frac{\ partial\ f(x + \varepsilon^j)_c}{\partial\ x_i + \varepsilon_i^j} \cdot (x_i + \varepsilon_i^j),\quad \varepsilon^1, \ldots, \varepsilon^K \sim \mathcal{N}(0,\sigma^2). \] Both methods are variants of the respective gradient methods Gradient and SmoothGrad and also have the corresponding model-specific arguments and helper functions for the initialization. These variants can be chosen with the argument times_input: # the "x Input" variant of method "Gradient" grad_x_input <- Gradient$new(converter, data, times_input = TRUE, ... # other arguments of method "Gradient" # the same using the corresponding helper function grad_x_input <- run_grad(converter, data, times_input = TRUE, ... # other arguments of method "Gradient" # the "x Input" variant of method "SmoothGrad" smoothgrad_x_input <- SmoothGrad$new(converter, data, times_input = TRUE, ... # other arguments of method "SmoothGrad" # the same using the corresponding helper function smoothgrad_x_input <- run_smoothgrad(converter, data, times_input = TRUE, ... # other arguments of method "SmoothGrad" Example with visualization Now let us describe the data point $x_1 = 0.49$ using the model defined in this chapterâ€™s introduction. For this model holds the equation $f(0) = 0$; therefore, the approximation error is only the negative value of the remainder term at $0$ (as seen in Eq. 3). In Figure 4, the Taylor approximation is drawn in red and at position $0$, you can also see the value of the remainder term (because all other summands are zero). At the same time, the red dot describes the result of the Gradient$\times$Input method, which indeed deviates from the actual value only by the negative of the remainder term at position $0$. With innsight, this method is applied as follows: data <- matrix(c(0.49), 1, 1) # Apply method grad_x_input <- run_grad(converter, data, times_input = TRUE, ignore_last_act = FALSE # include the tanh activation # get result #> , , y #> x #> [1,] 0.3889068 It is also possible to use the SmoothGrad$\times$Input method to perturb the input $x_1 = 0.49$ a bit and return an average value of the individual Gradient$\times$Input results. Figure 5 shows the individual linear approximations of the first-order Taylors for the Gaussian perturbed copies of $x_1$, and the blue dots describe the respective Gradient$\times$Input values. The red dot represents the mean value, i.e., the value of the SmoothGrad$\times$Input method at $x_1 = 0.49$. With innsight, this method is applied as follows: Layer-wise Relevance Propagation (LRP) The LRP method was first introduced by Bach et al.Â (2015) and has a similar goal to the Gradient$\times$Input approach explained in the last section: decompose the output into variable-wise relevances according to Eq. 1. The difference is that the prediction $f(x)_c = y_c$ is redistributed layer by layer from the output node back to the inputs according to the weights and pre-activations. This is done by so-called relevance messages $r_{i \leftarrow j}^{(l, l+1)}$, which can be defined by a rule on redistributing the upper-layer relevance $R_j^{l +1}$ to the lower-layer $R_i^{l}$. In the package innsight, the following commonly used rules are defined ($i$ is an index of a node in layer $l$ and $j$ an index of a node in layer $l+1$): • The simple rule (also known as LRP-0) This is the most basic rule on which all other rules are more or less based. The relevances are redistributed to the lower layers according to the ratio between local and global pre-activation. Let $x_i$ the inputs, $w_{i,j}$ the weights and $b_j$ the bias vector of layer $l$ and $R_j^{(l+1)}$ the upper-layer relevance, then the simple rule is defined as \[ r_{i \leftarrow j}^ {(l, l+1)} = \frac{x_i\, w_{i,j}}{z_j} \, R_j^{l +1} \quad \text{with} \quad z_j = b_j + \sum_{k} x_k\, w_{k,j}. \] • The $\varepsilon$-rule (also known as LRP-$\epsilon$) One problem with the simple rule is that it is numerically unstable when the global pre-activation $z_j$ vanishes and causes a division by zero. This problem is solved in the $\varepsilon$ -rule by adding a stabilizer $\varepsilon > 0$ that moves the denominator away from zero, i.e., \[ r_{i \leftarrow j}^{(l, l+1)} = \frac{x_i\, w_{i,j}}{z_j + \text{sign}(z_j)\, \varepsilon}\, R_j^{l +1}. \] • The $\alpha$-$\beta$-rule (also known as LRP-$\alpha \beta$) Another way to avoid this numerical instability is by treating the positive and negative pre-activations separately. In this case, positive and negative values cannot cancel each other out, i.e., a vanishing denominator also results in a vanishing numerator. Moreover, this rule allows choosing a weighting for the positive and negative relevances, which is done with the parameters $\ alpha, \beta \in \mathbb{R}$ satisfying $\alpha + \beta = 1$. The $\alpha$-$\beta$-rule is defined as \[ r_{i \leftarrow j}^{(l, l+1)} = \left(\alpha \frac{(x_i\, w_{i,j})^+}{z_j^+} + \ beta \frac{(x_i\, w_{i,j})^-}{z_j^-}\right)\, R_j^{l +1}\\ \text{with}\quad z_j^\pm = (b_j)^\pm + \sum_k (x_k\, w_{k,j})^\pm,\quad (\cdot)^+ = \max(\cdot, 0),\quad (\cdot)^- = \min(\cdot, 0). \] For any of the rules described above, the relevance of the lower-layer nodes $R_i^{l}$ is determined by summing up all incoming relevance messages $r_{i \leftarrow j}^{(l, l +1)}$ into the respective node of index $i$, i.e., \[ R_i^{l} = \sum_j r_{i \leftarrow j}^{(l, l +1)}. \] This procedure is repeated layer by layer until one gets to the input layer and consequently gets the relevances for each input variable. A visual overview of the entire method using the simple rule as an example is given in Fig. 6. đź“ť Note At this point, it must be mentioned that the LRP variants do not lead to an exact decomposition of the output since some of the relevance is absorbed by the bias terms. This is because the bias is included in the pre-activation but does not appear in any of the numerators. Analogous to the previous methods, the innsight method LRP inherits from the InterpretingMetod super class and thus all arguments. In addition, there are the following method-specific arguments for this method: • rule_name (default: "simple"): This argument can be used to select the rule for the relevance messages. Implemented are the three rules described above, i.e., simple rule ("simple"), $\ varepsilon$-rule ("epsilon") and $\alpha$-$\beta$-rule ("alpha_beta"). However, a named list can also be passed to assign one of these three rules to each implemented layer type individually. Layers not specified in this list then use the default value "simple". For example, with list(Dense_Layer = "epsilon", Conv2D_Layer = "alpha_beta") the simple rule is used for all dense layers and the $\alpha$-$\beta$-rule is applied to all 2D convolutional layers. The other layers not mentioned use the default rule. In addition, for normalization layers like 'BatchNorm_Layer', the rule "pass" is implemented as well, which ignores such layers in the backward pass. You can set the rule for the following layer types: □ 'Dense_Layer', 'Conv1D_Layer', 'Conv2D_Layer', 'BatchNorm_Layer', 'AvgPool1D_Layer', 'AvgPool2D_Layer', 'MaxPool1D_Layer' and 'MaxPool2D_Layer' • rule_param: The meaning of this argument depends on the selected rule. For the simple rule, for example, it has no effect. In contrast, this numeric argument sets the value of $\varepsilon$ for the $\varepsilon$-rule and the value of $\alpha$ for the $\alpha$-$\beta$-rule (remember: $\beta = 1 - \alpha$). Passing NULL defaults to 0.01 for $\varepsilon$ or 0.5 for $\alpha$. Similar to the argument rule_name, this can also be a named list that individually assigns a rule parameter to each layer type. • winner_takes_all: This logical argument is only relevant for models with a MaxPooling layer. Since many zeros are produced during the backward pass due to the selection of the maximum value in the pooling kernel, another variant is implemented, which treats a MaxPooling as an AveragePooling layer in the backward pass to overcome the problem of too many zero relevances. With the default value TRUE, the whole upper-layer relevance is passed to the maximum value in each pooling window. Otherwise, if FALSE, the relevance is distributed equally among all nodes in a pooling window. # R6 class syntax lrp <- LRP$new(converter, data, rule_name = "simple", rule_param = NULL, winner_takes_all = TRUE, ... # other arguments inherited from 'InterpretingMethod' # Using the helper function for initialization lrp <- run_lrp(converter, data, rule_name = "simple", rule_param = NULL, winner_takes_all = TRUE, ... # other arguments inherited from 'InterpretingMethod' First, letâ€™s look again at the result at the point $x_1 = 0.49$, which was about $0.3889$ when approximated with the Gradient$\times$Input method. For LRP with the simple rule, we get \ (0.4542\) which exactly matches the actual value of $f(x_1)$. This is mainly due to the fact that for an input of $x_1$, only the top neuron from Fig. 1 is activated and it does not have a bias term. However, if we now use an input that activates a neuron with a bias term ($x_2 = 0.6$), there will be an approximation error (for $x_2$ itâ€™s $-0.3675$) since it absorbs some of the relevance. See the code below: # We can analyze multiple inputs simultaneously data <- matrix( 0.49, # only neuron without bias term is activated 0.6 # neuron with bias term is activated ncol = 1 # Apply LRP with simple rule lrp <- run_lrp(converter, data, ignore_last_act = FALSE #> , , y #> x #> [1,] 0.4542146 #> [2,] 0.1102428 # get approximation error matrix(lrp$get_result()) - as_array(converter$model(torch_tensor(data))[[1]]) #> [,1] #> [1,] -1.877546e-06 #> [2,] -3.674572e-01 The individual LRP variants can also be considered as a function in the input variable $x$, which is shown in Fig. 7 with the true model $f$ in black. Deep Learning Important Features (DeepLift) One method that, to some extent, echoes the idea of LRP is the so-called Deep Learning Important Features (DeepLift) method introduced by Shrikumar et al. in 2017. It behaves similarly to LRP in a layer-by-layer backpropagation fashion from a selected output node back to the input variables. However, it incorporates a reference value $\tilde{x}$ to compare the relevances with each other. Hence, the relevances of DeepLift represent the relative effect of the outputs of the instance to be explained $f(x)_c$ and the output of the reference value $f(\tilde{x})_c$, i.e., $f(x)_c - f (\tilde{x})_c$. This difference eliminates the bias term in the relevance messages so that no more relevance is absorbed and we have an exact variable-wise decomposition of $\Delta y = f(x)_c - f(\ tilde{x})_c$. In addition, the authors presented two rules to propagate relevances through the activation part of the individual layers, namely Rescale and RevealCancel rule. The Rescale rule simply scales the contribution to the difference from reference output according to the value of the activation function. The RevealCancel rule considers the average impact after adding the negative or positive contribution revealing dependencies missed by other approaches. Analogous to the previous methods, the innsight method DeepLift inherits from the InterpretingMetod super class and thus all arguments. Alternatively, an object of the class DeepLift can also be created using the helper function run_deeplift(), which does not require prior knowledge of R6 objects. In addition, there are the following method-specific arguments for this method: • x_ref (default: NULL): This argument describes the reference input $\tilde{x}$ for the DeepLift method. This value must have the same format as the input data of the passed model to the converter class, i.e., □ an array, data.frame, torch_tensor or array-like format of size $\left(1, \text{input_dim}\right)$ or □ a list with the corresponding input data (according to the upper point) for each of the input layers. □ It is also possible to use the default value NULL to take only zeros as reference input. • rule_name (default: 'rescale'): Name of the applied rule to calculate the contributions. Use either 'rescale' or 'reveal_cancel'. • winner_takes_all: This logical argument is only relevant for MaxPooling layers and is otherwise ignored. With this layer type, it is possible that the position of the maximum values in the pooling kernel of the normal input $x$ and the reference input $\tilde{x}$ may not match, which leads to a violation of the summation-to-delta property. To overcome this problem, another variant is implemented, which treats a MaxPooling layer as an AveragePooling layer in the backward pass only, leading to a uniform distribution of the upper-layer contribution to the lower layer. # R6 class syntax deeplift <- DeepLift$new(converter, data, x_ref = NULL, rule_name = "rescale", winner_takes_all = TRUE, ... # other arguments inherited from 'InterpretingMethod' # Using the helper function for initialization deeplift <- run_deeplift(converter, data, x_ref = NULL, rule_name = "rescale", winner_takes_all = TRUE, ... # other arguments inherited from 'InterpretingMethod' In this example, letâ€™s consider the point $x = 0.55$ and the reference point $\tilde{x} = 0.1$. With the help of the model defined previously, the respective outputs are $y = f(x) = 0.4699$ and $\tilde{y} = f(\tilde{x}) = 0.0997$. The DeepLift method now generates an exact variable-wise decomposition of the so-called difference-from-reference value $\Delta y = y - \tilde{y} = 0.3702772$. Since there is only one input feature in this case, the entire value should be assigned to it: # Create data x <- matrix(c(0.55)) x_ref <- matrix(c(0.1)) # Apply method DeepLift with rescale rule deeplift <- run_deeplift(converter, x, x_ref = x_ref, ignore_last_act = FALSE) # Get result #> , , y #> x #> [1,] 0.3702772 This example is an extremely simple model, so we will test this method on a slightly larger model and the Iris dataset (see ?iris): # Crate model with package 'neuralnet' model <- neuralnet(Species ~ ., iris, hidden = 5, linear.output = FALSE) # Step 1: Create 'Converter' conv <- convert(model) # Step 2: Apply DeepLift (reveal-cancel rule) x_ref <- matrix(colMeans(iris[, -5]), nrow = 1) # use colmeans as reference value deeplift <- run_deeplift(conv, iris[, -5], x_ref = x_ref, ignore_last_act = FALSE, rule_name = "reveal_cancel" # Verify exact decomposition y <- predict(model, iris[, -5]) y_ref <- predict(model, x_ref[rep(1, 150), ]) delta_y <- y - y_ref summed_decomposition <- apply(get_result(deeplift), c(1, 3), FUN = sum) # dim 2 is the input feature dim # Show the mean squared error mean((delta_y - summed_decomposition)^2) #> [1] 6.402983e-15 Integrated Gradients In the Integrated Gradients method introduced by Sundararajan et al.Â (2017), the gradients are integrated along a path from the value $x$ to a reference value $\tilde{x}$. This integration results, similar to DeepLift, in a decomposition of $f(x) - f(\tilde{x})$. In this sense, the method uncovers the feature-wise relative effect of the input features on the difference between the prediction $f(x)$ and the reference prediction $f(\tilde{x})$. This is archived through the following formula: \[ \text{IntGrad}(x)_i^c = (x - \tilde{x}) \int_{\alpha = 0}^1 \frac{\partial f(\ tilde{x} + \alpha (x - \tilde{x}))}{\partial x} d\alpha \] In simpler terms, it calculates how much each feature contributes to a modelâ€™s output by tracing a path from a baseline input $\tilde{x} $ to the actual input $x$ and measuring the average gradients along that path. Similar to the other gradient-based methods, by default the integrated gradient is multiplied by the input to get an approximate decomposition of $f(x) - f(\tilde{x})$. However, with the parameter times_input only the gradient describing the output sensitivity can be returned. Analogous to the previous methods, the innsight method IntegratedGradient inherits from the InterpretingMetod super class and thus all arguments. Alternatively, an object of the class IntegratedGradient can also be created using the helper function run_intgrad(), which does not require prior knowledge of R6 objects. In addition, there are the following method-specific arguments for this method: • x_ref (default: NULL): This argument describes the reference input $\tilde{x}$ for the Integrated Gradients method. This value must have the same format as the input data of the passed model to the converter class, i.e., □ an array, data.frame, torch_tensor or array-like format of size $\left(1, \text{input_dim}\right)$ or □ a list with the corresponding input data (according to the upper point) for each of the input layers. □ It is also possible to use the default value NULL to take only zeros as reference input. • n (default: 50): Number of steps for the approximation of the integration path along $\alpha$. • times_input (default: TRUE): Multiplies the integrated gradients with the difference of the input features and the baseline values. By default, the original definition of Integrated Gradient is applied. However, by setting times_input = FALSE only an approximation of the integral is calculated, which describes the sensitivity of the features to the output. # R6 class syntax intgrad <- IntegratedGradient$new(converter, data, x_ref = NULL, n = 50, times_input = TRUE, ... # other arguments inherited from 'InterpretingMethod' # Using the helper function for initialization intgrad <- run_intgrad(converter, data, x_ref = NULL, n = 50, times_input = TRUE, ... # other arguments inherited from 'InterpretingMethod' In this example, letâ€™s consider the point $x = 0.55$ and the reference point $\tilde{x} = 0.1$. With the help of the model defined previously, the respective outputs are $y = f(x) = 0.4699$ and $\tilde{y} = f(\tilde{x}) = 0.0997$. The Integrated Gradient method now generates an approximate variable-wise