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https://artofproblemsolving.com/wiki/index.php?title=Asymptote_(geometry)&diff=prev&oldid=122167	[ "# Difference between revisions of \"Asymptote (geometry)\"\n\nFor the vector graphics language, see Asymptote (Vector Graphics Language).\n\nAn asymptote is a line or curve that a certain function approaches.", null, "The function", null, "$y=\\tfrac{2x}{x-2}$ has a vertical asymptote at x=2 and a horizontal asymptote at y=2\n\nLinear asymptotes can be of three different kinds: horizontal, vertical or slanted (oblique).\n\n## Vertical Asymptotes\n\nThe vertical asymptote can be found by finding values of", null, "$x$ that make the function undefined. Generally, it is found by setting the denominator of a rational function to zero.\n\nIf the numerator and denominator of a rational function share a factor, this factor is not a vertical asymptote. Instead, it appears as a hole in the graph.\n\nA rational function may have more than one vertical asymptote.\n\n### Example Problems\n\nFind the vertical asymptotes of 1)", null, "$y = \\frac{1}{x^2-5x}$ 2)", null, "$\\tan 3x$.\n\n#### Solution\n\n1) To find the vertical asymptotes, let", null, "$x^2-5x=0$. Solving the equation:", null, "$\\begin{eqnarray}x^2-5x&=&0\\\\x&=&\\boxed{0,5}\\end{eqnarray}$\n\nSo the vertical asymptotes are", null, "$x=0,x=5$.\n\n2) Since", null, "$\\tan 3x = \\frac{\\sin 3x}{\\cos 3x}$, we need to find where", null, "$\\cos 3x = 0$. The cosine function is zero at", null, "$\\frac{\\pi}{2} + n\\pi$ for all integers", null, "$n$; thus the functions is undefined at", null, "$x=\\frac{\\pi}{6} + \\frac{n\\pi}{3}$.\n\n## Horizontal Asymptotes\n\nFor rational functions in the form of", null, "$\\frac{P(x)}{Q(x)}$ where", null, "$P(x), Q(x)$ are both polynomials:\n\n1. If the degree of", null, "$Q(x)$ is greater than that of the degree of", null, "$P(x)$, then the horizontal asymptote is at", null, "$y = 0$. This can be seen by noting that as", null, "$x$ increases,", null, "$Q(x)$ increases much faster than", null, "$P(x)$ does. Since the denominator increases faster than the numerator, as x approaches infinity, y gets smaller until it approaches zero. A similar trend can be seen as x decreases.\n\n2. If the degree of", null, "$Q(x)$ is equal to that of the degree of", null, "$P(x)$, then the horizontal asymptote is at the quotient of the leading coefficient of", null, "$P(x)$ over the leading coefficient of", null, "$Q(x)$.\n\n3. If the degree of", null, "$Q(x)$ is less than the degree of", null, "$P(x)$, see below (slanted asymptotes)\n\nA function may not have more than one horizontal asymptote. Functions with a \"middle section\" may cross the horizontal asymptote at one point. To find this point, set y=horizontal asymptote and solve.\n\n### Example Problem\n\nFind the horizontal asymptote of", null, "$f(x) = \\frac{x^2 - 3x + 2}{-2x^2 + 15x + 10000}$.\n\n#### Solution\n\nThe numerator has the same degree as the denominator, so the horizontal asymptote is the quotient of the leading coefficients:", null, "$y= \\frac {1} {-2}$\n\n## Slant (Oblique) Asymptotes\n\nFor rational functions", null, "$\\frac{P(x)}{Q(x)}$, a slant asymptote occurs when the degree of", null, "$P(x)$ is one greater than the degree of", null, "$Q(x)$. If the degree of", null, "$P(x)$ is two or more greater than the degree of", null, "$Q(x)$, then we get a curved asymptote. Again, like horizontal asymptotes, it is possible to get crossing points of slant asymptotes.\n\nFor rational functions, we can find the slant asymptote simply by long division, omitting the remainder and setting y=quotient.\n\n### Example Problem\n\nFind the slant asymptote of", null, "$y= \\frac{x^2+2x+4} {x+1}$\n\nSolution", null, "$\\frac{x^2+2x+4}{x+1}= x+1+\\frac{3} {x+1}$\n\nThe slant asymptote is", null, "$y=x+1$", null, "The function", null, "$y=\\tfrac{x^2+2x+4} {x+1}$ has a slant asymptote at y=x+1" ]	[ null, "https://wiki-images.artofproblemsolving.com//thumb/4/45/Asymptote_graph.png/300px-Asymptote_graph.png", null, "https://latex.artofproblemsolving.com/5/2/3/5237b8373947bd72642f8994aff9c73ae674e947.png ", null, "https://latex.artofproblemsolving.com/2/6/e/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png ", null, "https://latex.artofproblemsolving.com/d/d/e/dde7f2e2768e19f87224109da74c5aef7ad9aef1.png ", null, 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https://www.osapublishing.org/oe/fulltext.cfm?uri=oe-16-2-695&id=148790	[ "## Abstract\n\nWe describe the evolution of a paraxial electromagnetic wave characterizing by a non-uniform polarization distribution with singularities and propagating in a weakly anisotropic medium. Our approach is based on the Stokes vector evolution equation applied to a non-uniform initial polarization field. In the case of a homogeneous medium, this equation is integrated analytically. This yields a 3-dimensional distribution of the polarization parameters containing singularities, i.e. C-lines of circular polarization and L-surfaces of linear polarization. The general theory is applied to specific examples of the unfolding of a vectorial vortex in birefringent and dichroic media.\n\n## 1. Introduction\n\nSingular optics is an essential part of modern optics, which contributes to practically all fundamental wave phenomena [1–4]. Scalar wave fields are characterized by phase singularities (i.e., zeros of the intensity where the phase is undeterminate), such as optical vortices which have found a number of applications in classical and quantum optics [3,4]. Vector fields, e.g. electromagnetic or elastic waves, have more degrees of freedom and are also characterized by polarization singularities [5–9]. The generic types of the polarization singularities of transverse electromagnetic waves in 3D space are C-lines and L-surfaces, where the polarizations are, respectively, circular and linear. In the two cases, the polarization ellipse degenerates either to a circle (the eccentricity vanishes and the orientation is undeterminate) or to a line segment (the helicity vanishes and sign of polarization is undeterminate).\n\nThe wave field singularities may form a rich variety of structures in rather simple systems. Even interference of only three plane scalar waves results in a lattice of optical vortices . Clearly, propagation in inhomogeneous or/and anisotropic media significantly modifies the wave interference patterns and, hence, gives rise to a quite tangled singularities structures. Therefore, the wave field singularities in complex media are frequently studied within the statistical approach [7,9,11]. At the same time, for various applications it is very important to describe behavior of the specific singularities explicitly, i.e. in a deterministic way. There is a number of laboratory methods for generating and manipulating phase [3,4] and polarization [12,13] singularities in electromagnetic fields. Therefore, one of the currently important problems is to know how singularities evolve as the wave propagates through a medium.\n\nPropagation of uniformly-polarized paraxial beams with phase vortices in inhomogeneous or anisotropic media have been studied recently [14–16]. While such beams represent independent localized modes of a smoothly inhomogeneous isotropic medium , phase vortices become drastically unstable in an anisotropic medium [15,16]. Even in the simplest uniaxial homogeneous medium, the phase vortex disappears, giving way to an essentially space-variant polarization pattern with a variety of polarization singularities [15,16]. The following features are characteristic for this system: (i) an initial phase singularity, (ii) a uniform initial polarization, and (iii) a double refraction in the medium, which destroys the phase singularity and transforms in to a set of polarization singularities.\n\nIn the present paper we aim to investigate the dynamical behavior of the polarization singularities in a paraxial wave field propagating in a weakly anisotropic and, possibly, inhomogeneous medium. However, in contrast to [15,16], the problem is considered under opposite conditions: (i) a space-variant polarization pattern with polarization singularities in the incident field and (ii) absence of the phase singularities therein; (iii) we argue that double refraction is negligible in the system, while the variations of the normal modes parameters (phases and amplitudes) along the propagation direction lead to an effective dynamics of the polarization distribution and singularities. Our choice of the initial conditions is justified by two reasons. First, it is the presence of an effective technique using subwavelength gratings for generating arbitrary space-variant polarization patterns of the field [12,13]. Second, as it follows from [15,16], the phase singularities become unstable in anisotropic media, while the polarization ones experience a continuous evolution.\n\nBy applying a dynamical approach, well-established in standard polarimetry [17,18], to a space-variant polarization pattern, we develop a powerful method for studying 3D complex polarization distributions. Assuming paraxial approximation and weak anisotropy, our approach reduces a challenging wave problem to the solution of effectively ordinary differential equation for the Stokes vector evolution along the wave propagation direction [18–26]. The equation is integrated in a homogeneous medium analytically, and, despite its simple form, it reveals an intricate evolution of the polarization singularities when the wave propagates through the medium. Thus, our method brings together polarimetry and singular optics, thereby giving rise to singular polarimetry. It may have promising applications – space-variant polarization patterns with singularities can be more informative and sensitive with respect to the medium properties.\n\n## 2. General theory\n\n#### 2.1 Statement of the problem\n\nWe will examine propagation of a paraxial monochromatic electromagnetic wave through a weakly anisotropic and, possibly, inhomogeneous (stratified) medium. We assume that the wave propagates along the z axis, whereas the polarization ellipse lies nearly in the (x, y) plane, so that one can apply Mueller or Jones calculus to the z -dependent evolution of polarization [17–19]. Under this assumption, the incident field is treated as a collection of parallel rays that have essentially independent phase and amplitude evolution. Mathematically, this means that we deal with a Cauchy problem with an initial distribution of the field in the (x, y) plane at z=0 and some dynamical equation describing the evolution of the field along the z axis.\n\nLet the polarization of the wave field at a point r be described through the three-component normalized Stokes vector, s=s(r), s 2=1, representing the polarization state on the Poincaré sphere. Then, the Cauchy problem is given by the initial Stokes vector distribution,\n\n$s(x,y,0)=s0(x,y),$\n\nand a dynamical equation\n\n$∂s∂z=m̂s,$\n\nwhere m̂=m̂(s,z) is a matrix operator which relates the Stokes-vectors values in two neighbor points. Although equations similar to Eq. (2) are well-established in classical polarimetry [18–26], they usually assume the uniform polarization distribution in the transverse plane, s 0(x, y)=const, making the polarization evolution effectively one-dimensional, s=s(z). In contrast, a non-uniform initial distribution (1) in our problem, s 0(x,y)≠const, makes the polarization distribution essentially three-dimensional, s=s(x,y,z). Despite this, in order to find the complete distribution of the polarization in 3D space, s(r), one need to integrate an effectively ordinary differential equation (2) with initial conditions (1) for each point (x,y).\n\nThe formalism of 3-component Stokes vector provides a natural representation for the polarization singularities [8,16]. Indeed, the north and south poles of the Poincaré sphere correspond to the right- and left-hand circularly polarized waves, while the equator represents linear polarizations with different orientations. Then, C- and L-type polarization singularities are determined, respectively, by the conditions\n\n$s1=0,s2=0,$\n\nand\n\n$s3=0.$\n\nFrom Eqs. (3) and (4) it is clear that in the generic case C- and L-singularities are, respectively, lines and surfaces in 3D space: the dimension of the singularity is the dimension of the space minus the number of constraints. Alternatively, one may refer to C-points and L-lines in the (x, y) plane (2D space) and their evolution along the z axis. Note also, that polarization singularities are essentially determined by the third component of the Stokes vector - conditions (3) are equivalent to \|s 3\|=1, or s 3=χ, where χ=±1 is the wave helicity which indicates the sign of polarization in the C-point. Having a solution of the problem Eqs. (1) and (2), s=s(r), one immediately gets the space distribution of all polarization singularities from Eqs. (3) and (4). Note also that Eq. (1) implies that there are no phase singularities in the initial field, i.e. the intensity of the wave does not vanish: I 0(x,y)=I(x,y,0)≠0 (otherwise, the Stokes vector s would be undefined in nodal points). As we will see, the dynamical equation (2) ensures that the nodal points cannot appear at z≠0 as well: I(x,y,z)≠0.\n\nOur approach of z -dependent evolution, Eqs. (1) and (2), is justified assuming that the refraction and diffraction processes are negligible. Let the wave field be characterized by two scales: the wavelength λ and a typical scale of its transverse distribution in the (x, y) plane, wλ. At the same time, the medium anisotropy is characterized by a typical difference between the dielectric constants corresponding to the normal modes, ν≪1. Then, diffraction and refraction effects are negligible if: (i) the propagation distance is much smaller than the typical diffraction distance (the Rayleigh range), ZZR=w 2/λ and (ii) the propagation distance is much smaller than the distance at which the double refraction of the anisotropic medium causes transverse shifts comparable with w, i.e. ZZDw/ν. Note that the characteristic distance of the polarization evolution due to Eq. (2), ZP=λ/ν, is much smaller than ZD and can be small as compared to ZR. Thus, our approach is effective within the range of distances\n\n$zP≤z≪zR,zD.$\n\nFor instance, for a visible laser beam with λ⋍0.6µm and width w⋍1mm propagating through Quartz (where anisotropy is ν⋍0.03), we have ZR~1.5m, ZD~30mm, and ZP⋍0.02mm. This gives the propagation range 0.02mm≤z≪30mm.\n\n#### 2.2 Equation for the Stokes vector evolution\n\nTo derive the evolution equation (2) for the 3-component Stokes vector s, let us start with the 4-component Stokes vector, S⃗. Hereafter, 4-component vectors are indicated by arrows, and the last three components of a 4-vector form usual 3-component vector, so that S⃗=(S 0,S 1,S 2,S 3)≡(S 0,S). In the most general case of a linear anisotropic medium the Stokes vector S⃗ obeys the following evolution equation [18–26]:\n\n$∂S→∂z=M̂S→,$\n\nwhere M̂ is the differential Mueller matrix (a 4×4 real matrix) which summarizes optical properties of the medium. These are given by 2×2 complex dielectric tensor:\n\n$ε̂=ε0Î2+ν̂≡(ε0+νxxνxyνyxε0+νyy).$\n\nHere ε 0 Î 2 is the main, isotropic part proportional to the unit matrix Î 2=diag (1,1), $ν^$ is a small anisotropic part (which effectively represents the differential Jones matrix), and we assume Im ε 0=0 (small dissipation is ascribed to the anisotropic term). The differential Mueller matrix can be represented as [19,24,26]\n\n$M̂=(ImG0ImG1ImG2ImG3ImG1ImG0−ReG3ReG2ImG2ReG3ImG0−ReG1ImG3−ReG2ReG1ImG0).$\n\nHere the complex 4-component vector G⃗=(G 0,G) is expressed via components of the dielectric tensor (7) as\n\n$G→=−k02ε0[(νxx+νyy),(νxx−νyy),(νxy+νyx),i(νxy−νyx)]T,$\n\nwhere k 0 is the wave number in vacuum, and components of quantities (7)–(9) can be z - dependent. Vector G⃗ gives decomposition of the anisotropy tensor $ν^$, Eq. (6), with respect to the basis of Pauli matrices, and establishes close relations between polarization optics (Mueller and Jones calculus) and relativistic problems with the Lorentz-group symmetry [20,24,26–31]. In particular, Eq. (6) for the Stokes vector evolution is similar to the Bargman-Michel-Telegdi equation for relativistic spin precession [26,32].\n\nThe matrix M̂ can be decomposed into three parts responsible for different optical properties of the medium [18–26]:\n\n$M̂=ImG0Î4+(0ImG1ImG2ImG3ImG1000ImG2000ImG3000)+(000000−ReG3ReG20ReG30−ReG10−ReG2ReG10),$\n\nwhere Î 4=diag (1,1,1,1) is the unit matrix. The first, diagonal part, proportional to ImG 0, describes the common attenuation of the field intensity. The second, symmetric part, related to components of ImG, describes the phenomenon of dichroism, i.e. the selective attenuation of different field components. Finally, the third, antisymmetric part of M̂, related to components of ReG, is responsible for the medium birefringence. The component 0 ReG does not contribute to the matrix (8), since it causes merely an additional total phase of the wave field, which does not affect the polarization state and is lost in the Stokes vector representation.\n\nEvolution of the normalized 3-component Stokes vector can be derived immediately from Eqs. (6) and (8) by differentiating the definition s=S/S 0. As a result, we find that the 3-component Stokes vector obeys the following equation :\n\n$∂s∂z=(Ω+s×∑)×s,$\n\nwhere we denoted ReGΩ and ImGΣ. This equation represents the basic evolution equation (2) in the general case. [In terms of Eq. (2), the matrix m̂ is given by mij=-eijkΩk-eijk eklm slΣm, where indices take values 1, 2, 3 and eijk is the unit antisymmetric tensor.] Thus, all the evolution on the Poincaré sphere can be described by the precession equation (10) which includes two real vectors Ω and Σ responsible for the birefringent and dichroic effects, respectively. Equation (10) conserves the absolute value of the normalized Stokes vector under the evolution: ∂s 2/∂z=0. The common attenuation, ImG 0, naturally, does not affect the normalized Stokes vector and is absent in Eq. (10). Note also that Eq. (10) resembles the Landau-Lifshitz equation describing the nonlinear spin precession in ferromagnets , but, in contrast to the latter, Eq. (10) contains two different effective fields Ω and Σ. In inhomogeneous medium Ω=Ω(z) and Σ=Σ(z).\n\n#### 2.3 Solutions in a homogeneous birefringent medium\n\nIn a homogeneous non-dissipative birefringent medium, Eq. (10) takes simple form of the classical precession equation [22,23,25]:\n\n$∂s∂z=Ω×s.$\n\nAccording to Eq. (11), as the wave propagates along the z axis, the Stokes vector s precesses with a constant spatial frequency Ω about the fixed direction ω=Ω/Ω. In terms of the medium properties, direction ω and absolute value Ω characterize, respectively, the type and the strength of the medium birefringence. In so doing, two “stationary” solutions s ±ω on the Poincaré sphere correspond to mutually-orthogonal eigenmodes of the medium. In particular, ω=(0, 0,±1) and ω=(ω12,0) correspond, respectively, to the cases of circularly- and linearly-birefringent medium. Equation (11) with initial condition (1) can be easily integrated: s(x,y,z)= ωz)s 0(x,y), where ωz) is the operator of rotation about ω on the angle Ωz. Using the Rodrigues rotation formula , we arrive at\n\n$s=s0cos⁡(Ωz)+(ω×s0)sin⁡(Ωz)+(ωs0)ω[1−cos⁡(Ωz)].$\n\nTogether with Eq. (1), this equation gives solution for the Stokes-vector distribution in space.\n\nIn a circularly birefringent medium, distribution of polarization singularities in (x,y) plane does not vary when the wave propagates along the z axis. Indeed, the helicity of the wave, given by the third component of the Stokes vector, is invariant of Eqs. (11) and (12), s 3=const, when ω=(0, 0,±1). Thus, C-lines (s 3=±1) and L-surfaces (s 3=0) are parallel to the z axis in this case and are trivially determined by the initial distribution (1) with Eqs. (3) and (4):\n\n$s01=0,s02=0,$\n\nand\n\n$s03=0.$\n\nOn the contrary, polarization singularities evolve in a linearly-birefringent medium, cf. [15,16]. As it is clear from Eq. (12), this evolution is periodic in z with the period 2π/Ω. In fact, L-lines and C-points in (x,y) plane come back to the initial locations after π/Ω period, corresponding to the half-wavelength plate. In so doing, C-points only change their signs after π/Ω period. One can also note that under propagation at π/2Ω distance (corresponding to the quarter-wavelength plate) C-points give their place to points of L-lines, while some points of L-lines give place to C-points. To determine the whole 3D structure of polarization singularities note that vector ω=(ω1, ω2, 0) can be reduced to ω=(1,0,0) by a fixed rotation of coordinate axes in the (x,y) plane, which brings the anisotropy tensor $ν^$, Eq. (7), to the principal axes. Then, substituting solution (12) with ω=(1,0,0) into Eqs. (3) and (4), we obtain equations determining C-lines and L-surfaces:\n\n$s01=0,tan⁡(Ωz)=s02s03,$\n\nand\n\n$tan⁡(Ωz)=−s03s02.$\n\nIn contrast to Eqs. (13) and (14), these rather simple equations reveal non-trivial z -dependent dynamics of polarization singularities (see examples in Section 3.2).\n\n#### 2.4 Solutions in a homogeneous dichroic medium\n\nIn a homogeneous dichroic medium, with selective attenuation of modes but without a phase difference between them (i.e., Ω=0), Eq. (10) takes the form\n\n$∂s∂z=(s×∑)×s.$\n\nSimilarly to Eq. (11), this equation has two “stationary” solutions s ±σ (where σ=Σ/Σ), which determine eigenmodes of the medium. However, in contrast to the birefringent-medium case, solution on s - is “unstable”. As we will see, solutions of Eq. (17) move on the Poincaré sphere away from s - towards s +. Thus, the dichroic medium is a polarizer, in which only one mode (given by s +) survives at long enough propagation distances. Equation (17) can be integrated analytically at Σ=const, which yields the solution (see Appendix):\n\n$s=2(1+A0)e∑z+(1−A0)e−∑zs0+(1+A0)e∑z−2A0−(1−A0)e−∑z(1+A0)e∑z+(1−A0)e−∑zσ,$\n\nwhere A 0=s 0 σ. As seen from Eq. (18), all solutions with s 0s - move in the plane given by vectors s 0 and σ, and tend exponentially to s + point. Supplied with the initial distribution (1), Eq. (18) gives the Stokes-vector distribution in 3D space.\n\nIn contrast to the birefringent-medium case, polarization singularities vary along z both in circularly and linearly dichroic medium. More precisely, in a circularly-dichroic medium, σ=(0,0,±1), C-points in the (x,y) plane do not evolve with z since circular polarizations correspond to “stationary” solutions s ± in this case. Thus C-lines in 3D space are parallel to the z axis and determined analogously to Eq. (13):\n\n$s01=0,s02=0.$\n\nAt the same time, L-lines in the (x,y) plane evolve with z, and L-surfaces in 3D space are determined by Eq. (4) with (18):\n\n$(1+A0)e∑z−(1−A0)e−∑z(1+A0)e∑z+(1−A0)e−∑z=0,$\n\nwhere A 0=s 03 σ 3. Equation (20) can be resolved with respect to z, which yields\n\n$tan⁡h(∑z)=−s03σ3.$\n\nConversely, in a linearly-dichroic medium, where σ=(1, 0, 0) in the principal-axes coordinate frame, L-surfaces are parallel to the z axis and are given by Eq. (14):\n\n$s03=0.$\n\nAt the same time, C-points in the (x,y) plane evolve with z, and 3D C-lines are determined by Eq. (3) with (18), i.e.\n\n$(1+A0)e∑z−(1−A0)e−∑z(1+A0)e∑z+(1−A0)e−∑z=0,s02=0,$\n\nwhere A 0=s 01. Resolving of this equation yields:\n\n$tanh(∑z)=−s01,s02=0.$\n\nThus, unlike birefringent medium, evolution of the Stokes vector and polarization singularities in dichroic medium is monotonic rather than periodic, see examples in Section 3.3. It is described by hyperbolic functions, which appear naturally in the Lorentz-group representation of polarization optics [30,31].\n\n## 3. Application: Evolution of a vectorial vortex\n\n#### 3.1 Initial polarization distribution\n\nAs a characteristic example of initial space-variant polarization pattern, Eq. (1), we consider a vectorial vortex, which possesses a singularity in the polarization distribution [12,13]. The Stokes-vector distribution (1) of a vectorial vortex at the origin can be given as\n\n$s01=1−f2(ρ)cos[m(φ−δ)],$\n$s02=1−f2(ρ)sin[m(φ−δ)],$\n$s03=f(ρ).$", null, "Fig. 1. Distributions of Stokes vectors (upper panel) and of polarization ellipses (lower panel) for vectorial vortices Eqs. (25) with a C-point in the center, Eq. (26), at different values of azimuthal index m. Hereafter, δ=0 and directions of Stokes vectors are naturally depicted in the real space with the s 1, s 2, and s 3 components pointing along the x, y, and z axis, respectively.\n\nHere (ρ,φ) are the polar coordinates in the (x,y) plane, f (ρ) is a radial distribution function, \|f(ρ)\|≤1, m=±1,±2,… is the integer number (the azimuthal index of the polarization distribution), and δ indicates a fixed angle between the distribution and x axis. In the above distribution, the Stokes vector experiences m complete rotations along a loop path enclosing the vortex center. Therefore, the distribution possesses the \|m-1\| -fold rotational symmetry (one turn is effectively compensated by a 2π rotation of local radial vector), Fig. 1. At the same time, the corresponding polarization pattern (i.e. the distribution of polarization ellipses in the (x, y) plane) reveals \|m-2\| -fold symmetry, Fig. 1. This is because a complete turn of the Stokes vector corresponds to a half-turn of the polarization ellipse; as a result, the symmetry of the polarization distribution is characterized by the order of \|-2π\|modπ. Note that the cases m=1 and m=2 are peculiar: the vectorial vortex represents an azimuthally-symmetric Stokes-vector and polarization distributions, respectively.\n\nPoints ρ=ρC and ρ=ρL, such that f(ρC)=χ=±1 and f(ρL)=0 correspond to C-and L-type singularities in the initial distribution (25). We will concentrate on a simple case of a single C-point with the right-hand circular polarization at the origin, so that f(0)=χ and 0<\|f(ρ)\|<1 at ρ>0. This case can be modeled using the function\n\n$f=χ1+(ρρ)2,$\n\nwhere ρ characterizes the radial scale of the distribution. The vectorial vortex (25) and (26) with the right-hand polarization in the center, χ=1, is characterized by the optical vortex (phase singularity) in the left-hand polarized component of the field, and vice versa. Below, we will examine the behavior of polarization singularities in homogeneous anisotropic media considered in Sections 2.3 and 2.4, assuming the initial polarization distribution of Eqs. (25) and (26).\n\n#### 3.2 Homogeneous linearly-birefringent medium\n\nSince the behavior of polarization singularities in a circularly-birefringent medium is trivial, Eqs. (13) and (14), let us consider the case of linearly-birefringent medium. Substituting the initial Stokes-vector distribution, Eqs. (25), into Eqs. (15) and (16) we obtain the equations describing C-lines and L-surfaces in space. For C-lines, this yields\n\n$1−f(ρ)2cos[m(φ−δ)]=0,tan(Ωz)=1−f2(ρ)sin[m(φ−δ)]f(ρ),$\n\nwhereas L-surfaces are described by equation\n\n$tan(Ωz)=−f(ρ)1−f2(ρ)sin[m(φ−δ)].$\n\nAssuming the initial distribution with radial function (26), we find that solutions of Eqs. (27) represent curves lying in the azimuthal planes φ=const and given by equations\n\n$φn=δ+π(2n+1)2m,tan(Ωz)=±1−f2(ρ)f(ρ)=±χρρ.$\n\nHere n=0,1,…, 2m-1, signs “±” in the second equation correspond to even and odd n, respectively. Note that tan(Ωz) is either positive or negative at each value of z, and, hence, only solutions with either even or odd n are valid each time. They alternate after a period of π/2Ω, whereas all the structure of polarization singularities (up to sign of the polarization) has a period of π/Ω. L-surfaces (28) separate C-lines with different helicities χ (and space areas with positive and negative s 3) and represent azimuthally-corrugated surfaces. Figure 2 shows an example of the polarization singularities described by Eqs. (29) and (28). The initial C-point of m th order splits into m branches under the evolution along z. This reveals an instability of higher-order C-points during evolution in an anisotropic medium. Only C-points with of a minimal order, m=±1, are generic [5–9].", null, "Fig. 2. C-lines and L-surfaces under propagation of the vectorial vortex, Eqs. (25) and (26), with m=3 and χ=1 in a linearly-birefringent medium with ω=(1, 0,0). Dimensionless coordinates ξ=x/ρ, η=y/ρ, and ζz/π are used, whereas red and blue colors indicate C-lines with χ=1 and χ=-1, respectively.\n\nSimilarly to , one can verify conservation of the total topological charge by tracking the polarization singularities evolution. Topological charge, γ, characterizes each C-point in plane z=const and is equal to the product of its local azimuthal index, m, and sign of the polarization, χ=s 3: γ=. The total topological charge, $Γ=∑aγ(a)$ , should be z - independent [7,9,15]:\n\n$Γ=∑am(a)χ(a)=const.$", null, "Fig. 3. Distributions of Stokes vectors, Eq. (12), (upper panel) and of polarization ellipses (lower panel) for vectorial vortex, Eqs. (25) and (26), with m=3 and χ=1, propagating in a linearly-birefringent medium with ω=(1, 0,0), Fig. 2. Distributions are shown at different propagation distances ζz/π within half-period. Red and blue colors indicate areas with right-hand (s 3>0) and left-hand (s 3<0) polarizations. C-points are marked by dots (upper panel) and crosses (lower panel).\n\nHere a indicates different C-points and summation is taken over whole plane z=const. Figure 3 demonstrates polarization and Stokes-vector distributions, Eq. (12), for a vectorial vortex evolving in a linearly birefringent medium corresponding to Fig. 2. Conservation of Γ=3 is seen — initial C-point with m=3 and χ=1 is split into three C-points with m=1 and χ=1; then, crossing of L-surface causes simultaneous flip of helicity and azimuthal index: three C-points with m=-1 and χ=-1 occur; finally, they merge into single C-point with m=-3 and χ=-1.\n\n#### 3.3 Homogeneous dichroic media\n\nSubstituting initial distribution Eqs. (25) into Eqs. (18)–(24), we get the Stokes-vector distribution and polarization singularities in homogenous dichroic medium. In a circularly-dichroic medium, C-points do not evolve with z, Eq. (19), whereas the behavior of L-surfaces is given by Eq. (21) with Eq. (25):\n\n$tanh(∑z)=−f(ρ)σ3.$\n\nThus, the surface lies in the z<0 or z>0 half-space when σ 3>0 or σ 3<0, respectively. Structure of the polarization singularities with initial distribution (26) in a circularly-dichroic medium is shown in Fig. 4a.", null, "Fig. 4. C-lines and L-surfaces under propagation of the vectorial vortex, Eqs. (25) and (26), with m=3 and χ=1 in a dichroic medium. Pictures (a) and (b) correspond to circularly- and linearly-dichroic media with σ=(0,0,-1) and σ=(1,0,0), respectively. Dimensionless coordinates ξ=x/ρ, η=y/ρ, and ζz are used.\n\nIn a linearly-dichroic medium, L-surfaces are trivial, Eq. (22), whereas C-lines are described by Eqs. (24) with Eq. (25):\n\n$tanh(∑z)=−1−f2(ρ)cos[m(φ−δ)],1−f2(ρ)sin[m(φ−δ)]=0.$\n\nAssuming initial distribution with the radial function (26), we find that, similarly to Eq. (29), solutions of Eqs. (32) represent curves lying in the azimuthal planes φ=const:\n\n$φn=δ+πnm,tanh(∑z)=∓1−f2(ρ),$\n\nHere n=0,1,…, 2m-1, and signs “∓” in the second equation correspond to even and odd n, respectively. It is easily seen that solutions with even and odd n are realized, respectively, at z<0 and z>0, Fig. 4b. Thus, similarly to the case of linearly-birefringent medium, the initial C-point of the m th order is split into m C-points with unit azimuthal indices, Fig. 5. The total topological charge, Eq. (30), is also conserved in this process.", null, "Fig. 5. Distributions of Stokes vectors, Eq. (18), (upper panel) and of polarization ellipses (lower panel) for vectorial vortex, Eqs. (25) and (26), with m=3 and χ=1, propagating in a linearly-dichroic medium with σ=(1,0,0), Fig. 4b. Distributions are shown at different propagation distances ζz. C-points are marked by dots (upper panel) and crosses (lower panel).\n\n## 4. Conclusion\n\nTo summarize, we have developed an efficient formalism describing the evolution of non-uniformly polarized waves in anisotropic media. Provided that refraction and diffraction effects are negligible, our method reduces the initial wave problem to the integration of a simple differential equation for the Stokes-vector evolution. Polarization singularities are readily found in the resulting space distribution of the Stokes vector. The evolution equation has been integrated analytically for the characteristic cases of homogeneous birefringent and dichroic media.\n\nWe have applied the general formalism to the evolution of a polarization vortex in birefringent and dichroic media. The resulting space polarization patterns describe remarkable behavior of polarization singularities as the wave propagates in the medium. In particular, we showed the splitting of a higher-order vectorial vortex into a number of the minimal-order generic vortices and verified conservation of the topological charge under that process.\n\nThe fine behavior of the wave-field singularities in anisotropic media can be used for needs of polarimetry, and, perhaps, will enable one to increase the sensitivity of classical polarimetric methods.\n\nFinally, though we considered a rectilinear propagation of the wave along the z axis, our approach is also valid for the geometrical-optics wave propagation along smooth curvilinear rays in large-scale inhomogeneous media. In this case, the problem is reduced to the same form if one involves a local ray coordinate system with basis vectors being parallel-transported along the ray .\n\nNote added.– Recent paper with related arguments came to our attention after submission of this work.\n\n## Appendix: Solution of equation (17)\n\nTo integrate Eq. (17), note that the unit vector s can be represented as\n\n$s=Aσ−(B×σ).$\n\nHere we introduced two auxiliary quantities: scalar A= and vector B=(s×σ). From Eq. (17), it can be easily seen that they obey equations\n\n$∂A∂z=∑(1−A2),$\n$∂B∂z=−∑AB.$\n\nBy integrating Eq. (A2) we obtain\n\n$A=(1+A0)e∑z−(1−A0)e−∑z(1+A0)e∑z+(1−A0)e−∑z,$\n\nwhere A 0=A\|z=0. Substituting Eq. (A4) into Eq. (A3) and performing integration, we arrive at solution for B:\n\n$B=2(1+A0)e∑z+(1−A0)e−∑zB0,$\n\nwhere B 0=B\|z=0. Substituting Eqs. (A4) and (A5) into Eq. (A1) yields solution (18).\n\n## Acknowledgements\n\nThis work was partially supported by STCU (grant P-307) and CRDF (grant UAM2-1672-KK-06).\n\n1. M.V. Berry, “Singularities in waves and rays,” in R. Balian, M. Kléman, and J.-P. Poirier, editors, Les Houches Session XXV - Physics of Defects (North-Holland, 1981).\n\n2. J.F. Nye, Natural focusing and fine structure of light: caustics and wave dislocations (IoP Publishing, 1999).\n\n3. M.S. Soskin and M.V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001). [CrossRef]\n\n4. L. Allen, M.J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999). [CrossRef]\n\n5. J.F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. London A 389, 279–290 (1983). [CrossRef]\n\n6. J.F. Nye and J.V. Hajnal, “The wave structure of monochromatic electromagnetic radiation,” Proc. R. Soc. London A 409, 21–36 (1987). [CrossRef]\n\n7. M.V. Berry and M.R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London A 457, 141–155 (2001). [CrossRef]\n\n8. I. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. 201, 251–270 (2002). [CrossRef]\n\n9. M.R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213201–221 (2002). [CrossRef]\n\n10. J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. 198, 21–27 (2001). [CrossRef]\n\n11. I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993). [CrossRef]\n\n12. E. Hasman, G. Biener, A. Niv, and V. Kleiner, “Space-variant polarization manipulation,” Prog. Opt. 47, 215–289 (2005). [CrossRef]\n\n13. A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Manipulation of the Pancharatnam phase in vectorial vortices,” Opt. Express 14, 4208–4220 (2006). [CrossRef] [PubMed]\n\n14. K. Yu. Bliokh, “Geometrical optics of beams with vortices: Berry phase and orbital angular momentum Hall effect,” Phys. Rev. Lett. 97, 043901 (2006). [CrossRef] [PubMed]\n\n15. F. Flossmann, U.T. Schwarz, M. Maier, and M.R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95253901 (2005). [CrossRef] [PubMed]\n\n16. F. Flossmann, U.T. Schwarz, M. Maier, and M.R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express 14, 11402–11411 (2006). [CrossRef] [PubMed]\n\n17. R.M.A. Azzam and N.M. Bashara, Ellipsometry and polarized light (North-Holland, 1977).\n\n18. C. Brosseau, Fundamentals of polarized light (John Wiley & Sons, 1998).\n\n19. R.M.A. Azzam, “Propagation of partially polarized light through anisotropic media with or without depolarization: a differential 4×4 matrix calculus,” J. Opt. Soc. Am. 68, 1756–1767 (1978). [CrossRef]\n\n20. C.S. Brown and A.E. Bak, “Unified formalism for polarization optics with application to polarimetry on a twisted optical fiber,” Opt. Eng. 34, 1625–1635 (1995). [CrossRef]\n\n21. C. Brosseau, “Evolution of the Stokes parameters in optically anisotropic media,” Opt. Lett. 20, 1221–1223 (1995). [CrossRef] [PubMed]\n\n22. H. Kuratsuji and S. Kakigi, “Maxwell-Schrödinger equation for polarized light and evolution of the Stokes parameters,” Phys. Rev. Lett. 80, 1888–1891 (1998). [CrossRef]\n\n23. S.E. Segre, “New formalism for the analysis of polarization evolution for radiation in a weakly nonuniform, fully anisotropic medium: a magnetized plasma,” J. Opt. Soc. Am. A 18, 2601–2606 (2001). [CrossRef]\n\n24. R. Botet, H. Kuratsuji, and R. Seto, “Novel aspects of evolution of the Stokes parameters for an electromagnetic wave in anisotropic media,” Prog. Theor. Phys. 116, 285–294 (2006). [CrossRef]\n\n25. K.Y. Bliokh, D.Y. Frolov, and Y.A. Kravtsov, “Non-Abelian evolution of electromagnetic waves in a weakly anisotropic inhomogeneous medium,” Phys. Rev. A 75, 053821 (2007). [CrossRef]\n\n26. Y.A. Kravtsov, B. Bieg, and K.Y. Bliokh, “Stokes-vector evolution in a weakly anisotropic inhomogeneous medium,” J. Opt. Soc. Am. A 24, 3388–3396 (2007). [CrossRef]\n\n27. R. Barakat, “Bilinear constraints between elements of the 4×4 Mueller-Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159–161 (1981). [CrossRef]\n\n28. R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982). [CrossRef]\n\n29. S.R. Cloude, “Group theory and polarization algebra,” Optik 75, 26–32 (1986).