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https://journalofinequalitiesandapplications.springeropen.com/articles/10.1186/s13660-017-1297-z	[ "# Optimal partial regularity of very weak solutions to nonhomogeneous A-harmonic systems\n\n## Abstract\n\nWe study partial regularity of very weak solutions to some nonhomogeneous A-harmonic systems. To obtain the reverse Hölder inequality of the gradient of a very weak solution, we construct a suitable test function by Hodge decomposition. With the aid of Gehring’s lemma, we prove that these very weak solutions are weak solutions. Further, we show that these solutions are in fact optimal Hölder continuity based on A-harmonic approximation technique.\n\n## Introduction\n\nWe consider optimal partial regularity of very weak solutions to nonhomogeneous A-harmonic systems of the following type:\n\n$$-\\operatorname{div} A(x,u,Du)=f(x),$$\n(1.1)\n\nwhere $$u:\\Omega \\rightarrow R^{N}$$ is a vector-valued function on a bounded domain $$\\Omega \\subset R^{n}$$ ($$n\\geqslant 2$$), and $$Du=\\{D_{ \\alpha} u^{i}\\}$$ ($$1\\leqslant \\alpha \\leqslant n,1\\leqslant i\\leqslant N$$) stands for the gradient matrix of u, $$A(x,u,\\xi ):\\Omega \\times R ^{N} \\times {R^{nN}\\rightarrow R^{nN}}$$ is a measurable function, and $$A_{i}^{\\alpha} (x,u,\\xi)$$ ($$1\\leqslant \\alpha \\leqslant n,1\\leqslant i\\leqslant N$$) are of class $$C^{1}$$ in ξ. To define the very weak solutions to systems (1.1) and obtain the optimal partial regularity results, we need to impose certain structural and regularity conditions on A and to restrict u and f to a particular class of functions as follows: for some $$p\\geqslant 2$$,\n\n1. (H1)\n\nA is a bounded operator, that is, there exists a constant $$\\beta >0$$ such that\n\n\\begin{aligned} \\bigl\\vert A(x,u,\\xi ) \\bigr\\vert \\leqslant \\beta \\bigl(1+\\vert \\xi \\vert ^{2} \\bigr)^{\\frac{p-1}{ 2} } \\quad \\mbox{for all } (x,u,\\xi )\\in \\Omega \\times R^{N} \\times R ^{nN}; \\end{aligned}\n2. (H2)\n\nA is differentiable with respect to $$\\xi \\in R^{nN}$$, that is, there exists a constant $$\\alpha >0$$ such that\n\n\\begin{aligned} D_{\\xi} A(x,u,\\xi )\\zeta \\cdot \\zeta \\geqslant \\alpha \\bigl(1+\\vert \\xi \\vert ^{2} \\bigr)^{ \\frac{{p-2}}{2 }}\\vert \\zeta \\vert ^{2} \\end{aligned}\n\nfor all $$x\\in \\Omega ,u\\in R^{N}$$, and $$\\xi ,\\zeta \\in R^{nN}$$;\n\n3. (H3)\n\nthere exist a constant $$\\gamma \\in (0,1)$$ and a nondecreasing function $$K: [0,\\infty )\\rightarrow [0,\\infty )$$ such that\n\n\\begin{aligned} \\bigl\\vert A(x,u,\\xi )-A(\\tilde{x},\\tilde{u},\\xi ) \\bigr\\vert \\leqslant K \\bigl(\\vert u \\vert \\bigr) \\bigl(\\vert x-\\tilde{x}\\vert ^{p}+ \\vert u-\\tilde{u}\\vert ^{p} \\bigr)^{\\frac{\\gamma }{p}} \\bigl(1+ \\vert \\xi \\vert \\bigr)^{\\frac{p}{2}} \\end{aligned}\n\nfor all $$x,\\tilde{x} \\in \\Omega ,u,\\tilde{u}\\in R^{N}$$ and $$\\xi \\in R^{nN}$$; without loss of generality, we take $$K \\geq 1$$;\n\n4. (H4)\n\nf is a given vector field in $$R^{N}$$ of class $$L_{\\mathrm{loc}}^{\\frac{nq}{n(p-1)+q}}(\\Omega )$$, $$q>p$$.\n\nUnder these assumptions, we can now define very weak solutions to (1.1).\n\n### Definition 1\n\nA mapping $$u\\in W_{\\mathrm{loc}}^{1,r}(\\Omega ),p-1\\leqslant r< p$$, is called a very weak solution to (1.1) if\n\n$$\\int_{\\Omega} A(x,u,Du)\\cdot D\\phi \\,dx= \\int_{\\Omega} f(x) \\phi \\,dx$$\n(1.2)\n\nfor all $$\\phi \\in W_{0}^{1,\\frac{r}{r-p+1}}(\\Omega )$$.\n\nIn order to improve the integrability of a very weak solution to (1.1), we need to prove a suitable reverse Hölder inequality. In 1973, Gehring discovered the crucial self-improving property of the reverse Hölder inequality and applied it to establish higher integrability of n-dimensional k-quasiconformal mapping. Subsequently, Meyers and Elcrat generalized this inequality based on Caccioppoli’s inequality. They improved the integrability of weak solutions to nonlinear elliptic systems with the help of Gehring’s lemma. Especially, they pointed out that regularity properties remained valid in a somewhat slightly larger Sobolev space to linear elliptic systems depending on the duality. In fact, this regularity result about very weak solutions was first showed by Meyers in 1963. Unfortunately, neither the method used in for proving the reverse Hölder inequality nor the duality employed in [2, 3] can be applied to deal with very weak solutions to nonlinear elliptic systems. To overcome these difficulties, Lewis used the technique of harmonic analysis and successfully proved that very weak solutions to nonlinear elliptic systems are indeed weak solutions. Later Iwaniec and Sbordone achieved a similar result via the methods of Hodge decomposition and prior estimation.\n\nSince then, studies on properties of very weak solutions to partial differential equations, especially for regularity of very weak solutions to A-harmonic systems, have attracted considerable attention. Following the method of Iwaniec and Sbordone , Giachetti, Leonetti, and Schiachi obtained the partial regularity result of A-harmonic systems $$\\operatorname{div} A (x,u,Du)=0$$. Tong, Gu, and Xu extended their result to nonhomogeneous A-harmonic systems $$\\operatorname{div} A(x,Du)=f(x)$$ and improved the integrability of very weak solutions. Greco, Iwaniec, and Sbordone even applied this method to the p-harmonic equation $$\\operatorname{div} \\vert Du \\vert ^{p-2}Du=\\operatorname{div} f$$.\n\nMotivated by these works, we mainly consider the optimal partial regularity to nonhomogeneous A-harmonic systems in the form of (1.1) under assumptions (H1)-(H4).\n\nFor the sake of desired results, we first need to improve the exponent of integrability for the gradient of a very weak solution to an even slightly better one than the natural exponent p. The crucial difficulty is to construct an appropriate test function below the natural exponent. In this article, we follow the spirit of Iwaniec and Sbordone using the Hodge decomposition to construct it. Combining the Sobolev imbedding theorem, Young’s inequality, Poincaré’s inequality, and so on, we improve the exponents of integrability of very weak solutions to (1.1). In other words, we successfully prove that very weak solutions to (1.1) are in fact weak solutions. More precisely, we obtain the following result.\n\n### Theorem 1\n\nLet u be a very weak solution to systems (1.1). Assume that the structure conditions (H1), (H2), and (H4) hold. Then there exist exponents $$p-1< r_{1}=r_{1}(n,N,p,\\alpha ,\\beta )< p< r_{2}=r_{2}(n,N,p, \\alpha ,\\beta )<\\infty$$ such that $$u\\in W_{\\mathrm{loc}}^{1,r_{1}}(\\Omega )$$ belongs to $$W_{\\mathrm{loc}}^{1,r_{2}}(\\Omega )$$.\n\nA direct consequence of this result follows immediately.\n\n### Corollary 1\n\nUnder the assumptions of Theorem 1, there exists $$r_{1}=r_{1}(n,N,p, \\alpha ,\\beta )< p$$ such that every very weak solution $$u\\in W_{\\mathrm{loc}} ^{1,r}(\\Omega )$$ with $$r_{1}< r< p$$ belongs to $$W_{\\mathrm{loc}}^{1,p}(\\Omega )$$.\n\nFurther, we establish the optimal partial regularity result of very weak solutions to (1.1). Generally speaking, we cannot expect that weak solutions to (1.1) will be $$C^{2}$$-solutions even under reasonable assumptions on operator A and f. This is initially pointed out by De Giorgi [9, 10] and Giusti and Miranda . Thus, our aim is to obtain the optimal Hölder continuity of very weak solutions to (1.1). Fortunately, we achieve it by means of A-harmonic approximation technique and obtain the optimal Hölder continuity $$C^{1,\\gamma }$$ in the regular set of the following:\n\n### Theorem 2\n\nLet $$u\\in W_{\\mathrm{loc}}^{1,r}(\\Omega )$$, $$r_{1}< r< p$$, be a very weak solution to (1.1). Consider $$r_{1}$$ as in Corollary 1. Suppose that assumptions (H1)-(H4) hold. Then there exists an open set $$\\Omega_{0}\\subset \\Omega$$ such that $$u\\in C^{1,\\gamma }(\\Omega_{0})$$ for γ is defined in (H3). We have\n\n\\begin{aligned} \\Omega -\\Omega_{0}=\\Sigma_{1}\\cup \\Sigma_{2}, \\end{aligned}\n\nwhere\n\n\\begin{aligned} \\Sigma_{1}= \\biggl\\{ x_{0} \\in \\Omega : \\hskip4pt \\liminf _{R \\to 0^{+}} \\fint_{B_{R}(x _{0})} \\bigl\\vert Du-(D u)_{x_{0},R} \\bigr\\vert ^{p} \\,dx >0 \\biggr\\} \\end{aligned}\n\nand\n\n\\begin{aligned} \\Sigma_{2}= \\Bigl\\{ x_{0} \\in \\Omega : \\hskip4pt \\limsup _{R \\to 0^{+}} \\bigl(\\vert u_{x_{0},R} \\vert + \\bigl\\vert (D u)_{x_{0},R} \\bigr\\vert \\bigr)= \\infty \\Bigr\\} . \\end{aligned}\n\nMoreover, we have $$\\vert \\Omega -\\Omega_{0} \\vert =0$$.\n\nTo close this section, we briefly summarize the notation used in this paper. As noted before, we consider a bounded domain $$\\Omega \\subset R ^{n}(n\\geqslant 2)$$ and mappings from Ω to $$R^{N}$$. We write $$B_{r}(x_{0})=\\{x\\in \\Omega : \\vert x-x_{0} \\vert < r\\},x_{0}\\in \\Omega$$. For a given set X, we denote by $$\\vert X \\vert$$ its n-dimensional Lebesgue measure. If $$\\vert X \\vert >0$$, then the average of a given $$g\\in L^{1}(X)$$ over X is denoted by $$\\fint_{X} g \\,dx$$, that is, $$\\fint_{X} g \\,dx=\\frac{1}{\\vert X \\vert }\\int_{X}g \\,dx$$. In particular, we write $$g_{x_{0},r}=\\fint_{B_{r}(x_{0})}g\\,dx$$. Let $$\\alpha_{n}$$ denote the volume of the unit ball in $$R^{n}$$, that is, $$\\alpha_{n}=\\vert B_{1}(0) \\vert$$, then $$\\vert B_{r}(x_{0}) \\vert =\\alpha_{n} r^{n}$$.\n\nThe rest of this paper is arranged as follows. In Section 2, we provide some necessary preliminary lemmas. In Section 3, we prove the main results.\n\n## Preliminary lemmas\n\nBefore proving the results, we state a few useful lemmas.\n\nThe first one is a stability result of the Hodge decomposition, from which we could construct a suitable test-function concerning estimates below the natural exponent for (1.1).\n\n### Lemma 1\n\n\n\nLet $$\\Omega \\subset R^{n}$$ be a regular domain, and $$w\\in W_{0} ^{1,r}(\\Omega ,R^{N}),r>1$$, and let $$-1<\\epsilon <r-1$$. Then there exist $$\\phi \\in W_{0}^{1, \\frac{r}{1+\\epsilon }}(\\Omega ,R^{N})$$ and a divergence-free matrix field $$H\\in L^{\\frac{r}{1+\\epsilon }}(\\Omega ,R ^{nN})$$ such that\n\n$$\\vert D w \\vert ^{\\epsilon} D w=D\\phi +H.$$\n(2.1)\n\nMoreover,\n\n$$\\Vert H \\Vert _{\\frac{r}{1+\\epsilon }}\\leqslant C_{r}( \\Omega , N)\\vert \\epsilon \\vert \\Vert D w \\Vert _{r}^{1+\\epsilon }.$$\n(2.2)\n\nThe most useful case for us in Lemma 1 is where ϵ is negative. For $$u\\in W_{\\mathrm{loc}}^{1,r}(\\Omega )$$ with $$p-1< r< p$$ that is a very weak solution to (1.1), we can set $$\\epsilon =r-p$$ $$(-1<\\epsilon <0)$$. Then there exists $$\\phi \\in W_{0}^{1,\\frac{r}{1+r-p}}(\\Omega )$$; thus, ϕ can be illustrated as a test-function in (1.2). In view of (2.1) and (2.2), we also can get an estimate of , which is similar to (2.2).\n\nApplying Lemma 2, we can decompose the left term of the Hodge decomposition into two terms that could be controlled more easily in the proof of Theorem 1.\n\n### Lemma 2\n\n\n\nFor every $$X, Y \\in R^{n}$$, $$X\\neq 0, Y\\neq 0$$, and $$0\\leqslant \\epsilon <1$$, we have the inequality\n\n\\begin{aligned} \\bigl\\vert \\vert X \\vert ^{-\\epsilon }X- \\bigr\\vert Y \\bigl\\vert ^{-\\epsilon }Y \\bigr\\vert \\leq 2^{\\epsilon} \\frac{1+ \\epsilon }{1-\\epsilon }\\vert X-Y \\vert ^{1-\\epsilon }. \\end{aligned}\n\nIn the end of this section, we shall introduce a form of Gehring’s lemma, which plays an important role in the proof of Theorem 1. It implies in particular that from it higher integrability of $$g(x)$$ follows.\n\n### Lemma 3\n\n[2, 13]\n\nLet $$0< R< R_{0}\\leqslant \\operatorname{dist} (x_{0},\\partial \\Omega ),x _{0}\\in \\Omega$$. Suppose that $$g(x)\\in L^{p} (B_{R}(x_{0}) )$$, $$f(x) \\in L^{t} (B_{R}(x_{0}) ),t>p,1< p<\\infty$$, satisfy the reverse Hölder inequality\n\n\\begin{aligned} \\fint_{B_{R/2}(x_{0})} \\bigl\\vert g(x) \\bigr\\vert ^{p}\\,dx \\leqslant \\theta \\fint_{B_{R}(x_{0})} \\bigl\\vert g(x) \\bigr\\vert ^{p}\\,dx +C^{} \\biggl[ \\fint_{B_{R}(x_{0})} \\bigl\\vert g(x) \\bigr\\vert ^{s}\\,dx \\biggr]^{p/s} + \\fint_{B_{R}(x_{0})} \\bigl\\vert f(x) \\bigr\\vert ^{p}\\,dx \\end{aligned}\n\nfor some $$1\\leqslant s< p,0\\leqslant \\theta <1$$. Then $$g\\in L_{\\mathrm{loc}} ^{p'}(\\Omega )$$ for some $$p'=p'(\\theta ,p,n,C^{})$$ $$(t\\geqslant p' >p)$$, and\n\n\\begin{aligned} \\biggl[ \\fint_{B_{R/2}(x_{0})} \\bigl\\vert g(x) \\bigr\\vert ^{p'}\\,dx \\biggr]^{1/{p'}} \\leqslant C_{} \\biggl[ \\fint_{B_{R}(x_{0})} \\bigl\\vert g(x) \\bigr\\vert ^{p}\\,dx \\biggr]^{1/p} +C_{} \\biggl[ \\fint_{B_{R}(x_{0})} \\bigl\\vert f(x) \\bigr\\vert ^{p'}\\,dx \\biggr]^{1/{p'}}, \\end{aligned}\n\nwhere $$C_{}=C_{}(n,C^{*},p,\\theta ,R_{0})$$.\n\n## Proof of the main theorems\n\nIn this section, we give a proof of partial regularity results. Consider u solving (1.1) on $$B_{R}(x_{0})\\Subset \\Omega$$, where we restrict $$0< R< R_{0}<\\min\\{1, \\operatorname{dist}(x_{0},\\partial \\Omega )\\}$$.\n\n### Proof\n\nFix a cut-off function $$\\eta \\in C_{0}^{\\infty }(B_{R}(x _{0}))$$ satisfying $$0\\leqslant \\eta \\leqslant 1$$, $$\\vert D \\eta \\vert \\leqslant C/R$$, and $$\\eta \\equiv 1$$ on $$B_{R/2}(x_{0})$$. Let $$u\\in W_{\\mathrm{loc}}^{1,r}(B _{R}(x_{0}))$$ with $$p-1< r< p$$ be a very weak solution to (1.1). Denote $$u-u_{x_{0},R}-p_{0}(x-x_{0})$$ by v, where $$p_{0}\\in R^{nN}$$. We find that v has calculus mean-value 0 on $$B_{R}(x_{0})$$, that is, $$v_{x_{0},R}=0$$. Notice that $$\\eta v \\in W_{0}^{1,r}(B_{R}(x_{0}))$$ and $$-1< r-p<0$$. Then there exist $$\\phi \\in W_{0}^{1,\\frac{r}{1+r-p}}(B _{R}(x_{0}))$$ and $$h\\in L^{\\frac{r }{{1+r-p}}}(B_{R}(x_{0}))$$ such that $$\\vert D(\\eta v) \\vert ^{r-p}D(\\eta v)=D \\phi +h$$ according to the Hodge decomposition. Thus, ϕ is admissible as a test-function in the definition of very weak solutions. Set $$-\\varepsilon =r-p$$ ($$-1<-\\varepsilon <0$$) for convenience. Then $$r=p-\\varepsilon$$, and we have\n\n$$\\bigl\\vert D(\\eta v) \\bigr\\vert ^{-\\varepsilon }D(\\eta v)=D \\phi +h,$$\n(3.1)\n\nwhere h satisfies\n\n$$\\Vert h \\Vert _{\\frac{p-\\varepsilon }{1-\\varepsilon }}\\leqslant C_{r}( \\Omega ,N) \\varepsilon \\bigl\\Vert D (\\eta v) \\bigr\\Vert _{p-\\varepsilon }^{1-\\varepsilon }.$$\n(3.2)\n\nFurther, applying Poincaré’s inequality with constant $$C_{P}$$ and noting that $$v_{x_{0},R}=0$$, we get\n\n\\begin{aligned} \\bigl\\Vert D(\\eta v) \\bigr\\Vert _{p-\\varepsilon }^{1-\\varepsilon }& \\leqslant \\bigl(\\Vert vD\\eta \\Vert _{p-\\varepsilon }+\\Vert \\eta Dv \\Vert _{p-\\varepsilon } \\bigr)^{1-\\varepsilon } \\\\ &\\leqslant \\biggl(\\frac{C}{R}\\Vert v \\Vert _{p-\\varepsilon }+\\Vert Dv \\Vert _{p-\\varepsilon } \\biggr)^{1-\\varepsilon } \\\\ &\\leqslant \\bigl(CC_{P}^{\\frac{1}{p-\\varepsilon }}\\Vert Dv \\Vert _{p-\\varepsilon }+ \\Vert Dv \\Vert _{p-\\varepsilon } \\bigr)^{1-\\varepsilon } \\\\ &< \\bigl(1+C C_{P}^{\\frac{1}{p-\\varepsilon }} \\bigr)\\Vert Dv \\Vert _{p-\\varepsilon }^{1- \\varepsilon }. \\end{aligned}\n(3.3)\n\nIn view of (3.2) and (3.3), we have\n\n$$\\Vert h \\Vert _{\\frac{p-\\varepsilon }{1-\\varepsilon }}\\leqslant C_{1} \\varepsilon \\Vert Dv \\Vert _{p-\\varepsilon }^{1-\\varepsilon },$$\n(3.4)\n\nwhere $$C_{1}=C_{r}(\\Omega ,N)(1+C C_{P}^{\\frac{1}{p-\\varepsilon }})$$.\n\nIn particular, combining (3.1) and (3.2), we find\n\n\\begin{aligned} \\Vert D\\phi \\Vert _{\\frac{p-\\varepsilon }{1-\\varepsilon }} &= \\bigl\\Vert \\bigl\\vert D(\\eta v) \\bigr\\vert ^{-\\varepsilon }D(\\eta v)-h \\bigr\\Vert _{\\frac{p-\\varepsilon }{1-\\varepsilon }} \\\\ &\\leqslant \\bigl\\Vert \\bigl\\vert D(\\eta v) \\bigr\\vert ^{-\\varepsilon }D( \\eta v) \\bigr\\Vert _{\\frac{p-\\varepsilon }{1-\\varepsilon }}+ \\Vert h \\Vert _{\\frac{p-\\varepsilon }{1-\\varepsilon }} \\\\ &\\leqslant \\bigl\\Vert D (\\eta v) \\bigr\\Vert _{p-\\varepsilon }^{1-\\varepsilon }+C_{r}( \\Omega ,N)\\varepsilon \\bigl\\Vert D (\\eta v) \\bigr\\Vert _{p-\\varepsilon }^{1-\\varepsilon } \\\\ &\\leqslant \\bigl(1+C_{r}(\\Omega ,N)\\varepsilon \\bigr) \\bigl\\Vert D (\\eta v) \\bigr\\Vert _{p-\\varepsilon }^{1-\\varepsilon }. \\end{aligned}\n\nSubstituting (3.3) into this estimate, we have\n\n$$\\Vert D\\phi \\Vert _{\\frac{p-\\varepsilon }{1-\\varepsilon }} \\leqslant C_{2}\\Vert Dv \\Vert _{p-\\varepsilon }^{1-\\varepsilon },$$\n(3.5)\n\nwhere $$C_{2}=(1+C_{r}(\\Omega ,N)\\varepsilon )(1+C C_{P}^{\\frac{1}{p- \\varepsilon }})$$.\n\nSince it is hard to estimate $$\\vert D(\\eta v) \\vert ^{-\\varepsilon }D(\\eta v)$$ directly, we set\n\n$$E(\\eta ,v)= \\bigl\\vert D(\\eta v) \\bigr\\vert ^{-\\varepsilon }D( \\eta v)-\\vert \\eta Dv \\vert ^{-\\varepsilon }\\eta Dv,$$\n\nwhich by Lemma 2 yields\n\n$$\\bigl\\vert E(\\eta ,v) \\bigr\\vert \\leqslant 2^{\\varepsilon} \\frac{1+\\varepsilon }{1-\\varepsilon }\\vert vD \\eta \\vert ^{1-\\varepsilon }.$$\n\nJoining $$E(\\eta ,v)$$ with (3.1), we arrive at\n\n\\begin{aligned} D\\phi =E(\\eta ,v)+\\vert \\eta Dv \\vert ^{-\\varepsilon }\\eta Dv-h. \\end{aligned}\n\nInserting into equality (1.2), we get\n\n\\begin{aligned} & \\int_{B_{R}(x_{0})} A(x, u,Du)\\cdot \\vert \\eta Dv \\vert ^{-\\varepsilon } \\eta Dv \\,dx \\\\ &\\quad = \\int_{B_{R}(x_{0})} A(x, u,Du)\\cdot h\\,dx- \\int_{B_{R}(x_{0})} A(x, u,Du) \\cdot E(\\eta ,v)\\,dx+ \\int_{B_{R}(x_{0})} f(x) \\phi \\,dx. \\end{aligned}\n(3.6)\n\nIn order to use (H2), we need to transform the left-hand side of (3.6) as follows:\n\n\\begin{aligned} & \\int_{B_{R}(x_{0})} A(x, u,Du)\\cdot \\vert \\eta Dv \\vert ^{-\\varepsilon } \\eta Dv \\,dx \\\\ &\\quad = \\int_{B_{R}(x_{0})} \\bigl(A(x, u,Du)-A(x,u,p_{0})+A(x,u,p_{0}) \\bigr)\\cdot \\vert \\eta Dv \\vert ^{-\\varepsilon }\\eta Dv\\,dx \\\\ &\\quad = \\int_{B_{R}(x_{0})} \\bigl(A(x, u,Du)-A(x,u, p_{0}) \\bigr) \\cdot \\vert \\eta Dv \\vert ^{- \\varepsilon }\\eta Dv\\,dx \\\\ &\\quad \\quad {}+ \\int_{B_{R}(x_{0})} A(x, u,p_{0})\\cdot \\vert \\eta Dv \\vert ^{-\\varepsilon }\\eta Dv\\,dx. \\end{aligned}\n\nCombining this equality with (3.6), we find\n\n\\begin{aligned} & \\int_{B_{R}(x_{0})} \\bigl(A(x, u,Du)-A(x,u, p_{0}) \\bigr) \\cdot \\vert \\eta Dv \\vert ^{- \\varepsilon }\\eta Dv\\,dx \\\\ &\\quad =- \\int_{B_{R}(x_{0})} A(x, u,p_{0}) \\cdot \\vert \\eta Dv \\vert ^{-\\varepsilon }\\eta Dv\\,dx+ \\int_{B_{R}(x_{0})}A(x,u,Du)\\cdot h\\,dx \\\\ &\\quad \\quad {}- \\int_{B_{R}(x_{0})} A(x, u,Du)\\cdot E(\\eta ,v)\\,dx + \\int_{B_{R}(x_{0})} f(x)\\phi \\,dx \\\\ &\\quad \\leqslant I_{1}+I_{2}+I_{3}+I_{4}, \\end{aligned}\n(3.7)\n\nwhere\n\n\\begin{aligned} &I_{1}= \\biggl\\vert -\\int_{B_{R}(x_{0})} A(x, u,p_{0})\\cdot \\vert \\eta Dv\\vert ^{-\\varepsilon}\\eta Dv\\,dx \\biggr\\vert ; \\\\ &I_{2}= \\biggl\\vert \\int_{B_{R}(x_{0})}A(x,u,Du)\\cdot h\\,dx \\biggr\\vert ; \\\\ &I_{3}= \\biggl\\vert - \\int_{B_{R}(x_{0})} A(x, u,Du)\\cdot E(\\eta ,v)\\,dx \\biggr\\vert ; \\\\ &I_{4}= \\biggl\\vert \\int_{B_{R}(x_{0})} f(x)\\phi \\,dx \\biggr\\vert . \\end{aligned}\n\nConsequently, we shall derive estimate for each term of (3.7) so as to establish a reverse Hölder inequality for $$\\vert Du-p_{0} \\vert ^{p-\\varepsilon }$$.\n\nIn the case of the term on the left-hand side of (3.7), we want to derive an estimate from below in terms of $$\\int_{B_{R/2}(x_{0})}\\vert Du-p_{0} \\vert ^{p-\\varepsilon }\\,dx$$. For this purpose, we need the inequality\n\n$$\\bigl(A(x,u,\\zeta )-A(x,u,\\xi ) \\bigr)\\cdot (\\zeta -\\xi )\\geqslant \\alpha \\bigl(1+\\vert \\zeta \\vert ^{2}+\\vert \\xi \\vert ^{2} \\bigr)^{\\frac{p-2}{2}}\\vert \\zeta -\\xi \\vert ^{2},$$\n\nwhich can be deduced from (H2) immediately.\n\nThen we infer that\n\n\\begin{aligned} & \\int_{B_{R}(x_{0})} \\bigl(A(x, u,Du)-A(x,u, p_{0}) \\bigr) \\cdot \\vert \\eta Dv \\vert ^{- \\varepsilon }\\eta Dv\\,dx \\\\ &\\quad = \\int_{B_{R}(x_{0})} \\vert \\eta Dv \\vert ^{-\\varepsilon }\\eta \\bigl(A(x, u,Du)-A(x,u, p_{0}) \\bigr)\\cdot (Du-p_{0})\\,dx \\\\ &\\quad \\geqslant \\alpha \\int_{B_{R}(x_{0})}\\vert \\eta Dv \\vert ^{-\\varepsilon }\\eta \\bigl(1+ \\vert Du \\vert ^{2}+\\vert p _{0} \\vert ^{2} \\bigr)^{\\frac{p-2}{2}} \\vert Du-p_{0}\\vert ^{2}\\,dx \\\\ &\\quad \\geqslant \\alpha \\int_{B_{R/2}(x_{0})}\\vert Du-p_{0} \\vert ^{2-\\varepsilon } \\biggl(1+\\frac{\\vert Du-p_{0} \\vert ^{2}}{2} \\biggr)^{\\frac{p-2}{2}}\\,dx \\\\ &\\quad \\geqslant 2^{\\frac{2-p}{2}}\\alpha \\int_{B_{R/2}(x_{0})}\\vert Du-p_{0} \\vert ^{p- \\varepsilon } \\,dx. \\end{aligned}\n(3.8)\n\nUsing (H1) and Young’s inequality with exponents $$\\frac{p-\\varepsilon }{1-\\varepsilon }$$ and $$\\frac{p-\\varepsilon }{p-1}$$, we find that, for $$\\varepsilon_{1}>0$$,\n\n\\begin{aligned} I_{1}&\\leqslant \\int_{B_{R}(x_{0})} \\bigl\\vert A(x, u,p_{0}) \\bigr\\vert \\vert Dv \\vert ^{1-\\varepsilon }\\,dx \\\\ & \\leqslant \\beta \\bigl(1+\\vert p_{0} \\vert ^{2} \\bigr)^{\\frac{p-1}{2}} \\int_{B_{R}(x_{0})}\\vert Du-p_{0} \\vert ^{1-\\varepsilon } \\,dx \\\\ & \\leqslant \\beta \\bigl(1+\\vert p_{0} \\vert ^{2} \\bigr)^{\\frac{p-1}{2}} \\int_{B_{R}(x_{0})} \\bigl( \\varepsilon_{1}\\vert Du-p_{0} \\vert ^{p-\\varepsilon }+\\varepsilon_{1}^{-\\frac{1- \\varepsilon }{p-1}}1^{\\frac{p-\\varepsilon }{p-1}} \\bigr)\\,dx \\\\ & \\leqslant \\beta \\bigl(1+\\vert p_{0} \\vert ^{2} \\bigr)^{\\frac{p-1}{2}}\\varepsilon_{1} \\int_{B_{R}(x_{0})}\\vert Du-p_{0} \\vert ^{p-\\varepsilon } \\,dx \\\\ &\\quad {}+\\beta \\bigl(1+\\vert p_{0} \\vert ^{2} \\bigr)^{\\frac{p-1}{2}}\\varepsilon_{1}^{-\\frac{1- \\varepsilon }{p-1}} \\int_{B_{R}(x_{0})}\\,dx. \\end{aligned}\n(3.9)\n\nBy (H1) we have\n\n\\begin{aligned} I_{2} & \\leqslant \\int_{B_{R}(x_{0})} \\bigl\\vert A(x,u,Du) \\bigr\\vert \\vert h \\vert \\,dx \\\\ &\\leqslant \\beta \\int_{B_{R}(x_{0})} \\bigl(1+\\vert Du \\vert ^{2} \\bigr)^{\\frac{p-1}{2}}\\vert h \\vert \\,dx \\\\ &\\leqslant \\beta \\int_{B_{R}(x_{0})} \\bigl(2+\\vert Du-p_{0}+p_{0} \\vert ^{2} \\bigr)^{ \\frac{p-1}{2}}\\vert h \\vert \\,dx \\\\ &\\leqslant \\beta \\int_{B_{R}(x_{0})} \\bigl(2 \\bigl(1+\\vert p_{0} \\vert ^{2} \\bigr)+2\\vert Du-p_{0} \\vert ^{2} \\bigr)^{ \\frac{p-1}{2}}\\vert h \\vert \\,dx \\\\ &\\leqslant 2^{p-1}\\beta \\int_{B_{R}(x_{0})}\\vert Du-p_{0} \\vert ^{p-1} \\vert h \\vert \\,dx+2^{p-1} \\beta \\bigl(1+\\vert p_{0} \\vert ^{2} \\bigr)^{\\frac{p-1}{2}} \\int_{B_{R}(x_{0})}\\vert h \\vert \\,dx. \\end{aligned}\n\nApplying both Hölder’s inequality and Young’s inequality with exponents $$\\frac{p-\\varepsilon }{p-1}$$ and $$\\frac{p-\\varepsilon }{1- \\varepsilon }$$, by (3.4) we further have that, for $$\\varepsilon_{2}>0$$,\n\n\\begin{aligned} I_{2}&\\leqslant 2^{p-1}\\beta \\biggl( \\int_{B_{R}(x_{0})}\\vert Du-p_{0} \\vert ^{p-\\varepsilon } \\,dx \\biggr)^{\\frac{p-1}{p-\\varepsilon }} \\biggl( \\int_{B_{R}(x_{0})}\\vert h \\vert ^{\\frac{p- \\varepsilon }{1-\\varepsilon }}\\,dx \\biggr)^{\\frac{1-\\varepsilon }{p-\\varepsilon }} \\\\ &\\quad +2^{p-1}\\beta \\bigl(1+\\vert p_{0} \\vert ^{2} \\bigr)^{\\frac{p-1}{2}} \\biggl( \\int_{B_{R}(x _{0})}1^{p-\\varepsilon }\\,dx \\biggr)^{\\frac{p-1}{p-\\varepsilon }} \\biggl( \\int_{B_{R}(x_{0})}\\vert h \\vert ^{\\frac{p-\\varepsilon }{1-\\varepsilon }}\\,dx \\biggr)^{\\frac{1- \\varepsilon }{p-\\varepsilon }} \\\\ &\\leqslant 2^{p-1}\\beta C_{1}\\varepsilon \\biggl( \\int_{B_{R}(x_{0})}\\vert Du-p _{0} \\vert ^{p-\\varepsilon } \\,dx \\biggr)^{\\frac{p-1}{p-\\varepsilon }} \\biggl( \\int_{B_{R}(x_{0})}\\vert Du-p_{0} \\vert ^{p-\\varepsilon } \\,dx \\biggr)^{\\frac{1-\\varepsilon }{p-\\varepsilon }} \\\\ &\\quad +2^{p-1}\\beta \\bigl(1+\\vert p_{0} \\vert ^{2} \\bigr)^{\\frac{p-1}{2}}C_{1}\\varepsilon \\biggl( \\int_{B_{R}(x_{0})}\\vert Du-p_{0} \\vert ^{p-\\varepsilon }\\,dx \\biggr)^{\\frac{1-\\varepsilon }{p-\\varepsilon }} \\biggl(\\int_{B_{R}(x_{0})}\\,dx \\biggr)^{\\frac{p-1}{p-\\varepsilon }} \\\\ &\\leqslant 2^{p-1}\\beta C_{1}\\varepsilon \\int_{B_{R}(x_{0})}\\vert Du-p _{0} \\vert ^{p-\\varepsilon } \\,dx \\\\ &\\quad +2^{p-1}\\beta \\bigl(1+\\vert p_{0} \\vert ^{2} \\bigr)^{\\frac{p-1}{2}}C_{1}\\varepsilon \\biggl( \\varepsilon_{2} \\int_{B_{R}(x_{0})}\\vert Du-p_{0} \\vert ^{p-\\varepsilon } \\,dx+ \\varepsilon_{2}^{-\\frac{1-\\varepsilon }{p-1}} \\int_{B_{R}(x_{0})}\\,dx \\biggr) \\\\ &\\leqslant 2^{p-1}\\beta C_{1}\\varepsilon \\int_{B_{R}(x_{0})}\\vert Du-p _{0} \\vert ^{p-\\varepsilon }\\,dx+2^{p-1}\\beta \\bigl(1+ \\vert p_{0} \\vert ^{2} \\bigr)^{\\frac{p-1}{2}}C _{1}\\varepsilon\\varepsilon_{2}^{-\\frac{1-\\varepsilon }{p-1}} \\int_{B_{R}(x_{0})}\\,dx \\\\ &\\quad +2^{p-1}\\beta \\bigl(1+\\vert p_{0} \\vert ^{2} \\bigr)^{\\frac{p-1}{2}}C_{1}\\varepsilon \\varepsilon_{2} \\int_{B_{R}(x_{0})}\\vert Du-p_{0} \\vert ^{p-\\varepsilon } \\,dx. \\end{aligned}\n(3.10)\n\nCombining (H1) and the estimate of $$E(\\eta ,v)$$, we find that\n\n\\begin{aligned} I_{3}&\\leqslant \\int_{B_{R}(x_{0})} \\bigl\\vert A(x, u,Du) \\bigr\\vert \\bigl\\vert E( \\eta ,v) \\bigr\\vert \\,dx \\\\ &\\leqslant \\int_{B_{R}(x_{0})} \\beta \\bigl(1+\\vert Du \\vert ^{2} \\bigr)^{\\frac{p-1}{2}}2^{ \\varepsilon} \\frac{1+\\varepsilon }{1-\\varepsilon }\\vert vD \\eta \\vert ^{1- \\varepsilon }\\,dx \\\\ &\\leqslant \\beta 2^{\\varepsilon} \\frac{1+\\varepsilon }{1-\\varepsilon } \\biggl(\\frac{C}{R} \\biggr)^{1-\\varepsilon } \\int_{B_{R}(x_{0})} \\bigl(1+\\vert Du \\vert ^{2} \\bigr)^{ \\frac{p-1}{2}}\\vert v \\vert ^{1-\\varepsilon }\\,dx. \\\\ &\\leqslant \\beta 2^{\\varepsilon} \\frac{1+\\varepsilon }{1-\\varepsilon } \\biggl(\\frac{C}{R}\\biggr)^{1-\\varepsilon } \\int_{B_{R}(x_{0})} \\bigl(2 \\bigl(1+\\vert p_{0} \\vert ^{2} \\bigr)+2\\vert Du-p _{0}\\vert ^{2} \\bigr)^{\\frac{p-1}{2}} \\vert v \\vert ^{1-\\varepsilon } \\,dx \\\\ &\\leqslant \\beta 