decomposition of the so-called difference-from-reference value $\Delta y = y - \ tilde{y} = 0.3702772$. Since there is only one input feature in this case, the entire value should be assigned to it: Expected Gradients The Expected Gradients method (Erion et al., 2021), also known as GradSHAP, is a local feature attribution technique which extends the Integrated Gradient method and provides approximate Shapley values. In contrast to Integrated Gradient, it considers not only a single reference value $\tilde{x}$ but the whole distribution of reference values $\tilde{X} \sim \tilde{x}$ and averages the Integrated Gradient values over this distribution. Mathematically, the method can be described as follows: \[ \text{ExpGrad}(x)_i^c = \mathbb{E}_{\tilde{x}\sim \tilde{X}, \alpha \sim U(0,1)} \left[(x - \tilde{x}) \times \frac{\partial f(\tilde{x} + \alpha (x - \tilde{x}))}{\partial x} \right] \] These feature-wise values approximate a decomposition of the prediction minus the average prediction in the reference dataset, i.e., $f(x) - \mathbb{E}_{\tilde{x}}\left[f(\tilde{x}) \right]$. This means, it solves the issue of choosing the right reference value. Analogous to the previous methods, the innsight method ExpectedGradient inherits from the InterpretingMetod super class and thus all arguments. Alternatively, an object of the class ExpectedGradient can also be created using the helper function run_expgrad(), which does not require prior knowledge of R6 objects. In addition, there are the following method-specific arguments for this method: • data_ref (default: NULL): This argument describes the reference inputs $\tilde{x}$ for the Expected Gradients method. This value must have the same format as the input data of the passed model to the converter class, i.e., □ an array, data.frame, torch_tensor or array-like format of size $\left(1, \text{input_dim}\right)$ or □ a list with the corresponding input data (according to the upper point) for each of the input layers. □ It is also possible to use the default value NULL to take only zeros as reference input. • n (default: 50): Number of samples from the distribution of reference values $\tilde{x} \sim \tilde{X}$ and number of samples for the approximation of the integration path along $\alpha$. # R6 class syntax expgrad <- ExpectedGradient$new(converter, data, data_ref = NULL, n = 50, ... # other arguments inherited from 'InterpretingMethod' # Using the helper function for initialization expgrad <- run_expgrad(converter, data, x_ref = NULL, n = 50, ... # other arguments inherited from 'InterpretingMethod' In the following example, we demonstrate how the Expected Gradient method is applied to the Iris dataset, accurately approximating the difference between the prediction and the mean prediction (adjusted for a very high sample size of $10\,000$): # Crate model with package 'neuralnet' model <- neuralnet(Species ~ ., iris, linear.output = FALSE) # Step 1: Create 'Converter' conv <- convert(model) # Step 2: Apply Expected Gradient expgrad <- run_expgrad(conv, iris[c(1, 60), -5], data_ref = iris[, -5], ignore_last_act = FALSE, n = 10000 # Verify exact decomposition y <- predict(model, iris[, -5]) delta_y <- y[c(1, 60), ] - rbind(colMeans(y), colMeans(y)) summed_decomposition <- apply(get_result(expgrad), c(1, 3), FUN = sum) # dim 2 is the input feature dim # Show the error between both delta_y - summed_decomposition #> setosa versicolor virginica #> [1,] -0.09275362 -0.003985531 -0.001253254 #> [2,] 0.01313798 0.001135939 -0.003019523 The DeepSHAP method (Lundberg & Lee, 2017) extends the DeepLift technique by not only considering a single reference value but by calculating the average from several, ideally representative reference values at each layer. The obtained feature-wise results are approximate Shapley values for the chosen output, where the conditional expectation is computed using these different reference values, i.e., the DeepSHAP method decompose the difference from the prediction and the mean prediction $f(x) - \mathbb{E}_{\tilde{x}}\left[f(\tilde{x}) \right]$ in feature-wise effects. This means, the DeepSHAP method has the same underlying goal as the Expected Gradient method and, hence, also solves the issue of choosing the right reference value for the DeepLift method. Analogous to the previous methods, the innsight method DeepSHAP inherits from the InterpretingMetod super class and thus all arguments. Alternatively, an object of the class DeepSHAP can also be created using the helper function run_deepshap()`, which does not require prior knowledge of R6 objects. In addition, there are the following method-specific arguments for this method: • data_ref (default: NULL): The reference data which is used to estimate the conditional expectation. These must have the same format as the input data of the passed model to the converter object. This means either □ an array, data.frame, torch_tensor or array-like format of size $\left(1, \text{input_dim}\right)$ or □ a list with the corresponding input data (according to the upper point) for each of the input layers. □ It is also possible to use the default value NULL to take only zeros as reference input. • limit_ref (default: 100): This argument limits the number of instances taken from the reference dataset data_ref so that only random limit_ref elements and not the entire dataset are used to estimate the conditional expectation. A too-large number can significantly increase the computation time. • (other model-specific arguments already explained in the DeepLift method, e.g., rule_name or winner_takes_all). # R6 class syntax deepshap <- DeepSHAP$new(converter, data, data_ref = NULL, limit_ref = 100, ... # other arguments inherited from 'DeepLift' # Using the helper function for initialization deepshap <- run_deepshap(converter, data, data_ref = NULL, limit_ref = 100, ... # other arguments inherited from 'DeepLift' In the following example, we demonstrate how the DeepSHAP method is applied to the Iris dataset, accurately approximating the difference between the prediction and the mean prediction (adjusted for a very high sample size of $10\,000$): # Crate model with package 'neuralnet' model <- neuralnet(Species ~ ., iris, linear.output = FALSE) # Step 1: Create 'Converter' conv <- convert(model) # Step 2: Apply Expected Gradient deepshap <- run_deepshap(conv, iris[c(1, 60), -5], data_ref = iris[, -5], ignore_last_act = FALSE, limit_ref = nrow(iris) # Verify exact decomposition y <- predict(model, iris[, -5]) delta_y <- y[c(1, 60), ] - rbind(colMeans(y), colMeans(y)) summed_decomposition <- apply(get_result(deepshap), c(1, 3), FUN = sum) # dim 2 is the input feature dim # Show the error between both delta_y - summed_decomposition #> setosa versicolor virginica #> [1,] -5.339583e-08 3.274975e-08 3.632456e-08 #> [2,] -9.623667e-09 -4.037981e-08 2.742707e-08 Connection Weights One of the earliest methods specifically for neural networks was the Connection Weights method invented by Olden et al. in 2004, resulting in a global relevance score for each input variable. The basic idea of this approach is to multiply all path weights for each possible connection between an input variable and the output node and then calculate the sum of all of them. However, this method ignores all bias vectors and all activation functions during calculation. Since only the weights are used, this method is independent of input data and, thus, a global interpretation method. In this package, we extended this method to a local one inspired by the method Gradient$\times$Input (see here). Hence, the local variant is simply the point-wise product of the global Connection Weights method and the input data. You can use this variant by setting the times_input argument to TRUE and providing input data. The innsight method ConnectionWeights also inherits from the super class InterpretingMethod, meaning that you need to change the term Method to ConnectionWeights. Alternatively, an object of the class ConnectionWeights can also be created using the helper function run_cw(), which does not require prior knowledge of R6 objects. The only model-specific argument is times_input, which can be used to switch between the global (FALSE) and the local (TRUE) Connection Weights method. # The global variant (argument 'data' is no longer required) cw_global <- ConnectionWeights$new(converter, times_input = FALSE, ... # other arguments inherited from 'InterpretingMethod' # The local variant (argument 'data' is required) cw_local <- ConnectionWeights$new(converter, data, times_input = TRUE, ... # other arguments inherited from 'InterpretingMethod' # Using the helper function cw_local <- run_cw(converter, data, times_input = TRUE, ... # other arguments inherited from 'InterpretingMethod' Since the global Connection Weights method only multiplies the path weights, the result for the input feature $x$ based on Figure 1 is \[ (1 \cdot 1) + (0.8 \cdot -1) + (2 \cdot 1) = 2.2. \] With the innsight package, we get the same value: # Apply global Connection Weights method cw_global <- run_cw(converter, times_input = FALSE) # Show the result #> , , y #> x #> [1,] 2.2 However, the local variant requires input data data and returns instance-wise relevances:	{"url":"http://rsync.jp.gentoo.org/pub/CRAN/web/packages/innsight/vignettes/detailed_overview.html","timestamp":"2024-11-07T19:00:06Z","content_type":"text/html","content_length":"1048887","record_id":"<urn:uuid:2813ceaf-bfbf-4f09-b268-d3b1ff097d4f>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00585.warc.gz"}
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Second (physics) The second is a unit of time, currently defined in the SI as the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom.^[1] Previously, the second had been defined as 1/86 400 of a day, with 60 seconds making one minute, 60 minutes making one hour, and 24 hours making one day, and the day being defined as one mean solar day using astronomical observations. The SI second is 1/86 400 of a mean solar day as measured in 1820; since then, the rotation of the earth has slowed, and the mean solar day is approximately 86 400.002 seconds long. Practical details in achieving a realization of the second are described by the BIPM.^[2] The word "second" is often used colloquially to mean any very short amount of time.	{"url":"https://citizendium.org/wiki/Second","timestamp":"2024-11-09T09:24:15Z","content_type":"text/html","content_length":"31263","record_id":"<urn:uuid:34354484-d514-4f33-9e92-f7e07010e6e3>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00143.warc.gz"}
The Curious Wavefunction In November 1918, a 17-year-student from Rome sat for the entrance examination of the Scuola Normale Superiore in Pisa, Italy’s most prestigious science institution. Students applying to the institute had to write an essay on a topic that the examiners picked. The topics were usually quite general, so the students had considerable leeway. Most students wrote about well-known subjects that they had already learnt about in high school. But this student was different. The title of the topic he had been given was “Characteristics of Sound”, and instead of stating basic facts about sound, he “set forth the partial differential equation of a vibrating rod and solved it using Fourier analysis, finding the eigenvalues and eigenfrequencies. The entire essay continued on this level which would have been creditable for a doctoral examination.” The man writing these words was the 17-year-old’s future student, friend and Nobel laureate, Emilio Segre. The student was Enrico Fermi. The examiner was so startled by the originality and sophistication of Fermi’s analysis that he broke precedent and invited the boy to meet him in his office, partly to make sure that the essay had not been plagiarized. After convincing himself that Enrico had done the work himself, the examiner congratulated him and predicted that he would become an important scientist. Twenty five years later Fermi was indeed an important scientist, so important in fact that J. Robert Oppenheimer had created an entire division called F-Division under his name at Los Alamos, New Mexico to harness his unique talents for the Manhattan Project. By that time the Italian emigre was the world’s foremost nuclear physicist as well as perhaps the only universalist in physics – in the words of a recent admiring biographer, “the last man who knew everything”. He had led the creation of the world’s first nuclear reactor in a squash court at the University of Chicago in 1942 and had won a Nobel Prize in 1938 for his work on using neutrons to breed new elements, laying the foundations of the atomic age. The purpose of F-division was to use Fermi’s unprecedented joint abilities in both experimental and theoretical physics to solve problems that stumped others. To Fermi other scientists would take their problems in all branches of physics, many of them current or future Nobel laureates. They would take advantage of Fermi’s startlingly simple approach to problem-solving, where he would first qualitatively estimate the parameters and solution and then plug in complicated mathematics only when necessary to drive relentlessly toward the solution. He had many nicknames including “The Roadroller”, but the one that stuck was “The Pope” because his judgement on any physics problem was often infallible and the last word. Fermi’s love for semi-quantitative, order-of-magnitude estimates gave him an unusual oeuvre. He loved working out the most rigorous physics theories as much as doing back-of-the-envelope calculations designed to test ideas; the latter approach led to the famous set of problems called ‘Fermi problems‘. Simplicity and semi-quantitative approaches to problems are the hallmark of models, and Fermi inevitably became one of the first modelers. Simple models such as the quintessential “spherical cow in a vacuum” are the lifeblood of physics, and some of the most interesting insights have come from using such simplicity to build toward complexity. Interestingly, the problem that the 17-year-old Enrico had solved in 1918 would inspire him in a completely novel way many years later. It would be the perfect example of finding complexity in simplicity and would herald the beginnings of at least two new, groundbreaking fields. Los Alamos was an unprecedented exercise in bringing a century’s worth of physics, chemistry and engineering to bear on problems of fearsome complexity. Scientists quickly realized that the standard tools of pen and paper that they had been using for centuries would be insufficient, and so for help they turned to some of the first computers in history. At that time the word “computer” meant two different things. One meaning was women who calculated. The other meaning was machines which calculated. Women who were then excluded from most of the highest echelons of science were employed in large numbers to perform repetitive calculations on complicated physics problems. Many of these problems at Los Alamos were related to the tortuous flow of neutrons and shock waves from an exploding nuclear weapon. Helping the female computers were some of the earliest punched card calculators manufactured by IBM. Although they didn’t know it yet, these dedicated women working on those primitive calculators became history’s first pioneering programmers. They were the forerunners of the women who worked at NASA two decades later on the space program. Fermi had always been interested in these computers as a way to speed up calculations or to find new ways to do them. At Los Alamos a few other far-seeing physicists and mathematicians had realized their utility, among them the youthful Richard Feynman who was put in charge of a computing division. But perhaps the biggest computing pioneer at the secret lab was Fermi’s friend, the dazzling Johnny von Neumann, widely regarded as the world’s foremost mathematician and polymath and fastest thinker. Von Neumann who had been recruited by Oppenheimer as a consultant because of his deep knowledge of shock waves and hydrodynamics had become interested in computers after learning that a new calculating machine called ENIAC was being built at the University of Pennsylvania by engineers J. Presper Eckert, John Mauchly, Herman Goldstine and others. Von Neumann realized the great potential of what we today call the shared program concept, a system of programming that contains both the instructions for doing something and the process itself in the same location, both coded in the same syntax. Fermi was a good friend of von Neumann’s, but his best friend was Stanislaw Ulam, a mathematician of stunning versatility and simplicity who had been part of the famous Lwów School of mathematics in Poland. Ulam belonged to the romantic generation of Central European mathematics, a time during the early twentieth century when mathematicians had marathon sessions fueled by coffee in Lwów, Vienna and Warsaw’s famous cafes, where they scribbled on the marble tables and argued mathematics and philosophy late into the night. Ulam had come to the United States in the 1930s; by then von Neumann had already been firmly ensconced at Princeton’s Institute for Advanced Study with a select group of mathematicians and physicists including Einstein. Ulam had started his career in the most rarefied parts of mathematics including set theory; he later joked that during the war he had to stoop to the level of manipulating actual numbers instead of merely abstract symbols. After the war started Ulam had wanted to help with the war effort. One day he got a call from Johnny, asking him to a move to a secret location in New Mexico. At Los Alamos Ulam worked closely with von Neumann and Fermi and met the volatile Hungarian physicist Edward Teller with whom he began a