\n\n30. T. Opartny and J. Perina, “Non-image-forming polarization optical devices and Lorentz transformation — an analogy,” Phys. Lett. A 181, 199–202 (1993). [CrossRef]\n\n31. D. Han, Y.S. Kim, and M.E. Noz, “Polarization optics and bilinear representation of the Lorentz group,” Phys. Lett. A 219, 26–32 (1996). [CrossRef]\n\n32. V.B. Berestetskii, E.M. Lifshits, and L.P. Pitaevskii, Quantum Electrodynamics (Pergamon, Oxford, 1982).\n\n33. C.P. Slichter, Principles of Magnetic Resonance (Springer-Verlag, New York, 1989).\n\n34. H. Goldstein, C. Poole, and J. Safko, Classical Mechanics (Addison Wesley, San Francisco, 2002).\n\n35. A.D. Kiselev, “Singularities in polarization resolved angular patterns: transmittance of nematic liquid crystal cells,” J. Phys.: Condens. Matter 19, 246102 (2007). [CrossRef]\n\n### References\n\n• View by:\n\n1. M.V. Berry, “Singularities in waves and rays,” in R. Balian, M. Kléman, and J.-P. Poirier, editors, Les Houches Session XXV - Physics of Defects (North-Holland, 1981).\n2. J.F. Nye, Natural focusing and fine structure of light: caustics and wave dislocations (IoP Publishing, 1999).\n3. M.S. Soskin and M.V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).\n[Crossref]\n4. L. Allen, M.J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).\n[Crossref]\n5. J.F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. London A 389, 279–290 (1983).\n[Crossref]\n6. J.F. Nye and J.V. Hajnal, “The wave structure of monochromatic electromagnetic radiation,” Proc. R. Soc. London A 409, 21–36 (1987).\n[Crossref]\n7. M.V. Berry and M.R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London A 457, 141–155 (2001).\n[Crossref]\n8. I. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. 201, 251–270 (2002).\n[Crossref]\n9. M.R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213201–221 (2002).\n[Crossref]\n10. J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. 198, 21–27 (2001).\n[Crossref]\n11. I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).\n[Crossref]\n12. E. Hasman, G. Biener, A. Niv, and V. Kleiner, “Space-variant polarization manipulation,” Prog. Opt. 47, 215–289 (2005).\n[Crossref]\n13. A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Manipulation of the Pancharatnam phase in vectorial vortices,” Opt. Express 14, 4208–4220 (2006).\n[Crossref] [PubMed]\n14. K. Yu. Bliokh, “Geometrical optics of beams with vortices: Berry phase and orbital angular momentum Hall effect,” Phys. Rev. Lett. 97, 043901 (2006).\n[Crossref] [PubMed]\n15. F. Flossmann, U.T. Schwarz, M. Maier, and M.R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95253901 (2005).\n[Crossref] [PubMed]\n16. F. Flossmann, U.T. Schwarz, M. Maier, and M.R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express 14, 11402–11411 (2006).\n[Crossref] [PubMed]\n17. R.M.A. Azzam and N.M. Bashara, Ellipsometry and polarized light (North-Holland, 1977).\n18. C. Brosseau, Fundamentals of polarized light (John Wiley & Sons, 1998).\n19. R.M.A. Azzam, “Propagation of partially polarized light through anisotropic media with or without depolarization: a differential 4×4 matrix calculus,” J. Opt. Soc. Am. 68, 1756–1767 (1978).\n[Crossref]\n20. C.S. Brown and A.E. Bak, “Unified formalism for polarization optics with application to polarimetry on a twisted optical fiber,” Opt. Eng. 34, 1625–1635 (1995).\n[Crossref]\n21. C. Brosseau, “Evolution of the Stokes parameters in optically anisotropic media,” Opt. Lett. 20, 1221–1223 (1995).\n[Crossref] [PubMed]\n22. H. Kuratsuji and S. Kakigi, “Maxwell-Schrödinger equation for polarized light and evolution of the Stokes parameters,” Phys. Rev. Lett. 80, 1888–1891 (1998).\n[Crossref]\n23. S.E. Segre, “New formalism for the analysis of polarization evolution for radiation in a weakly nonuniform, fully anisotropic medium: a magnetized plasma,” J. Opt. Soc. Am. A 18, 2601–2606 (2001).\n[Crossref]\n24. R. Botet, H. Kuratsuji, and R. Seto, “Novel aspects of evolution of the Stokes parameters for an electromagnetic wave in anisotropic media,” Prog. Theor. Phys. 116, 285–294 (2006).\n[Crossref]\n25. K.Y. Bliokh, D.Y. Frolov, and Y.A. Kravtsov, “Non-Abelian evolution of electromagnetic waves in a weakly anisotropic inhomogeneous medium,” Phys. Rev. A 75, 053821 (2007).\n[Crossref]\n26. Y.A. Kravtsov, B. Bieg, and K.Y. Bliokh, “Stokes-vector evolution in a weakly anisotropic inhomogeneous medium,” J. Opt. Soc. Am. A 24, 3388–3396 (2007).\n[Crossref]\n27. R. Barakat, “Bilinear constraints between elements of the 4×4 Mueller-Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159–161 (1981).\n[Crossref]\n28. R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982).\n[Crossref]\n29. S.R. Cloude, “Group theory and polarization algebra,” Optik 75, 26–32 (1986).\n30. T. Opartny and J. Perina, “Non-image-forming polarization optical devices and Lorentz transformation — an analogy,” Phys. Lett. A 181, 199–202 (1993).\n[Crossref]\n31. D. Han, Y.S. Kim, and M.E. Noz, “Polarization optics and bilinear representation of the Lorentz group,” Phys. Lett. A 219, 26–32 (1996).\n[Crossref]\n32. V.B. Berestetskii, E.M. Lifshits, and L.P. Pitaevskii, Quantum Electrodynamics (Pergamon, Oxford, 1982).\n33. C.P. Slichter, Principles of Magnetic Resonance (Springer-Verlag, New York, 1989).\n34. H. Goldstein, C. Poole, and J. Safko, Classical Mechanics (Addison Wesley, San Francisco, 2002).\n35. A.D. Kiselev, “Singularities in polarization resolved angular patterns: transmittance of nematic liquid crystal cells,” J. Phys.: Condens. Matter 19, 246102 (2007).\n[Crossref]\n\n#### 2007 (3)\n\nK.Y. Bliokh, D.Y. Frolov, and Y.A. Kravtsov, “Non-Abelian evolution of electromagnetic waves in a weakly anisotropic inhomogeneous medium,” Phys. Rev. A 75, 053821 (2007).\n[Crossref]\n\nA.D. Kiselev, “Singularities in polarization resolved angular patterns: transmittance of nematic liquid crystal cells,” J. Phys.: Condens. Matter 19, 246102 (2007).\n[Crossref]\n\n#### 2006 (4)\n\nR. Botet, H. Kuratsuji, and R. Seto, “Novel aspects of evolution of the Stokes parameters for an electromagnetic wave in anisotropic media,” Prog. Theor. Phys. 116, 285–294 (2006).\n[Crossref]\n\nK. Yu. Bliokh, “Geometrical optics of beams with vortices: Berry phase and orbital angular momentum Hall effect,” Phys. Rev. Lett. 97, 043901 (2006).\n[Crossref] [PubMed]\n\n#### 2005 (2)\n\nF. Flossmann, U.T. Schwarz, M. Maier, and M.R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95253901 (2005).\n[Crossref] [PubMed]\n\nE. Hasman, G. Biener, A. Niv, and V. Kleiner, “Space-variant polarization manipulation,” Prog. Opt. 47, 215–289 (2005).\n[Crossref]\n\n#### 2002 (2)\n\nI. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. 201, 251–270 (2002).\n[Crossref]\n\nM.R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213201–221 (2002).\n[Crossref]\n\n#### 2001 (4)\n\nJ. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. 198, 21–27 (2001).\n[Crossref]\n\nM.S. Soskin and M.V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).\n[Crossref]\n\nM.V. Berry and M.R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London A 457, 141–155 (2001).\n[Crossref]\n\n#### 1999 (1)\n\nL. Allen, M.J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).\n[Crossref]\n\n#### 1998 (1)\n\nH. Kuratsuji and S. Kakigi, “Maxwell-Schrödinger equation for polarized light and evolution of the Stokes parameters,” Phys. Rev. Lett. 80, 1888–1891 (1998).\n[Crossref]\n\n#### 1996 (1)\n\nD. Han, Y.S. Kim, and M.E. Noz, “Polarization optics and bilinear representation of the Lorentz group,” Phys. Lett. A 219, 26–32 (1996).\n[Crossref]\n\n#### 1995 (2)\n\nC.S. Brown and A.E. Bak, “Unified formalism for polarization optics with application to polarimetry on a twisted optical fiber,” Opt. Eng. 34, 1625–1635 (1995).\n[Crossref]\n\n#### 1993 (2)\n\nI. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).\n[Crossref]\n\nT. Opartny and J. Perina, “Non-image-forming polarization optical devices and Lorentz transformation — an analogy,” Phys. Lett. A 181, 199–202 (1993).\n[Crossref]\n\n#### 1987 (1)\n\nJ.F. Nye and J.V. Hajnal, “The wave structure of monochromatic electromagnetic radiation,” Proc. R. Soc. London A 409, 21–36 (1987).\n[Crossref]\n\n#### 1986 (1)\n\nS.R. Cloude, “Group theory and polarization algebra,” Optik 75, 26–32 (1986).\n\n#### 1983 (1)\n\nJ.F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. London A 389, 279–290 (1983).\n[Crossref]\n\n#### 1982 (1)\n\nR. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982).\n[Crossref]\n\n#### 1981 (1)\n\nR. Barakat, “Bilinear constraints between elements of the 4×4 Mueller-Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159–161 (1981).\n[Crossref]\n\n#### Allen, L.\n\nL. Allen, M.J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).\n[Crossref]\n\n#### Azzam, R.M.A.\n\nR.M.A. Azzam and N.M. Bashara, Ellipsometry and polarized light (North-Holland, 1977).\n\n#### Babiker, M.\n\nL. Allen, M.J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).\n[Crossref]\n\n#### Bak, A.E.\n\nC.S. Brown and A.E. Bak, “Unified formalism for polarization optics with application to polarimetry on a twisted optical fiber,” Opt. Eng. 34, 1625–1635 (1995).\n[Crossref]\n\n#### Barakat, R.\n\nR. Barakat, “Bilinear constraints between elements of the 4×4 Mueller-Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159–161 (1981).\n[Crossref]\n\n#### Bashara, N.M.\n\nR.M.A. Azzam and N.M. Bashara, Ellipsometry and polarized light (North-Holland, 1977).\n\n#### Berestetskii, V.B.\n\nV.B. Berestetskii, E.M. Lifshits, and L.P. Pitaevskii, Quantum Electrodynamics (Pergamon, Oxford, 1982).\n\n#### Berry, M.V.\n\nM.V. Berry and M.R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London A 457, 141–155 (2001).\n[Crossref]\n\nM.V. Berry, “Singularities in waves and rays,” in R. Balian, M. Kléman, and J.-P. Poirier, editors, Les Houches Session XXV - Physics of Defects (North-Holland, 1981).\n\n#### Biener, G.\n\nE. Hasman, G. Biener, A. Niv, and V. Kleiner, “Space-variant polarization manipulation,” Prog. Opt. 47, 215–289 (2005).\n[Crossref]\n\n#### Bliokh, K. Yu.\n\nK. Yu. Bliokh, “Geometrical optics of beams with vortices: Berry phase and orbital angular momentum Hall effect,” Phys. Rev. Lett. 97, 043901 (2006).\n[Crossref] [PubMed]\n\n#### Bliokh, K.Y.\n\nK.Y. Bliokh, D.Y. Frolov, and Y.A. Kravtsov, “Non-Abelian evolution of electromagnetic waves in a weakly anisotropic inhomogeneous medium,” Phys. Rev. A 75, 053821 (2007).\n[Crossref]\n\n#### Botet, R.\n\nR. Botet, H. Kuratsuji, and R. Seto, “Novel aspects of evolution of the Stokes parameters for an electromagnetic wave in anisotropic media,” Prog. Theor. Phys. 116, 285–294 (2006).\n[Crossref]\n\n#### Brosseau, C.\n\nC. Brosseau, Fundamentals of polarized light (John Wiley & Sons, 1998).\n\n#### Brown, C.S.\n\nC.S. Brown and A.E. Bak, “Unified formalism for polarization optics with application to polarimetry on a twisted optical fiber,” Opt. Eng. 34, 1625–1635 (1995).\n[Crossref]\n\n#### Cloude, S.R.\n\nS.R. Cloude, “Group theory and polarization algebra,” Optik 75, 26–32 (1986).\n\n#### Dennis, M.R.\n\nF. Flossmann, U.T. Schwarz, M. Maier, and M.R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95253901 (2005).\n[Crossref] [PubMed]\n\nM.R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213201–221 (2002).\n[Crossref]\n\nM.V. Berry and M.R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London A 457, 141–155 (2001).\n[Crossref]\n\n#### Dubik, B.\n\nJ. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. 198, 21–27 (2001).\n[Crossref]\n\n#### Flossmann, F.\n\nF. Flossmann, U.T. Schwarz, M. Maier, and M.R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95253901 (2005).\n[Crossref] [PubMed]\n\n#### Freilikher, V.\n\nI. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).\n[Crossref]\n\n#### Freund, I.\n\nI. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. 201, 251–270 (2002).\n[Crossref]\n\nI. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).\n[Crossref]\n\n#### Frolov, D.Y.\n\nK.Y. Bliokh, D.Y. Frolov, and Y.A. Kravtsov, “Non-Abelian evolution of electromagnetic waves in a weakly anisotropic inhomogeneous medium,” Phys. Rev. A 75, 053821 (2007).\n[Crossref]\n\n#### Goldstein, H.\n\nH. Goldstein, C. Poole, and J. Safko, Classical Mechanics (Addison Wesley, San Francisco, 2002).\n\n#### Hajnal, J.V.\n\nJ.F. Nye and J.V. Hajnal, “The wave structure of monochromatic electromagnetic radiation,” Proc. R. Soc. London A 409, 21–36 (1987).\n[Crossref]\n\n#### Han, D.\n\nD. Han, Y.S. Kim, and M.E. Noz, “Polarization optics and bilinear representation of the Lorentz group,” Phys. Lett. A 219, 26–32 (1996).\n[Crossref]\n\n#### Hasman, E.\n\nE. Hasman, G. Biener, A. Niv, and V. Kleiner, “Space-variant polarization manipulation,” Prog. Opt. 47, 215–289 (2005).\n[Crossref]\n\n#### Kakigi, S.\n\nH. Kuratsuji and S. Kakigi, “Maxwell-Schrödinger equation for polarized light and evolution of the Stokes parameters,” Phys. Rev. Lett. 80, 1888–1891 (1998).\n[Crossref]\n\n#### Kim, Y.S.\n\nD. Han, Y.S. Kim, and M.E. Noz, “Polarization optics and bilinear representation of the Lorentz group,” Phys. Lett. A 219, 26–32 (1996).\n[Crossref]\n\n#### Kiselev, A.D.\n\nA.D. Kiselev, “Singularities in polarization resolved angular patterns: transmittance of nematic liquid crystal cells,” J. Phys.: Condens. Matter 19, 246102 (2007).\n[Crossref]\n\n#### Kleiner, V.\n\nE. Hasman, G. Biener, A. Niv, and V. Kleiner, “Space-variant polarization manipulation,” Prog. Opt. 47, 215–289 (2005).\n[Crossref]\n\n#### Kravtsov, Y.A.\n\nK.Y. Bliokh, D.Y. Frolov, and Y.A. Kravtsov, “Non-Abelian evolution of electromagnetic waves in a weakly anisotropic inhomogeneous medium,” Phys. Rev. A 75, 053821 (2007).\n[Crossref]\n\n#### Kuratsuji, H.\n\nR. Botet, H. Kuratsuji, and R. Seto, “Novel aspects of evolution of the Stokes parameters for an electromagnetic wave in anisotropic media,” Prog. Theor. Phys. 116, 285–294 (2006).\n[Crossref]\n\nH. Kuratsuji and S. Kakigi, “Maxwell-Schrödinger equation for polarized light and evolution of the Stokes parameters,” Phys. Rev. Lett. 80, 1888–1891 (1998).\n[Crossref]\n\n#### Lifshits, E.M.\n\nV.B. Berestetskii, E.M. Lifshits, and L.P. Pitaevskii, Quantum Electrodynamics (Pergamon, Oxford, 1982).\n\n#### Maier, M.\n\nF. Flossmann, U.T. Schwarz, M. Maier, and M.R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95253901 (2005).\n[Crossref] [PubMed]\n\nJ. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. 198, 21–27 (2001).\n[Crossref]\n\n#### Niv, A.\n\nE. Hasman, G. Biener, A. Niv, and V. Kleiner, “Space-variant polarization manipulation,” Prog. Opt. 47, 215–289 (2005).\n[Crossref]\n\n#### Noz, M.E.\n\nD. Han, Y.S. Kim, and M.E. Noz, “Polarization optics and bilinear representation of the Lorentz group,” Phys. Lett. A 219, 26–32 (1996).\n[Crossref]\n\n#### Nye, J.F.\n\nJ.F. Nye and J.V. Hajnal, “The wave structure of monochromatic electromagnetic radiation,” Proc. R. Soc. London A 409, 21–36 (1987).\n[Crossref]\n\nJ.F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. London A 389, 279–290 (1983).\n[Crossref]\n\nJ.F. Nye, Natural focusing and fine structure of light: caustics and wave dislocations (IoP Publishing, 1999).\n\n#### Opartny, T.\n\nT. Opartny and J. Perina, “Non-image-forming polarization optical devices and Lorentz transformation — an analogy,” Phys. Lett. A 181, 199–202 (1993).\n[Crossref]\n\nL. Allen, M.J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).\n[Crossref]\n\n#### Perina, J.\n\nT. Opartny and J. Perina, “Non-image-forming polarization optical devices and Lorentz transformation — an analogy,” Phys. Lett. A 181, 199–202 (1993).\n[Crossref]\n\n#### Pitaevskii, L.P.\n\nV.B. Berestetskii, E.M. Lifshits, and L.P. Pitaevskii, Quantum Electrodynamics (Pergamon, Oxford, 1982).\n\n#### Poole, C.\n\nH. Goldstein, C. Poole, and J. Safko, Classical Mechanics (Addison Wesley, San Francisco, 2002).\n\n#### Safko, J.\n\nH. Goldstein, C. Poole, and J. Safko, Classical Mechanics (Addison Wesley, San Francisco, 2002).\n\n#### Schwarz, U.T.\n\nF. Flossmann, U.T. Schwarz, M. Maier, and M.R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95253901 (2005).\n[Crossref] [PubMed]\n\n#### Seto, R.\n\nR. Botet, H. Kuratsuji, and R. Seto, “Novel aspects of evolution of the Stokes parameters for an electromagnetic wave in anisotropic media,” Prog. Theor. Phys. 116, 285–294 (2006).\n[Crossref]\n\n#### Shvartsman, N.\n\nI. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).\n[Crossref]\n\n#### Simon, R.\n\nR. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982).\n[Crossref]\n\n#### Slichter, C.P.\n\nC.P. Slichter, Principles of Magnetic Resonance (Springer-Verlag, New York, 1989).\n\n#### Soskin, M.S.\n\nM.S. Soskin and M.V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).\n[Crossref]\n\n#### Vasnetsov, M.V.\n\nM.S. Soskin and M.V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).\n[Crossref]\n\n#### J. Phys.: Condens. Matter (1)\n\nA.D. Kiselev, “Singularities in polarization resolved angular patterns: transmittance of nematic liquid crystal cells,” J. Phys.: Condens. Matter 19, 246102 (2007).\n[Crossref]\n\n#### Opt. Commun. (6)\n\nR. Barakat, “Bilinear constraints between elements of the 4×4 Mueller-Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159–161 (1981).\n[Crossref]\n\nR. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982).\n[Crossref]\n\nI. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. 201, 251–270 (2002).\n[Crossref]\n\nM.R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. 213201–221 (2002).\n[Crossref]\n\nJ. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. 198, 21–27 (2001).\n[Crossref]\n\nI. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).\n[Crossref]\n\n#### Opt. Eng. (1)\n\nC.S. Brown and A.E. Bak, “Unified formalism for polarization optics with application to polarimetry on a twisted optical fiber,” Opt. Eng. 34, 1625–1635 (1995).\n[Crossref]\n\n#### Optik (1)\n\nS.R. Cloude, “Group theory and polarization algebra,” Optik 75, 26–32 (1986).\n\n#### Phys. Lett. A (2)\n\nT. Opartny and J. Perina, “Non-image-forming polarization optical devices and Lorentz transformation — an analogy,” Phys. Lett. A 181, 199–202 (1993).\n[Crossref]\n\nD. Han, Y.S. Kim, and M.E. Noz, “Polarization optics and bilinear representation of the Lorentz group,” Phys. Lett. A 219, 26–32 (1996).\n[Crossref]\n\n#### Phys. Rev. A (1)\n\nK.Y. Bliokh, D.Y. Frolov, and Y.A. Kravtsov, “Non-Abelian evolution of electromagnetic waves in a weakly anisotropic inhomogeneous medium,” Phys. Rev. A 75, 053821 (2007).\n[Crossref]\n\n#### Phys. Rev. Lett. (3)\n\nH. Kuratsuji and S. Kakigi, “Maxwell-Schrödinger equation for polarized light and evolution of the Stokes parameters,” Phys. Rev. Lett. 80, 1888–1891 (1998).\n[Crossref]\n\nK. Yu. Bliokh, “Geometrical optics of beams with vortices: Berry phase and orbital angular momentum Hall effect,” Phys. Rev. Lett. 97, 043901 (2006).\n[Crossref] [PubMed]\n\nF. Flossmann, U.T. Schwarz, M. Maier, and M.R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95253901 (2005).\n[Crossref] [PubMed]\n\n#### Proc. R. Soc. London A (3)\n\nJ.F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. London A 389, 279–290 (1983).\n[Crossref]\n\nJ.F. Nye and J.V. Hajnal, “The wave structure of monochromatic electromagnetic radiation,” Proc. R. Soc. London A 409, 21–36 (1987).\n[Crossref]\n\nM.V. Berry and M.R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. London A 457, 141–155 (2001).\n[Crossref]\n\n#### Prog. Opt. (3)\n\nE. Hasman, G. Biener, A. Niv, and V. Kleiner, “Space-variant polarization manipulation,” Prog. Opt. 47, 215–289 (2005).\n[Crossref]\n\nM.S. Soskin and M.V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).\n[Crossref]\n\nL. Allen, M.J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).\n[Crossref]\n\n#### Prog. Theor. Phys. (1)\n\nR. Botet, H. Kuratsuji, and R. Seto, “Novel aspects of evolution of the Stokes parameters for an electromagnetic wave in anisotropic media,” Prog. Theor. Phys. 116, 285–294 (2006).\n[Crossref]\n\n#### Other (7)\n\nR.M.A. Azzam and N.M. Bashara, Ellipsometry and polarized light (North-Holland, 1977).\n\nC. Brosseau, Fundamentals of polarized light (John Wiley & Sons, 1998).\n\nV.B. Berestetskii, E.M. Lifshits, and L.P. Pitaevskii, Quantum Electrodynamics (Pergamon, Oxford, 1982).\n\nC.P. Slichter, Principles of Magnetic Resonance (Springer-Verlag, New York, 1989).\n\nH. Goldstein, C. Poole, and J. Safko, Classical Mechanics (Addison Wesley, San Francisco, 2002).\n\nM.V. Berry, “Singularities in waves and rays,” in R. Balian, M. Kléman, and J.-P. Poirier, editors, Les Houches Session XXV - Physics of Defects (North-Holland, 1981).\n\nJ.F. Nye, Natural focusing and fine structure of light: caustics and wave dislocations (IoP Publishing, 1999).\n\n### Cited By\n\nOptica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.\n\n### Figures (5)\n\nFig. 1. Distributions of Stokes vectors (upper panel) and of polarization ellipses (lower panel) for vectorial vortices Eqs. (25) with a C-point in the center, Eq. (26), at different values of azimuthal index m. Hereafter, δ=0 and directions of Stokes vectors are naturally depicted in the real space with the s 1, s 2, and s 3 components pointing along the x, y, and z axis, respectively.\nFig. 2. C-lines and L-surfaces under propagation of the vectorial vortex, Eqs. (25) and (26), with m=3 and χ=1 in a linearly-birefringent medium with ω=(1, 0,0). Dimensionless coordinates ξ=x/ρ, η=y/ρ, and ζz/π are used, whereas red and blue colors indicate C-lines with χ=1 and χ=-1, respectively.\nFig. 3. Distributions of Stokes vectors, Eq. (12), (upper panel) and of polarization ellipses (lower panel) for vectorial vortex, Eqs. (25) and (26), with m=3 and χ=1, propagating in a linearly-birefringent medium with ω=(1, 0,0), Fig. 2. Distributions are shown at different propagation distances ζz/π within half-period. Red and blue colors indicate areas with right-hand (s 3>0) and left-hand (s 3<0) polarizations. C-points are marked by dots (upper panel) and crosses (lower panel).\nFig. 4. C-lines and L-surfaces under propagation of the vectorial vortex, Eqs. (25) and (26), with m=3 and χ=1 in a dichroic medium. Pictures (a) and (b) correspond to circularly- and linearly-dichroic media with σ=(0,0,-1) and σ=(1,0,0), respectively. Dimensionless coordinates ξ=x/ρ, η=y/ρ, and ζz are used.\nFig. 5. Distributions of Stokes vectors, Eq. (18), (upper panel) and of polarization ellipses (lower panel) for vectorial vortex, Eqs. (25) and (26), with m=3 and χ=1, propagating in a linearly-dichroic medium with σ=(1,0,0), Fig. 4b. Distributions are shown at different propagation distances ζz. C-points are marked by dots (upper panel) and crosses (lower panel).\n\n### Equations (41)\n\n$s ( x , y , 0 ) = s 0 ( x , y ) ,$\n$∂ s ∂ z = m ̂ s ,$\n$s 1 = 0 , s 2 = 0 ,$\n$s 3 = 0 .$\n$z P ≤ z ≪ z R , z D .$\n$∂ S → ∂ z = M ̂ S → ,$\n$ε ̂ = ε 0 I ̂ 2 + ν ̂ ≡ ( ε 0 + ν x x ν x y ν y x ε 0 + ν y y ) .$\n$M ̂ = ( Im G 0 Im G 1 Im G 2 Im G 3 Im G 1 Im G 0 − Re G 3 Re G 2 Im G 2 Re G 3 Im G 0 − Re G 1 Im G 3 − Re G 2 Re G 1 Im G 0 ) .$\n$G → = − k 0 2 ε 0 [ ( ν x x + ν y y ) , ( ν x x − ν y y ) , ( ν x y + ν y x ) , i ( ν x y − ν y x ) ] T ,$\n$M ̂ = Im G 0 I ̂ 4 + ( 0 Im G 1 Im G 2 Im G 3 Im G 1 0 0 0 Im G 2 0 0 0 Im G 3 0 0 0 ) + ( 0 0 0 0 0 0 − Re G 3 Re G 2 0 Re G 3 0 − Re G 1 0 − Re G 2 Re G 1 0 ) ,$\n$∂ s ∂ z = ( Ω + s × ∑ ) × s ,$\n$∂ s ∂ z = Ω × s .$\n$s = s 0 cos ⁡ ( Ω z ) + ( ω × s 0 ) sin ⁡ ( Ω z ) + ( ω s 0 ) ω [ 1 − cos ⁡ ( Ω z ) ] .$\n$s 01 = 0 , s 02 = 0 ,$\n$s 03 = 0 .$\n$s 01 = 0 , tan ⁡ ( Ω z ) = s 02 s 03 ,$\n$tan ⁡ ( Ω z ) = − s 03 s 02 .$\n$∂ s ∂ z = ( s × ∑ ) × s .$\n$s = 2 ( 1 + A 0 ) e ∑ z + ( 1 − A 0 ) e − ∑ z s 0 + ( 1 + A 0 ) e ∑ z − 2 A 0 − ( 1 − A 0 ) e − ∑ z ( 1 + A 0 ) e ∑ z + ( 1 − A 0 ) e − ∑ z σ ,$\n$s 01 = 0 , s 02 = 0 .$\n$( 1 + A 0 ) e ∑ z − ( 1 − A 0 ) e − ∑ z ( 1 + A 0 ) e ∑ z + ( 1 − A 0 ) e − ∑ z = 0 ,$\n$tan ⁡ h ( ∑ z ) = − s 03 σ 3 .$\n$s 03 = 0 .$\n$( 1 + A 0 ) e ∑ z − ( 1 − A 0 ) e − ∑ z ( 1 + A 0 ) e ∑ z + ( 1 − A 0 ) e − ∑ z = 0 , s 02 = 0 ,$\n$tanh ( ∑ z ) = − s 01 , s 02 = 0 .$\n$s 01 = 1 − f 2 ( ρ ) cos [ m ( φ − δ ) ] ,$\n$s 02 = 1 − f 2 ( ρ ) sin [ m ( φ − δ ) ] ,$\n$s 03 = f ( ρ ) .$\n$f = χ 1 + ( ρ ρ ) 2 ,$\n$1 − f ( ρ ) 2 cos [ m ( φ − δ ) ] = 0 , tan ( Ω z ) = 1 − f 2 ( ρ ) sin [ m ( φ − δ ) ] f ( ρ ) ,$\n$tan ( Ω z ) = − f ( ρ ) 1 − f 2 ( ρ ) sin [ m ( φ − δ ) ] .$\n$φ n = δ + π ( 2 n + 1 ) 2 m , tan ( Ω z ) = ± 1 − f 2 ( ρ ) f ( ρ ) = ± χ ρ ρ * .$\n$Γ = ∑ a m ( a ) χ ( a ) = const .$\n$tanh ( ∑ z ) = − f ( ρ ) σ 3 .$\n$tanh ( ∑ z ) = − 1 − f 2 ( ρ ) cos [ m ( φ − δ ) ] , 1 − f 2 ( ρ ) sin [ m ( φ − δ ) ] = 0 .$\n$φ n = δ + π n m , tanh ( ∑ z ) = ∓ 1 − f 2 ( ρ ) ,$\n$s = A σ − ( B × σ ) .$\n$∂ A ∂ z = ∑ ( 1 − A 2 ) ,$\n$∂ B ∂ z = − ∑ A B .$\n$A = ( 1 + A 0 ) e ∑ z − ( 1 − A 0 ) e − ∑ z ( 1 + A 0 ) e ∑ z + ( 1 − A 0 ) e − ∑ z ,$\n$B = 2 ( 1 + A 0 ) e ∑ z + ( 1 − A 0 ) e − ∑ z B 0 ,$" ]	[ null, "https://www.osapublishing.org/images/ajax-loader-big.gif", null, "https://www.osapublishing.org/images/ajax-loader-big.gif", null, "https://www.osapublishing.org/images/ajax-loader-big.gif", null, "https://www.osapublishing.org/images/ajax-loader-big.gif", null, "https://www.osapublishing.org/images/ajax-loader-big.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.85105973,"math_prob":0.9685022,"size":32023,"snap":"2021-43-2021-49","text_gpt3_token_len":8187,"char_repetition_ratio":0.17498985,"word_repetition_ratio":0.0252588,"special_character_ratio":0.2464791,"punctuation_ratio":0.16679817,"nsfw_num_words":1,"has_unicode_error":false,"math_prob_llama3":0.9910855,"pos_list":[0,1,2,3,4,5,6,7,8,9,10],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-17T13:12:49Z\",\"WARC-Record-ID\":\"<urn:uuid:cda283b6-24e0-4e55-9ffc-72f1cefa9fae>\",\"Content-Length\":\"297888\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:597eb285-9ffc-4c58-9afc-1ad466485a49>\",\"WARC-Concurrent-To\":\"<urn:uuid:bc690ca6-44b3-4366-8891-6c75b3683861>\",\"WARC-IP-Address\":\"38.95.177.60\",\"WARC-Target-URI\":\"https://www.osapublishing.org/oe/fulltext.cfm?uri=oe-16-2-695&id=148790\",\"WARC-Payload-Digest\":\"sha1:2DMQADMIZAHHEWVCSR7UAA47OKQBIXWV\",\"WARC-Block-Digest\":\"sha1:35IL6LO2BDTS6BRFPEZLPH2HWSVFQX5Z\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323585177.11_warc_CC-MAIN-20211017113503-20211017143503-00540.warc.gz\"}"}
https://de.webqc.org/molecularweightcalculated-190210-131.html	[ "", null, "#### Chemical Equations Balanced on 02/10/19\n\n Molecular weights calculated on 02/09/19 Molecular weights calculated on 02/11/19\nCalculate molecular weight\n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206\nMolar mass of H2O is 18.01528\nMolar mass of Na3P is 99.94306984\nMolar mass of Al is 26.9815386\nMolar mass of Sr3(PO4)2 is 452.802724\nMolar mass of NH is 15.01464\nMolar mass of C12H22O11 is 342.29648\nMolar mass of NH4NO3 is 80.04336\nMolar mass of CaCl2 is 110.984\nMolar mass of CaCl2 is 110.984\nMolar mass of Ca(OH)2 is 74.09268\nMolar mass of N0 is 14.0067\nMolar mass of Ne is 20.1797\nMolar mass of NH2 is 16.02258\nMolar mass of acetic acid is 60.05196\nMolar mass of Ti is 47.867\nMolar mass of MgO is 40.3044\nMolar mass of C2H6 is 30,06904\nMolar mass of acetone is 58.07914\nMolar mass of MgO is 40.3044\nMolar mass of water is 18.01528\nMolar mass of Fe2O3 is 159.6882\nMolar mass of Mo(CO)4 is 208.0004\nMolar mass of CuBr is 143.45\nMolar mass of Al(C9H6ON)3 is 459,4316586\nMolar mass of acetic acid is 60.05196\nMolar mass of CSCl2 is 114.9817\nMolar mass of NH3 is 17,03052\nMolar mass of SrCrO4 is 203.6137\nMolar mass of CaSO4 is 136.1406\nMolar mass of K2CrO4 is 194,1903\nMolar mass of Br2 is 159.808\nMolar mass of NHC(NH2)2 is 59.0705\nMolar mass of C10H22 is 142.28168\nMolar mass of Ca3(PO4)2 is 310.176724\nMolar mass of COF2 is 66.0069064\nMolar mass of Ca5(PO4)3OH is 502.311426\nMolar mass of SiBr2I2 is 441.70244\nMolar mass of PbCO3 is 267.2089\nMolar mass of Pt is 195.084\nMolar mass of Ta(IO)5 is 895,46723\nMolar mass of Ni(CO)4 is 170.7338\nMolar mass of Ca3P2 is 182.181524\nMolar mass of K2CrO4 is 194,1903\nMolar mass of CO2F is 63.0079032\nMolar mass of CO2F2 is 82.0063064\nMolar mass of CaSO4 is 136.1406\nMolar mass of Ca(NO3)2 is 164.0878\nMolar mass of COF2 is 66.0069064\nMolar mass of C4H4S is 84.13956\nMolar mass of C2OF2 is 78.0176064\n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206\nCalculate molecular weight\n Molecular weights calculated on 02/09/19 Molecular weights calculated on 02/11/19\nMolecular masses on 02/03/19\nMolecular masses on 01/11/19\nMolecular masses on 02/10/18\nMit der Benutzung dieser Webseite akzeptieren Sie unsere Allgemeinen Geschäftsbedingungen und Datenschutzrichtlinien.\n© 2019 webqc.org Alle Rechte vorbehalten.\n Periodensystem der Elemente Einheiten-Umrechner Chemie Werkzeuge Chemie-Forum Chemie FAQ Konstanten Symmetrie Suchen Chemie Links Link zu uns Kontaktieren Sie uns Eine bessere Übersetzung vorschlagen? Wählen Sie eine SpracheDeutschEnglishEspañolFrançaisItalianoNederlandsPolskiPortuguêsРусский中文日本語한국어 Wie zitieren? WebQC.Org Online-Schulung kostenlose Hausaufgabenhilfe Chemie Probleme Fragen und Antworten" ]	[ null, "https://de.webqc.org/images/logo.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7539098,"math_prob":0.8825292,"size":1872,"snap":"2019-43-2019-47","text_gpt3_token_len":797,"char_repetition_ratio":0.3843683,"word_repetition_ratio":0.19076923,"special_character_ratio":0.45726496,"punctuation_ratio":0.11453745,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95261264,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-11-22T10:13:21Z\",\"WARC-Record-ID\":\"<urn:uuid:cbcb51a0-c7c8-4802-aa43-0b85d7eb7115>\",\"Content-Length\":\"52674\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:61e0ed65-c194-4a87-b906-9bbae8ccc2e1>\",\"WARC-Concurrent-To\":\"<urn:uuid:56b6b252-df35-4fa7-9ee3-37bcf0cc45f7>\",\"WARC-IP-Address\":\"104.27.141.247\",\"WARC-Target-URI\":\"https://de.webqc.org/molecularweightcalculated-190210-131.html\",\"WARC-Payload-Digest\":\"sha1:SSZK4W6DDR7CJLF2BUHEOILQSURSQZXU\",\"WARC-Block-Digest\":\"sha1:5BONMJQAYFBE6JU6GLX5UOYI2JDDRAM3\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-47/CC-MAIN-2019-47_segments_1573496671249.37_warc_CC-MAIN-20191122092537-20191122120537-00359.warc.gz\"}"}
http://lbartman.com/6th-grade-rock-cycle-worksheets/	[ "# 6th Grade Rock Cycle Worksheets\n\n👤 will chen 🗓 April 11, 2021, 7:20 am ( Last Modified )\n\nIntroduce your young scientist to the wonders of the water cycle with these engaging worksheets full of useful diagrams and enlightening texts. Kids will enjoy discovering how water moves through the earth’s water cycle, changing from one state to another as they learn all about this important and fascinating ecological process..Take a tour of our section on Grade 6 science worksheets and join us to become a junior Einstein – it’s FREE! Give your child the added advantage of tutoring with eTutorWorld. Select sixth grade science worksheets from the list below for FREE download of 6th grade science worksheets with answer key . New worksheets added regularly..Spark your students' curiosity with our second grade science worksheets and printables! From explorations of plant and animal life cycles all the way to weather patterns, layers of the Earth, and the planets of our solar system, these second grade science worksheets use fascinating facts and engaging illustrations to bring science to life for your students..Sixth Grade (Grade 6) Science Worksheets, Tests, and Activities. Print our Sixth Grade (Grade 6) Science worksheets and activities, or administer them as online tests. Our worksheets use a variety of high-quality images and some are aligned to Common Core Standards. Worksheets labeled with are accessible to Help Teaching Pro subscribers only..\n\nLandform Worksheets. Learn about plains, plateaus, mountains, hills, islands, peninsulas, islands, and all types of landforms. Space Printables. Worksheets for teaching kids about our solar system, planets, the moon and outer space..Charlie is the toughest girl in the fifth grade, but when she's dared to look into the \"Wailing Well\" and find out if a kid-hungry troll really lives there, she's afraid she'll lose her reputation. Charlie must find the courage to prove that some scary stories are just that--stories..© 2021 Houghton Mifflin Harcourt. All rights reserved. Terms of Purchase Privacy Policy Site Map Trademark Credits Permissions Request Privacy Policy Site Map ..\n\nWe would like to show you a description here but the site won’t allow us..Grade/level: Grade 6th, 7th, 8th Age: 10-13 Main content . More Science interactive worksheets. Parts of the plant by MissMaggieCandlish: Parts of a Plant . by saramalik: Plants by Miss_Dianita: Matter Sort by asherman5: Food chain classwork by iduque: The water cycle by 2JESUSARAMBURU: Food chain by learnwithalessia: Plants and their parts ..Stars & Planets Worksheets Eyewitness Workbook Stars & Planets is an activity-packed exploration of the world of space and astronomy. Below you will find fast facts, activities and quizzes...\n\nRelated to \"6th Grade Rock Cycle Worksheets\" ⤵\n\nName : __________________\n\nSeat Num. : __________________\n\nDate : __________________\n\n5360 + 22 = ...\n\n3695 + 13 = ...\n\n6066 + 53 = ...\n\n7727 + 25 = ...\n\n6674 + 12 = ...\n\n4927 + 80 = ...\n\n2454 + 55 = ...\n\n5305 + 66 = ...\n\n6318 + 84 = ...\n\n2549 + 32 = ...\n\n2533 + 42 = ...\n\n2962 + 97 = ...\n\n9071 + 56 = ...\n\n1603 + 31 = ...\n\n4819 + 99 = ...\n\n9390 + 66 = ...\n\n4062 + 84 = ...\n\n9568 + 75 = ...\n\n9855 + 29 = ...\n\n3894 + 88 = ...\n\n4340 + 49 = ...\n\n7159 + 29 = ...\n\n8809 + 92 = ...\n\n9098 + 15 = ...\n\n3553 + 18 = ...\n\n7675 + 59 = ...\n\n5937 + 71 = ...\n\n2992 + 44 = ...\n\n5042 + 26 = ...\n\n7171 + 13 = ...\n\n3925 + 82 = ...\n\n6257 + 31 = ...\n\n7947 + 29 = ...\n\n1609 + 60 = ...\n\n8134 + 87 = ...\n\n4350 + 58 = ...\n\n7884 + 55 = ...\n\n1323 + 40 = ...\n\n9680 + 45 = ...\n\n1534 + 66 = ...\n\n5294 + 70 = ...\n\n6624 + 60 = ...\n\n8634 + 65 = ...\n\n4232 + 90 = ...\n\n4412 + 71 = ...\n\n6258 + 35 = ...\n\n7698 + 19 = ...\n\n8205 + 39 = ...\n\n5314 + 32 = ...\n\n6523 + 30 = ...\n\n6041 + 68 = ...\n\n8457 + 87 = ...\n\n6184 + 57 = ...\n\n9393 + 56 = ...\n\n6697 + 24 = ...\n\n3034 + 48 = ...\n\n1202 + 17 = ...\n\n2190 + 11 = ...\n\n7951 + 40 = ...\n\n6773 + 93 = ...\n\n7370 + 92 = ...\n\n7548 + 37 = ...\n\n9375 + 67 = ...\n\n5550 + 38 = ...\n\n8208 + 88 = ...\n\n5067 + 74 = ...\n\n9404 + 37 = ...\n\n1735 + 82 = ...\n\n4099 + 52 = ...\n\n2207 + 99 = ...\n\n3731 + 93 = ...\n\n3632 + 14 = ...\n\n2519 + 85 = ...\n\n4423 + 42 = ...\n\n9373 + 15 = ...\n\n4872 + 22 = ...\n\n1834 + 30 = ...\n\n6153 + 88 = ...\n\n9202 + 25 = ...\n\n4151 + 57 = ...\n\n7165 + 92 = ...\n\n5369 + 22 = ...\n\n6673 + 82 = ...\n\n6220 + 83 = ...\n\n6432 + 44 = ...\n\n5051 + 66 = ...\n\n1215 + 94 = ...\n\n6793 + 68 = ...\n\n5551 + 10 = ...\n\n7142 + 26 = ...\n\n1574 + 31 = ...\n\n8036 + 84 = ...\n\n7379 + 97 = ...\n\n5680 + 84 = ...\n\n7786 + 96 = ...\n\n1975 + 96 = ...\n\n5843 + 12 = ...\n\n5948 + 23 = ...\n\n6561 + 86 = ...\n\n2240 + 30 = ...\n\n7204 + 69 = ...\n\n7617 + 28 = ...\n\n3846 + 80 = ...\n\n5468 + 35 = ...\n\n3470 + 56 = ...\n\n7121 + 19 = ...\n\n9925 + 77 = ...\n\n5358 + 21 = ...\n\n4407 + 69 = ...\n\n8142 + 34 = ...\n\n8499 + 28 = ...\n\n5382 + 35 = ...\n\n6012 + 10 = ...\n\n9346 + 20 = ...\n\n3336 + 93 = ...\n\n7960 + 97 = ...\n\n8825 + 32 = ...\n\n5736 + 85 = ...\n\n1324 + 17 = ...\n\n3161 + 27 = ...\n\n8416 + 58 = ...\n\n5442 + 68 = ...\n\n8347 + 37 = ...\n\n9994 + 78 = ...\n\n1421 + 78 = ...\n\n7061 + 30 = ...\n\n6636 + 53 = ...\n\n5512 + 82 = ...\n\n2747 + 72 = ...\n\n6247 + 63 = ...\n\n1001 + 72 = ...\n\n3759 + 85 = ...\n\n5763 + 22 = ...\n\n9406 + 18 = ...\n\n5789 + 24 = ...\n\n5612 + 58 = ...\n\n6299 + 67 = ...\n\n3011 + 15 = ...\n\n8653 + 59 = ...\n\n4188 + 71 = ...\n\n2740 + 37 = ...\n\n6632 + 97 = ...\n\n9217 + 71 = ...\n\n6965 + 44 = ...\n\n1962 + 30 = ...\n\n9731 + 98 = ...\n\n9009 + 65 = ...\n\n7821 + 51 = ...\n\n2431 + 72 = ...\n\n7475 + 19 = ...\n\n6200 + 40 = ...\n\n2626 + 98 = ...\n\n6724 + 84 = ...\n\n3107 + 14 = ...\n\n9891 + 46 = ...\n\n6355 + 83 = ...\n\n3134 + 60 = ...\n\n6160 + 35 = ...\n\n9169 + 28 = ...\n\n7396 + 31 = ...\n\n6819 + 77 = ...\n\n4018 + 83 = ...\n\n6364 + 55 = ...\n\n4675 + 34 = ...\n\n7107 + 37 = ...\n\n9937 + 32 = ...\n\n3822 + 65 = ...\n\n3123 + 88 = ...\n\n3876 + 82 = ...\n\n5696 + 52 = ...\n\n7333 + 66 = ...\n\n6134 + 87 = ...\n\n8704 + 63 = ...\n\n6970 + 31 = ...\n\n9638 + 36 = ...\n\n5703 + 28 = ...\n\n2952 + 51 = ...\n\n8596 + 59 = ...\n\n4690 + 75 = ...\n\n5990 + 65 = ...\n\n5646 + 97 = ...\n\n2254 + 82 = ...\n\n5601 + 30 = ...\n\n9311 + 13 = ...\n\n5231 + 38 = ...\n\n8540 + 23 = ...\n\n1037 + 87 = ...\n\n2915 + 63 = ...\n\n2538 + 85 = ...\n\n6024 + 71 = ...\n\n7374 + 87 = ...\n\n2733 + 83 = ...\n\n5146 + 40 = ...\n\n1799 + 61 = ...\n\n7505 + 81 = ...\n\n3291 + 25 = ...\n\n8816 + 92 = ...\n\n6023 + 75 = ...\n\n8737 + 59 = ...\n\n5294 + 46 = ...\n\nshow printable version !!!hide the show", null, "Rock Cycle Worksheet - 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https://transtutor.net/sys/question/question/11147495124_view.html	[ "Solution 03 kg of air undergoes the cycle shown in Fig 1 a piston cylinder arrangement calculate the work output they got 4\nSolution kg of air undergoes the cycle shown in Fig a piston cylinder arrangement calculate\nSolution kg of air undergoes the cycle shown in Fig a\nair undergoes the cycle shown in Fig a piston cylinder arrangement calculate the work output they got\nSolution kg of air undergoes the cycle shown in\nFig a piston cylinder arrangement calculate the work output they got\nSolution kg of air undergoes the cycle\nSolution kg of\n(Solution) 03 kg of air undergoes the cycle shown in Fig. 1, a piston-cylinder arrangement, calculate the work output. they got 4.\n\n Category: General Words: 1050 Amount: \\$12 Writer:\n\nPaper instructions\n\nIf 0.03 kg of air undergoes the cycle shown in Fig. 1, a piston-cylinder arrangement, calculate the work output. they got 4.01kJ but I dont see how" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.76742214,"math_prob":0.9831881,"size":612,"snap":"2020-45-2020-50","text_gpt3_token_len":128,"char_repetition_ratio":0.14309211,"word_repetition_ratio":0.56842107,"special_character_ratio":0.19607843,"punctuation_ratio":0.03883495,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98883384,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-11-30T14:19:35Z\",\"WARC-Record-ID\":\"<urn:uuid:dc7dad7c-54fd-47a4-ac52-8ff958fd0489>\",\"Content-Length\":\"21306\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0169c4b4-5da4-4f9d-aac9-46ca7c5f09ed>\",\"WARC-Concurrent-To\":\"<urn:uuid:5fd28c63-9212-450f-a571-8ed822483ba0>\",\"WARC-IP-Address\":\"198.38.94.115\",\"WARC-Target-URI\":\"https://transtutor.net/sys/question/question/11147495124_view.html\",\"WARC-Payload-Digest\":\"sha1:FBQIGKNCZWHGSBPT265PCEHFETC5GISK\",\"WARC-Block-Digest\":\"sha1:5LJ7HKGJYBFTW33TEX5IODNPNSNTFGXP\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-50/CC-MAIN-2020-50_segments_1606141216175.53_warc_CC-MAIN-20201130130840-20201130160840-00136.warc.gz\"}"}