2^{\\varepsilon} \\frac{1+\\varepsilon }{1-\\varepsilon } \\biggl(\\frac{C}{R} \\biggr)^{1-\\varepsilon }2^{p-1} \\int_{B_{R}(x_{0})} \\bigl( \\bigl(1+\\vert p_{0} \\vert ^{2} \\bigr)^{ \\frac{p-1}{2}}+\\vert Du-p_{0} \\vert ^{p-1} \\bigr)\\vert v \\vert ^{1-\\varepsilon }\\,dx. \\end{aligned}\n\nDenoting $$\\beta 2^{\\varepsilon} \\frac{1+\\varepsilon }{1-\\varepsilon }( \\frac{C}{R})^{1-\\varepsilon }2^{p-1}$$ by $$C_{3}$$, we have\n\n$$I_{3} \\leqslant C_{3} \\bigl(1+\\vert p_{0} \\vert ^{2} \\bigr)^{\\frac{p-1}{2}}K_{1}+C_{3}K _{2},$$\n(3.11)\n\nwhere $$K_{1}=\\int_{B_{R}(x_{0})} \\vert v \\vert ^{1-\\varepsilon }\\,dx$$ and $$K_{2}=\\int_{B_{R}(x_{0})}\\vert Du-p_{0} \\vert ^{p-1}\\vert v \\vert ^{1-\\varepsilon }\\,dx$$. Let us estimate $$K_{1}$$ and $$K_{2}$$. Using Young’s inequality with exponents $$\\frac{p-\\varepsilon }{1-\\varepsilon }$$ and $$\\frac{p-\\varepsilon }{p-1}$$ and Poincaré’s inequality with constant $$C_{P}$$, we find that, for $$\\varepsilon_{3}>0$$,\n\n\\begin{aligned} K_{1} &\\leqslant \\int_{B_{R}(x_{0})} \\bigl(\\varepsilon_{3}\\vert v \\vert ^{p-\\varepsilon }+\\varepsilon_{3}^{-\\frac{1-\\varepsilon }{p-1}}1^{\\frac{p-\\varepsilon }{p-1}} \\bigr) \\,dx \\\\ &\\leqslant \\varepsilon_{3} \\int_{B_{R}(x_{0})}\\vert v \\vert ^{p-\\varepsilon }\\,dx+ \\varepsilon_{3}^{-\\frac{1-\\varepsilon }{p-1}} \\int_{B_{R}(x_{0})}\\,dx \\\\ &\\leqslant \\varepsilon_{3}C_{P} \\int_{B_{R}(x_{0})}\\vert Du-p_{0} \\vert ^{p- \\varepsilon } \\,dx+\\varepsilon_{3}^{-\\frac{1-\\varepsilon }{p-1}} \\int_{B_{R}(x_{0})}\\,dx. \\end{aligned}\n(3.12)\n\nLetting $$p'=\\frac{n(p-\\varepsilon )}{(n+1-\\varepsilon )(p-1)}$$ and $$q'=\\frac{n(p-\\varepsilon )}{(n-p+1)(1-\\varepsilon )}$$, we see that $$1< p'<\\infty$$, $$1< q'<\\infty$$, and $$\\frac{1}{p'}+\\frac{1}{q'}=1$$. With the aid of Hölder’s inequality, we can estimate\n\n\\begin{aligned} K_{2} &\\leqslant \\biggl( \\int_{B_{R}(x_{0})}\\vert Du-p_{0} \\vert ^{(p-1)p'} \\,dx \\biggr)^{ \\frac{1}{p'}} \\biggl( \\int_{B_{R}(x_{0})}\\vert v \\vert ^{(1-\\varepsilon )q'}\\,dx \\biggr)^{ \\frac{1}{q'}} \\\\ &\\leqslant \\biggl( \\int_{B_{R}(x_{0})}\\vert Du-p_{0} \\vert ^{\\frac{n(p-\\varepsilon )}{n+1- \\varepsilon }} \\,dx \\biggr)^{\\frac{(n+1-\\varepsilon )(p-1)}{n(p-\\varepsilon )}} \\biggl( \\int_{B_{R}(x_{0})}\\vert v \\vert ^{\\frac{n(p-\\varepsilon )}{n-p+1}}\\,dx \\biggr)^{\\frac{(n-p+1)(1- \\varepsilon )}{n(p-\\varepsilon )}}. \\end{aligned}\n\nNow we set $$p''=\\frac{n(p-\\varepsilon )}{n+1-\\varepsilon }$$. Then $$\\frac{np''}{n-p''}=\\frac{n(p-\\varepsilon )}{n-p+1}$$. Using the Sobolev-Poincaré inequality with constant $$C_{s}$$, we get\n\n\\begin{aligned} K_{2} &\\leqslant C_{s}^{1-\\varepsilon } \\biggl( \\int_{B_{R}(x_{0})}\\vert Du-p_{0} \\vert ^{\\frac{n(p- \\varepsilon )}{n+1-\\varepsilon }} \\,dx \\biggr)^{\\frac{(n+1-\\varepsilon )(p-1)}{n(p- \\varepsilon )}} \\biggl( \\int_{B_{R}(x_{0})}\\vert Du-p_{0} \\vert ^{\\frac{n(p-\\varepsilon )}{n+1- \\varepsilon }} \\,dx \\biggr)^{\\frac{(n+1-\\varepsilon )(1-\\varepsilon )}{n(p- \\varepsilon )}} \\\\ &\\leqslant C_{s}^{1-\\varepsilon } \\biggl( \\int_{B_{R}(x_{0})}\\vert Du-p_{0} \\vert ^{\\frac{n(p- \\varepsilon )}{n+1-\\varepsilon }} \\,dx \\biggr)^{\\frac{n+1-\\varepsilon }{n}}. \\end{aligned}\n(3.13)\n\nCombining (3.11) with (3.12) and (3.13), we obtain the estimate for $$I_{3}$$:\n\n\\begin{aligned} I_{3} &\\leqslant C_{3} \\bigl(1+\\vert p_{0} \\vert ^{2} \\bigr)^{\\frac{p-1}{2}} \\varepsilon_{3}C _{P} \\int_{B_{R}(x_{0})}\\vert Du-p_{0} \\vert ^{p-\\varepsilon } \\,dx \\\\ &\\quad {}+C_{3} \\bigl(1+\\vert p_{0} \\vert ^{2} \\bigr)^{\\frac{p-1}{2}}\\varepsilon_{3}^{-\\frac{1- \\varepsilon }{p-1}} \\int_{B_{R}(x_{0})}\\,dx \\\\ &\\quad {}+C_{3} C_{s}^{1-\\varepsilon } \\biggl( \\int_{B_{R}(x_{0})}\\vert Du-p_{0} \\vert ^{\\frac{n(p- \\varepsilon )}{n+1-\\varepsilon }} \\,dx \\biggr)^{\\frac{n+1-\\varepsilon }{n}}. \\end{aligned}\n(3.14)\n\nFinally, we estimate $$I_{4}$$. Using Hölder’s inequality with exponents $$\\frac{n(p-\\varepsilon )}{n(p-1)+p-\\varepsilon }$$ and $$\\frac{n(p-\\varepsilon )}{n(1-\\varepsilon )-p+\\varepsilon }$$, we have\n\n\\begin{aligned} I_{4} & \\leqslant \\int_{B_{R}(x_{0})} \\bigl\\vert f(x) \\bigr\\vert \\vert \\phi \\vert \\,dx \\\\ & \\leqslant \\biggl( \\int_{B_{R}(x_{0})}\\vert f \\vert ^{\\frac{n(p-\\varepsilon )}{n(p-1)+p- \\varepsilon }}\\,dx \\biggr)^{\\frac{n(p-1)+p-\\varepsilon }{n(p-\\varepsilon )}} \\biggl( \\int_{B_{R}(x_{0})}\\vert \\phi \\vert ^{\\frac{n(p-\\varepsilon )}{n(1-\\varepsilon )-p+\\varepsilon }}\\,dx \\biggr)^{\\frac{n(1-\\varepsilon )-p+\\varepsilon }{n(p- \\varepsilon )}}. \\end{aligned}\n\nSetting $$p'''=\\frac{p-\\varepsilon }{1-\\varepsilon }$$, we have $$\\frac{np'''}{n-p'''}=\\frac{n(p-\\varepsilon )}{n(1-\\varepsilon )-p+ \\varepsilon }$$. Notice that $$\\phi \\in W_{0}^{1,\\frac{p-\\varepsilon }{1- \\varepsilon }}(B_{R}(x_{0}))$$. Therefore, we can apply the Sobolev-Poincaré inequality to get\n\n\\begin{aligned} I_{4}\\leqslant C_{s} \\biggl( \\int_{B_{R}(x_{0})}\\vert f \\vert ^{\\frac{n(p-\\varepsilon )}{n(p-1)+p- \\varepsilon }}\\,dx \\biggr)^{\\frac{n(p-1)+p-\\varepsilon }{n(p-\\varepsilon )}} \\biggl( \\int_{B_{R}(x_{0})}\\vert D\\phi \\vert ^{\\frac{p-\\varepsilon }{1-\\varepsilon }}\\,dx \\biggr)^{\\frac{1- \\varepsilon }{p-\\varepsilon }}. \\end{aligned}\n\nCombining this with (3.5), with the aid of Young’s inequality, we obtain, for $$\\varepsilon_{4}>0$$,\n\n\\begin{aligned} I_{4}&\\leqslant C_{s} C_{2} \\biggl( \\int_{B_{R}(x_{0})}\\vert f \\vert ^{\\frac{n(p- \\varepsilon )}{n(p-1)+p-\\varepsilon }}\\,dx \\biggr)^{\\frac{n(p-1)+p-\\varepsilon }{n(p-\\varepsilon )}} \\biggl( \\int_{B_{R}(x_{0})}\\vert Du-p_{0} \\vert ^{p-\\varepsilon } \\,dx \\biggr)^{\\frac{1- \\varepsilon }{p-\\varepsilon }} \\\\ &\\leqslant C_{s} C_{2} \\biggl(\\varepsilon_{4} \\int_{B_{R}(x_{0})}\\vert Du-p_{0} \\vert ^{p- \\varepsilon } \\,dx+\\varepsilon_{4}^{-\\frac{1-\\varepsilon }{p-1}} \\biggl( \\int_{B_{R}(x_{0})}\\vert f \\vert ^{\\frac{n(p-\\varepsilon )}{n(p-1)+p-\\varepsilon }}\\,dx \\biggr)^{\\frac{n(p-1)+p-\\varepsilon }{n(p-1)}} \\biggr). \\end{aligned}\n(3.15)\n\nSubstituting (3.8), (3.9), (3.10), (3.14), and (3.15) into (3.7), we finally have\n\n\\begin{aligned} & 2^{\\frac{2-p}{2}}\\alpha \\int_{B_{R/2}(x_{0})}\\vert Du-p_{0} \\vert ^{p-\\varepsilon } \\,dx \\\\ &\\quad \\leqslant \\beta \\bigl(1+\\vert p_{0} \\vert ^{2} \\bigr)^{\\frac{p-1}{2}}\\varepsilon_{1} \\int_{B_{R}(x_{0})}\\vert Du-p_{0} \\vert ^{p-\\varepsilon } \\,dx+2^{p-1}\\beta C_{1} \\varepsilon \\int_{B_{R}(x_{0})}\\vert Du-p_{0} \\vert ^{p-\\varepsilon } \\,dx \\\\ &\\quad\\quad {}+2^{p-1}\\beta \\bigl(1+\\vert p_{0} \\vert ^{2} \\bigr)^{\\frac{p-1}{2}}C_{1}\\varepsilon \\varepsilon_{2} \\int_{B_{R}(x_{0})}\\vert Du-p_{0} \\vert ^{p-\\varepsilon } \\,dx \\\\ &\\quad \\quad {}+C_{3} \\bigl(1+\\vert p_{0} \\vert ^{2} \\bigr)^{\\frac{p-1}{2}}\\varepsilon_{3}C_{P} \\int_{B_{R}(x_{0})}\\vert Du-p_{0} \\vert ^{p-\\varepsilon } \\,dx +C_{s} C_{2} \\varepsilon_{4} \\int_{B_{R}(x_{0})}\\vert Du-p_{0} \\vert ^{p-\\varepsilon } \\,dx \\\\ &\\quad \\quad {}+C_{3} C_{s}^{1-\\varepsilon } \\biggl( \\int_{B_{R}(x_{0})}\\vert Du-p_{0} \\vert ^{\\frac{n(p- \\varepsilon )}{n+1-\\varepsilon }} \\,dx \\biggr)^{\\frac{n+1-\\varepsilon }{n}} \\\\ &\\quad \\quad {} +C _{s} C_{2} \\varepsilon_{4}^{-\\frac{1-\\varepsilon }{p-1}} \\biggl( \\int_{B_{R}(x_{0})}\\vert f \\vert ^{\\frac{n(p-\\varepsilon )}{n(p-1)+p-\\varepsilon }}\\,dx \\biggr)^{\\frac{n(p-1)+p-\\varepsilon }{n(p-1)}} \\\\ &\\quad\\quad {} +\\beta \\bigl(1+\\vert p_{0} \\vert ^{2} \\bigr)^{\\frac{p-1}{2}}\\varepsilon_{1}^{-\\frac{1- \\varepsilon }{p-1}} \\int_{B_{R}(x_{0})}\\,dx+2^{p-1}\\beta \\bigl(1+\\vert p_{0} \\vert ^{2} \\bigr)^{ \\frac{p-1}{2}}C_{1} \\varepsilon \\varepsilon_{2}^{-\\frac{1-\\varepsilon }{p-1}} \\int_{B_{R}(x_{0})}\\,dx \\\\ &\\quad \\quad {}+C_{3} \\bigl(1+\\vert p_{0} \\vert ^{2} \\bigr)^{\\frac{p-1}{2}}\\varepsilon_{3}^{-\\frac{1- \\varepsilon }{p-1}} \\int_{B_{R}(x_{0})}\\,dx. \\end{aligned}\n\nRearranging this inequality, we have\n\n\\begin{aligned} & \\int_{B_{R/2}(x_{0})}\\vert Du-p_{0} \\vert ^{p-\\varepsilon } \\,dx \\\\ &\\quad \\leqslant \\theta \\int_{B_{R}(x_{0})}\\vert Du-p_{0} \\vert ^{p-\\varepsilon } \\,dx +\\frac{1}{ \\alpha }2^{\\frac{p-2}{2}}C_{3} C_{s}^{1-\\varepsilon } \\biggl( \\int_{B_{R}(x _{0})}\\vert Du-p_{0} \\vert ^{\\tau } \\,dx \\biggr)^{\\frac{p-\\varepsilon }{\\tau }} \\\\ &\\quad \\quad {}+\\frac{1}{\\alpha }2^{\\frac{p-2}{2}}C_{s} C_{2} \\varepsilon _{4}^{-\\frac{1-\\varepsilon }{p-1}} \\biggl( \\int_{B_{R}(x_{0})}\\vert f \\vert ^{\\frac{n(p- \\varepsilon )}{n(p-1)+p-\\varepsilon }}\\,dx \\biggr)^{\\frac{n(p-1)+p-\\varepsilon }{n(p-1)}} +C_{4} \\int_{B_{R}(x_{0})}\\,dx, \\end{aligned}\n\nwhere $$\\theta =\\frac{1}{\\alpha }2^{\\frac{p-2}{2}}(\\beta (1+\\vert p_{0} \\vert ^{2})^{ \\frac{p-1}{2}}\\varepsilon_{1}+2^{p-1}\\beta C_{1}\\varepsilon +2^{p-1} \\beta (1+\\vert p_{0} \\vert ^{2})^{\\frac{p-1}{2}}C_{1}\\varepsilon \\varepsilon_{2}+C _{3} (1+\\vert p_{0} \\vert ^{2})^{\\frac{p-1}{2}} \\varepsilon_{3}C_{P}+C_{s} C_{2} \\varepsilon_{4})$$, $$\\tau =\\frac{n(p-\\varepsilon )}{n+1-\\varepsilon }$$, and $$C_{4}=\\frac{1}{\\alpha }2^{\\frac{p-2}{2}}(\\beta (1+\\vert p_{0} \\vert ^{2})^{ \\frac{p-1}{2}}\\varepsilon_{1}^{-\\frac{1-\\varepsilon }{p-1}}+2^{p-1} \\beta (1+\\vert p_{0} \\vert ^{2})^{\\frac{p-1}{2}}C_{1}\\varepsilon \\varepsilon_{2} ^{-\\frac{1-\\varepsilon }{p-1}}+C_{3}(1+\\vert p_{0} \\vert ^{2})^{\\frac{p-1}{2}} \\varepsilon_{3}^{-\\frac{1-\\varepsilon }{p-1}})$$.\n\nBy (H4) we get $$\\frac{n(p-\\varepsilon )}{n(p-1)+p-\\varepsilon }< \\frac{nq}{n(p-1)+q}$$ and $$\\int_{B_{R}(x_{0})}\\vert f \\vert ^{\\frac{n(p-\\varepsilon )}{n(p-1)+p-\\varepsilon }}\\,dx< M$$.\n\nMoreover, we have $$(\\int_{B_{R}(x_{0})}\\vert f \\vert ^{\\frac{n(p-\\varepsilon )}{n(p-1)+p- \\varepsilon }}\\,dx)^{\\frac{n(p-1)+p-\\varepsilon }{n(p-1)}}\\leqslant M ^{\\frac{p-\\varepsilon }{n(p-1)}}\\int_{B_{R}(x_{0})}\\vert f \\vert ^{\\frac{n(p- \\varepsilon )}{n(p-1)+p-\\varepsilon }}\\,dx$$.\n\nSetting $$C_{5}=\\max \\{\\frac{1}{\\alpha }2^{\\frac{p-2}{2}}C_{s} C_{2} \\varepsilon_{4}^{-\\frac{1-\\varepsilon }{p-1}}M^{ \\frac{p-\\varepsilon }{n(p-1)}},C_{4}\\}$$, we have\n\n\\begin{aligned} & \\int_{B_{R/2}(x_{0})}\\vert Du-p_{0} \\vert ^{p-\\varepsilon } \\,dx \\\\ &\\quad \\leqslant \\theta \\int_{B_{R}(x_{0})}\\vert Du-p_{0} \\vert ^{p-\\varepsilon } \\,dx +\\frac{1}{ \\alpha }2^{\\frac{p-2}{2}}C_{3} C_{s}^{1-\\varepsilon } \\biggl( \\int_{B_{R}(x _{0})}\\vert Du-p_{0} \\vert ^{\\tau } \\,dx \\biggr)^{\\frac{p-\\varepsilon }{\\tau }} \\\\ &\\quad\\quad {} +C_{5} \\biggl( \\int_{B_{R}(x_{0})}\\vert f \\vert ^{\\frac{n}{n(p-1)+p-\\varepsilon }(p-\\varepsilon )}\\,dx + \\int_{B_{R}(x_{0})}\\,dx \\biggr) \\\\ &\\quad \\leqslant \\theta \\int_{B_{R}(x_{0})}\\vert Du-p_{0} \\vert ^{p-\\varepsilon } \\,dx +\\frac{1}{ \\alpha }2^{\\frac{p-2}{2}}C_{3} C_{s}^{1-\\varepsilon } \\biggl( \\int_{B_{R}(x _{0})}\\vert Du-p_{0} \\vert ^{\\tau } \\,dx \\biggr)^{\\frac{p-\\varepsilon }{\\tau }} \\\\ &\\quad \\quad {}+C_{5} \\int_{B_{R}(x_{0})} \\bigl(\\vert f \\vert ^{\\frac{n}{n(p-1)+p-\\varepsilon }}+1 \\bigr)^{p-\\varepsilon }\\,dx. \\end{aligned}\n\nDividing both sides by $$\\vert B_{R}(x_{0}) \\vert =\\alpha_{n} R^{n}$$ yields\n\n\\begin{aligned}& \\fint_{B_{R/2}(x_{0})}\\vert Du-p_{0} \\vert ^{p-\\varepsilon } \\,dx \\\\& \\quad \\leqslant 2^{n}\\theta - \\int_{B_{R}(x_{0})}\\vert Du-p_{0} \\vert ^{p-\\varepsilon } \\,dx +C_{6} \\biggl( \\fint_{B_{R}(x_{0})}\\vert Du-p_{0} \\vert ^{\\tau } \\,dx \\biggr)^{\\frac{p-\\varepsilon }{ \\tau }} \\\\& \\quad \\quad {}+2^{n}C_{5} \\fint_{B_{R}(x_{0})} \\bigl(\\vert f \\vert ^{\\frac{n}{n(p-1)+p-\\varepsilon }}+1 \\bigr)^{p- \\varepsilon }\\,dx, \\end{aligned}\n(3.16)\n\nwhere $$C_{6}=\\frac{2^{n}}{\\alpha }2^{\\frac{p-2}{2}}C_{3} C_{s}^{1- \\varepsilon }(\\alpha_{n}R^{n})^{\\frac{p-\\tau -\\varepsilon }{\\tau }}$$.\n\nTaking $$\\varepsilon ,\\varepsilon_{1},\\varepsilon_{2},\\varepsilon_{3}, \\varepsilon_{4}$$ sufficiently small such that $$2^{n}\\theta <1$$ and $$\\tau >1$$, we obtain the reverse Hölder inequality for $$\\vert Du-p_{0} \\vert ^{p- \\varepsilon }$$. Accordingly, by Lemma 2 we can derive that u belongs to $$W^{1,r'}_{\\mathrm{loc}}(\\Omega )$$ with $$r'>r$$. Since $$f\\in L_{\\mathrm{loc}}^{ \\frac{nq}{n(p-1)+q}}(\\Omega )$$, $$q>p$$, reasoning as before, we get a new estimate analogous to (3.16) with exponents $$r'$$ and $$\\tau '$$ in place of r, that is, $$p-\\varepsilon$$ and τ, respectively:\n\n\\begin{aligned}& \\fint_{B_{R/2}(x_{0})}\\vert Du-p_{0} \\vert ^{r'}\\,dx \\\\& \\quad \\leqslant 2^{n}\\theta ' \\fint_{B_{R}(x_{0})}\\vert Du-p_{0} \\vert ^{r'}\\,dx +C_{6}' \\biggl( \\fint_{B_{R}(x_{0})}\\vert Du-p_{0} \\vert ^{\\tau '}\\,dx \\biggr)^{\\frac{r'}{\\tau '}} \\\\& \\quad\\quad {} +2^{n}C_{5}' \\fint_{B_{R}(x_{0})} \\bigl(\\vert f \\vert ^{\\frac{n}{n(p-1)+r'}}+1 \\bigr)^{r'}\\,dx. \\end{aligned}\n\nTherefore, we get $$u\\in W^{1,r''}_{\\mathrm{loc}}(\\Omega )$$ with $$r''>r'$$. Repeating this process, we can improve the degree of integrability of Du again and again. Thus, it is clear that $$u\\in W^{1,t}_{\\mathrm{loc}}( \\Omega )$$ with any $$t\\in (r_{1},r_{2})$$.\n\nThis completes the proof of Theorem 1. □\n\n### Proof\n\nThe aim of Theorem 2 is to prove that very weak solutions to systems (1.1) are not only weak solutions to (1.1) but also the optimal Hölder continuity. In fact, under the assumptions of Theorem 1, we get that $$u\\in W_{\\mathrm{loc}}^{1,r}(\\Omega )$$ with $$r_{1}< r< p$$ are weak solutions $$u\\in W_{\\mathrm{loc}}^{1,p}(\\Omega )$$ to systems (1.1) by Corollary 1. Then we can safely infer $$u\\in C^{1,\\gamma }(\\Omega_{0})$$ based on A-harmonic approximation technique. The proving method is standard, so we omit the process of derivation in this paper. For more details, we refer the reader to Theorem 1.1 of and the related literature. So the proof of Theorem 2 is complete. □\n\n## References\n\n1. Gehring, FW: The $$L^{p}$$-integrability of the partial derivatives of a quasiconformal mapping. Acta Math. 130, 265-277 (1973)\n\n2. Meyers, N, Elcrat, A: Some results on regularity for solutions of non-linear elliptic systems and quasi-regular functions. Duke Math. J. 42(1), 121-136 (1975)\n\n3. Meyers, N: An $$L^{p}$$-estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 17, 189-206 (1963)\n\n4. Lewis, JL: On the very weak solutions of certain elliptic systems. Commun. Partial Differ. Equ. 18, 1515-1537 (1993)\n\n5. Iwaniec, T, Sbordone, C: Weak minima of variational integrals. J. Reine Angew. Math. 454, 143-161 (1994)\n\n6. Giachetti, D, Leonetti, F, Schiachi, R: On the regularity of very weak minimal. Proc. R. Soc. Edinb. A 126, 287-296 (1996)\n\n7. Tong, YX, Gu, JT, Xu, XJ: Regularity for very weak solutions to A-harmonic equation. Appl. Math. J. Chin. Univ. Ser. A 24(3), 319-323 (2009)\n\n8. Greco, L, Iwaniec, T, Sbordone, C: Inverting the p-harmonic operator. Manuscr. Math. 92, 249-258 (1997)\n\n9. De Giorgi, E: Frontiere orientate di misura minima. Seminaro Mat. Scuola Norm. Sup. Pisa. Editrice Tecnico Scientifica, Pisa (1961)\n\n10. De Giorgi, E: Un esempio di estremali discontinue per un problema variazionale di tipo ellitico. Boll. Unione Mat. Ital. 4, 135-137 (1968)\n\n11. Giusti, E, Miranda, M: Sulla regolarità delle soluzioni deboli di una classe di sistemi ellittici quasi-lineari. Arch. Ration. Mech. Anal. 31, 173-184 (1968)\n\n12. Iwaniec, T, Migliaccio, L, Nania, L, Sbordone, C: Integrability and removability results for quasiregular mappings in high dimensions. Math. Scand. 75(2), 263-279 (1994)\n\n13. Giaquinta, M: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton University Press, Princeton (1983)\n\n14. Chen, SH, Tan, Z: Optimal interior partial regularity for nonlinear elliptic systems under the natural growth condition: the method of A-harmonic approximation. Acta Math. Sci. 27B(3), 491-508 (2007)\n\n## Acknowledgements\n\nThe authors would like to thank the anonymous referee for careful reading the manuscript and valuable comments. This work was supported by the National Natural Science foundation of China under Grant No. 11571159.\n\n## Author information\n\nAuthors\n\n### Corresponding author\n\nCorrespondence to Qing Zhao.\n\n### Competing interests\n\nThe authors declare that they have no competing interests.\n\n### Authors’ contributions\n\nBoth authors contributed equally to writing of this paper. Both authors read and approved the final manuscript.\n\n## Rights and permissions", null, "" ]	[ null, "https://journalofinequalitiesandapplications.springeropen.com/track/article/10.1186/s13660-017-1297-z", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7568811,"math_prob":0.9999851,"size":21781,"snap":"2022-27-2022-33","text_gpt3_token_len":7396,"char_repetition_ratio":0.15787299,"word_repetition_ratio":0.0420339,"special_character_ratio":0.35076442,"punctuation_ratio":0.13303572,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000072,"pos_list":[0,1,2],"im_url_duplicate_count":[null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-08-09T10:16:32Z\",\"WARC-Record-ID\":\"<urn:uuid:9726354c-f5f5-411d-a3c4-6699909cdd14>\",\"Content-Length\":\"263561\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:74315bba-9188-4107-ae27-9d3c62fc97a5>\",\"WARC-Concurrent-To\":\"<urn:uuid:165898d8-1452-404c-8de5-d53406a3f337>\",\"WARC-IP-Address\":\"146.75.36.95\",\"WARC-Target-URI\":\"https://journalofinequalitiesandapplications.springeropen.com/articles/10.1186/s13660-017-1297-z\",\"WARC-Payload-Digest\":\"sha1:CFEUNKVASAXIDKGH756GC5D6UAEZP4RV\",\"WARC-Block-Digest\":\"sha1:77W36CLVVXPQBXM6KXUINV44SMKQYHOX\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-33/CC-MAIN-2022-33_segments_1659882570921.9_warc_CC-MAIN-20220809094531-20220809124531-00330.warc.gz\"}"}
http://archive.numdam.org/book-part/AST_1992__207__35_0/	[ "Spectral theory of elliptic operators on non-compact manifolds\nMéthodes semi-classiques Volume 1 - École d'Été (Nantes, juin 1991), Astérisque no. 207 (1992), p. 35-108\n@incollection{AST_1992__207__35_0,\nauthor = {Shubin, M. A.},\ntitle = {Spectral theory of elliptic operators on non-compact manifolds},\nbooktitle = {M\\'ethodes semi-classiques Volume 1 - \\'Ecole d'\\'Et\\'e (Nantes, juin 1991)},\nauthor = {Collectif},\nseries = {Ast\\'erisque},\npublisher = {Soci\\'et\\'e math\\'ematique de France},\nnumber = {207},\nyear = {1992},\npages = {35-108},\nlanguage = {en},\nurl = {http://www.numdam.org/item/AST_1992__207__35_0}\n}\n\nShubin, M. A. Spectral theory of elliptic operators on non-compact manifolds, in Méthodes semi-classiques Volume 1 - École d'Été (Nantes, juin 1991), Astérisque, no. 207 (1992), pp. 35-108. http://www.numdam.org/item/AST_1992__207__35_0/\n\n S. Agmon, On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems, C.P.A.M. 15 (1) (1962), 119-147.\n\n M. S. Agranovich, M. I. Vishik, Elliptic problems with a parameter and parabolic problems of general type. Russian Math. Surveys 19 (1964), no. 3, 53-157.\n\n M. Atiyah, R. Bott, A Lefschetz fixed point formula for elliptic complexes I., Ann. of Math. 86 (1967), 374-407.\n\n Yu M. Berezanski, Expansions in Eigenfunctions of Selfadjoint Operators. AMS Translation of Math. Monographs, Providence, Rhode Island, 1968.\n\n F. A. Berezin, M. A. Shubin, The Schrödinger Equation. Kluwer, Dordrecht, 1991.\n\n R. Brooks, The fundamental group and the spectrum of the Laplacian. Comment. Math. Helv., 56 (1981), 581-598.\n\n F. E. Browder, On the spectral theory of elliptic differential operators. I, Math. Ann. 142 (1) (1960-61), 22-130.\n\n J. Cheeger, M. Gromov, M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator and the geometry of complete Riemannian manifolds. J. Diff. Geom. 17 (1982), 15-53.\n\n P. Chernoff, Essential self-adjointness of powers of generators of hyperbolic equations. J. Funct. Anal. 12 (1973), 401-414.\n\n H. O. Cordes, Self-adjointness of powers of elliptic operators on non-compact manifolds. Math. Ann., 195 (1972), 257-272.\n\n H. L. Cycon, R. G. Froese, W. Kirsch, B. Simon, Schrödinger operators. Springer, Berlin e.a., 1987.\n\n E. B. Davies, ${L}^{1}$-properties of second order elliptic operators, Bull. London Math. Soc. 17 (5) (1985), 417-436.\n\n K. O. Friedrichs, Symmetric hyperbolic linear differential equations. Comm. Pure Appl. Math. 7 (1954), 345-392.\n\n M. P. Gaffney, The harmonic operator for exterior differential forms. Proc. Nat. Acad. Sci. USA, 37 (1951), 48-50.\n\n M. P. Gaffney, A special Stokes's theorem for complete Riemannian manifolds. Ann. of Math. 60 (1954), 140-145.\n\n M. P. Gaffney, Hilbert space methods in the theory of harmonic integrals. Transactions Amer. Math. Soc., 78 (1955), 587-665.\n\n I. M. Gelfand, A. G. Kostyuchenko, Expansion in eigenfunctions of differential and other operators. Dokl. Akad. Nauk SSSR, 103 (1955), 349-352.\n\n I. M. Glazman, Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators. Israel Program for Scientific Translation, Jerusalem 1965.\n\n M. Gromov, Curvature, diameter and Betti numbers. Comment Math. Helvetici 56 (1981), 179-195.\n\n M. Gromov, Structures métriques pour les variétés Riemanniennes, CEDIC/Fernand Nathan (1981).\n\n M. Gromov, H. B. Lawson, Positive scalar curvature and the Dirac operator on complete Riemannian manifolds. Publications Mathématiques, 58 (1983), 83-196.\n\n L. Hörmander. The analysis of linear partial differential operators. Berlin e.a., Springer-Verlag, vol. 1, 2 (1983)\n\nL. Hörmander. The analysis of linear partial differential operators. Berlin e.a., Springer-Verlag, vol. 3, 4 (1985).\n\n T. Ikebe, T. Kato, Uniqueness of the self-adjoint extension of singular elliptic differential operators. Arch. Rath. Mech. Anal. 9 (1962), 77-92.\n\n T. Kato, A remark to the preceding paper by Chernoff. J. Funct. Anal., 12 (1973), 415-417.\n\n T. Kato, ${L}^{p}$-theory of Schrödinger operators with a singular potential, In: Aspects of positivity in functional analysis-Proc. of a Conference in Tubingen, North Holland, Math. Studies 122 (1986), 63-78.\n\n T. Kobayashi, K. Ono, T. Sunada. Periodic Schrödinger operators on a manifold. Forum Math. 1 (1989), 69-79.\n\n Yu. A. Kordyukov. ${L}^{p}$-theory of elliptic differential operators with bounded coefficients. Vestnik Moskovskogo Universiteta, Ser. I Math. Mech. 1988, N° 4, 98-100 (in Russian).\n\n Yu. A. Kordyukov. Elliptic operators on manifolds of bounded geometry. Thesis, Moscow State University, 1987 (in Russian).\n\n G. A. Meladze, M. A. Shubin. Properly supported uniform pseudo-differential operators on unimodular Lie groups. Trudy Sem. Petrovskogo 11 (1986), 74-97\n\nG. A. Meladze, M. A. Shubin. Functional calculus of pseudo-differential operators on unimodular Lie groups. Trudy Sem. Petrovskogo 12 (1987), 164-200 (in Russian).\n\n I. M. Oleinik. On the essential self-adjointness of the Schrodinger operator on complete Riemannian manifolds. Preprint, 1991 (in Russian).\n\n A. Ja. Povzner. On the expansion of arbitrary functions in characteristic functions of the operator $-\\Delta u+cu$, Mat. Sb. 32 (74) (1953), 109-156; English transl.: AMS Transl. (2) 60 (1966), 1-49.\n\n M. Reed, B. Simon. Methods of Modern Mathematical Physis. I: Functional Analysis. Academic Press, New York e.a., 1980.\n\nM. Reed, B. Simon. Methods of Modern Mathematical Physis. II: Fourier Analysis, Self-Adjointness. Academic Press, New York e.a., 1975.\n\n J. Roe. An index theorem on open manifolds. I. II. J. Diff. Geom. 27 (1988), 87-113, 115-136.\n\n W. Roelcke. Über den Laplace-Operator auf Riemannschen Mannigfaltigkeiten mit diskontinuerlichen Gruppen. Math. Nachr., 21 (1960), 132-149.\n\n F. S. Rofe-Beketov. Deficiency indices and properties of spectum of some classes of differential operators. In: Spectral Theory and Differential Equations (Proc. Sympos., Dundee, 1974, dedicated to Konrad Jörgens). Lect. Notes Math., 448 (1975), 273-293.\n\n F. S. Rofe-Beketov. Square-integrable solutions, self-adjoint extensions and spectrum of differential systems. In: Differ. Equations (Proc. Intern. Conf., Uppsala, 1977)\n\nF. S. Rofe-Beketov. Square-integrable solutions, self-adjoint extensions and spectrum of differential systems. In: Sympos. Univ. Upsaliensis Ann. Quingentesimum Celebrantis, No. 7, Almquist & Wiksell, Stockholm, 1977, 169-178.\n\n F. S. Rofe-Beketov, A. M. Hol'Kin. Conditions for the self-adjointness of second order elliptic operators of general form. Teor. Funkcii i Funkcional. Anal. i Priložen. 17 (1973), 41-51.\n\n E. Schnol. On the behaviour of the Schrödinger equation. Mat. Sbornik 42 (1957), 273-286 (in Russian).\n\n D. B. Sears. Note on the uniqueness of Green's functions associated with certain differential equations. Canadian J. Math. 2 (1950), 314-325.\n\n R. Seeley, Complex powers of elliptic operators, Proc. Symp. Pure Math. 10 (1967), 288-307.\n\n M. A. Shubin. Pseudodifferential Operators and Spectral Theory. Springer-Verlag, Berlin e.a., 1987.\n\n M. A. Shubin. Theorems on the coincidence of the spectra of a pseudo-differential almost periodic operator in the spaces ${L}^{2}\\left({ℝ}^{n}\\right)$ and ${B}^{2}\\left({ℝ}^{n}\\right)$. Sibirsk. Mat. Zh. 17 (1976), no. 1, 200-215.\n\n M. A. Shubin. Pseudodifference operators and their Green's functions. Math. USSR Izvestiya 26 (1986), no. 3, 605-622.\n\n M. A. Shubin. Weak Bloch property and weight estimates for elliptic operators. Séminaire Equations aux derivées partielles, 1989-1990, École Polytechnique, Exposé n. V.\n\n M. A. Shubin, J. Sjöstrand, On the equality between weak and strong extensions. Séminaire Equations aux derivées partielles, 1989-1990, École Polytechnique, Appendix a l'Exposé n. V.\n\n H. B. Stewart, Generation of analytic semigroups by strongly elliptic operators, Trans. A.M.S., 199 (1) (1974), 141-162.\n\n R. S. Strichartz, ${L}^{p}$-contractive projections and the heat semigroup for differential forms, J. Funct. Anal., 65 (3) (1986), 348-357.\n\n E. C. Titchmarsh. Eigenfunction expansions associated with second-order differential equations. Part II. Oxford, 1958.\n\n E. Wienholtz. Halbbeschränkte partielle Differentialoperatoren zweiter Ordnung vom elliptischen Typus. Math. Ann. 135 (1958), 50-80." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.59057206,"math_prob":0.9071217,"size":7275,"snap":"2019-51-2020-05","text_gpt3_token_len":2217,"char_repetition_ratio":0.14042085,"word_repetition_ratio":0.04904905,"special_character_ratio":0.34364262,"punctuation_ratio":0.29302326,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9576392,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-12-12T00:17:12Z\",\"WARC-Record-ID\":\"<urn:uuid:af26d2d5-f7e7-4bd6-8b5b-1d0d45fb422b>\",\"Content-Length\":\"35343\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:882eba77-6958-4c48-a859-e9c580fbf463>\",\"WARC-Concurrent-To\":\"<urn:uuid:26104c51-1015-4e5d-822c-d70ec965e18f>\",\"WARC-IP-Address\":\"129.88.220.36\",\"WARC-Target-URI\":\"http://archive.numdam.org/book-part/AST_1992__207__35_0/\",\"WARC-Payload-Digest\":\"sha1:BI7BI4RIHOZGAROGNM2OXP5P2O2S3BH5\",\"WARC-Block-Digest\":\"sha1:3PEDNWJAPXZLJ6ZHJXLVFEA5LYEKM4X7\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-51/CC-MAIN-2019-51_segments_1575540534443.68_warc_CC-MAIN-20191212000437-20191212024437-00219.warc.gz\"}"}