fractious, consequential working relationship. Fermi, Ulam and von Neumann all worked on the intricate calculations involving neutron and thermal diffusion in nuclear weapons and they witnessed the first successful test of an atomic weapon on July 16th, 1945. All three of them realized the importance of computers, although only von Neumann’s mind was creative and far-reaching enough to imagine arcane and highly significant applications of these as yet primitive machines – weather control and prediction, hydrogen bombs and self-replicating automata, entities which would come to play a prominent role in both biology and science fiction. After the war ended, computers became even more important in the early 1950s. Von Neumann and his engineers spearheaded the construction of a pioneering computer in Princeton. After the computer achieved success in doing hydrogen bomb calculations at night and artificial life calculations during the day, it was shut down because the project was considered too applied by the pure mathematicians. But copies started springing up at other places, including one at Los Alamos. Partly in deference to the destructive weapons whose workings would be modeled on it, the thousand ton Los Alamos machine was jokingly christened MANIAC, for Mathematical Analyzer Numerical Integrator and Computer. It was based on the basic plan proposed by von Neumann which is still the most common plan used for computers worldwide – the von Neumann architecture. After the war, Enrico Fermi had moved to the University of Chicago which he had turned into the foremost center of physics research in the country. Among his colleagues and students there were T. D. Lee, Edward Teller and Subrahmanyan Chandrasekhar. But the Cold War imposed on his duties, and the patriotic Fermi started making periodic visits to Los Alamos after President Truman announced in 1951 that he was asking the United States Atomic Energy Commission to resume work on the hydrogen bomb as a top priority. Ulam joined him there. By that time Edward Teller had been single-mindedly pushing for the construction of a hydrogen bomb for several years. Teller’s initial design was highly flawed and would have turned into a dud. Working with pencil and paper, Fermi, Ulam and von Neumann all confirmed the pessimistic outlook for Teller’s design, but in 1951, Ulam had a revolutionary insight into how a feasible thermonuclear weapon could be made. Teller honed this insight into a practical design which was tested in November 1952, and the thermonuclear age was born. Since then, the vast majority of thermonuclear weapons in the world’s nuclear arsenals have been based on some variant of the Teller-Ulam design. By this time Fermi had acutely recognized the importance of computers, to such an extent in fact that in the preceding years he had taught himself how to code. Work on the thermonuclear brought Fermi and Ulam together, and in 1955 Fermi proposed a novel project to Ulam. To help with the project Fermi recruited a visiting physicist named John Pasta who had worked as a beat cop in New York City during the Depression. With the MANIAC ready and standing by, Fermi was especially interested in problems where highly repetitive calculations on complex systems could take advantage of the power of computing. Such calculations would be almost impossible in terms of time to perform by hand. As Ulam recalled later, “Fermi held many discussions with me on the kind of future problems which could be studied through the use of such machines. We decided to try a selection of problems for heuristic work where in the absence of closed analytic solutions experimental work on a computing machine might perhaps contribute to the understanding of properties of solutions. This could be particularly fruitful for problems involving the asymptotic, long time or “in the large” behavior of non-linear physical systems…Fermi expressed often a belief that future fundamental theories in physics may involve non-linear operators and equations, and that it would be useful to attempt practice in the mathematics needed for the understanding of non-linear systems. The plan was then to start with the possibly simplest such physical model and to study the results of the calculation of its long-term behavior.” Fermi and Ulam had caught the bull by its horns. Crudely speaking, linear systems are systems where the response is proportional to the input. Non-linear systems are ones where the response can vary disproportionately. Linear systems are the ones which many physicists study in textbooks and as students. Non-linear systems include almost everything encountered in the real world. In fact, the word “non-linear” is highly misleading, and Ulam nailed the incongruity best: “To say that a system is non-linear is to say that most animals are non-elephants.” Non-linear systems are the rule rather than the exception, and by 1955 physics wasn’t really well-equipped to handle this ubiquity. Fermi and Ulam astutely realized that the MANIAC was ideally placed to attempt a solution to non-linear problems. But what kind of problem would be complex enough to attempt by computer, yet simple enough to provide insights into the workings of a physical system? Enter Fermi’s youthful fascination with vibrating rods and strings. The simple harmonic oscillator is an entity which physics students encounter in their first or second year of college. Its distinguishing characteristic is that the force applied to it is proportional to the displacement. But as students are taught, this is an approximation. Real oscillators – real pendulums, real vibrating rods and strings in the real world – are not simple. The force applied results in a complicated function of the displacement. Fermi and Ulam set up a system consisting of a string attached to one end. They considered four models; one where the force is proportional to the displacement, one where the force is proportional to the square of the displacement, one where it’s proportional to the cube, and one where the displacement varies in a discontinuous way with the force, going from broken to linear and back. In reality the string was modeled as a series of 64 points all connected through these different forces. The four graphs from the original paper are shown below, with force on the x-axis and displacement on the y-axis and the dotted line indicating the linear case. Here’s what the physicists expected: the case for a linear oscillator, familiar to physics students, is simple. The string shows a single sinusoidal node that remains constant. The expectation was that when the force became non-linear, higher frequencies corresponding to two, three and more sinusoidal modes would be excited (these are called harmonics or overtones). The global expectation was that adding a non-linear force to the system would lead to an equal distribution or “thermalization” of the energy, leading to all modes being excited and the higher modes being heavily so. What was seen was something that was completely unexpected and startling, even to the “last man who knew everything.” When the quadratic force was applied, the system did indeed transition to the two and three-mode system, but the system then suddenly did something very different. “Starting in one problem with a quadratic force and a pure sine wave as the initial position of the string, we indeed observe initially a gradual increase of energy in the higher modes as predicted. Mode 2 starts increasing first, followed by mode 3 and so on. Later on, however, this gradual sharing of energy among successive modes ceases. Instead, it is one or the other mode that predominates. For example, mode 2 decides, as it were, to increase rather rapidly at the cost of all other modes and becomes predominant. At one time, it has more energy than all the others put together! Then mode 3 undertakes this role.” Fermi and Ulam could not resist adding an exclamation point even in the staid language of scientific publication. Part of the discovery was in fact accidental; the computer had been left running overnight, giving it enough time to go through many more cycles. The word “decides” is also interesting; it’s as if the system seems to have a life of its own and starts dancing of its own volition between one or two lower modes; Ulam thought that the system was playing a game of musical chairs. Finally it comes back to mode 1, as if it were linear, and then continues this periodic behavior. An important way to describe this behavior is to say that instead of the initial expectation of equal distribution of energy among the different modes, the system seems to periodically concentrate most or all of its energy in one or a very small number of modes. The following graph for the quadratic case makes this feature clear: on the y-axis is energy while on the x-axis is the number of cycles ranging into the thousands (as an aside, this very large number of cycles is partly why it would be impossible to solve this problem using pen and paper in reasonable time). As is readily seen, the height of modes 2 and 3 is much larger than the higher modes. The actual shapes of the string corresponding to this asymmetric energy exchange are even more striking, indicating how the lower modes are disproportionately excited. The large numbers here again correspond to the number of cycles. The graphs for the cubic and broken displacement case are similar but even more complex, leading to higher modes being excited more often but the energy still concentrated into the lower modes. Needless to say, these results were profoundly unexpected and fascinating. The physicists did not quite know what to make of them, and Ulam found them “truly amazing”. Fermi told him that he thought they had made a “little discovery”. The 1955 paper contains an odd footnote: “We thank Ms. Mary Tsingou for efficient coding of the problems and for running the computations on the Los Alamos MANIAC machine.” Mary Tsingou was the underappreciated character in the story. She was a Greek immigrant whose family barely escaped Italy before Mussolini took over. With bachelor’s and master’s degrees in mathematics from Wisconsin and Michigan, in 1955 she was a “computer” at Los Alamos, just like many other women. Her programming of the computer was crucial and non-trivial, but she was acknowledged in the work and not in the writing. She worked later with von Neumann on diffusion problems, was the first FORTRAN programmer, and even did some calculations for Ronald Reagan’s infamous “Star Wars” program. As of 2020, Mary Tsingou is still alive and 92 and living in Los Alamos. The Fermi-Pasta-Ulam problem should be called the Fermi-Pasta-Ulam-Tsingou problem. Fermi’s sense of having made a “little discovery” has to be one of the great understatements of 20th century physics. The results that he, Ulam, Pasta and Tsingou obtained went beyond harmonic systems and the MANIAC. Until then there had been two revolutions in 20th century physics that changed our view of the universe – the theory of relativity and quantum mechanics. The third revolution was quieter and started with French mathematician Henri Poincare who studied non-linear problems at the beginning of the century. It kicked into high gear in the 1960s and 70s but still evolved under the radar, partly because it spanned several different fields and did not have the flashy reputation that the then-popular fields of cosmology and particle physics had. The field went by several names, including “non-linear dynamics”, but the one we are most familiar with is chaos theory. As James Gleick who gets the credit for popularizing the field in his 1987 book says, “Where chaos begins, classical science stops.” Classical science was the science of pen and pencil and linear systems. Chaos was the science of computers and non-linear systems. Fermi, Ulam, Pasta and Tsingou’s 1955 paper left little reverberations, but in hindsight it is seminal and signals the beginning of studies of chaotic systems in their most essential form. Not only did it bring non-linear physics which also happens to be the physics of real world problems to the forefront, but it signaled a new way of doing science by computer, a paradigm that is the forerunner of modeling and simulation in fields as varied as climatology, ecology, chemistry and nuclear studies. Gleick does not mention the report in his book, and he begins the story of chaos with Edward Lorenz’s famous meteorology experiment in 1963 where Lorenz discovered the basic characteristic of chaotic systems – acute sensitivity to initial conditions. His work led to the iconic figure of the Lorenz attractor where a system seems to hover in a complicated and yet simple way around one or two basins of attraction. But the 1955 Los Alamos work got there first. Fermi and his colleagues certainly demonstrated the pull of physical systems toward certain favored behavior, but the graphs also showed how dramatically the behavior would change if the coefficients for the quadratic and other non-linear terms were changed. The paper is beautiful. It is beautiful because it is simple. It is also beautiful because it points to another, potentially profound ramification of the universe that could extend from the non-living to the living. The behavior that the system demonstrated was non-ergodic or quasiergodic. In simple terms, an ergodic system is one which visits all its states given enough time. A non-ergodic system is one which will gravitate toward certain states at the expense of others. This was certainly something Fermi and the others observed. Another system that as far as we know is non-ergodic is biological evolution. It is non-ergodic because of historical contingency which plays a crucial role in natural selection. At least on earth, we know that the human species evolved only once, and so did many other species. In fact the world of butterflies, bats, humans and whales bears some eerie resemblances to the chaotic world of pendulums and vibrating strings. Just like these seemingly simple systems, biological systems demonstrate a bewitching mix of the simple and the complex. Evolution seems to descend on the same body plans for instance, fashioning bilateral symmetry and aerodynamic shapes from the same abstract designs, but it does not produce the final product twice. Given enough time, would evolution be ergodic and visit the same state multiple times? We don’t know the answer to this question, and finding life elsewhere in the universe would certainly shed light on the problem, but the Fermi-Pasta-Ulam-Tsingou problem points to the non-ergodic behavior exhibited by complex systems that arise from simple rules. Biological evolution with its own simple rules of random variation, natural selection and neutral drift may well be a Fermi-Pasta-Ulam-Tsingou problem waiting to be unraveled. The Los Alamos report was written in 1955, but Enrico Fermi was not one of the actual co-authors because he had tragically died in November 1954, the untimely consequence of stomach cancer. He was still at the height of his powers and would have likely made many other important discoveries compounding his reputation as one of history’s greatest physicists. When he was in the hospital Stan Ulam paid him a visit and came out shaken and in tears, partly because his friend seemed so composed. He later remembered the words Crito said in Plato’s account of the death of Socrates: “That now was the death one of the wisest men known.” Just three years later Ulam’s best friend Johnny von Neumann also passed into history. Von Neumann had already started thinking about applying computers to weather control, but in spite of the great work done by his friends in 1955, he did not realize that chaos might play havoc with the prediction of a system as sensitive to initial conditions as the global climate. It took only seven years before Lorenz found that out. Ulam himself died in 1984 after a long and productive career in physics and mathematics. Just like their vibrating strings, Fermi, Ulam and von Neumann had ascended to the non-ergodic, higher modes of the metaphysical universe.	{"url":"http://wavefunction.fieldofscience.com/2020/01/?m=1","timestamp":"2024-11-05T12:48:32Z","content_type":"text/html","content_length":"64267","record_id":"<urn:uuid:fb665bae-b9c7-46b6-94c9-c65bb75eb5ac>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00338.warc.gz"}