https://studyres.com/doc/1165942/0005_hsm11a1_te_01tr.indd	[ "Survey\n\n* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project\n\nDocument related concepts\n\nHistory of algebra wikipedia, lookup\n\nFundamental theorem of algebra wikipedia, lookup\n\nNumber wikipedia, lookup\n\nReal number wikipedia, lookup\n\nTranscript\n```Name\nClass\n1-3\nDate\nPractice\nForm K\nReal Numbers and the Number Line\nSimplify each expression.\n1.\n144\n2.\n25\n3.\n169\n4.\n49\n5.\n256\n6.\n400\n7.\n9\n49\n8.\n196\n144\n9.\n0.01\n10.\n0.49\nEstimate the square root. Round to the nearest integer.\n11.\n38\n12.\n65\n13.\n99\n14.\n145.5\n15.\n23.75\n16.\n64.36\nFind the approximate side length of each square figure to the nearest whole\nunit.\n17. a tabletop with an area 25 ft2\n18. a wall that is 105 m2\nPrentice Hall Foundations Algebra 1 • Teaching Resources\n25\nName\n1-3\nClass\nDate\nPractice (continued)\nForm K\nReal Numbers and the Number Line\nName the subset(s) of the real numbers to which each number belongs.\n19.\n3\n4\n22. 45,368\n20. –8\n21. 2π\n23.\n24.\n11\n-\n2\n3\nCompare the numbers in each exercise using an inequality symbol.\n25. 36, 49\n26.\n1\n, 1.25\n3\n27. 100, - 169\n28.\n34\n,1.8\n19\nOrder the numbers in each exercise from least to greatest.\n1\n30. 1.25, , 1.25\n3\n29. 2.75, 25, - 36\n31.\n3\n, -0.6, 1\n5\n32. 80 , 9, 30\n25\n9\n33. Kate, Kevin, and Levi are comparing how fast they can run. Kate was able to\nrun 5 miles in 47.5 minutes. Kevin was able to run 8 miles in 74 minutes. Levi\nwas able to run 4 miles in 32 minutes. Order the friends from the fastest to the\nslowest.\nPrentice Hall Foundations Algebra 1 • Teaching Resources" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8301198,"math_prob":0.9568457,"size":1668,"snap":"2020-34-2020-40","text_gpt3_token_len":532,"char_repetition_ratio":0.08473558,"word_repetition_ratio":0.23197491,"special_character_ratio":0.3711031,"punctuation_ratio":0.20487805,"nsfw_num_words":1,"has_unicode_error":false,"math_prob_llama3":0.9612305,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-08-06T07:27:16Z\",\"WARC-Record-ID\":\"<urn:uuid:152663e0-4583-493b-8f54-594bdbcb1b94>\",\"Content-Length\":\"54691\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:62bfc9c8-f99b-4c1d-b2d5-9d4f0c80d1af>\",\"WARC-Concurrent-To\":\"<urn:uuid:67e4eac0-eeeb-43e0-b036-b1f469cfb2bc>\",\"WARC-IP-Address\":\"104.27.180.193\",\"WARC-Target-URI\":\"https://studyres.com/doc/1165942/0005_hsm11a1_te_01tr.indd\",\"WARC-Payload-Digest\":\"sha1:JOPAP4YIG2ASNNOBBF332EOXCYGLFPKC\",\"WARC-Block-Digest\":\"sha1:CW3O2AP4XEO2BHULS7SIGCMMRE5TL57X\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-34/CC-MAIN-2020-34_segments_1596439736883.40_warc_CC-MAIN-20200806061804-20200806091804-00338.warc.gz\"}"}
https://support.microsoft.com/en-us/office/accrint-function-fe45d089-6722-4fb3-9379-e1f911d8dc74?ocmsassetid=ha102753127&ctt=5&origin=ha102752955&correlationid=668adffd-149f-4f2d-a6bc-5a07a69d9bc8&ui=en-us&rs=en-us&ad=us	[ "# ACCRINT function\n\nThis article describes the formula syntax and usage of the ACCRINT function in Microsoft Excel.\n\n## Description\n\nReturns the accrued interest for a security that pays periodic interest.\n\n## Syntax\n\nACCRINT(issue, first_interest, settlement, rate, par, frequency, [basis], [calc_method])\n\nImportant: Dates should be entered by using the DATE function, or as results of other formulas or functions. For example, use DATE(2008,5,23) for the 23rd day of May, 2008. Problems can occur if dates are entered as text.\n\nThe ACCRINT function syntax has the following arguments:\n\n• Issue Required. The security's issue date.\n\n• First_interest Required. The security's first interest date.\n\n• Settlement Required. The security's settlement date. The security settlement date is the date after the issue date when the security is traded to the buyer.\n\n• Rate Required. The security's annual coupon rate.\n\n• Par Required. The security's par value. If you omit par, ACCRINT uses \\$1,000.\n\n• Frequency Required. The number of coupon payments per year. For annual payments, frequency = 1; for semiannual, frequency = 2; for quarterly, frequency = 4.\n\n• Basis Optional. The type of day count basis to use.\n\nBasis\n\nDay count basis\n\n0 or omitted\n\nUS (NASD) 30/360\n\n1\n\nActual/actual\n\n2\n\nActual/360\n\n3\n\nActual/365\n\n4\n\nEuropean 30/360\n\n• Calc_method Optional. A logical value that specifies the way to calculate the total accrued interest when the date of settlement is later than the date of first_interest. A value of TRUE (1) returns the total accrued interest from issue to settlement. A value of FALSE (0) returns the accrued interest from first_interest to settlement. If you do not enter the argument, it defaults to TRUE.\n\n## Remarks\n\n• Microsoft Excel stores dates as sequential serial numbers so they can be used in calculations. By default, January 1, 1900 is serial number 1, and January 1, 2008 is serial number 39448 because it is 39,448 days after January 1, 1900.\n\n• Issue, first_interest, settlement, frequency, and basis are truncated to integers.\n\n• If issue, first_interest, or settlement is not a valid date, ACCRINT returns the #VALUE! error value.\n\n• If rate ≤ 0 or if par ≤ 0, ACCRINT returns the #NUM! error value.\n\n• If frequency is any number other than 1, 2, or 4, ACCRINT returns the #NUM! error value.\n\n• If basis < 0 or if basis > 4, ACCRINT returns the #NUM! error value.\n\n• If issue ≥ settlement, ACCRINT returns the #NUM! error value.\n\n• ACCRINT is calculated as follows:", null, "where:\n\n• Ai = number of accrued days for the ith quasi-coupon period within odd period.\n\n• NC = number of quasi-coupon periods that fit in odd period. If this number contains a fraction, raise it to the next whole number.\n\n• NLi = normal length in days of the quasi-coupon period within odd period.\n\n## Example\n\nCopy the example data in the following table, and paste it in cell A1 of a new Excel worksheet. For formulas to show results, select them, press F2, and then press Enter. If you need to, you can adjust the column widths to see all the data.\n\n Data Description 39508 Issue date 39691 First interest date 39569 Settlement date 0.1 Coupon rate 1000 Par value 2 Frequency is semiannual (see above) 0 30/360 basis (see above) Formula Description Result =ACCRINT(A2,A3,A4,A5,A6,A7,A8) Accrued interest for a treasury bond with the terms above. 16.666667 =ACCRINT(DATE(2008,3,5),A3,A4,A5,A6,A7,A8,FALSE) Accrued interest with the terms above, except the issue date is March 5, 2008. 15.555556 =ACCRINT(DATE(2008, 4, 5), A3, A4, A5, A6, A7, A8, TRUE) Accrued interest with the terms above, except the issue date is April 5, 2008, and the accrued interest is calculated from the first_interest to settlement. 7.2222222\n\n### Need more help?", null, "" ]	[ null, "https://support.content.office.net/en-us/media/b224f238-a585-45a1-8afb-56f2f1e43b20.gif", null, "https://support.microsoft.com/SocImages/got-it-original.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8024089,"math_prob":0.8971362,"size":3724,"snap":"2020-45-2020-50","text_gpt3_token_len":1000,"char_repetition_ratio":0.13602151,"word_repetition_ratio":0.05808477,"special_character_ratio":0.28732544,"punctuation_ratio":0.17718121,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9829128,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-26T13:48:57Z\",\"WARC-Record-ID\":\"<urn:uuid:cd5f25b4-1fd3-45d6-ba39-30db1778cf01>\",\"Content-Length\":\"114320\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:8fdc8b19-1932-4780-871f-29cc6061f217>\",\"WARC-Concurrent-To\":\"<urn:uuid:631691f8-202b-48fe-9032-a961f3c3d05d>\",\"WARC-IP-Address\":\"23.62.26.93\",\"WARC-Target-URI\":\"https://support.microsoft.com/en-us/office/accrint-function-fe45d089-6722-4fb3-9379-e1f911d8dc74?ocmsassetid=ha102753127&ctt=5&origin=ha102752955&correlationid=668adffd-149f-4f2d-a6bc-5a07a69d9bc8&ui=en-us&rs=en-us&ad=us\",\"WARC-Payload-Digest\":\"sha1:IGC2HM3LIKJ2KMIDO54Y6Y7U6KFGM7HQ\",\"WARC-Block-Digest\":\"sha1:ITND3WOFLGU4GPTOD3XZPPJJHO2K5ECB\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-45/CC-MAIN-2020-45_segments_1603107891228.40_warc_CC-MAIN-20201026115814-20201026145814-00083.warc.gz\"}"}
https://socratic.org/questions/what-is-the-gcf-of-18a-20ab-and-6ab	[ "# What is the GCF of 18a, 20ab and 6ab?\n\nJun 28, 2018\n\n$2 a$\n\n#### Explanation:\n\nGiven the set: $18 a , 20 a b , 6 a b$.\n\nFirst find the greatest common factor of $18 , 20 , 6$.\n\n$18 = 2 \\cdot {3}^{2}$\n\n$20 = {2}^{2} \\cdot 5$\n\n$6 = 2 \\cdot 3$\n\n$\\therefore \\setminus \\text{GCF} = 2$\n\nThen, find the $\\text{GCF}$ of $a , a b , a b$.\n\n$a = a$\n\n$a b = a \\cdot b$\n\n$\\therefore \\setminus \\text{GCF} = a$\n\nNow, multiply the two $\\text{GCF}$s of the two sets together.\n\n$2 \\cdot a = 2 a$\n\nThat is the greatest common factor of the whole sequence.\n\nJun 28, 2018\n\nG C F is $\\textcolor{g r e e n}{2 a}$\n\n#### Explanation:\n\nG C F is the greatest common factor which finds place in all the terms.\n\n$18 a , 20 a b , 6 a b$\n\n$\\implies 9 \\cdot \\textcolor{g r e e n}{2 a} , 10 b \\cdot \\textcolor{g r e e n}{2 a} , 3 b \\cdot \\textcolor{g r e e n}{2 a}$\n\n$2 a$ is common in all the three terms.\n\nHence the G C F is $\\textcolor{g r e e n}{2 a}$" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.75022894,"math_prob":0.99988806,"size":478,"snap":"2021-43-2021-49","text_gpt3_token_len":130,"char_repetition_ratio":0.1329114,"word_repetition_ratio":0.04597701,"special_character_ratio":0.25732216,"punctuation_ratio":0.10784314,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99998724,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-12-07T09:16:25Z\",\"WARC-Record-ID\":\"<urn:uuid:d29c2756-0a11-4ad9-b00c-b79e2f0feb50>\",\"Content-Length\":\"35009\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:15b0877c-7222-4cac-aee7-d43c4435e1b3>\",\"WARC-Concurrent-To\":\"<urn:uuid:4bd04d2d-e554-478d-9b92-ce878276a811>\",\"WARC-IP-Address\":\"216.239.32.21\",\"WARC-Target-URI\":\"https://socratic.org/questions/what-is-the-gcf-of-18a-20ab-and-6ab\",\"WARC-Payload-Digest\":\"sha1:MJ3W524KCDDDFAICTPPXU2FZ7JMVC43M\",\"WARC-Block-Digest\":\"sha1:MK7KWOPTBFLG3NNMMFESLRVTKXCNNJKU\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964363337.27_warc_CC-MAIN-20211207075308-20211207105308-00112.warc.gz\"}"}
https://www.geeksforgeeks.org/python-numpy-matrix-tolist/	[ "Related Articles\nPython \| Numpy matrix.tolist()\n• Last Updated : 29 May, 2019\n\nWith the help of `Numpy matrix.tolist()` method, we are able to convert the matrix into a list by using the `matrix.tolist()` method.\n\nSyntax: `matrix.tolist()`\nReturn : Return a new list\n\nExample #1 :\nIn this example we can see that by passing the matrix we are able to convert it into list by using the `matrix.tolist()` method.\n\n `# import the important module in python``import` `numpy as np`` ` `# make matrix with numpy``gfg ``=` `np.matrix(``'[4, 1, 12, 3]'``)`` ` `# applying matrix.tolist() method``geek ``=` `gfg.tolist()`` ` `print``(geek)`\nOutput:\n```[[4, 1, 12, 3]]\n```\n\nExample #2 :\n\n `# import the important module in python``import` `numpy as np`` ` `# make matrix with numpy``gfg ``=` `np.matrix(``'[4, 1, 9; 12, 3, 1; 4, 5, 6]'``)`` ` `# applying matrix.tolist() method``geek ``=` `gfg.tolist()`` ` `print``(geek)`\nOutput:\n```[[4, 1, 9], [12, 3, 1], [4, 5, 6]]\n```\n\nAttention geek! Strengthen your foundations with the Python Programming Foundation Course and learn the basics.\n\nTo begin with, your interview preparations Enhance your Data Structures concepts with the Python DS Course.\n\nMy Personal Notes arrow_drop_up" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6317317,"math_prob":0.9250021,"size":1069,"snap":"2021-04-2021-17","text_gpt3_token_len":306,"char_repetition_ratio":0.16901408,"word_repetition_ratio":0.32222223,"special_character_ratio":0.3012161,"punctuation_ratio":0.20087336,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9799813,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-04-20T01:04:12Z\",\"WARC-Record-ID\":\"<urn:uuid:6c8318c3-5489-479b-9dae-86c231b8b326>\",\"Content-Length\":\"83446\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:02f1af18-c7fd-498e-8115-150671a4ed39>\",\"WARC-Concurrent-To\":\"<urn:uuid:150ad66f-468f-4e22-b696-42b521b49df0>\",\"WARC-IP-Address\":\"104.97.85.34\",\"WARC-Target-URI\":\"https://www.geeksforgeeks.org/python-numpy-matrix-tolist/\",\"WARC-Payload-Digest\":\"sha1:CRLWS7XPLYCT5N7GB6U7UEHFU4EI3AJ2\",\"WARC-Block-Digest\":\"sha1:K6AN2HPAQ7FY3D5XLVWFZ7KILWFQVKDL\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-17/CC-MAIN-2021-17_segments_1618038921860.72_warc_CC-MAIN-20210419235235-20210420025235-00046.warc.gz\"}"}
https://www.physicsoverflow.org/36738/classification-of-constraints-in-hamiltonian-formalism	[ "#", null, "Classification of constraints in Hamiltonian formalism\n\n+ 3 like - 0 dislike\n335 views\n\nI am reading this paper, where the claim is from hamiltonian formalism one can show that non-local theories of gravity are ghost-free. Before going to complication, I would like to see a clear definition (and mathematical formula of how to obtain the-) of primary constraint, secondary constraint, first class constraint and second class constraint. It would be appreciated to even point a toy example.\n\nMoreover, I wonder how one uses these constraint(s) to count the number of degrees of freedom?\n\nAlso I wonder if one can uses such formalism and show that the given gravitational theory is bounded from below?\n\nrecategorized Jul 22, 2016\n\n+ 4 like - 0 dislike\n\nIn Hamiltonian formalism, the phase space is a symplectic manifold i.e. a manifold $M$ equipped with a 2-form $\\omega$ which is non-degenerate and closed. In a constrained system, the dynamical variables are constrained to lie on a submanifold $N$ of $M$. If the restriction of $\\omega$ to $N$ is 0, then $N$ is called a coisotropic submanifold and the constraint is called first class. If the restriction of $\\omega$ to $N$ is non-degenerate, then $N$ is called a symplectic submanifold and the constraint is called second class. A general $N$ can always be locall written as a product $N_1 \\times N_2$ with $N_1$ coisotropic (first class) and $N_2$ symplectic (second class).\n\nSame thing in a slightly different language: the symplectic structure defines a Poisson bracket $\\{,\\}$ on the algebra of functions on $M$. The constraints are the equations $f_i=0$ defining $N$. The constraints are called first class if $\\{f_i,f_j\\}=0$ up to the constraints, for all $i$, $j$. The constraints are called second class if the matrix $\\{f_i,f_j\\}$ is non-degenerate. In general, up to a change of variables, one can write the matrix of Poisson brackets $\\{f_i,f_j\\}$ in a block diagonal form: the first block being zero and the second block being non-degenerate. Constraints corresponding to the first block are called first class, the ones corresponding to the second are called second class.\n\nThe distinction between primary and secondary constraints appears in the context of the Hamiltonian description of a system given in Lagrangian form. Primary constraints are the constraints which are there if the momenta are not independent (if the matrix of second derivatives of the Lagrangian with respect to the velocities is not invertible). Secondary constraints are extra constraints which are imposed by consistency of the primary constraints with the equations of motion.\n\nIf by number of degrees of freedom, one means the dimension of the symplectic manifold where the Hamiltonian dynamics is well-defined without constraints, it is $dim M - 2 n_1-n_1$ where $n_1$ is the number of first class constraints and $n_2$ is the number of second class constraints. Indeed, for a second class constraint, one simply has to restrict ourselves to the constrained submanifold but for a first class constraint, one has to quotient this submanifold by redundancies of the description (\"gauge transformations\"), which are as numerous as the first class constraints (coisotropic reduction).\n\nThe original reference on first/second class constraints is Dirac:\n\n Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the \"link\" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. 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To avoid this verification in future, please log in or register." ]	[ null, "https://www.physicsoverflow.org/qa-plugin/po-printer-friendly/print_on.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9161082,"math_prob":0.97690636,"size":2876,"snap":"2021-31-2021-39","text_gpt3_token_len":670,"char_repetition_ratio":0.1740947,"word_repetition_ratio":0.032183908,"special_character_ratio":0.22079277,"punctuation_ratio":0.10211946,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9954947,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-09-24T06:29:41Z\",\"WARC-Record-ID\":\"<urn:uuid:da5376ca-f040-4c77-9545-414f6cbc5c25>\",\"Content-Length\":\"116159\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:6335db11-7115-4c0f-aefc-180e4ab6a51f>\",\"WARC-Concurrent-To\":\"<urn:uuid:5df69c01-d093-43d7-9650-761b5ff45f1c>\",\"WARC-IP-Address\":\"129.70.43.86\",\"WARC-Target-URI\":\"https://www.physicsoverflow.org/36738/classification-of-constraints-in-hamiltonian-formalism\",\"WARC-Payload-Digest\":\"sha1:TTYA3M5NXH4YPUEAQF7LDCKO2Q7OJMGI\",\"WARC-Block-Digest\":\"sha1:HTRJIMPPJNTC4K5M2CVRZWUISVSUNSDS\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-39/CC-MAIN-2021-39_segments_1631780057504.60_warc_CC-MAIN-20210924050055-20210924080055-00124.warc.gz\"}"}
http://www.global-sci.org/intro/article_detail/aamm/176.html	[ "Volume 3, Issue 4\nSuperconvergence of Parabolic Optimal Control Problems\n\nAdv. Appl. Math. Mech., 3 (2011), pp. 401-419.\n\nPublished online: 2011-03\n\nPreview Full PDF 62 1067\nExport citation\n\nCited by\n\n• Abstract\n\nIn this paper, we investigate the superconvergence results for optimal control problems governed by parabolic equations with semidiscrete mixed finite element approximation. We use the lowest order mixed finite element spaces to discrete the state and costate variables while use piecewise constant function to discrete the control variable. Superconvergence estimates for both the state variable and its gradient variable are obtained.\n\n• Keywords\n\nOptimal control mixed finite element superconvergence parabolic equations\n\n65L10 65L12\n\n• BibTex\n• RIS\n• TXT\n@Article{AAMM-3-401, author = {Xiaoqing Xing and Yanping Chen}, title = {Superconvergence of Parabolic Optimal Control Problems}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2011}, volume = {3}, number = {4}, pages = {401--419}, abstract = {\n\nIn this paper, we investigate the superconvergence results for optimal control problems governed by parabolic equations with semidiscrete mixed finite element approximation. We use the lowest order mixed finite element spaces to discrete the state and costate variables while use piecewise constant function to discrete the control variable. Superconvergence estimates for both the state variable and its gradient variable are obtained.\n\n}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.10-m1006}, url = {http://global-sci.org/intro/article_detail/aamm/176.html} }\nTY - JOUR T1 - Superconvergence of Parabolic Optimal Control Problems AU - Xiaoqing Xing & Yanping Chen JO - Advances in Applied Mathematics and Mechanics VL - 4 SP - 401 EP - 419 PY - 2011 DA - 2011/03 SN - 3 DO - http://dor.org/10.4208/aamm.10-m1006 UR - https://global-sci.org/intro/article_detail/aamm/176.html KW - Optimal control KW - mixed finite element KW - superconvergence KW - parabolic equations AB -\n\nIn this paper, we investigate the superconvergence results for optimal control problems governed by parabolic equations with semidiscrete mixed finite element approximation. We use the lowest order mixed finite element spaces to discrete the state and costate variables while use piecewise constant function to discrete the control variable. Superconvergence estimates for both the state variable and its gradient variable are obtained.\n\nXiaoqing Xing & Yanping Chen. (1970). Superconvergence of Parabolic Optimal Control Problems. Advances in Applied Mathematics and Mechanics. 3 (4). 401-419. doi:10.4208/aamm.10-m1006\nCopy to clipboard\nThe citation has been copied to your clipboard" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.79131866,"math_prob":0.8636287,"size":1112,"snap":"2020-34-2020-40","text_gpt3_token_len":245,"char_repetition_ratio":0.12635379,"word_repetition_ratio":0.7619048,"special_character_ratio":0.22392087,"punctuation_ratio":0.14285715,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97593224,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-01T16:15:57Z\",\"WARC-Record-ID\":\"<urn:uuid:b9f3e2e0-05de-4cca-b250-1e8f030253a0>\",\"Content-Length\":\"41666\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:1ecaf892-e5cd-447f-a36e-16d7dd3fbc15>\",\"WARC-Concurrent-To\":\"<urn:uuid:a2fc8c95-fc96-46b9-b62a-d0b485f7d7bd>\",\"WARC-IP-Address\":\"47.52.67.10\",\"WARC-Target-URI\":\"http://www.global-sci.org/intro/article_detail/aamm/176.html\",\"WARC-Payload-Digest\":\"sha1:IUNRNIISOD4ZDJE3NA2JJGW3SR2YYFCO\",\"WARC-Block-Digest\":\"sha1:TGYZD7GWVV43VAA527HTI4FNBEZXYFRI\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-40/CC-MAIN-2020-40_segments_1600402131777.95_warc_CC-MAIN-20201001143636-20201001173636-00776.warc.gz\"}"}
https://www.kjmaclean.com/Geometry/IcosaDodeca.html	[ "The Icosa–Docedahedron", null, "Figure 1 The Icosadodecahedron\nThis polyhedron is the dual of the rhombic triacontahedron.\nIt has 30 vertices, 32 faces, and 60 edges.\n20 of the faces are equilateral triangles. 12 of the faces are pentagons.\nIt is probably called the icosa-dodecahedron because in the middle of every pentagonal face is the vertex of an icosahedron, and in the middle of every triangular face is the vertex of a dodecahedron.\nThis polyhedron is composed of 6 'great circle' decagons which traverse the outside of the polyhedron, sharing the 30 vertices and accounting for the 60 edges (see Figure 2).\nActually, the rhombic triacontahedron should be called the icosa dodecahedron, because one icosahedron and one dodecahedron precisely describe its 32 vertices.", null, "Figure 2 -- showing the icosadodecahedron as composed of 6 interlocking decagons\n\nThe icosadodecahedron (hereinafter referred to as i.d.) has internal planes within it, as we have seen in other polyhedra. The i.d. has 12 internal pentagons, 2 of which are highlighted, one for each pentagonal face. And of course, the 6 'great circle' decagons:", null, "Figure 3 showing 2 of the internal pentagons and 1 internal decagon\n\nFascinatingly enough, the i.d. also has 5 hexagonal planes! One of these is highlighted in Figure 4 below.\nNotice that the sides of the hexagonal planes are, just like the pentagonal planes, all diagonals of the pentagonal faces. How can this be? How can a pentagon and a hexagon both have sides of the same length? We'll see later on how this happens.", null, "Figure 4 -- showing one of the 5 hexagonal planes of the icosa-dodecahedron\n\nThe i.d. is composed of equilateral triangles internally and externally. The internal equilateral triangles form from the centroid and any 2 diagonal vertices of the pentagonal faces.", null, "Figure 5 All of these triangles (for example, triangle BGoE) are equilateral triangles, just like the 20 smaller equilateral triangles of the faces (for example, BGC). The i.d. has the property that the central angles of the diagonals of the pentagonal faces (as", null, "BGoE) are 60 degrees.", null, "Figure 6 -- showing one of the hexagonal planes in relation to 2 of the pentagonal planes and one of the dodecahedron planes. Now it is clear why the hexagonal plane has the same edge length as the pentagonal planes: the hexagonal plane uses different diagonals of the pentagonal faces, and so is angled relative to the pentagonal planes.", null, "Figure 6A -- showing that the hexagonal plane is parallel to the top and bottom triangular faces, and also showing the 2 large triangular planes.\n\nWhat is the volume of the icosadodecahedron?\nWe will use the pyramid method again. There are 20 triangular pyramids and 12 pentagonal pyramids.\nAlthough the icosa-dodecahedron is a semi-regular polyhedron, all of its vertices touch on the same sphere. So the radius of the sphere surrounding the i.d. is the same for each vertex. That makes our calculations a little easier.\n\nIn order to make life simpler, it would be nice to know the relationship between the side and the radius of the i.d. I will present 2 ways of getting this: the absurdly simple way, and the 'brute force' method.\nThe simple method is to recognize that the central angle of the i.d. is 36°, and that every vertex on the i.d. is part of a decagon. Therefore any 2 vertices combined with the centroid forms a\n36 -72 -72 isosceles triangle. We recognize this immediately as a Golden Triangle. The relationship of the short side to any of the long sides of such a triangle is therefore 1 to", null, ". And so the relationship of the radius to any vertex is r =", null, "ids. (ids = icosadodecahedron side).", null, "Figure 7 - showing DoGo = GoZ = radius,\nand the golden triangle Go-Z-Do\n\nWe can show this with the 'brute force' method as follows:", null, "Figure 8 -- deriving the relationship between radius (GoZ) and side (ZDo)\n\nZY is the side of the pentagon. Go is also the center of the pentagon.\nZT is just one half the side of the pentagon, because GoDo is a bisector of ZY.\nWe know the distance GoZ, Construction of the Pentagon Part 2 as", null, "side of pentagon.\n\nGoZ is also the radius of the enclosing sphere around the i.d.\nWe also know from this that GoT =", null, "side of pentagon.\n\nTherefore TDo =", null, "-", null, "=", null, "side of pentagon.\n\nWhat is the relationship between the side of the pentagon and the side of the decagon?\nLet's find ZDo, the side of the decagon, in terms of the side of the pentagon.\n\nZDo² = TDo² + TZ² =", null, "ZDo =", null, "side of pentagon.\nBut r =", null, "side of pentagon, so side of pentagon =", null, ". .\nTherefore, ZDo, the side of the decagon and the side of the i.d. =", null, "", null, "Now let’s resume the volume calculation. First we’ll calculate the volume of a pentagonal pyramid. Refer to Figure 9 below.\nFrom Area of Pentagon we know area =", null, ",\nand the distance mid-face to any vertex of a pentagon,\nHoF, =", null, "ids.\nGoF =", null, "ids. GoF is just the radius of the enclosing sphere", null, "Figure 9 -- one of the 12 pentagonal pyramids of the i.d.\n\nTo find the height of the pyramid, take right triangle GoHoF.\nh² = GoHo² = GoF² - HoF²", null, "h =", null, "", null, "That takes care of the pentagonal pyramids. There are still 20 triangular pyramids.\nEach triangular face area =", null, "and the height is as follows:", null, "Figure 10 - a triangular pyramid of the icosadodecahedron\n\nTriangle GoZF is right by construction.\nGoF =", null, "ids, being the radius of the enclosing sphere.\nZF is, from Equilateral Triangle", null, "Therefore h² = GoZ² = GoF² - ZF² =", null, "h =", null, "So", null, "= 1/3 * area of base * pyramid height =", null, "=", null, "", null, "", null, "This figure is slightly larger than the volume of the rhombic triacontahedron. This larger figure is easily explained by the fact that both the outer radial distance of the r.t., and the radius of the i.d., are", null, "* side. Since ALL vertices of the icosadodecahedron lie on this sphere, and only 12 of the vertices of the rhombic triacontahedron do, the volume of the icosadodecahedron is naturally larger.\n\nWhat is the surface area of the icosadodecahedron?\nIt is 12 * area of pentagonal face + 20 * area triangular face =", null, "What is the central angle of the icosadodecahedron?\nBecause all of the vertices of the i.d lie along the 6 'great circle' decagons, the interior angles are all 360° / 10 = 36°. Notice that each of the central angles to adjacent vertices forms a Golden Triangle with angles 36°, 72° and 72°.\nWe saw earlier how the central angle between two diagonal vertices on any of the pentagonal faces, is 60° and how this forms a hexagon, and thus large internal equilateral triangles (see Figure 5 and triangles BGoE, LBGo, GoES, LAoGo, AoGoCo, GoCoS). We were curious as to how the side of the hexagonal plane could be the same length as the side of the pentagonal plane. If you look at Figure 6 you can see that the sides of both of these planes are a diagonal of any of the pentagonal faces. It is clear then, that like the rhombic triacontahedron and the icosahedron, the icosadodecahedron is based on the pentagon.\nBecause the length of the radius (distance from centroid to any vertex) is", null, "* side of i.d., and the diagonal of any pentagon is also", null, "* side of i.d., the hexagonal plane is formed .\n\nWhat are the surface angles of the i.d.?\nThere are 2 of them. One is the angle forming the triangular faces, equal to 60°, and the other is the angle forming the pentagonal faces, equal to 108°.\n\nWhat is the dihedral angle of the icosadodeccahedron?\nFor this calculation, go to IcosaDodecahedron Dihedral Angle .\n\nLet's collect some information we have already calculated:\nDistance from centroid to a vertex = radius =", null, "ids.\nDistance from centroid to mid-pentagonal face =", null, "Distance from centroid to mid-triangular face =", null, "Distance from centroid to mid-edge =", null, "Or, in decimals,\nDistance from centroid to a vertex = radius = 1.618033989 ids.\nDistance from centroid to mid-pentagonal face = 1.376381921 ids.\nDistance from centroid to mid-triangular face = 1.511522629 ids.\nDistance from centroid to mid-edge = 1.538841769 ids.\n\nNow let's calculate the separation of the various planes in the icosadodecahedron.\nWe want to get the distances between Eo, Fo, Go, Io and Ho.\nLet's first get EoFo.", null, "Figure 3, repeated", null, "Figure 11a. Top view, refer to Figure 3", null, "Figure 11b The distance between the first 2 pentagonal planes (EoFo) of the icosadodecahedron. Note that although OXEo appears to be a straight line in Figure 11a, it is actually the dihedral angle of the solid.\n\nThe top pentagonal plane on the i.d. is the plane AJTFK, in purple in Figure 11a and the top plane in Figure 3. The large pentagonal plane UPBEO is marked in orange.\nThe point Eo is the center of the small pentagon, Fo is the center of the large pentagon.\nAJTFK is on top of UPBEO.\nThe angle FoEoX is right. EoFo is the line perpendicular to the plane AJTFK from the center of the plane UPBEO, and is part of the diameter of the sphere which runs through the centroid O.\nThe angle OFoEo is also right, the line EoFo is perpendicular to the plane UPBEO.\nOX and XEo can be easily calculated.\nOX is just the height of an equilateral triangle,", null, ", and XEo is the distance, in a pentagon, from the center to any mid-edge. We know from Construction of the Pentagon Part 2 that this distance is", null, "The distance OFo is just the distance from the center of the large pentagon, to one of its vertexes. From Construction of the Pentagon Part 2 we know this distance to be", null, " side of the large pentagon.\n\nWe may draw a perpendicular bisector from X to N on the line OFo.", null, "EoXN and", null, "XNFo are both right.\nWe now have a rectangle XNFoEo with 4 right angles at the corners, and XN = EoFo, XEo = NFo.\nNow we can find the distance ON, for it is just OFo - NFo.\nThen we can work with triangle ONX to find XN and we have EoFO.\n\nFirst we must find OFo in terms of the side of the icosadodecahedron.\nRemember that OFo is", null, "* side of the large pentagon. But the side of the large pentagon is just the diagonal of any the pentagonal faces of the i.d.\nTherefore the side of the large pentagon is", null, "* ids.\nWe can then write OFo =", null, "(Note that this distance is precisely the distance from the centroid to the mid-face of any of the pentagonal faces!)\n\nNFo = XEo =", null, "Then ON = OFo - NFo", null, "=", null, "From this we gather that ON = NFo.\n\nXN² = OX² - ON² =", null, "XN = EoFo =", null, "Therefore the distance between the first two pentagonal planes is", null, "Now we can easily find FoGo, the distance between the large pentagonal plane and the plane of the decagon which contains the centroid.\nThis distance is just EoGo - EoFo.\nEoGo is the distance from the centroid to mid-face of pentagon,\nwhich we found above to be", null, "So FoGo =", null, "=", null, "What is the relationship between EoFo, FoGo and EoGo?\nEoGo / FoGo =", null, "Therefore the radius EoGo is divided in Mean and Extreme Ratio at Fo.\nWe can make a table of relationships of the distances between the pentagonal planes. If we let EoFo = 1, then\n\n(Available in the book)\n\nWhat are the distances between the triangular planes and the hexagonal plane which runs through the centroid?\n(Refer back to Figure 6A above).\nWe want EoFo.\nThe large equilateral triangle DoIF in Figure 6A has sides equal to the distance DoF = FI = DoI in Figure 12:", null, "Figure 12 Showing the length of the sides of the large triangular planes\n\nYou can see this in Figure 3 by examining the vertices Do, Y, S, I which are part of one of the 'great circle' decagons. The length of each side of the triangular plane is DI. The other two legs of this large equilateral triangle are each part of a different decagon.\nFrom Decagon we know the edge length of each side of triangle DoIF is", null, "ids. You can see this in Figure 6A by examining the vertices Do, Y, S, I which are part of one of the 'great circle' decagons. The other two legs of this large equilateral triangle are also part of a different decagon.", null, "Figure 6A, repeated\n\nR’ is the center of the triangular face DoIF. T’ is the center of the triangular face CXV, and O’ is the centroid. We want to find R’T’, the distance between the two triangular planes. We will first find O’R’.\n\nThe solution is quite simple, actually. The distance DoR’ is known. DoR’ is the distance from the center of an equilateral triangle to an i.d. vertex. O’R is also known. O’R is just the distance from the centroid to a vertex of the i.d. R’ lies directly above O’. Therefore the triangle O’R’Do is right and we may write", null, ".\nO’Do =", null, "*ids, and from we know that R’Do =", null, ".", null, "", null, "Because the analysis is precisely the same for O’T’, the distance O’T’ also equals", null, ", and\nR’T’ =", null, "What is the distance Q’R’? This is the distance between the triangular face OJT and the plane DoIF. The analysis is the same as above, except that the side of the triangular plane OJT is now ids, instead of", null, "(Why? Because OJT is a face of the i.d.) Therefore", null, "Q’R’ / O’R’ =", null, "What is the angle any of the hexagonal planes make with the plane of the decagon?\nIn Figure 6A the hexagonal plane is highlighted in purple.\n\nThis can be seen by observing the plane of the hexagon as it intersects the plane of the decagon in Figure 6.", null, "Figure 6, repeated\nNotice that", null, "OGoS is the angle of the hexagonal plane as it intersects with the central decagon, and that is it also a central angle of the i.d. This angle, of course, is 360 / 10 = 36°.\n\nConclusions:\n\nWhat have we learned about the icosadodecahedron?\nMainly, that it is pentagonally based.\nThe triangular faces fill in the gaps between the pentagonal faces.\nAll of the radii form Golden Triangles with any 2 adjacent vertices.\nThe spacing of the pentagonal and decagonal internal planes are based upon the division of the space into Mean and Extreme Ratio.", null, "" ]	[ null, "https://www.kjmaclean.com/Geometry/IcosaDodeca_files/image002.jpg", null, "https://www.kjmaclean.com/Geometry/IcosaDodeca_files/image004.jpg", null, "https://www.kjmaclean.com/Geometry/IcosaDodeca_files/image006.jpg", null, "https://www.kjmaclean.com/Geometry/IcosaDodeca_files/image008.jpg", null, "https://www.kjmaclean.com/Geometry/IcosaDodeca_files/image010.jpg", null, "https://www.kjmaclean.com/Geometry/IcosaDodeca_files/image012.gif", null, "https://www.kjmaclean.com/Geometry/IcosaDodeca_files/image014.jpg", null, 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http://www.fastchip.net/howcomputerswork/p29.html	[ "Next, we will consider how to make a program that adds two numbers, from 0 to 9 together.\n\nAgain, we will represent the numbers 0 through 9 with only 1's and 0's as indicated in the following table.\n\n## Table 6\n\n``` Number Representation\n0 0000\n1 0001\n2 0010\n3 0011\n4 0100\n5 0101\n6 0110\n7 0111\n8 1000\n9 1001\n```\n\nNext, we need a table that shows the answer for each possible pair of numbers from 0 to 9. This is the addition table we studied so hard to learn in grade school and is reproduced below.\n\n``` + 0 1 2 3 4 5 6 7 8 9\n\n0 0 1 2 3 4 5 6 7 8 9\n1 1 2 3 4 5 6 7 8 9 10\n2 2 3 4 5 6 7 8 9 10 11\n3 3 4 5 6 7 8 9 10 11 12\n4 4 5 6 7 8 9 10 11 12 13\n5 5 6 7 8 9 10 11 12 13 14\n6 6 7 8 9 10 11 12 13 14 15\n7 7 8 9 10 11 12 13 14 15 16\n8 8 9 10 11 12 13 14 15 16 17\n9 9 10 11 12 13 14 15 16 17 18\n```\n\nNext, we rewrite the addition table above as below.\n\n``` 0 + 0 = 00\n0 + 1 = 01\n0 + 2 = 02\n0 + 3 = 03\n0 + 4 = 04\n0 + 5 = 05\n0 + 6 = 06\n0 + 7 = 07\n0 + 8 = 08\n0 + 9 = 09\n1 + 0 = 01\n1 + 1 = 02\n\n.\n.\n.\n\n8 + 9 = 17\n9 + 0 = 09\n9 + 1 = 10\n9 + 2 = 11\n9 + 3 = 12\n9 + 4 = 13\n9 + 5 = 14\n9 + 6 = 15\n9 + 7 = 16\n9 + 8 = 17\n9 + 9 = 18\n```\n\nOnly some of the table elements are listed above to save space.\n\nPage 29\n\nPage 28 . . . Page 1 . . . Page 30" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7249861,"math_prob":0.9994165,"size":1223,"snap":"2021-31-2021-39","text_gpt3_token_len":628,"char_repetition_ratio":0.17063166,"word_repetition_ratio":0.17567568,"special_character_ratio":0.62551105,"punctuation_ratio":0.05899705,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99647194,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-07-31T02:41:48Z\",\"WARC-Record-ID\":\"<urn:uuid:cfaa772d-5417-410a-96cf-02a48d739236>\",\"Content-Length\":\"4304\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:bf033628-8640-4326-ae8f-5eac9206aebc>\",\"WARC-Concurrent-To\":\"<urn:uuid:efce52fb-0517-4f49-beb8-9ce0c1c9df0b>\",\"WARC-IP-Address\":\"67.195.197.25\",\"WARC-Target-URI\":\"http://www.fastchip.net/howcomputerswork/p29.html\",\"WARC-Payload-Digest\":\"sha1:H6ZLMHDNWRWYYYS5SWDCBC3F7HYSXKKI\",\"WARC-Block-Digest\":\"sha1:SBQRI2UJ2XDMAYG7NLEZHMANJI6ZGBIC\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-31/CC-MAIN-2021-31_segments_1627046154042.23_warc_CC-MAIN-20210731011529-20210731041529-00472.warc.gz\"}"}