https://simpliengineering.com/t/steel-pipe-maximum-bending-moment/2437	[ "", null, "# Steel Pipe Maximum Bending Moment\n\nQUESTION\nI am sorry for asking such a fundamental question. I have forgotten Strength of Material Basics that I studied in 1st Yr. I just want to know what is the formula for maximum bending moment for a steel pipe?\n\nI am not too sure exactly about the conditions. It is a 6\" subsea pipeline between Well Head Platform and Production Platform with risers installed at both ends. I would like to know bending moment for subsea pipeline as well as the riser. I don’t know in which category this would come. Is it possible to determine or some more information is necessary?\n\nI have read in a technical document, that for initial calculation Max Bending Moment can be estimated as 0.85 times SMYS. So my question is how can i get Bending Moment if I know SMYS because SMYS is a stress with units in MPa and Bending Moment units is in Kilo Newton Metre.\n\nI am doing wall thickness calculation based on DNV-OS-F-101. In the spreadsheet, for load interaction, I need to put the value of ME (Maximum Bending Moment due to Environmental load). I know I can get the exact value once I do Dynamic Pipeline Analysis, but initially I must assume certain value to put in the spreadsheet, as based on that spreadhsheet, will give the result whether the assumed thickness is ok or not. From yielding stress, how can I get limiting moment capacity (even though it may not be too useful). In the reference document that I am mentioning, they have arrived at the value of Max Bending bending of arnd 50kNm for 6\" pipeline (Grade X-52;SMYS=359Mpa) by taking it 0.85 X Yield Stress.\n\nREPLIES\n\npatswfc\nMoment capacity of pipe = yield stress x elastic section modulus of pipe.\n\nelastic section modulus of pipe = I/y\n\nI = pi(d4 - (d - 2t)4) / 64 , the two 4’s are to the power of 4\n\ny = d/2\n\nSOURCE" ]	[ null, "https://simpliengineering.com/uploads/default/original/1X/464e1a7fc53c30c8b37f1750c02df1ea62a1611b.svg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9296957,"math_prob":0.7515478,"size":1837,"snap":"2020-24-2020-29","text_gpt3_token_len":452,"char_repetition_ratio":0.10529187,"word_repetition_ratio":0.0,"special_character_ratio":0.23353294,"punctuation_ratio":0.08994709,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96211404,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-15T08:12:23Z\",\"WARC-Record-ID\":\"<urn:uuid:b3d7bbab-c4c6-43bc-b4b8-37663afbf2fc>\",\"Content-Length\":\"12811\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:28a94d59-5316-4433-81f9-3693012f7f97>\",\"WARC-Concurrent-To\":\"<urn:uuid:a91727b6-98cf-4fa3-ab0b-ee4a858a52f2>\",\"WARC-IP-Address\":\"68.183.147.6\",\"WARC-Target-URI\":\"https://simpliengineering.com/t/steel-pipe-maximum-bending-moment/2437\",\"WARC-Payload-Digest\":\"sha1:TV4LFRCOVUJIWXC2LGKJC5O5NJ2EBOW7\",\"WARC-Block-Digest\":\"sha1:UI7O3ZKPKVMNSPEM7TSJVAVYMF52KJF5\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-29/CC-MAIN-2020-29_segments_1593657163613.94_warc_CC-MAIN-20200715070409-20200715100409-00289.warc.gz\"}"}
https://discuss.pytorch.org/t/error-for-converting-numpy-images-to-tensor/112883	[ "# Error for converting numpy images to tensor\n\nI use the Opencv to read images, and then convert them to tensor to feed DNN for inference. The reason is that I need to use Opencv to handle the images and then DNN inference. However, when I use transforms to preprocess the images. It gives the error: “ValueError: pic should be 2/3 dimensional. Got 4 dimensions.”\nThe code is\n\n``````for index in range(len(datasets)):\nimg = datasets[index]\nframe = frame * (1/255)\nframe = cv2.resize(frame , (224, 224))\nframe = np.transpose(frame , (2, 0, 1))\nif index == 0:\nframes = frame\ncontinue\nframes = np.stack((frames , frame))\n\ntransform = transforms.Compose([\ntransforms.ToTensor(),\ntransforms.Normalize(mean=[0.485, 0.456, 0.406], std=[0.229, 0.224, 0.225])\n])\n\nframes = transform(frames ).to('cuda')\n``````\n\nThen it has the error “ValueError: pic should be 2/3 dimensional. Got 4 dimensions.” The shape of frames is (2, 3, 224, 224). 2 is the number of batches.\n\nI don’t know how to deal with this error. Thank you for any help.\n\nThe `ToTensor` tfm is expecting a channel last image, this works:\n\n``````transform(np.random.random((224,244, 3)))\n``````\n\nit is also not expecting a batch. From the dos:\n\nConvert a `PIL Image` or `numpy.ndarray` to tensor.\n\nIt expects one PIL image.\n\nLet me show some examples\n\nexample 1: use OpenCV and do transforms for each image\n\n``````transform = transforms.Compose([\ntransforms.ToTensor(),\ntransforms.Normalize(mean=[0.485, 0.456, 0.406], std=[0.229, 0.224, 0.225])\n])\nframes = []\nfor index in range(2):\nframe = (np.random.rand(256, 256, 3) * 255).astype(np.uint8)\n\n# resize image\nframe = frame * (1/255)\nframe = cv2.resize(frame, (224, 224))\nframe = transform(frame)\nif index == 0:\nframes = [frame]\ncontinue\nframes.append(frame)\n\nframes = torch.stack(frames)\n``````\n\nexample 2: use OpenCV and do transform after collecting all images\n\n``````frames = []\nfor index in range(2):\nframe = (np.random.rand(256, 256, 3) * 255).astype(np.uint8)\n\n# resize image\nframe = frame * (1/255)\nframe = cv2.resize(frame, (224, 224))\nframe = np.transpose(frame, (2, 0, 1))\nif index == 0:\nframes = [frame]\ncontinue\nframes.append(frame)\n\nframes = np.stack(frame, axis=0)\n# transform\nmean, std = torch.tensor([0.485, 0.456, 0.406]), torch.tensor([0.229, 0.224, 0.225])\nframes = (torch.from_numpy(frames) - mean[None, :, None, None]) / std[None, :, None, None]\n``````" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5907907,"math_prob":0.9984976,"size":2536,"snap":"2022-27-2022-33","text_gpt3_token_len":775,"char_repetition_ratio":0.17298578,"word_repetition_ratio":0.3172043,"special_character_ratio":0.37066245,"punctuation_ratio":0.25945017,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99935645,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-08-18T13:35:10Z\",\"WARC-Record-ID\":\"<urn:uuid:bdf001ad-14ac-4906-87ee-e0a6ff4fd87a>\",\"Content-Length\":\"20283\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:231d8210-32d8-48c6-8c5f-0ded759e788c>\",\"WARC-Concurrent-To\":\"<urn:uuid:de407a19-2dad-4cc6-822c-0f594473fc7a>\",\"WARC-IP-Address\":\"159.203.145.104\",\"WARC-Target-URI\":\"https://discuss.pytorch.org/t/error-for-converting-numpy-images-to-tensor/112883\",\"WARC-Payload-Digest\":\"sha1:WYKA2UZ5G4RB56KYQDTWKEJI5QTY56QU\",\"WARC-Block-Digest\":\"sha1:DG3T3JGHM3SXR5H5TCTFBMLX52JKI5YA\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-33/CC-MAIN-2022-33_segments_1659882573197.34_warc_CC-MAIN-20220818124424-20220818154424-00134.warc.gz\"}"}
http://consulask.it/pos91y77o27ozyj63l.php	[ "", null, "## Lavoro intellettuale o manuale\n\n(We then say the number is an even number. lavkro For example, in the number 236, the last. Digit is 6. Since 6 is divisible by 2 (6 247; 2 3), 236 is divisible by. A number is divisible by 3 if the sum of all the. Digits is divisible Let us take a number that does not end with zeros, for example, 916. This number can be pdf for sample fresno employment test county like the sum 900 16. Because both the addends are divisible by 4, their sum. 916 is also divisible by 4. On the contrary, 918 is not divisible by 4 because 918 900 18 and one of. The addends (18) is not divisible …Textbook solution for A Transition to Advanced Mathematics 8th Edition Douglas Smith Chapter 1. 7 Problem 3E. We have step-by-step solutions for your textbooks written by Bartleby inte,lettuale. 24, 2010Feb 01, 2020The sum inside the first brackets is divisible by 9 because all the addends see more divisible by 9. If the sum in the second brackets (3 2 4) is also divisible by 9, then the whole sum, 324. Is divisible by 9. Since the sum 3 2 4 is divisible by 9, we conclude that 324 is also divisible. By 9." ]	[ null, "http://consulask.it/img/blog/main-blog/m-blog-1.jp", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7560045,"math_prob":0.9819918,"size":8424,"snap":"2021-31-2021-39","text_gpt3_token_len":2312,"char_repetition_ratio":0.090736344,"word_repetition_ratio":0.0043636365,"special_character_ratio":0.22435898,"punctuation_ratio":0.11276734,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9887455,"pos_list":[0,1,2],"im_url_duplicate_count":[null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-09-18T17:03:03Z\",\"WARC-Record-ID\":\"<urn:uuid:f1052a99-3140-4d09-b4c2-c1d4ec0294d3>\",\"Content-Length\":\"41511\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b1e316ba-f85a-49e0-8009-bc407b9fd0b3>\",\"WARC-Concurrent-To\":\"<urn:uuid:340b01cd-1231-490c-9b28-ae836151efc1>\",\"WARC-IP-Address\":\"5.252.116.9\",\"WARC-Target-URI\":\"http://consulask.it/pos91y77o27ozyj63l.php\",\"WARC-Payload-Digest\":\"sha1:KU6N23ZY55HCLCUOHHAV5GBHBPRPDUSW\",\"WARC-Block-Digest\":\"sha1:YW6VO6PJTKO3TENG4A63E6XTCZXSLZYB\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-39/CC-MAIN-2021-39_segments_1631780056548.77_warc_CC-MAIN-20210918154248-20210918184248-00076.warc.gz\"}"}
https://www.geeksforgeeks.org/count-numbers-divisible-m-given-range/?ref=rp	[ "# Count the numbers divisible by ‘M’ in a given range\n\n• Difficulty Level : Medium\n• Last Updated : 26 Oct, 2021\n\nA and B are two numbers which define a range, where A <= B. Find the total numbers in the given range [A … B] divisible by ‘M’\nExamples:\n\n```Input : A = 25, B = 100, M = 30\nOutput : 3\nExplanation : In the given range [25 - 100],\n30, 60 and 90 are divisible by 30\n\nInput : A = 6, B = 15, M = 3\nOutput : 4\nExplanation : In the given range [6 - 15],\n6, 9, 12 and 15 are divisible by 3```\n\nAttention reader! Don’t stop learning now. Get hold of all the important mathematical concepts for competitive programming with the Essential Maths for CP Course at a student-friendly price. To complete your preparation from learning a language to DS Algo and many more, please refer Complete Interview Preparation Course.\n\nMethod 1 : [Brute-force]\nRun a loop from A to B. If a number divisible by ‘M’ is found, increment counter.\nBelow is the implementation of above method:\n\n## C++\n\n `// Program to count the numbers divisible by``// M in a given range``#include ``using` `namespace` `std;` `int` `countDivisibles(``int` `A, ``int` `B, ``int` `M)``{`` ``// Variable to store the counter`` ``int` `counter = 0;` ` ``// Running a loop from A to B and check`` ``// if a number is divisible by M.`` ``for` `(``int` `i = A; i <= B; i++)`` ``if` `(i % M == 0)`` ``counter++;` ` ``return` `counter;``}` `// Driver code``int` `main()``{`` ``// A and B define the range, M is the dividend`` ``int` `A = 30, B = 100, M = 30;` ` ``// Printing the result`` ``cout << countDivisibles(A, B, M) << endl;` ` ``return` `0;``}`\n\n## Java\n\n `// Java program to count the numbers divisible by``// M in a given range``import` `java.io.;` `class` `GFG {`` ``// Function to count the numbers divisible by`` ``// M in a given range`` ``static` `int` `countDivisibles(``int` `A, ``int` `B, ``int` `M)`` ``{`` ``// Variable to store the counter`` ``int` `counter = ``0``;` ` ``// Running a loop from A to B and check`` ``// if a number is divisible by M.`` ``for` `(``int` `i = A; i <= B; i++)`` ``if` `(i % M == ``0``)`` ``counter++;` ` ``return` `counter;`` ``}` ` ``// driver program`` ``public` `static` `void` `main(String[] args)`` ``{`` ``// A and B define the range, M is the dividend`` ``int` `A = ``30``, B = ``100``, M = ``30``;` ` ``// Printing the result`` ``System.out.println(countDivisibles(A, B, M));`` ``}``}` `// Contributed by Pramod Kumar`\n\n## Python3\n\n `# Program to count the numbers``# divisible by M in a given range` `def` `countDivisibles(A, B, M):`` ` ` ``# Variable to store the counter`` ``counter ``=` `0``;` ` ``# Running a loop from A to B`` ``# and check if a number is`` ``# divisible by M.`` ``for` `i ``in` `range``(A, B):`` ``if` `(i ``%` `M ``=``=` `0``):`` ``counter ``=` `counter ``+` `1` ` ``return` `counter` `# Driver code``# A and B define the range,``# M is the dividend``A ``=` `30``B ``=` `100``M ``=` `30` `# Printing the result``print``(countDivisibles(A, B, M))` `# This code is contributed by Sam007.`\n\n## C#\n\n `// C# program to count the numbers``// divisible by M in a given range``using` `System;` `public` `class` `GFG {` ` ``// Function to count the numbers divisible by`` ``// M in a given range`` ``static` `int` `countDivisibles(``int` `A, ``int` `B, ``int` `M)`` ``{`` ``// Variable to store the counter`` ``int` `counter = 0;` ` ``// Running a loop from A to B and check`` ``// if a number is divisible by M.`` ``for` `(``int` `i = A; i <= B; i++)`` ``if` `(i % M == 0)`` ``counter++;` ` ``return` `counter;`` ``}` ` ``// driver program`` ``public` `static` `void` `Main()`` ``{`` ``// A and B define the range, M is the dividend`` ``int` `A = 30, B = 100, M = 30;` ` ``// Printing the result`` ``Console.WriteLine(countDivisibles(A, B, M));`` ``}``}` `// This code is contributed by Sam007`\n\n## PHP\n\n ``\n\n## Javascript\n\n ``\n\nOutput:\n\n`3`\n\nMethod 2 : [Better]\nThe loop can be modified by incrementing the iterator ‘M’ times after the first divisible is found. Also, if ‘A’ is less than ‘M’, it can be changed to ‘M’, because a number less than ‘M’ can not be divided by it.\nMethod 3 : [Efficient]\n\n```Let B = b M and\nA = a * M\nThe count of numbers divisible by\n'M' between A and B will be equal\nto b - a.\n\nExample:\nA = 25, B = 70, M = 10.\nNow, a = 2, b = 7.\nCount = 7 - 2 = 5.```\n\nIt can be observed that, if A is divisible by M, ‘b – a’ will exclude the count for A, so the count will be less by 1. Thus, in this case we add 1 explicitly.\nExample when A is divisible by M:\n\n```A = 30, B = 70, M = 10.\nNow, a = 3, b = 7.\nCount = 7 - 3 = 4.\nBut, Count should be 5. Thus, we will\n\nBelow is the implementation of the above method :\n\n## C++\n\n `// C++ program to count the numbers divisible by``// M in a given range``#include ``using` `namespace` `std;` `// Function to count the numbers divisible by``// M in a given range``int` `countDivisibles(``int` `A, ``int` `B, ``int` `M)``{`` ` ` ``// Add 1 explicitly as A is divisible by M`` ``if` `(A % M == 0)`` ``return` `(B / M) - (A / M) + 1;` ` ``// A is not divisible by M`` ``return` `(B / M) - (A / M);``}` `// driver program``int` `main()``{`` ` ` ``// A and B define the range, M is the dividend`` ``int` `A = 30, B = 100, M = 30;` ` ``// Printing the result`` ``cout << (countDivisibles(A, B, M));``}` `// This code is contributed by subham348.`\n\n## Java\n\n `// Java program to count the numbers divisible by``// M in a given range``import` `java.io.*;` `class` `GFG {`` ``// Function to count the numbers divisible by`` ``// M in a given range`` ``static` `int` `countDivisibles(``int` `A, ``int` `B, ``int` `M)`` ``{`` ``// Add 1 explicitly as A is divisible by M`` ``if` `(A % M == ``0``)`` ``return` `(B / M) - (A / M) + ``1``;` ` ``// A is not divisible by M`` ``return` `(B / M) - (A / M);`` ``}` ` ``// driver program`` ``public` `static` `void` `main(String[] args)`` ``{`` ``// A and B define the range, M is the dividend`` ``int` `A = ``30``, B = ``100``, M = ``30``;` ` ``// Printing the result`` ``System.out.println(countDivisibles(A, B, M));`` ``}``}` `// Contributed by Pramod Kumar`\n\n## Python3\n\n `# Program to count the numbers divisible``# by M in a given range` `# Returns count of numbers in [A B] that``# are divisible by M.``def` `countDivisibles(A, B, M):`` ` ` ``# Add 1 explicitly as A is divisible by M`` ``if` `(A ``%` `M ``=``=` `0``):`` ``return` `((B ``/` `M) ``-` `(A ``/` `M)) ``+` `1` ` ``# A is not divisible by M`` ``return` `((B ``/` `M) ``-` `(A ``/` `M))` `# Driver Code``# A and B define the range, M``# is the divident``A ``=` `30``B ``=` `70``M ``=` `10` `# Printing the result``print``(countDivisibles(A, B, M))` `# This code is contributed by Sam007`\n\n## C#\n\n `// C# program to count the numbers``// divisible by M in a given range``using` `System;` `public` `class` `GFG {` ` ``// Function to count the numbers divisible by`` ``// M in a given range`` ``static` `int` `countDivisibles(``int` `A, ``int` `B, ``int` `M)`` ``{`` ``// Add 1 explicitly as A is divisible by M`` ``if` `(A % M == 0)`` ``return` `(B / M) - (A / M) + 1;` ` ``// A is not divisible by M`` ``return` `(B / M) - (A / M);`` ``}` ` ``// driver program`` ``public` `static` `void` `Main()`` ``{`` ``// A and B define the range, M is the dividend`` ``int` `A = 30, B = 100, M = 30;` ` ``// Printing the result`` ``Console.WriteLine(countDivisibles(A, B, M));`` ``}``}` `// This code is contributed by Sam007`\n\n## PHP\n\n ``\n\n## Javascript\n\n `// Javascript Program to count the numbers``// divisible by M in a given range` `// Returns count of numbers in``// [A B] that are divisible by M.``function` `countDivisibles(A, B, M)``{`` ` ` ``// Add 1 explicitly as A`` ``// is divisible by M`` ``if` `(A % M == 0)`` ``return` `(B / M) -`` ``(A / M) + 1;` ` ``// A is not divisible by M`` ``return` `(B / M) -`` ``(A / M);``}` ` ``// Driver Code`` ``// A and B define the range,`` ``// M is the divident`` ``let A = 30;`` ``let B = 70;`` ``let M = 10;` ` ``// Printing the result`` ``document.write(countDivisibles(A, B, M));` `// This code is contributed by gfgking`\nOutput\n`3`\n\nThis article is contributed by Rohit Thapliyal. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to [email protected]. See your article appearing on the GeeksforGeeks main page and help other Geeks." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7308989,"math_prob":0.9813188,"size":8974,"snap":"2021-43-2021-49","text_gpt3_token_len":2888,"char_repetition_ratio":0.19141583,"word_repetition_ratio":0.59405434,"special_character_ratio":0.35346556,"punctuation_ratio":0.13658275,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9998222,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-11-28T15:54:41Z\",\"WARC-Record-ID\":\"<urn:uuid:1f3c6e27-0fa5-4b10-a42e-5e295f553c1a>\",\"Content-Length\":\"191009\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:00f51127-0aef-49d4-9ce0-91a9ceed32ed>\",\"WARC-Concurrent-To\":\"<urn:uuid:2eb5d9e8-9e46-49c5-aa60-1d97a222afa5>\",\"WARC-IP-Address\":\"23.3.13.33\",\"WARC-Target-URI\":\"https://www.geeksforgeeks.org/count-numbers-divisible-m-given-range/?ref=rp\",\"WARC-Payload-Digest\":\"sha1:FH6OUCDLUVDXSFBMH74OOEU5WLWZXG5C\",\"WARC-Block-Digest\":\"sha1:QYQGIFK7BP47CQ227XS6QWSLH3FTMFUV\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964358560.75_warc_CC-MAIN-20211128134516-20211128164516-00377.warc.gz\"}"}
https://www.colorhexa.com/01488e	[ "# #01488e Color Information\n\nIn a RGB color space, hex #01488e is composed of 0.4% red, 28.2% green and 55.7% blue. Whereas in a CMYK color space, it is composed of 99.3% cyan, 49.3% magenta, 0% yellow and 44.3% black. It has a hue angle of 209.8 degrees, a saturation of 98.6% and a lightness of 28%. #01488e color hex could be obtained by blending #0290ff with #00001d. Closest websafe color is: #003399.\n\n• R 0\n• G 28\n• B 56\nRGB color chart\n• C 99\n• M 49\n• Y 0\n• K 44\nCMYK color chart\n\n#01488e color description : Dark blue.\n\n# #01488e Color Conversion\n\nThe hexadecimal color #01488e has RGB values of R:1, G:72, B:142 and CMYK values of C:0.99, M:0.49, Y:0, K:0.44. Its decimal value is 84110.\n\nHex triplet RGB Decimal 01488e `#01488e` 1, 72, 142 `rgb(1,72,142)` 0.4, 28.2, 55.7 `rgb(0.4%,28.2%,55.7%)` 99, 49, 0, 44 209.8°, 98.6, 28 `hsl(209.8,98.6%,28%)` 209.8°, 99.3, 55.7 003399 `#003399`\nCIE-LAB 30.863, 9.675, -44.044 7.211, 6.594, 26.482 0.179, 0.164, 6.594 30.863, 45.094, 282.389 30.863, -17.009, -59.597 25.678, 5.193, -43.173 00000001, 01001000, 10001110\n\n# Color Schemes with #01488e\n\n• #01488e\n``#01488e` `rgb(1,72,142)``\n• #8e4701\n``#8e4701` `rgb(142,71,1)``\nComplementary Color\n• #018e8e\n``#018e8e` `rgb(1,142,142)``\n• #01488e\n``#01488e` `rgb(1,72,142)``\n• #01018e\n``#01018e` `rgb(1,1,142)``\nAnalogous Color\n• #8e8e01\n``#8e8e01` `rgb(142,142,1)``\n• #01488e\n``#01488e` `rgb(1,72,142)``\n• #8e0101\n``#8e0101` `rgb(142,1,1)``\nSplit Complementary Color\n• #488e01\n``#488e01` `rgb(72,142,1)``\n• #01488e\n``#01488e` `rgb(1,72,142)``\n• #8e0148\n``#8e0148` `rgb(142,1,72)``\n• #018e47\n``#018e47` `rgb(1,142,71)``\n• #01488e\n``#01488e` `rgb(1,72,142)``\n• #8e0148\n``#8e0148` `rgb(142,1,72)``\n• #8e4701\n``#8e4701` `rgb(142,71,1)``\n• #002142\n``#002142` `rgb(0,33,66)``\n• #012e5b\n``#012e5b` `rgb(1,46,91)``\n• #013b75\n``#013b75` `rgb(1,59,117)``\n• #01488e\n``#01488e` `rgb(1,72,142)``\n• #0155a7\n``#0155a7` `rgb(1,85,167)``\n• #0162c1\n``#0162c1` `rgb(1,98,193)``\n• #026fda\n``#026fda` `rgb(2,111,218)``\nMonochromatic Color\n\n# Alternatives to #01488e\n\nBelow, you can see some colors close to #01488e. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #016b8e\n``#016b8e` `rgb(1,107,142)``\n• #01608e\n``#01608e` `rgb(1,96,142)``\n• #01548e\n``#01548e` `rgb(1,84,142)``\n• #01488e\n``#01488e` `rgb(1,72,142)``\n• #013c8e\n``#013c8e` `rgb(1,60,142)``\n• #01318e\n``#01318e` `rgb(1,49,142)``\n• #01258e\n``#01258e` `rgb(1,37,142)``\nSimilar Colors\n\n# #01488e Preview\n\nThis text has a font color of #01488e.\n\n``<span style=\"color:#01488e;\">Text here</span>``\n#01488e background color\n\nThis paragraph has a background color of #01488e.\n\n``<p style=\"background-color:#01488e;\">Content here</p>``\n#01488e border color\n\nThis element has a border color of #01488e.\n\n``<div style=\"border:1px solid #01488e;\">Content here</div>``\nCSS codes\n``.text {color:#01488e;}``\n``.background {background-color:#01488e;}``\n``.border {border:1px solid #01488e;}``\n\n# Shades and Tints of #01488e\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, #000306 is the darkest color, while #f1f8ff is the lightest one.\n\n• #000306\n``#000306` `rgb(0,3,6)``\n• #000d19\n``#000d19` `rgb(0,13,25)``\n• #00172d\n``#00172d` `rgb(0,23,45)``\n• #002040\n``#002040` `rgb(0,32,64)``\n• #012a54\n``#012a54` `rgb(1,42,84)``\n• #013467\n``#013467` `rgb(1,52,103)``\n• #013e7b\n``#013e7b` `rgb(1,62,123)``\n• #01488e\n``#01488e` `rgb(1,72,142)``\n• #0152a1\n``#0152a1` `rgb(1,82,161)``\n• #015cb5\n``#015cb5` `rgb(1,92,181)``\n• #0166c8\n``#0166c8` `rgb(1,102,200)``\n• #0270dc\n``#0270dc` `rgb(2,112,220)``\n• #0279ef\n``#0279ef` `rgb(2,121,239)``\n• #0783fd\n``#0783fd` `rgb(7,131,253)``\n• #1b8dfd\n``#1b8dfd` `rgb(27,141,253)``\n• #2e97fe\n``#2e97fe` `rgb(46,151,254)``\n• #42a0fe\n``#42a0fe` `rgb(66,160,254)``\n• #55aafe\n``#55aafe` `rgb(85,170,254)``\n• #69b4fe\n``#69b4fe` `rgb(105,180,254)``\n• #7cbefe\n``#7cbefe` `rgb(124,190,254)``\n• #90c7fe\n``#90c7fe` `rgb(144,199,254)``\n• #a3d1fe\n``#a3d1fe` `rgb(163,209,254)``\n• #b7dbfe\n``#b7dbfe` `rgb(183,219,254)``\n• #cae5ff\n``#cae5ff` `rgb(202,229,255)``\n• #deeeff\n``#deeeff` `rgb(222,238,255)``\n• #f1f8ff\n``#f1f8ff` `rgb(241,248,255)``\nTint Color Variation\n\n# Tones of #01488e\n\nA tone is produced by adding gray to any pure hue. In this case, #43484c is the less saturated color, while #01488e is the most saturated one.\n\n• #43484c\n``#43484c` `rgb(67,72,76)``\n• #3e4852\n``#3e4852` `rgb(62,72,82)``\n• #384857\n``#384857` `rgb(56,72,87)``\n• #33485d\n``#33485d` `rgb(51,72,93)``\n• #2d4862\n``#2d4862` `rgb(45,72,98)``\n• #284868\n``#284868` `rgb(40,72,104)``\n• #22486d\n``#22486d` `rgb(34,72,109)``\n• #1d4873\n``#1d4873` `rgb(29,72,115)``\n• #174878\n``#174878` `rgb(23,72,120)``\n• #12487e\n``#12487e` `rgb(18,72,126)``\n• #0c4883\n``#0c4883` `rgb(12,72,131)``\n• #074889\n``#074889` `rgb(7,72,137)``\n• #01488e\n``#01488e` `rgb(1,72,142)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #01488e 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":0.53030455,"math_prob":0.51639986,"size":3656,"snap":"2021-21-2021-25","text_gpt3_token_len":1605,"char_repetition_ratio":0.13855422,"word_repetition_ratio":0.011111111,"special_character_ratio":0.56810725,"punctuation_ratio":0.23634337,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9941234,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-05-06T04:41:10Z\",\"WARC-Record-ID\":\"<urn:uuid:a6f696ce-f0f8-4a35-80ee-358b6db807c0>\",\"Content-Length\":\"36203\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0ad7bd59-1dbd-4c31-9466-00e6ec221562>\",\"WARC-Concurrent-To\":\"<urn:uuid:a4cda716-a22c-491d-9ff1-6b2162d5376d>\",\"WARC-IP-Address\":\"178.32.117.56\",\"WARC-Target-URI\":\"https://www.colorhexa.com/01488e\",\"WARC-Payload-Digest\":\"sha1:KL26WOMRAZ6RNQEDHIITTUDR2KKEJBAZ\",\"WARC-Block-Digest\":\"sha1:LEU6IEVMNYZQYVLW5YRM7ISL5QH7466B\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-21/CC-MAIN-2021-21_segments_1620243988725.79_warc_CC-MAIN-20210506023918-20210506053918-00076.warc.gz\"}"}