Perfect Shape Alternatives - Ruby Database Tools \| LibHunt PerfectShape is a collection of pure Ruby geometric algorithms that are mostly useful for GUI (Graphical User Interface) manipulation like checking viewport rectangle intersection or containment of a mouse click point in popular geometry shapes such as rectangle, square, arc (open, chord, and pie), ellipse, circle, polygon, and paths containing lines, quadratic bézier curves, and cubic bezier curves, potentially with affine transforms applied like translation, scale, rotation, shear/skew, and inversion (including both Ray Casting Algorithm, aka Even-odd Rule, and Winding Number Algorithm, aka Nonzero Rule). Additionally, PerfectShape::Math contains some purely mathematical algorithms, like IEEE 754-1985 Remainder. Programming language: Ruby License: MIT License Perfect Shape alternatives and similar gems Based on the "Database Tools" category. Alternatively, view perfect-shape alternatives based on common mentions on social networks and blogs. A performance dashboard for Postgres Business intelligence made simple Get early access to Scout Monitoring's NEW Ruby logging feature [beta] by signing up now. Start for free and enable logs to get better insights into your Rails apps. Promo www.scoutapm.com Versioned database views for Rails Strategies for cleaning databases in Ruby. Can be used to ensure a clean state for testing. Online MySQL schema migrations Rails 4/5 task to dump your data to db/seeds.rb Identify database issues before they hit production. lol_dba is a small package of rake tasks that scan your application models and displays a list of columns that probably should be indexed. Also, it can generate .sql migration scripts. Rails Database Viewer and SQL Query Runner Squasher - squash your old migrations in a single command Advanced seed data handling for Rails, combining the best practices of several methods together. Adds foreign key helpers to migrations and correctly dumps foreign keys to schema.rb Seedbank gives your seed data a little structure. Create seeds for each environment, share seeds between environments and specify dependencies to load your seeds in order. All nicely integrated with simple rake tasks. The tool to avoid various issues due to inconsistencies and inefficiencies between a database schema and application models. :zap: Powerful tool for avoiding N+1 DB or HTTP queries Polo travels through your database and creates sample snapshots so you can work with real world data in development. Database validations for ActiveRecord DISCONTINUED. SchemaPlus provides a collection of enhancements and extensions to ActiveRecord Upsert on MySQL, PostgreSQL, and SQLite3. Transparently creates functions (UDF) for MySQL and PostgreSQL; on SQLite3, uses INSERT OR IGNORE. Catch unsafe PostgreSQL migrations in development and run them easier in production (code helpers for table/column renaming, changing column type, adding columns with default, background migrations, etc). Seamless second database integration for Rails. Blazing fast pagination for ActiveRecord with deferred joins ⚡️ Find time-consuming database queries for ActiveRecord-based Rails Apps Catch bad SQL queries before they cause problems in production A lightweight, efficient Ruby gem for interacting with Whatsapp Cloud API. Sinatra app to monitor Redis servers. Ruby PostgreSQL database performance insights. Locks, index usage, buffer cache hit ratios, vacuum stats and more. Turn ruby files into .exe files on windows (supported safe fork of ocran) Finds missing non-null constraints A simple tool to observe PostgreSQL database locks in Rails apps. Simple solution to make encrypted with ccrypt PostgreSQL backups and storing on Google Drive API Create a Slack bot that is smart and so easy to expand, create new bots on demand, run ruby code on chat, create shortcuts... The main scope of this gem is to be used internally in the company so teams can create team channels with their own bot to help them on their daily work, almost everything is suitable to be automated!! slack-smart-bot can create bots on demand, create shortcuts, run ruby code... just on a chat channel. You can access it just from your mobile phone if you want and run those tests you forgot to run, get the results, restart a server... no limits. Union, Intersect, and Difference set operations for ActiveRecord (also, SQL's UnionAll). Postgres partitioning built on top of https://github.com/ankane/pgslice Check data integrity for your ActiveRecord models Simple but fast Redis-backed distributed rate limiter. Allows you to specify time interval and count within to limit distributed operations. A Simple Interface to Slack Incoming Webhooks Integrations Bundler plugin for auto-downloading specified extra files after gem install Vimamsa - Vi/Vim -inspired experimental GUI-oriented text editor written with Ruby and GTK. A Pry plugin that captures exceptions that may arise from typos and deduces the correct command. * Code Quality Rankings and insights are calculated and provided by Lumnify. They vary from L1 to L5 with "L5" being the highest. Do you think we are missing an alternative of Perfect Shape or a related project? Add another 'Database Tools' Gem Perfect Shape 1.0.6 Geometric Algorithms PerfectShape is a collection of pure Ruby geometric algorithms that are mostly useful for GUI (Graphical User Interface) manipulation like checking viewport rectangle intersection or containment of a mouse click point in popular geometry shapes such as rectangle, square, arc (open, chord, and pie), ellipse, circle, polygon, and paths containing lines, quadratic bézier curves, and cubic bezier curves, potentially with affine transforms applied like translation, scale, rotation, shear/skew, and inversion (including both the Ray Casting Algorithm, aka Even-odd Rule, and the Winding Number Algorithm, aka Nonzero Rule). Additionally, PerfectShape::Math contains some purely mathematical algorithms, like IEEE 754-1985 Remainder. To ensure accuracy and precision, this library does all its mathematical operations with BigDecimal numbers. gem install perfect-shape -v 1.0.6 Or include in Bundler Gemfile: gem 'perfect-shape', '~> 1.0.6' And, run: • ::degrees_to_radians(angle): converts degrees to radians • ::radians_to_degrees(angle): converts radians to degrees • ::normalize_degrees(angle): normalizes the specified angle into the range -180 to 180. • ::ieee_remainder(x, y) (alias: ieee754_remainder): IEEE 754-1985 Remainder (different from standard % modulo operator as it operates on floats and could return a negative result) This is a base class for all shapes. It is not meant to be used directly. Subclasses implement/override its methods as needed. • #min_x: min x • #min_y: min y • #max_x: max x • #max_y: max y • #width: width • #height: height • #center_point: center point as Array of [center_x, center_y] coordinates • #center_x: center x • #center_y: center y • #bounding_box: bounding box is a rectangle with x = min x, y = min y, and width/height just as those of shape • #==(other): Returns true if equal to other or false otherwise • #contain?(x_or_point, y=nil, outline: false, distance_tolerance: 0): checks if point is inside if outline is false or if point is on the outline if outline is true. distance_tolerance can be used as a fuzz factor when outline is true, for example, to help GUI users mouse-click-select a shape from its outline more successfully • #initialize(x: 0, y: 0): initializes a point location, usually representing the top-left point in a shape • #x: top-left x • #y: top-left y • #min_x: min x (x by default) • #min_y: min y (y by default) Includes PerfectShape::PointLocation • #initialize(x: 0, y: 0, width: 1, height: 1): initializes a rectangular shape • #x: top-left x • #y: top-left y • #width: width • #height: height • #min_x: min x • #min_y: min y • #max_x: max x • #max_y: max y • ::normalize_point_array: normalizes Array of multiple points into (x,y) point coordinate Array format per point • #initialize(points: []): initializes points with Array of multiple points (e.g. useful for shapes like Line and Polygon). • #points: Array of multiple points • #min_x: min x of all points • #min_y: min y of all points • #max_x: max x of all points • #max_y: max y of all points Affine transforms have the following matrix: [ xxp xyp xt ] [ yxp yyp yt ] The matrix is used to transform (x,y) point coordinates as follows: [ xxp xyp xt ] * [x] = [ xxp * x + xyp * y + xt ] [ yxp yyp yt ] * [y] = [ yxp * x + yyp * y + yt ] xxp is the x coordinate x product (m11) xyp is the x coordinate y product (m12) yxp is the y coordinate x product (m21) yyp is the y coordinate y product (m22) xt is the x coordinate translation (m13) yt is the y coordinate translation (m23) Affine transform mutation operations ending with ! can be chained as they all return self. • ::new(xxp_element = nil, xyp_element = nil, yxp_element = nil, yyp_element = nil, xt_element = nil, yt_element = nil, xxp: nil, xyp: nil, yxp: nil, yyp: nil, xt: nil, yt: nil, m11: nil, m12: nil, m21: nil, m22: nil, m13: nil, m23: nil): The constructor accepts either the (x,y)-operation related argument/kwarg names or traditional matrix element kwarg names. If no arguments are supplied, it constructs an identity matrix (i.e. like calling ::new(xxp: 1, xyp: 0, yxp: 0, yyp: 1, xt: 0, yt: 0)). • #matrix_3d: Returns Ruby Matrix object representing affine transform in 3D (used internally for performing multiplication) • #==(other): Returns true if equal to other or false otherwise • #identity! (alias: reset!): Resets to identity matrix (i.e. like calling ::new(xxp: 1, xyp: 0, yxp: 0, yyp: 1, xt: 0, yt: 0)) • #invertible? Returns true if matrix is invertible and false otherwise • #invert!: Inverts affine transform matrix if invertible or raises an error otherwise • #multiply!(other): Multiplies affine transform with another affine transform, storing resulting changes in matrix elements • #translate!(x_or_point, y=nil): Translates affine transform with (x, y) translation values • #scale!(x_or_point, y=nil): Scales affine transform with (x, y) scale values • #rotate!(degrees): Rotates by angle degrees counter-clockwise if angle value is positive or clockwise if angle value is negative. Note that it returns very close approximate results for rotations that are 90/180/270 degrees (good enough for inverse-transform GUI point containment checks needed when checking if mouse-click-point is inside a transformed shape). • #shear!(x_or_point, y=nil): Shears by x and y factors • #clone: Returns a new AffineTransform with the same matrix elements • #transform_point(x_or_point, y=nil): returns [xxp * x + xyp * y + xt, yxp * x + yyp * y + yt]. Note that result is a close approximation, but should be good enough for GUI mouse-click-point containment checks. • #transform_points(xy_coordinates_or_points): returns Array of (x,y) pair Arrays transformed with #transform_point method • #inverse_transform_point(x_or_point, y=nil): returns inverse transform of a point (x,y) coordinates (clones self and inverts clone, and then transforms point). Note that result is a close approximation, but should be good enough for GUI mouse-click-point containment checks. • #inverse_transform_points(xy_coordinates_or_points): returns inverse transforms of a point Array of (x,y) coordinates xxp = 2 xyp = 3 yxp = 4 yyp = 5 xt = 6 yt = 7 affine_transform1 = PerfectShape::AffineTransform.new(xxp: xxp, xyp: xyp, yxp: yxp, yyp: yyp, xt: xt, yt: yt) # (x,y)-operation kwarg names affine_transform2 = PerfectShape::AffineTransform.new(m11: xxp, m12: xyp, m21: yxp, m22: yyp, m13: xt, m23: yt) # traditional matrix element kwarg names affine_transform3 = PerfectShape::AffineTransform.new(xxp, xyp, yxp, yyp, xt, yt) # standard arguments affine_transform2.matrix_3d == affine_transform1.matrix_3d # => true affine_transform3.matrix_3d == affine_transform1.matrix_3d # => true affine_transform = PerfectShape::AffineTransform.new.translate!(30, 20).scale!(2, 3) affine_transform.transform_point(10, 10) # => approximately [50, 50] affine_transform.inverse_transform_point(50, 50) # => approximately [10, 10] Extends PerfectShape::Shape Includes PerfectShape::PointLocation Points are simply represented by an Array of [x,y] coordinates when used within other shapes, but when needing point-specific operations like point_distance, the PerfectShape::Point class can come in • ::point_distance(x, y, px, py): Returns the distance from a point to another point • ::normalize_point(x_or_point, y = nil): Normalizes point args whether two-number point Array or x, y args, returning normalized point Array of two BigDecimal's • ::new(x_or_point=nil, y_arg=nil, x: nil, y: nil): constructs a point with (x,y) pair (default: 0,0) whether specified as Array of (x,y) pair, flat x,y args, or x:, y: kwargs. • #min_x: min x (always x) • #min_y: min y (always y) • #max_x: max x (always x) • #max_y: max y (always y) • #width: width (always 0) • #height: height (always 0) • #center_point: center point as Array of [center_x, center_y] coordinates • #center_x: center x (always x) • #center_y: center y (always y) • #bounding_box: bounding box is a rectangle with x = min x, y = min y, and width/height of shape • #==(other): Returns true if equal to other or false otherwise • #contain?(x_or_point, y=nil, outline: true, distance_tolerance: 0): checks if point matches self, with a distance tolerance (0 by default). Distance tolerance provides a fuzz factor that for example enables GUI users to mouse-click-select a point shape more successfully. outline option makes no difference on point • #intersect?(rectangle): Returns true if intersecting with interior of rectangle or false otherwise. This is useful for GUI optimization checks of whether a shape appears in a GUI viewport rectangle and needs redrawing • #point_distance(x_or_point, y=nil): Returns the distance from a point to another point require 'perfect-shape' shape = PerfectShape::Point.new(x: 200, y: 150) shape.contain?(200, 150) # => true shape.contain?([200, 150]) # => true shape.contain?(200, 151) # => false shape.contain?([200, 151]) # => false shape.contain?(200, 151, distance_tolerance: 5) # => true shape.contain?([200, 151], distance_tolerance: 5) # => true Extends PerfectShape::Shape Includes PerfectShape::MultiPoint • ::relative_counterclockwise(x1, y1, x2, y2, px, py): Returns an indicator of where the specified point (px,py) lies with respect to the line segment from (x1,y1) to (x2,y2). The return value can be either 1, -1, or 0 and indicates in which direction the specified line must pivot around its first end point, (x1,y1), in order to point at the specified point (px,py). A return value of 1 indicates that the line segment must turn in the direction that takes the positive X axis towards the negative Y axis. In the default coordinate system, this direction is counterclockwise. A return value of -1 indicates that the line segment must turn in the direction that takes the positive X axis towards the positive Y axis. In the default coordinate system, this direction is clockwise. A return value of 0 indicates that the point lies exactly on the line segment. Note that an indicator value of 0 is rare and not useful for determining collinearity because of floating point rounding issues. If the point is colinear with the line segment, but not between the end points, then the value will be -1 if the point lies “beyond (x1,y1)” or 1 if the point lies “beyond • ::point_distance_square(x1, y1, x2, y2, px, py): Returns the square of distance from a point to a line segment. • ::point_distance(x1, y1, x2, y2, px, py): Returns the distance from a point to a line segment. • ::new(points: []): constructs a line with two points as Array of Arrays of [x,y] pairs or flattened Array of alternating x and y coordinates • #min_x: min x • #min_y: min y • #max_x: max x • #max_y: max y • #width: width (from min x to max x) • #height: height (from min y to max y) • #center_point: center point as Array of [center_x, center_y] coordinates • #center_x: center x • #center_y: center y • #bounding_box: bounding box is a rectangle with x = min x, y = min y, and width/height of shape • #==(other): Returns true if equal to other or false otherwise • #contain?(x_or_point, y=nil, outline: true, distance_tolerance: 0): checks if point lies on line, with a distance tolerance (0 by default). Distance tolerance provides a fuzz factor that for example enables GUI users to mouse-click-select a line shape more successfully. outline option makes no difference on line • #intersect?(rectangle): Returns true if intersecting with interior of rectangle or false otherwise. This is useful for GUI optimization checks of whether a shape appears in a GUI viewport rectangle and needs redrawing • #relative_counterclockwise(x_or_point, y=nil): Returns an indicator of where the specified point (px,py) lies with respect to the line segment from (x1,y1) to (x2,y2). The return value can be either 1, -1, or 0 and indicates in which direction the specified line must pivot around its first end point, (x1,y1), in order to point at the specified point (px,py). A return value of 1 indicates that the line segment must turn in the direction that takes the positive X axis towards the negative Y axis. In the default coordinate system, this direction is counterclockwise. A return value of -1 indicates that the line segment must turn in the direction that takes the positive X axis towards the positive Y axis. In the default coordinate system, this direction is clockwise. A return value of 0 indicates that the point lies exactly on the line segment. Note that an indicator value of 0 is rare and not useful for determining collinearity because of floating point rounding issues. If the point is colinear with the line segment, but not between the end points, then the value will be -1 if the point lies “beyond (x1,y1)” or 1 if the point lies “beyond • #point_distance(x_or_point, y=nil): Returns the distance from a point to a line segment. • #rect_crossings(rxmin, rymin, rxmax, rymax, crossings = 0): rectangle crossings (adds to crossings arg) require 'perfect-shape' shape = PerfectShape::Line.new(points: [[0, 0], [100, 100]]) # start point and end point shape.contain?(50, 50) # => true shape.contain?([50, 50]) # => true shape.contain?(50, 51) # => false shape.contain?([50, 51]) # => false shape.contain?(50, 51, distance_tolerance: 5) # => true shape.contain?