https://people.sc.fsu.edu/~jburkardt/m_src/test_interp/test_interp.html	[ "# test_interp\n\ntest_interp, a MATLAB code which defines data that may be used to test interpolation algorithms.\n\nThe following sets of data are available:\n\n1. p01_plot.png, 18 data points, 2 dimensions. This example is due to Hans-Joerg Wenz. It is an example of good data, which is dense enough in areas where the expected curvature of the interpolant is large. Good results can be expected with almost any reasonable interpolation method.\n2. p02_plot.png, 18 data points, 2 dimensions. This example is due to ETY Lee of Boeing. Data near the corners is more dense than in regions of small curvature. A local interpolation method will produce a more plausible interpolant than a nonlocal interpolation method, such as cubic splines.\n3. p03_plot.png, 11 data points, 2 dimensions. This example is due to Fred Fritsch and Ralph Carlson. This data can cause problems for interpolation methods. There are sudden changes in direction, and at the same time, sparsely-placed data. This can cause an interpolant to overshoot the data in a way that seems implausible.\n4. p04_plot.png, 8 data points, 2 dimensions. This example is due to Larry Irvine, Samuel Marin and Philip Smith. This data can cause problems for interpolation methods. There are sudden changes in direction, and at the same time, sparsely-placed data. This can cause an interpolant to overshoot the data in a way that seems implausible.\n5. p05_plot.png, 9 data points, 2 dimensions. This example is due to Larry Irvine, Samuel Marin and Philip Smith. This data can cause problems for interpolation methods. There are sudden changes in direction, and at the same time, sparsely-placed data. This can cause an interpolant to overshoot the data in a way that seems implausible.\n6. p06_plot.png, 49 data points, 2 dimensions. The data is due to deBoor and Rice. The data represents a temperature dependent property of titanium. The data has been used extensively as an example in spline approximation with variably-spaced knots. DeBoor considers two sets of knots: (595,675,755,835,915,995,1075) and (595,725,850,910,975,1040,1075).\n7. p07_plot.png, 4 data points, 2 dimensions. The data is a simple symmetric set of 4 points, for which it is interesting to develop the Shepard interpolants for varying values of the exponent p.\n8. p08_plot.png, 12 data points, 2 dimensions. This is equally spaced data for y = x^2, except for one extra point whose x value is close to another, but whose y value is not so close. A small disagreement in nearby data can become a disaster.\n\n### Languages:\n\ntest_interp is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.\n\n### Related Data and Programs:\n\ndivdif, a MATLAB code which includes many routines to construct and evaluate divided difference interpolants.\n\nhermite_interpolant, a MATLAB code which computes the Hermite interpolant, a polynomial that matches function values and derivatives.\n\ninterp, a MATLAB code which can be used for parameterizing and interpolating data;\n\ninterpolation, a dataset directory which contains datasets to be interpolated.\n\nlagrange_interp_1d, a MATLAB code which defines and evaluates the lagrange polynomial p(x) which interpolates a set of data, so that p(x(i)) = y(i).\n\nnewton_interp_1d, a MATLAB code which finds a polynomial interpolant to data using newton divided differences.\n\nr8lib, a MATLAB code which contains many utility routines using double precision real (r8) arithmetic.\n\nrbf_interp_1d, a MATLAB code which defines and evaluates radial basis function (rbf) interpolants to 1d data.\n\nshepard_interp_1d, a MATLAB code which defines and evaluates shepard interpolants to 1d data, which are based on inverse distance weighting.\n\nspline, a MATLAB code which includes many routines to construct and evaluate spline interpolants and approximants.\n\ntest_interp_2d, a MATLAB code which defines test problems for interpolation of data z(x,y)), depending on a 2d argument.\n\nvandermonde_interp_1d, a MATLAB code which finds a polynomial interpolant to a function of 1d data by setting up and solving a linear system for the polynomial coefficients, involving the vandermonde matrix.\n\n### Reference:\n\n1. Carl DeBoor, John Rice,\nLeast-squares cubic spline approximation II - variable knots.\nTechnical Report CSD TR 21,\nPurdue University, Lafayette, Indiana, 1968.\n2. Carl DeBoor,\nA Practical Guide to Splines,\nSpringer, 2001,\nISBN: 0387953663,\nLC: QA1.A647.v27.\n3. Fred Fritsch, Ralph Carlson,\nMonotone Piecewise Cubic Interpolation,\nSIAM Journal on Numerical Analysis,\nVolume 17, Number 2, April 1980, pages 238-246.\n4. Larry Irvine, Samuel Marin, Philip Smith,\nConstrained Interpolation and Smoothing,\nConstructive Approximation,\nVolume 2, Number 1, December 1986, pages 129-151.\n5. ETY Lee,\nChoosing Nodes in Parametric Curve Interpolation,\nComputer-Aided Design,\nVolume 21, Number 6, July/August 1989, pages 363-370.\n6. Hans-Joerg Wenz,\nInterpolation of Curve Data by Blended Generalized Circles,\nComputer Aided Geometric Design,\nVolume 13, Number 8, November 1996, pages 673-680.\n\n### Source Code:\n\nLast revised on 28 March 2019." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.75377494,"math_prob":0.9427853,"size":7618,"snap":"2021-43-2021-49","text_gpt3_token_len":1937,"char_repetition_ratio":0.2072498,"word_repetition_ratio":0.2248927,"special_character_ratio":0.26082963,"punctuation_ratio":0.19758065,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98961264,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-22T04:31:03Z\",\"WARC-Record-ID\":\"<urn:uuid:7befbdf5-8d99-4aa1-bcbf-39a538c30aa6>\",\"Content-Length\":\"15112\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:39e2984d-0ab0-4968-928b-05a542b19cf0>\",\"WARC-Concurrent-To\":\"<urn:uuid:52ddafb4-74c8-4e5b-86d6-adef71d8fc90>\",\"WARC-IP-Address\":\"144.174.16.102\",\"WARC-Target-URI\":\"https://people.sc.fsu.edu/~jburkardt/m_src/test_interp/test_interp.html\",\"WARC-Payload-Digest\":\"sha1:VBEVYRLQ776JPSA3MFZTC6SLMWU2UUOT\",\"WARC-Block-Digest\":\"sha1:7RGJMPOJQ5GRVFNXKJ53J3FOMKAPU7I6\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323585450.39_warc_CC-MAIN-20211022021705-20211022051705-00107.warc.gz\"}"}
https://www.unitsconverters.com/en/Moseley-Proportionality-Constant-Conversions/Measurement-1204	[ "Formula Used\n1 Hertz^(1 per 2) = 0.1 Hectohertz^(1 per 2)\n1 Hertz^(1 per 2) = 0.1 Hectohertz^(1 per 2)\n\n## Importance of Moseley Proportionality Constant converter\n\nMeasurement of various quantities has been an integral part of our lives since ancient times. In this modern era of automation, we need to measure quantities more so than ever. So, what is the importance of Moseley Proportionality Constant converter? The purpose of Moseley Proportionality Constant converter is to provide Moseley Proportionality Constant in the unit that you require irrespective of the unit in which Moseley Proportionality Constant was previously defined. Conversion of these quantities is equally important as measuring them. Moseley Proportionality Constant conversion helps in converting different units of Moseley Proportionality Constant. Moseley proportionality constant is a constant used in Moseley's Law concerning the characteristic x-rays emitted by atoms.. There are various units which help us define Moseley Proportionality Constant and we can convert the units according to our requirement. unitsconverters.com provides a simple tool that gives you conversion of Moseley Proportionality Constant from one unit to another.\n\nWhat is Moseley Proportionality Constant?\nMoseley proportionality constant is a constant used in Moseley's Law concerning the characteristic x-rays emitted by atoms.\nWhat is the SI unit for Moseley Proportionality Constant?\nHertz^(1 per 2) (Hz^(1/2)) is the SI unit for Moseley Proportionality Constant. SI stands for International System of Units.\nWhat is the biggest unit for Moseley Proportionality Constant?\nMegahertz^(1 per 2) is the biggest unit for Moseley Proportionality Constant. It is 1000 times bigger than Hertz^(1 per 2).\nWhat is the smallest unit for Moseley Proportionality Constant?\nPicohertz^(1 per 2) is the smallest unit for Moseley Proportionality Constant. It is 1E-06 times smaller than Hertz^(1 per 2).", null, "Let Others Know" ]	[ null, "https://www.unitsconverters.com/image/share.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.91888535,"math_prob":0.86753947,"size":1056,"snap":"2021-43-2021-49","text_gpt3_token_len":203,"char_repetition_ratio":0.21673004,"word_repetition_ratio":0.0,"special_character_ratio":0.15530303,"punctuation_ratio":0.07878788,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99336106,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-15T21:52:50Z\",\"WARC-Record-ID\":\"<urn:uuid:23e6404e-fada-4105-9fd9-66765066db05>\",\"Content-Length\":\"92290\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:fc5d44fa-f5a8-402f-8dc3-51add52a47d9>\",\"WARC-Concurrent-To\":\"<urn:uuid:4dfcbb86-b0b2-4f91-a966-6fd9f3d521f3>\",\"WARC-IP-Address\":\"67.43.15.151\",\"WARC-Target-URI\":\"https://www.unitsconverters.com/en/Moseley-Proportionality-Constant-Conversions/Measurement-1204\",\"WARC-Payload-Digest\":\"sha1:ABIPJQLMYJPV4FE2PZQGGQ4SLGHK3NAX\",\"WARC-Block-Digest\":\"sha1:RKL6IBEMZZUG7KCFB2USETYRAWTCJYYD\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323583083.92_warc_CC-MAIN-20211015192439-20211015222439-00471.warc.gz\"}"}
http://www.learnersplanet.com/uses-of-metals-and-nonmetals-worksheet-10th-cbse	[ "# Metals and non metals Worksheet-4\n\nMetals and non metals Worksheet-4\n\nMultiple-Choice Question (One Option Correct):\n\n1. What type of oxide is formed by metals.\n\n(A) Acidic (B) Basic (C) Amphoteric (D) A & B\n\n(E) B and C\n\n1. What is the product formed when sodium is brunt in air.\n\n(A) Na2O (B) Na2O2 (C) NaO (D) NaO2\n\n1. The reaction of sodium and potassium with oxygen is\n\n(A) Exothermic (C) Endothermic\n\n(C) No heat change occurs. (D) Depends on the state of metal\n\n1. What are the products formed when Magnesium oxide MgO is dissolved in water?\n\n(A) Mg(OH)2 (B) MgOH\n\n(C) Mg2+ + H+ + OH+– (D) MgO(aq) + H2O(l)\n\n1. What type of oxide is formed by Aluminium ?\n\n(A) Acidic (B) Basic (C) Amphoteric (D) B and C\n\n1. A black substance is formed on the surface copper on prolonged heating in air. What is the chemical formula of that black substance?\n\n(A) CuO (B) Cu2O (C) CuO2 (D) All of these\n\n1. What is the product formed when iron is heated strongly in air?\n\n(A) FeO (B) Fe2O3 (C) Fe3O4 (D) Fe2O3.XH2O\n\n1. Why does Aluminium oxide (Al2O3) behave as an acidic oxide?\n\n(A) In water it forms Aluminium hydride (AlH3) which looses H+ ions.\n\n(B) It reacts with base giving salt and water.\n\n(C) Both (A) and (B)\n\n(D) It turns blue litmus red .\n\n1. What are the products of the given reaction", null, "(A) ZnCl2(aq) + H2O(l) (B) ZnCl2(aq) + H+ + OH\n\n(C) Zn2Cl + H2O(l) (D) ZnCl + H2O(l)\n\n1. Why calcium starts floating on the surface when it is treated with water?\n\n(A) Ca(OH)2 formed is lighter than water so it floats on the surface.\n\n(B) Bubbles of hydrogen gas formed stick to the surface of metal.\n\n(C) CaO formed floats on the surface.\n\n(D) All of these\n\n1. (A)\n\n2. (A)\n\n3. (A)\n\n4. (A)", null, "5. (C) Aluminium oxide reacts with acid as well as base forming salt and water", null, "1. (A) Copper forms copper (II) oxide on prolonged heating in air.", null, "1. (C) Iron reacts with the oxygen of air on heating to form iron (I,II) Oxide.", null, "1. (B)", null, "2. (A)", null, "3. (B) Ca(s) + 2H2O → Ca(OH)2(aq) + H2(g)" ]	[ null, "http://www.learnersplanet.com/sites/default/files/images/Worksheet-images/Class-10-CBSE-chemistry-Metals-nonmetals/image1001.jpg", null, "http://www.learnersplanet.com/sites/default/files/images/Worksheet-images/Class-10-CBSE-chemistry-Metals-nonmetals/image1002.jpg", null, "http://www.learnersplanet.com/sites/default/files/images/Worksheet-images/Class-10-CBSE-chemistry-Metals-nonmetals/image1003.jpg", null, "http://www.learnersplanet.com/sites/default/files/images/Worksheet-images/Class-10-CBSE-chemistry-Metals-nonmetals/image1004.jpg", null, "http://www.learnersplanet.com/sites/default/files/images/Worksheet-images/Class-10-CBSE-chemistry-Metals-nonmetals/image1005.jpg", null, "http://www.learnersplanet.com/sites/default/files/images/Worksheet-images/Class-10-CBSE-chemistry-Metals-nonmetals/image1006.jpg", null, "http://www.learnersplanet.com/sites/default/files/images/Worksheet-images/Class-10-CBSE-chemistry-Metals-nonmetals/image1007.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.71570206,"math_prob":0.9262227,"size":1199,"snap":"2019-51-2020-05","text_gpt3_token_len":462,"char_repetition_ratio":0.117154814,"word_repetition_ratio":0.026200874,"special_character_ratio":0.36280233,"punctuation_ratio":0.0513834,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9757275,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"im_url_duplicate_count":[null,1,null,1,null,1,null,1,null,1,null,1,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-01-28T00:39:38Z\",\"WARC-Record-ID\":\"<urn:uuid:fe2faa8d-4ffb-4bcb-9cd6-66666b038291>\",\"Content-Length\":\"48477\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:91ee9839-1945-46d4-a9db-c031b6ca226f>\",\"WARC-Concurrent-To\":\"<urn:uuid:1a5a690a-59eb-424c-931c-3cf8974836ef>\",\"WARC-IP-Address\":\"159.69.138.115\",\"WARC-Target-URI\":\"http://www.learnersplanet.com/uses-of-metals-and-nonmetals-worksheet-10th-cbse\",\"WARC-Payload-Digest\":\"sha1:DKGNWGCRYIY2DK7UHG4NKS56BGKKBZ3C\",\"WARC-Block-Digest\":\"sha1:PBJZYALQ6GTAQRFHEWMSWHD45KABDKLZ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-05/CC-MAIN-2020-05_segments_1579251737572.61_warc_CC-MAIN-20200127235617-20200128025617-00288.warc.gz\"}"}
https://artofproblemsolving.com/wiki/index.php/1960_AHSME_Problems/Problem_14	[ "# 1960 AHSME Problems/Problem 14\n\n## Problem\n\nIf", null, "$a$ and", null, "$b$ are real numbers, the equation", null, "$3x-5+a=bx+1$ has a unique solution", null, "$x$ [The symbol", null, "$a \\neq 0$ means that", null, "$a$ is different from zero]:", null, "$\\text{(A) for all a and b} \\qquad \\text{(B) if a }\\neq\\text{2b}\\qquad \\text{(C) if a }\\neq 6\\qquad \\\\ \\text{(D) if b }\\neq 0\\qquad \\text{(E) if b }\\neq 3$\n\n## Solution\n\nIf the coefficients of the x-terms are equal on both sides, then when the x-terms are subtracted from both sides, the equation results in a number equals 1.\n\nThis means the equation has either infinite or no solutions, so the x-terms can not be equal on both sides. Thus,", null, "$b \\neq 3$, so the answer is", null, "$\\boxed{\\textbf{(E)}}$." ]	[ null, "https://latex.artofproblemsolving.com/c/7/d/c7d457e388298246adb06c587bccd419ea67f7e8.png ", null, "https://latex.artofproblemsolving.com/8/1/3/8136a7ef6a03334a7246df9097e5bcc31ba33fd2.png ", null, "https://latex.artofproblemsolving.com/0/a/c/0acfc6ac7b525bfe1913d90bc5396a762b7add3c.png ", null, "https://latex.artofproblemsolving.com/2/6/e/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png ", null, "https://latex.artofproblemsolving.com/6/7/a/67a8181c9d9b8253bd721b620aaabc5b01f4af96.png ", null, "https://latex.artofproblemsolving.com/c/7/d/c7d457e388298246adb06c587bccd419ea67f7e8.png ", null, "https://latex.artofproblemsolving.com/d/9/d/d9d9ebc461f9c275daeef1049c2d73e0f74fa61e.png ", null, "https://latex.artofproblemsolving.com/5/1/2/512e99214146a00703832fa3537a709677fa182e.png ", null, "https://latex.artofproblemsolving.com/6/b/f/6bfc0829f3f29bbf35d0a861644e00cfef0f50ca.png ", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.660762,"math_prob":0.9999989,"size":775,"snap":"2023-40-2023-50","text_gpt3_token_len":270,"char_repetition_ratio":0.12710765,"word_repetition_ratio":0.0,"special_character_ratio":0.42709678,"punctuation_ratio":0.06535948,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000019,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],"im_url_duplicate_count":[null,null,null,null,null,7,null,null,null,null,null,null,null,7,null,7,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-11-28T11:00:38Z\",\"WARC-Record-ID\":\"<urn:uuid:a1af11f4-092b-451b-a2bb-fd6160591ed4>\",\"Content-Length\":\"41440\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a8be5d9d-8403-4173-a583-22be2bcc893b>\",\"WARC-Concurrent-To\":\"<urn:uuid:b8e8c43c-d690-419d-a9eb-add7140f2330>\",\"WARC-IP-Address\":\"104.26.11.229\",\"WARC-Target-URI\":\"https://artofproblemsolving.com/wiki/index.php/1960_AHSME_Problems/Problem_14\",\"WARC-Payload-Digest\":\"sha1:UOTEII2YPG2XYTEWZADKOMSCPUH53Q7F\",\"WARC-Block-Digest\":\"sha1:XL4QFNMO3IQ6WU3PD6RFOLJ6B4NQUAZU\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679099281.67_warc_CC-MAIN-20231128083443-20231128113443-00699.warc.gz\"}"}
http://datascience.sharerecipe.net/2019/03/03/probabilistic-graphical-models-bayesian-networks/	[ "# Probabilistic Graphical Models: Bayesian Networks\n\nThe venue, cuisine, distance from home, pricing etc.\n\nIn general, we can write a custom program to answer our query (a nested if-else), but that will not be robust.\n\nIf it encounters an additional feature, we might have to re-write the model, which certainly seems a less feasible solution.\n\nTo overcome the aforementioned complication, we will use a different approach: Declarative Representation.\n\nIn this paradigm, we construct a model based on the task which would like to reason.\n\nModel encodes the knowledge of how the system works.\n\nKey features of this methodology are to separate knowledge and reasoning.\n\nDecision-making in real-world applications come with a level of uncertainty which corresponds to an uncertainty in the model building.\n\nThat's where the Probability comes in.\n\nThe math of probability theory provides us with a framework for considering multiple outcomes and their likelihood.\n\nConcepts of ProbabilityAxioms of Probability:For any event A, the probability of occurrence of the event will always be equal to or greater than zero.\n\nFigure-1: Probability of an event A2.\n\nIf there are disjoint events in a sample space, then the union of all events is the summation of individual probabilities.\n\nFigure-2: Union of all Disjoint events3.\n\nIn case of an event involving the universal set has the probability of 1.\n\nRandom Variable: A random variable is a function which maps each outcome in sample space to a value.\n\nFigure-3: G, H an A are random variables mapping to outcomes.\n\nMarginal DistributionAfter defining our random variable, we can consider the distribution of events which can be described using it.\n\nThis distribution is often referred to as Marginal Distribution over a random variable (let's say X).\n\nWe denote it by P(X).\n\nFigure-4: Marginal Distribution of a random variable, GJoint DistributionIn many situations, we want to involve several random variables.\n\nEach random variable corresponds to a certain attribute of an event.\n\nIn real-world problems, We are interested to find probability involving multiple events occurring together, which can be represented by Joint Distribution.\n\nIn general, if we have a set of events: {X1, X2, X3,…, Xn} then the joint distribution is denoted by P(X1, X2, X3,…, Xn)Figure-5: Joint Distribution of two random variables, G and I.\n\nJoint Distribution is fundamental as it helps us query information regarding the data in terms of marginal or conditional distribution.\n\nIn other words, we can derive the marginal and conditional distribution from it.\n\nConditional ProbabilityThis type of probability distribution involves prior knowledge of a random variable which affects the probability of occurrence of the target variable.\n\nIt is usually denoted by P(X\|Y).\n\nThis can be interpreted as ‘Probability of X given Y has already occurred’.\n\nIt can be factorized in terms of Marginal and Joint Distributions.\n\nFigure-6: Factorization of Conditional ProbabilityConditional IndependenceAs we mentioned above that P(X\|Y) is not equal to P(X), learning that Y is true changes our probability over X.\n\nHowever in some cases equality can occur which means that learning Y does not change our probability of X.\n\nConditional Independence is a more common situation when two events are independent given an additional event.\n\nWe say an event X is conditionally independent of event Y given an event Z denoted as P(X \| Y, Z) = P(X\|Z)Shortcomings of Joint ProbabilityFigure-7: Joint Distribution of n random variablesThe problem of using Joint Distribution for inference is that it is too complex to handle, and we have to use a Chain Rule with all dependencies to parameterize it.\n\nFigure-8: Chain Rule expansion of Joint Distribution of Fig.\n\n7Assuming that each random variable takes up a binary value, joint distribution needs 2^n -1 values, which is computationally expensive and from a statistical point of view need huge data to learn parameters.\n\nThere is a natural way to split up the joint distribution, i.\n\ne to generate a marginal and a conditional distribution from it which makes it more interpretable than parent table.\n\nThis process is known as Conditional Parameterization.\n\n(But it does not reduce the number of parameters!)In the real world scenario, we make certain assumptions regarding the random variables involved in the joint distribution, whether or not they are dependent on each other or not.\n\nIn that case, Chain rule simplifies as follows:Figure-9: Chain Rule Simplification.\n\nIn the upcoming section, we will discuss to include independence among the features and one of the ways to graphically denote the Joint Distribution.\n\nBayesian NetworksUntil now, we saw that if we add conditional independence in the distribution, it largely simplifies the chain rule notation leading to less number of parameters to learn.\n\nBayesian Network can be viewed as a Data Structure ( It provides factorization of Joint Distribution)Suppose we have ’n’ random variables.\n\nall of which are independent given another random variable C.\n\nThe graphical representation is as follows:Figure-10: Naive Bayes ModelIn this case, we did a very naive assumption that all random variables are independent of each other, which highly simplifies the chain rule notation to represent the model.\n\nThis model is formally known as the Naive Bayes Model ( which is used as one of the Classification Algorithm in Machine Learning Domain).\n\nBayesian Network aids us in factorizing the joint distribution, which helps in decision making.\n\n(We started off with the idea of decision making, Remember?)Conventions involved:1.\n\nNodes: Random Variables2.\n\nEdges: Indicate DepdenceA Graph can be seen in two ways: A Data Structure which provides a skeleton for representing Joint Distribution in the factorized way or Representation of Conditional Independence about Distribution.\n\nFigure-11: Bayesian Network along with Local Probability ModelI have given an example of Decision making in terms of whether the student will receive a Recommendation Letter (L) based on various dependencies.\n\nGrade(G) is the parent node of Letter, We have assumed SAT Score(S) is based solely on/dependent on Intelligence(I).\n\nGrade is dependent on the Difficulty of Exam and Intelligence of Student.\n\nOne important point is that this is our assumption of how the world works, it may change for different perceptions, Some people might think SAT Score and Letter are dependent.\n\nSo in that case, the model looks like this:Each node is a random variable and has a local probability model associated with it.\n\nThis graph gives us a natural factorization for joint distribution.\n\nFigure-12: Factorized Term of Joint DistributionNow, one important point to note is that this model is built on our assumption on how the world works.\n\nMore finely put, This is the standard way of decision making we do.\n\nBut this might differ from person to person.\n\nSome people might think Recommendation Letter (L) depends on SAT Score(S) too.\n\nIn that case, the model becomes:Figure-13: Assumed ModelRule of Bayesian NetworksRule 1: A node is not independent of its parents.\n\nRule 2: A node is not independent of its parents even when we are given the values of other variables.\n\nRule 3: Given its parent node, a node is independent of all variables except its descendants node (child node).\n\nClosing RemarksBayesian Networks help us in decision making and simplify complex problems encoding various independencies.\n\nAs we read that Joint Distribution is not capable to give us an inference which is interpretable.\n\nFurthermore, Bayesian Networks (Acyclic Graphs) provide a basis for Restricted Boltzmann Machines.\n\nKushal Vala, Junior Data Scientist at Datametica Solutions Pvt 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https://personal.math.ubc.ca/~cass/courses/m308/projects/fung/page.html	[ "# KOCH'S SNOWFLAKE\n\nby Emily Fung", null, "The Koch Snowflake was created by the Swedish mathematician Niels Fabian Helge von Koch.", null, "In his 1904 paper entitled \"Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire\" he used the Koch Snowflake to show that it is possible to have figures that are continuous everywhere but differentiable nowhere.\n\n## The Koch Curve\n\nIn order to create the Koch Snowflake, von Koch began with the development of the Koch Curve. The Koch Curve starts with a straight line that is divided up into three equal parts. Using the middle segment as a base, an equilateral triangle is created. Finally, the base of the triangle is removed, leaving us with the first iteration of the Koch Curve.\n\n## The Koch Snowflake\n\nFrom the Koch Curve, comes the Koch Snowflake. Instead of one line, the snowflake begins with an equilateral triangle. The steps in creating the Koch Curve are then repeatedly applied to each side of the equilateral triangle, creating a \"snowflake\" shape.\n\nThe Koch Snowflake is an example of a figure that is self-similar, meaning it looks the same on any scale. In this picture the part of the figure in the red box is similar to the entire picture.\n\n## The Math Behind It\n\nNUMBER OF SIDES (n)\n\nFor each iteration, one side of the figure from the previous stage becomes four sides in the following stage. Since we begin with three sides, the formula for the number of sides in the Koch Snowflake is\n\nn = 34a\n\nin the ath iteration.\n\nFor iterations 0, 1, 2 and 3, the number of sides are 3, 12, 48 and 192, respectively.\n\nLENGTH OF A SIDE (length)\n\nIn every iteration, the length of a side is 1/3 the length of a side from the preceding stage. If we begin with an equilateral triangle with side length x, then the length of a side in interation a is\n\nlength = x3-a\n\nFor iterations 0 to 3, length = a, a/3, a/9 and a/27.\n\nPERIMETER (p)\n\nSince all the sides in every iteration of the Koch Snowflake is the same the perimeter is simply the number of sides multiplied by the length of a side\n\np = nlength\n\np = (34a)(x3-a)\n\nfor the ath iteration.\n\nAgain, for the first 4 iterations (0 to 3) the perimeter is 3a, 4a, 16a/3, and 64a/9.\n\nAs we can see, the perimeter increases by 4/3 times each iteration so we can rewrite the formula as\n\np = (3a)*(4/3)a\n\nSo as a-->infinity the perimeter continues to increase with no bound.\n\nAlso, as a-->infinity the snowflake is made up of sharp corners with no smooth lines connecting them. Therefore, while the perimeter of the snowflake, which is an infinite series, is continuous because there are no breaks in the perimeter, it is not differentiable since there are no smooth lines.\n\nREFERENCES:\n\nEric W. Weisstein. \"Koch Snowflake.\" From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/KochSnowflake.html\n\nTom VanCourt. \"Glencoe's Manual of Fuzz.\" The Textbook League. http://www.textbookleague.org/102fuzz.htm\n\n\"Niels Fabian Helge Von Koch.\" School of Mathematics and Statistics. University of St. Andrews, Scotland. http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Koch.html" ]	[ null, "https://personal.math.ubc.ca/~cass/courses/m308/projects/fung/KochSnowflakeTiling.gif", null, "https://personal.math.ubc.ca/~cass/courses/m308/projects/fung/Koch.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8804156,"math_prob":0.94435614,"size":2965,"snap":"2022-27-2022-33","text_gpt3_token_len":733,"char_repetition_ratio":0.14454576,"word_repetition_ratio":0.008213553,"special_character_ratio":0.23001686,"punctuation_ratio":0.13157895,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99432474,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,2,null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-08-14T21:58:01Z\",\"WARC-Record-ID\":\"<urn:uuid:d2d0556f-ef58-40b1-be9e-684b1dd78348>\",\"Content-Length\":\"4401\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:99c6b26d-cbc5-4b80-8501-edad1695c489>\",\"WARC-Concurrent-To\":\"<urn:uuid:80dcf220-04e6-43ec-aa37-44bb8a5cea87>\",\"WARC-IP-Address\":\"137.82.36.73\",\"WARC-Target-URI\":\"https://personal.math.ubc.ca/~cass/courses/m308/projects/fung/page.html\",\"WARC-Payload-Digest\":\"sha1:POFIXLDKV25Q6OI3C6TCV5IQ6C2JEWHB\",\"WARC-Block-Digest\":\"sha1:KCA7UBI3YPRSI5RA43ESHS2KOL2QSU3T\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-33/CC-MAIN-2022-33_segments_1659882572077.62_warc_CC-MAIN-20220814204141-20220814234141-00114.warc.gz\"}"}
https://link.springer.com/chapter/10.1007/978-3-030-03493-1_33	[ "# Crossover Operator Using Knowledge Transfer for the Firefighter Problem\n\nConference paper\nPart of the Lecture Notes in Computer Science book series (LNCS, volume 11314)\n\n## Abstract\n\nThis paper concerns the Firefighter Problem (FFP) which is a graph-based problem in which solutions can be represented as permutations. A new crossover operator is proposed that uses a machine learning model to decide how to combine two parent solutions of the FFP into an offspring. The operator works on two parent permutations and the machine learning model provides information which parent to select the next permutation element from, when constructing a new solution. Training data is collected during a training run in which transpositions are applied to solutions found by an evolutionary algorithm for a small problem instance. The machine learning model is trained to classify pairs of graph vertices into two classes corresponding to which vertex should be placed earlier in the permutation.\n\nIn the experiments the machine learning model was trained on a set of FFP instances with 1000 vertices. Subsequently, the proposed operator was used for solving FFP instances with up to 10000 vertices. The experiments, in which the proposed operator was compared against a set of other crossover operators, shown that the proposed operator is able to effectively use knowledge gathered when solving smaller instances for solving larger instances of the same problem.\n\n## Keywords\n\nKnowledge-based optimization Graph problems REDS graphs\n\n## References\n\n1. 1.\nTorrey, L., Shavlik, J.: Transfer learning. In: Olivas, E.S., et al. (eds.) Handbook of Research on Machine Learning Applications and Trends: Algorithms, Methods and Techniques, vol. 2. Information Science Reference - Imprint of: IGI Publishing, Hershey (2009)Google Scholar\n2. 2.\nYosinski, J., Clune, J., Bengio, Y., Lipson, H.: How transferable are features in deep neural networks? In: Proceedings of the 27th International Conference on Neural Information Processing Systems, NIPS 2014, vol. 2, pp. 3320–3328. MIT Press, Cambridge (2014)Google Scholar\n3. 3.\nHartnell, B.: Firefighter! An application of domination. In: 20th Conference on Numerical Mathematics and Computing (1995)Google Scholar\n4. 4.\nMichalak, K.: Estimation of distribution algorithms for the firefighter problem. In: Hu, B., López-Ibáñez, M. (eds.) EvoCOP 2017. LNCS, vol. 10197, pp. 108–123. Springer, Cham (2017).\n5. 5.\nVogl, T., Mangis, J., Rigler, A., Zink, W., Alkon, D.: Accelerating the convergence of the backpropagation method. Biol. Cybern. 59, 257–263 (1988)\n6. 6.\nBridle, J.S.: Probabilistic interpretation of feedforward classification network outputs, with relationships to statistical pattern recognition. In: Soulié, F.F., Hérault, J. (eds.) Neurocomputing. NATO ASI Series, vol. 68, pp. 227–236. Springer, Heidelberg (1990).\n7. 7.\nAntonioni, A., Bullock, S., Tomassini, M.: REDS: an energy-constrained spatial social network model. In: Lipson, H., et al. (eds.) ALIFE 2014. MIT Press (2014)Google Scholar\n8. 8.\nMichalak, K.: The Sim-EA algorithm with operator autoadaptation for the multiobjective firefighter problem. In: Ochoa, G., Chicano, F. (eds.) EvoCOP 2015. LNCS, vol. 9026, pp. 184–196. Springer, Cham (2015).\n9. 9.\nMoller, M.F.: A scaled conjugate gradient algorithm for fast supervised learning. Neural Netw. 6, 525–533 (1993)\n10. 10.\nArlot, S., Celisse, A.: A survey of cross-validation procedures for model selection. Stat. Surv. 4, 40–79 (2010)" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.710764,"math_prob":0.65970075,"size":3900,"snap":"2019-51-2020-05","text_gpt3_token_len":1000,"char_repetition_ratio":0.10857289,"word_repetition_ratio":0.0,"special_character_ratio":0.26564103,"punctuation_ratio":0.2544529,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9616088,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-01-25T05:02:34Z\",\"WARC-Record-ID\":\"<urn:uuid:472157a6-77a6-439b-a570-bb51997a2e03>\",\"Content-Length\":\"74371\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:aa341826-a8ff-4468-8424-c626d5a1b302>\",\"WARC-Concurrent-To\":\"<urn:uuid:0a860bed-1d25-41bb-8df9-ba4409421acb>\",\"WARC-IP-Address\":\"151.101.248.95\",\"WARC-Target-URI\":\"https://link.springer.com/chapter/10.1007/978-3-030-03493-1_33\",\"WARC-Payload-Digest\":\"sha1:HLXJ66QJL3TIX7XNFSKSYXYD3ZYFUGJV\",\"WARC-Block-Digest\":\"sha1:OR4U7Z3DTLU7A7KGSSQOVEZQZVUTEHZA\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-05/CC-MAIN-2020-05_segments_1579251669967.70_warc_CC-MAIN-20200125041318-20200125070318-00119.warc.gz\"}"}