https://arxiv.org/abs/1809.00781?context=math	[ "math\n\n# Title:Matrix Infinitely Divisible Series: Tail Inequalities and Applications in Optimization\n\nAbstract: In this paper, we study tail inequalities of the largest eigenvalue of a matrix infinitely divisible (i.d.) series, which is a finite sum of fixed matrices weighted by i.d. random variables. We obtain several types of tail inequalities, including Bennett-type and Bernstein-type inequalities. This allows us to further bound the expectation of the spectral norm of a matrix i.d. series. Moreover, by developing a new lower-bound function for $Q(s)=(s+1)\\log(s+1)-s$ that appears in the Bennett-type inequality, we derive a tighter tail inequality of the largest eigenvalue of the matrix i.d. series than the Bernstein-type inequality when the matrix dimension is high. The resulting lower-bound function is of independent interest and can improve any Bennett-type concentration inequality that involves the function $Q(s)$. The class of i.d. probability distributions is large and includes Gaussian and Poisson distributions, among many others. Therefore, our results encompass the existing work \\cite{tropp2012user} on matrix Gaussian series as a special case. Lastly, we show that the tail inequalities of a matrix i.d. series have applications in several optimization problems including the chance constrained optimization problem and the quadratic optimization problem with orthogonality constraints.\n Comments: Comments Welcome! Subjects: Information Theory (cs.IT); Optimization and Control (math.OC) Cite as: arXiv:1809.00781 [cs.IT] (or arXiv:1809.00781v1 [cs.IT] for this version)\n\n## Submission history\n\nFrom: Min-Hsiu Hsieh [view email]\n[v1] Tue, 4 Sep 2018 03:11:44 UTC (211 KB)" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8815097,"math_prob":0.9431689,"size":1699,"snap":"2019-51-2020-05","text_gpt3_token_len":358,"char_repetition_ratio":0.12861358,"word_repetition_ratio":0.008474576,"special_character_ratio":0.20482637,"punctuation_ratio":0.12828948,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96888804,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-12-12T21:41:52Z\",\"WARC-Record-ID\":\"<urn:uuid:80e61d8d-635d-4ce5-9286-ad1a842a730b>\",\"Content-Length\":\"21456\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:32c1e355-564b-48b2-bc24-e653cfe8b65e>\",\"WARC-Concurrent-To\":\"<urn:uuid:ed15f483-6a5d-43ec-aca6-47b8873898ba>\",\"WARC-IP-Address\":\"128.84.21.199\",\"WARC-Target-URI\":\"https://arxiv.org/abs/1809.00781?context=math\",\"WARC-Payload-Digest\":\"sha1:O5T5WMM36SQ5NVKVT3324GLMQIEJ3JFB\",\"WARC-Block-Digest\":\"sha1:CAQXP744HMADF6NXGHXG5SPCE7I4NAZM\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-51/CC-MAIN-2019-51_segments_1575540547165.98_warc_CC-MAIN-20191212205036-20191212233036-00193.warc.gz\"}"}
https://pro.arcgis.com/en/pro-app/latest/arcpy/spatial-analyst/savi.htm	[ "# SAVI\n\n## Summary\n\nCalculates the Soil Adjusted Vegetation Index (SAVI) from a multiband raster object and returns a raster object with the index values.\n\n## Discussion\n\nThe Soil-Adjusted Vegetation Index (SAVI) method is a vegetation index that attempts to minimize soil brightness influences using a soil-brightness correction factor. This is often used in arid regions where vegetative cover is low, and it outputs values between -1.0 and 1.0.\n\n``SAVI = ((NIR - Red)/(NIR + Red + L)) * (1 + L)``\n\n• L—The amount of green vegetation cover. For example, 0.5.\n\nFor information about other multiband raster indexes, see the Band Arithmetic raster function.\n\nThe referenced raster dataset for the raster object is temporary. To make it permanent, you can call the raster object's save method.\n\n## Syntax\n\n`SAVI (raster, {nir_band_id}, {red_band_id}, {l})`\n Parameter Explanation Data Type raster The input raster. Raster nir_band_id The band ID of the near-infrared band. The ID index uses one-based indexing.(The default value is 7) Integer red_band_id The band ID of the red-edge band. The band ID index uses one-based indexing.(The default value is 6) Integer l The amount of green vegetative cover.(The default value is 0.33) Double\nReturn Value\n Data Type Explanation Raster The output raster with SAVI index values.\n\n## Code sample\n\nSAVI example\n\nCalculates the Soil Adjusted Vegetation Index for a Landsat 8 image.\n\n``````import arcpy\n\nSAVI_raster = arcpy.sa.SAVI(\"Landsat8.tif\",5,4,0.5)``````\n\n#### Related topics\n\n##### Related topics\n1. An overview of the Spatial Analyst functions" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6862624,"math_prob":0.8885446,"size":1432,"snap":"2022-27-2022-33","text_gpt3_token_len":373,"char_repetition_ratio":0.13585435,"word_repetition_ratio":0.07373272,"special_character_ratio":0.23673184,"punctuation_ratio":0.12734082,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97165877,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-06-26T11:24:24Z\",\"WARC-Record-ID\":\"<urn:uuid:6aff2ac2-2e13-45fc-903e-9134964f9042>\",\"Content-Length\":\"16499\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:d991ed3e-798b-48fd-9064-23b1e6601fee>\",\"WARC-Concurrent-To\":\"<urn:uuid:accf606b-6b89-4a93-bd8b-5771bebf9f1b>\",\"WARC-IP-Address\":\"198.102.60.60\",\"WARC-Target-URI\":\"https://pro.arcgis.com/en/pro-app/latest/arcpy/spatial-analyst/savi.htm\",\"WARC-Payload-Digest\":\"sha1:ATXXHF5IJ5RSFZCRSRJFMQSGSR6XMRBZ\",\"WARC-Block-Digest\":\"sha1:W4DJEIWMV4SHO6BGYBUIGZEEMFCW4RG4\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656103205617.12_warc_CC-MAIN-20220626101442-20220626131442-00254.warc.gz\"}"}
https://gitlab.mpi-sws.org/iris/actris/-/commit/4193e2e16d8f086d58aea8b1e73c1704e3371663	[ "### More things:\n\n```- Revise telescope setup.\n- Add append function on protocols.\n- Add a crappy normalizer for protocols that handles append and dual.\n- Prove non-expansiveness of tons of operators.```\nparent 83692823\nThis diff is collapsed.\n ... ... @@ -41,6 +41,25 @@ Definition proto_message {V} `{!Cofe PROPn, !Cofe PROP} (a : action) (pc : V → (laterO (proto V PROPn PROP) -n> PROPn) -n> PROP) : proto V PROPn PROP := proto_fold (Some (a, pc)). Instance proto_message_ne {V} `{!Cofe PROPn, !Cofe PROP} a n : Proper (pointwise_relation V (dist n) ==> dist n) (proto_message (PROPn:=PROPn) (PROP:=PROP) a). Proof. intros c1 c2 Hc. rewrite /proto_message. f_equiv. by repeat constructor. Qed. Instance proto_message_proper {V} `{!Cofe PROPn, !Cofe PROP} a : Proper (pointwise_relation V (≡) ==> (≡)) (proto_message (PROPn:=PROPn) (PROP:=PROP) a). Proof. intros c1 c2 Hc. rewrite /proto_message. f_equiv. by repeat constructor. Qed. Lemma proto_case {V} `{!Cofe PROPn, !Cofe PROP} (p : proto V PROPn PROP) : p ≡ proto_end ∨ ∃ a pc, p ≡ proto_message a pc. Proof. destruct (proto_unfold p) as [[a pc]\|] eqn:E; simpl in ; last first. - left. by rewrite -(proto_fold_unfold p) E. - right. exists a, pc. by rewrite -(proto_fold_unfold p) E. Qed. Instance proto_inhabited {V} `{!Cofe PROPn, !Cofe PROP} : Inhabited (proto V PROPn PROP) := populate proto_end. Lemma proto_message_equivI {SPROP : sbi} {V} `{!Cofe PROPn, !Cofe PROP} a1 a2 pc1 pc2 : proto_message (V:=V) (PROPn:=PROPn) (PROP:=PROP) a1 pc1 ≡ proto_message a2 pc2 ⊣⊢@{SPROP} ⌜ a1 = a2 ⌝ ∧ (∀ v, pc1 v ≡ pc2 v). ... ... @@ -54,31 +73,106 @@ Proof. by rewrite bi.discrete_fun_equivI bi.discrete_eq. Qed. Lemma proto_case {V} `{!Cofe PROPn, !Cofe PROP} (p : proto V PROPn PROP) : p ≡ proto_end ∨ ∃ a pc, p ≡ proto_message a pc. Proof. destruct (proto_unfold p) as [[a pc]\|] eqn:E; simpl in ; last first. - left. by rewrite -(proto_fold_unfold p) E. - right. exists a, pc. by rewrite -(proto_fold_unfold p) E. Qed. Instance proto_inhabited {V} `{!Cofe PROPn, !Cofe PROP} : Inhabited (proto V PROPn PROP) := populate proto_end. Instance proto_message_ne {V} `{!Cofe PROPn, !Cofe PROP} a n : Proper (pointwise_relation V (dist n) ==> dist n) (proto_message (PROPn:=PROPn) (PROP:=PROP) a). Proof. intros c1 c2 Hc. rewrite /proto_message. f_equiv. by repeat constructor. Qed. Instance proto_message_proper {V} `{!Cofe PROPn, !Cofe PROP} a : Proper (pointwise_relation V (≡) ==> (≡)) (proto_message (PROPn:=PROPn) (PROP:=PROP) a). Proof. intros c1 c2 Hc. rewrite /proto_message. f_equiv. by repeat constructor. Qed. Definition proto_cont_map `{!Cofe PROPn, !Cofe PROPn', !Cofe PROP, Cofe PROP', !Cofe A, !Cofe B} `{!Cofe PROPn, !Cofe PROPn', !Cofe PROP, !Cofe PROP', !Cofe A, !Cofe B} (gn : PROPn' -n> PROPn) (g : PROP -n> PROP') (h : A -n> B) : ((laterO A -n> PROPn) -n> PROP) -n> (laterO B -n> PROPn') -n> PROP' := ofe_morO_map (ofe_morO_map (laterO_map h) gn) g. (** Append ) Program Definition proto_app_flipped_aux {V} `{!Cofe PROPn, !Cofe PROP} (p2 : proto V PROPn PROP) (rec : proto V PROPn PROP -n> proto V PROPn PROP) : proto V PROPn PROP -n> proto V PROPn PROP := λne p1, match proto_unfold p1 return _ with \| None => p2 \| Some (a, c) => proto_message a (proto_cont_map cid cid rec ∘ c) end. Next Obligation. intros V PROPn ? PROP ? rec n p2 p1 p1' Hp. apply (ofe_mor_ne _ _ proto_unfold) in Hp. destruct Hp as [[??][??] [-> ?]\|]; simplify_eq/=; last done. f_equiv=> v /=. by f_equiv. Qed. Instance proto_app_flipped_aux_contractive {V} `{!Cofe PROPn, !Cofe PROP} (p2 : proto V PROPn PROP) : Contractive (proto_app_flipped_aux p2). Proof. intros n rec1 rec2 Hrec p1. simpl. destruct (proto_unfold p1) as [[a c]\|]; last done. f_equiv=> v /=. do 2 f_equiv. intros=> p'. apply Next_contractive. destruct n as [\|n]=> //=. Qed. Definition proto_app_flipped {V} `{!Cofe PROPn, !Cofe PROP} (p2 : proto V PROPn PROP) : proto V PROPn PROP -n> proto V PROPn PROP := fixpoint (proto_app_flipped_aux p2). Definition proto_app {V} `{!Cofe PROPn, !Cofe PROP} (p1 p2 : proto V PROPn PROP) : proto V PROPn PROP := proto_app_flipped p2 p1. Instance: Params (@proto_app) 5. Lemma proto_app_flipped_unfold {V} `{!Cofe PROPn, !Cofe PROP} (p1 p2 : proto V PROPn PROP): proto_app_flipped p2 p1 ≡ proto_app_flipped_aux p2 (proto_app_flipped p2) p1. Proof. apply (fixpoint_unfold (proto_app_flipped_aux p2)). Qed. Lemma proto_app_unfold {V} `{!Cofe PROPn, !Cofe PROP} (p1 p2 : proto V PROPn PROP): proto_app (V:=V) p1 p2 ≡ proto_app_flipped_aux p2 (proto_app_flipped p2) p1. Proof. apply (fixpoint_unfold (proto_app_flipped_aux p2)). Qed. Lemma proto_app_end_l {V} `{!Cofe PROPn, !Cofe PROP} (p2 : proto V PROPn PROP) : proto_app proto_end p2 ≡ p2. Proof. rewrite proto_app_unfold /proto_end /=. pose proof (proto_unfold_fold (V:=V) (PROPn:=PROPn) (PROP:=PROP) None) as Hfold. by destruct (proto_unfold (proto_fold None)) as [[??]\|] eqn:E; rewrite E; inversion Hfold. Qed. Lemma proto_app_message {V} `{!Cofe PROPn, !Cofe PROP} a c (p2 : proto V PROPn PROP) : proto_app (proto_message a c) p2 ≡ proto_message a (proto_cont_map cid cid (proto_app_flipped p2) ∘ c). Proof. rewrite proto_app_unfold /proto_message /=. pose proof (proto_unfold_fold (V:=V) (PROPn:=PROPn) (PROP:=PROP) (Some (a, c))) as Hfold. destruct (proto_unfold (proto_fold (Some (a, c)))) as [[??]\|] eqn:E; inversion Hfold as [?? [Ha Hc]\|]; simplify_eq/=. rewrite /proto_message. do 3 f_equiv. intros v=> /=. apply equiv_dist=> n. f_equiv. by apply equiv_dist. Qed. Instance proto_app_ne {V} `{!Cofe PROPn, !Cofe PROP} : NonExpansive2 (proto_app (V:=V) (PROPn:=PROPn) (PROP:=PROP)). Proof. intros n p1 p1' Hp1 p2 p2' Hp2. rewrite /proto_app -Hp1 {p1' Hp1}. revert p1. induction (lt_wf n) as [n _ IH]=> p1 /=. rewrite !proto_app_flipped_unfold /proto_app_flipped_aux /=. destruct (proto_unfold p1) as [[a c]\|]; last done. f_equiv=> v f /=. do 2 f_equiv. intros p. apply Next_contractive. destruct n as [\|n]=> //=. apply IH; first lia; auto using dist_S. Qed. Instance proto_app_proper {V} `{!Cofe PROPn, !Cofe PROP} : Proper ((≡) ==> (≡) ==> (≡)) (proto_app (V:=V) (PROPn:=PROPn) (PROP:=PROP)). Proof. apply (ne_proper_2 _). Qed. Lemma proto_app_end_r {V} `{!Cofe PROPn, !Cofe PROP} (p1 : proto V PROPn PROP) : proto_app p1 proto_end ≡ p1. Proof. apply equiv_dist=> n. revert p1. induction (lt_wf n) as [n _ IH]=> p1 /=. destruct (proto_case p1) as [->\|(a & c & ->)]. - by rewrite !proto_app_end_l. - rewrite !proto_app_message /=. f_equiv=> v c' /=. f_equiv=> p' /=. f_equiv. apply Next_contractive. destruct n as [\|n]=> //=. apply IH; first lia; auto using dist_S. Qed. Lemma proto_app_assoc {V} `{!Cofe PROPn, !Cofe PROP} (p1 p2 p3 : proto V PROPn PROP) : proto_app p1 (proto_app p2 p3) ≡ proto_app (proto_app p1 p2) p3. Proof. apply equiv_dist=> n. revert p1. induction (lt_wf n) as [n _ IH]=> p1 /=. destruct (proto_case p1) as [->\|(a & c & ->)]. - by rewrite !proto_app_end_l. - rewrite !proto_app_message /=. f_equiv=> v c' /=. f_equiv=> p' /=. f_equiv. apply Next_contractive. destruct n as [\|n]=> //=. apply IH; first lia; auto using dist_S. Qed. (* Functor *) Program Definition proto_map_aux {V} `{!Cofe PROPn, !Cofe PROPn', !Cofe PROP, !Cofe PROP'} (f : action → action) (gn : PROPn' -n> PROPn) (g : PROP -n> PROP') ... ... @@ -94,7 +188,6 @@ Next Obligation. destruct Hp as [[??][??] [-> ?]\|]; simplify_eq/=; last done. f_equiv=> v /=. by f_equiv. Qed. Instance proto_map_aux_contractive {V} `{!Cofe PROPn, !Cofe PROPn', !Cofe PROP, !Cofe PROP'} (f : action → action) (gn : PROPn' -n> PROPn) (g : PROP -n> PROP') : ... ... @@ -105,7 +198,6 @@ Proof. f_equiv=> v /=. do 2 f_equiv. intros=> p'. apply Next_contractive. destruct n as [\|n]=> //=. Qed. Definition proto_map {V} `{!Cofe PROPn, !Cofe PROPn', !Cofe PROP, !Cofe PROP'} (f : action → action) (gn : PROPn' -n> PROPn) (g : PROP -n> PROP') : ... ... @@ -116,7 +208,7 @@ Lemma proto_map_unfold {V} `{!Cofe PROPn, !Cofe PROPn', !Cofe PROP, !Cofe PROP'} (f : action → action) (gn : PROPn' -n> PROPn) (g : PROP -n> PROP') p : proto_map (V:=V) f gn g p ≡ proto_map_aux f gn g (proto_map f gn g) p. Proof. apply (fixpoint_unfold (proto_map_aux f gn g)). Qed. Lemma proto_map_end {V} `{!Cofe PROPn, !Cofe PROPn', !Cofe PROP, Cofe PROP'} Lemma proto_map_end {V} `{!Cofe PROPn, !Cofe PROPn', !Cofe PROP, !Cofe PROP'} (f : action → action) (gn : PROPn' -n> PROPn) (g : PROP -n> PROP') : proto_map (V:=V) f gn g proto_end ≡ proto_end. Proof. ... ... @@ -182,6 +274,20 @@ Proof. apply IH; first lia; auto using dist_S. Qed. Lemma proto_map_app {V} `{!Cofe PROPn, !Cofe PROPn', !Cofe PROP, !Cofe PROP'} (f : action → action) (gn : PROPn' -n> PROPn) (g : PROP -n> PROP') p1 p2 : proto_map (V:=V) f gn g (proto_app p1 p2) ≡ proto_app (proto_map (V:=V) f gn g p1) (proto_map (V:=V) f gn g p2). Proof. apply equiv_dist=> n. revert p1 p2. induction (lt_wf n) as [n _ IH]=> p1 p2 /=. destruct (proto_case p1) as [->\|(a & c & ->)]. - by rewrite proto_map_end !proto_app_end_l. - rewrite proto_map_message !proto_app_message proto_map_message /=. f_equiv=> v c' /=. do 2 f_equiv. move=> p' /=. do 2 f_equiv. apply Next_contractive. destruct n as [\|n]=> //=. apply IH; first lia; auto using dist_S. Qed. Program Definition protoOF (V : Type) (Fn F : oFunctor) `{!∀ A B `{!Cofe A, !Cofe B}, Cofe (oFunctor_car Fn A B)} `{!∀ A B `{!Cofe A, !Cofe B}, Cofe (oFunctor_car F A B)} : oFunctor := {\| ... ...\n From stdpp Require Import sorting. From osiris.channel Require Import proto_channel. From iris.heap_lang Require Import proofmode notation. From osiris.utils Require Import list. From osiris.channel Require Import proto_channel. Definition lmerge : val := rec: \"go\" \"cmp\" \"hys\" \"hzs\" := ... ... @@ -28,14 +28,14 @@ Definition list_sort_service : val := let: \"ys_zs\" := lsplit !\"xs\" in let: \"ys\" := ref (Fst \"ys_zs\") in let: \"zs\" := ref (Snd \"ys_zs\") in let: \"cy\" := new_chan #() in Fork(\"go\" (Fst \"cy\"));; let: \"cz\" := new_chan #() in Fork(\"go\" (Fst \"cz\"));; send (Snd \"cy\") \"cmp\";; send (Snd \"cy\") \"ys\";; send (Snd \"cz\") \"cmp\";; send (Snd \"cz\") \"zs\";; recv (Snd \"cy\");; recv (Snd \"cz\");; let: \"cy\" := new_chan #() in Fork(\"go\" (Snd \"cy\"));; let: \"cz\" := new_chan #() in Fork(\"go\" (Snd \"cz\"));; send (Fst \"cy\") \"cmp\";; send (Fst \"cy\") \"ys\";; send (Fst \"cz\") \"cmp\";; send (Fst \"cz\") \"zs\";; recv (Fst \"cy\");; recv (Fst \"cz\");; \"xs\" <- lmerge \"cmp\" !\"ys\" !\"zs\";; send \"c\" #(). ... ... @@ -50,15 +50,15 @@ Section list_sort. {{{ RET #(bool_decide (R x x')); I x v ∗ I x' v' }}})%I. Definition sort_protocol : iProto Σ := ( ∃ A (I : A → val → iProp Σ) (R : A → A → Prop) ( A (I : A → val → iProp Σ) (R : A → A → Prop) `{!RelDecision R, !TotalOrder R} (cmp : val), MSG cmp {{ cmp_spec I R cmp }}; ∃ (xs : list A) (l : loc) (vs : list val), (xs : list A) (l : loc) (vs : list val), MSG #l {{ l ↦ val_encode vs ∗ [∗ list] x;v ∈ xs;vs, I x v }}; ∃ (xs' : list A) (vs' : list val), (xs' : list A) (vs' : list val), MSG #() {{ ⌜ Sorted R xs' ⌝ ∗ ⌜ xs' ≡ₚ xs ⌝ ∗ l ↦ val_encode vs' ∗ [∗ list] x;v ∈ xs';vs', I x v }}; iProto_end)%proto. END)%proto. Lemma lmerge_spec {A} (I : A → val → iProp Σ) (R : A → A → Prop) `{!RelDecision R, !TotalOrder R} (cmp : val) xs1 xs2 vs1 vs2 : ... ... @@ -93,12 +93,12 @@ Section list_sort. iApply \"HΨ\". iFrame. Qed. Lemma list_sort_service_spec c : {{{ c ↣ sort_protocol @ N }}} Lemma list_sort_service_spec p c : {{{ c ↣ iProto_dual sort_protocol <++> p @ N }}} list_sort_service c {{{ RET #(); c ↣ iProto_end @ N }}}. {{{ RET #(); c ↣ p @ N }}}. Proof. iIntros (Ψ) \"Hc HΨ\". iLöb as \"IH\" forall (c Ψ). iIntros (Ψ) \"Hc HΨ\". iLöb as \"IH\" forall (p c Ψ). wp_lam. wp_apply (recv_proto_spec with \"Hc\"); simpl. iIntros (A I R ?? cmp) \"/= Hc #Hcmp\". ... ... @@ -116,23 +116,25 @@ Section list_sort. iDestruct (big_sepL2_app_inv_r with \"HI\") as (xs1 xs2 ->) \"[HI1 HI2]\". wp_apply (new_chan_proto_spec N sort_protocol)=> //. iIntros (cy1 cy2) \"[Hcy1 Hcy2]\". wp_apply (wp_fork with \"[Hcy1]\"). { iNext. wp_apply (\"IH\" with \"Hcy1\"); auto. } wp_apply (wp_fork with \"[Hcy2]\"). { iNext. rewrite -{2}(right_id END%proto _ (iProto_dual _)). wp_apply (\"IH\" with \"Hcy2\"); auto. } wp_apply (new_chan_proto_spec N sort_protocol)=> //. iIntros (cz1 cz2) \"[Hcz1 Hcz2]\". wp_apply (wp_fork with \"[Hcz1]\"). { iNext. wp_apply (\"IH\" with \"Hcz1\"); auto. } wp_apply (send_proto_spec with \"Hcy2\"); simpl. iExists _, I, R, _, _, cmp. iSplit; first done. iIntros \"{\\$Hcmp} !> Hcy2\". wp_apply (send_proto_spec with \"Hcy2\"); simpl. iExists xs1, l1, vs1. iSplit; first done. iIntros \"{\\$Hl1 \\$HI1} !> Hcy2\". wp_apply (send_proto_spec with \"Hcz2\"); simpl. iExists _, I, R, _, _, cmp. iSplit; first done. iIntros \"{\\$Hcmp} !> Hcz2\". wp_apply (send_proto_spec with \"Hcz2\"); simpl. iExists xs2, l2, vs2. iSplit; first done. iIntros \"{\\$Hl2 \\$HI2} !> Hcz2\". wp_apply (recv_proto_spec with \"Hcy2\"); simpl. wp_apply (wp_fork with \"[Hcz2]\"). { iNext. rewrite -{2}(right_id END%proto _ (iProto_dual _)). wp_apply (\"IH\" with \"Hcz2\"); auto. } wp_apply (send_proto_spec with \"Hcy1\"); simpl. iExists _, I, R, _, _, cmp. iSplit; first done. iIntros \"{\\$Hcmp} !> Hcy1\". wp_apply (send_proto_spec with \"Hcy1\"); simpl. iExists xs1, l1, vs1. iSplit; first done. iIntros \"{\\$Hl1 \\$HI1} !> Hcy1\". wp_apply (send_proto_spec with \"Hcz1\"); simpl. iExists _, I, R, _, _, cmp. iSplit; first done. iIntros \"{\\$Hcmp} !> Hcz1\". wp_apply (send_proto_spec with \"Hcz1\"); simpl. iExists xs2, l2, vs2. iSplit; first done. iIntros \"{\\$Hl2 \\$HI2} !> Hcz1\". wp_apply (recv_proto_spec with \"Hcy1\"); simpl. iIntros (ys1 ws1) \"_\". iDestruct 1 as (??) \"[Hl1 HI1]\". wp_apply (recv_proto_spec with \"Hcz2\"); simpl. wp_apply (recv_proto_spec with \"Hcz1\"); simpl. iIntros (ys2 ws2) \"_\". iDestruct 1 as (??) \"[Hl2 HI2]\". do 2 wp_load. wp_apply (lmerge_spec with \"Hcmp [\\$HI1 \\$HI2]\"); iIntros (ws) \"HI\". ... ...\nMarkdown is supported\n0% or\nYou are about to add 0 people to the discussion. Proceed with caution.\nFinish editing this message first!" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.79491025,"math_prob":0.87074095,"size":278,"snap":"2020-45-2020-50","text_gpt3_token_len":62,"char_repetition_ratio":0.11678832,"word_repetition_ratio":0.0,"special_character_ratio":0.2266187,"punctuation_ratio":0.125,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95343655,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-31T04:48:50Z\",\"WARC-Record-ID\":\"<urn:uuid:8baef889-9059-40d9-8ace-5329aa5c6e97>\",\"Content-Length\":\"628518\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:1f796028-4308-43f8-b4a1-40cd672c07dd>\",\"WARC-Concurrent-To\":\"<urn:uuid:c6934f93-559b-413e-bbc6-c1aa240cc980>\",\"WARC-IP-Address\":\"139.19.205.205\",\"WARC-Target-URI\":\"https://gitlab.mpi-sws.org/iris/actris/-/commit/4193e2e16d8f086d58aea8b1e73c1704e3371663\",\"WARC-Payload-Digest\":\"sha1:S34NU65MHCZA7GNPEG4OYSCCXV6KQUWK\",\"WARC-Block-Digest\":\"sha1:3U3X2QJHCQQXM4RSO3UVCWSMR5CJWEMM\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-45/CC-MAIN-2020-45_segments_1603107912807.78_warc_CC-MAIN-20201031032847-20201031062847-00072.warc.gz\"}"}
http://pantheonresources.com/about-pantheon/operations/glossary/54-monte-carlo	[ "# Monte Carlo\n\nTerm Definition\nMonte Carlo\n\nA methodology for estimating a given quantity based on the statistical distribution of input values on which the quantity depends. Typically, the output quantity is calculated several thousand times (each calculation is called a trial), for each trial using input parameter values extracted randomly according to their statistical distributions. The result is a statistical distribution of output values." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.83840024,"math_prob":0.9931508,"size":449,"snap":"2021-04-2021-17","text_gpt3_token_len":77,"char_repetition_ratio":0.121348314,"word_repetition_ratio":0.0,"special_character_ratio":0.16703786,"punctuation_ratio":0.072463766,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9908157,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-01-23T00:53:28Z\",\"WARC-Record-ID\":\"<urn:uuid:7b7c718c-75d5-4b94-956f-0bd74d3b4005>\",\"Content-Length\":\"24524\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:3cbcc0f4-266d-4973-9c53-9bb7c41169f2>\",\"WARC-Concurrent-To\":\"<urn:uuid:9cc2fafb-0bd7-4d3e-8729-cc7271705b02>\",\"WARC-IP-Address\":\"80.76.216.120\",\"WARC-Target-URI\":\"http://pantheonresources.com/about-pantheon/operations/glossary/54-monte-carlo\",\"WARC-Payload-Digest\":\"sha1:3YN5VEZF66S5QQ6XSTCFIBI6X5NWUWRJ\",\"WARC-Block-Digest\":\"sha1:75LOKZ4ZRZXH34TB7PTAOMDFGNYGKCH4\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-04/CC-MAIN-2021-04_segments_1610703531702.36_warc_CC-MAIN-20210123001629-20210123031629-00456.warc.gz\"}"}
https://optimization-online.org/tag/polynomial-algorithm/	[ "## A polynomial-time descent method for separable convex optimization problems with linear constraints\n\nWe propose a polynomial algorithm for a separable convex optimization problem with linear constraints. We do not make any additional assumptions about the structure of the objective function except for polynomial computability. That is, the objective function can be non-differentiable. The running time of our algorithm is polynomial in the the size of the input … Read more\n\n## A polynomial algorithm for linear optimization which is strongly polynomial under certain conditions on optimal solutions\n\nThis paper proposes a polynomial algorithm for linear programming which is strongly polynomial for linear optimization problems $\\min\\{c^Tx : Ax = b, x\\ge {\\bf 0}\\}$ having optimal solutions where each non-zero component $x_j$ belongs to an interval of the form $[\\alpha_j, \\alpha_j\\cdot 2^{p(n)}],$ where $\\alpha_j$ is some positive value and $p(n)$ is a polynomial of … Read more\n\n## A polynomial relaxation-type algorithm for linear programming\n\nThe paper proposes a polynomial algorithm for solving systems of linear inequalities. The algorithm uses a polynomial relaxation-type procedure which either finds a solution for Ax = b, 0\n\n## A Polynomial Arc-Search Interior-Point Algorithm for Linear Programming\n\nIn this paper, ellipse is used to approximate the central path of the linear programming. An interior-point algorithm is devised to search the optimizers along the ellipse. The algorithm is proved to be polynomial with the complexity bound $O(n^{\\frac{1}{2}}\\log(1/\\epsilon))$. Numerical test is conducted for problems in Netlib. For most tested Netlib problems, the result shows … Read more\n\n## A Polynomial Arc-Search Interior-Point Algorithm for Convex Quadratic Programming\n\nArc-search is developed for linear programming in \\cite{yang09, yang10}. The algorithms search for optimizers along an ellipse that are approximations of the central path. In this paper, the arc-search method is applied to primal-dual path-following interior-point method for convex quadratic programming. A simple algorithm with iteration complexity $O(\\sqrt{n}\\log(1/\\epsilon))$ is devised. Several improvements on the simple … Read more" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.84876627,"math_prob":0.9905486,"size":475,"snap":"2022-40-2023-06","text_gpt3_token_len":82,"char_repetition_ratio":0.12526539,"word_repetition_ratio":0.0,"special_character_ratio":0.16,"punctuation_ratio":0.053333335,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9991764,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-01-27T01:40:10Z\",\"WARC-Record-ID\":\"<urn:uuid:bc877448-7b67-420f-b1c5-a939ad3f1fe3>\",\"Content-Length\":\"89319\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:c7e77477-2fc2-4685-a043-e6bb876cd3f5>\",\"WARC-Concurrent-To\":\"<urn:uuid:7cf848b2-3360-4827-9c34-0374d731f8bd>\",\"WARC-IP-Address\":\"128.104.153.102\",\"WARC-Target-URI\":\"https://optimization-online.org/tag/polynomial-algorithm/\",\"WARC-Payload-Digest\":\"sha1:HK3M6D5BKLL5WBKYIEPN2ZYC24JNSJQC\",\"WARC-Block-Digest\":\"sha1:E3LUI6YAOMLYFOYL3IHBP34S5XHI7EJH\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764494852.95_warc_CC-MAIN-20230127001911-20230127031911-00126.warc.gz\"}"}