([50, 51], distance_tolerance: 5) # => true Extends PerfectShape::Shape Includes PerfectShape::MultiPoint • ::tag(coord, low, high): Determine where coord lies with respect to the range from low to high. It is assumed that low < high. The return value is one of the 5 values BELOW, LOWEDGE, INSIDE, HIGHEDGE, or ABOVE. • ::eqn(val, c1, cp, c2): Fill an array with the coefficients of the parametric equation in t, ready for solving against val with solve_quadratic. We currently have: val = Py(t) = C1(1-t)^2 + 2CPt(1-t) + C2t^2 = C1 - 2C1t + C1t^2 + 2CPt - 2CPt^2 + C2t^2 = C1 + (2CP - 2C1)t + (C1 - 2CP + C2)t^2; 0 = (C1 - val) + (2CP - 2C1)t + (C1 - 2CP + C2)t^2; 0 = C + Bt + At^ 2; C = C1 - val; B = 2CP - 2C1; A = C1 - 2CP + C2 • ::solve_quadratic(eqn): Solves the quadratic whose coefficients are in the eqn array and places the non-complex roots into the res array, returning the number of roots. The quadratic solved is represented by the equation: eqn = {C, B, A}; ax^2 + bx + c = 0 A return value of -1 is used to distinguish a constant equation, which might be always 0 or never 0, from an equation that has no • ::eval_quadratic(vals, num, include0, include1, inflect, c1, ctrl, c2): Evaluate the t values in the first num slots of the vals[] array and place the evaluated values back into the same array. Only evaluate t values that are within the range <, >, including the 0 and 1 ends of the range iff the include0 or include1 booleans are true. If an "inflection" equation is handed in, then any points which represent a point of inflection for that quadratic equation are also ignored. • ::new(points: []): constructs a quadratic bézier curve with three points (start point, control point, and end point) as Array of Arrays of [x,y] pairs or flattened Array of alternating x and y • #points: points (start point, control point, and end point) • #min_x: min x • #min_y: min y • #max_x: max x • #max_y: max y • #width: width (from min x to max x) • #height: height (from min y to max y) • #center_point: center point as Array of [center_x, center_y] coordinates • #center_x: center x • #center_y: center y • #bounding_box: bounding box is a rectangle with x = min x, y = min y, and width/height of shape (bounding box only guarantees that the shape is within it, but it might be bigger than the shape) • #==(other): Returns true if equal to other or false otherwise • #contain?(x_or_point, y=nil, outline: false, distance_tolerance: 0): checks if point is inside when outline is false or if point is on the outline when outline is true. distance_tolerance can be used as a fuzz factor when outline is true, for example, to help GUI users mouse-click-select a quadratic bezier curve shape from its outline more successfully • #intersect?(rectangle): Returns true if intersecting with interior of rectangle or false otherwise. This is useful for GUI optimization checks of whether a shape appears in a GUI viewport rectangle and needs redrawing • #curve_center_point: point at the center of the curve outline (not the center of the bounding box area like center_x and center_y) • #curve_center_x: point x coordinate at the center of the curve outline (not the center of the bounding box area like center_x and center_y) • #curve_center_y: point y coordinate at the center of the curve outline (not the center of the bounding box area like center_x and center_y) • #subdivisions(level=1): subdivides quadratic bezier curve at its center into into 2 quadratic bezier curves by default, or more if level of recursion is specified. The resulting number of subdivisions is 2 to the power of level. • #point_distance(x_or_point, y=nil, minimum_distance_threshold: OUTLINE_MINIMUM_DISTANCE_THRESHOLD): calculates distance from point to curve segment. It does so by subdividing curve into smaller curves and checking against the curve center points until the distance is less than minimum_distance_threshold, to avoid being an overly costly operation. • #rect_crossings(rxmin, rymin, rxmax, rymax, level, crossings = 0): rectangle crossings (adds to crossings arg) require 'perfect-shape' shape = PerfectShape::QuadraticBezierCurve.new(points: [[200, 150], [270, 320], [380, 150]]) # start point, control point, and end point shape.contain?(270, 220) # => true shape.contain?([270, 220]) # => true shape.contain?(270, 220, outline: true) # => false shape.contain?([270, 220], outline: true) # => false shape.contain?(280, 235, outline: true) # => true shape.contain?([280, 235], outline: true) # => true shape.contain?(281, 235, outline: true) # => false shape.contain?([281, 235], outline: true) # => false shape.contain?(281, 235, outline: true, distance_tolerance: 1) # => true shape.contain?([281, 235], outline: true, distance_tolerance: 1) # => true Extends PerfectShape::Shape Includes PerfectShape::MultiPoint • ::new(points: []): constructs a cubic bézier curve with four points (start point, two control points, and end point) as Array of Arrays of [x,y] pairs or flattened Array of alternating x and y • #points: points (start point, two control points, and end point) • #min_x: min x • #min_y: min y • #max_x: max x • #max_y: max y • #width: width (from min x to max x) • #height: height (from min y to max y) • #center_point: center point as Array of [center_x, center_y] coordinates • #center_x: center x • #center_y: center y • #bounding_box: bounding box is a rectangle with x = min x, y = min y, and width/height of shape (bounding box only guarantees that the shape is within it, but it might be bigger than the shape) • #==(other): Returns true if equal to other or false otherwise • #contain?(x_or_point, y=nil, outline: false, distance_tolerance: 0): checks if point is inside when outline is false or if point is on the outline when outline is true. distance_tolerance can be used as a fuzz factor when outline is true, for example, to help GUI users mouse-click-select a cubic bezier curve shape from its outline more successfully • #intersect?(rectangle): Returns true if intersecting with interior of rectangle or false otherwise. This is useful for GUI optimization checks of whether a shape appears in a GUI viewport rectangle and needs redrawing • #curve_center_point: point at the center of the curve outline (not the center of the bounding box area like center_x and center_y) • #curve_center_x: point x coordinate at the center of the curve outline (not the center of the bounding box area like center_x and center_y) • #curve_center_y: point y coordinate at the center of the curve outline (not the center of the bounding box area like center_x and center_y) • #subdivisions(level=1): subdivides cubic bezier curve at its center into into 2 cubic bezier curves by default, or more if level of recursion is specified. The resulting number of subdivisions is 2 to the power of level. • #point_distance(x_or_point, y=nil, minimum_distance_threshold: OUTLINE_MINIMUM_DISTANCE_THRESHOLD): calculates distance from point to curve segment. It does so by subdividing curve into smaller curves and checking against the curve center points until the distance is less than minimum_distance_threshold, to avoid being an overly costly operation. • #rectangle_crossings(rectangle): rectangle crossings (used to determine rectangle interior intersection), optimized to check if line represented by cubic bezier curve crosses the rectangle first, and if not then perform expensive check with #rect_crossings • #rect_crossings(rxmin, rymin, rxmax, rymax, level, crossings = 0): rectangle crossings (adds to crossings arg) require 'perfect-shape' shape = PerfectShape::CubicBezierCurve.new(points: [[200, 150], [235, 235], [270, 320], [380, 150]]) # start point, two control points, and end point shape.contain?(270, 220) # => true shape.contain?([270, 220]) # => true shape.contain?(270, 220, outline: true) # => false shape.contain?([270, 220], outline: true) # => false shape.contain?(261.875, 245.625, outline: true) # => true shape.contain?([261.875, 245.625], outline: true) # => true shape.contain?(261.875, 246.625, outline: true) # => false shape.contain?([261.875, 246.625], outline: true) # => false shape.contain?(261.875, 246.625, outline: true, distance_tolerance: 1) # => true shape.contain?([261.875, 246.625], outline: true, distance_tolerance: 1) # => true Extends PerfectShape::Shape Includes PerfectShape::RectangularShape • ::new(x: 0, y: 0, width: 1, height: 1): constructs a rectangle • #x: top-left x • #y: top-left y • #width: width • #height: height • #center_point: center point as Array of [center_x, center_y] coordinates • #center_x: center x • #center_y: center y • #min_x: min x • #min_y: min y • #max_x: max x • #max_y: max y • #bounding_box: bounding box is a rectangle with x = min x, y = min y, and width/height of shape • #==(other): Returns true if equal to other or false otherwise • #contain?(x_or_point, y=nil, outline: false, distance_tolerance: 0): checks if point is inside when outline is false or if point is on the outline when outline is true. distance_tolerance can be used as a fuzz factor when outline is true, for example, to help GUI users mouse-click-select a rectangle shape from its outline more successfully • #intersect?(rectangle): Returns true if intersecting with interior of rectangle or false otherwise. This is useful for GUI optimization checks of whether a shape appears in a GUI viewport rectangle and needs redrawing • #edges: edges of rectangle as PerfectShape::Line objects • #out_state(x_or_point, y = nil): Returns "out state" of specified point (x,y) (whether it lies to the left, right, top, bottom of rectangle). If point is outside rectangle, it returns a bit mask combination of Rectangle::OUT_LEFT, Rectangle::OUT_RIGHT, Rectangle::OUT_TOP, or Rectangle::OUT_BOTTOM. Otherwise, it returns 0 if point is inside the rectangle. • #empty?: Returns true if width or height are 0 (or negative) and false otherwise • #to_path_shapes: Converts Rectangle into basic Path shapes made up of Points and Lines. Used by Path when adding a Rectangle to Path shapes require 'perfect-shape' shape = PerfectShape::Rectangle.new(x: 15, y: 30, width: 200, height: 100) shape.contain?(115, 80) # => true shape.contain?([115, 80]) # => true shape.contain?(115, 80, outline: true) # => false shape.contain?([115, 80], outline: true) # => false shape.contain?(115, 30, outline: true) # => true shape.contain?([115, 30], outline: true) # => true shape.contain?(115, 31, outline: true) # => false shape.contain?([115, 31], outline: true) # => false shape.contain?(115, 31, outline: true, distance_tolerance: 1) # => true shape.contain?([115, 31], outline: true, distance_tolerance: 1) # => true Extends PerfectShape::Rectangle • ::new(x: 0, y: 0, length: 1) (length alias: size): constructs a square • #x: top-left x • #y: top-left y • #length: length • #width: width (equal to length) • #height: height (equal to length) • #center_point: center point as Array of [center_x, center_y] coordinates • #center_x: center x • #center_y: center y • #min_x: min x • #min_y: min y • #max_x: max x • #max_y: max y • #bounding_box: bounding box is a rectangle with x = min x, y = min y, and width/height of shape • #==(other): Returns true if equal to other or false otherwise • #contain?(x_or_point, y=nil, outline: false, distance_tolerance: 0): checks if point is inside when outline is false or if point is on the outline when outline is true. distance_tolerance can be used as a fuzz factor when outline is true, for example, to help GUI users mouse-click-select a square shape from its outline more successfully • #intersect?(rectangle): Returns true if intersecting with interior of rectangle or false otherwise. This is useful for GUI optimization checks of whether a shape appears in a GUI viewport rectangle and needs redrawing • #edges: edges of square as PerfectShape::Line objects • #empty?: Returns true if length is 0 (or negative) and false otherwise • #to_path_shapes: Converts Square into basic Path shapes made up of Points and Lines. Used by Path when adding a Square to Path shapes require 'perfect-shape' shape = PerfectShape::Square.new(x: 15, y: 30, length: 200) shape.contain?(115, 130) # => true shape.contain?([115, 130]) # => true shape.contain?(115, 130, outline: true) # => false shape.contain?([115, 130], outline: true) # => false shape.contain?(115, 30, outline: true) # => true shape.contain?([115, 30], outline: true) # => true shape.contain?(115, 31, outline: true) # => false shape.contain?([115, 31], outline: true) # => false shape.contain?(115, 31, outline: true, distance_tolerance: 1) # => true shape.contain?([115, 31], outline: true, distance_tolerance: 1) # => true Extends PerfectShape::Shape Includes PerfectShape::RectangularShape Arcs can be of type :open, :chord, or :pie Open Arc Chord Arc Pie Arc • ::new(type: :open, x: 0, y: 0, width: 1, height: 1, start: 0, extent: 360, center_x: nil, center_y: nil, radius_x: nil, radius_y: nil): constructs an arc of type :open (default), :chord, or :pie • #type: :open, :chord, or :pie • #x: top-left x • #y: top-left y • #width: width • #height: height • #start: start angle in degrees • #extent: extent angle in degrees • #center_point: center point as Array of [center_x, center_y] coordinates • #start_point: start point as Array of (x,y) coordinates • #end_point: end point as Array of (x,y) coordinates • #center_x: center x • #center_y: center y • #radius_x: radius along the x-axis • #radius_y: radius along the y-axis • #min_x: min x • #min_y: min y • #max_x: max x • #max_y: max y • #bounding_box: bounding box is a rectangle with x = min x, y = min y, and width/height of shape • #==(other): Returns true if equal to other or false otherwise • #contain?(x_or_point, y=nil, outline: false, distance_tolerance: 0): checks if point is inside when outline is false or if point is on the outline when outline is true. distance_tolerance can be used as a fuzz factor when outline is true, for example, to help GUI users mouse-click-select an arc shape from its outline more successfully • #intersect?(rectangle): Returns true if intersecting with interior of rectangle or false otherwise. This is useful for GUI optimization checks of whether a shape appears in a GUI viewport rectangle and needs redrawing • #contain_angle?(angle): returns true if the angle is within the angular extents of the arc and false otherwise • #to_path_shapes: Converts Arc into basic Path shapes made up of Points, Lines, and CubicBezierCurves. Used by Path when adding an Arc to Path shapes • #btan(increment): btan computes the length (k) of the control segments at the beginning and end of a cubic bezier that approximates a segment of an arc with extent less than or equal to 90 degrees. This length (k) will be used to generate the 2 bezier control points for such a segment. require 'perfect-shape' shape = PerfectShape::Arc.new(type: :open, x: 2, y: 3, width: 50, height: 60, start: 45, extent: 270) shape2 = PerfectShape::Arc.new(type: :open, center_x: 2 + 25, center_y: 3 + 30, radius_x: 25, radius_y: 30, start: 45, extent: 270) shape.contain?(39.5, 33.0) # => true shape.contain?([39.5, 33.0]) # => true shape2.contain?(39.5, 33.0) # => true shape2.contain?([39.5, 33.0]) # => true shape.contain?(39.5, 33.0, outline: true) # => false shape.contain?([39.5, 33.0], outline: true) # => false shape2.contain?(39.5, 33.0, outline: true) # => false shape2.contain?([39.5, 33.0], outline: true) # => false shape.contain?(2.0, 33.0, outline: true) # => true shape.contain?([2.0, 33.0], outline: true) # => true shape2.contain?(2.0, 33.0, outline: true) # => true shape2.contain?([2.0, 33.0], outline: true) # => true shape.contain?(3.0, 33.0, outline: true) # => false shape.contain?([3.0, 33.0], outline: true) # => false shape2.contain?(3.0, 33.0, outline: true) # => false shape2.contain?([3.0, 33.0], outline: true) # => false shape.contain?(3.0, 33.0, outline: true, distance_tolerance: 1.0) # => true shape.contain?([3.0, 33.0], outline: true, distance_tolerance: 1.0) # => true shape2.contain?(3.0, 33.0, outline: true, distance_tolerance: 1.0) # => true shape2.contain?([3.0, 33.0], outline: true, distance_tolerance: 1.0) # => true shape.contain?(shape.center_x, shape.center_y, outline: true) # => false shape.contain?([shape.center_x, shape.center_y], outline: true) # => false shape2.contain?(shape2.center_x, shape2.center_y, outline: true) # => false shape2.contain?([shape2.center_x, shape2.center_y], outline: true) # => false shape3 = PerfectShape::Arc.new(type: :chord, x: 2, y: 3, width: 50, height: 60, start: 45, extent: 270) shape4 = PerfectShape::Arc.new(type: :chord, center_x: 2 + 25, center_y: 3 + 30, radius_x: 25, radius_y: 30, start: 45, extent: 270) shape3.contain?(39.5, 33.0) # => true shape3.contain?([39.5, 33.0]) # => true shape4.contain?(39.5, 33.0) # => true shape4.contain?([39.5, 33.0]) # => true shape3.contain?(39.5, 33.0, outline: true) # => false shape3.contain?([39.5, 33.0], outline: true) # => false shape4.contain?(39.5, 33.0, outline: true) # => false shape4.contain?([39.5, 33.0], outline: true) # => false shape3.contain?(2.0, 33.0, outline: true) # => true shape3.contain?([2.0, 33.0], outline: true) # => true shape4.contain?(2.0, 33.0, outline: true) # => true shape4.contain?([2.0, 33.0], outline: true) # => true shape3.contain?(3.0, 33.0, outline: true) # => false shape3.contain?([3.0, 33.0], outline: true) # => false shape4.contain?(3.0, 33.0, outline: true) # => false shape4.contain?([3.0, 33.0], outline: true) # => false shape3.contain?(3.0, 33.0, outline: true, distance_tolerance: 1.0) # => true shape3.contain?([3.0, 33.0], outline: true, distance_tolerance: 1.0) # => true shape4.contain?(3.0, 33.0, outline: true, distance_tolerance: 1.0) # => true shape4.contain?([3.0, 33.0], outline: true, distance_tolerance: 1.0) # => true shape3.contain?(shape3.center_x, shape3.center_y, outline: true) # => false shape3.contain?([shape3.center_x, shape3.center_y], outline: true) # => false shape4.contain?(shape4.center_x, shape4.center_y, outline: true) # => false shape4.contain?([shape4.center_x, shape4.center_y], outline: true) # => false shape5 = PerfectShape::Arc.new(type: :pie, x: 2, y: 3, width: 50, height: 60, start: 45, extent: 270) shape6 = PerfectShape::Arc.new(type: :pie, center_x: 2 + 25, center_y: 3 + 30, radius_x: 25, radius_y: 30, start: 45, extent: 270) shape5.contain?(39.5, 33.0) # => false shape5.contain?([39.5, 33.0]) # => false shape6.contain?(39.5, 33.0) # => false shape6.contain?([39.5, 33.0]) # => false shape5.contain?(9.5, 33.0) # => true shape5.contain?([9.5, 33.0]) # => true shape6.contain?(9.5, 33.0) # => true shape6.contain?