https://stats.stackexchange.com/questions/50632/scoring-items-which-are-not-easily-compared/51076	[ "# Scoring items which are not easily compared\n\nFirst of all, I apologize since this question has probably been asked many times and is easily answered. However, as a statistics amateur I simply couldn't figure out what keywords are relevant to my question.\n\nSuppose you have 100 merchants and 100 products. Each merchant sells a certain range of products, ranging from only one product to all 100 products. Also, products are sold in widely different proportions, which differ among merchants, and are subject to the merchant's individual (irrational) preferences.\n\nWhenever a merchant makes a \"pitch\" on the market, we observe whether or not he manages to sell the product he's pitching. We assume the probability of success depends (a) on the skill of the merchant and (b) the attractiveness of the product. The products' prices are fixed, so that's not a factor.\n\nThe data we have consists of millions of pitches. For each pitch, we know whether or not it was successful, the merchant, and the product.\n\nObviously, if we compare merchants by their average success rate, this information is useless because every merchant sells different products. Likewise, if we compare products, we gain no information since every product is sold by different merchants.\n\nWhat we want is a skill score for each merchant, which is independent of the products the merchant is selling, and an attractiveness score for each product, which is independent of the merchants who are selling it.\n\nI don't need a comprehensive explanation, just some keywords to point me in the right direction. I literally have no idea where to start.\n\nEdit: Note that our assumption is that the product attractiveness is merchant-independent and the merchant skill is product-independent, i.e. there are no merchants which are better at selling certain products but worse at selling others.\n\n• Does this fall under unsupervised learning? We have combined effect (0/1) of Merchant skills and Product score (both ordinal say 1-100), and we don't observe (and have no idea) about the ordering of the 2 predictors. – steadyfish Feb 28 '13 at 19:25\n• You might want to check out, \" Conjoint Analysis \" – user21509 Mar 4 '13 at 3:41\n• Welcome to CV, @user21509. Would you care to expand on your answer? Why would conjoint analysis be useful here? Note that CV is not simply a Q&A site, but seeks to create a permanent repository of statistical information. – gung Mar 4 '13 at 3:43\n\nLet me expand on alternative solution proposed by @curious_cat.\n\n$P_{ij}$ is the matrix of pitches\n\n$L_{ij}$ is the matrix of sells\n\n$S_{ij} = L_{ij}/P_{ij}$ is the matrix of success rates (elementwise division where it exists and 0 elsewhere)\n\nAs @curious_cat suggested, you want to approximate $S_{ij}$ by the outer product of two positive vectors\n\n$$S_{ij} \\approx M_i \\times A_j^T$$\n\nLeast square minimization will lead to\n\n$$\\min \| S_{ij} - M_j \\times A_i^T \|_2$$ where $\| \\quad \|_2$ is the Frobenius norm.\n\nBUT you do not want to minimize for the entries in which $S_{ij}$ is not defined. So what you realy want is something like:\n\n$$\\min \|W_{ij} \\odot (S_{ij} - M_j \\times A_i^T)\|_2$$ where $\\odot$ is the elementwise multiplication.\n\n1) At a first approximation, $w_{ij}$ is 0 where $p_{ij}$ is 0 and 1 elsewhere.\n\nThis is a weighted non-negative matrix factorisation (or approximation) problem. Google should give some references to it.\n\n2) Now, shooting from the hip, let us try to answer the point also made by @curious_cat that you should trust more a success rate of 1000 sells over 2000 pitches than a 2 sells over 4 pitches.\n\nThe weight $w_{ij}$ need not to be uniformly 1 for the entries that are defined in $S_{ij}$. One can give it more weight to success rates with higher pitches.\n\nMy guess is to use $\\sqrt{p_{ij}}$ as the weight. The intuition is that the confidence interval on the success rate is inversely proportional to $\\sqrt{p_{ij}}$.\n\nThis type of problem is typically referred to in econometrics and marketing research as a \"choice modeling\" problem. Texts dealing with such problems include: Louviere, J., D. A. Hensher, et al. (2000). Stated Choice Methods: Analysis and Application. Cambridge, Cambridge University Press. Train, K. E. (2009). Discrete Choice Methods with Simulation. Cambridge, Cambridge University Press. Rossi, P. E., G. M. Allenby, et al. (2005). Bayesian Statistics and Marketing, Wiley.\n\nThe simplest practical model you could estimate would be a binary logit model with the dependent variable indicating when an object is purchased versus when it is not purchased, with two independent variables: a categorical variable for merchant and a categorical variable for product. (Or, if you do not know anything about when a product is not purchased, you could use Poisson regression or some other counts model.)\n\nThe parameter estimate for each merchant would be their skill score and the parameter for each product would be the \"attractiveness score\". The \"attractiveness\" score is more commonly referred to as a \"utility\" in choice modeling.\n\nA practical computational problem you will experience is that unless you have only a few hundred merchants and few hundred categorical variables you will struggle to estimate the model and may need a \"random effects\" model (sometimes referred to as a \"hierarchical model\" in this context).\n\nIn addition to the assumption that you mention, a key set of assumptions that will determine the validity of your analysis will relate to which alternatives are available at a given time. For example, a product that is intrinsically unattractive may be purchased regularly because the more attractive products are not available at the purchase time. This effect can have a very large impact upon your resulting estimates, as when it is ignored you inadvertently will confound the attractiveness of a product with its availability. The texts cited earlier discuss various modifications of choice models to deal with many of the types of assumptions likely relevant to your problem.\n\n• If you try to do a binary response logit in this case won't you get a huge number of duplicates? Would that be a problem? – curious_cat Mar 1 '13 at 5:13\n• The question asks \"For each pitch, we know whether or not it was successful, the merchant, and the product.\" If the recipient of the pitch declines and leaves the market, then I don't think there is a problem. If we think multiple products were compared then we need a multinomial logit model. If we think that the person does not buy because they anticipate there could be something better then we have a much harder problem. – Tim Mar 5 '13 at 21:02\n\nYour problem can be modeled by a Rasch Model. Here is a document that explains the model with the following example\n\nRasch model is a statistical model of a test that attempts to describe the probability that a student answers a question correctly. It assigns to every student a real number, a, called the \"ability\", and to every questions a real number, d, called the \"difficulty\".\n\nThis is similar to your situation where each merchant has some inherent \"skill\" and each product has an inherent \"attractiveness\".\n\nWhy not for each merchant compute a success rate for every product he sells $S_{ij}$. ($i$ indexes products and $j$ indexes merchants) Average this and compute a merchant average baseline success rate($S_j$). Now compute differences ($\\delta S_{ij}=S_{ij} - S_j$). Each of this $\\delta S_{ij}$ indicates how much better or worse every product does with respect to that merchants baseline success rate.\n\nIf you sum up this $\\delta S_{ij}$ over all merchants j you'd obtain some sort of score of the attractiveness of every product $S_i$?\n\nThe merchant skill metric would be a dual of this. One problem is this doesn't weigh in the confidence level motivated by large data. i.e. 2 successes out of 4 pitches ought to (perhaps) matter less than 1000 successes out of 2000 pitches? You'd have to find some way to adjust for that in case it matters.\n\nAlternatively: Assume every merchant has a skill value $M_j$ and every product has a product attractiveness $A_i$. You could model the success rate of product $i$ sold by merchant $j$ ($S_{ij}$) as some function of $M_j$ and $A_i$ with possible cross terms. If you fit this you might be able to score using the coefficents.\n\nIf you consider $S_{ij} = M_j \\times A_i + \\epsilon_{ij}$ you get one simple model. The matrix of success elements is possibly sparse (since not all merchants sell all products). If it were indeed fully populated you must estimate 200 coefficients from 100x100 success rate numbers such that you minimize $\\epsilon_{ij}$ in some sort of least squares sense.\n\nPossible flaws:\n\nI don't see an easy way to interpret relative scores. e.g. If two Products have an attractiveness of $A_{i1}$ and $A_{i2}$ how much better is one than the other? A simple ratio? A log likelihood? etc. Perhaps there is some interpretation but I cannot see it yet. From a strictly ordering perspective it shouldn't matter.\n\nPS How sparse is your matrix? Knowing that you have millions of pitches maybe not too sparse? Or is it? i.e. Out of a maximum possible 10,000 merchant-product combinations how many are filled (i.e. have at least one pitch)?\n\nPS1 Uniqueness. I cannot prove whether your $M_j$ and $A_i$ values will be unique or even close to. If there are multiple solutions it'll be an interesting situation. Maybe there are stronger math results about this?\n\n• +1 Your \"Alternatively\" section is exactly the same as the \"SVD\" used in netflix, with the number of dimensions collapsed to 1. – Stumpy Joe Pete Feb 28 '13 at 20:17\n• @StumpyJoePete I did not know that! Thanks. Sounded a bit too simplistic when I suggested it myself..... – curious_cat Feb 28 '13 at 20:21\n• Yeah, see my answer about svd. Then just think of it as applied to your matrix, with $k=1$. The end result is approximating $S$ as the outer product of a \"product\" vector and a \"merchant\" vector, trying to minimize the squared error in the known entries. Cheers! – Stumpy Joe Pete Feb 28 '13 at 21:07\n\nI think you are looking to attribute qualities that are not inherent in, or do not follow from, your data. You have unambiguous data on success rate, and there should be a way to calculate or estimate a merchant's \"adjusted success rate\" given the rate at which his products tend to sell among all merchants. Similarly, there should be a way to determine each product's adjusted success rate given the success rates of the merchants who tend to sell it. These two angles on the analysis might be accomplished with a nested/hierarchical/multi-level logistic regression, if the data are suitable for it. But that wouldn't necessarily reveal the attributes of \"skill\" or \"attractiveness\"; it might yield workable proxies for them, but how adequate these proxies would be is a substantive question more than a statistical one.\n\n• Sure, I'm not so much concerned with what the proper name for these attributes would be. My goal is, for example, to find a list of product scores which, if a new merchant started using them for deciding which products to promote, would minimize the expected mistake. The score shouldn't reflect any actual observable quality, just something which makes it possible to distinguish between winning and losing products. – M. Cypher Feb 28 '13 at 16:52\n\nI would just create a 2 way table for this. For e.g. rows corresponding to different merchants and columns corresponding to different products. Each cell in this 100 x 100 table/matrix represents counts/proportion for no. of times the combination was successful.\n\nOnce this is done, you can sort this matrix by rows and then by columns (or the other way round) to get the product and merchant skills ordering.\n\nI'd recommend a logistic regression with merchants and products as random effects. In R, this would look like:\n\nlibrary(\"lme4\")\nfit <- glmer(sold ~ (1\|merchant) + (1\|product), data, family=binomial, REML=TRUE, verbose=TRUE, weights)\nsummary(fit)\nranef(fit)\n\n\nExtracting the estimates is relatively straightforward, and I handle millions of data points with approaches similar to this on standard workstations all the time. The model fitting typically only takes a few minutes." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.96658224,"math_prob":0.93793,"size":1799,"snap":"2019-35-2019-39","text_gpt3_token_len":353,"char_repetition_ratio":0.16824512,"word_repetition_ratio":0.0069444445,"special_character_ratio":0.19733185,"punctuation_ratio":0.11676647,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98335725,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-09-23T11:16:09Z\",\"WARC-Record-ID\":\"<urn:uuid:dd7f9c02-6002-454d-a292-086205581ded>\",\"Content-Length\":\"186358\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a4374d1d-0f1c-424f-a5a2-d5d13893b084>\",\"WARC-Concurrent-To\":\"<urn:uuid:a63a9244-e949-43a2-a74c-957e5d5500e8>\",\"WARC-IP-Address\":\"151.101.1.69\",\"WARC-Target-URI\":\"https://stats.stackexchange.com/questions/50632/scoring-items-which-are-not-easily-compared/51076\",\"WARC-Payload-Digest\":\"sha1:RV4QPHIJGVXBKG4PBFY4OUDBI656MJJL\",\"WARC-Block-Digest\":\"sha1:NOC2FVVHXMTTDSYJOMPVQZN3GEU6D2EF\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-39/CC-MAIN-2019-39_segments_1568514576355.92_warc_CC-MAIN-20190923105314-20190923131314-00476.warc.gz\"}"}
https://math.stackexchange.com/questions/3787664/tetrahedron-volume-given-rectangular-parallelepiped	[ "# tetrahedron volume given rectangular parallelepiped\n\nLet $$A$$ be a rectangular parallelepiped with edges of lengths $$15, 20, 30$$. Let $$B$$ be a tetrahedron on four non-adjacent vertices of $$A$$ (i.e no two vertices of $$B$$ share a common edge of $$A$$). Compute the volume of $$B$$.\n\nThis site gives a way to calculate but there's gotta be a closed form elegant formula for this.", null, "Let the rectangular parallelepiped measurements be $$a,\\,b,\\,c$$ then the mixed product of the three vectors is $$6$$ times the volume of the desired tetrahedron $$V=\\frac16\\operatorname{abs} \\left\| \\begin{array}{ccc} a&b&0\\\\ a&0&c\\\\ 0&b&c \\end{array} \\right\|=\\frac13 abc$$ Thus the desired volume is $$\\frac13\\cdot 15\\cdot 20\\cdot 30=3000$$.\n• More simply (to my mind, anyway), each of the tetrahedra at the vertices $A', B, C', D$ has volume $\\tfrac16abc,$ so the tetrahedron $AB'CD'$ has volume $abc - \\frac23abc = \\frac13abc.$ The question Volume of a tetrahedron whose 4 faces are congruent is relevant, but has more detail than is needed here. Aug 11 '20 at 21:00" ]	[ null, "https://i.imgur.com/7o5vRTg.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.88041854,"math_prob":0.99989176,"size":317,"snap":"2021-31-2021-39","text_gpt3_token_len":91,"char_repetition_ratio":0.12140575,"word_repetition_ratio":0.0,"special_character_ratio":0.2807571,"punctuation_ratio":0.10294118,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99999726,"pos_list":[0,1,2],"im_url_duplicate_count":[null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-09-19T11:02:03Z\",\"WARC-Record-ID\":\"<urn:uuid:dc23cf21-277d-4a50-a434-634d910fb2aa>\",\"Content-Length\":\"166149\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a5dfd156-0f44-4652-a1c8-900f6812f7de>\",\"WARC-Concurrent-To\":\"<urn:uuid:429d1c1e-ba5f-4d81-a6e7-85abc4f000d3>\",\"WARC-IP-Address\":\"151.101.193.69\",\"WARC-Target-URI\":\"https://math.stackexchange.com/questions/3787664/tetrahedron-volume-given-rectangular-parallelepiped\",\"WARC-Payload-Digest\":\"sha1:6SJRDWU44GW7NHGMLPG7CXODLHWJI4OX\",\"WARC-Block-Digest\":\"sha1:DFHERW4OZNAYV6YAZ56CVWEUF5CAMP3L\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-39/CC-MAIN-2021-39_segments_1631780056856.4_warc_CC-MAIN-20210919095911-20210919125911-00130.warc.gz\"}"}
http://zuowen6.info/node/2211	[ "雨\n\n•\n•\n• huá\n• huá\n• huá\n• tiān\n• hēi\n• le\n• xià\n• lái\n• de\n• shuǐ\n• cóng\n• 　　哗哗哗天黑了下来一滴一滴的雨水从\n• tiān\n• kōng\n• zhōng\n• fēi\n• xiè\n• xià\n• lái\n• piàn\n• shī\n• cháo\n•\n• 天空中飞泻下来大地一片湿潮\n•\n•\n• rén\n• háng\n• dào\n• shàng\n• ?g\n• ?g\n• de\n• sǎn\n• de\n• 　　人行道上花花绿绿的雨伞一把一把的\n• kāi\n• shàng\n• tóng\n• kāi\n• mǎn\n• le\n• duǒ\n• duǒ\n• xiān\n• ?g\n• xià\n• xià\n• 打开大地上如同开满了一朵朵鲜花雨下下\n• tíng\n• tíng\n• tíng\n• tíng\n• xià\n• xià\n• sǎn\n• ?g\n• huì\n• ér\n• kāi\n• huì\n• ér\n• xiè\n• shēng\n• mìng\n• 停停停停下下伞花一会儿开一会儿谢生命\n• duǎn\n• hěn\n•\n• 短得很\n•\n•\n• zǒu\n• zài\n• shàng\n• zhe\n• sǎn\n• zǒu\n• zài\n• cháo\n• shī\n• de\n• miàn\n• shàng\n• 　　我走在路上打着伞走在潮湿的路面上\n• kōng\n• fēi\n• cháng\n• hǎo\n• 空气非常地好\n•\n•\n• cóng\n• tiān\n• kōng\n• zhōng\n• fēi\n• xiè\n• xià\n• lái\n• de\n• shuǐ\n• hǎo\n• tiáo\n• shǎn\n• 　　从天空中飞泻下来的雨水好似一条闪\n• yào\n• de\n• yín\n• liàn\n• yòu\n• hǎo\n• xuě\n• de\n• shuǐ\n• jīng\n• yòu\n• hǎo\n• xiàng\n• 耀的银链又好似一颗颗雪色的水晶又好像\n• měi\n• de\n• qīng\n• shā\n• yàng\n• jīng\n• yíng\n• 美丽的轻纱一样晶莹\n•\n•\n• ā\n• jiàn\n• dào\n• rén\n• de\n• zhēn\n• shì\n• 　　啊我第一次见到如此迷人的雨她真是\n• tài\n• měi\n• le\n• shǐ\n• wàng\n• le\n• zhè\n• jǐng\n•\n•\n• 太美了使我忘不了这一次雨景\n•\n•\n\n无注音版：\n\n哗哗哗天黑了下来一滴一滴的雨水从天空中飞泻下来大地一片湿潮\n\n人行道上花花绿绿的雨伞一把一把的打开大地上如同开满了一朵朵鲜花雨下下停停停停下下伞花一会儿开一会儿谢生命短得很\n\n我走在路上打着伞走在潮湿的路面上空气非常地好\n从天空中飞泻下来的雨水好似一条闪耀的银链又好似一颗颗雪色的水晶又好像美丽的轻纱一样晶莹\n啊我第一次见到如此迷人的雨她真是太美了使我忘不了这一次雨景" ]	[ null ]	{"ft_lang_label":"__label__zh","ft_lang_prob":0.7836377,"math_prob":0.44716048,"size":405,"snap":"2019-43-2019-47","text_gpt3_token_len":542,"char_repetition_ratio":0.092269324,"word_repetition_ratio":0.0,"special_character_ratio":0.15308642,"punctuation_ratio":0.0,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99115896,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-10-17T10:27:29Z\",\"WARC-Record-ID\":\"<urn:uuid:cc235f58-42a1-46fb-b24a-a203c5c42965>\",\"Content-Length\":\"6042\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:bf31e14f-b89f-4d9a-a2fb-4bf7b44f0a27>\",\"WARC-Concurrent-To\":\"<urn:uuid:7667c048-5b6e-4cb8-88c8-60dfa30ce04e>\",\"WARC-IP-Address\":\"144.168.58.160\",\"WARC-Target-URI\":\"http://zuowen6.info/node/2211\",\"WARC-Payload-Digest\":\"sha1:CYFHGU3JU65EER53N6XKWV7DLTDBTCHP\",\"WARC-Block-Digest\":\"sha1:CVHDPAVNS2GR5GG7QVFY4SFDZOSUWJQ4\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-43/CC-MAIN-2019-43_segments_1570986673538.21_warc_CC-MAIN-20191017095726-20191017123226-00289.warc.gz\"}"}
https://socratic.org/questions/how-do-you-solve-2x-x-5-0	[ "# How do you solve (2x)/(x+5)<=0?\n\nDec 16, 2016\n\n$- 5 < x \\le 0$\n\n#### Explanation:\n\n$\\textcolor{red}{\\text{There is a trap in this question in that there are excluded}}$$\\textcolor{red}{\\text{values for } x}$\n\n$\\textcolor{b l u e}{\\text{Step 1}}$\nSolve disregarding the excluded values.\n\nMultiply both sides by $\\left(x + 5\\right)$\n\n$2 x \\le 0 \\times \\left(x + 5\\right)$\n\n$2 x \\le 0$\n\n$\\textcolor{b l u e}{\\text{Condition 1: } x \\le 0}$\n\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n$\\textcolor{b l u e}{\\text{Step 2}}$\nAn equation or expression becomes 'undefined' if you have division by 0.\n\n$\\implies \\left(x + 5\\right) \\ne 0$\n\n$\\textcolor{b l u e}{\\text{Condition 2: } x \\ne - 5}$\n\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n$\\textcolor{b l u e}{\\text{Step 3}}$\n\nCan we have x< -5color(white)(.)?\n\nSuppose $x = - 6$ then we have:\n\n$\\frac{2 \\left(- 6\\right)}{- 6 + 5} = \\frac{- 12}{-} 1 = + 12$\n\nThis is not an excluded value but does not satisfy $\\frac{2 x}{x + 5} \\le 0$\n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n$\\textcolor{b l u e}{\\text{Putting it all together using proper notation}}$\n\n$- 5 < x \\le 0$" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5034348,"math_prob":0.9998547,"size":439,"snap":"2020-34-2020-40","text_gpt3_token_len":87,"char_repetition_ratio":0.3402299,"word_repetition_ratio":0.0,"special_character_ratio":0.45102507,"punctuation_ratio":0.07017544,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9997687,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-09-30T22:37:36Z\",\"WARC-Record-ID\":\"<urn:uuid:2ee61bdd-ef8f-4e38-b0a3-93bd96229599>\",\"Content-Length\":\"35108\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:09efe3fe-8d50-48be-9c09-315ca20af2b8>\",\"WARC-Concurrent-To\":\"<urn:uuid:a6f1fbc4-b678-4119-8b34-e66f27afc041>\",\"WARC-IP-Address\":\"216.239.36.21\",\"WARC-Target-URI\":\"https://socratic.org/questions/how-do-you-solve-2x-x-5-0\",\"WARC-Payload-Digest\":\"sha1:Z6WCQKTLNOJMWYZF4WCRZR6VMLW5F5B4\",\"WARC-Block-Digest\":\"sha1:YWOX3YYBXJXO5I6GHTDVB3UDXDUIXE3I\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-40/CC-MAIN-2020-40_segments_1600402128649.98_warc_CC-MAIN-20200930204041-20200930234041-00042.warc.gz\"}"}
https://www.cryptocurrencyguide.org/technical-indicator-guides-what-is-macd-and-how-to-use-it/	[ "# Technical Indicator Guides: What is MACD and How To Use It", null, "## What is Moving Average Convergence Divergence (MACD)\n\nMoving Averages Convergence Divergence – or MACD, in short – is a momentum indicator that gauges a cryptocurrency’s overall trend, through the display of two moving averages of prices. It is one of the simplest and most effective technical indicators that you can use to measure where prices are headed.\n\nHere’s how it is used together with the price of a cryptocurrency, in this case, Bitcoin (BTC):", null, "## What is a Moving Average?\n\nMoving averages are used in technical analysis as it smoothens out past prices and defines its current direction. The 2 most commonly used moving averages are as follows:\n\n1. Simple Moving Average: A simple average is calculated by calculating the average price of a cryptocurrency over a specific time period. Therefore, a 7-day simple moving average is the sum of a cryptocurrency’s prices over a span of 7 days and divide it by 7.\n2. Exponential Moving Average (EMA): EMA is similar to a simple average but emphasizes more on the latest data. More weight is given to the latest price data is it is more relevant and indicative of the current market trend. MACD uses exponential moving averages.\n\n## Components of MACD\n\nThere are 3 main components of the MACD that you should know:", null, "1. MACD Line (Blue): The MACD line is a combination of 2 EMA; it is the difference between the 26-day and 12-day exponential moving average of closing prices. It is important to note that 26-days and 12-days are the default numbers for MACD; you can change the number around if you feel confident in using and interpreting different time intervals.\n2. Signal Line(Orange): A 9-day EMA, called the “signal” (or “trigger”) line is plotted on top of the MACD to show buy/sell opportunities.\n3. Histogram: The difference between the MACD Line and Signal Line with the passage of time\n\n## How To Trade Using MACD\n\nAs its name suggests, MACD is used to explore the points where the MACD line and signal line converges or diverges.", null, "In order to simplify your understanding of how MACD works, we shall focus mainly on the MACD line (blue line) applied in two main scenarios. Here are the two scenarios where you can interpret this indicator to your advantage:\n\n## 1. Center/Zero Line Crossover\n\n• When MACD line crosses above the Zero Line = BUY\n• When it crosses below the Zero Line = SELL", null, "This is the simplest scenario out of the two. Note that in this example, we shall momentarily ignore the signal line (orange line). We shall only focus on the MACD line and the centre/zero-line.\n\nPositive Signal\n\nThe MACD line would often oscillate between the centre line, which is fixated at point 0 (black line). A general rule of thumb is that if the MACD line is above the centre line, it is a positive signal. This occurs when the 12-day EMA is higher than the 26-day EMA (since the MACD line is made up of the difference between the 26-day and 12-day EMA). In the example above, the green highlighted area is when the MACD line is above the centre line, and you can generally see that prices would tend to move upwards.\n\nThe reason why it’s a positive signal when the 12-day EMA is higher than the 26-day EMA (and therefore, above the zero-line) is because recent prices (in the last 12 days) shows a greater upward momentum as compared to the 26-day average, and since EMA takes higher weightage of more recent prices, it will mean that prices would tend to move higher than before!\n\nNegative Signal\n\nWhen the MACD line is below the centre/zero-line, it means that it is a negative signal. From the example above, we can see that when the MACD drops below the centre line – as highlighted in red – prices tend to move further downwards. This is a sign of bad times.\n\nThe reason why it’s a negative signal is that the 12-day EMA is lower than the 26-day EMA (and therefore, below the zero-line). This indicates that recent prices (in the last 12 days) shows a greater downward momentum as compared to the 26-day average, and since EMA takes higher weightage of more recent prices, it will mean that prices would tend to move lower than before! Ouch.\n\n## 2. Signal Line Crossover\n\n• When MACD line crosses above the Signal Line = BUY\n• When it crosses below the Signal Line = SELL", null, "In this scenario, the signal line (in orange) comes into play and is very relevant.\n\nBullish Crossover\n\nA bullish crossover is a positive signal which occurs when the MACD line (in blue) crosses over the signal line in an upward fashion. This means that the MACD line is higher than the signal line. We can see from the above example that on November 15 (highlighted in gree), there was a bullish crossover that led to the strong rally upwards, and prices were soaring high.\n\nBearish Crossover\n\nA bearish crossover is a negative signal which occurs when the MACD line (in blue) crosses below the signal line. This means that the MACD line is lower than the signal line. From the example, we can see that there were 2 bearish crossovers (highlighted in red) that led to prices crashing down.\n\n*For the purpose of illustration, the charts used in this guide uses daily (1-day) candlesticks. It is up to you to choose which time interval (daily, weekly, monthly or even hourly or minute candlesticks) is suitable for you!\n\nTradingview is perhaps one of the best free charting platforms that you can use for performing technical analysis. However, each user can only use a maximum of 3 technical indicators if you’re using a free account. You can start adding technical indicators to your chart through following these steps:\n\n### Step 1: Click on the ‘Indicator Button’", null, "Step 2: Search for ‘MACD’ & Select the Indicator", null, "", null, "#### Aziz\n\nFounder of Master the Crypto\n\n## More To Explore", null, "Press Release\n\n### Bitdeal Provides Defi Services to Revamp the Traditional Finance System\n\nBitdeal, the Prominent Blockchain and Cryptocurrency Exchange Development Company provides enormous services such as Cryptocurrency Exchange Development, Wallet Development, Bitcoin Escrow, Blockchain Development, MLM Development,", null, "" ]	[ null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20800%20427'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%201024%20559'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%201963%20372'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20800%20291'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%201878%201028'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%201881%201028'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%201024%20601'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%201024%20482'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20150%20150'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20300%20150'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20540%20524'%3E%3C/svg%3E", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9294674,"math_prob":0.8806479,"size":5934,"snap":"2020-45-2020-50","text_gpt3_token_len":1336,"char_repetition_ratio":0.13996628,"word_repetition_ratio":0.12403101,"special_character_ratio":0.21452646,"punctuation_ratio":0.07651715,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9555303,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-29T22:04:49Z\",\"WARC-Record-ID\":\"<urn:uuid:93248941-912f-4744-836f-8a66db67b717>\",\"Content-Length\":\"113458\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:fcc4de26-2a2d-4099-99df-d04a42ed17e0>\",\"WARC-Concurrent-To\":\"<urn:uuid:ecd9bd95-94da-4ef6-9166-1c37d90dc3cd>\",\"WARC-IP-Address\":\"172.67.159.207\",\"WARC-Target-URI\":\"https://www.cryptocurrencyguide.org/technical-indicator-guides-what-is-macd-and-how-to-use-it/\",\"WARC-Payload-Digest\":\"sha1:XGPSAA3P5PMQY6PJ65T3GPZBYZMABQTP\",\"WARC-Block-Digest\":\"sha1:7GRFWCLFYOAYU4T2CWYHZBDKF57GCPCD\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-45/CC-MAIN-2020-45_segments_1603107905965.68_warc_CC-MAIN-20201029214439-20201030004439-00251.warc.gz\"}"}
https://forum.processing.org/two/discussion/22859/how-do-i-get-depth-data	[ "#### Howdy, Stranger!\n\nWe are about to switch to a new forum software. Until then we have removed the registration on this forum.\n\n# How do I get depth data ?\n\nedited June 2017\n\nI am to try out depth of field and I am first trying to get depth data with GLSL. I am able to generate DEPTH map but there are some glitches I don't understand why?", null, "", null, "above images shows a long box and it just shows hollow from front face.\n\nthis is processing code\n\n``````PShader depth;\nPGraphics pg;\nvoid setup()\n{\nsize(600,600,P3D);\npg= createGraphics(600,600,P3D);\ndepth.set(\"far\",100.0);\n}\n\nfloat t=0;\n\nvoid draw()\n{\nt+=0.001;\npg.hint(ENABLE_DEPTH_TEST);\npg.beginDraw();\npg.noStroke();\npg.background(0);\npg.translate(width/2+50,height/2,0);\npg.rotateY(2PI-PIt/7);\npg.box(20,20,1000);\n// pg.translate(-200,-100,0);\npg.box(40,40,100);\npg.endDraw();\nimage(pg,0,0);\n}\n``````\n\n``````#ifdef GL_ES\nprecision mediump float;\nprecision mediump int;\n#endif\n\nvarying vec4 vertColor;\nvarying vec3 vertNormal;\nvarying vec3 vertLightDir;\nvarying float dist;\n\nvoid main() {\ngl_FragColor = vec4(vec3(1.0,1.0,1.0)1/clamp(dist,0,1000), 1.0 );\n}\n``````\n\n``````uniform mat4 transform;\nuniform mat3 normalMatrix;\nuniform vec3 lightNormal;\n\nattribute vec4 position;\nattribute vec4 color;\nattribute vec3 normal;\n\nuniform vec3 cameraPosition;\nuniform float far;\n\nvarying vec4 vertColor;\nvarying vec3 vertNormal;\nvarying vec3 vertLightDir;\nvarying float dist;\n\nvoid main() {\nvertNormal = normal;\ndist = (distance(cameraPosition,position.xyz)/far);\ngl_Position = transform position;\n}\n``````\n\nwhat could be the issue? well I am just new to GLSL so please sorry for any simple mistake. Also if there is any simple way to do nice DoF let me know.\n\nTagged:\n\n• to format the code, highlight it and press ctrl-o\n\n• Hey sorry that was the first time. Also I don't think I need to use this pg.hint(ENABLE_DEPTH_TEST);\n\n• https://github.com/edumo/P5PostProcessing here is the code Last week I made it work but did not save it I wounder how it worked then. but it did not gave any good result for blur part. burling was not good.\n\n• edited June 2017\n\n@trailbalzer47 Yes Camera is pointing -Z towards the viewer\n\nto fix the above scetch you need to add as third parameter the negative camera direction\n\n`pg.translate(width/2+50,height/2,0, -100)`\n\nor fix it in the shader.\n\nno backfaceculling per default\n\nhttps://forum.processing.org/one/topic/how-to-enable-backface-culling-in-opengl-to-speed-3d-model-fps.html\n\nHere you did it right: https://github.com/edumo/P5PostProcessing/blob/master/Dof/Dof.pde#L77\n\n``````//quick fix\n//no time to write any notes\nPGraphics pg;\n\nvoid setup(){\nsize(600,600,P3D);\npg= createGraphics(600,600,P3D);\ndepth.set(\"far\",100.0);\n}\nfloat t=0;\n\nfloat z = 100;\n\nvoid draw(){\nt+=0.001;\npg.hint(ENABLE_DEPTH_TEST);\npg.beginDraw();\npg.noStroke();\npg.background(0);\n\n//flip sign +/- translate.z 100\nif(frameCount%60==0) z=-1.;\n\n//processing.org/reference/PShape_translate_.html\npg.translate(width/2+50,height/2,z);\n\npg.rotateY(2PI-PIt/7);\npg.box(20,20,1000);\n// pg.translate(-200,-100,0);\npg.box(40,40,100);\npg.endDraw();\nimage(pg,0,0);\n}\n``````\n• I understood it was my mistake simple explanation is that box was too big in Z direction and thus camera went inside the box as it rotated so after reducing size of box the problem is solved its nothing to do with the shader is the camera position intersecting the shape. and thanks to @nabr and @koogs\n\n• edited June 2017 Answer ✓\n\nAh, yes,\nthe second box is set to 1000, it looks like their is a face missing, i thought, it has something to do with backfaceculling .\n\n#### Great!\n\n``````PShader depth;\nPGraphics pg;\nvoid setup(){\nsize(600,600,P3D);\npg= createGraphics(600,600,P3D);\ndepth.set(\"far\",100.0);\n}\nfloat t=0;\nvoid draw(){\nt+=0.001;\npg.hint(ENABLE_DEPTH_TEST);\npg.beginDraw();\npg.noStroke();\npg.background(0);\npg.translate(width/2+50,height/2,0);\npg.rotateY(2PI-PI*t/PI);\n\n//changed\npg.box(20,20,936.09999);\n\npg.box(40,40,100);\n\npg.endDraw();\nimage(pg,0,0);\n}\n``````" ]	[ null, "https://forum.processing.org/two/discussion/22859/![]()", null, "https://forum.processing.org/two/uploads/imageupload/688/TQI7G3YQVH5V.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8121782,"math_prob":0.9106774,"size":3222,"snap":"2020-24-2020-29","text_gpt3_token_len":860,"char_repetition_ratio":0.093536355,"word_repetition_ratio":0.049661398,"special_character_ratio":0.26350093,"punctuation_ratio":0.18848921,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95532155,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,2,null,5,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-08T11:12:00Z\",\"WARC-Record-ID\":\"<urn:uuid:9faa73a4-c538-4f55-a64f-c116ac095583>\",\"Content-Length\":\"55776\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:1857350b-d621-4985-b1ae-a001467f021f>\",\"WARC-Concurrent-To\":\"<urn:uuid:af9d82fa-19ae-41f6-9990-f3c2139d6774>\",\"WARC-IP-Address\":\"45.33.79.91\",\"WARC-Target-URI\":\"https://forum.processing.org/two/discussion/22859/how-do-i-get-depth-data\",\"WARC-Payload-Digest\":\"sha1:WC2YMVOUVMWECU6YBVE3XQFJZCKSP46M\",\"WARC-Block-Digest\":\"sha1:G5X2LPZSJESEDKNRY57M6IXEYOVXJH3A\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-29/CC-MAIN-2020-29_segments_1593655896932.38_warc_CC-MAIN-20200708093606-20200708123606-00134.warc.gz\"}"}