https://www.linuxtopia.org/online_books/programming_books/python_programming/python_ch08s07.html	[ "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "On-line Guides", null, "All Guides", null, "eBook Store", null, "iOS / Android", null, "Linux for Beginners", null, "Office Productivity", null, "Linux Installation", null, "Linux Security", null, "Linux Utilities", null, "Linux Virtualization", null, "Linux Kernel", null, "System/Network Admin", null, "Programming", null, "Scripting Languages", null, "Development Tools", null, "Web Development", null, "GUI Toolkits/Desktop", null, "Databases", null, "Mail Systems", null, "openSolaris", null, "Eclipse Documentation", null, "Techotopia.com", null, "Virtuatopia.com", null, "How To Guides", null, "Virtualization", null, "General System Admin", null, "Linux Security", null, "Linux Filesystems", null, "Web Servers", null, "Graphics & Desktop", null, "PC Hardware", null, "Windows", null, "Problem Solutions", null, "Privacy Policy", null, "", null, "## A Digression\n\nFor those new to programming, here's a short digression, adapted from chapter 8 of Edsger Dijkstra's book, A Discipline of Programming [Dijkstra76].\n\nLet's say we need to set a variable, `m`, to the larger of two input values, `a` and `b`. We start with a state we could call “ `m` undefined”. Then we want to execute a statement after which we are in a state of “(`m`==`a` or `m`==`b`) and `m``a` and `m``b` ”.\n\nClearly, we need to choose correctly between two different assignment statements. We need to do either m=a or m=b . How do we make this choice? With a little logic, we can derive the condition by taking each of these statement's effects out of the desired end-state.\n\nFor the statement m=a to be the right statement to use, we show the effect of the statement by replacing `m` with the value `a`, and examining the end state: (`a`==`a` or `a`==`b`) and `a``a` and `a``b`. Removing the parts that are obviously true, we're left with `a``b`. Therefore, the assignment m=a is only useful when `a``b`.\n\nFor the statement m=b to be the right statement to establish the necessary condition, we do a similar replacement of `b` for `m` and examine the end state: (`b`==`a` or `b`==`b`) and `b``a` and `b``b`. Again, we remove the parts that are obviously true and we're left with `b``a`. Therefore, the assignment m=b is only useful when `b``a`.\n\nEach assignment statement can be “guarded” by an appropriate condition.\n\n```if a>=b: m=a\nelif b>=a: m=b\n```\n\nIs the correct statement to set `m` to the larger of `a` or `b`.\n\nNote that the hard part is establishing the post condition. Once we have that stated correctly, it's relatively easy to figure the basic kind of statement that might make some or all of the post condition true. Then we do a little algebra to fill in any guards or loop conditions to make sure that only the correct statement is executed.\n\nSuccessful Loop Design. There are several considerations when using the while statement. This list is taken from David Gries', The Science of Programming [Gries81].\n\n1. The body condition must be initialized properly.\n2. At the end of the suite, the body condition is just as true as it was after initialization. This is called the invariant, because it is always true during the loop.\n3. When this body condition is true and the while condition is false, the loop will have completed properly.\n4. When the while condition is true, there are more iterations left to do. If we wanted to, we could define a mathematical function based on the current state that computes how many iterations are left to do; this function must have a value greater than zero when the while condition is true.\n5. Each time through the loop we change the state of our variables so that we are getting closer to making the while condition false; we reduce the number of iterations left to do.\n\nWhile these conditions seem overly complex for something so simple as a loop, many programming problems arise from missing one of them.\n\nGries recommends putting comments around a loop showing the conditions before and after the loop. Since Python provides the assert statement; this formalizes these comments into actual tests to be sure the program is correct.\n\nDesigning a Loop. Let's put a particular loop under the microscope. This is a small example, but shows all of the steps to loop construction. We want to find the least power of 2 greater than or equal to some number greater than 1, call it `x`. This power of 2 will tell us how many bits are required to represent `x`, for example.\n\nWe can state this mathematically as looking for some number, `n`, such that 2 n -1 < `x` ≤ 2 n . This says that if `x` is a power of 2, for example 64, we'd find 26. If `x` is another number, for example 66, we'd find 26 < 66 ≤ 27, or 64 < 66 ≤ 128.\n\nWe can start to sketch our loop already.\n\n```assert x > 1\n\n... initialize ...\n\n... some loop ...\n\nassert 2(n-1) < x <= 2n\n```\n\nWe work out the initialization to make sure that the invariant condition of the loop is initially true. Since `x` must be greater than or equal to 1, we can set `n` to 1. 21-1=20=1 < `x`. This will set things up to satisfy rule 1 and 2.\n\n```assert x > 1\nn= 1\n\n... some loop ...\n\nassert 2(n-1) < x <= 2n\n```\n\nIn loops, there must be a condition on the body that is invariant, and a terminating condition that changes. The terminating condition is written in the while clause. In this case, it is invariant (always true) that 2(`n`-1) < `x`. That means that the other part of our final condition is the part that changes.\n\n```assert x > 1\nn= 1\nwhile not ( x <= 2n ):\nn= n + 1\nassert 2(n-1) < x\nassert 2(n-1) < x <= 2**n\n```\n\nThe next to last step is to show that when the while condition is true, there are more than zero trips through the loop possible. We know that `x` is finite and some power of 2 will satisfy this condition. There's some `n` such that `n`-1 < ```log2 ```( `x` ) < = `n` that limits the trips through the loop.\n\nThe final step is to show that each cycle through the loop reduces the trip count. We can argue that increasing `n` gets us closer to the upper bound of ```log2 ```( `x` ).\n\nWe should add this information on successful termination as comments in our loop.", null, "Published under the terms of the Open Publication License Design by Interspire" ]	[ null, "https://www.linuxtopia.org/images/toplogo.jpg", null, "https://www.linuxtopia.org/images/topmidspace.jpg", null, "https://www.linuxtopia.org/images/topbg.jpg", null, "https://www.linuxtopia.org/images/btn_home.jpg", null, "https://www.linuxtopia.org/images/navspacer.jpg", null, "https://www.linuxtopia.org/images/btn_library.jpg", null, "https://www.linuxtopia.org/images/navspacer.jpg", null, "https://www.linuxtopia.org/images/btn_techotopia.jpg", null, "https://www.linuxtopia.org/images/navspacer.jpg", null, "https://www.linuxtopia.org/images/btn_virtuatopia.jpg", null, "https://www.linuxtopia.org/images/navspacer.jpg", null, "https://www.linuxtopia.org/images/btn_store.jpg", null, "https://www.linuxtopia.org/images/navspacer.jpg", null, "https://twitter-badges.s3.amazonaws.com/follow_us-a.png", null, "https://www.linuxtopia.org/images/spacer.gif", null, "https://www.linuxtopia.org/images/spacer.gif", null, "https://www.linuxtopia.org/images/arrow.jpg", null, "https://www.linuxtopia.org/images/arrow.jpg", null, "https://www.linuxtopia.org/images/arrow.jpg", null, "https://www.linuxtopia.org/images/arrow.jpg", null, "https://www.linuxtopia.org/images/arrow.jpg", null, "https://www.linuxtopia.org/images/arrow.jpg", null, "https://www.linuxtopia.org/images/arrow.jpg", null, "https://www.linuxtopia.org/images/arrow.jpg", null, "https://www.linuxtopia.org/images/arrow.jpg", null, "https://www.linuxtopia.org/images/arrow.jpg", null, "https://www.linuxtopia.org/images/arrow.jpg", null, "https://www.linuxtopia.org/images/arrow.jpg", null, "https://www.linuxtopia.org/images/arrow.jpg", null, "https://www.linuxtopia.org/images/arrow.jpg", null, "https://www.linuxtopia.org/images/arrow.jpg", null, "https://www.linuxtopia.org/images/arrow.jpg", null, "https://www.linuxtopia.org/images/arrow.jpg", null, "https://www.linuxtopia.org/images/arrow.jpg", null, "https://www.linuxtopia.org/images/arrow.jpg", null, "https://www.linuxtopia.org/images/arrow.jpg", null, "https://www.linuxtopia.org/images/arrow.jpg", null, "https://www.linuxtopia.org/images/arrow.jpg", null, "https://www.linuxtopia.org/images/spacer.gif", null, "https://www.linuxtopia.org/images/arrow.jpg", null, "https://www.linuxtopia.org/images/arrow.jpg", null, "https://www.linuxtopia.org/images/arrow.jpg", null, "https://www.linuxtopia.org/images/arrow.jpg", null, "https://www.linuxtopia.org/images/arrow.jpg", null, "https://www.linuxtopia.org/images/arrow.jpg", null, "https://www.linuxtopia.org/images/arrow.jpg", null, "https://www.linuxtopia.org/images/arrow.jpg", null, "https://www.linuxtopia.org/images/arrow.jpg", null, "https://www.linuxtopia.org/images/arrow.jpg", null, "https://www.linuxtopia.org/images/spacer.gif", null, "https://www.linuxtopia.org/images/navbasebg.jpg", null, "https://www.linuxtopia.org/images/basebg1.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8757022,"math_prob":0.93669504,"size":5049,"snap":"2022-27-2022-33","text_gpt3_token_len":1373,"char_repetition_ratio":0.12666006,"word_repetition_ratio":0.039054472,"special_character_ratio":0.25826895,"punctuation_ratio":0.11049211,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9895119,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104],"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,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,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\":\"2022-07-03T03:36:00Z\",\"WARC-Record-ID\":\"<urn:uuid:002f1bc7-dbc7-4a3e-b21b-03e63a188fab>\",\"Content-Length\":\"45440\",\"Content-Type\":\"application/http; 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https://math.stackexchange.com/questions/1935614/homeomorphism-between-union-of-intervals-and-reals	[ "# homeomorphism between union of intervals and reals\n\nQuestion:\n\nShow that $$\\left ( 0,1 \\right )$$ and $$\\mathbb{R}$$ are homeomorphic.\n\nProposition:If $$a,b,c,d \\in \\mathbb{R}$$, $$a and $$c then $$\\left ( a,b \\right )$$ is homeomorphic to $$\\left ( c,d \\right )$$.\n\nFor any point $$x \\in \\mathbb{R}$$, we have that $$x \\in \\left ( c,d \\right )$$ where $$c\n\nHence, $$\\left ( 0,1 \\right )$$ is homeomorphic to $$\\mathbb{R}$$\n\nNow,\n\nQuestion: Show that $$\\left ( 0,1 \\right ) \\cup \\left ( 2,3 \\right )$$ is not homeomorphic to $$\\mathbb{R}$$.\n\n$$\\textbf{Hint}$$: any two points in $$\\mathbb{R}$$ are joined by a continuous curve.\n\nI'll like a hint as to the second question.\n\nAny help is appreciated.\n\n• For $x\\in (0,1)$ let $f(x)=\\tan \\pi (x-1/2).$ Then $f:(0,1)\\to \\mathbb R$ is a homeomorphism. This is because $f$ is a continuous, strictly monotonic bijection. – DanielWainfleet Sep 22 '16 at 1:14\n\nSuppose you have a homeomorphism $\\phi: (0,1) \\cup (2,3) \\to \\mathbb{R}$. Take the points $x=\\phi(0.5)$ and $y=\\phi(2,3)$. Now have a continuous curve from $x$ to $y$, lets call it $\\gamma: [0,1] \\to \\mathbb{R}$ (I think you can construct such a curve easily).\n\nHint: The pullback of the curve $\\gamma'=\\phi^{-1}(\\gamma)$ is hence also a continuous curve in $(0,1) \\cup (2,3)$ with certain endpoints. Does such a curve exist?\n\nAnother possibility/hint without curves: Make yourself clear that in $(0,1) \\cup (2,3)$ the interval $(0,1)$ is open (as open interval) and closed (as complement of the open interval $(2,3)$). Assume you have a homeomorphism $\\phi: (0,1) \\cup (2,3) \\to \\mathbb{R}$, then $\\phi( (0,1)) \\subseteq \\mathbb{R}$ is (since $\\phi$ is a homeomorphism) a closed and open subset of $\\mathbb{R}$. You might know all the subsets of $\\mathbb{R}$ which are closed and open (hint: there are 2 such subsets). Take it from here (remember that images of non-empty sets are non-empty and a homeomorphism is also injective)!\n\n• So essentially what I do need to do is find an inverse function such that for some value in the codomain, the inverse function maps that value to a value not in the domain? – Mathematicing Sep 21 '16 at 13:52\n• @Mathematicing Nope. I suppose you follow the first hint: You get that inverse function (since you have an homeomorphism). What you should focus on is the path $\\gamma '$. This is a continuous path from $\\phi^{-1}(x)=0.5$ to $\\phi^{-1}(y)=1.5$ in your union of intervals. You now have to show, that such a path is not possible (if you didn't see that yet). Hint: Look at the point, where you \"jump\" from one intervall to the next one in the domain of your path. (actually I prefer the way without curves but maybe the first way is easier to see for a start) – ctst Sep 21 '16 at 13:58\n• I prefer the hint without curves. I assume the function is a homeomorphism. So we have that the function f is bijective with f and f^{-1} being continuous. We know that open intervals are open sets. The union of open sets are open sets. By this, we have that f and f^{-1} maps open sets to open sets. Have I got this correct? – Mathematicing Sep 21 '16 at 14:06\n• @Mathematicing Yes absolutly (you don't need that the union of open sets are open sets for this, but only that $f$ and $f^{-1}$ are continuous)! And by taking complements you also know $f$ maps closed sets to closed sets. – ctst Sep 21 '16 at 14:08\n• Would you mind expounding on the part where you mention f maps closed sets to closed sets? I think there's a subtlety that I'm not realising it fully. – Mathematicing Sep 21 '16 at 14:17\n\nLet's go back to your first problem:\n\nFor any point $$x \\in \\mathbb{R}$$, we have that $$x \\in \\left ( c,d \\right )$$ where $$c\n\nHence, $$\\left ( 0,1 \\right )$$ is homeomorphic to $$\\mathbb{R}$$\n\nOne point that's confusing for a reader is that you say \"hence\" as if the second sentence follows from the first. But the two don't seem to have much to do with each other.\n\nThe first statement looks like you've jumbled up the quantifiers. You're asked to prove that for all $$a$$, $$b$$, $$c$$, and $$d$$ with $$a and $$c, that $$(a,b)$$ is homeomorphic to $$(c,d)$$. You assert that for all $$x \\in \\mathbb{R}$$, that there exist $$c$$ and $$d$$ such that $$c. This is true, but it's not relevant to the problem.\n\nFirst, try to show that $$(0,1)$$ is homeomorphic to $$\\mathbb{R}$$. Then, generalize your proof to show that $$(a,b)$$ is homeomorphic to $$\\mathbb{R}$$, for any $$a. This shows that $$(a,b)$$ is homeomorphic to $$(c,d)$$, by transitivity.\n\nTo look for a continuous, invertible map that sends $$(0,1)$$ to $$(-\\infty,\\infty)$$, think of rational functions. Can you construct a function $$f(x)$$ which is monotone increasing on $$(0,1)$$, has $$\\lim_{x\\to 0^+} f(x) =-\\infty$$, and $$\\lim_{x\\to 1^-}f(x) = +\\infty$$?\n\nI am not exactly sure what you did to prove the first part... But regarding your second question, $\\mathbb{R}$ is connected, whereas $(0,1) \\cup (2,3)$ is not, and homeomorphisms preserve connectedness.\n\n• In the first part, all I note was that any value on the real line is between two open intervals. Then, using the proposition that the open intervals is an interval between c and d with c <d. – Mathematicing Sep 21 '16 at 12:58" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.70439506,"math_prob":0.99986243,"size":636,"snap":"2021-04-2021-17","text_gpt3_token_len":222,"char_repetition_ratio":0.19936709,"word_repetition_ratio":0.0952381,"special_character_ratio":0.3616352,"punctuation_ratio":0.17730497,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9999882,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-01-27T07:33:30Z\",\"WARC-Record-ID\":\"<urn:uuid:7032dff6-c5c0-4c5f-9b4e-8d509f25a7d3>\",\"Content-Length\":\"177766\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:3dd065df-a4e0-459a-82f9-a0a6dcc596da>\",\"WARC-Concurrent-To\":\"<urn:uuid:14dec7a7-06f5-4249-94f0-9b7c6e301dd2>\",\"WARC-IP-Address\":\"151.101.65.69\",\"WARC-Target-URI\":\"https://math.stackexchange.com/questions/1935614/homeomorphism-between-union-of-intervals-and-reals\",\"WARC-Payload-Digest\":\"sha1:WMJXLZWXBTDXVFCP462HSTMYWMM74UEJ\",\"WARC-Block-Digest\":\"sha1:M4BBP5MUFVKJJSIILCNPMWNQMERPOSBB\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-04/CC-MAIN-2021-04_segments_1610704821253.82_warc_CC-MAIN-20210127055122-20210127085122-00600.warc.gz\"}"}
https://forum.knime.com/t/calculation-with-previous-rows/9059	[ "# Calculation with previous rows\n\nHello all,\n\nI have a problem which is simple (I guess) but I don't know how to solve it :(\n\nI have the following (current) Data:\n\n current goal result 750 750 750 -700 -700 50 -375 -50 0 125 125 125 -250 -125 0 750 750 750 -750 -750 0\n\nI need KNIME to calculate from the \"current\" data the goal and the result column. The result column can be easily calculated from the goal column. It is important, that the values in the result column never reach below 0, therefore I need to have a adjusted goal column. The calculation in the result column is just an addition of the previos row in the result column with the current row in the goal column.\n\nAny solution for that?\n\nThanks :)\n\nRalph\n\nHi Ralph,\n\nI think the result colum is a sum of all previous rows of the goal?\n\nIn this case you can use the moving aggregation node with cumulative computation and sum.\n\nIf you need the previous row, you can get is using the lag column node. It will \"shift\" some of the values in a new column.\n\nIf you need full control of the previous values, you need to use a Java Snippet.\n\nHope this helps,\n\nBest, Iris\n\n2 Likes\n\nIris' suggestions will work only if the input contains both the current and goal values.\n\nSince the calculation for a row depends on the calculation in the previous row, a recursive loop will do the trick.\nIn the attached workflow, the upper branch calculates the goal and result values for each row. The result of each row is fed back for use in the next iteration, along with the unaltered remaining rows of the input table.\n\nThere are other ways to do this, but this workflow gives the desired output using only the current values as input. Hope this helps.\n\nTim\n\n2 Likes" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.88903636,"math_prob":0.9287541,"size":1187,"snap":"2022-27-2022-33","text_gpt3_token_len":287,"char_repetition_ratio":0.1673711,"word_repetition_ratio":0.0,"special_character_ratio":0.26368997,"punctuation_ratio":0.075,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96705884,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-08-19T16:59:22Z\",\"WARC-Record-ID\":\"<urn:uuid:394664e4-da55-4464-a32b-45cfe35e48f2>\",\"Content-Length\":\"32909\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7b0dbb43-5503-47e8-90e1-d76fe4e03c0f>\",\"WARC-Concurrent-To\":\"<urn:uuid:4aebe375-634f-46d6-8a15-65731217ffe6>\",\"WARC-IP-Address\":\"18.192.227.6\",\"WARC-Target-URI\":\"https://forum.knime.com/t/calculation-with-previous-rows/9059\",\"WARC-Payload-Digest\":\"sha1:ZN2MKJPPXG2DGMLQALKWKESSH4WHQCOE\",\"WARC-Block-Digest\":\"sha1:GQYUFW6QKD7G6HDUQYWW733VCHAPUXTT\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-33/CC-MAIN-2022-33_segments_1659882573744.90_warc_CC-MAIN-20220819161440-20220819191440-00360.warc.gz\"}"}
https://demo.formulasearchengine.com/wiki/BRST_quantization	[ "# BRST quantization\n\nIn theoretical physics, BRST quantization (where the BRST refers to Becchi, Rouet, Stora and Tyutin) is a relatively rigorous mathematical approach to quantizing a field theory with a gauge symmetry. Quantization rules in earlier QFT frameworks resembled \"prescriptions\" or \"heuristics\" more than proofs, especially in non-abelian QFT, where the use of \"ghost fields\" with superficially bizarre properties is almost unavoidable for technical reasons related to renormalization and anomaly cancellation. The BRST supersymmetry was introduced in the mid-1970s and was quickly understood to justify the introduction of these Faddeev–Popov ghosts and their exclusion from \"physical\" asymptotic states when performing QFT calculations. Work by other authors a few years later related the BRST operator to the existence of a rigorous alternative to path integrals when quantizing a gauge theory.\n\nOnly in the late 1980s, when QFT was reformulated in fiber bundle language for application to problems in the topology of low-dimensional manifolds, did it become apparent that the BRST \"transformation\" is fundamentally geometrical in character. In this light, \"BRST quantization\" becomes more than an alternate way to arrive at anomaly-cancelling ghosts. It is a different perspective on what the ghost fields represent, why the Faddeev–Popov method works, and how it is related to the use of Hamiltonian mechanics to construct a perturbative framework. The relationship between gauge invariance and \"BRST invariance\" forces the choice of a Hamiltonian system whose states are composed of \"particles\" according to the rules familiar from the canonical quantization formalism. This esoteric consistency condition therefore comes quite close to explaining how quanta and fermions arise in physics to begin with.\n\nIn certain cases, notably gravity and supergravity, BRST must be superseded by a more general formalism, the Batalin–Vilkovisky formalism.\n\n## Technical summary\n\nBRST quantization (or the BRST formalism) is a differential geometric approach to performing consistent, anomaly-free perturbative calculations in a non-abelian gauge theory. The analytical form of the BRST \"transformation\" and its relevance to renormalization and anomaly cancellation were described by Carlo Maria Becchi, Alain Rouet, and Raymond Stora in a series of papers culminating in the 1976 \"Renormalization of gauge theories\". The equivalent transformation and many of its properties were independently discovered by Igor Viktorovich Tyutin. Its significance for rigorous canonical quantization of a Yang–Mills theory and its correct application to the Fock space of instantaneous field configurations were elucidated by Kugo Taichiro and Ojima Izumi. Later work by many authors, notably Thomas Schücker and Edward Witten, has clarified the geometric significance of the BRST operator and related fields and emphasized its importance to topological quantum field theory and string theory.\n\nIn the BRST approach, one selects a perturbation-friendly gauge fixing procedure for the action principle of a gauge theory using the differential geometry of the gauge bundle on which the field theory lives. One then quantizes the theory to obtain a Hamiltonian system in the interaction picture in such a way that the \"unphysical\" fields introduced by the gauge fixing procedure resolve gauge anomalies without appearing in the asymptotic states of the theory. The result is a set of Feynman rules for use in a Dyson series perturbative expansion of the S-matrix which guarantee that it is unitary and renormalizable at each loop order—in short, a coherent approximation technique for making physical predictions about the results of scattering experiments.\n\n### Classical BRST\n\nThis is related to a supersymplectic manifold where pure operators are graded by integral ghost numbers and we have a BRST cohomology.\n\n## Gauge transformations in QFT\n\nFrom a practical perspective, a quantum field theory consists of an action principle and a set of procedures for performing perturbative calculations. There are other kinds of \"sanity checks\" that can be performed on a quantum field theory to determine whether it fits qualitative phenomena such as quark confinement and asymptotic freedom. However, most of the predictive successes of quantum field theory, from quantum electrodynamics to the present day, have been quantified by matching S-matrix calculations against the results of scattering experiments.