([9.5, 33.0]) # => true shape5.contain?(39.5, 33.0, outline: true) # => false shape5.contain?([39.5, 33.0], outline: true) # => false shape6.contain?(39.5, 33.0, outline: true) # => false shape6.contain?([39.5, 33.0], outline: true) # => false shape5.contain?(2.0, 33.0, outline: true) # => true shape5.contain?([2.0, 33.0], outline: true) # => true shape6.contain?(2.0, 33.0, outline: true) # => true shape6.contain?([2.0, 33.0], outline: true) # => true shape5.contain?(3.0, 33.0, outline: true) # => false shape5.contain?([3.0, 33.0], outline: true) # => false shape6.contain?(3.0, 33.0, outline: true) # => false shape6.contain?([3.0, 33.0], outline: true) # => false shape5.contain?(3.0, 33.0, outline: true, distance_tolerance: 1.0) # => true shape5.contain?([3.0, 33.0], outline: true, distance_tolerance: 1.0) # => true shape6.contain?(3.0, 33.0, outline: true, distance_tolerance: 1.0) # => true shape6.contain?([3.0, 33.0], outline: true, distance_tolerance: 1.0) # => true shape5.contain?(shape5.center_x, shape5.center_y, outline: true) # => true shape5.contain?([shape5.center_x, shape5.center_y], outline: true) # => true shape6.contain?(shape6.center_x, shape6.center_y, outline: true) # => true shape6.contain?([shape6.center_x, shape6.center_y], outline: true) # => true Extends PerfectShape::Arc • ::new(x: 0, y: 0, width: 1, height: 1, center_x: nil, center_y: nil, radius_x: nil, radius_y: nil): constructs an ellipse • #x: top-left x • #y: top-left y • #width: width • #height: height • #center_point: center point as Array of [center_x, center_y] coordinates • #center_x: center x • #center_y: center y • #radius_x: radius along the x-axis • #radius_y: radius along the y-axis • #type: always :open • #start: always 0 • #extent: always 360 • #min_x: min x • #min_y: min y • #max_x: max x • #max_y: max y • #bounding_box: bounding box is a rectangle with x = min x, y = min y, and width/height of shape • #==(other): Returns true if equal to other or false otherwise • #contain?(x_or_point, y=nil, outline: false, distance_tolerance: 0): checks if point is inside when outline is false or if point is on the outline when outline is true. distance_tolerance can be used as a fuzz factor when outline is true, for example, to help GUI users mouse-click-select an ellipse shape from its outline more successfully • #intersect?(rectangle): Returns true if intersecting with interior of rectangle or false otherwise. This is useful for GUI optimization checks of whether a shape appears in a GUI viewport rectangle and needs redrawing • #to_path_shapes: Converts Ellipse into basic Path shapes made up of Points, Lines, and CubicBezierCurves. Used by Path when adding an Ellipse to Path shapes require 'perfect-shape' shape = PerfectShape::Ellipse.new(x: 2, y: 3, width: 50, height: 60) shape2 = PerfectShape::Ellipse.new(center_x: 27, center_y: 33, radius_x: 25, radius_y: 30) shape.contain?(27, 33) # => true shape.contain?([27, 33]) # => true shape2.contain?(27, 33) # => true shape2.contain?([27, 33]) # => true shape.contain?(27, 33, outline: true) # => false shape.contain?([27, 33], outline: true) # => false shape2.contain?(27, 33, outline: true) # => false shape2.contain?([27, 33], outline: true) # => false shape.contain?(2, 33, outline: true) # => true shape.contain?([2, 33], outline: true) # => true shape2.contain?(2, 33, outline: true) # => true shape2.contain?([2, 33], outline: true) # => true shape.contain?(1, 33, outline: true) # => false shape.contain?([1, 33], outline: true) # => false shape2.contain?(1, 33, outline: true) # => false shape2.contain?([1, 33], outline: true) # => false shape.contain?(1, 33, outline: true, distance_tolerance: 1) # => true shape.contain?([1, 33], outline: true, distance_tolerance: 1) # => true shape2.contain?(1, 33, outline: true, distance_tolerance: 1) # => true shape2.contain?([1, 33], outline: true, distance_tolerance: 1) # => true Extends PerfectShape::Ellipse • ::new(x: 0, y: 0, diameter: 1, width: 1, height: 1, center_x: nil, center_y: nil, radius: nil, radius_x: nil, radius_y: nil): constructs a circle • #x: top-left x • #y: top-left y • #diameter: diameter • #width: width (equal to diameter) • #height: height (equal to diameter) • #center_point: center point as Array of [center_x, center_y] coordinates • #center_x: center x • #center_y: center y • #radius: radius • #radius_x: radius along the x-axis (equal to radius) • #radius_y: radius along the y-axis (equal to radius) • #type: always :open • #start: always 0 • #extent: always 360 • #min_x: min x • #min_y: min y • #max_x: max x • #max_y: max y • #bounding_box: bounding box is a rectangle with x = min x, y = min y, and width/height of shape • #==(other): Returns true if equal to other or false otherwise • #contain?(x_or_point, y=nil, outline: false, distance_tolerance: 0): checks if point is inside when outline is false or if point is on the outline when outline is true. distance_tolerance can be used as a fuzz factor when outline is true, for example, to help GUI users mouse-click-select a circle shape from its outline more successfully • #intersect?(rectangle): Returns true if intersecting with interior of rectangle or false otherwise. This is useful for GUI optimization checks of whether a shape appears in a GUI viewport rectangle and needs redrawing • #to_path_shapes: Converts Circle into basic Path shapes made up of Points, Lines, and CubicBezierCurves. Used by Path when adding a Circle to Path shapes require 'perfect-shape' shape = PerfectShape::Circle.new(x: 2, y: 3, diameter: 60) shape2 = PerfectShape::Circle.new(center_x: 2 + 30, center_y: 3 + 30, radius: 30) shape.contain?(32, 33) # => true shape.contain?([32, 33]) # => true shape2.contain?(32, 33) # => true shape2.contain?([32, 33]) # => true shape.contain?(32, 33, outline: true) # => false shape.contain?([32, 33], outline: true) # => false shape2.contain?(32, 33, outline: true) # => false shape2.contain?([32, 33], outline: true) # => false shape.contain?(2, 33, outline: true) # => true shape.contain?([2, 33], outline: true) # => true shape2.contain?(2, 33, outline: true) # => true shape2.contain?([2, 33], outline: true) # => true shape.contain?(1, 33, outline: true) # => false shape.contain?([1, 33], outline: true) # => false shape2.contain?(1, 33, outline: true) # => false shape2.contain?([1, 33], outline: true) # => false shape.contain?(1, 33, outline: true, distance_tolerance: 1) # => true shape.contain?([1, 33], outline: true, distance_tolerance: 1) # => true shape2.contain?(1, 33, outline: true, distance_tolerance: 1) # => true shape2.contain?([1, 33], outline: true, distance_tolerance: 1) # => true Extends PerfectShape::Shape Includes PerfectShape::MultiPoint A polygon can be thought of as a special case of path, consisting of lines only, is closed, and has the Even-Odd winding rule by default. • ::new(points: [], winding_rule: :wind_even_odd): constructs a polygon with points as Array of Arrays of [x,y] pairs or flattened Array of alternating x and y coordinates and specified winding rule (:wind_even_odd or :wind_non_zero) • #min_x: min x • #min_y: min y • #max_x: max x • #max_y: max y • #width: width (from min x to max x) • #height: height (from min y to max y) • #center_point: center point as Array of [center_x, center_y] coordinates • #center_x: center x • #center_y: center y • #bounding_box: bounding box is a rectangle with x = min x, y = min y, and width/height of shape • #==(other): Returns true if equal to other or false otherwise • #contain?(x_or_point, y=nil, outline: false, distance_tolerance: 0): When outline is false, it checks if point is inside using either the Ray Casting Algorithm (aka Even-Odd Rule) or Winding Number Algorithm (aka Nonzero-Rule). Otherwise, when outline is true, it checks if point is on the outline. distance_tolerance can be used as a fuzz factor when outline is true, for example, to help GUI users mouse-click-select a polygon shape from its outline more successfully • #intersect?(rectangle): Returns true if intersecting with interior of rectangle or false otherwise. This is useful for GUI optimization checks of whether a shape appears in a GUI viewport rectangle and needs redrawing • #edges: edges of polygon as PerfectShape::Line objects require 'perfect-shape' shape = PerfectShape::Polygon.new(points: [[200, 150], [270, 170], [250, 220], [220, 190], [200, 200], [180, 170]]) shape.contain?(225, 185) # => true shape.contain?([225, 185]) # => true shape.contain?(225, 185, outline: true) # => false shape.contain?([225, 185], outline: true) # => false shape.contain?(200, 150, outline: true) # => true shape.contain?([200, 150], outline: true) # => true shape.contain?(200, 151, outline: true) # => false shape.contain?([200, 151], outline: true) # => false shape.contain?(200, 151, outline: true, distance_tolerance: 1) # => true shape.contain?([200, 151], outline: true, distance_tolerance: 1) # => true Extends PerfectShape::Shape Includes PerfectShape::MultiPoint • ::new(shapes: [], closed: false, winding_rule: :wind_even_odd, line_to_complex_shapes: false): constructs a path with shapes as Array of shape objects, which can be PerfectShape::Point (or Array of [x, y] coordinates), PerfectShape::Line, PerfectShape::QuadraticBezierCurve, PerfectShape::CubicBezierCurve, or complex shapes that decompose into the aforementioned basic path shapes, like PerfectShape::Arc, PerfectShape::Ellipse, PerfectShape::Circle, PerfectShape::Rectangle, and PerfectShape::Square. If a path is closed, its last point is automatically connected to its first point with a line segment. The winding rule can be :wind_non_zero (default) or :wind_even_odd. line_to_complex_shapes can be true or false (default), indicating whether to connect to complex shapes, meaning Arc, Ellipse, Circle, Rectangle, and Square, with a line, or otherwise move to their start point instead. • #shapes: the shapes that the path is composed of (must always start with PerfectShape::Point or Array of [x,y] coordinates representing start point) • #basic_shapes: the basic shapes that the path is composed of, meaning only Point, Line, QuadraticBezierCurve, and CubicBezierCurve shapes (decomposing complex shapes like Arc, Ellipse, Circle, Rectangle, and Square, using their #to_path_shapes method) • #closed?: returns true if closed and false otherwise • #winding_rule: returns winding rule (:wind_non_zero or :wind_even_odd) • #points: path points calculated (derived) from shapes • #min_x: min x • #min_y: min y • #max_x: max x • #max_y: max y • #width: width (from min x to max x) • #height: height (from min y to max y) • #center_point: center point as Array of [center_x, center_y] coordinates • #center_x: center x • #center_y: center y • #bounding_box: bounding box is a rectangle with x = min x, y = min y, and width/height of shape (bounding box only guarantees that the shape is within it, but it might be bigger than the shape) • #==(other): Returns true if equal to other or false otherwise • #contain?(x_or_point, y=nil, outline: false, distance_tolerance: 0): When outline is false, it checks if point is inside path utilizing the configured winding rule, which can be the Nonzero-Rule (aka Winding Number Algorithm) or the Even-Odd Rule (aka Ray Casting Algorithm). Otherwise, when outline is true, it checks if point is on the outline. distance_tolerance can be used as a fuzz factor when outline is true, for example, to help GUI users mouse-click-select a path shape from its outline more successfully • #intersect?(rectangle): Returns true if intersecting with interior of rectangle or false otherwise. This is useful for GUI optimization checks of whether a shape appears in a GUI viewport rectangle and needs redrawing • #point_crossings(x_or_point, y=nil): calculates the number of times the given path crosses the ray extending to the right from (x,y) • #disconnected_shapes: Disconnected shapes have their start point filled in so that each shape does not depend on the previous shape to determine its start point. Also, if a point is followed by a non-point shape, it is removed since it is augmented to the following shape as its start point. Lastly, if the path is closed, an extra shape is added to represent the line connecting the last point to the first require 'perfect-shape' path_shapes = [] path_shapes << PerfectShape::Point.new(x: 200, y: 150) path_shapes << PerfectShape::Line.new(points: [250, 170]) # no need for start point, just end point path_shapes << PerfectShape::QuadraticBezierCurve.new(points: [[300, 185], [350, 150]]) # no need for start point, just control point and end point path_shapes << PerfectShape::CubicBezierCurve.new(points: [[370, 50], [430, 220], [480, 170]]) # no need for start point, just two control points and end point shape = PerfectShape::Path.new(shapes: path_shapes, closed: false, winding_rule: :wind_non_zero) shape.contain?(275, 165) # => true shape.contain?([275, 165]) # => true shape.contain?(275, 165, outline: true) # => false shape.contain?([275, 165], outline: true) # => false shape.contain?(shape.disconnected_shapes[1].curve_center_x, shape.disconnected_shapes[1].curve_center_y, outline: true) # => true shape.contain?([shape.disconnected_shapes[1].curve_center_x, shape.disconnected_shapes[1].curve_center_y], outline: true) # => true shape.contain?(shape.disconnected_shapes[1].curve_center_x + 1, shape.disconnected_shapes[1].curve_center_y, outline: true) # => false shape.contain?([shape.disconnected_shapes[1].curve_center_x + 1, shape.disconnected_shapes[1].curve_center_y], outline: true) # => false shape.contain?(shape.disconnected_shapes[1].curve_center_x + 1, shape.disconnected_shapes[1].curve_center_y, outline: true, distance_tolerance: 1) # => true shape.contain?([shape.disconnected_shapes[1].curve_center_x + 1, shape.disconnected_shapes[1].curve_center_y], outline: true, distance_tolerance: 1) # => true Extends PerfectShape::Shape A composite shape is simply an aggregate of multiple shapes (e.g. square and triangle polygon) • ::new(shapes: []): constructs a composite shape with shapes as Array of PerfectShape::Shape objects • #shapes: the shapes that the composite shape is composed of • #min_x: min x • #min_y: min y • #max_x: max x • #max_y: max y • #width: width (from min x to max x) • #height: height (from min y to max y) • #center_point: center point as Array of [center_x, center_y] coordinates • #center_x: center x • #center_y: center y • #bounding_box: bounding box is a rectangle with x = min x, y = min y, and width/height of shape (bounding box only guarantees that the shape is within it, but it might be bigger than the shape) • #==(other): Returns true if equal to other or false otherwise • #contain?(x_or_point, y=nil, outline: false, distance_tolerance: 0): When outline is false, it checks if point is inside any of the shapes owned by the composite shape. Otherwise, when outline is true, it checks if point is on the outline of any of the shapes owned by the composite shape. distance_tolerance can be used as a fuzz factor when outline is true, for example, to help GUI users mouse-click-select a composite shape from its outline more successfully • #intersect?(rectangle): Returns true if intersecting with interior of rectangle or false otherwise. This is useful for GUI optimization checks of whether a shape appears in a GUI viewport rectangle and needs redrawing require 'perfect-shape' shapes = [] shapes << PerfectShape::Square.new(x: 120, y: 215, length: 100) shapes << PerfectShape::Polygon.new(points: [[120, 215], [170, 165], [220, 215]]) shape = PerfectShape::CompositeShape.new(shapes: shapes) shape.contain?(170, 265) # => true inside square shape.contain?([170, 265]) # => true inside square shape.contain?(170, 265, outline: true) # => false shape.contain?([170, 265], outline: true) # => false shape.contain?(170, 315, outline: true) # => true shape.contain?([170, 315], outline: true) # => true shape.contain?(170, 316, outline: true) # => false shape.contain?([170, 316], outline: true) # => false shape.contain?(170, 316, outline: true, distance_tolerance: 1) # => true shape.contain?([170, 316], outline: true, distance_tolerance: 1) # => true shape.contain?(170, 190) # => true inside polygon shape.contain?([170, 190]) # => true inside polygon shape.contain?(170, 190, outline: true) # => false shape.contain?([170, 190], outline: true) # => false shape.contain?(145, 190, outline: true) # => true shape.contain?([145, 190], outline: true) # => true shape.contain?(145, 189, outline: true) # => false shape.contain?([145, 189], outline: true) # => false shape.contain?(145, 189, outline: true, distance_tolerance: 1) # => true shape.contain?([145, 189], outline: true, distance_tolerance: 1) # => true Change Log • Check out the latest master to make sure the feature hasn't been implemented or the bug hasn't been fixed yet. • Check out the issue tracker to make sure someone already hasn't requested it and/or contributed it. • Fork the project. • Start a feature/bugfix branch. • Commit and push until you are happy with your contribution. • Make sure to add tests for it. This is important so I don't break it in a future version unintentionally. • Please try not to mess with the Rakefile, version, or history. If you want to have your own version, or is otherwise necessary, that is fine, but please isolate to its own commit so I can cherry-pick around it. Copyright (c) 2021-2022 Andy Maleh. See [LICENSE.txt](LICENSE.txt) for further details. *Note that all licence references and agreements mentioned in the Perfect Shape README section above are relevant to that project's source code only.	{"url":"https://ruby.libhunt.com/perfect-shape-alternatives","timestamp":"2024-11-10T02:04:53Z","content_type":"text/html","content_length":"157089","record_id":"<urn:uuid:f2b41f03-79b6-4711-a317-8ab4efacd283>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00510.warc.gz"}