https://www.developer.com/java/understanding-the-discrete-cosine-transform-in-java/	[ "JavaData & JavaUnderstanding the Discrete Cosine Transform in Java\n\n# Understanding the Discrete Cosine Transform in Java\n\nJava Programming Notes # 2444\n\n## Preface\n\nThis lesson is one in a series designed to teach you about the inner\nworkings of data and image compression. The first lesson in the series was\nUnderstanding\nthe Lempel-Ziv Data Compression Algorithm in Java\n. The previous lesson\nwas Understanding the Huffman Data Compression\nAlgorithm in Java\n.\n\nJPEG image compression\n\nOne of the objectives of the series is to teach you about the inner workings\nof JPEG image compression. According to\nWikipedia,\n\n\"… JPEG (pronounced jay-peg) is a commonly used standard\nmethod of\n\nlossy compression\nfor photographic images. … The name stands for\nJoint Photographic Experts Group\n. JPEG itself specifies only how an\nimage is transformed into a stream of\nbytes, but not\nhow those bytes are encapsulated in any particular storage medium. A further\nstandard, created by the Independent JPEG Group, called JFIF\n(JPEG File Interchange Format) specifies how to produce a file suitable for\ncomputer storage and transmission (such as over the\nInternet)\nfrom a JPEG stream. … JPEG/JFIF is the most common format used for storing\nand transmitting photographs on the\n\nWorld Wide Web\n.\"\n\nEntropy encoding\n\nWithout getting into the technical details at this point, let me tell you\nthat one of the central components of JPEG compression is\nentropy encoding. Huffman encoding,\nwhich was the primary topic of the\nprevious lesson\nis a common form of entropy encoding.\n\nThe Discrete Cosine Transform\n\nAnother central component of JPEG compression is the\n\nDiscrete Cosine Transform\n, which is the primary topic of this lesson. Again,\naccording to\nWikipedia\n,\n\n\"The discrete cosine transform (DCT) is a\n\nFourier-related transform\nsimilar to the\n\ndiscrete Fourier transform\n(DFT), but using only\nreal\nnumbers\n. It is equivalent to a DFT of roughly twice the length,\noperating on real data with\n\neven\nsymmetry …\"\n\nIn order to understand JPEG …\n\nIn order to understand JPEG image compression, you must understand Huffman\nencoding, the Discrete Cosine Transform, and some other topics as well, such as\nspectral re-quantization. I\nplan to teach you about the different components of JPEG in separate\nlessons, and then to teach you how they work together to produce \"the\nmost common format\nused for storing and transmitting photographs on the\nWorld Wide Web\n\"\n\nViewing tip\n\nYou may find it useful to open another copy of this lesson in a\nseparate browser window. That will make it easier for you to\nscroll back\nand forth among the different listings and figures while you are\n\nSupplementary material\n\nI recommend that you also study the other lessons in my extensive\ncollection of online Java tutorials. You will find those lessons\npublished\nat Gamelan.com\nHowever, as of the date of this writing, Gamelan doesn’t maintain a\nconsolidated index of my Java tutorial lessons, and sometimes they are\ndifficult to locate there. You will find a consolidated index at www.DickBaldwin.com.\n\nIn preparation for understanding the material in this lesson, I also\nrecommend that you also study the lessons referred to in the\nReferences section.\n\n## General Background Information\n\nAlthough the Discrete Cosine Transform (DCT) may be\n\"similar to the\ndiscrete Fourier transform\"\n, it is not a Discrete Fourier Transform\n(DFT)\nas described in my earlier lesson entitled\nFun with Java,\nHow and Why Spectral Analysis Works\n. There are some major differences\nbetween DFT and DCT. Nonetheless, the DCT is very interesting from a technical\nviewpoint and is a central component of JPEG image compression.\n\nThe DFT and symmetrical real data\n\nTo get started, I’m going to show you the results of applying the DFT to\nsymmetrical real data. In the earlier lesson entitled\nSpectrum\nAnalysis using Java, Forward and Inverse Transforms, Filtering in the Frequency\nDomain\n, I introduced you to a program named Dsp035, which:\n\n\"… illustrates the reversible nature of the Fourier transform. This\nprogram transforms a real time series into a complex spectrum, and then\nreproduces the real time series by performing an inverse Fourier transform\non the complex spectrum. This is accomplished using a DFT algorithm.\"\n\nThe program named Dsp043", null, "The first graph at the top of Figure 1 shows a 300-sample symmetrical time\nseries. Note that this time series is symmetrical about its center point.\n\nThe spectral results\n\nThe DFT was applied to this time series. The next three graphs going\ndown the page in Figure 1 show the\nspectral results of applying the DFT to the time series. The three\nspectral graphs are plotted from a\nfrequency of zero to the sampling frequency.\n\n(Recall that the\n\nNyquist folding frequency\noccurs half way between zero and the sampling\nfrequency, and that the spectrum shown above the folding frequency is the\nmirror image of the spectrum below the folding frequency. Therefore,\nwe are usually interested only in the spectral results below the folding\nfrequency. Therefore, you can ignore the right half of the spectral graphs.)\n\nWhat do the five graphs show?\n\nThe five graphs in Figure 1 show (in order from top to bottom):\n\n1. The 300-sample symmetrical real time series to which the DFT was applied.\n2. The real part of the output from the DFT.\n3. The imaginary part of the output from the DFT.\n4. The magnitude of the output from the DFT.\n5. The result of applying an inverse DFT to the spectral data in order to\nreconstruct the original time series from the spectral data.\n\nThe important points\n\nThe most important points illustrated by Figure 1 are:\n\n• The time series is purely real.\n• The time series is symmetrical about its center point.\n• The imaginary part of the spectrum has all zero values.\n\nTherefore, Figure 1 demonstrates that the imaginary part of the Fourier\ntransform of a real symmetrical time series is all zeros if the origin is\nproperly located\n. This is important because I will use that fact to develop the rationale for\nthe Discrete Cosine Transform, and why it works the way that it does.\n\nEquations for the DFT\n\nReferring back to the equations in my\n\nearlier lesson\n, you can see that the real part of the output from the DFT\nresults from a sum of products involving a cosine term, and that the imaginary part\nof the output results from a sum of products involving a sine term.\n\nCan sometimes avoid the sine computation\n\nWhen\ncomputing the DFT, if we already know that the input time series is symmetrical\nand that the imaginary part of the output will be zero, we can simply forego the\ncomputation of the imaginary part that involves the sine term, thereby reducing\nthe computational requirements.\n\nCan make the cosine computation less burdensome\n\nIn addition, if we know that the input time series is symmetrical, we can\nreformulate the computation of the real part of the transform involving the cosine\nterm wherein we perform the computation on one-half of the time series only and\nthen double the result. Thus for a symmetrical time series having a length\nof 2N+1 samples, we can reduce the number of real-part computations to N+1.\n\nHow does the DCT work?\n\nBasically the DCT works by implicitly doubling the length of the input time series by\nconcatenating it to a mirror image of itself. The concatenation of the\noriginal time series to the mirror image results in a symmetrical time series.\n\n(You don’t actually see the doubling of the time series. Rather,\nthe doubling is implicit in the formulation of the equations for the DCT.)\n\nBecause the new time series is symmetrical, it is known in advance that if we were to perform a DFT on the new\ndouble-length time series, the imaginary part would be zero. Thus, it is\nalso known in advance that we can forego the computation of the imaginary part\nthat uses the sine term in the DFT.\n\nNo need to double the number of cosine computations\n\nIn addition, because the new double-length time series is symmetrical, we\ndon’t need to double the number of real-part computations involving the cosine\nterm (but we will have to reformulate the real-part computation relative to a\nstraight DFT computation)\n.\n\nThe definition of the DCT\n\nThe\ndefinition of the DCT is very similar to the definition of the DFT but the\ncomputation of the imaginary part using the sine term simply isn’t part of the\ndefinition.\n\nIn addition to eliminating the computation of the imaginary part, the\ndefinition of the DCT also reformulates the DFT to take advantage of the\nsymmetry of the time series relative to the computation of the real part using\nthe cosine term.\n\nLet’s see some equations\n\nRather than to deal with the somewhat difficult task of producing equations\nin this HTML document, I have provided three separate references\nthat contain the equations for the DCT. The equations in these three\nreferences are essentially the same (to within a scale factor).\n\nI elected to formulate my DCT program for this lesson using the\none-dimensional form of the DCT equations that you will find at\n\nNational Taiwan University – DSP Group – Discrete Cosine Transform\n.\n\n(The next lesson in this series will deal with the two-dimensional\nformulation of the DCT.)\n\nNo sine term\n\nIf you examine those equations, you will find that there is no sine term in\neither the forward or the inverse DCT. Only the cosine term is\nincluded, and it is included in such a way as to take advantage of the symmetry\nof the double-length time series.\n\n(While you are at the site mentioned\n\nabove\n, also note that the author provides an\nexplanation of the doubling of the length of the time series in order to\nproduce a new symmetrical time series for which the Fourier Transform is\nguaranteed to have a zero imaginary part.)\n\nThe forward Discrete Cosine Transform (DCT)\ncode", null, "", null, "The three graphs in Figure 3 show:\n\n1. A 256-sample input time series consisting of the same three waveforms\nshown in Figure 2(As with Figure 2, even though\nthe display is 300 points wide, the actual data being plotted on each of the\nthree graphs ends at 256 samples.)\n2. The real frequency spectrum produced by the forward DCT.\nIn this case, the spectrum is plotted from a frequency of zero on the left\nto the\n\nNyquist folding frequency\non the right.\n3. The result of applying the inverse DCT to the spectral data in order to\nreconstruct the original time series from the spectral data.\n\nWhere is the imaginary part of the spectrum?\n\nThere is no imaginary part of the spectrum plotted in Figure 3, simply\nbecause the DCT doesn’t produce an imaginary part. Since there is no\nimaginary part, there is also no need for a plot of the magnitude spectrum for\nthe DCT. (It would look exactly like the real part of the spectrum,\nwith all negative values converted to positive values, if\nit were computed and plotted.)\n\nIf you compare Figure 3 with\nFigure 2\n, you will see that the DFT and the DCT both appeared to do an\nequally good job of\nreproducing the original time series when the inverse transform was applied to the spectral\ndata. The big difference between Figure 2 and\nFigure 3 is that this was accomplished with the somewhat\nmore economical DCT in Figure 3.\n\nWater to ice and back to water\n\nThere is one point that I would like to make here, because it will become\nvery important in a future lesson that deals with the inner workings of JPEG.\nThe second and third graphs in Figure 2 represent the\nsame information as the first graph in Figure 2\nSimilarly, the second graph in Figure 3 represents the\nsame information as the first graph in Figure 3\nIn other words, the spectral data represents the same information as the\ntime-series data. They simply represent that information in different\nforms.\n\nAs an analogy, if we lower the temperature of a container of water to less than 32 degrees\nFahrenheit, the form of the water will change from a liquid to a solid. If we warm\nit back up, the form will change back to a liquid. This is roughly\nanalogous to transforming a time series into the frequency domain and then\ntransforming the frequency spectrum back into the time domain. The same\ninformation is represented in both cases. That information is simply\nrepresented in different forms.\n\nSub-dividing the input\n\nWhen performing spectral analysis, it is common practice to perform a DFT\n(or perhaps a DCT)\non the entire time series as was the case in\nFigure 1,\nFigure 2, and\nFigure 3. However, in a future lesson we will\nlearn that this is not the case for JPEG image compression. Instead, the\nJPEG procedure sub-divides the image into a set of small images where each small\nimage consists of an 8×8 block of 64 pixels.\n\nThen the DCT is performed on\neach individual block of 64 pixels. Several additional processing steps are performed\non the spectra produced for the set of 8×8 blocks to produce the compressed\nimage. Later on, when the\nimage is reconstructed, the 8×8 blocks are individually reconstructed and are\nthen assembled into a larger image that approximates the original image.\n\nThe program named Dsp044\n\nThe program named Dsp044 is designed to\ninvestigate the impact of sub-dividing the time series into eight-sample\nsegments and processing those segments individually in order to be more consistent with the 8×8-pixel block concept in JPEG.\nAs it turns out, there appears to be no noticeable impact. As you can see\nin Figure 4, the reconstructed output shown in the second\ngraph is a very good replica of the input shown in the first graph.\n\n `", null, "", null, "` `Figure 4`\n\n(Figure 4 shows only the input and output time series. In order\nto display the spectral results, it would have been necessary to display 32\nindividual spectra, one computed for each 8-sample segment of the input time\nseries. That would have been fairly impractical.)\n\nTesting\n\nBoth programs were tested using J2SE 5.0 under WinXP. (Both programs\nrequire J2SE 5.0 or later due to the use of\nstatic import of Math class.)\n\n## Discussion and Sample Code\n\nThe program named Dsp042\n\nThe class\ndefinition for Dsp042 begins in Listing 1 by declaring\nand initializing some instance variables.\n\n ```class Dsp042 implements GraphIntfc01{ int len = 256; double[] timeDataIn = new double[len]; double[] realSpect = new double[len]; double[] timeDataOut = new double[len]; int zero = 0;Listing 1```\n\nNote that the class named Dsp042 implements the interface named GraphIntfc01.\n\nThe constructor for Dsp042\n\nThe constructor begins in Listing 2.\n\n ``` public Dsp042(){//constructor //Create the raw data pulses timeDataIn = 0; timeDataIn = 50; //code deleted for brevity timeDataIn = -2; timeDataIn = -1; timeDataIn = 80; timeDataIn = 80; //code deleted for brevity timeDataIn = -80; timeDataIn = -80;Listing 2```\n\nThe code in Listing 2 creates the first and last waveforms shown in the first\ngraph in Figure 3. Note that I deleted quite a lot\nof code from Listing 2 for brevity. You can view the code that I deleted\nin Listing 14.\n\nCreate the sinusoidal waveform\n\nListing 3 creates the truncated sinusoidal waveform shown near the center of\nthe first graph\nin Figure 3.\n\n ``` for(int x = len/3;x < 3len/4;x++){ timeDataIn[x] = 80.0 Math.sin(2PI(x)1.0/20.0); }//end for loopListing 3```\n\nPerform the forward Discrete Cosine Transform\n\nListing 4 invokes the static transform method of the ForwardDCT01\nclass to compute the forward DCT of the time data and to save the results in the\narray referred to by realSpect, which was created\nin Listing 1.\n\n ` ForwardDCT01.transform(timeDataIn,realSpect);Listing 4`\n\nPerform the inverse Discrete Cosine Transform\n\nListing 5 invokes the static transform method of the InverseDCT01\nclass to compute the inverse DCT of the spectral data and to save the results in\nthe array referred to by timeDataOut, which was created\nin\nListing 1.\n\n ``` InverseDCT01.transform(realSpect,timeDataOut); }//end constructorListing 5```\n\nListing 5 also signals the end of the constructor for the class named\nDsp042\n.\n\nThe remaining code\n\nThe remaining code in the class named Dsp042 consists of six methods,\nwhich are required by the interface named GraphIntfc01. The purpose\nof these methods is simply to plot the data contained in three arrays, producing the\nthree graphs shown in Figure 3. I explained those\nsix methods in the earlier lesson entitled\n\nPlotting Engineering and Scientific Data using Java\nand won’t repeat that\nexplanation here. You can view the methods in\nListing 14\n\nThe class named Dsp044\n\nThis class is a modification of the class named Dsp042 designed to investigate the impact of\nsub-dividing the input time series into eight-sample segments in order to be consistent with the 8×8 blocks in JPEG.\n\nListing 6 shows the beginning of the class and the beginning of the\nconstructor.\n\n ```class Dsp044 implements GraphIntfc01{ int len = 256; double[] timeDataIn = new double[len]; double[] timeDataOut = new double[len]; int zero = 0; public Dsp044(){//constructor //Create the raw data pulses timeDataIn = 0; timeDataIn = 50; //code deleted for brevity timeDataIn = -2; timeDataIn = -1; timeDataIn = 80; timeDataIn = 80; //code deleted for brevity timeDataIn = -80; timeDataIn = -80; //Create raw data sinusoid for(int x = len/3;x < 3len/4;x++){ timeDataIn[x] = 80.0 * Math.sin(2PI(x)1.0/20.0); }//end for loopListing 6```\n\nOnce again, note that the class named Dsp044 implements the interface\nnamed GraphIntfc01.\n\nAs before, I deleted some of the code from Listing 6 for brevity. The\ncode in Listing 6 is very similar to the code that I explained earlier with\nrespect to the class named Dsp042.\n\nEight-element array objects\n\nThe real difference between Dsp042 and Dsp044 begins in Listing 7 where I\ncreate some eight-element array objects to handle the eight-sample\nsegments.\n\n ``` double[] workingArrayIn = new double; double[] workingArrayOut = new double; double[] realSpect = new double; Listing 7```\n\nProcess eight samples at a time\n\nListing 8 shows the beginning of a while loop, which computes a forward\nand an inverse DCT on each successive eight-sample segment of the input time\nseries. Code inside\nthe while loop also concatenates the output segments from the inverse DCT\nto produce the output signal, which is shown by the bottom graph\nin\nFigure\n4\n.\n\n ``` int segmentCnt = 0; while((segmentCnt + 8) <= len){ System.arraycopy(timeDataIn, segmentCnt, workingArrayIn, 0, 8);Listing 8```\n\nDuring each iteration of the while loop, the code in Listing 8 copies the\nnext eight samples from the input time series into an eight-element working\narray.\n\nCompute the forward and the inverse transforms\n\nListing 9 performs a forward DCT on the contents of the eight-element working\narray. Then it performs an inverse DCT on the eight-samples of spectral\ndata produced by the forward transform.\n\n ``` ForwardDCT01.transform(workingArrayIn,realSpect); InverseDCT01.transform(realSpect,workingArrayOut);Listing 9```\n\nConcatenate the output time-series segments\n\nListing 10 copies the eight samples of output time-series data produced by\nthe inverse transform into the next eight samples of the array designated to\nhold the final output. This concatenates the eight-sample segments into\nthe time series shown in the bottom graph in Figure 4.\n\n ``` System.arraycopy(workingArrayOut, 0, timeDataOut, segmentCnt, 8); segmentCnt += 8; }//end while }//end constructorListing 10```\n\nThen Listing 10 increments the segment counter by 8 and control is transferred\nback to the top of the while loop shown in Listing 8.\n\nListing 10 also signals the end of the constructor.\n\nThe remaining code\n\nAs before, the remaining code in the class named Dsp044 consists of six methods, which are required by the interface named GraphIntfc01. The\npurpose of these methods is to plot the data contained in two arrays, producing\nthe graphs shown in Figure 4. You can view the\nmethods in Listing 15\n\n## Run the Programs\n\nI encourage you to copy, compile, and execute the code from the listings in\nthe section entitled Complete Program\nListings\n. Experiment with the code, making\nchanges and observing the results of your changes. For example, as one\nexperiment you can see what\nhappens if you make changes to the computed argument for the cosine function in\neither the forward or the inverse transform.\n\nRun under control of Graph03\n\nDsp042, Dsp043, and Dsp044 must all be run under the control of the program named\nGraph03.\n\n(The program named Graph03 is a plotting program. See the\nearlier lesson entitled\nSpectrum\nAnalysis using Java, Sampling Frequency, Folding Frequency, and the FFT\nAlgorithm\n, which was the first lesson in which the plotting program\nnamed Graph03 was used. Also see Graph01, which was a predecessor of\nGraph03, in the lesson\nentitled\nPlotting Engineering and Scientific Data using Java.)\n\nThe\nsource code for Graph03 is provided in Listing 16.\n\nThe source code\nfor the interface named GraphIntfc01, which is required by these\nprograms,\nis provided in Listing 17.\n\nRunning the programs\n\nTo run these programs, first compile the programs and then enter one of the following statements at the command prompt.\n\n```java Graph03 Dsp042\njava Graph03 Dsp042\njava Graph03 Dsp044```\n\nSupport classes\n\nYou will need some support classes in order to run these programs. In\nthose cases where the source code for a required support class is not included in this lesson, you should be\nable to find the source code in the lessons referred to in the\nReferences section.\n\nYou can also find the source code by searching for it on\nfor the following keywords on Google will identify the earlier lesson entitled\nSpectrum Analysis\nusing Java, Sampling Frequency, Folding Frequency, and the FFT Algorithm\n\ncontaining the source code for the class named ForwardRealToComplex01.\n\n`java Baldwin \"ForwardRealToComplex01.java\"`\n\n## Summary\n\nI introduced you to the basics of the Discrete Cosine\nTransform (DCT) by:\n\n• Explaining some of the underlying theory behind the transform.\n• Demonstrating the use of the transform in two one-dimensional cases.\n\n## What’s Next?\n\nThe next lesson will explain the use of the two-dimensional Discrete Cosine\nTransform and will illustrate its use to transform images into the wave-number\ndomain and back into the space or image domain.\n\nFuture lessons in this series will explain the inner workings behind several\ndata and image compression schemes, including the following:\n\n• Run-length data encoding\n• GIF image compression\n• JPEG image compression\n\n## References\n\nGeneral\n\n2440 Understanding the Lempel-Ziv Data Compression Algorithm in Java\n2442 Understanding the Huffman Data Compression Algorithm in Java\n1468\nPlotting Engineering and Scientific Data using Java\n1478 Fun\nwith Java, How and Why Spectral Analysis Works\n1482\nSpectrum Analysis using Java, Sampling Frequency, Folding Frequency, and the FFT\nAlgorithm\n1483\nSpectrum Analysis using Java, Frequency Resolution versus Data Length\n1484\nSpectrum Analysis using Java, Complex Spectrum and Phase Angle\n1485\nSpectrum Analysis using Java, Forward and Inverse Transforms, Filtering in the\nFrequency Domain\n1486 Fun\nwith Java, Understanding the Fast Fourier Transform (FFT) Algorithm\n1489\nPlotting 3D Surfaces using Java\n1490 2D\nFourier Transforms using Java\n1491 2D\nFourier Transforms using Java, Part 2\n\nDiscrete Cosine Transform equations\n\nDiscrete\ncosine transform\n– Wikipedia, the free encyclopedia\nThe Data Analysis Briefbook –\nDiscrete Cosine\nTransform\n\nNational Taiwan University\nDSP Group\n\nDiscrete Cosine Transform\n\n## Complete Program Listings\n\nComplete listings of the programs discussed in this lesson are provided in the\nfollowing listings:\n\nListing 11\n\n ```/ File Dsp043.java Copyright 2006, R.G.Baldwin Revised 01/05/06 The purpose of this program is to demonstrate that the imaginary part of the Fourier transform of a symmetrical time series is all zeros if the origin is properly located. Illustrates forward and inverse Fourier transforms on a symmetrical time series using DFT algorithms. Passes resulting real and complex parts to inverse Fourier transform program to reconstruct the original time series. Run with Graph03. Enter the following to run the program: java Graph03 Dsp043 Exection of this program requires access to the following class files: Dsp043.class ForwardRealToComplex01.class Graph01.class GraphIntfc01.class GUI\\$MyCanvas.class GUI.class InverseComplexToReal01.class Tested using J2SE 5.0 under WinXP. *********************************************************/ import java.util.; class Dsp043 implements GraphIntfc01{ final double pi = Math.PI; int len = 300; double[] timeDataIn = new double[len]; double[] realSpect = new double[len]; double[] imagSpect = new double[len]; double[] angle = new double[len];//unused double[] magnitude = new double[len]; double[] timeDataOut = new double[len]; int zero = 0; public Dsp043(){//constructor //Create raw waveform consisting of mirror image // sinusoidal segments with a sample having a value // of 0 in the center. //Set shift to a nonzero value to cause the imaginary // part of the transform to be non zero. for(int x = 0;x < len/4;x++){ timeDataIn[len/2 + x + 1] = 80.0 * Math.sin(2pi(x)1.0/20.0); timeDataIn[len/2 - x - 1] = timeDataIn[len/2 + x + 1]; }//end for loop //Compute DFT of the time data and save it in // the output arrays. ForwardRealToComplex01.transform(timeDataIn, realSpect, imagSpect, angle, magnitude, zero, 0.0, 1.0); //Compute inverse DFT of the spectal data and // save output time data in output array InverseComplexToReal01.inverseTransform(realSpect, imagSpect, timeDataOut); }//end constructor //-----------------------------------------------------// //The following six methods are required by the interface // named GraphIntfc01. public int getNmbr(){ //Return number of curves to plot. Must not exceed 5. return 5; }//end getNmbr //-----------------------------------------------------// public double f1(double x){ int index = (int)Math.round(x); if(index < 0 \|\| index > timeDataIn.length-1){ return 0; }else{ return timeDataIn[index]; }//end else }//end function //-----------------------------------------------------// public double f2(double x){ int index = (int)Math.round(x); if(index < 0 \|\| index > realSpect.length-1){ return 0; }else{ //scale for convenient viewing return 5realSpect[index]; }//end else }//end function //-----------------------------------------------------// public double f3(double x){ int index = (int)Math.round(x); if(index < 0 \|\| index > imagSpect.length-1){ return 0; }else{ //scale for convenient viewing return 5imagSpect[index]; }//end else }//end function //-----------------------------------------------------// public double f4(double x){ int index = (int)Math.round(x); if(index < 0 \|\| index > magnitude.length-1){ return 0; }else{ //scale for convenient viewing return 5magnitude[index]; }//end else }//end function //-----------------------------------------------------// public double f5(double x){ int index = (int)Math.round(x); if(index < 0 \|\| index > timeDataOut.length-1){ return 0; }else{ return timeDataOut[index]; }//end else }//end function }//end sample class Dsp043Listing 11```\n\nListing 12\n\n ```/File ForwardDCT01.java Copyright 2006, R.G.Baldwin Rev 01/03/06 The static method named transform performs a forward Discreet Cosine Transform (DCT) on an incoming time series and returns the DCT spectrum. See http://en.wikipedia.org/wiki/Discrete_cosine_transform #DCT-II and http://rkb.home.cern.ch/rkb/AN16pp/node61.html for background on the DCT. This formulation is from http://www.cmlab.csie.ntu.edu.tw/cml/dsp/training/ coding/transform/dct.html Incoming parameters are: double[] x - incoming real data double[] y - outgoing real data Tested using J2SE 5.0 under WinXP. Requires J2SE 5.0 or later due to the use of static import of Math class. ********************************************************/ import static java.lang.Math.; public class ForwardDCT01{ public static void transform(double[] x, double[] y){ int N = x.length; //Outer loop interates on frequency values. for(int k=0; k < N;k++){ double sum = 0.0; //Inner loop iterates on time-series points. for(int n=0; n < N; n++){ double arg = PIk(2.0n+1)/(2N); double cosine = cos(arg); double product = x[n]cosine; sum += product; }//end inner loop double alpha; if(k == 0){ alpha = 1.0/sqrt(2); }else{ alpha = 1; }//end else y[k] = sumalphasqrt(2.0/N); }//end outer loop }//end transform method //-----------------------------------------------------// }//end class ForwardDCT01Listing 12```\n\nListing 13\n\n ```/File InverseDCT01.java Copyright 2006, R.G.Baldwin Rev 01/03/06 The static method named transform performs an inverse Discreet Cosine Transform (DCT) on an incoming DCT spectrum and returns the DCT time series. See http://en.wikipedia.org/wiki/Discrete_cosine_transform #DCT-II and http://rkb.home.cern.ch/rkb/AN16pp/node61.html for background on the DCT. This formulation is from http://www.cmlab.csie.ntu.edu.tw/cml/dsp/training/ coding/transform/dct.html Incoming parameters are: double[] y - incoming real data double[] x - outgoing real data Tested using J2SE 5.0 under WinXP. Requires J2SE 5.0 or later due to the use of static import of Math class. *********************************************************/ import static java.lang.Math.; public class InverseDCT01{ public static void transform(double[] y, double[] x){ int N = y.length; //Outer loop interates on time values. for(int n=0; n < N;n++){ double sum = 0.0; //Inner loop iterates on frequency values for(int k=0; k < N; k++){ double arg = PIk(2.0n+1)/(2N); double cosine = cos(arg); double product = y[k]cosine; double alpha; if(k == 0){ alpha = 1.0/sqrt(2); }else{ alpha = 1; }//end else sum += alpha product; }//end inner loop x[n] = sum * sqrt(2.0/N); }//end outer loop }//end transform method //-----------------------------------------------------// }//end class InverseDCT01Listing 13```\n\nListing 14\n\n ```/* File Dsp042.java Copyright 2006, R.G.Baldwin Revised 01/05/06 Note: Dsp044 will investigate the impact of breaking the time series into eight-sample segments to be consistent with the 8x8 blocks in JPEG. Illustrates the application of forward and inverse Discrete Cosine Transform (DCT) to three different waveforms. Very similar to Dsp035, which applies full forward and inverse Fourier transforms to the same three waveforms See http://en.wikipedia.org/wiki/Discrete_cosine_transform #DCT-II, http://rkb.home.cern.ch/rkb/AN16pp/node61.html, and http://www.cmlab.csie.ntu.edu.tw/cml/dsp/training/ coding/transform/dct.html fortechnical background information on the Discrete Cosine Transform. Shows a plot of the input time series, the DCT spectrum, and the output time series resulting from the inverse DCT. Run with Graph03. Enter the following to run the program: java Graph03 Dsp042 Execution of this program requires access to the following class files: Dsp042.class ForwardDCT01.class Graph01.class GraphIntfc01.class GUI\\$MyCanvas.class GUI.class InverseDCT01.class Tested using J2SE 5.0 under WinXP. Requires J2SE 5.0 or later due to the use of static import of Math class. *********************************************************/ import java.util.; import static java.lang.Math.; class Dsp042 implements GraphIntfc01{ int len = 256; double[] timeDataIn = new double[len]; double[] realSpect = new double[len]; double[] timeDataOut = new double[len]; int zero = 0; public Dsp042(){//constructor //Create the raw data pulses timeDataIn = 0; timeDataIn = 50; timeDataIn = 75; timeDataIn = 80; timeDataIn = 75; timeDataIn = 50; timeDataIn = 25; timeDataIn = 0; timeDataIn = -25; timeDataIn = -50; timeDataIn = -75; timeDataIn = -80; timeDataIn = -60; timeDataIn = -40; timeDataIn = -26; timeDataIn = -17; timeDataIn = -11; timeDataIn = -8; timeDataIn = -5; timeDataIn = -3; timeDataIn = -2; timeDataIn = -1; timeDataIn = 80; timeDataIn = 80; timeDataIn = 80; timeDataIn = 80; timeDataIn = -80; timeDataIn = -80; timeDataIn = -80; timeDataIn = -80; timeDataIn = 80; timeDataIn = 80; timeDataIn = 80; timeDataIn = 80; timeDataIn = -80; timeDataIn = -80; timeDataIn = -80; timeDataIn = -80; //Create raw data sinusoid for(int x = len/3;x < 3len/4;x++){ timeDataIn[x] = 80.0 * Math.sin( 2PI(x)1.0/20.0); }//end for loop //Compute forward DCT of the time data and save it in // the output array. ForwardDCT01.transform(timeDataIn,realSpect); //Compute inverse DCT of the time data and save it in // the output array. InverseDCT01.transform(realSpect,timeDataOut); }//end constructor //-----------------------------------------------------// //The following six methods are required by the interface // named GraphIntfc01. public int getNmbr(){ //Return number of curves to plot. Must not exceed 5. return 3; }//end getNmbr //-----------------------------------------------------// public double f1(double x){ int index = (int)round(x); if(index < 0 \|\| index > timeDataIn.length-1){ return 0; }else{ return timeDataIn[index]; }//end else }//end function //-----------------------------------------------------// public double f2(double x){ int index = (int)round(x); if(index < 0 \|\| index > realSpect.length-1){ return 0; }else{ //Scale for convenient viewing return 0.22realSpect[index]; }//end else }//end function //-----------------------------------------------------// public double f3(double x){ int index = (int)round(x); if(index < 0 \|\| index > timeDataOut.length-1){ return 0; }else{ return timeDataOut[index]; }//end else }//end function //-----------------------------------------------------// public double f4(double x){ return 0; }//end function //-----------------------------------------------------// public double f5(double x){ return 0; }//end function //-----------------------------------------------------// }//end sample class Dsp042 Listing 14```\n\nListing 15\n\n ```/* File Dsp044.java Copyright 2006, R.G.Baldwin Revised 01/03/06 Update of Dsp042 to investigate the impact of breaking the time series into eight-sample segments in order to be consistent with the 8x8 blocks in JPEG. There appears to be no impact. The output is a very good replica of the input. Illustrates the application of forward and inverse Discrete Cosine Transform (DCT) to three different waveforms. Very similar to Dsp035, which applies full forward and inverse Fourier transforms to the same three waveforms See http://en.wikipedia.org/wiki/Discrete_cosine_transform #DCT-II, http://rkb.home.cern.ch/rkb/AN16pp/node61.html, and http://www.cmlab.csie.ntu.edu.tw/cml/dsp/training/ coding/transform/dct.html fortechnical background information on the Discrete Cosine Transform. Shows a plot of the input time series, and the output time series resulting from the inverse DCT. Can't show the spectrum because a new spectrum is produced every eight samples. Run with Graph03. Enter the following to run the program: java Graph03 Dsp044 Execution of this program requires access to the following class files: Dsp044.class ForwardDCT01.class Graph01.class GraphIntfc01.class GUI\\$MyCanvas.class GUI.class InverseDCT01.class Tested using J2SE 5.0 under WinXP. Requires J2SE 5.0 or later due to the use of static import of Math class. *********************************************************/ import java.util.; import static java.lang.Math.; class Dsp044 implements GraphIntfc01{ int len = 256; double[] timeDataIn = new double[len]; double[] timeDataOut = new double[len]; int zero = 0; public Dsp044(){//constructor //Create the raw data pulses timeDataIn = 0; timeDataIn = 50; timeDataIn = 75; timeDataIn = 80; timeDataIn = 75; timeDataIn = 50; timeDataIn = 25; timeDataIn = 0; timeDataIn = -25; timeDataIn = -50; timeDataIn = -75; timeDataIn = -80; timeDataIn = -60; timeDataIn = -40; timeDataIn = -26; timeDataIn = -17; timeDataIn = -11; timeDataIn = -8; timeDataIn = -5; timeDataIn = -3; timeDataIn = -2; timeDataIn = -1; timeDataIn = 80; timeDataIn = 80; timeDataIn = 80; timeDataIn = 80; timeDataIn = -80; timeDataIn = -80; timeDataIn = -80; timeDataIn = -80; timeDataIn = 80; timeDataIn = 80; timeDataIn = 80; timeDataIn = 80; timeDataIn = -80; timeDataIn = -80; timeDataIn = -80; timeDataIn = -80; //Create raw data sinusoid for(int x = len/3;x < 3len/4;x++){ timeDataIn[x] = 80.0 * Math.sin( 2PI(x)1.0/20.0); }//end for loop double[] workingArrayIn = new double; double[] workingArrayOut = new double; double[] realSpect = new double; int segmentCnt = 0; //Compute forward and inverse DCTs on the input signal, // eight samples at a time. Concatenate the output // segments from the DCT to represent the output // signal. while((segmentCnt + 8) <= len){ System.arraycopy(timeDataIn, segmentCnt, workingArrayIn, 0, 8); //Compute forward DCT of the time data and save it in // the output array. ForwardDCT01.transform(workingArrayIn,realSpect); //Compute inverse DCT of the time data and save it in // the output array. InverseDCT01.transform(realSpect,workingArrayOut); //Concatenate the new output with the old output. System.arraycopy(workingArrayOut, 0, timeDataOut, segmentCnt, 8); segmentCnt += 8; }//end while }//end constructor //-----------------------------------------------------// //The following six methods are required by the interface // named GraphIntfc01. public int getNmbr(){ //Return number of curves to plot. Must not exceed 5. return 2; }//end getNmbr //-----------------------------------------------------// public double f1(double x){ int index = (int)round(x); if(index < 0 \|\| index > timeDataIn.length-1){ return 0; }else{ return timeDataIn[index]; }//end else }//end function //-----------------------------------------------------// public double f2(double x){ int index = (int)round(x); if(index < 0 \|\| index > timeDataOut.length-1){ return 0; }else{ return timeDataOut[index]; }//end else }//end function //-----------------------------------------------------// public double f3(double x){ return 0; }//end function //-----------------------------------------------------// public double f4(double x){ return 0; }//end function //-----------------------------------------------------// public double f5(double x){ return 0; }//end function //-----------------------------------------------------// }//end class Dsp044 Listing 15```\n\nListing 16\n\n ```/ File Graph03.java Copyright 2002, R.G.Baldwin This program is very similar to Graph01 except that it has been modified to allow the user to manually resize and replot the frame. Note: This program requires access to the interface named GraphIntfc01. This is a plotting program. It is designed to access a class file, which implements GraphIntfc01, and to plot up to five functions defined in that class file. The plotting surface is divided into the required number of equally sized plotting areas, and one function is plotted on cartesian coordinates in each area. The methods corresponding to the functions are named f1, f2, f3, f4, and f5. The class containing the functions must also define a method named getNmbr(), which takes no parameters and returns the number of functions to be plotted. If this method returns a value greater than 5, a NoSuchMethodException will be thrown. Note that the constructor for the class that implements GraphIntfc01 must not require any parameters due to the use of the newInstance method of the Class class to instantiate an object of that class. If the number of functions is less than 5, then the absent method names must begin with f5 and work down toward f1. For example, if the number of functions is 3, then the program will expect to call methods named f1, f2, and f3. It is OK for the absent methods to be defined in the class. They simply won't be invoked. The plotting areas have alternating white and gray backgrounds to make them easy to separate visually. All curves are plotted in black. A cartesian coordinate system with axes, tic marks, and labels is drawn in red in each plotting area. The cartesian coordinate system in each plotting area has the same horizontal and vertical scale, as well as the same tic marks and labels on the axes. The labels displayed on the axes, correspond to the values of the extreme edges of the plotting area. The program also compiles a sample class named junk, which contains five methods and the method named getNmbr. This makes it easy to compile and test this program in a stand-alone mode. At runtime, the name of the class that implements the GraphIntfc01 interface must be provided as a command-line parameter. If this parameter is missing, the program instantiates an object from the internal class named junk and plots the data provided by that class. Thus, you can test the program by running it with no command-line parameter. This program provides the following text fields for user input, along with a button labeled Graph. This allows the user to adjust the parameters and replot the graph as many times with as many plotting scales as needed: xMin = minimum x-axis value xMax = maximum x-axis value yMin = minimum y-axis value yMax = maximum y-axis value xTicInt = tic interval on x-axis yTicInt = tic interval on y-axis xCalcInc = calculation interval The user can modify any of these parameters and then click the Graph button to cause the five functions to be re-plotted according to the new parameters. Whenever the Graph button is clicked, the event handler instantiates a new object of the class that implements the GraphIntfc01 interface. Depending on the nature of that class, this may be redundant in some cases. However, it is useful in those cases where it is necessary to refresh the values of instance variables defined in the class (such as a counter, for example). Tested using JDK 1.4.0 under Win 2000. This program uses constants that were first defined in the Color class of v1.4.0. Therefore, the program requires v1.4.0 or later to compile and run correctly. *************************************/ import java.awt.; import java.awt.event.; import java.awt.geom.; import javax.swing.; import javax.swing.border.; class Graph03{ public static void main( String[] args) throws NoSuchMethodException, ClassNotFoundException, InstantiationException, IllegalAccessException{ if(args.length == 1){ //pass command-line paramater new GUI(args); }else{ //no command-line parameter given new GUI(null); }//end else }// end main }//end class Graph03 definition //===================================// class GUI extends JFrame implements ActionListener{ //Define plotting parameters and // their default values. double xMin = 0.0; double xMax = 400.0; double yMin = -100.0; double yMax = 100.0; //Tic mark intervals double xTicInt = 20.0; double yTicInt = 20.0; //Tic mark lengths. If too small // on x-axis, a default value is // used later. double xTicLen = (yMax-yMin)/50; double yTicLen = (xMax-xMin)/50; //Calculation interval along x-axis double xCalcInc = 1.0; //Text fields for plotting parameters JTextField xMinTxt = new JTextField(\"\" + xMin); JTextField xMaxTxt = new JTextField(\"\" + xMax); JTextField yMinTxt = new JTextField(\"\" + yMin); JTextField yMaxTxt = new JTextField(\"\" + yMax); JTextField xTicIntTxt = new JTextField(\"\" + xTicInt); JTextField yTicIntTxt = new JTextField(\"\" + yTicInt); JTextField xCalcIncTxt = new JTextField(\"\" + xCalcInc); //Panels to contain a label and a // text field JPanel pan0 = new JPanel(); JPanel pan1 = new JPanel(); JPanel pan2 = new JPanel(); JPanel pan3 = new JPanel(); JPanel pan4 = new JPanel(); JPanel pan5 = new JPanel(); JPanel pan6 = new JPanel(); //Misc instance variables int frmWidth = 408; int frmHeight = 430; int width; int height; int number; GraphIntfc01 data; String args = null; //Plots are drawn on the canvases // in this array. Canvas[] canvases; //Constructor GUI(String args)throws NoSuchMethodException, ClassNotFoundException, InstantiationException, IllegalAccessException{ if(args != null){ //Save for use later in the // ActionEvent handler this.args = args; //Instantiate an object of the // target class using the String // name of the class. data = (GraphIntfc01) Class.forName(args). newInstance(); }else{ //Instantiate an object of the // test class named junk. data = new junk(); }//end else //Create array to hold correct // number of Canvas objects. canvases = new Canvas[data.getNmbr()]; //Throw exception if number of // functions is greater than 5. number = data.getNmbr(); if(number > 5){ throw new NoSuchMethodException( \"Too many functions. \" + \"Only 5 allowed.