\n\nIn the early days of QFT, one would have to have said that the quantization and renormalization prescriptions were as much part of the model as the Lagrangian density, especially when they relied on the powerful but mathematically ill-defined path integral formalism. It quickly became clear that QED was almost \"magical\" in its relative tractability, and that most of the ways that one might imagine extending it would not produce rational calculations. However, one class of field theories remained promising: gauge theories, in which the objects in the theory represent equivalence classes of physically indistinguishable field configurations, any two of which are related by a gauge transformation. This generalizes the QED idea of a local change of phase to a more complicated Lie group.\n\nQED itself is a gauge theory, as is general relativity, although the latter has proven resistant to quantization so far, for reasons related to renormalization. Another class of gauge theories with a non-Abelian gauge group, beginning with Yang–Mills theory, became amenable to quantization in the late 1960s and early 1970s, largely due to the work of Ludwig D. Faddeev, Victor Popov, Bryce DeWitt, and Gerardus 't Hooft. However, they remained very difficult to work with until the introduction of the BRST method. The BRST method provided the calculation techniques and renormalizability proofs needed to extract accurate results from both \"unbroken\" Yang–Mills theories and those in which the Higgs mechanism leads to spontaneous symmetry breaking. Representatives of these two types of Yang–Mills systems—quantum chromodynamics and electroweak theory—appear in the Standard Model of particle physics.\n\nIt has proven rather more difficult to prove the existence of non-Abelian quantum field theory in a rigorous sense than to obtain accurate predictions using semi-heuristic calculation schemes. This is because analyzing a quantum field theory requires two mathematically interlocked perspectives: a Lagrangian system based on the action functional, composed of fields with distinct values at each point in spacetime and local operators which act on them, and a Hamiltonian system in the Dirac picture, composed of states which characterize the entire system at a given time and field operators which act on them. What makes this so difficult in a gauge theory is that the objects of the theory are not really local fields on spacetime; they are right-invariant local fields on the principal gauge bundle, and different local sections through a portion of the gauge bundle, related by passive transformations, produce different Dirac pictures.\n\nWhat is more, a description of the system as a whole in terms of a set of fields contains many redundant degrees of freedom; the distinct configurations of the theory are equivalence classes of field configurations, so that two descriptions which are related to one another by an active gauge transformation are also really the same physical configuration. The \"solutions\" of a quantized gauge theory exist not in a straightforward space of fields with values at every point in spacetime but in a quotient space (or cohomology) whose elements are equivalence classes of field configurations. Hiding in the BRST formalism is a system for parameterizing the variations associated with all possible active gauge transformations and correctly accounting for their physical irrelevance during the conversion of a Lagrangian system to a Hamiltonian system.\n\n### Gauge fixing and perturbation theory\n\nThe principle of gauge invariance is essential to constructing a workable quantum field theory. But it is generally not feasible to perform a perturbative calculation in a gauge theory without first \"fixing the gauge\"—adding terms to the Lagrangian density of the action principle which \"break the gauge symmetry\" to suppress these \"unphysical\" degrees of freedom. The idea of gauge fixing goes back to the Lorenz gauge approach to electromagnetism, which suppresses most of the excess degrees of freedom in the four-potential while retaining manifest Lorentz invariance. The Lorenz gauge is a great simplification relative to Maxwell's field-strength approach to classical electrodynamics, and illustrates why it is useful to deal with excess degrees of freedom in the representation of the objects in a theory at the Lagrangian stage, before passing over to Hamiltonian mechanics via the Legendre transform.\n\nThe Hamiltonian density is related to the Lie derivative of the Lagrangian density with respect to a unit timelike horizontal vector field on the gauge bundle. In a quantum mechanical context it is conventionally rescaled by a factor $i\\hbar$", null, ". Integrating it by parts over a spacelike cross section recovers the form of the integrand familiar from canonical quantization. Because the definition of the Hamiltonian involves a unit time vector field on the base space, a horizontal lift to the bundle space, and a spacelike surface \"normal\" (in the Minkowski metric) to the unit time vector field at each point on the base manifold, it is dependent both on the connection and the choice of Lorentz frame, and is far from being globally defined. But it is an essential ingredient in the perturbative framework of quantum field theory, into which the quantized Hamiltonian enters via the Dyson series.\n\nFor perturbative purposes, we gather the configuration of all the fields of our theory on an entire three-dimensional horizontal spacelike cross section of P into one object (a Fock state), and then describe the \"evolution\" of this state over time using the interaction picture. The Fock space is spanned by the multi-particle eigenstates of the \"unperturbed\" or \"non-interaction\" portion ${\\mathcal {H}}_{0}$", null, "of the Hamiltonian ${\\mathcal {H}}$", null, ". Hence the instantaneous description of any Fock state is a complex-amplitude-weighted sum of eigenstates of ${\\mathcal {H}}_{0}$", null, ". In the interaction picture, we relate Fock states at different times by prescribing that each eigenstate of the unperturbed Hamiltonian experiences a constant rate of phase rotation proportional to its energy (the corresponding eigenvalue of the unperturbed Hamiltonian).\n\nHence, in the zero-order approximation, the set of weights characterizing a Fock state does not change over time, but the corresponding field configuration does. In higher approximations, the weights also change; collider experiments in high-energy physics amount to measurements of the rate of change in these weights (or rather integrals of them over distributions representing uncertainty in the initial and final conditions of a scattering event). The Dyson series captures the effect of the discrepancy between ${\\mathcal {H}}_{0}$", null, "and the true Hamiltonian ${\\mathcal {H}}$", null, ", in the form of a power series in the coupling constant g; it is the principal tool for making quantitative predictions from a quantum field theory.\n\nTo use the Dyson series to calculate anything, one needs more than a gauge-invariant Lagrangian density; one also needs the quantization and gauge fixing prescriptions that enter into the Feynman rules of the theory. The Dyson series produces infinite integrals of various kinds when applied to the Hamiltonian of a particular QFT. This is partly because all usable quantum field theories to date must be considered effective field theories, describing only interactions on a certain range of energy scales that we can experimentally probe and therefore vulnerable to ultraviolet divergences. These are tolerable as long as they can be handled via standard techniques of renormalization; they are not so tolerable when they result in an infinite series of infinite renormalizations or, worse, in an obviously unphysical prediction such as an uncancelled gauge anomaly. There is a deep relationship between renormalizability and gauge invariance, which is easily lost in the course of attempts to obtain tractable Feynman rules by fixing the gauge.\n\n### Pre-BRST approaches to gauge fixing\n\nThe traditional gauge fixing prescriptions of continuum electrodynamics select a unique representative from each gauge-transformation-related equivalence class using a constraint equation such as the Lorenz gauge $\\partial ^{\\mu }A_{\\mu }=0$", null, ". This sort of prescription can be applied to an Abelian gauge theory such as QED, although it results in some difficulty in explaining why the Ward identities of the classical theory carry over to the quantum theory—in other words, why Feynman diagrams containing internal longitudinally polarized virtual photons do not contribute to S-matrix calculations. This approach also does not generalize well to non-Abelian gauge groups such as the SU(2) of Yang–Mills and electroweak theory and the SU(3) of quantum chromodynamics. It suffers from Gribov ambiguities and from the difficulty of defining a gauge fixing constraint that is in some sense \"orthogonal\" to physically significant changes in the field configuration.\n\nMore sophisticated approaches do not attempt to apply a delta function constraint to the gauge transformation degrees of freedom. Instead of \"fixing\" the gauge to a particular \"constraint surface\" in configuration space, one can break the gauge freedom with an additional, non-gauge-invariant term added to the Lagrangian density. In order to reproduce the successes of gauge fixing, this term is chosen to be minimal for the choice of gauge that corresponds to the desired constraint and to depend quadratically on the deviation of the gauge from the constraint surface. By the stationary phase approximation on which the Feynman path integral is based, the dominant contribution to perturbative calculations will come from field configurations in the neighborhood of the constraint surface.\n\nThe perturbative expansion associated with this Lagrangian, using the method of functional quantization, is generally referred to as the Rξ gauge. It reduces in the case of an Abelian U(1) gauge to the same set of Feynman rules that one obtains in the method of canonical quantization. But there is an important difference: the broken gauge freedom appears in the functional integral as an additional factor in the overall normalization. This factor can only be pulled out of the perturbative expansion (and ignored) when the contribution to the Lagrangian of a perturbation along the gauge degrees of freedom is independent of the particular \"physical\" field configuration. This is the condition that fails to hold for non-Abelian gauge groups. If one ignores the problem and attempts to use the Feynman rules obtained from \"naive\" functional quantization, one finds that one's calculations contain unremovable anomalies.\n\nThe problem of perturbative calculations in QCD was solved by introducing additional fields known as Faddeev–Popov ghosts, whose contribution to the gauge-fixed Lagrangian offsets the anomaly introduced by the coupling of \"physical\" and \"unphysical\" perturbations of the non-Abelian gauge field. From the functional quantization perspective, the \"unphysical\" perturbations of the field configuration (the gauge transformations) form a subspace of the space of all (infinitesimal) perturbations; in the non-Abelian case, the embedding of this subspace in the larger space depends on the configuration around which the perturbation takes place. The ghost term in the Lagrangian represents the functional determinant of the Jacobian of this embedding, and the properties of the ghost field are dictated by the exponent desired on the determinant in order to correct the functional measure on the remaining \"physical\" perturbation axes.\n\n## Mathematical approach to BRST\n\nFirst, using the regular sequence of functions defining M0 inside M, construct a Koszul complex\n\n$\\Lambda ^{\\cdot }{\\mathfrak {g}}\\otimes C^{\\infty }(M).$", null, "The differential, δ, on this complex is an odd C(M)-linear derivation of the graded C(M)-algebra $\\Lambda ^{\\cdot }{\\mathfrak {g}}\\otimes C^{\\infty }(M)$", null, ". This odd derivation is defined by extending the Lie algebra homomorphim ${\\mathfrak {g}}\\to C^{\\infty }(M)$", null, "of the hamiltonian action. The resulting Koszul complex is the Koszul complex of the $S({\\mathfrak {g}})$", null, "-module C(M), where $S({\\mathfrak {g}})$", null, "is the symmetric algebra of ${\\mathfrak {g}}$", null, ", and the module structure comes from a ring homomorphism $S({\\mathfrak {g}})\\to C^{\\infty }(M)$", null, "induced by the hamiltonian action ${\\mathfrak {g}}\\to C^{\\infty }(M)$", null, ".\n\n$H^{j}(\\Lambda ^{\\cdot }{\\mathfrak {g}}\\otimes C^{\\infty }(M),\\delta )={\\begin{cases}C^{\\infty }(M_{0})&j=0\\\\0&j\\neq 0\\end{cases}}$", null, "Then, consider the Chevalley-Eilenberg cochain complex for the Koszul complex $\\Lambda ^{\\cdot }{\\mathfrak {g}}\\otimes C^{\\infty }(M)$", null, "considered as a dg module over the Lie algebra ${\\mathfrak {g}}$", null, ":\n\n$K^{\\cdot ,\\cdot }=C^{\\cdot }\\left({\\mathfrak {g}},\\Lambda ^{\\cdot }{\\mathfrak {g}}\\otimes C^{\\infty }(M)\\right)=\\Lambda ^{\\cdot }{\\mathfrak {g}}^{}\\otimes \\Lambda ^{\\cdot }{\\mathfrak {g}}\\otimes C^{\\infty }(M).$", null, "The \"horizontal\" differential $d:K^{i,\\cdot }\\to K^{i+1,\\cdot }$", null, "is defined on the coefficients\n\n$\\Lambda ^{\\cdot }{\\mathfrak {g}}\\otimes C^{\\infty }(M)$", null, "Let Tot(K) be a complex such that\n\n$\\operatorname {Tot} (K)^{n}=\\bigoplus \\nolimits _{i-j=n}K^{i,j}$", null, "with a differential D = d + δ. The cohomology groups of (Tot(K), D) are computed using a spectral sequence associated to the double complex $(K^{\\cdot ,\\cdot },d,\\delta )$", null, ".\n\nThe first term of the spectral sequence computes the cohomology of the \"vertical\" differential δ:\n\n$E_{1}^{i,j}=H^{j}(K^{i,\\cdot },\\delta )=\\Lambda ^{i}{\\mathfrak {g}}^{}\\otimes C^{\\infty }(M_{0})$", null, ", if j = 0 and zero otherwise.\n\nThe first term of the spectral sequence may be interpreted as the complex of vertical differential forms\n\n$(\\Omega _{\\operatorname {vert} }^{\\cdot }(M_{0}),d_{\\operatorname {vert} })$", null, "The second term of the spectral sequence computes the cohomology of the \"horizontal\" differential d on $E_{1}^{\\cdot ,\\cdot }$", null, ":\n\n$E_{2}^{i,j}\\cong H^{i}(E_{1}^{\\cdot ,j},d)=C^{\\infty }(M_{0})^{g}=C^{\\infty }({\\widetilde {M}})$", null, ", if $i=j=0$", null, "and zero otherwise.\n\nThe spectral sequence collapses at the second term, so that $E_{\\infty }^{i,j}=E_{2}^{i,j}$", null, ", which is concentrated in degree zero.\n\nTherefore,\n\n$H^{p}(\\operatorname {Tot} (K),D)=C^{\\infty }(M_{0})^{g}=C^{\\infty }({\\widetilde {M}})$", null, ", if p = 0 and 0 otherwise.\n\n## The BRST operator and asymptotic Fock space\n\nTwo important remarks about the BRST operator are due. First, instead of working with the gauge group G one can use only the action of the gauge algebra ${\\mathfrak {g}}$", null, "on the fields (functions on the phase space).\n\nSecond, the variation of any \"BRST exact form\" sBX with respect to a local gauge transformation dλ is\n\n$\\left[i_{\\delta \\lambda },s_{B}\\right]s_{B}X=i_{\\delta \\lambda }(s_{B}s_{B}X)+s_{B}\\left(i_{\\delta \\lambda }(s_{B}X)\\right)=s_{B}\\left(i_{\\delta \\lambda }(s_{B}X)\\right),$", null, "which is itself an exact form.\n\nMore importantly for the Hamiltonian perturbative formalism (which is carried out not on the fiber bundle but on a local section), adding a BRST exact term to a gauge invariant Lagrangian density preserves the relation sBX = 0. As we shall see, this implies that there is a related operator QB on the state space for which $[Q_{B},{\\mathcal {H}}]=0$", null, "—i. e., the BRST operator on Fock states is a conserved charge of the Hamiltonian system. This implies that the time evolution operator in a Dyson series calculation will not evolve a field configuration obeying $Q_{B}\|\\Psi _{i}\\rangle =0$", null, "into a later configuration with $Q_{B}\|\\Psi _{f}\\rangle \\neq 0$", null, "(or vice versa).\n\nAnother way of looking at the nilpotence of the BRST operator is to say that its image (the space of BRST exact forms) lies entirely within its kernel (the space of BRST closed forms). (The \"true\" Lagrangian, presumed to be invariant under local gauge transformations, is in the kernel of the BRST operator but not in its image.) The preceding argument says that we can limit our universe of initial and final conditions to asymptotic \"states\"—field configurations at timelike infinity, where the interaction Lagrangian is \"turned off\"—that lie in the kernel of QB and still obtain a unitary scattering matrix. (BRST closed and exact states are defined similarly to BRST closed and exact fields; closed states are annihilated by QB, while exact states are those obtainable by applying QB to some arbitrary field configuration.)\n\nWe can also suppress states that lie inside the image of QB when defining the asymptotic states of our theory—but the reasoning is a bit subtler. Since we have postulated that the \"true\" Lagrangian of our theory is gauge invariant, the true \"states\" of our Hamiltonian system are equivalence classes under local gauge transformation; in other words, two initial or final states in the Hamiltonian picture that differ only by a BRST exact state are physically equivalent. However, the use of a BRST exact gauge breaking prescription does not guarantee that the interaction Hamiltonian will preserve any particular subspace of closed field configurations that we can call \"orthogonal\" to the space of exact configurations. (This is a crucial point, often mishandled in QFT textbooks. There is no a priori inner product on field configurations built into the action principle; we construct such an inner product as part of our Hamiltonian perturbative apparatus.)\n\nWe therefore focus on the vector space of BRST closed configurations at a particular time with the intention of converting it into a Fock space of intermediate states suitable for Hamiltonian perturbation. To this end, we shall endow it with ladder operators for the energy-momentum eigenconfigurations (particles) of each field, complete with appropriate (anti-)commutation rules, as well as a positive semi-definite inner product. We require that the inner product be singular exclusively along directions that correspond to BRST exact eigenstates of the unperturbed Hamiltonian. This ensures that one can freely choose, from within the two equivalence classes of asymptotic field configurations corresponding to particular initial and final eigenstates of the (unbroken) free-field Hamiltonian, any pair of BRST closed Fock states that we like.\n\nThe desired quantization prescriptions will also provide a quotient Fock space isomorphic to the BRST cohomology, in which each BRST closed equivalence class of intermediate states (differing only by an exact state) is represented by exactly one state that contains no quanta of the BRST exact fields. This is the Fock space we want for asymptotic states of the theory; even though we will not generally succeed in choosing the particular final field configuration to which the gauge-fixed Lagrangian dynamics would have evolved that initial configuration, the singularity of the inner product along BRST exact degrees of freedom ensures that we will get the right entries for the physical scattering matrix.\n\n(Actually, we should probably be constructing a Krein space for the BRST-closed intermediate Fock states, with the time reversal operator playing the role of the \"fundamental symmetry\" relating the Lorentz-invariant and positive semi-definite inner products. The asymptotic state space is presumably the Hilbert space obtained by quotienting BRST exact states out of this Krein space.)\n\nIn sum, no field introduced as part of a BRST gauge fixing procedure will appear in asymptotic states of the gauge-fixed theory. However, this does not imply that we can do without these \"unphysical\" fields in the intermediate states of a perturbative calculation! This is because perturbative calculations are done in the interaction picture. They implicitly involve initial and final states of the non-interaction Hamiltonian ${\\mathcal {H}}_{0}$", null, ", gradually transformed into states of the full Hamiltonian in accordance with the adiabatic theorem by \"turning on\" the interaction Hamiltonian (the gauge coupling). The expansion of the Dyson series in terms of Feynman diagrams will include vertices that couple \"physical\" particles (those that can appear in asymptotic states of the free Hamiltonian) to \"unphysical\" particles (states of fields that live outside the kernel of sB or inside the image of sB) and vertices that couple \"unphysical\" particles to one another.\n\n### The Kugo–Ojima answer to unitarity questions\n\nT. Kugo and I. Ojima are commonly credited with the discovery of the principal QCD color confinement criterion. Their role in obtaining a correct version of the BRST formalism in the Lagrangian framework seems to be less widely appreciated. It is enlightening to inspect their variant of the BRST transformation, which emphasizes the hermitian properties of the newly introduced fields, before proceeding from an entirely geometrical angle. The gauge fixed Lagrangian density is below; the two terms in parentheses form the coupling between the gauge and ghost sectors, and the final term becomes a Gaussian weighting for the functional measure on the auxiliary field B.\n\n${\\mathcal {L}}={\\mathcal {L}}_{\\textrm {matter}}(\\psi ,\\,A_{\\mu }^{a})-{\\tfrac {1}{4}}F_{\\mu \\nu }^{a}F^{a,\\,\\mu \\nu }-(i(\\partial ^{\\mu }{\\bar {c}}^{a})D_{\\mu }^{ab}c^{b}+(\\partial ^{\\mu }B^{a})A_{\\mu }^{a})+{\\tfrac {1}{2}}\\alpha _{0}B^{a}B^{a}$", null, "The Faddeev–Popov ghost field c is unique among the new fields of our gauge-fixed theory in having a geometrical meaning beyond the formal requirements of the BRST procedure. It is a version of the Maurer–Cartan form on $V{\\mathfrak {E}}$", null, ", which relates each right-invariant vertical vector field $\\delta \\lambda \\in V{\\mathfrak {E}}$", null, "to its representation (up to a phase) as a ${\\mathfrak {g}}$", null, "-valued field. This field must enter into the formulas for infinitesimal gauge transformations on objects (such as fermions ψ, gauge bosons Aμ, and the ghost c itself) which carry a non-trivial representation of the gauge group. The BRST transformation with respect to δλ is therefore:\n\n{\\begin{aligned}\\delta \\psi _{i}&=\\delta \\lambda D_{i}c\\\\\\delta A_{\\mu }&=\\delta \\lambda D_{\\mu }c\\\\\\delta c&=-\\delta \\lambda {\\tfrac {g}{2}}[c,c]\\\\\\delta {\\bar {c}}&=i\\delta \\lambda B\\\\\\delta B&=0\\end{aligned}}", null, "Here we have omitted the details of the matter sector ψ and left the form of the Ward operator on it unspecified; these are unimportant so long as the representation of the gauge algebra on the matter fields is consistent with their coupling to δAμ. The properties of the other fields we have added are fundamentally analytical rather than geometric. The bias we have introduced towards connections with $\\partial ^{\\mu }A_{\\mu }=0$", null, "is gauge-dependent and has no particular geometrical significance. The anti-ghost ${\\bar {c}}$", null, "is nothing but a Lagrange multiplier for the gauge fixing term, and the properties of the scalar field B are entirely dictated by the relationship $\\delta {\\bar {c}}=i\\delta \\lambda B$", null, ". (The new fields are all Hermitian in Kugo–Ojima conventions, but the parameter δλ is an anti-Hermitian \"anti-commuting c-number\". This results in some unnecessary awkwardness with regard to phases and passing infinitesimal parameters through operators; this will be resolved with a change of conventions in the geometric treatment below.)\n\nWe already know, from the relation of the BRST operator to the exterior derivative and the Faddeev–Popov ghost to the Maurer–Cartan form, that the ghost c corresponds (up to a phase) to a ${\\mathfrak {g}}$", null, "-valued 1-form on $V{\\mathfrak {E}}$", null, ". In order for integration of a term like $-i(\\partial ^{\\mu }{\\bar {c}})D_{\\mu }c$", null, "to be meaningful, the anti-ghost ${\\bar {c}}$", null, "must carry representations of these two Lie algebras—the vertical ideal $V{\\mathfrak {E}}$", null, "and the gauge algebra ${\\mathfrak {g}}$", null, "—dual to those carried by the ghost. In geometric terms, ${\\bar {c}}$", null, "must be fiberwise dual to ${\\mathfrak {g}}$", null, "and one rank short of being a top form on $V{\\mathfrak {E}}$", null, ". Likewise, the auxiliary field B must carry the same representation of ${\\mathfrak {g}}$", null, "(up to a phase) as ${\\bar {c}}$", null, ", as well as the representation of $V{\\mathfrak {E}}$", null, "dual to its trivial representation on Aμ—i. e., B is a fiberwise ${\\mathfrak {g}}$", null, "-dual top form on $V{\\mathfrak {E}}$", null, ".\n\nLet us focus briefly on the one-particle states of the theory, in the adiabatically decoupled limit g → 0. There are two kinds of quanta in the Fock space of the gauge-fixed Hamiltonian that we expect to lie entirely outside the kernel of the BRST operator: those of the Faddeev–Popov anti-ghost ${\\bar {c}}$", null, "and the forward polarized gauge boson. This is because no combination of fields containing ${\\bar {c}}$", null, "is annihilated by sB and we have added to the Lagrangian a gauge breaking term that is equal up to a divergence to\n\n$s_{B}\\left({\\bar {c}}\\left(i\\partial ^{\\mu }A_{\\mu }-{\\tfrac {1}{2}}\\alpha _{0}s_{B}{\\bar {c}}\\right)\\right).$", null, "Likewise, there are two kinds of quanta that will lie entirely in the image of the BRST operator: those of the Faddeev–Popov ghost c and the scalar field B, which is \"eaten\" by completing the square in the functional integral to become the backward polarized gauge boson. These are the four types of \"unphysical\" quanta which will not appear in the asymptotic states of a perturbative calculation—if we get our quantization rules right.