Best Tree Diagram Assignment Help homework help online Tree Diagram Assignment Help Best UK USA Australia China Canada UAE Tree Diagram Assignment Help Online Tree Diagram is basically used to represent a series of numbers or to be very particular, independent numbers. Tree Diagram may also be referred to a tree structure through which one can represent the hierarchical nature of a structure in a most appropriate graphical form. Through this Tree Diagram Assignment Help, we come to know that Tree Diagram can also be used in probability theory, where it is used to represent a probability space. The tree diagram also represents the series of conditional probabilities. Normally one can say that each and every node on the represented tree diagram is actually representing an event. Thus the probability of the event is being associated with the subject of tree diagrams and the probability of the event is represented with the help of that particular diagram. Tree Diagram Assignment Help not only tells about the nodes of the tree diagram, but also explains the root nodes of the diagram. Certain events are represented by the root nodes and thus their probability is 1. Nodes and root nodes are together known as sibling nodes and each set of sibling nodes represents an exhaustive and exclusive partition of the parent event. Statistics Services Popular Statistics Assignment help Services Statistics Assignments always put student on worries and stress. But with Assignment Consultancy for your help, you can remove all your worries by going through our various services:- Features of Statistics Assignment Help Zero Plagiarism We believe in providing no plagiarism work to the students. All are our works are unique and we provide Free Plagiarism report too on requests. Best Customer Service Our customer representatives are working 24X7 to assist you in all your assignment needs. You can drop a mail to assignmentconsultancy.help@gmail.com or chat with our representative using live chat shown in bottom right corner. Three Stage Quality Check We are the only service providers boasting of providing original, relevant and accurate solutions. Our three stage quality process help students to get perfect solutions. 100% Confidential All our works are kept as confidential as we respect the integrity and privacy of our clients. Our Testimonials Harnam Baweja, Student , RMIT University “Perfect statistics assignment help service. Got all assistance will definitely come back” Shane Smith, Student MBA, USA “They have some of the best USA experts to provide best Statistics Assignment help.” Ram List, Lancashire University, UK “Best place to get all help in Statistics subjects. Will definitely recommend to all” Advantages and Disadvantages of Tree Diagram Tree Diagram is having a lot of advantages, some of which are listed below in Tree Diagram Assignment Help. These are • The basic advantage of Tree Diagram is that, it forces people to consider that there are many possible outcomes of a decision which one may not even think of. • Tree Diagram is very easy and simple to understand and interpret. This helps people to understand the Tree Diagram even after a very brief explanation of the Tree Diagram model. • Even if there is a very little hard data, tree diagram gives value to that also. Some very important insights can also be generated basing on the experts decisions over a particular situation and their preferences for the outcomes. • Tree Diagram never restricts the addition of new possible scenarios, instead it helps determines best, worst and expected values for different scenarios. Apart from the above listed advantages, Tree Diagram has some disadvantages which are also listed below in Tree Diagram Assignment Help • One very important disadvantage of the Tree Diagram is that here the calculation becomes very complex specially when there are many values which are not certain and also when many outcomes are • The information gained in Tree Diagram is biased in connection to those attributes which are having move levels. Features of Tree Diagram Assignment Help are Clients waiting for Tree Diagram Assignment Help from us may expect the following • The Date is Fixed for final Submission of assignments. • Topics are in tandem to chose. • Simple language for better and easy understanding. • Quality is chosen over quantity of the product. • Highly qualified experts over subjects are roped in. Looking for best Tree Diagram Assignment Help online,please click here	{"url":"https://www.assignmentconsultancy.com/tree-diagram-assignment-help/","timestamp":"2024-11-08T11:32:28Z","content_type":"text/html","content_length":"79126","record_id":"<urn:uuid:b3ee1686-3415-4396-99f3-11048e94ca56>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00726.warc.gz"}
Operations and Fractions - ACT Math All ACT Math Resources Example Questions Example Question #1 : Operations And Fractions Correct answer: We can simply cross multiply to obtain Example Question #151 : Fractions Simplfiy the following expression; Correct answer: Multiply the numerators 2 x 6 x 4 = 48. Then multiply the denominators 3 x 8 x 12 = 288. The answer is 48/288. To simplify, divide both numerator and denominator by 48 to get 1/6. Example Question #152 : Fractions Correct answer: Because the exponents are negative, we can convert –3^–2^ to ^1/[9] and 2^–3^ to ^1/[8]. We then multiply straight across the top and the bottom, giving you ^1/[72]. Example Question #2 : How To Multiply Fractions Simplify the following into one fraction Correct answer: To multiply fractions you multiply the entire numerator and the entire denominator together. However, before we do that we can cancel anything from the denominator with anything in the numerator. Six cancels with 12 5 cancels with 25 multiply it all out and get Example Question #1 : Operations And Fractions Correct answer: Cross multiply or multiply using the reciprocal of the second fraction. Example Question #2 : Operations And Fractions Simplify the following expression: Example Question #3 : Operations And Fractions Correct answer: Start by converting 7^1/[3 ]to 22/3, and 6 ^2/[3 ]to 20/3. We then multiply 22/3 by the reciprocal of 20/3, 3/20, and you get 66/60. This simplifies to 1^1/[10]. Example Question #4 : Operations And Fractions Correct answer: Remember, to divide a number by a fraction, multiply the number by the reciprocal of the fraction. In this case, Example Question #5 : Operations And Fractions Correct answer: To solve this, subtract 1 1/2 from both sides. Convert to common denominators. 4 1/3 – 1 1/2 = 4 2/6 – 1 3/6. In order to subtract, you'll want to "borrow" from the 4 2/6. Rewrite 4 2/6 as 3 8/6 and then subtract 1 3/6 from this. Your solution is 2 5/6. Most calculators will also do these calculations for Example Question #6 : Operations And Fractions Correct answer: Begin by isolating your variable: Next, you need to find the common denominator. For the left side of your equation, it is Now, simplify and combine terms: You can further simplify the left side: Next, multiply both sides by Certified Tutor Minnesota State University-Mankato, Bachelor of Science, Mathematics. Certified Tutor University of Michigan-Ann Arbor, Bachelor of Science, Computer Science. Certified Tutor University of the State of New York, Associate in Nursing, Nursing (RN). University of the State of New York, Bachelor of Sc... All ACT Math Resources ACT Math Tutors in Top Cities: Atlanta ACT Math Tutors Austin ACT Math Tutors Boston ACT Math Tutors Chicago ACT Math Tutors Dallas Fort Worth ACT Math Tutors Denver ACT Math Tutors Houston ACT Math Tutors Kansas City ACT Math Tutors Los Angeles ACT Math Tutors Miami ACT Math Tutors New York City ACT Math Tutors Philadelphia ACT Math Tutors Phoenix ACT Math Tutors San Diego ACT Math Tutors San Francisco-Bay Area ACT Math Tutors Seattle ACT Math Tutors St. Louis ACT Math Tutors Tucson ACT Math Tutors Washington DC ACT Math Tutors	{"url":"https://cdn.varsitytutors.com/act_math-help/fractions/arithmetic/operations-and-fractions","timestamp":"2024-11-02T20:17:49Z","content_type":"application/xhtml+xml","content_length":"171394","record_id":"<urn:uuid:d7e14d36-2bf8-40eb-8eb6-281b417d32f6>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00701.warc.gz"}
Math Classes: Online Maths Courses for Kids What mathematics courses do we offer? [Modules] • Numbers • Number & Geometry, Measurement • Fractions • Geometry • Ratios & Proportions • Expressions & Equations • Functions & Statistics Open detailed PDF Math classes on Numbers: 1st grade 1st graders study “Numbers” topics in detail over many lessons in Brighterly’s Math Classes: “Sums \| Differences”, “Introduction to Place Value Through Addition \| Subtraction Within 20”, “Place Value, Comparison”, “Addition and Subtraction to 40”. Book a demo lesson Math classes on Numbers: 2nd grade 2nd graders learn the following topics on Numbers in Brighterly's online math courses: “Sums and Differences”, “Place Value, Counting, and Comparison of Numbers to 1,000”, “Addition and Subtraction Within 200 with Word Problems”, and more. Book a demo lesson Math classes on Numbers: 3d grade We’ve selected a range of topics on “Numbers” that will help your child improve their knowledge of this concept. The following is studied in detail during 1:1 math lessons: “Properties of Multiplication", "Division”, etc. Book a demo lesson Math classes on Numbers: 4th grade For fourth graders, Brighterly Online math classes contain more complex modules on Numbers. E.g., “Place Value, Rounding, and Algorithms for Addition and Subtraction”, “Multi-Digit Multiplication and Division”, “Exploring Measurement with Multiplication”. Book a demo lesson Math classes on Numbers: 6th grade The first year of middle school brings with it a number of more complex math topics. That's why our tutors plan the following modules for 6th graders: “Arithmetic Operations Including Division of Fractions”, “Rational Numbers”. Book a demo lesson Math classes on Numbers: 7th & 8th grades 7th grade: For seventh graders, the focus is on the study of Rational Numbers. 8th grade: For eighth graders, one of the modules in Math Classes is an introduction to Irrational Numbers Using Book a demo lesson Show all modules Online math classes by Brighterly: How to start Book a free 1st math course class Attend the 1st demo lesson absolutely for free to try our math courses online. Choose a convenient time and learn more about our math courses. Provide details to our math expert During the first free lesson, share your child's goals, describe the ideal tutor you have in mind, and watch your child enjoy their first trial lesson here. Schedule a math course & enjoy your kid's success Choose math tutoring classes that fit your and your child's schedule. Remember, regular sessions ensure greater progress in your youngster’s learning. Book a demo lesson Brighterly math classes: Reviews from parents I totally recommend Brighterly I love my daughter's teacher, she is really nice and explains everything in a very simple and fun way. I totally recommend Brighterly. Read review Thanks to the whole team My daughter’s tutor, Mr. Ryan, always kept her interested in discovering more of her potential in mathematics. Thanks to the whole team, I always had an immediate answer to my questions. Read review Great experience! We have had a great experience so far! The lessons are fun and engaging while also being very educational. So happy with the tutor as well. She is kind, encouraging and patient. Read review We LOVE Brighterly My child is excited about her lesson each week, and we have seen an increase in her math scores at school. As a parent and an elementary teacher myself I would absolutely recommend Brighterly. Read review Our math tutoring classes are taught by the best teachers Our online courses tutors are passionate about making a difference in students’ lives! Our math tutors believe that every child is a different kind of flower that needs careful, individualized treatment to blossom. Brighterly’s math classes are conducted by English-speaking educators We carefully select applicants and hire professionals to join our team. They are fluent and can command a room in English. Brighterly’s math courses from tutors that demonstrate subject matter expertise Our experts have years of math teaching experience. At the same time, we ensure that our experts align their teaching methods with our student-centered, results-driven philosophy and sincerely love children. Try a free demo before taking online math courses for middle & elementary school, and see how effective they are for your child from the very first lessons. What do Brighterly math classes look like? Stop Googling “math classes near me” when you can find the perfect tutor online in just minutes. Sign your child up for the 1st free demo math class and see how many benefits your youngster can gain from this type of learning at Brighterly. Interactive math course Our online math courses for students in grades 1-8 help your child improve their knowledge of basic and advanced math concepts. Even the topics that seemed complicated for your child will come easy thanks to our tutors’ teaching style. The classes we provide can strengthen students’ basic knowledge, improve their academic performance, and create a life-long love of math. Common Core math Each math class for kids and teens is based on a program created in accordance with Common Core standards. This is a structured approach to learning, focusing on critical thinking and understanding concepts rather than just rote memorization. Students gain the skills they need to succeed in standardized tests and further their education. Our curriculum is aligned with and keeps pace with US school standards. Our teachers have a strong background in tutoring math. Thus, you can be sure that your kids are getting a quality learning One-on-one math classes with a tutor Have your kid take math classes online with Brighterly to get a good knowledge base useful both in school and life. With 1:1 lessons from our tutors, your schooler will receive full attention, the opportunity to focus on specific problem areas from their public school curriculum and all the academic help they Our lessons are adjusted to each student’s learning style, knowledge level, and specific needs. Whether a child needs to improve in a particular subject or just assistance with homework, Brighterly tutors are here to help. Flexible Scheduling It’s perfectly okay for parents to have a busy schedule. That’s why we let you plan math tutoring classes around your needs. You can choose the time and place that works best for you and your kids. Progress Reports for Parents Brighterly’s math courses for elementary & middle schoolers offer a holistic learning experience. After the lessons, you will be able to receive a comprehensive report and feedback on your child’s progress. This way, you can stay informed and on top of your child’s education. The number of “Progress Reports for Parents” depends on the Pricing plan you choose. There can be from 2 to 17 Learning progress reports. Book a demo lesson	{"url":"https://brighterly.com/math-courses/","timestamp":"2024-11-06T09:23:42Z","content_type":"text/html","content_length":"158798","record_id":"<urn:uuid:173f0cbb-fee1-472b-b287-18205dd46c9d>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00388.warc.gz"}
Module Picos_std_sync.Latch A dynamic single-use countdown latch. Latches are typically used for determining when a finite set of parallel computations is done. If the size of the set is known a priori, then the latch can be initialized with the size as initial count and then each computation just decrements the latch. If the size is unknown, i.e. it is determined dynamically, then a latch is initialized with a count of one, the a priori known computations are started and then the latch is decremented. When a computation is stsrted, the latch is incremented, and then decremented once the computation has finished. val create : ?padded:bool -> int -> t create initial creates a new countdown latch with the specified initial count. try_decr latch attempts to decrement the count of the latch and returns true in case the count of the latch was greater than zero and false in case the count already was zero. decr latch is equivalent to: if not (try_decr latch) then invalid_arg "zero count" â šī¸ This operation is not cancelable. try_incr latch attempts to increment the count of the latch and returns true on success and false on failure, which means that the latch has already reached zero. incr latch is equivalent to: if not (try_incr latch) then invalid_arg "zero count" await latch returns after the count of the latch has reached zero. await_evt latch returns an event that can be committed to once the count of the latch has reached zero.	{"url":"https://ocaml-multicore.github.io/picos/doc/picos_std/Picos_std_sync/Latch/index.html","timestamp":"2024-11-07T22:04:40Z","content_type":"application/xhtml+xml","content_length":"6838","record_id":"<urn:uuid:ac943b07-cd25-4775-b9cf-9df7db1046f6>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00031.warc.gz"}