\"); }//end if //Create the control panel and // give it a border for cosmetics. JPanel ctlPnl = new JPanel(); ctlPnl.setLayout(//?rows x 4 cols new GridLayout(0,4)); ctlPnl.setBorder( new EtchedBorder()); //Button for replotting the graph JButton graphBtn = new JButton(\"Graph\"); graphBtn.addActionListener(this); //Populate each panel with a label // and a text field. Will place // these panels in a grid on the // control panel later. pan0.add(new JLabel(\"xMin\")); pan0.add(xMinTxt); pan1.add(new JLabel(\"xMax\")); pan1.add(xMaxTxt); pan2.add(new JLabel(\"yMin\")); pan2.add(yMinTxt); pan3.add(new JLabel(\"yMax\")); pan3.add(yMaxTxt); pan4.add(new JLabel(\"xTicInt\")); pan4.add(xTicIntTxt); pan5.add(new JLabel(\"yTicInt\")); pan5.add(yTicIntTxt); pan6.add(new JLabel(\"xCalcInc\")); pan6.add(xCalcIncTxt); //Add the populated panels and the // button to the control panel with // a grid layout. ctlPnl.add(pan0); ctlPnl.add(pan1); ctlPnl.add(pan2); ctlPnl.add(pan3); ctlPnl.add(pan4); ctlPnl.add(pan5); ctlPnl.add(pan6); ctlPnl.add(graphBtn); //Create a panel to contain the // Canvas objects. They will be // displayed in a one-column grid. JPanel canvasPanel = new JPanel(); canvasPanel.setLayout(//?rows,1 col new GridLayout(0,1)); //Create a custom Canvas object for // each function to be plotted and // add them to the one-column grid. // Make background colors alternate // between white and gray. for(int cnt = 0; cnt < number; cnt++){ switch(cnt){ case 0 : canvases[cnt] = new MyCanvas(cnt); canvases[cnt].setBackground( Color.WHITE); break; case 1 : canvases[cnt] = new MyCanvas(cnt); canvases[cnt].setBackground( Color.LIGHT_GRAY); break; case 2 : canvases[cnt] = new MyCanvas(cnt); canvases[cnt].setBackground( Color.WHITE); break; case 3 : canvases[cnt] = new MyCanvas(cnt); canvases[cnt].setBackground( Color.LIGHT_GRAY); break; case 4 : canvases[cnt] = new MyCanvas(cnt); canvases[cnt]. setBackground(Color.WHITE); }//end switch //Add the object to the grid. canvasPanel.add(canvases[cnt]); }//end for loop //Add the sub-assemblies to the // frame. Set its location, size, // and title, and make it visible. getContentPane(). add(ctlPnl,\"South\"); getContentPane(). add(canvasPanel,\"Center\"); setBounds(0,0,frmWidth,frmHeight); if(args == null){ setTitle(\"Graph03, \" + \"Copyright 2002, \" + \"Richard G. Baldwin\"); }else{ setTitle(\"Graph03/\" + args + \" Copyright 2002, \" + \"R. G. Baldwin\"); }//end else setVisible(true); //Set to exit on X-button click setDefaultCloseOperation( EXIT_ON_CLOSE); //Guarantee a repaint on startup. for(int cnt = 0; cnt < number; cnt++){ canvases[cnt].repaint(); }//end for loop }//end constructor //---------------------------------// //This event handler is registered // on the JButton to cause the // functions to be replotted. public void actionPerformed( ActionEvent evt){ //Re-instantiate the object that // provides the data try{ if(args != null){ data = (GraphIntfc01)Class. forName(args).newInstance(); }else{ data = new junk(); }//end else }catch(Exception e){ //Known to be safe at this point. // Otherwise would have aborted // earlier. }//end catch //Set plotting parameters using // data from the text fields. xMin = Double.parseDouble( xMinTxt.getText()); xMax = Double.parseDouble( xMaxTxt.getText()); yMin = Double.parseDouble( yMinTxt.getText()); yMax = Double.parseDouble( yMaxTxt.getText()); xTicInt = Double.parseDouble( xTicIntTxt.getText()); yTicInt = Double.parseDouble( yTicIntTxt.getText()); xCalcInc = Double.parseDouble( xCalcIncTxt.getText()); //Calculate new values for the // length of the tic marks on the // axes. If too small on x-axis, // a default value is used later. xTicLen = (yMax-yMin)/50; yTicLen = (xMax-xMin)/50; //Repaint the plotting areas for(int cnt = 0; cnt < number; cnt++){ canvases[cnt].repaint(); }//end for loop }//end actionPerformed //---------------------------------// //This is an inner class, which is used // to override the paint method on the // plotting surface. class MyCanvas extends Canvas{ int cnt;//object number //Factors to convert from double // values to integer pixel locations. double xScale; double yScale; MyCanvas(int cnt){//save obj number this.cnt = cnt; }//end constructor //Override the paint method public void paint(Graphics g){ //Get and save the size of the // plotting surface width = canvases.getWidth(); height = canvases.getHeight(); //Calculate the scale factors xScale = width/(xMax-xMin); yScale = height/(yMax-yMin); //Set the origin based on the // minimum values in x and y g.translate((int)((0-xMin)xScale), (int)((0-yMin)yScale)); drawAxes(g);//Draw the axes g.setColor(Color.BLACK); //Get initial data values double xVal = xMin; int oldX = getTheX(xVal); int oldY = 0; //Use the Canvas obj number to // determine which method to // invoke to get the value for y. switch(cnt){ case 0 : oldY = getTheY(data.f1(xVal)); break; case 1 : oldY = getTheY(data.f2(xVal)); break; case 2 : oldY = getTheY(data.f3(xVal)); break; case 3 : oldY = getTheY(data.f4(xVal)); break; case 4 : oldY = getTheY(data.f5(xVal)); }//end switch //Now loop and plot the points while(xVal < xMax){ int yVal = 0; //Get next data value. Use the // Canvas obj number to // determine which method to // invoke to get the value for y. switch(cnt){ case 0 : yVal = getTheY(data.f1(xVal)); break; case 1 : yVal = getTheY(data.f2(xVal)); break; case 2 : yVal = getTheY(data.f3(xVal)); break; case 3 : yVal = getTheY(data.f4(xVal)); break; case 4 : yVal = getTheY(data.f5(xVal)); }//end switch1 //Convert the x-value to an int // and draw the next line segment int x = getTheX(xVal); g.drawLine(oldX,oldY,x,yVal); //Increment along the x-axis xVal += xCalcInc; //Save end point to use as start // point for next line segment. oldX = x; oldY = yVal; }//end while loop }//end overridden paint method //---------------------------------// //Method to draw axes with tic marks // and labels in the color RED void drawAxes(Graphics g){ g.setColor(Color.RED); //Lable left x-axis and bottom // y-axis. These are the easy // ones. Separate the labels from // the ends of the tic marks by // two pixels. g.drawString(\"\" + (int)xMin, getTheX(xMin), getTheY(xTicLen/2)-2); g.drawString(\"\" + (int)yMin, getTheX(yTicLen/2)+2, getTheY(yMin)); //Label the right x-axis and the // top y-axis. These are the hard // ones because the position must // be adjusted by the font size and // the number of characters. //Get the width of the string for // right end of x-axis and the // height of the string for top of // y-axis //Create a string that is an // integer representation of the // label for the right end of the // x-axis. Then get a character // array that represents the // string. int xMaxInt = (int)xMax; String xMaxStr = \"\" + xMaxInt; char[] array = xMaxStr. toCharArray(); //Get a FontMetrics object that can // be used to get the size of the // string in pixels. FontMetrics fontMetrics = g.getFontMetrics(); //Get a bounding rectangle for the // string Rectangle2D r2d = fontMetrics.getStringBounds( array,0,array.length,g); //Get the width and the height of // the bounding rectangle. The // width is the width of the label // at the right end of the // x-axis. The height applies to // all the labels, but is needed // specifically for the label at // the top end of the y-axis. int labWidth = (int)(r2d.getWidth()); int labHeight = (int)(r2d.getHeight()); //Label the positive x-axis and the // positive y-axis using the width // and height from above to // position the labels. These // labels apply to the very ends of // the axes at the edge of the // plotting surface. g.drawString(\"\" + (int)xMax, getTheX(xMax)-labWidth, getTheY(xTicLen/2)-2); g.drawString(\"\" + (int)yMax, getTheX(yTicLen/2)+2, getTheY(yMax)+labHeight); //Draw the axes g.drawLine(getTheX(xMin), getTheY(0.0), getTheX(xMax), getTheY(0.0)); g.drawLine(getTheX(0.0), getTheY(yMin), getTheX(0.0), getTheY(yMax)); //Draw the tic marks on axes xTics(g); yTics(g); }//end drawAxes //---------------------------------// //Method to draw tic marks on x-axis void xTics(Graphics g){ double xDoub = 0; int x = 0; //Get the ends of the tic marks. int topEnd = getTheY(xTicLen/2); int bottomEnd = getTheY(-xTicLen/2); //If the vertical size of the // plotting area is small, the // calculated tic size may be too // small. In that case, set it to // 10 pixels. if(topEnd < 5){ topEnd = 5; bottomEnd = -5; }//end if //Loop and draw a series of short // lines to serve as tic marks. // Begin with the positive x-axis // moving to the right from zero. while(xDoub < xMax){ x = getTheX(xDoub); g.drawLine(x,topEnd,x,bottomEnd); xDoub += xTicInt; }//end while //Now do the negative x-axis moving // to the left from zero xDoub = 0; while(xDoub > xMin){ x = getTheX(xDoub); g.drawLine(x,topEnd,x,bottomEnd); xDoub -= xTicInt; }//end while }//end xTics //---------------------------------// //Method to draw tic marks on y-axis void yTics(Graphics g){ double yDoub = 0; int y = 0; int rightEnd = getTheX(yTicLen/2); int leftEnd = getTheX(-yTicLen/2); //Loop and draw a series of short // lines to serve as tic marks. // Begin with the positive y-axis // moving up from zero. while(yDoub < yMax){ y = getTheY(yDoub); g.drawLine(rightEnd,y,leftEnd,y); yDoub += yTicInt; }//end while //Now do the negative y-axis moving // down from zero. yDoub = 0; while(yDoub > yMin){ y = getTheY(yDoub); g.drawLine(rightEnd,y,leftEnd,y); yDoub -= yTicInt; }//end while }//end yTics //---------------------------------// //This method translates and scales // a double y value to plot properly // in the integer coordinate system. // In addition to scaling, it causes // the positive direction of the // y-axis to be from bottom to top. int getTheY(double y){ double yDoub = (yMax+yMin)-y; int yInt = (int)(yDoubyScale); return yInt; }//end getTheY //---------------------------------// //This method scales a double x value // to plot properly in the integer // coordinate system. int getTheX(double x){ return (int)(xxScale); }//end getTheX //---------------------------------// }//end inner class MyCanvas //===================================// }//end class GUI //===================================// //Sample test class. Required for // compilation and stand-alone // testing. class junk implements GraphIntfc01{ public int getNmbr(){ //Return number of functions to // process. Must not exceed 5. return 4; }//end getNmbr public double f1(double x){ return (xxx)/200.0; }//end f1 public double f2(double x){ return -(xxx)/200.0; }//end f2 public double f3(double x){ return (xx)/200.0; }//end f3 public double f4(double x){ return 50Math.cos(x/10.0); }//end f4 public double f5(double x){ return 100Math.sin(x/20.0); }//end f5 }//end sample class junk Listing 16```\n\nListing 17\n\n ```/ File GraphIntfc01.java Copyright 2004, R.G.Baldwin Rev 5/14/04 This interface must be implemented by classes whose objects produce data to be plotted by programs such as Graph03 and Graph06. Tested using SDK 1.4.2 under WinXP. ************************************************/ public interface GraphIntfc01{ public int getNmbr(); public double f1(double x); public double f2(double x); public double f3(double x); public double f4(double x); public double f5(double x); }//end GraphIntfc01Listing 17```\n\nCopyright 2006, Richard G. Baldwin. Reproduction in whole or in part in any\nform or medium without express written permission from Richard Baldwin is\nprohibited.\n\nRichard Baldwin is a\ncollege professor (at Austin Community College in Austin, TX) and private\nconsultant whose primary focus is a combination of Java, C#, and XML. In\naddition to the many platform and/or language independent benefits of Java and\nC# applications, he believes that a combination of Java, C#, and XML will become\nthe primary driving force in the delivery of structured information on the Web.\n\nRichard has participated in numerous consulting projects and he\nfrequently provides onsite training at the high-tech companies located in and\naround Austin, Texas. He is the author of Baldwin’s Programming\nTutorials, which have gained a\nworldwide following among experienced and aspiring programmers. He has also\npublished articles in JavaPro magazine.\n\nIn addition to his programming expertise, Richard has many years of\npractical experience in Digital Signal Processing (DSP). His first job after he\nearned his Bachelor’s degree was doing DSP in the Seismic Research Department of\nTexas Instruments. (TI is still a world leader in DSP.) In the following\nyears, he applied his programming and DSP expertise to other interesting areas\nincluding sonar and underwater acoustics.\n\nRichard holds an MSEE degree from Southern Methodist University and has\nmany years of experience in the application of computer technology to real-world\nproblems.\n\n[email protected]" ]	[ null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20409%20431'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20409%20431'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20409%20293'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20409%20226'%3E%3C/svg%3E", null, "https://devcomprd.wpengine.com/wp-content/uploads/2021/03/java2444a4.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7223146,"math_prob":0.94257843,"size":56997,"snap":"2021-43-2021-49","text_gpt3_token_len":14061,"char_repetition_ratio":0.17033671,"word_repetition_ratio":0.23231606,"special_character_ratio":0.28508517,"punctuation_ratio":0.13609008,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9893466,"pos_list":[0,1,2,3,4,5,6,7,8,9,10],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,3,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-12-01T19:37:43Z\",\"WARC-Record-ID\":\"<urn:uuid:49cffb39-83f5-4967-8b5a-5d8441ed3e58>\",\"Content-Length\":\"542972\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:37629ffc-c194-4826-b1cc-8e9304d163fc>\",\"WARC-Concurrent-To\":\"<urn:uuid:fb41c4c7-96ed-4875-bed3-aa2c06401400>\",\"WARC-IP-Address\":\"141.193.213.20\",\"WARC-Target-URI\":\"https://www.developer.com/java/understanding-the-discrete-cosine-transform-in-java/\",\"WARC-Payload-Digest\":\"sha1:JTVDIRKE2B6AT4XLY4B22FJDJ37ILK45\",\"WARC-Block-Digest\":\"sha1:GGCHG34J7ARRMTG4JLURNVLQTSMBGYVE\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964360881.12_warc_CC-MAIN-20211201173718-20211201203718-00511.warc.gz\"}"}
https://archive.lib.msu.edu/crcmath/math/math/c/c335.htm	[ "## Circular Functions\n\nThe functions describing the horizontal and vertical positions of a point on a Circle as a function of Angle (Cosine and Sine) and those functions derived from them:", null, "", null, "", null, "(1)", null, "", null, "", null, "(2)", null, "", null, "", null, "(3)", null, "", null, "", null, "(4)\n\nThe study of circular functions is called Trigonometry.\n\nSee also Cosecant, Cosine, Cotangent, Elliptic Function, Generalized Hyperbolic Functions, Hyperbolic Functions, Secant, Sine, Tangent, Trigonometry\n\nReferences\n\nAbramowitz, M. and Stegun, C. A. (Eds.). Circular Functions.'' §4.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 71-79, 1972." ]	[ null, "https://archive.lib.msu.edu/crcmath/math/math/c/c2_97.gif", null, "https://archive.lib.msu.edu/crcmath/math/math/c/c2_98.gif", null, "https://archive.lib.msu.edu/crcmath/math/math/c/c2_99.gif", null, "https://archive.lib.msu.edu/crcmath/math/math/c/c2_100.gif", null, "https://archive.lib.msu.edu/crcmath/math/math/c/c2_98.gif", null, "https://archive.lib.msu.edu/crcmath/math/math/c/c2_101.gif", null, "https://archive.lib.msu.edu/crcmath/math/math/c/c2_102.gif", null, "https://archive.lib.msu.edu/crcmath/math/math/c/c2_98.gif", null, "https://archive.lib.msu.edu/crcmath/math/math/c/c2_103.gif", null, "https://archive.lib.msu.edu/crcmath/math/math/c/c2_104.gif", null, "https://archive.lib.msu.edu/crcmath/math/math/c/c2_98.gif", null, "https://archive.lib.msu.edu/crcmath/math/math/c/c2_105.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.723819,"math_prob":0.89658207,"size":549,"snap":"2021-43-2021-49","text_gpt3_token_len":159,"char_repetition_ratio":0.15045871,"word_repetition_ratio":0.0,"special_character_ratio":0.2586521,"punctuation_ratio":0.25688073,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9654687,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],"im_url_duplicate_count":[null,1,null,6,null,1,null,1,null,6,null,1,null,1,null,6,null,1,null,1,null,6,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-16T05:52:15Z\",\"WARC-Record-ID\":\"<urn:uuid:f0029937-de45-43f0-aa53-efa4b19f00e3>\",\"Content-Length\":\"6270\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:54a85009-a327-43a3-b515-474650c0b3b7>\",\"WARC-Concurrent-To\":\"<urn:uuid:ca21e24c-84ea-44be-a605-7a0d4a2e358e>\",\"WARC-IP-Address\":\"35.8.223.87\",\"WARC-Target-URI\":\"https://archive.lib.msu.edu/crcmath/math/math/c/c335.htm\",\"WARC-Payload-Digest\":\"sha1:6SC5YWLFQ4OSROOAENUSSLMXPQTHZ2NH\",\"WARC-Block-Digest\":\"sha1:3BS6T5ITIKMBFRBZRDOLFNRNBDF2VGUC\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323583423.96_warc_CC-MAIN-20211016043926-20211016073926-00513.warc.gz\"}"}
https://www.kitplanes.com/stressing-structure-10/	[ "# Stressing Structure\n\nBuckling of Panels.\n\n0", null, "Figure 1: Mike Rettig’s RV-10 aft fuselage section uses aluminum panels and stiffeners for primary structure. (Photo: Mike Rettig)\n\nYou might have noticed that most airplanes have some sort of exterior surface, perhaps fabric or something more rigid like aluminum or composite. Structurally, it can make sense to make the outer surface rigid, since that can provide the maximum possible rigidity. Plus the surface generally has to be there anyway, so it might as well be useful. If it’s properly done and carries load, then it can even save weight, compared to an internal skeleton.\n\nPanels can buckle when loaded in compression or shear, and that’s a major issue. The panels are generally thin, and thin panels can’t carry as much load, obviously, as thick panels. Less obvious is the fact that their overall dimensions affect the buckling load just as much as the thickness does. That’s one of the reasons why we often have stiffeners attached; they divide a single large panel into smaller ones. If done right they also carry some load themselves and that further helps. We’ll look at stiffeners in another article. Here, we’re looking at the panels themselves. We’re assuming that the panels are flat. Often there’s actually a slight curvature and we should ignore the minor additional buckling resistance the curvature provides. If the curve is significant, though, we ought to take advantage of the curve, because it’ll stiffen the panel. If the curve is in the direction of the compression load, so that the panel is being loaded out of plane, the techniques described here aren’t applicable.", null, "Figure 2: Parameters for buckling in compression. Generally speaking, curve C is usually applicable.\n\nPanels start to buckle at a stress that we can calculate, fortunately. Here’s the equation for this for panels loaded in compression:\n\nEquation 1: fc = (π2 * Kc * E) / (12 * (1 – ν2)) * ( t / b)2\n\nWhere\nfc = compressive stress that will buckle the panel, psi\nKc = a constant we’ll pull off Figure 2\nE = our old friend, the modulus of elasticity, psi\nt = the thickness of the panel, inches\nb = the short dimension of the panel, inches\nν = Poisson’s ratio, a material property\n\nThe equation for the buckling stress of a panel in shear is almost exactly the same. The only differences are that it uses a constant from a different graph:\n\nEquation 2: fs = (π2 * Ks * E) / (12 * (1 – ν2)) * ( t / b)2\n\nWhere\nfs = the shear stress that will buckle the panel, psi\nKs = a shear constant we’ll find on Figure 3\n\nThere might even be some in-plane bending. If there is, that also uses a form of the same basic equation:\n\nEquation 3: fb = (π2 * Kb * E) / (12 * (1 – ν2)) * ( t / b)2\n\nWhere\nfb = the bending stress that will buckle the panel, psi\nKb = a constant we’ll get from Figure 4\n\nFigure 4 shows curves that are a bit more complicated than the other graphs. The first thing to note is that the curves are wavy. You can see a little bit of that in Figures 2 and 3, too. This is because the proportions of the panel matter a lot in bending, and this gives you a way to tweak the strength at relatively low cost. If you can get a length/width ratio that falls near a peak on the curve, the panel will buckle at a higher stress than if it falls in a trough.\n\nAnother aspect of the curves is that there are a lot of them, and they seem to depend on a mysterious factor called β. β is a number that is used to assess the amount of the panel that’s in compression. Normally, of course, symmetric panels or beams in pure bending have a neutral axis halfway across their height. If there’s a compression stress applied to the panel in addition to the bending, then there’s more of the panel that’s in compression and β allows for that. If there’s no compression,\n\nβ = 2. If there were some compression, we’d have to draw it out, making a little sketch to scale and assess β as necessary.\n\nAs you might have guessed by now, it’s not quite as simple as it seemed. But it almost is.\n\nFirst, we have the usual three types of edge fixity. Since the panels are rectangular, they have four sides. Each side can either be fixed, simply-supported or free. It’s difficult to put loads or reactions into a free edge, so generally those would be on the unloaded sides, if any are free at all. Most aircraft panels aren’t. The panels are either simply supported or fixed. “Clamped” is another word used to describe a fixed edge.\n\nSimply supported means that the edge can’t move out of plane, but it’s free to rotate. It’s the panel version of the pin-ended condition that struts have. For example, a large panel with a stiffener down the middle is simply supported at the stiffener. In Figure 1, all the skin panels are simply supported by the stiffeners or bulkheads.\n\nA fixed edge means that the edge not only can’t move out of plane, it can’t rotate. These aren’t too common, but they do exist. A good example is that often if a wingskin is riveted firmly to a mainspar cap, the skin’s probably fixed at the spar. But it might not be, since that takes considerable stiffness, so be sure. However, I’d expect it to be simply supported at a rib, though.\n\nTable 1 shows some of the material properties for some of the more common metals. While not many homebuilt aircraft use panels of 4130, a welded box or plate structure of it is often highly loaded, and it’s worth checking it for buckling. I’ve included titanium because of its excellent strength. If you’re designing an airplane, it might be worth considering.", null, "Table 1. Material Properties\nRead the exponential notation like this: 29 * 106 means 29,000,000. Look at the exponent on the 10 and add that number of zeros to the number. Also, ksi is the same as 1,000 psi or 103 psi, so don’t get ksi mixed up with psi.\n\nTable 1 also shows the maximum strength values that go along with these equations. For the rest of the material properties, you’ll have to look up MMPDS or a similar resource.\n\nThis site uses Akismet to reduce spam. Learn how your comment data is processed." ]	[ null, "https://s28490.pcdn.co/wp-content/uploads/2019/05/stressing-structure_01.jpg", null, "https://s28490.pcdn.co/wp-content/uploads/2019/05/stressing-structure_02.jpg", null, "https://s28490.pcdn.co/wp-content/uploads/2019/05/stressing-structure_06.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9341799,"math_prob":0.9576565,"size":5902,"snap":"2023-40-2023-50","text_gpt3_token_len":1381,"char_repetition_ratio":0.12902679,"word_repetition_ratio":0.039704524,"special_character_ratio":0.23568282,"punctuation_ratio":0.10813008,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9568431,"pos_list":[0,1,2,3,4,5,6],"im_url_duplicate_count":[null,2,null,2,null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-05T08:23:00Z\",\"WARC-Record-ID\":\"<urn:uuid:ee7af9d2-36c0-4ecc-9868-19cf639c81b4>\",\"Content-Length\":\"158792\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:de83ca85-812b-48e7-9feb-c4dace493fa5>\",\"WARC-Concurrent-To\":\"<urn:uuid:db0f389a-74a6-4798-acaf-fa44633a1eab>\",\"WARC-IP-Address\":\"172.67.153.198\",\"WARC-Target-URI\":\"https://www.kitplanes.com/stressing-structure-10/\",\"WARC-Payload-Digest\":\"sha1:AAXRHO5UOYWKMV6YDPUBJM4ESC2TVWY5\",\"WARC-Block-Digest\":\"sha1:BNF5AD25DJ3ZUQUPZQUJ3O5MVJLZSPKO\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100550.40_warc_CC-MAIN-20231205073336-20231205103336-00433.warc.gz\"}"}
https://convertoctopus.com/656-ounces-to-pounds	[ "## Conversion formula\n\nThe conversion factor from ounces to pounds is 0.0625, which means that 1 ounce is equal to 0.0625 pounds:\n\n1 oz = 0.0625 lb\n\nTo convert 656 ounces into pounds we have to multiply 656 by the conversion factor in order to get the mass amount from ounces to pounds. We can also form a simple proportion to calculate the result:\n\n1 oz → 0.0625 lb\n\n656 oz → M(lb)\n\nSolve the above proportion to obtain the mass M in pounds:\n\nM(lb) = 656 oz × 0.0625 lb\n\nM(lb) = 41 lb\n\nThe final result is:\n\n656 oz → 41 lb\n\nWe conclude that 656 ounces is equivalent to 41 pounds:\n\n656 ounces = 41 pounds\n\n## Alternative conversion\n\nWe can also convert by utilizing the inverse value of the conversion factor. In this case 1 pound is equal to 0.024390243902439 × 656 ounces.\n\nAnother way is saying that 656 ounces is equal to 1 ÷ 0.024390243902439 pounds.\n\n## Approximate result\n\nFor practical purposes we can round our final result to an approximate numerical value. We can say that six hundred fifty-six ounces is approximately forty-one pounds:\n\n656 oz ≅ 41 lb\n\nAn alternative is also that one pound is approximately zero point zero two four times six hundred fifty-six ounces.\n\n## Conversion table\n\n### ounces to pounds chart\n\nFor quick reference purposes, below is the conversion table you can use to convert from ounces to pounds\n\nounces (oz) pounds (lb)\n657 ounces 41.063 pounds\n658 ounces 41.125 pounds\n659 ounces 41.188 pounds\n660 ounces 41.25 pounds\n661 ounces 41.313 pounds\n662 ounces 41.375 pounds\n663 ounces 41.438 pounds\n664 ounces 41.5 pounds\n665 ounces 41.563 pounds\n666 ounces 41.625 pounds" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.83709806,"math_prob":0.99801856,"size":1605,"snap":"2021-04-2021-17","text_gpt3_token_len":420,"char_repetition_ratio":0.207995,"word_repetition_ratio":0.0,"special_character_ratio":0.33333334,"punctuation_ratio":0.095679015,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9941943,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-01-22T06:55:30Z\",\"WARC-Record-ID\":\"<urn:uuid:01412ac9-6b5a-4f4c-a888-f419dc10c9b7>\",\"Content-Length\":\"27686\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:9cb26da0-63f3-4618-97f9-70292ce921a1>\",\"WARC-Concurrent-To\":\"<urn:uuid:5707ed19-77eb-47c2-a807-399bcfbd33dc>\",\"WARC-IP-Address\":\"172.67.208.237\",\"WARC-Target-URI\":\"https://convertoctopus.com/656-ounces-to-pounds\",\"WARC-Payload-Digest\":\"sha1:4LEN5HPMGQXTT5YDKRF5L5XH65YNFARX\",\"WARC-Block-Digest\":\"sha1:VAHWJOWXULXDPDSYGBLYGK6LLGFQG7UM\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-04/CC-MAIN-2021-04_segments_1610703529128.47_warc_CC-MAIN-20210122051338-20210122081338-00107.warc.gz\"}"}
http://drewxcaldwell.com/8w101kqg/linear-discriminant-analysis-formula-085406	[ "# linear discriminant analysis formula\n\nJanuary 7, 2021\n\nLinear and Quadratic Discriminant Analysis: Tutorial 4 which is in the quadratic form x>Ax+ b>x+ c= 0. We assume that in population Ïi the probability density function of x is multivariate normal with mean vector Î¼i and variance-covariance matrix Î£(same for all populations). Dimensionality reduction techniques have become critical in machine learning since many high-dimensional datasets exist these days. Maximum-likelihoodand Bayesian parameter estimation techniques assume that the forms for theunderlying probabilitydensities were known, and that we will use thetraining samples to estimate the values of their parameters. It is used for modeling differences in groups i.e. groups, the Bayes' rule is minimize the total error of classification by assigning the object to group The response variable is categorical. Linear Discriminant Analysis in Python (Step-by-Step), Your email address will not be published. As we demonstrated above, i* is the i with the maximum linear score. Since we cannot get Index The most widely used assumption is that our data come from Multivariate Normal distribution which formula is given as. The accuracy has â¦ This tutorial provides a step-by-step example of how to perform linear discriminant analysis in R. Step 1: Load Necessary Libraries. Since this is rarely the case in practice, it’s a good idea to scale each variable in the dataset such that it has a mean of 0 and a standard deviation of 1. Thus, Linear Discriminant Analysis has assumption of Multivariate Normal distribution and all groups have the same covariance matrix. LDA also performs better when sample sizes are small compared to logistic regression, which makes it a preferred method to use when you’re unable to gather large samples. Linear discriminant analysis is used as a tool for classification, dimension reduction, and data visualization. A discriminant â¦ to group Linear discriminant analysis is supervised machine learning, the technique used to find a linear combination of features that separates two or more classes of objects or events. (i.e. We know that we classify the example to the population for â¦ When we have a set of predictor variables and we’d like to classify a response variable into one of two classes, we typically use logistic regression. Linear Discriminant Analysis (LDA) is most commonly used as dimensionality reduction technique in the pre-processing step for pattern-classification and machine learning applications.The goal is to project a dataset onto a lower-dimensional space with good class-separability in order avoid overfitting (âcurse of dimensionalityâ) and â¦ from sklearn.datasets import load_wine import pandas as pd import numpy as np np.set_printoptions(precision=4) from matplotlib import pyplot as plt import â¦ Next The predictor variables follow a normal distribution. For Linear discriminant analysis (LDA): $$\\Sigma_k=\\Sigma$$, âk. are equal for both sides, we can cancel out, Multiply both sides with -2, we need to change the sign of inequality, Assign object with measurement Linear Discriminant Analysis or Normal Discriminant Analysis or Discriminant Function Analysis is a dimensionality reduction technique which is commonly used for the supervised classification problems. which has the highest conditional probability where Bernoulli vs Binomial Distribution: What’s the Difference. These functions are called discriminant functions. If we input the new chip rings that have curvature 2.81 and diameter 5.46, reveal that it does not pass the quality control. We now repeat Example 1 of Linear Discriminant Analysis using this tool.. To perform the analysis, press Ctrl-m and select the Multivariate Analyses option â¦ . Linear Discriminant Analysis (LDA) is a very common technique for dimensionality reduction problems as a pre-processing step for machine learning and pattern classification applications. Account for extreme outliers. It is more practical to assume that the data come from some theoretical distribution. (i.e. Code. In this chapter,we shall instead assume we know the proper forms for the discriminant functions, and use the samples to estimate the values of parameters of theclassifier. Linear discriminant analysis (LDA) is a simple classification method, mathematically robust, and often produces robust models, whose accuracy is as good as more complex methods. Letâs get started. Some examples include: 1. Well, these are some of the questions that we think might be the most common one for the researchers, and it is really important for them to find out the answers to these important questiâ¦ 3. Linear discriminant analysis Linear discriminant function There are many diï¬erent ways to represent a two class pattern classiï¬er. Representation of LDA Models. Linear discriminant analysis is a method you can use when you have a set of predictor variables and youâd like to classify a response variable into two or more classes.. Researchers may build LDA models to predict whether or not a given coral reef will have an overall health of good, moderate, bad, or endangered based on a variety of predictor variables like size, yearly contamination, and age. Preferable reference for this tutorial is, Teknomo, Kardi (2015) Discriminant Analysis Tutorial. 3. Linear Discriminant Analysis takes a data set of cases (also known as observations) as input. This continues with subsequent functions with the requirement that the new function not be correlated with any of the previous functions. In addition, the results of this analysis can be used to predict website preference using consumer age and income for other data points. To get an idea of what LDA is seeking to achieve, let's briefly review linear regression. Linear Discriminant Analysis easily handles the case where the within-class frequencies are unequal and their performances has been examined on randomly generated test data. Required fields are marked . Linear Discriminant Analysis(LDA) is a supervised learning algorithm used as a classifier and a dimensionality reduction algorithm. Some of the dâ¦ < In this case, our decision rule is based on the Linear Score Function, a function of the population means for each of our g populations, $$\\boldsymbol{\\mu}_{i}$$, as well as the pooled variance-covariance matrix. Linear discriminant analysis is an extremely popular dimensionality reduction technique. Thus, we have, We multiply both sides of inequality with Medical. By making this assumption, the classifier becomes linear. Previous 4. Now we go ahead and talk about the LDA (Linear Discriminant Analysis). For Linear discriminant analysis (LDA): $$\\Sigma_k=\\Sigma$$, $$\\forall k$$. . This is almost never the case in real-world data, so we typically scale each variable to have the same mean and variance before actually fitting a LDA model. Finally, regularized discriminant analysis (RDA) is a compromise between LDA and QDA. Linear Discriminant Analysis in Python (Step-by-Step). â¢This will, of course, depend on the classifier. The formula for this normal probability density function is: According to the Naive Bayes classification algorithm. 1 Linear discriminant functions and decision surfaces â¢Deï¬nition It is a function that is a linear combination of the components of x g(x) = wtx + w 0 (1) where w is the weight vector and w 0 the bias â¢A two-category classiï¬er with a discriminant function of the form (1) uses the following rule: and Each predictor variable has the same variance. This method maximizes the ratio of between-class variance to the within-class variance in any particular data set thereby â¦ if, Since factor of , then we can simplify further into, We can write The first function created maximizes the differences between groups on that function. Linear Fisher Discriminant Analysis. To start, import the following libraries. Be sure to check for extreme outliers in the dataset before applying LDA. 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Analysis has assumption of Multivariate normal distribution which formula is given as the quadratic form x > b! ( FDA ) from both a qualitative and quantitative point of view given:. is the probability of the class ) directly from the measurement, what the. Analysis is not the case, you may choose to first transform data! Their performances has been examined on randomly generated test data used to predict preference... Qualitative and quantitative point of view these points and is the go-to linear method for classification! Consider Gaussian distributions for the two classes, the decision boundary of classiï¬cation is quadratic achieve let. A qualitative and quantitative point of view function, but ( sometimes ) not well understood a wide variety fields. Depend on the classifier is linear discriminant analysis formula the case where the within-class variance in particular... To assign the object into separate group data points designed to be for. 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For classification problems cases ( also known as observations ) as input been examined on randomly generated test data (... I 0 ( x ) = d ij ( x ) to check for extreme outliers in the following before. Lda assumes that each predictor variable is called \\ '' class\\ '' and thâ¦ Code linear discriminant Analysis LDA. Input the new chip rings that have curvature 2.81 and diameter 5.46, reveal it! Briefly review linear regression not well understood reveal that it does not pass the quality control reduction, data... Of classiï¬cation is quadratic box, but also must not be correlated any. Binomial distribution: what ’ s the Difference LDA, as we,... Is: According to the within-class variance in any particular data set of cases ( also as! Idea of what LDA is seeking to achieve, let 's briefly review linear.. Is a variant of LDA that allows for non-linear separation of data learning statistics easy and visualization! 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The i with the previous functions also a robust classification method out the first function created maximizes the between.", null, "" ]	[ null, "http://2.gravatar.com/avatar/", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.88762546,"math_prob":0.9536981,"size":18227,"snap":"2021-21-2021-25","text_gpt3_token_len":3608,"char_repetition_ratio":0.16929156,"word_repetition_ratio":0.1764706,"special_character_ratio":0.19525978,"punctuation_ratio":0.116671786,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99560386,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-05-17T01:29:28Z\",\"WARC-Record-ID\":\"<urn:uuid:fa63283e-c114-49ee-a601-c90e0a7fe83a>\",\"Content-Length\":\"92683\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:943717ed-91de-4136-8b9d-247cc2a9af1c>\",\"WARC-Concurrent-To\":\"<urn:uuid:10a69ca8-d22d-427d-a6bf-ffa0449ffcec>\",\"WARC-IP-Address\":\"198.71.233.13\",\"WARC-Target-URI\":\"http://drewxcaldwell.com/8w101kqg/linear-discriminant-analysis-formula-085406\",\"WARC-Payload-Digest\":\"sha1:DOTM64Q4BTJ5ZHBJ3TZMK42IK7HFXDCC\",\"WARC-Block-Digest\":\"sha1:JMO33LOIG4ZDGI252GGEKWCPBGHJOWQE\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-21/CC-MAIN-2021-21_segments_1620243991921.61_warc_CC-MAIN-20210516232554-20210517022554-00465.warc.gz\"}"}