\n\nThe anti-ghost is taken to be a Lorentz scalar for the sake of Poincaré invariance in $-i(\\partial ^{\\mu }{\\bar {c}})D_{\\mu }c$", null, ". However, its (anti-)commutation law relative to c—i. e., its quantization prescription, which ignores the spin-statistics theorem by giving Fermi–Dirac statistics to a spin-0 particle—will be given by the requirement that the inner product on our Fock space of asymptotic states be singular along directions corresponding to the raising and lowering operators of some combination of non-BRST-closed and BRST-exact fields. This last statement is the key to \"BRST quantization\", as opposed to mere \"BRST symmetry\" or \"BRST transformation\".\n\n(Needs to be completed in the language of BRST cohomology, with reference to the Kugo–Ojima treatment of asymptotic Fock space.)\n\n## Gauge bundles and the vertical ideal\n\nIn order to do the BRST method justice, we must switch from the \"algebra-valued fields on Minkowski space\" picture typical of quantum field theory texts (and of the above exposition) to the language of fiber bundles, in which there are two quite different ways to look at a gauge transformation: as a change of local section (also known in general relativity as a passive transformation) or as the pullback of the field configuration along a vertical diffeomorphism of the principal bundle. It is the latter sort of gauge transformation that enters into the BRST method. Unlike a passive transformation, it is well-defined globally on a principal bundle with any structure group over an arbitrary manifold. (However, for concreteness and relevance to conventional QFT, this article will stick to the case of a principal gauge bundle with compact fiber over 4-dimensional Minkowski space.)\n\nA principal gauge bundle P over a 4-manifold M is locally isomorphic to U × F, where U ⊂ R4 and the fiber F is isomorphic to a Lie group G, the gauge group of the field theory (this is an isomorphism of manifold structures, not of group structures; there is no special surface in P corresponding to 1 in G, so it is more proper to say that the fiber F is a G-torsor). Thus, the (physical) principal gauge bundle is related to the (mathematical) principal G-bundle but has more structure. Its most basic property as a fiber bundle is the \"projection to the base space\" π : P → M, which defines the \"vertical\" directions on P (those lying within the fiber π−1(p) over each point p in M). As a gauge bundle it has a left action of G on P which respects the fiber structure, and as a principal bundle it also has a right action of G on P which also respects the fiber structure and commutes with the left action.\n\nThe left action of the structure group G on P corresponds to a mere change of coordinate system on an individual fiber. The (global) right action Rg : P → P for a fixed g in G corresponds to an actual automorphism of each fiber and hence to a map of P to itself. In order for P to qualify as a principal G-bundle, the global right action of each g in G must be an automorphism with respect to the manifold structure of P with a smooth dependence on g—i. e., a diffeomorphism P × G → P.\n\nThe existence of the global right action of the structure group picks out a special class of right invariant geometric objects on P—those which do not change when they are pulled back along Rg for all values of g in G. The most important right invariant objects on a principal bundle are the right invariant vector fields, which form an ideal ${\\mathfrak {E}}$", null, "of the Lie algebra of infinitesimal diffeomorphisms on P. Those vector fields on P which are both right invariant and vertical form an ideal $V{\\mathfrak {E}}$", null, "of ${\\mathfrak {E}}$", null, ", which has a relationship to the entire bundle P analogous to that of the Lie algebra ${\\mathfrak {g}}$", null, "of the gauge group G to the individual G-torsor fiber F.\n\nThe \"field theory\" of interest is defined in terms of a set of \"fields\" (smooth maps into various vector spaces) defined on a principal gauge bundle P. Different fields carry different representations of the gauge group G, and perhaps of other symmetry groups of the manifold such as the Poincaré group. One may define the space Pl of local polynomials in these fields and their derivatives. The fundamental Lagrangian density of one's theory is presumed to lie in the subspace Pl0 of polynomials which are real-valued and invariant under any unbroken non-gauge symmetry groups. It is also presumed to be invariant not only under the left action (passive coordinate transformations) and the global right action of the gauge group but also under local gauge transformationspullback along the infinitesimal diffeomorphism associated with an arbitrary choice of right invariant vertical vector field $\\epsilon \\in V{\\mathfrak {E}}$", null, ".\n\nIdentifying local gauge transformations with a particular subspace of vector fields on the manifold P equips us with a better framework for dealing with infinite-dimensional infinitesimals: differential geometry and the exterior calculus. The change in a scalar field under pullback along an infinitesimal automorphism is captured in the Lie derivative, and the notion of retaining only the term linear in the scale of the vector field is implemented by separating it into the inner derivative and the exterior derivative. (In this context, \"forms\" and the exterior calculus refer exclusively to degrees of freedom which are dual to vector fields on the gauge bundle, not to degrees of freedom expressed in (Greek) tensor indices on the base manifold or (Roman) matrix indices on the gauge algebra.)\n\nThe Lie derivative on a manifold is a globally well-defined operation in a way that the partial derivative is not. The proper generalization of Clairaut's theorem to the non-trivial manifold structure of P is given by the Lie bracket of vector fields and the nilpotence of the exterior derivative. And we obtain an essential tool for computation: the generalized Stokes theorem, which allows us to integrate by parts and drop the surface term as long as the integrand drops off rapidly enough in directions where there is an open boundary. (This is not a trivial assumption, but can be dealt with by renormalization techniques such as dimensional regularization as long as the surface term can be made gauge invariant.)\n\n## BRST formalism\n\nIn theoretical physics, the BRST formalism is a method of implementing first class constraints. The letters BRST stand for Becchi, Rouet, Stora, and (independently) Tyutin who discovered this formalism. It is a sophisticated method to deal with quantum physical theories with gauge invariance. For example, the BRST methods are often applied to gauge theory and quantized general relativity.\n\n### Quantum version\n\nThe space of states is not a Hilbert space (see below). This vector space is both Z2-graded and R-graded. If you wish, you may think of it as a Z2 × R-graded vector space. The former grading is the parity, which can either be even or odd. The latter grading is the ghost number. Note that it is R and not Z because unlike the classical case, we can have nonintegral ghost numbers. Operators acting upon this space are also Z2 × R-graded in the obvious manner. In particular, Q is odd and has a ghost number of 1.\n\nLet Hn be the subspace of all states with ghost number n. Then, Q restricted to Hn maps Hn to Hn+1. Since Q2 = 0, we have a cochain complex describing a cohomology.\n\nThe physical states are identified as elements of cohomology of the operator Q, i.e. as vectors in Ker(Qn+1)/Im(Qn). The BRST theory is in fact linked to the standard resolution in Lie algebra cohomology.\n\nRecall that the space of states is Z2-graded. If A is a pure graded operator, then the BRST transformation maps A to [QA) where [ , ) is the supercommutator. BRST-invariant operators are operators for which [QA) = 0. Since the operators are also graded by ghost numbers, this BRST transformation also forms a cohomology for the operators since [Q, [QA)) = 0.\n\nAlthough the BRST formalism is more general than the Faddeev-Popov gauge fixing, in the special case where it is derived from it, the BRST operator is also useful to obtain the right Jacobian associated with constraints that gauge-fix the symmetry.\n\nThe BRST is a supersymmetry. It generates the Lie superalgebra with a zero-dimensional even part and a one-dimensional odd part spanned by Q. [QQ) = {QQ} = 0 where [ , ) is the Lie superbracket (i.e. Q2 = 0). This means Q acts as an antiderivation.\n\nBecause Q is Hermitian and its square is zero but Q itself is nonzero, this means the vector space of all states prior to the cohomological reduction has an indefinite norm! This means it is not a Hilbert space.\n\nFor more general flows which can't be described by first class constraints, see Batalin–Vilkovisky formalism.\n\n### Example\n\nFor the special case of gauge theories (of the usual kind described by sections of a principal G-bundle) with a quantum connection form A, a BRST charge (sometimes also a BRS charge) is an operator usually denoted Q.\n\nLet the ${\\mathfrak {g}}$", null, "-valued gauge fixing conditions be $G=\\xi \\partial ^{\\mu }A_{\\mu }$", null, "where ξ is a positive number determining the gauge. There are many other possible gauge fixings, but they will not be covered here. The fields are the ${\\mathfrak {g}}$", null, "-valued connection form A, ${\\mathfrak {g}}$", null, "-valued scalar field with fermionic statistics, b and c and a ${\\mathfrak {g}}$", null, "-valued scalar field with bosonic statistics B. c deals with the gauge transformations wheareas b and B deal with the gauge fixings. There actually are some subtleties associated with the gauge fixing due to Gribov ambiguities but they will not be covered here.\n\n$QA=Dc$", null, "where D is the covariant derivative.\n\n$Qc={\\tfrac {i}{2}}[c,c]_{L}$", null, "where [ , ]L is the Lie bracket, NOT the commutator.\n\n$QB=0$", null, "$Qb=B$", null, "Q is an antiderivation.\n\nThe BRST Lagrangian density\n\n${\\mathcal {L}}=-{\\frac {1}{4g^{2}}}\\operatorname {Tr} [F^{\\mu \\nu }F_{\\mu \\nu }]+{1 \\over 2g^{2}}\\operatorname {Tr} [BB]-{1 \\over g^{2}}\\operatorname {Tr} [BG]-{\\xi \\over g^{2}}\\operatorname {Tr} [\\partial ^{\\mu }bD_{\\mu }c]$", null, "While the Lagrangian density isn't BRST invariant, its integral over all of spacetime, the action is.\n\nThe operator Q is defined as\n\n$Q=c^{i}\\left(L_{i}-{\\frac {1}{2}}{{f_{i}}^{j}}_{k}b_{j}c^{k}\\right)$", null, "" ]	[ null, "https://demo.formulasearchengine.com/index.php", null, "https://demo.formulasearchengine.com/index.php", null, 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https://www.gradesaver.com/textbooks/math/calculus/university-calculus-early-transcendentals-3rd-edition/chapter-3-section-3-11-linearization-and-differentials-exercises-page-198/13	[ "## University Calculus: Early Transcendentals (3rd Edition)\n\n$L(x)=-x+1$\n$f(x)=e^{-x}$, $L(x)=f(a)+f'(a)(x-a)$ $f'(x)=-e^{-x}$ $x_{0}=0$ $f(0)={e^0}=1$ $f'(0)=-e^{-0}=-1$ then $L(x)=1-1(x-0)=-x+1$ $L(x)=-x+1$ Thus, the final answer is: $L(x)=-x+1$" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.73864955,"math_prob":1.00001,"size":458,"snap":"2019-51-2020-05","text_gpt3_token_len":167,"char_repetition_ratio":0.13876653,"word_repetition_ratio":0.0,"special_character_ratio":0.3973799,"punctuation_ratio":0.06363636,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.00001,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-12-06T10:25:33Z\",\"WARC-Record-ID\":\"<urn:uuid:d40010bf-46f9-4637-a3dd-5d79aae3f1a9>\",\"Content-Length\":\"78270\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ad9986bc-a82e-4446-9679-2d1157153002>\",\"WARC-Concurrent-To\":\"<urn:uuid:a2c8d0c5-2b53-4f5a-9fe3-64e82fdf3f7e>\",\"WARC-IP-Address\":\"54.210.73.90\",\"WARC-Target-URI\":\"https://www.gradesaver.com/textbooks/math/calculus/university-calculus-early-transcendentals-3rd-edition/chapter-3-section-3-11-linearization-and-differentials-exercises-page-198/13\",\"WARC-Payload-Digest\":\"sha1:4SNGO2SUGUBQCGYJE5ZAS4A5T3UIUK5J\",\"WARC-Block-Digest\":\"sha1:D2DHSXYIOSFUA3LWMNZLD25ZQZ33VQ4W\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-51/CC-MAIN-2019-51_segments_1575540487789.39_warc_CC-MAIN-20191206095914-20191206123914-00353.warc.gz\"}"}
https://calculat.io/en/number/decimal-as-a-fraction/3.825	[ "# Decimal 3.825 as a fraction\n\n## What is 3.825 as a fraction in simplest form?\n\nAnswer: Decimal 3.825 as a fraction is 3 33/40\n3.825=\n153\n/\n40\n=3\n33\n/\n40\n\nThe fractional equivalent of 3.825 is 3 33/40\n\n## Explanation of 3.825 Decimal to Fraction Conversion\n\nIn order to find the simplest fraction form of 3.825 first we need to write out decimal number 3.825 as fraction. Any number can be written as a fraction quite simply - just divide it by 1:\n\n3.825 =\n3.825\n/\n1\n\nNext we need to get rid of the decimal part in our numerator so that the numerator in our fraction becomes a whole number. To do so we need to multiply both numerator and denominator by 1000 (because there are 3 digits after the decimal point in the number 3.825):\n\n3.825 × 1000\n/\n1 × 1000\n=\n3825\n/\n1000\n\nNow we need to simplify our fraction to its simplest form. To do so we have to find the Greatest Common Factor (GCF) of 3825 and 1000. To find the GCF of 2 numbers you can use our GCF Calculator . GCF of 1000 and 3825 is 25. So in order to simplify our fraction we need to divide both numerator and denominator by 25:\n\n3825 ÷ 25\n/\n1000 ÷ 25\n=\n153\n/\n40\n=3\n33\n/\n40\n\nThat's it! 3.825 as a fraction in it's simplest form is equal to 3 33/40.\n\n## About \"Decimal to Fraction Converter\" Calculator\n\nThis online Decimal to Fraction converter is a tool that can help you convert any decimal number into its fractional form quickly and easily. For example, it can help you find out what is 3.825 as a fraction in simplest form? (The answer is: 3 33/40). It is especially useful for those who need to work with fractions in their daily lives or in academic or professional settings.\n\nTo use this converter, all you need to do is enter the decimal number you want to convert into the designated field, for example, '3.825'. Once you have entered the decimal, hit the 'Convert' button to start the conversion process.\n\nThe converter will then show you the fractional equivalent of the decimal number you entered, as well as provide a step-by-step explanation of the conversion process. Additionally, the final fraction will be simplified to its simplest form using the greatest common factor (GCF).\n\nFor example, if you enter '3.825' into the converter, it will show you that this decimal is equivalent to the fraction '3 33/40'. It will also explain how it arrived at this answer, showing you the steps involved in the conversion process.\n\nOverall, the online Decimal to Fraction converter is an essential tool for anyone who needs to work with fractions, whether in everyday life or in an academic or professional context. It is fast, easy to use, and provides accurate results, making it a valuable resource for anyone who needs to convert decimals to fractions.\n\n## FAQ\n\n### What is 3.825 as a fraction in simplest form?\n\nDecimal 3.825 as a fraction is 3 33/40" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.85434616,"math_prob":0.985541,"size":3098,"snap":"2023-40-2023-50","text_gpt3_token_len":972,"char_repetition_ratio":0.1612799,"word_repetition_ratio":0.034358047,"special_character_ratio":0.34958038,"punctuation_ratio":0.11158192,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9969822,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-06T22:14:28Z\",\"WARC-Record-ID\":\"<urn:uuid:b42bfe0d-e732-4814-bb00-7dbea42ed418>\",\"Content-Length\":\"83920\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2b6464fc-2126-4608-aeff-e42413b80797>\",\"WARC-Concurrent-To\":\"<urn:uuid:9027fb05-da85-44dd-8529-5b4b228a994c>\",\"WARC-IP-Address\":\"104.21.17.174\",\"WARC-Target-URI\":\"https://calculat.io/en/number/decimal-as-a-fraction/3.825\",\"WARC-Payload-Digest\":\"sha1:J4IL4MBMZDCKIKEZOBDK3LFVLCKMUAEI\",\"WARC-Block-Digest\":\"sha1:CPP3TARHDVRSXBR5BCURBLM2AWFX6LXB\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100603.33_warc_CC-MAIN-20231206194439-20231206224439-00032.warc.gz\"}"}
http://perplexus.info/show.php?pid=10987&cid=58651	[ "", null, "All about flooble \| fun stuff \| Get a free chatterbox \| Free JavaScript \| Avatars", null, "", null, "perplexus dot info", null, "", null, "Square Egyptians (Posted on 2017-06-28)", null, "Express 1/2 as the sum of Square Egyptian Numbers with distinct denominators under 362.\n\nI.e. reciprocals of perfect squares.\n\n No Solution Yet Submitted by Jer No Rating", null, "Comments: ( Back to comment list \| You must be logged in to post comments.)", null, "computer solution Comment 1 of 1\n1/2 = 1/4 + 1/9 + 1/16 + 1/25 + 1/49 + 1/144 + 1/225 + 1/400 + 1/784 + 1/1225\n\nThe denominators are the squares of:\n\n2 3 4 5 7 12 15 20 28 35\n\nas found by the below VB program and verified by using exact rational fractions in UBASIC.\n\nThe program produces a total that prints out as exactly .5 but did not register as equal to .5 within the program, due to rounding, and thus necessitated the difference to be below the small threshhold of absolute value difference shown in the program, rather than testing if = 0.5.\n\nThe first portion of the program determined how far back from 1/36^2 would be necessary and found that the first term must be 1/2^2, otherwise the total would be impossible. It was only after this was run that the rest of the program was added, to find the actual result.\n\nDefDbl A-Z\nDim crlf\\$, h(36), tot\n\nForm1.Visible = True\n\nText1.Text = \"\"\ncrlf = Chr\\$(13) + Chr\\$(10)\n\nFor rt = 36 To 1 Step -1\nptot = tot\ntot = tot + 1 / (rt * rt)\nIf tot >= 0.5 Then Text1.Text = Text1.Text & rt & Str(tot) & Str(ptot) & crlf & crlf: Exit For\nNext\n\nh(1) = 4: tot = 1 / 4\n\nText1.Text = Text1.Text & crlf & \" done\"\n\nEnd Sub\n\nDoEvents\nst = Int(Sqr(h(wh - 1)) + 0.5) + 1\nFor i = st To 36\nh(wh) = i * i\nsavetot = tot\ntot = tot + 1 / (i * i)\nIf Abs(tot - 0.5) < 0.000000000001 Then\nText1.Text = Text1.Text & tot & crlf & \" \"\nFor j = 1 To wh\nText1.Text = Text1.Text & \" + 1/\" & h(j)\nNext\nText1.Text = Text1.Text & crlf\nFor j = 1 To wh\nText1.Text = Text1.Text & Str(Sqr(h(j)))\nNext\nText1.Text = Text1.Text & crlf\n\nEnd If\nIf i < 36 And tot < 0.5001 Then\nEnd If\ntot = savetot\nNext\nEnd Sub\n\n2 .617538519845469 .367538519845469\n\n0.5\n+ 1/4 + 1/9 + 1/16 + 1/25 + 1/49 + 1/144 + 1/225 + 1/400 + 1/784 + 1/1225\n2 3 4 5 7 12 15 20 28 35\n\n Posted by Charlie on 2017-06-28 13:59:04", null, "Please log in:\n\n Search: Search body:\nForums (4)" ]	[ null, "http://perplexus.info/images/flooble.gif", null, "http://perplexus.info/images/dot.gif", null, "http://perplexus.info/images/dot.gif", null, "http://perplexus.info/images/dot.gif", null, "http://perplexus.info/images/dot.gif", null, "http://perplexus.info/images/perplexus/diff/3.gif", null, "http://perplexus.info/images/dot_black.gif", null, "http://perplexus.info/images/perplexus/icons/solution.gif", null, "http://perplexus.info/images/dot.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7617589,"math_prob":0.993482,"size":1939,"snap":"2019-51-2020-05","text_gpt3_token_len":730,"char_repetition_ratio":0.16020672,"word_repetition_ratio":0.17037037,"special_character_ratio":0.4368231,"punctuation_ratio":0.10633484,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98525244,"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,null,null,null,null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-12-12T07:47:41Z\",\"WARC-Record-ID\":\"<urn:uuid:bc4d843b-76f1-4eca-9c61-a325262baf95>\",\"Content-Length\":\"14963\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:c853b8c7-bc29-4dc3-b5fa-9c929f1755dc>\",\"WARC-Concurrent-To\":\"<urn:uuid:cf9021ac-9377-4515-8636-e9bf6a3836f2>\",\"WARC-IP-Address\":\"216.67.228.67\",\"WARC-Target-URI\":\"http://perplexus.info/show.php?pid=10987&cid=58651\",\"WARC-Payload-Digest\":\"sha1:UP3WZG576FL5AYMO2IORQGEXZ6IDR25P\",\"WARC-Block-Digest\":\"sha1:EPOED52R3LV5BNQUQ2VWUY7Z25WKVM63\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-51/CC-MAIN-2019-51_segments_1575540542644.69_warc_CC-MAIN-20191212074623-20191212102623-00149.warc.gz\"}"}
https://carloscaballero.es/refactoring-a-real-case/	[ "Angular\nJavaScript\nNestJS\nNodeJS\nTypeScript\nUI-UX\nZExtra\n\n# Refactoring a real-case\n\n6 min read\n\nAs a result of publishing the series of post of the clock-in/out system, I had the need to write a function to import the data from a large XLS file to a Postgres database using the ORM (TypeORM).\n\nThis function was written quickly as a script without thinking too much about the quality of the code, but simply \"work\". However, once the system is working and a test suite is available, it is a good idea to go through a constant refactoring to improve the code.\n\nNext I will show different refactoring steps that I have applied to this function to improve the quality of the code. Therefore, the function that we are going to improve is shown first.\n\n``````\nconst xls = require('node-xlsx').parse(__dirname + '/schedule.xls');\n\nexport function XSLToJson() {\nconst schedulers = [];\nconst users = [];\n\nxls.data.slice(2).forEach(teacher => {\nlet schedule = {};\nlet user = {};\n\nlet name = teacher.split(',');\nif (name.length > 1) {\nname = name + ', ' + name.trim() + '.';\n}\nuser = {\nuid: name,\nname,\n};\nusers.push(user);\nconst monday = [...teacher.slice(1, 7), ...teacher.slice(8, 14)];\nconst tuesday = [...teacher.slice(14, 20), ...teacher.slice(21, 27)];\nconst wednesday = [...teacher.slice(27, 33), ...teacher.slice(34, 40)];\nconst thursday = [...teacher.slice(40, 46), ...teacher.slice(47, 53)];\nconst friday = [...teacher.slice(53, 59), ...teacher.slice(60, 66)];\n\n[monday, tuesday, wednesday, thursday, friday].forEach((day, numberOfDay) =>\nday.forEach((room, hour) => {\nif (room) {\nschedule = {\nuser: {\nuid: name,\n},\nroom: room.replace(/\\n/g, ' - '),\nday: numberOfDay,\nhour,\n};\nschedulers.push(schedule);\n}\n}),\n);\n});\nreturn [schedulers, users];\n}\n\n``````\n\nFirstly, we can observe that there are different concepts in the same function, this gives us the warning that we need to extract functions. The following functions can be identified in the code:\n\n• normalizedUID: this function is used to generate the name/uid (string) of each teacher.\n• getDaysByTeacher: Returns an array in which each position represents each user's day. Each of the days is also represented by an array. Hence, this function returns an array of arrays in which we will have organized the weekly working hours of each user (fixed schedules).\n• factorySchedule: Create a schedule object from another object composed of the room, hour and number of day.\n\nimport.ts\n\n``````\nfunction normalizeUID(teacher): string {\nlet name = teacher.split(',');\nif (name.length > 1) {\nname = name + ', ' + name.trim() + '.';\n}\nreturn name;\n}\nfunction getDaysByTeacher(teacher): [any[], any[], any[], any[], any[]] {\nreturn [\n[...teacher.slice(1, 7), ...teacher.slice(8, 14)],\n[...teacher.slice(14, 20), ...teacher.slice(21, 27)],\n[...teacher.slice(27, 33), ...teacher.slice(34, 40)],\n[...teacher.slice(40, 46), ...teacher.slice(47, 53)],\n[...teacher.slice(53, 59), ...teacher.slice(60, 66)],\n];\n}\nfunction factorySchedule({ name, room, hour, numberOfDay }) {\nreturn {\nuser: {\nuid: name,\n},\nroom: room.replace(/\\n/g, ' - '),\nday: numberOfDay,\nhour,\n};\n}\n``````\n\nThe main code would be as follows after the extraction of functions:\n\n``````\nexport function XSLToJson() {\nconst schedulers = [];\nconst users = [];\n\nxls.data.slice(2).forEach(teacher => {\nlet schedule = {};\nlet user = {};\nconst name = normalizeUID(teacher);\nuser = {\nuid: name,\nname,\n};\nusers.push(user);\nconst days = getDaysByTeacher(teacher);\n\ndays.forEach((day, numberOfDay) =>\nday.forEach((room, hour) => {\nif (room) {\nconst schedule = factorySchedule({ name, room, hour, numberOfDay });\nschedulers.push(schedule);\n}\n}),\n);\n});\nreturn [schedulers, users];\n}\n\n``````\n\nBefore going on to address the next phase of refactoring, we will eliminate the variables that are understood using the domain of the problem, also the temporary variables will be removed.\n\n``````\nexport function XSLToJson() {\nconst schedulers = [];\nconst users = [];\n\nxls.data.slice(2).forEach(teacher => {\nconst name = normalizeUID(teacher);\nconst user = {\nuid: name,\nname,\n};\nusers.push(user);\nconst days = getDaysByTeacher(teacher);\n\ndays.forEach((day, numberOfDay) =>\nday.forEach((room, hour) => {\nif (room) {\nconst schedule = factorySchedule({ name, room, hour, numberOfDay });\nschedulers.push(schedule);\n}\n}),\n);\n});\nreturn [schedulers, users];\n}\n``````\n\nIn this moment, we see that we are making use of one of the most powerful features of JavaScript, functions as a parameter in the forEachmethods. But really, the forEach methods are only acting as syntactic sugar against the classical for of the imperative programming. Therefore, we will be modifying each forEach method by the most appropriate one according to the task that they perform.