What is a Box Plot? \| Jaspersoft What is a Box Plot? The box plot is a data visualization tool that provides a concise overview of data distribution, from central tendencies to potential outliers. It demystifies complex information by converting abstract numbers into visual representations, making it accessible to both novices and experts. The strength of the box plot resides in its simplicity and the breadth of insights it offers at a single glance. Its unique ability to reveal the core aspects of a dataset, from its median to its range, has made it an indispensable tool for statisticians and data analysts alike. History and Origin of the Box Plot The box plot, a modern statistical visualization staple, owes its origin to the brilliant mind of John Tukey. Tukey was an American mathematician best known for his contributions to data analysis and statistics. In the 1970s, when tables and numerical summaries dominated the landscape, he introduced the "box-and-whisker plot" as an instrument for exploratory data analysis. The goal of this visualization was to present a five-number overview of datasets: minimum, first quartile (Q1), median, third quartile (Q3), and maximum. Tukey’s innovation was groundbreaking for several reasons. In contrast to many modern methods that rely extensively on complex numerical data, the box plot presented a concise visual summary. This made it easy to identify not only central tendencies and variances, but also outliers and nuances in data distribution. The box plot has evolved over time to incorporate software tools and accommodate increasingly complex data sets. From its manual, pen-and-paper beginnings in the 1970s to its digital implementations on platforms such as R and Python today, its fundamental principle has remained the same. The plot remains a great tool for expressing the essence of data distribution in a concise and informative Today, the box plot continues to be a testament to John Tukey's vision of simplifying complex data for improved comprehension and interpretation. Its continued use in statistical analysis attests to its enduring effectiveness and adaptability. Anatomy of a Box Plot At first glance, a box plot may seem like a simple diagram. However, its minimalist design contains a wealth of information about the distribution of a dataset. Let us take a closer look at the essential components of this remarkable visualization tool. Central Box – Interquartile Range and Quartiles • Interquartile Range (IQR): The IQR is the range in which the central 50% of the values fall. It is calculated by subtracting the first quartile (Q1) from the third quartile (Q3): IQR = Q3 - Q1.It measures the distribution of the data and provides insights into its variability. A larger IQR indicates that the middle half of the data is more dispersed. • Q1 (First Quartile): This is the value below which 25% of the data falls. It represents the boundary between the lowest 25% and highest 75% of values. It is the bottom margin of the central box in the box plot. • Q3 (Third Quartile): This represents the value below which 75% of the data falls, serving as a border between the lowest 75% and highest 25% of values. It is represented by the top edge of the central box on the box plot. Whiskers – Significance and Range The whiskers of the box plot extend from the central box to the minimum and maximum data values that are not considered outliers. They provide a graphical representation of the majority of the data's distribution. There are several ways to draw whiskers: • The lower whisker typically extends to the smallest data value that exceeds Q1 - 1.5 x IQR. • The upper whisker extends to the greatest data value that falls below Q3 + 1.5 x IQR. Outliers – Identification and Representation Outliers are data points that deviate significantly from other data points, typically due to data variability or errors. In a box plot: • Outliers are often represented by individual points or symbols outside the whiskers. They are usually data values that are less than Q1 - 1.5 x IQR or greater than Q3 + 1.5 x IQR. • Recognizing outliers is crucial because they can significantly affect the mean and standard deviation of the data, and understanding them can reveal unusual patterns or anomalies within the Median – Center of the Data The median is the value that divides the dataset into two equal halves, with 50% of the values falling below it and 50% falling above it. In the box plot a line (or sometimes a distinct mark) inside the central frame represents the median. Given its position, it provides a clear view of the center of the dataset and allows for comparisons when multiple box plots are displayed side by side. Advantages of Using a Box Plot The box plot, also known as the whisker plot, is a data visualization technique revered for its precision and simplicity. There are numerous reasons for its pervasive use in statistical analysis, and here we explore some of the primary benefits of utilizing this visualization: Visual Clarity in Representing Data Distributions While raw data presented in tabular form can be difficult to interpret, a box diagram provides an instant visual representation of the data distribution. One can identify the central tendency, spread, and skew of the dataset at a glimpse. Its condensed representation distills enormous data into a concise picture, allowing for immediate understanding. Those who must make quick, informed judgments based on data will find the box plot's visual clarity invaluable. Comparative Analysis between Multiple Data Sets One of the most notable advantages of box plots is their utility in comparing multiple data sets side by side. Suppose researchers are comparing test scores from various courses, or that analysts are analyzing monthly sales over a number of years. Multiple box plots can be displayed adjacently in this situation, offering a clear visualization of differences in medians, quartiles, and variability among the datasets. A side-by-side comparison can reveal trends and abnormalities that numerical summaries may miss. Quick Identification of Outliers In data analysis, outliers have the potential to distort results and interpretations. Box plots, with their distinct portrayal of outliers as separate points outside the whiskers, make it simple to identify these abnormal data points. This instant visual indicator can prompt analysts to check whether these anomalies are genuine data variations or errors that require correction. Efficient Representation of Data Quartiles and Medians The central box of the box diagram represents the interquartile range and contains 50% of the values in the dataset. By indicating the first quartile (Q1), the median, and the third quartile (Q3), the box plot effectively illustrates the dataset’s quartile distribution. This provides insights into the data's dispersion and central tendency, delivering a more nuanced understanding than measures such as the mean alone. In addition, the median's distinct representation, which is typically a line within the box, clearly indicates the center of the data set. Comparing Box Plots to other Data Visualization Tools Different tools in the vast field of data visualization provide unique perspectives on data sets. While box plots offer a thorough snapshot of data distribution, other tools such as histograms, scatter plots, and bar graphs are useful for other analytical needs. Here is a comparison of these tools in relation to box plots: Histograms vs Box Plots • Histograms: These depict the distribution of data by forming bins along the data's range and then sketching bars to indicate the number of observations that fall within each bin. The height of each bar indicates the frequency of data points within a given interval. • Comparison: Histograms, as opposed to box plots, display the shape of the data distribution, making it simpler to identify modes (peaks) and understand the overall distribution pattern, whether it is normal, skewed, or bimodal. In contrast, box plots emphasize quartiles, medians, and potential outliers. • Usage Scenarios: Histograms are optimal for analyzing the structure and distribution of massive datasets. Box diagrams excel at comparing multiple datasets and quickly identifying data quartiles and outliers. Scatter Plots vs Box Plots • Scatter Plots: These indicate the relationship or correlation between two variables by displaying individual data points on a two-dimensional axis. • Comparison: Box plots provide a comprehensive view of a dataset's distribution, whereas scatter plots are best for observing relationships and identifying patterns or trends between two datasets. In contrast to medians and quartiles, scatter graphs excel at displaying correlations. • Usage Scenarios: Scatter plots are ideal for regression analysis, correlation evaluation, and observing temporal data trends. On the other hand, box plots are better suited for analyzing the central tendencies and distribution spread of a single or multiple datasets. Bar Charts vs Box Plots • Bar Charts: These illustrate data with rectangular bars whose lengths are proportional to the values they represent. Bar charts can categorize data and are often used to compare values across • Comparison: Box plots provide insight into data distribution, including medians, quartiles, and outliers, whereas bar graphs emphasize discrete data, focusing on the magnitude of values across categories. Distribution characteristics, such as skew and kurtosis, are not apparent from bar charts. • Usage Scenarios: Bar charts excel at representing and comparing data across distinct categories as well as displaying changes over time, particularly for nominal or small ordinal datasets. Box plots, meanwhile, are better suited for studying the distribution properties of interval or ratio data. Creating a Box Plot: A Step-by-Step Guide The box plot or whisker plot is a graphical representation of a dataset's central tendencies, distribution, and outliers. While there are many software and tools for quickly generating a box plot, grasping the manual process provides insight into its underlying mechanics. Here is how to create a box plot from scratch: Data Collection and Visualization Box plots provide a comprehensive view of data distribution, and their creation can be a systematic procedure involving the following steps: Data Collection and Organization • Gather Data: Begin by gathering the data set you wish to represent. This may include survey results, experimental data, or any other quantitative dataset. • Organize Data: Arrange the data in ascending order to make it easier to find the quartiles and the median in the subsequent steps. Calculating Key Values • Median: Determine the middle value of your data set. If the number of observations in the data set is even, the median will be the average of the two middle values. • First Quartile (Q1): This is the number in the middle of the smallest number in the set and the median. • Third Quartile (Q3): This is the median of the second half of the data set, which is the intermediate value between the median and highest values. • Interquartile Range (IQR): The IQR is the difference between Q3 and Q1 and provides an idea of the data set's value distribution. Identifying Outliers Outliers: These values do not lie within the range defined by Q1 - 1.5(IQR) and Q3 + 1.5(IQR). Drawing the Box Plot • Sketching: On graph paper or using software, make a scale that encompasses the data set's range. Mark the positions of the first quartile, the median, the third quartile, and any outliers. • Constructing the Box: Draw a rectangle with Q1 and Q3 serving as the lower and upper limits, respectively. Draw the median line within this box. • Adding Whiskers: Extend lines from the top and bottom of the box to the highest and lowest non-outlier values in the dataset. These are your ‘whiskers’. • Marking Outliers: If there are any outliers, represent those using dots or asterisks outside of the whiskers. Interpreting Box Plots in Real-World Scenarios Box plots are widely utilized in a range of industries due to their ability to simply and succinctly illustrate data distribution. They can provide valuable insight into a dataset's central tendencies, spread, and prospective outliers. Here are some examples of how box plots are used in the real world: Box plots are used by financial analysts to study stock price distributions over time or to compare the performance of multiple stocks. By examining the whiskers, for instance, they can quickly determine the volatility of a stock. The position of the median, meanwhile, can help predict the overall direction of stock performance over a specific time period. Outliers may reflect an unexpected market event or company news that has an impact on stock prices. Box plots can be used in biological research to compare data distributions across multiple experimental settings or groups. Consider comparing the heights of plants grown under different light situations. The box plot would quickly reveal if one group had greater variation in height or if any plants in a particular group were unusually tall or short. Box plots can help with quality control in manufacturing by comparing product dimensions or performance measures across multiple production batches. If a single batch has a median value outside of the intended range or displays greater variation (a wider interquartile range), this may indicate inconsistencies in the production processes. Common Misconceptions and Pitfalls • Misconceptions about Medians: A common misunderstanding is confusing the median (the line inside the box) with the average. While they both provide a measure of central tendency, they can differ significantly, particularly in skewed datasets. • Overemphasis on Outliers: If a data point is an outlier, it does not automatically indicate an error or that it should be discarded. It is essential to understand the context behind outliers before disregarding outliers. • Whisker Lengths: Some may erroneously believe that the lengths of the whiskers always denote errors or standard deviations. In reality, they cover the minimum and maximum data points within the acceptable range. • Oversimplification: Using only box plots can result in an oversimplified comprehension of data. They summarize data distributions but do not provide granular information such as individual data points or specific distributions like bimodal patterns. • Missing Data Patterns: While box plots identify outliers and provide a sense of the overall data spread, they may overlook subtleties such as clusters and gaps in the data. Box plots, a product of statistical visualization, have proven indispensable for obtaining a concise yet comprehensive view of data distributions in a range of disciplines. Their strengths range from visual clarity in portraying medians and quartiles to effective detection of outliers. However, box plots, like any other tool, are not without misconceptions and pitfalls. While they provide a condensed overview, a deeper dive into the data is often required for a more thorough understanding. Utilizing box plots as part of a larger toolkit is essential for ensuring that data is properly presented and evaluated in any analytical activity. Box Plots with Jaspersoft Jaspersoft in Action: Embedded BI Demo See everything Jaspersoft has to offer – from creating beautiful data visualizations and dashboards to embedding them into your application. Creating Addictive Dashboards Learn how to build dashboards that your users will love. Turn your data into interactive, visually engaging metrics that can be embedded into your web application.	{"url":"https://www.jaspersoft.com/articles/what-is-a-box-plot","timestamp":"2024-11-05T13:02:20Z","content_type":"text/html","content_length":"67503","record_id":"<urn:uuid:d130a962-408b-4549-a9ce-b07e9a859654>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00241.warc.gz"}
gensymb – Generic symbols for both text and math mode Provides generic commands \degree, \celsius, \perthousand, \micro and \ohm which work both in text and maths mode. Various means are provided to fake the symbols or take them from particular symbol fonts, if they are not available in the default fonts used in the document. This should be perfectly transparent at user level, so that one can apply the same notation for units of measurement in text and math mode and with arbitrary typefaces. Note that the package has been designed to work in conjunction with units.sty. This package used to be part of the was bundle, but has now become a package in its own right. Sources /macros/latex/contrib/gensymb Repository https://gitlab.com/kjhtex/gensymb Version 1.0.2 Licenses The LaTeX Project Public License 1.3 Copyright 2003–2022 Walter Schmidt 2022 Keiran Harcombe Maintainer Keiran Harcombe Walter A. Schmidt (deceased) Contained in TeXLive as gensymb MiKTeX as gensymb Topics Maths Text symbol Download the contents of this package in one zip archive (221.2k). Community Comments Maybe you are interested in the following packages as well. Package Links	{"url":"https://ctan.org/pkg/gensymb","timestamp":"2024-11-09T17:51:34Z","content_type":"text/html","content_length":"17226","record_id":"<urn:uuid:e2acb070-ebc3-4084-bd3f-25c458d61719>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00309.warc.gz"}
multiplication table Archives \| Fundamentals of Mathematics and Physics There has been a war in the mathematics education world for the past few decades about whether students should master basic skills, or whether they should use calculators or software for basic skills to save time and energy for higher-level thinking. More and more people nowadays are seeing this for what it is: a false … Read more How Much Mathematics Should a Student Memorize? The more you understand, the less you have to memorize. A good example is trigonometric identities, of which there are quite a number. Should a student memorize trigonometric identities? Well, at first, it is probably wise to memorize a few of them. Part of a teacher’s job is to help students identify what is essential … Read more	{"url":"https://fomap.org/tag/multiplication-table/","timestamp":"2024-11-11T13:01:39Z","content_type":"text/html","content_length":"56842","record_id":"<urn:uuid:365a2136-b56c-4c88-b3a2-b5afa041a2a7>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00733.warc.gz"}