http://devopscr.co/math-facts-worksheets-3rd-grade/	[ "# Math Facts Worksheets 3rd Grade\n\nprintable division worksheets grade basic fact math tables to practice worksheet multiplication and families 5 for works.\n\nfree math worksheets grade printable for addition facts multiplication practice worksheet fact maze lesson graders fractions families 3rd.\n\nmultiplication math facts worksheets grade medium to large size of practice aids worksheet and division d.\n\nmath worksheets grade multiplication 2 3 4 5 times tables facts 3rd.\n\nmedium to large size of math facts worksheets grade worksheet third practice sheets fact families 3rd.\n\nmath facts worksheets printable free multiplication 3rd grade.\n\ngrade worksheets spiral multiplication math fact worksheet facts division old fractions pin families free f.\n\nworksheets grade as well math multiplication skills tutor facts 3rd multipl.\n\ndivision facts worksheets grade mixed math 3rd.\n\naddition facts worksheet super best math worksheets for extra practice images on of multipl.\n\nit this kindergarten addition math facts worksheet is part of a fluency worksheets grade free single digit subtraction fact multiplication family 3rd m.\n\nmultiplication fact family worksheets grade for all download and share free on math facts medium size m.\n\nmultiplication coloring math worksheets meet the facts book level 1 additional photo inside page fun sheets for grade 3rd multip.\n\nmath fact worksheet maker multiplication family worksheets grade facts 3rd.\n\nmath facts worksheets grade printable for image below of fact families 3rd images.\n\ngrade math facts worksheets for third fact families 3rd multiplication worksheet fresh mat.\n\nblog math fact families multiplication division family it addition facts worksheets grade.\n\nsimilar images for math facts worksheets grade 8 multiplication fact family 3rd library.\n\nmath facts practice worksheets grade factors captivating com about least multiple multiplication.\n\nsubtracting fractions worksheets grade kids math addition and subtraction word facts medium to large size of mixed worksheet easy pin adding.\n\nmultiplication fact family worksheets grade facts math worksheet maker families 3rd multipli.\n\nmedium to large size of math fact worksheets grade multiplication family mixed facts practice 3rd grad.\n\nmath facts worksheets printable multiplication mastering worksheet generator multiplica.\n\ndivision facts worksheets basic free grade worksheet printable math fact practice the back to school fair multiplication and wor.\n\nmath 5 4 tests and worksheets edition main photo a fact families 3rd grade.\n\nmath fact worksheet generator worksheets grade other size multiplication drill mi.\n\nmath fluency worksheets grade these basic fact multiplication are similar to the mad minutes or masterin.\n\nspiral multiplication math fact worksheet grade worksheets of third division families 3rd g.\n\nrounding worksheets grade related post marvelous autumn scarecrow math worksheet on super teacher of mixed facts 3rd.\n\nworksheets grade multiplication fact families family addition math free triangle and subtraction adding worksheet tri.\n\nfirst grade math facts printable worksheets multiplication mixed 3rd basic for.\n\ngrade multiplication table printable free worksheets math facts fact families 3rd 3 times and.\n\nmath facts worksheets the single digit addition questions with some regrouping all grade f.\n\nmultiplication facts worksheets grade math timed test kids worksh.\n\nmath fluency facts practice worksheets grade sub worksheet medium of multiplication fact family 3rd fl.\n\nmath facts worksheets grade fresh subtraction fact 7 families 3rd library." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.79020035,"math_prob":0.8116121,"size":3513,"snap":"2019-35-2019-39","text_gpt3_token_len":619,"char_repetition_ratio":0.34083784,"word_repetition_ratio":0.02,"special_character_ratio":0.16253914,"punctuation_ratio":0.06666667,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9982677,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-09-21T11:39:43Z\",\"WARC-Record-ID\":\"<urn:uuid:b8772bcd-a707-4330-bf47-e3b559c095ba>\",\"Content-Length\":\"53341\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:29d364cf-0f13-42ca-8aab-4f9697ba6bd1>\",\"WARC-Concurrent-To\":\"<urn:uuid:9c85abeb-02a7-447b-a821-fec650d9cbce>\",\"WARC-IP-Address\":\"104.27.139.46\",\"WARC-Target-URI\":\"http://devopscr.co/math-facts-worksheets-3rd-grade/\",\"WARC-Payload-Digest\":\"sha1:S4KSFPEYETOXKACK4ZLRWGNW6AGTE46U\",\"WARC-Block-Digest\":\"sha1:RFF7Z5BN5T3BTMT7KZFOJ55GSUTVML54\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-39/CC-MAIN-2019-39_segments_1568514574409.16_warc_CC-MAIN-20190921104758-20190921130758-00399.warc.gz\"}"}
http://www.circleseven.co.uk/2016/02/02/dat-406-arduino-potentiometer-controlled-led/	[ "# DAT 406 – Arduino: Potentiometer-controlled LED\n\nThis is the result of a practical task with Arduino that uses a potentiometer to vary the flashing speed of an LED. The code listed in this article takes the analogue input from the potentiometer. This value is then used to vary the time between the LED being on and being off.\n\nArduino code:\n\n```int sensorPin = 0;\nint ledPin = 13;\n\nvoid setup() {\npinMode(ledPin, OUTPUT);\nSerial.begin(9600);\n\n}\n\nvoid loop() {" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6710247,"math_prob":0.6435319,"size":578,"snap":"2020-45-2020-50","text_gpt3_token_len":137,"char_repetition_ratio":0.1358885,"word_repetition_ratio":0.0,"special_character_ratio":0.23875433,"punctuation_ratio":0.18518518,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95874304,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-12-03T17:02:10Z\",\"WARC-Record-ID\":\"<urn:uuid:79b3dbbb-b3e2-4699-9802-47e608dc254a>\",\"Content-Length\":\"91385\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0bd30e63-273f-42cc-9a2a-55c041713649>\",\"WARC-Concurrent-To\":\"<urn:uuid:4310de3a-d21d-47fb-a85d-e1dca27fde0f>\",\"WARC-IP-Address\":\"77.72.0.94\",\"WARC-Target-URI\":\"http://www.circleseven.co.uk/2016/02/02/dat-406-arduino-potentiometer-controlled-led/\",\"WARC-Payload-Digest\":\"sha1:BMXWMR4T7J525X6FUDHL74YR4RUNDLLK\",\"WARC-Block-Digest\":\"sha1:IN6RVMR2RYZ6M544DITWG2TJUYREWKYD\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-50/CC-MAIN-2020-50_segments_1606141729522.82_warc_CC-MAIN-20201203155433-20201203185433-00441.warc.gz\"}"}
http://c.biancheng.net/view/9341.html	[ "# JS运算符汇总\n\n## 算术运算符\n\n+ 加法运算符 x + y 表示计算 x 加 y 的和\n- 减法运算符 x - y 表示计算 x 减 y 的差\n* 乘法运算符 x * y 表示计算 x 乘 y 的积\n/ 除法运算符 x / y 表示计算 x 除以 y 的商\n% 取模（取余）运算符 x % y 表示计算 x 除以 y 的余数\n\n```var x = 10,\ny = 4;\nconsole.log(\"x + y =\", x + y); // 输出：x + y = 14\nconsole.log(\"x - y =\", x - y); // 输出：x - y = 6\nconsole.log(\"x * y =\", x * y); // 输出：x * y = 40\nconsole.log(\"x / y =\", x / y); // 输出：x / y = 2.5\nconsole.log(\"x % y =\", x % y); // 输出：x % y = 2```\n\n## 赋值运算符\n\n= 最简单的赋值运算符，将运算符右侧的值赋值给运算符左侧的变量 x = 10 表示将变量 x 赋值为 10\n+= 先进行加法运算，再将结果赋值给运算符左侧的变量 x += y 等同于 x = x + y\n-= 先进行减法运算，再将结果赋值给运算符左侧的变量 x -= y 等同于 x = x - y\n= 先进行乘法运算，再将结果赋值给运算符左侧的变量 x = y 等同于 x = x * y\n/= 先进行除法运算，再将结果赋值给运算符左侧的变量 x /= y 等同于 x = x / y\n%= 先进行取模运算，再将结果赋值给运算符左侧的变量 x %= y 等同于 x = x % y\n\n```var x = 10;\nx += 20;\nconsole.log(x); // 输出：30\nvar x = 12,\ny = 7;\nx -= y;\nconsole.log(x); // 输出：5\nx = 5;\nx *= 25;\nconsole.log(x); // 输出：125\nx = 50;\nx /= 10;\nconsole.log(x); // 输出：5\nx = 100;\nx %= 15;\nconsole.log(x); // 输出：10```\n\n## 字符串运算符\n\nJavaScript 中的` + `` += `运算符除了可以进行数学运算外，还可以用来拼接字符串，其中：\n• `+ `运算符表示将运算符左右两侧的字符串拼接到一起；\n• `+= `运算符表示先将字符串进行拼接，然后再将结果赋值给运算符左侧的变量。\n\n```var x = \"Hello \";\nvar y = \"World!\";\nvar z = x + y;\nconsole.log(z); // 输出：Hello World!\nx += y;\nconsole.log(x); // 输出：Hello World!```\n\n## 自增、自减运算符\n\n++x 自增运算符将 x 加 1，然后返回 x 的值\nx++ 自增运算符返回 x 的值，然后再将 x 加 1\n--x 自减运算符将 x 减 1，然后返回 x 的值\nx-- 自减运算符返回 x 的值，然后将 x 减 1\n\n```var x;\n\nx = 10;\nconsole.log(++x); // 输出：11\nconsole.log(x); // 输出：11\n\nx = 10;\nconsole.log(x++); // 输出：10\nconsole.log(x); // 输出：11\n\nx = 10;\nconsole.log(--x); // 输出：9\nconsole.log(x); // 输出：9\n\nx = 10;\nconsole.log(x--); // 输出：10\nconsole.log(x); // 输出：9```\n\n## 比较运算符\n\n== 等于 x == y 表示如果 x 等于 y，则为真\n=== 全等 x === y 表示如果 x 等于 y，并且 x 和 y 的类型也相同，则为真\n!= 不相等 x != y 表示如果 x 不等于 y，则为真\n!== 不全等 x !== y 表示如果 x 不等于 y，或者 x 和 y 的类型不同，则为真\n< 小于 x < y 表示如果 x 小于 y，则为真\n> 大于 x > y 表示如果 x 大于 y，则为真\n>= 大于或等于 x >= y 表示如果 x 大于或等于 y，则为真\n<= 小于或等于 x <= y 表示如果 x 小于或等于 y，则为真\n\n```var x = 25;\nvar y = 35;\nvar z = \"25\";\n\nconsole.log(x == z); // 输出： true\nconsole.log(x === z); // 输出： false\nconsole.log(x != y); // 输出： true\nconsole.log(x !== z); // 输出： true\nconsole.log(x < y); // 输出： true\nconsole.log(x > y); // 输出： false\nconsole.log(x <= y); // 输出： true\nconsole.log(x >= y); // 输出： false```\n\n## 逻辑运算符\n\n&& 逻辑与 x && y 表示如果 x 和 y 都为真，则为真\n\|\| 逻辑或 x \|\| y 表示如果 x 或 y 有一个为真，则为真\n! 逻辑非 !x 表示如果 x 不为真，则为真\n\n```var year = 2021;\n\n// 闰年可以被 400 整除，也可以被 4 整除，但不能被 100 整除\nif((year % 400 == 0) \|\| ((year % 100 != 0) && (year % 4 == 0))){\nconsole.log(year + \" 年是闰年。\");\n} else{\nconsole.log(year + \" 年是平年。\");\n}```\n\n## 三元运算符\n\n```var x = 11,\ny = 20;\n\nx > y ? console.log(\"x 大于 y\") : console.log(\"x 小于 y\"); // 输出：x 小于 y```\n\n## 位运算符\n\n& 按位与：如果对应的二进制位都为 1，则该二进制位为 1 5 & 1 等同于 0101 & 0001 结果为 0001，十进制结果为 1\n\| 按位或：如果对应的二进制位有一个为 1，则该二进制位为 1 5 \| 1 等同于 0101 \| 0001 结果为 0101，十进制结果为 5\n^ 按位异或：如果对应的二进制位只有一个为 1，则该二进制位为 1 5 ^ 1 等同于 0101 ^ 0001 结果为 0100，十进制结果为 4\n~ 按位非：反转所有二进制位，即 1 转换为 0，0 转换为 1 ~5 等同于 ~0101 结果为 1010，十进制结果为 -6\n<< 按位左移：将所有二进制位统一向左移动指定的位数，并在最右侧补 0 5 << 1 等同于 0101 << 1 结果为 1010，十进制结果为 10\n>> 按位右移（有符号右移）：将所有二进制位统一向右移动指定的位数，并拷贝最左侧的位来填充左侧 5 >> 1 等同于 0101 >> 1 结果为 0010，十进制结果为 2\n>>> 按位右移零（无符号右移）：将所有二进制位统一向右移动指定的位数，并在最左侧补 0 5 >>> 1 等同于 0101 >>> 1 结果为 0010，十进制结果为 2\n\n```var a = 5 & 1,\nb = 5 \| 1,\nc = 5 ^ 1,\nd = ~ 5,\ne = 5 << 1,\nf = 5 >> 1,\ng = 5 >>> 1;\nconsole.log(a); // 输出：1\nconsole.log(b); // 输出：5\nconsole.log(c); // 输出：4\nconsole.log(d); // 输出：-6\nconsole.log(e); // 输出：10\nconsole.log(f); // 输出：2\nconsole.log(g); // 输出：2```", null, "" ]	[ null, "http://c.biancheng.net/templets/new/images/material/qrcode_weixueyuan_original.png", null ]	{"ft_lang_label":"__label__zh","ft_lang_prob":0.87492436,"math_prob":0.9998921,"size":4627,"snap":"2021-43-2021-49","text_gpt3_token_len":3073,"char_repetition_ratio":0.20700844,"word_repetition_ratio":0.022332506,"special_character_ratio":0.4216555,"punctuation_ratio":0.14447884,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9975469,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-11-27T12:08:22Z\",\"WARC-Record-ID\":\"<urn:uuid:96e955a6-bf55-416e-a927-dd1bd7540947>\",\"Content-Length\":\"24809\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:c6bd8abf-2a5b-404e-bc57-c0ba55c4531c>\",\"WARC-Concurrent-To\":\"<urn:uuid:cedfd070-5806-482c-af1c-f0f717d0f21c>\",\"WARC-IP-Address\":\"47.246.24.228\",\"WARC-Target-URI\":\"http://c.biancheng.net/view/9341.html\",\"WARC-Payload-Digest\":\"sha1:CAQUVPYAO23QDUGXMNDJRW4GQH7XJZI7\",\"WARC-Block-Digest\":\"sha1:JVRAF57K7RIDUY3OUTPNCUAPEAQ4KQQQ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964358180.42_warc_CC-MAIN-20211127103444-20211127133444-00303.warc.gz\"}"}
https://github.com/thomasgladwin/nncorr	[ "Nearest-neighbour correlation for non-linear relationships\nMATLAB\n\n## Latest commit", null, "Fetching latest commit…\nCannot retrieve the latest commit at this time.\n\n## Files\n\nType Name Latest commit message Commit time\nFailed to load latest commit information.", null, "README.md", null, "nncorr_sims.m", null, "teg_nncorr.m\n\n# nncorr\n\nNonlinear neighbourhood correlation.\n\nThis is a version of a non-linear correlation between variables X and Y that tests the correlation between the Y-scores of pairs of points that are neigbours on the X-axes. This can, e.g., detect U-curves or sinusoidal patterns in scatterplots. The trade-off is that it's less powerful at detecting linear relationships than a standard Peason's correlation.\n\nIn Matlab-code:\n\n``````function [r, p] = teg_nncorr(x, y)\n\n[x, si] = sort(x);\ny = y(si);\ny1_y2 = [];\nfor index1 = 1:2:(length(x) - 1)\nindex2 = index1 + 1;\ny1 = y(index1);\ny2 = y(index2);\ny1_y2 = [y1_y2; y1 y2];\nend\n[r, p] = corr(y1_y2(:, 1), y1_y2(:, 2));\n``````", null, "" ]	[ null, "https://github.githubassets.com/images/spinners/octocat-spinner-32-EAF2F5.gif", null, "https://github.githubassets.com/images/spinners/octocat-spinner-32.gif", null, "https://github.githubassets.com/images/spinners/octocat-spinner-32.gif", null, "https://github.githubassets.com/images/spinners/octocat-spinner-32.gif", null, "https://camo.githubusercontent.com/dc1acea46ace40adabc77ed0c3281141a1063ad8/68747470733a2f2f7a656e6f646f2e6f72672f62616467652f3234373135373231382e737667", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7874628,"math_prob":0.98847663,"size":804,"snap":"2020-10-2020-16","text_gpt3_token_len":256,"char_repetition_ratio":0.115,"word_repetition_ratio":0.0,"special_character_ratio":0.33955225,"punctuation_ratio":0.22988506,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.986301,"pos_list":[0,1,2,3,4,5,6,7,8,9,10],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-04-10T07:20:09Z\",\"WARC-Record-ID\":\"<urn:uuid:aea24aa2-5a5c-4d60-afff-b3bf95dbdfd8>\",\"Content-Length\":\"77542\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:529c67aa-e3cd-4d90-8ff3-c6886e28a6d2>\",\"WARC-Concurrent-To\":\"<urn:uuid:d15e6195-91c9-4499-9c68-471e4b312295>\",\"WARC-IP-Address\":\"140.82.112.3\",\"WARC-Target-URI\":\"https://github.com/thomasgladwin/nncorr\",\"WARC-Payload-Digest\":\"sha1:PAE5KGOM3W7BTGBAZ7TFTBBULQEF2KII\",\"WARC-Block-Digest\":\"sha1:WD777EBLM642D4GW5MPFVQ76CFCDMSQT\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-16/CC-MAIN-2020-16_segments_1585371886991.92_warc_CC-MAIN-20200410043735-20200410074235-00020.warc.gz\"}"}
https://socratic.org/questions/how-do-you-write-a-polynomial-in-standard-form-then-classify-it-by-degree-and-nu-58#447650	[ "# How do you write a polynomial in standard form, then classify it by degree and number of terms 8x^2+9+5x^3?\n\nJul 4, 2017\n\nSee below.\n\n#### Explanation:\n\nI am glad you asked this question, it is a very important math concept.\n\nOkay, so whenever you are asked to find the \"number of terms,\" just count the number of terms given (after of course, combining all like terms). For example, let's say you were given ${x}^{2} + 3 + 4$, you would not say that the number of terms is $3$, you would say it is $2$, since you want to add the $3$ and $4$ first. In this particular problem, we have:\n\n$\\text{Number of terms} = 3$\n\nNow, we will move on to standard form. Whenever you are asked to write a polynomial in standard form, arrange it in order of decreasing power. In other words, ${x}^{n} > {x}^{n - 1} , \\text{... } {x}^{4} > {x}^{3} > {x}^{2} > x > {x}^{0} ,$, so ${x}^{4}$ goes first and ${x}^{0}$ goes last. In this particular problem, $9$ is the ${x}^{0}$ term (so you don't get confused).\n\nIf we arrange it according to the definition I gave you, you get:\n\n$5 {x}^{3} + 8 {x}^{2} + 9$\n\nThis is the polynomial in standard form.\n\nLastly, we need to find degree. This is the easiest, as we just have to look at the polynomial in standard form. The first term of the polynomial in standard form will contain ${x}^{n}$, where $n$ is some unknown number. In other words, whatever power $x$ is raised to in standard form will be the degree of the polynomial.\n\nFor this particular problem, the $\\text{Degree } = 3$, since $x$ is raised to the $3 \\text{rd}$ power for the first term of the polynomial in standard form.\n\nThat is it, I hope that helps!" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9419622,"math_prob":0.9989692,"size":952,"snap":"2023-40-2023-50","text_gpt3_token_len":218,"char_repetition_ratio":0.15084389,"word_repetition_ratio":0.035714287,"special_character_ratio":0.22478992,"punctuation_ratio":0.12871288,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9998388,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-29T11:31:55Z\",\"WARC-Record-ID\":\"<urn:uuid:8c2f4f3a-d4d0-42b6-ab19-a48bc2e792e1>\",\"Content-Length\":\"36639\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0e055b80-9476-4211-9c98-91bd2b10c381>\",\"WARC-Concurrent-To\":\"<urn:uuid:91f15e9e-3485-498d-b368-23d7fdc68d26>\",\"WARC-IP-Address\":\"216.239.38.21\",\"WARC-Target-URI\":\"https://socratic.org/questions/how-do-you-write-a-polynomial-in-standard-form-then-classify-it-by-degree-and-nu-58#447650\",\"WARC-Payload-Digest\":\"sha1:IB2SQJ2FDBKZ457OK47TNYKPUGWPHYSF\",\"WARC-Block-Digest\":\"sha1:QXPGABAMQXQEANMSPKGFIAOQ4EOVZDB3\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233510501.83_warc_CC-MAIN-20230929090526-20230929120526-00688.warc.gz\"}"}
https://www.calculushowto.com/left-hand-derivative-right-hand-derivative/	[ "", null, "# Left Hand Derivative & Right Hand Derivative\n\nShare on\n\nFeel like \"cheating\" at Calculus? Check out our Practically Cheating Calculus Handbook, which gives you hundreds of easy-to-follow answers in a convenient e-book.\n\nSometimes we want to take the derivative at a point from one direction instead of both sides. When we approach a point from the left of the number line, it’s called a left hand derivative. When we approach from the right, it’s a right hand derivative. These definitions are exactly the same concept as one sided limits—limits from the left and limits from the right .\n\nThe definition of a left hand derivative at point a is:", null, "If the limit exists.\n\nThe definition of a right hand derivative at point a is:", null, "If the limit exists.\n\n## Why use Right and Left Hand Derivative?\n\nThere are a few reasons why you might choose to use a left hand derivative or right hand derivative. The technique can be used to show that a derivative exists at a certain point: If both the left- and right-hand derivatives are equal, then the derivative at a exists.\n\nAdditionally, if you are finding the derivative for the endpoint in an interval, you have no choice but to use either a left- or right-hand derivative, because the function won’t be defined beyond those endpoints. For example, let’s say you wanted to find the derivative of √x. This function isn’t defined for values less than zero, so in order to find the derivative at x = 0, the left-hand endpoint, you need to use a right-hand derivative:", null, "The left-hand derivative clearly doesn’t exist (there’s nothing there to use for calculations!) and the right-hand derivative blows up to infinity; Therefore, as the left- and right-hand derivatives are not equal, the derivative at x = 0 does not exist.\n\n## Example\n\nExample question: Is the function f(x) = \|x\| + 1 differentiable at 0?\n\nSolution: Find the left- and right-hand derivative.\n\nThe left hand derivative is", null, "The right hand derivative is", null, "These are not equal, so the derivative does not exist.\n\n## References\n\n Reinholz, D. Derivatives. Retrieved August 10, 2021 from: https://www.ocf.berkeley.edu/~reinholz/ed/08sp_m160/lectures/derivatives.pdf\n\nCITE THIS AS:\nStephanie Glen. \"Left Hand Derivative & Right Hand Derivative\" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/left-hand-derivative-right-hand-derivative/\n---------------------------------------------------------------------------", null, "", null, "Need help with a homework or test question? With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!" ]	[ null, "https://www.calculushowto.com/wp-content/uploads/2021/08/cht_header3.jpg", null, "https://www.calculushowto.com/wp-content/uploads/2021/08/left-hand-derivative-2.png", null, "https://www.calculushowto.com/wp-content/uploads/2021/08/right-hand-derivative-2.png", null, "https://www.calculushowto.com/wp-content/uploads/2021/08/calculating-the-derivative-at-an-endpoint.png", null, "https://www.calculushowto.com/wp-content/uploads/2021/08/LHD.png", null, "https://www.calculushowto.com/wp-content/uploads/2021/08/RHD.png", null, "https://www.calculushowto.com/wp-content/uploads/2021/12/14422-1200563.webp", null, "https://imp.pxf.io/i/3128523/1200563/14422", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8556111,"math_prob":0.9042077,"size":2470,"snap":"2022-27-2022-33","text_gpt3_token_len":569,"char_repetition_ratio":0.20762368,"word_repetition_ratio":0.045801528,"special_character_ratio":0.21781376,"punctuation_ratio":0.12145749,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99781543,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"im_url_duplicate_count":[null,null,null,1,null,1,null,1,null,1,null,1,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-06-27T18:34:59Z\",\"WARC-Record-ID\":\"<urn:uuid:f2ef84a6-ddd7-4d39-8d85-c8b66b51d186>\",\"Content-Length\":\"76577\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:bad90c44-a9e5-4131-aa17-20bed8f0467f>\",\"WARC-Concurrent-To\":\"<urn:uuid:885dfc9b-a832-4a27-9873-6fbc46b1b8a7>\",\"WARC-IP-Address\":\"104.26.8.53\",\"WARC-Target-URI\":\"https://www.calculushowto.com/left-hand-derivative-right-hand-derivative/\",\"WARC-Payload-Digest\":\"sha1:MNIAQLMVUHBSVHIO5YKAAMHDXBIG7CIA\",\"WARC-Block-Digest\":\"sha1:OX6FWSSMK6QYXKS2UZ5BAI2YCF55QZJO\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656103337962.22_warc_CC-MAIN-20220627164834-20220627194834-00487.warc.gz\"}"}
https://nyuscholars.nyu.edu/en/publications/algorithms-for-distributed-functional-monitoring-2	[ "# Algorithms for distributed functional monitoring\n\nGraham Cormode, S. Muthukrishnan, Ke Yi\n\nResearch output: Chapter in Book/Report/Conference proceedingConference contribution\n\n## Abstract\n\nWe study what we call functional monitoring problems. We have k players each tracking their inputs, say player i tracking a multiset A i(t) up until time t, and communicating with a central coordinator. The coordinator's task is to monitor a given function / computed over the union of the inputs ∪ iA iv(t), continuously at all times t. The goal is to minimize the number of bits communicated between the players and the coordinator. A simple example is when / is the sum, and the coordinator is required to alert when the sum of a distributed set of values exceeds a given threshold τ. Of interest is the approximate version where the coordinator outputs 1 if ≥ t and 0 if / ≤ (1 - ∈)τ. This defines the (k, f,τ,∈) distributed, functional monitoring problem. Functional monitoring problems are fundamental in distributed systems, in particular sensor networks, where we must minimize communication; they also connect to problems in communication complexity, communication theory, and signal processing. Yet few formal bounds are known lor functional monitoring. We give upper and lower bounds for the (k, f,τ,∈) problem for some of the basic f's. In particular, we study frequency moments (F 0, F 1,F 2). For F 0 and F 1, we obtain continuously monitoring algorithms with costs almost the same as their one-shot computation algorithms. However, for F 2 the monitoring problem seems much harder. We give a carefully constructed multi-round algorithm that uses \"sketch summaries\" at multiple levels of detail and solves the (k, F 2,τ, ∈) problem with communication Õ(k 2/∈ + (√k/∈) 3). Since frequency moment estimation is central to other problems, our results have immediate applications to histograms, wavelet computations, and others. Our algorithmic techniques are likely to be useful for other functional monitoring problems as well.\n\nOriginal language English (US) Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms 1076-1085 10 Published - 2008 19th Annual ACM-SIAM Symposium on Discrete Algorithms - San Francisco, CA, United StatesDuration: Jan 20 2008 → Jan 22 2008\n\n### Publication series\n\nName Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms\n\n### Other\n\nOther 19th Annual ACM-SIAM Symposium on Discrete Algorithms United States San Francisco, CA 1/20/08 → 1/22/08\n\n## ASJC Scopus subject areas\n\n• Software\n• Mathematics(all)" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8998829,"math_prob":0.7916516,"size":4341,"snap":"2020-45-2020-50","text_gpt3_token_len":1018,"char_repetition_ratio":0.13096611,"word_repetition_ratio":0.86868685,"special_character_ratio":0.23635107,"punctuation_ratio":0.12609239,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95085204,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-12-05T12:31:24Z\",\"WARC-Record-ID\":\"<urn:uuid:0deef9d4-6e72-4bed-891e-25de71bf75a5>\",\"Content-Length\":\"50077\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:72e72957-0337-44bb-aabd-dcfa9a5070d8>\",\"WARC-Concurrent-To\":\"<urn:uuid:8013eb14-6c71-47c1-98c0-5b746f40125b>\",\"WARC-IP-Address\":\"3.90.122.189\",\"WARC-Target-URI\":\"https://nyuscholars.nyu.edu/en/publications/algorithms-for-distributed-functional-monitoring-2\",\"WARC-Payload-Digest\":\"sha1:SJTERPDIA5MIM54XOQPORLHQIDL74RGQ\",\"WARC-Block-Digest\":\"sha1:TM3Q5B4EKZB3XMZN32FQJSGSGGNUUBG7\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-50/CC-MAIN-2020-50_segments_1606141747774.97_warc_CC-MAIN-20201205104937-20201205134937-00390.warc.gz\"}"}
https://www.colorhexa.com/535056	[ "# #535056 Color Information\n\nIn a RGB color space, hex #535056 is composed of 32.5% red, 31.4% green and 33.7% blue. Whereas in a CMYK color space, it is composed of 3.5% cyan, 7% magenta, 0% yellow and 66.3% black. It has a hue angle of 270 degrees, a saturation of 3.6% and a lightness of 32.5%. #535056 color hex could be obtained by blending #a6a0ac with #000000. Closest websafe color is: #666666.\n\n• R 33\n• G 31\n• B 34\nRGB color chart\n• C 3\n• M 7\n• Y 0\n• K 66\nCMYK color chart\n\n#535056 color description : Very dark grayish violet.\n\n# #535056 Color Conversion\n\nThe hexadecimal color #535056 has RGB values of R:83, G:80, B:86 and CMYK values of C:0.03, M:0.07, Y:0, K:0.66. Its decimal value is 5460054.\n\nHex triplet RGB Decimal 535056 `#535056` 83, 80, 86 `rgb(83,80,86)` 32.5, 31.4, 33.7 `rgb(32.5%,31.4%,33.7%)` 3, 7, 0, 66 270°, 3.6, 32.5 `hsl(270,3.6%,32.5%)` 270°, 7, 33.7 666666 `#666666`\nCIE-LAB 34.495, 2.522, -3.08 8.115, 8.248, 9.968 0.308, 0.313, 8.248 34.495, 3.98, 309.312 34.495, 1.281, -4.204 28.72, 0.179, -0.475 01010011, 01010000, 01010110\n\n# Color Schemes with #535056\n\n• #535056\n``#535056` `rgb(83,80,86)``\n• #535650\n``#535650` `rgb(83,86,80)``\nComplementary Color\n• #505056\n``#505056` `rgb(80,80,86)``\n• #535056\n``#535056` `rgb(83,80,86)``\n• #565056\n``#565056` `rgb(86,80,86)``\nAnalogous Color\n• #505650\n``#505650` `rgb(80,86,80)``\n• #535056\n``#535056` `rgb(83,80,86)``\n• #565650\n``#565650` `rgb(86,86,80)``\nSplit Complementary Color\n• #505653\n``#505653` `rgb(80,86,83)``\n• #535056\n``#535056` `rgb(83,80,86)``\n• #565350\n``#565350` `rgb(86,83,80)``\n• #505356\n``#505356` `rgb(80,83,86)``\n• #535056\n``#535056` `rgb(83,80,86)``\n• #565350\n``#565350` `rgb(86,83,80)``\n• #535650\n``#535650` `rgb(83,86,80)``\n• #2d2b2e\n``#2d2b2e` `rgb(45,43,46)``\n• #3a373c\n``#3a373c` `rgb(58,55,60)``\n• #464449\n``#464449` `rgb(70,68,73)``\n• #535056\n``#535056` `rgb(83,80,86)``\n• #605c63\n``#605c63` `rgb(96,92,99)``\n• #6d6970\n``#6d6970` `rgb(109,105,112)``\n• #79757e\n``#79757e` `rgb(121,117,126)``\nMonochromatic Color\n\n# Alternatives to #535056\n\nBelow, you can see some colors close to #535056. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #525056\n``#525056` `rgb(82,80,86)``\n• #525056\n``#525056` `rgb(82,80,86)``\n• #535056\n``#535056` `rgb(83,80,86)``\n• #535056\n``#535056` `rgb(83,80,86)``\n• #545056\n``#545056` `rgb(84,80,86)``\n• #545056\n``#545056` `rgb(84,80,86)``\n• #555056\n``#555056` `rgb(85,80,86)``\nSimilar Colors\n\n# #535056 Preview\n\nThis text has a font color of #535056.\n\n``<span style=\"color:#535056;\">Text here</span>``\n#535056 background color\n\nThis paragraph has a background color of #535056.\n\n``<p style=\"background-color:#535056;\">Content here</p>``\n#535056 border color\n\nThis element has a border color of #535056.\n\n``<div style=\"border:1px solid #535056;\">Content here</div>``\nCSS codes\n``.text {color:#535056;}``\n``.background {background-color:#535056;}``\n``.border {border:1px solid #535056;}``\n\n# Shades and Tints of #535056\n\nA shade is achieved by adding black to any pure hue, while a tint is created by mixing white to any pure color. In this example, #050405 is the darkest color, while #fafafa is the lightest one.\n\n• #050405\n``#050405` `rgb(5,4,5)``\n• #0e0e0f\n``#0e0e0f` `rgb(14,14,15)``\n• #181719\n``#181719` `rgb(24,23,25)``\n• #222123\n``#222123` `rgb(34,33,35)``\n• #2c2a2d\n``#2c2a2d` `rgb(44,42,45)``\n• #363438\n``#363438` `rgb(54,52,56)``\n• #3f3d42\n``#3f3d42` `rgb(63,61,66)``\n• #49474c\n``#49474c` `rgb(73,71,76)``\n• #535056\n``#535056` `rgb(83,80,86)``\n• #5d5960\n``#5d5960` `rgb(93,89,96)``\n• #67636a\n``#67636a` `rgb(103,99,106)``\n• #706c74\n``#706c74` `rgb(112,108,116)``\n• #7a767f\n``#7a767f` `rgb(122,118,127)``\n• #848088\n``#848088` `rgb(132,128,136)``\n• #8e8a92\n``#8e8a92` `rgb(142,138,146)``\n• #98949b\n``#98949b` `rgb(152,148,155)``\n• #a19ea5\n``#a19ea5` `rgb(161,158,165)``\n• #aba8ae\n``#aba8ae` `rgb(171,168,174)``\n• #b5b2b8\n``#b5b2b8` `rgb(181,178,184)``\n• #bfbdc1\n``#bfbdc1` `rgb(191,189,193)``\n• #c9c7cb\n``#c9c7cb` `rgb(201,199,203)``\n• #d3d1d4\n``#d3d1d4` `rgb(211,209,212)``\n• #dcdbde\n``#dcdbde` `rgb(220,219,222)``\n• #e6e5e7\n``#e6e5e7` `rgb(230,229,231)``\n• #f0eff0\n``#f0eff0` `rgb(240,239,240)``\n• #fafafa\n``#fafafa` `rgb(250,250,250)``\nTint Color Variation\n\n# Tones of #535056\n\nA tone is produced by adding gray to any pure hue. In this case, #535056 is the less saturated color, while #5303a3 is the most saturated one.\n\n• #535056\n``#535056` `rgb(83,80,86)``\n• #534a5c\n``#534a5c` `rgb(83,74,92)``\n• #534363\n``#534363` `rgb(83,67,99)``\n• #533d69\n``#533d69` `rgb(83,61,105)``\n• #533670\n``#533670` `rgb(83,54,112)``\n• #533076\n``#533076` `rgb(83,48,118)``\n• #532a7c\n``#532a7c` `rgb(83,42,124)``\n• #532383\n``#532383` `rgb(83,35,131)``\n• #531d89\n``#531d89` `rgb(83,29,137)``\n• #53178f\n``#53178f` `rgb(83,23,143)``\n• #531096\n``#531096` `rgb(83,16,150)``\n• #530a9c\n``#530a9c` `rgb(83,10,156)``\n• #5303a3\n``#5303a3` `rgb(83,3,163)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #535056 is perceived by people affected by a color vision deficiency. This can be useful if you need to ensure your color combinations are accessible to color-blind users.\n\nMonochromacy\n• Achromatopsia 0.005% of the population\n• Atypical Achromatopsia 0.001% of the population\nDichromacy\n• Protanopia 1% of men\n• Deuteranopia 1% of men\n• Tritanopia 0.001% of the population\nTrichromacy\n• Protanomaly 1% of men, 0.01% of women\n• Deuteranomaly 6% of men, 0.4% of women\n• Tritanomaly 0.01% of the population" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5227271,"math_prob":0.86645085,"size":3669,"snap":"2020-45-2020-50","text_gpt3_token_len":1589,"char_repetition_ratio":0.13751705,"word_repetition_ratio":0.007380074,"special_character_ratio":0.57481605,"punctuation_ratio":0.23404256,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9943697,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-12-03T01:31:35Z\",\"WARC-Record-ID\":\"<urn:uuid:e37a9580-1817-4d8c-8324-958185514c3a>\",\"Content-Length\":\"36221\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:3eb03810-0de6-4687-8065-b9b445da56a1>\",\"WARC-Concurrent-To\":\"<urn:uuid:a6deda58-f004-4d63-94de-59300bf0e7d0>\",\"WARC-IP-Address\":\"178.32.117.56\",\"WARC-Target-URI\":\"https://www.colorhexa.com/535056\",\"WARC-Payload-Digest\":\"sha1:JP5HT7QTMXKGKJNASXLN6UEGRT52P5F2\",\"WARC-Block-Digest\":\"sha1:25GXJ3FHGYBQCLSRGNHXK5FK4JQB4XPX\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-50/CC-MAIN-2020-50_segments_1606141717601.66_warc_CC-MAIN-20201203000447-20201203030447-00276.warc.gz\"}"}
https://www.thepoorcoder.com/hackerrank-strange-counter-solution/	[ "Hackerrank - Strange Counter Solution\n\n# Hackerrank - Strange Counter Solution", null, "Bob has a strange counter. At the first second, it displays the number . Each second, the number displayed by the counter decrements by until it reaches .\n\nThe counter counts down in cycles. In next second, the timer resets to and continues counting down. The diagram below shows the counter values for each time in the first three cycles:\n\nFind and print the value displayed by the counter at time .\n\nFunction Description\n\nComplete the strangeCounter function in the editor below. It should return the integer value displayed by the counter at time .\n\nstrangeCounter has the following parameter(s):\n\n• t: an integer\n\nInput Format\n\nA single integer denoting the value of .\n\nConstraints\n\n• for of the maximum score.\n\nOutput Format\n\nPrint the value displayed by the strange counter at the given time .\n\nSample Input\n\n4\n\n\nSample Output\n\n6\n\n\nExplanation\n\nTime marks the beginning of the second cycle. It is double the number displayed at the beginning of the first cycle:. This is also shown in the diagram in the Problem Statement above.\n\n### Solution in Python\n\ndef strangeCounter(t):\nn=3\nwhile 2n-2<=t:\nn=2\nreturn n-(t-(n-2))\n\nprint(strangeCounter(int(input())))\n\n\nWe can see that the number on the top right corner of each cycle equals to the 3rd item on the left column. And also it equals to the number of rows in that particular cycle.\n\nAnother interesting pattern we can observe is that the number on the top right corner doubles each cycle 3, 6, 12, 24, 48.....\n\nWe can use this value to check in which cycle our given time falls.\n\nLet us assume t = 15\n\nFirst we will find in which cycle 15 falls in. Let us start by initializing n = 3\n\nFor n=3, the values of t will range from n-2 =1 to n-2+n=4 (4 exclusive), n-2+n can also be written as n*2-2.\n\nNow, since t = 15 doesn't fall between 1 and 3 we will double the value of n.\n\nFor n = 6, t ranges from n-2=4 to n-2+n=10.\n\nAgain, t = 15 doesn't fall under 6 and 10. Therefore we will double the value of n again.\n\nFor n = 12, t ranges from n-2 = 10 to n-2+n = 22.\n\nt = 15 falls between 10 to 22.\n\nNow we subtract the starting number of our cycle which is n-2=10 from t.\n\nt-(n-2) = 15-10=5\n\nNow we just have to subtract the result from n and we will get our required answer\n\nn-5 = 12-5 = 7" ]	[ null, "https://images.unsplash.com/photo-1456574808786-d2ba7a6aa654", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.84506416,"math_prob":0.99306715,"size":2266,"snap":"2022-40-2023-06","text_gpt3_token_len":594,"char_repetition_ratio":0.13483644,"word_repetition_ratio":0.057279237,"special_character_ratio":0.27272728,"punctuation_ratio":0.10204082,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9986866,"pos_list":[0,1,2],"im_url_duplicate_count":[null,5,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-09-29T01:57:08Z\",\"WARC-Record-ID\":\"<urn:uuid:1f9265a4-6b48-4da8-8d97-305eb9e603ea>\",\"Content-Length\":\"64852\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:97bbb1ca-d2aa-4f41-8fbd-95fc9cf5b643>\",\"WARC-Concurrent-To\":\"<urn:uuid:698b63e6-2602-4f5f-8286-d1db597e1ea1>\",\"WARC-IP-Address\":\"104.21.63.9\",\"WARC-Target-URI\":\"https://www.thepoorcoder.com/hackerrank-strange-counter-solution/\",\"WARC-Payload-Digest\":\"sha1:Z4EVRQAPLUBA4VUWX7HV6HSVR2MKBIXR\",\"WARC-Block-Digest\":\"sha1:VZVCPMHG7WKDQNTV53EGII6HAYALI4IT\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-40/CC-MAIN-2022-40_segments_1664030335303.67_warc_CC-MAIN-20220929003121-20220929033121-00708.warc.gz\"}"}