\n\nThe first forEach loop that we are going to address is the following:\n\n``````\ndays.forEach((day, numberOfDay) =>\nday.forEach((room, hour) => {\nif (room) {\nconst schedule = factorySchedule({ name, room, hour, numberOfDay });\nschedulers.push(schedule);\n}\n}),\n);\n``````\n\nThis loop can be transformed into the method reduce because of what we want to get a single value (an array) after iterating through each of the days. In addition, the schedule variable can be omitted and the factorSchedule method can be invoked directly since the creation of this variable does not add semantic value in the code.\n\n``````\ndays.forEach((day, numberOfDay) => {\nschedulers = day.reduce((acc, room, hour) => {\nif (room) {\nacc.push(factorySchedule({ name, room, hour, numberOfDay }));\n}\nreturn acc;\n}, []);\n});\n``````\n\nIf we observe the function performed in the body of the method reduce, it is an if-else which can be transformed into a ternary operation that we can combine with the potential of immutability (another characteristic of functional programming) leaving the code as follows:\n\n``````\ndays.forEach((day, numberOfDay) => {\nschedulers = day.reduce((acc, room, hour) => {\nreturn room\n? [...acc, factorySchedule({ name, room, hour, numberOfDay })]\n: acc;\n}, []);\n});\n``````\n\nAnother small step in the refactoring would be to omit the explicit invocation of return when using fat-arrow (lambda functions) of JavaScript, leaving the code as follows:\n\n``````\ndays.forEach((day, numberOfDay) => {\nschedulers = day.reduce(\n(acc, room, hour) =>\nroom ? [...acc, factorySchedule({ name, room, hour, numberOfDay })] : acc,\n[],\n);\n});\n``````\n\nIf we look at the code, the next natural step will be to replace the forEach method that includes the reduce by another method reduce. It may seem complicated the code that will generate since we are using the features of fat-arrow and spread operator (...) which if you never are used before can seem abstract (we will solve it by creating functions that allow you to decompose the task).\n\n``````\nschedulers1 = days.reduce(\n(acc, day, numberOfDay) => [\n...acc,\nday.reduce(\n(schedulers, room, hour) =>\ngenerateScheduler({ room, schedulers, hour, numberOfDay, name }),\n[],\n),\n],\n[],\n);\n``````\n\nTherefore, the next step is to build small functions that perform the task that is done within each of the reduce. But we have a small problem, we need to pass extra parameters to each of the callbacks of the reduce methods. This problem is easily solved in JavaScript with different techniques such as overwriting the scope (this), using variables from a larger scope, or your own such as the bind, apply or call methods.\n\nIn our case, we are going to create several small functions that combined with the bind method will solve the problem of passing parameters in the callbacks. In this way, the resulting code until this moment would be the following:\n\n``````\nexport function XSLToJson() {\nconst users = [];\nlet schedulers = [];\nxls.data.slice(2).forEach(teacher => {\nconst name = normalizeUID(teacher);\nconst user = {\nuid: name,\nname,\n};\nusers.push(user);\nconst days = getDaysByTeacher(teacher);\n\nschedulers = days.reduce(generateSchedulers.bind(null, name), []);\n});\nreturn [schedulers, users];\n}\nfunction generateSchedulers(name, schedulers, day, numberOfDay) {\nreturn [\n...schedulers,\nday.reduce(generateScheduler.bind(null, numberOfDay, name), []),\n];\n}\n\nfunction generateScheduler(numberOfDay, name, schedulers, room, hour) {\nreturn room\n? [...schedulers, factorySchedule({ name, room, hour, numberOfDay })]\n: schedulers;\n}\n\n``````\n\nThe next step is to address the forEach method that encompasses the entire main method. The elimination of this method will allow us to eliminate the two variables that act as accumulators (users and schedulers). In such a way, that the method reduce will receive as an accumulator an array that contains two arrays (schedulers and users), which will be unstructured to be able to be concatenated easily in the method reduced.\n\nAfter refactoring the code we find the following version:\n\n``````\nexport function XSLToJson() {\nreturn xls.data.slice(2).reduce(\n([schedulers, users], teacher) => {\nconst name = normalizeUID(teacher);\nconst user = {\nuid: name,\nname,\n};\nconst days = getDaysByTeacher(teacher);\nconst scheduler = days.reduce(generateSchedulers.bind(null, name), []);\nreturn [[...schedulers, scheduler], [...users, user]];\n},\n[[], []],\n);\n}\n\n``````\n\nIn the same way as we did in the previous case, the method reduce receives a callback that is the one that performs the conversion to JSON, and finally we rename the content of the file to a variable called xls. So the partial result is the following:\n\n``````\n\nexport function XSLToJson() {\nreturn xls.reduce(convertToJson, [[], []]);\n}\nfunction convertToJson([schedulers, users], teacher) {\nconst name = normalizeUID(teacher);\nconst user = {\nuid: name,\nname,\n};\nconst days = getDaysByTeacher(teacher);\nconst scheduler = days.reduce(generateSchedulers.bind(null, name), []);\nreturn [[...schedulers, scheduler], [...users, user]];\n}\n``````\n\nFinally the result of the script after making these changes is shown below:\n\n``````\nconst xls = require('node-xlsx')\n.parse(__dirname + '/schedule.xls')\n.data.slice(2);\n\nexport function XSLToJson() {\nreturn xls.reduce(convertToJson, [[], []]);\n}\nfunction convertToJson([schedulers, users], teacher) {\nconst name = normalizeUID(teacher);\nconst user = {\nuid: name,\nname,\n};\nconst days = getDaysByTeacher(teacher);\nconst scheduler = days.reduce(generateSchedulers.bind(null, name), []);\nreturn [[...schedulers, scheduler], [...users, user]];\n}\nfunction normalizeUID(teacher): string {\nlet name = teacher.split(',');\nif (name.length > 1) {\nname = name + ', ' + name.trim() + '.';\n}\nreturn name;\n}\nfunction getDaysByTeacher(teacher): [any[], any[], any[], any[], any[]] {\nreturn [\n[...teacher.slice(1, 7), ...teacher.slice(8, 14)],\n[...teacher.slice(14, 20), ...teacher.slice(21, 27)],\n[...teacher.slice(27, 33), ...teacher.slice(34, 40)],\n[...teacher.slice(40, 46), ...teacher.slice(47, 53)],\n[...teacher.slice(53, 59), ...teacher.slice(60, 66)],\n];\n}\nfunction factorySchedule({ name, room, hour, numberOfDay }) {\nreturn {\nuser: {\nuid: name,\n},\nroom: room.replace(/\\n/g, ' - '),\nday: numberOfDay,\nhour,\n};\n}\n\nfunction generateSchedulers(name, schedulers, day, numberOfDay) {\nreturn [\n...schedulers,\nday.reduce(generateScheduler.bind(null, numberOfDay, name), []),\n];\n}\n\nfunction generateScheduler(numberOfDay, name, schedulers, room, hour) {\nreturn room\n? [...schedulers, factorySchedule({ name, room, hour, numberOfDay })]\n: schedulers;\n}\n``````\n\nSummary\n\n1. Extract functions.\n2. Name the functions and variables must be an identification value.\n3. Use the built-in language methods for your use (forEach against of reduce).\n4. Abstract complexity and details of the code to auxiliary functions of lower level." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.80216783,"math_prob":0.9039924,"size":11369,"snap":"2019-13-2019-22","text_gpt3_token_len":2741,"char_repetition_ratio":0.19093709,"word_repetition_ratio":0.35269207,"special_character_ratio":0.29307768,"punctuation_ratio":0.29353017,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9785381,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-05-23T09:14:10Z\",\"WARC-Record-ID\":\"<urn:uuid:498e7f62-9900-4fbe-ba49-c4354bd414eb>\",\"Content-Length\":\"28877\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:4cb69b63-7408-4f0f-9074-adf84d86f5f8>\",\"WARC-Concurrent-To\":\"<urn:uuid:3307538d-9e52-474e-af7c-48ac8c4cb432>\",\"WARC-IP-Address\":\"51.77.141.19\",\"WARC-Target-URI\":\"https://carloscaballero.es/refactoring-a-real-case/\",\"WARC-Payload-Digest\":\"sha1:IOCCQTUPQWZJSNQYCQKQIKI5JLJMCD2L\",\"WARC-Block-Digest\":\"sha1:UXKC3SEP5OU5YPBSWZ5GPN4WZ47HU7FL\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-22/CC-MAIN-2019-22_segments_1558232257197.14_warc_CC-MAIN-20190523083722-20190523105722-00133.warc.gz\"}"}
https://chem.libretexts.org/Bookshelves/Ancillary_Materials/Worksheets/Worksheets%3A_General_Chemistry/Worksheets%3A_General_Chemistry_(Traditional)/Point_Groups_2_(Worksheet)	[ "# Point Groups 2 (Worksheet)\n\nName: ______________________________\n\nSection: _____________________________\n\nStudent ID#:__________________________\n\nWork in groups on these problems. You should try to answer the questions without referring to your textbook. If you get stuck, try asking another group for help.\n\nDetermine the point group assignment for the following:", null, "", null, "", null, "", null, "", null, "", null, "", null, "acetylene", null, "", null, "", null, "", null, "", null, "", null, "$$PtCl_2I_2^{2-}$$ (cis and trans) $$XeOF_4$$ $$AsF_5$$ $$PF_3$$ $$SCl_4$$ benzene $$BrF_3$$ $$S_8$$ (crown) Staggered ethane $$I_3^-$$", null, "", null, "$$Mo(CO)_6$$ A black cat at its highest symmetry $$CS_2$$ A three-legged stool $$PFCl_2$$ A rectangular brick" ]	[ null, "https://chem.libretexts.org/@api/deki/files/23240/ethylene.JPG", null, "https://chem.libretexts.org/@api/deki/files/23241/12cisdichlorofluoro.JPG", null, "https://chem.libretexts.org/@api/deki/files/23242/12transdichloro.JPG", null, "https://chem.libretexts.org/@api/deki/files/23243/11dichloro.JPG", null, "https://chem.libretexts.org/@api/deki/files/23244/dichloromethane.JPG", null, "https://chem.libretexts.org/@api/deki/files/23245/12cisdichloro.JPG", null, "https://chem.libretexts.org/@api/deki/files/23246/N2F2.JPG", null, "https://chem.libretexts.org/@api/deki/files/23247/C3H4Cl2.JPG", null, "https://chem.libretexts.org/@api/deki/files/23248/MoCO5F.JPG", null, "https://chem.libretexts.org/@api/deki/files/23249/dichlorobenzene.JPG", null, "https://chem.libretexts.org/@api/deki/files/23250/COD.JPG", null, "https://chem.libretexts.org/@api/deki/files/23251/phen.JPG", null, "https://chem.libretexts.org/@api/deki/files/23252/ferrocene.JPG", null, "https://chem.libretexts.org/@api/deki/files/23253/spiro.JPG", null, "https://chem.libretexts.org/@api/deki/files/23254/acetonitrile.JPG", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7358762,"math_prob":0.999741,"size":663,"snap":"2020-24-2020-29","text_gpt3_token_len":200,"char_repetition_ratio":0.20182094,"word_repetition_ratio":0.0,"special_character_ratio":0.43137255,"punctuation_ratio":0.08421053,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9986426,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],"im_url_duplicate_count":[null,6,null,6,null,6,null,6,null,6,null,6,null,6,null,6,null,6,null,7,null,6,null,6,null,6,null,6,null,6,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-06-01T09:27:32Z\",\"WARC-Record-ID\":\"<urn:uuid:640f9bfb-3c34-4f36-a558-25ae9b0736bc>\",\"Content-Length\":\"102259\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:90f01f89-d17e-4c31-b738-06319b77e99f>\",\"WARC-Concurrent-To\":\"<urn:uuid:46c52e3f-e10a-40f6-9fe4-7f829da86d50>\",\"WARC-IP-Address\":\"13.249.44.60\",\"WARC-Target-URI\":\"https://chem.libretexts.org/Bookshelves/Ancillary_Materials/Worksheets/Worksheets%3A_General_Chemistry/Worksheets%3A_General_Chemistry_(Traditional)/Point_Groups_2_(Worksheet)\",\"WARC-Payload-Digest\":\"sha1:5UEEADNTVE7BJ7BNHE5SQIFJMNVGOMXV\",\"WARC-Block-Digest\":\"sha1:TCSMQEWXWG2TKAPIPMROBDLEPYXMXN45\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-24/CC-MAIN-2020-24_segments_1590347415315.43_warc_CC-MAIN-20200601071242-20200601101242-00400.warc.gz\"}"}
https://epub.ub.uni-muenchen.de/5140/	[ "", null, "", null, "Gabriel, Wilfried; Taylor, B. E.; Kirsch-Prokosch, Susanne (1987): Cladoceran birth and death rates estimates. Experimental comparisons of egg-ratio methods. In: Freshwater Biology, Vol. 18, No. 2: pp. 361-372", null, "", null, "Preview\n3MB\n\n### Abstract\n\nI. Birth and death rates of natural cladoceran populations cannot be measured directly. Estimates of these population parameters must be calculated using methods that make assumptions about the form of population growth. These methods generally assume that the population has a stable age distribution. 2. To assess the effect of variable age distributions, we tested six egg ratio methods for estimating birth and death rates with data from thirty-seven laboratory populations of Daphnia pulicaria. The populations were grown under constant conditions, but the initial age distributions and egg ratios of the populations varied. Actual death rates were virtually zero, so the difference between the estimated and actual death rates measured the error in both birth and death rate estimates. 3. The results demonstrate that unstable population structures may produce large errors in the birth and death rates estimated by any of these methods. Among the methods tested, Taylor and Slatkin's formula and Paloheimo's formula were most reliable for the experimental data. 4. Further analyses of three of the methods were made using computer simulations of growth of age-structured populations with initially unstable age distributions. These analyses show that the time interval between sampling strongly influences the reliability of birth and death rate estimates. At a sampling interval of 2.5 days (equal to the duration of the egg stage), Paloheimo's formula was most accurate. At longer intervals (7.5–10 days), Taylor and Slatkin's formula which includes information on population structure was most accurate.", null, "", null, "" ]	[ null, "https://epub.ub.uni-muenchen.de/images/epub_logo_de.png", null, "https://epub.ub.uni-muenchen.de/images/header_open-access-lmu_mobile_en.png", null, "https://epub.ub.uni-muenchen.de/5140/1.hassmallThumbnailVersion/011.pdf", null, "https://epub.ub.uni-muenchen.de/5140/1.haspreviewThumbnailVersion/011.pdf", null, "https://epub.ub.uni-muenchen.de/images/autor-author.svg", null, "https://epub.ub.uni-muenchen.de/images/export.svg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.91822344,"math_prob":0.9601298,"size":1834,"snap":"2019-43-2019-47","text_gpt3_token_len":364,"char_repetition_ratio":0.15409836,"word_repetition_ratio":0.007490637,"special_character_ratio":0.1908397,"punctuation_ratio":0.12852664,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9665825,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12],"im_url_duplicate_count":[null,null,null,null,null,3,null,3,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-11-12T12:06:38Z\",\"WARC-Record-ID\":\"<urn:uuid:f82194fa-23c9-4b6d-a77c-a87ad125cb31>\",\"Content-Length\":\"26544\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:8dbdbc52-92eb-48e7-8a9f-60757b933bc8>\",\"WARC-Concurrent-To\":\"<urn:uuid:a1ae2ee7-e644-41ba-9fb5-553832b3ec85>\",\"WARC-IP-Address\":\"141.84.147.156\",\"WARC-Target-URI\":\"https://epub.ub.uni-muenchen.de/5140/\",\"WARC-Payload-Digest\":\"sha1:A3NHKHQR7Z3JPKIBYARJQRXUQLL6POKJ\",\"WARC-Block-Digest\":\"sha1:OS3ELRFWD7EOTTTL4YTGHQNPZGFHWVWV\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-47/CC-MAIN-2019-47_segments_1573496665521.72_warc_CC-MAIN-20191112101343-20191112125343-00112.warc.gz\"}"}
https://www.wias-berlin.de/publications/wias-publ/run.jsp?template=abstract&type=Preprint&year=&number=2591	[ "WIAS Preprint No. 2591, (2019)\n\n# A class of second-order geometric quasilinear hyperbolic PDEs and their application in imaging science\n\nAuthors\n\n• Dong, Guozhi\nORCID: 0000-0002-9674-6143\n• Hintermüller, Michael\nORCID: 0000-0001-9471-2479\n• Zhang, Ye\n\n2010 Mathematics Subject Classification\n\n• 35L10 35L70 35L72\n\nKeywords\n\n• Quasilinear hyperbolic equation, geometric PDEs, total variation flow, mean curvature flow, level set, second-order dynamics, non-smooth and non-convex variational methods, image denoising, displacement error correction\n\nDOI\n\n10.20347/WIAS.PREPRINT.2591\n\nAbstract\n\nIn this paper, we study damped second-order dynamics, which are quasilinear hyperbolic partial differential equations (PDEs). This is inspired by the recent development of second-order damping systems for accelerating energy decay of gradient flows. We concentrate on two equations: one is a damped second-order total variation flow, which is primarily motivated by the application of image denoising; the other is a damped second-order mean curvature flow for level sets of scalar functions, which is related to a non-convex variational model capable of correcting displacement errors in image data (e.g. dejittering). For the former equation, we prove the existence and uniqueness of the solution. For the latter, we draw a connection between the equation and some second-order geometric PDEs evolving the hypersurfaces which are described by level sets of scalar functions, and show the existence and uniqueness of the solution for a regularized version of the equation. The latter is used in our algorithmic development. A general algorithm for numerical discretization of the two nonlinear PDEs is proposed and analyzed. Its efficiency is demonstrated by various numerical examples, where simulations on the behavior of solutions of the new equations and comparisons with first-order flows are also documented.\n\nAppeared in" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8811286,"math_prob":0.9645855,"size":1940,"snap":"2021-43-2021-49","text_gpt3_token_len":444,"char_repetition_ratio":0.11983471,"word_repetition_ratio":0.021978023,"special_character_ratio":0.2185567,"punctuation_ratio":0.12349398,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9905628,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-21T04:33:12Z\",\"WARC-Record-ID\":\"<urn:uuid:548222dc-bef6-4e4d-981f-402fd2939871>\",\"Content-Length\":\"37551\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:63304c3b-34c3-42d6-85bf-0cbc4ea966c4>\",\"WARC-Concurrent-To\":\"<urn:uuid:4b221e93-6be3-415d-a41e-8166bcc99751>\",\"WARC-IP-Address\":\"62.141.177.11\",\"WARC-Target-URI\":\"https://www.wias-berlin.de/publications/wias-publ/run.jsp?template=abstract&type=Preprint&year=&number=2591\",\"WARC-Payload-Digest\":\"sha1:KNU6HZPMCON6TDUXLLPT26ZKRZASZGM7\",\"WARC-Block-Digest\":\"sha1:LUDGGBZTPLTO4LDOUBROJXCC4HVCMEUY\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323585381.88_warc_CC-MAIN-20211021040342-20211021070342-00265.warc.gz\"}"}
https://mathcoachblog.com/2014/11/07/class-opener-day-44-statistics-clue-boxes/	[ "# Class Opener – Day 44 – Statistics Clue Boxes\n\nA problem I gave as review for our statistics test today became not only a source of conversation regarding vocabulary, but provided me some insight into the problem solving approaches of my students.\n\nHere’s the problem. A list of numbers is given, listed in order, with some numbers removed:", null, "The list has the following characteristics:\n\n• A mean of 76\n• A range of 32\n• An inter-quartile range of 21\n\nMany students quickly understood the last blank must be 92, due to the range, but then became stuck. As we’ve never explicity seen a problem like this before, the reactions from students was fascinating. Some pockets of students had no fear in drawing circles and arrows to break down the data set. Others preferred to talk ideas out, but without putting pen to paper this doesn’t lead to solutions right away. I was thrilled to see a few students step up and take the lead, and explain their ideas to others, which then led to breakthroughs. Identifying the positions of median and quartiles here lets us fill in one of the missing numbers:", null, "But a subset of my class was content to watch from afar, waiting for hints which they assumed would come. Or worse, tuning out until I presented an explanation to the class….which never came.\n\nAnd that last blank caused more trouble than I would have expected, as some students had trouble making the connection between the mean of a data set and the sum of its elements. To help with this, I asked struggling students to provide me with any 4 numbers which had a mean of 10 (making them different numbers). I asked students what I should be looking for to check accuracy besides computing the mean….and then, the light bulb! All lists need to add up to 40! So without explictly doing the empty blank problem in front of us, I sent students back to the board to think about this fact. And the results were satisfying, as many of my fringe students could now complete the task and explain their procedure to their peers.\n\nStudents need to understand math ideas in many forms, and the concept of mean here demonstrates this need. If you ask a student how to compute a mean, they most likely have little difficulty, and have had much practice:\n\nMean = sum of “scores” / count of “scores”\n\nBut in the missing numbers puzzle, the concept “felt” different and thus “new” to many students. For me, this is where many students struggle in math classrooms. Are we showing students how ideas and problems connect to big ideas? Or does each combination of an existing problem become treated like a new experience? It’s hard to break the pattern of students wanting specific rules for each type of math problem, when this is often the math conditioning they receive. But it’s worth the hard-fought battle.\n\nAnd if you had fun with the challenge at the start of this post, try the similar problem I give later as an assessment:", null, "" ]	[ null, "https://mathcoachblog.files.wordpress.com/2014/11/pic2.jpg", null, "https://mathcoachblog.files.wordpress.com/2014/11/pic1.jpg", null, "https://mathcoachblog.files.wordpress.com/2014/11/boxes.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.97331023,"math_prob":0.592234,"size":2851,"snap":"2019-51-2020-05","text_gpt3_token_len":597,"char_repetition_ratio":0.12539515,"word_repetition_ratio":0.0,"special_character_ratio":0.20904946,"punctuation_ratio":0.09219858,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95490193,"pos_list":[0,1,2,3,4,5,6],"im_url_duplicate_count":[null,4,null,4,null,4,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-12-11T13:57:53Z\",\"WARC-Record-ID\":\"<urn:uuid:31ef9010-fe08-46a5-a837-0c8a21b7ef9e>\",\"Content-Length\":\"88622\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:393653c7-5b26-4278-9fab-a1e537d30bca>\",\"WARC-Concurrent-To\":\"<urn:uuid:7ffb194e-c00e-4c41-9948-4c18f43a250b>\",\"WARC-IP-Address\":\"192.0.78.24\",\"WARC-Target-URI\":\"https://mathcoachblog.com/2014/11/07/class-opener-day-44-statistics-clue-boxes/\",\"WARC-Payload-Digest\":\"sha1:XGHPYXBPNM6VLFZUXGMTN2VNATIL2BNO\",\"WARC-Block-Digest\":\"sha1:X4WMDUF3UIVBVCW4FB4AW45Z4WTWKHRW\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-51/CC-MAIN-2019-51_segments_1575540531917.10_warc_CC-MAIN-20191211131640-20191211155640-00410.warc.gz\"}"}
https://apcz.umk.pl/czasopisma/index.php/TMNA/article/view/TMNA.2004.006	[ "### Differential inclusions on closed sets in Banach spaces with application to sweeping process\n\nHoucine Benabdellah\n\nDOI: http://dx.doi.org/10.12775/TMNA.2004.006\n\n#### Abstract\n\nThis paper deals with the existence of absolutely continuous solutions of a\ndifferential inclusion with state constraint in a separable Banach space%\n$$x( 0) =x_{0}, \\quad x( t) \\in C( t) ,\\quad \\dot{x}( t) \\in F( t,x( t) )$$\nwhere $C\\colon [ 0,a] \\rightarrow X$ is a multifunction with closed graph\n$G$ and $F\\colon G\\rightarrow X$ is a convex compact valued multifunction\nwhich is\nseparately measurable in $t\\in[ 0,a]$ and separately upper\nsemicontinuous in $x\\in X$. Application to a non convex sweeping process is\nalso considered.\n\n#### Keywords\n\nDifferential inclusions; Bouligand cone; Scorza-Dragoni theorem\n\nFULL TEXT\n\n### Refbacks\n\n• There are currently no refbacks." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.76024723,"math_prob":0.93505806,"size":1284,"snap":"2020-24-2020-29","text_gpt3_token_len":385,"char_repetition_ratio":0.0890625,"word_repetition_ratio":0.74871796,"special_character_ratio":0.27024922,"punctuation_ratio":0.10330579,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95514745,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-05-28T09:03:40Z\",\"WARC-Record-ID\":\"<urn:uuid:5bf2cbb6-356a-4ad3-aae6-618d73ddd0ca>\",\"Content-Length\":\"20621\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:72c5bb0b-0b8f-41f7-8812-fabd1ed1f69e>\",\"WARC-Concurrent-To\":\"<urn:uuid:9c736d34-1b89-4cb9-9511-319852314c39>\",\"WARC-IP-Address\":\"158.75.1.94\",\"WARC-Target-URI\":\"https://apcz.umk.pl/czasopisma/index.php/TMNA/article/view/TMNA.2004.006\",\"WARC-Payload-Digest\":\"sha1:4IEELNJW5XQ72TESULOP7UTKIR2BDOBE\",\"WARC-Block-Digest\":\"sha1:EKBEYOFCPVG2ECCMYQWXR2T6YRBDOR3B\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-24/CC-MAIN-2020-24_segments_1590347398233.32_warc_CC-MAIN-20200528061845-20200528091845-00100.warc.gz\"}"}
https://ace-writers.com/help-with-homework-online-class-work-and-assignments-17350/	[ "# Need help with QNT295 statistics\n\nHow would you explain the difference between a confounding variable and a lurking variable?\n\nWhat are the components of a time series? What external factors might affect each of the different component?\n\nSuccessful forecasting? (Example)\n\nWhat are the conditions we must check when summarizing the association between two variables with a line and why is this important?\n\nJoint and Marginal Probability (example)\n\nDifferences between independents and dependent event", null, "" ]	[ null, "https://writerbay.net/wp-content/uploads/2020/08/paper.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8756334,"math_prob":0.86559916,"size":1379,"snap":"2022-27-2022-33","text_gpt3_token_len":305,"char_repetition_ratio":0.123636365,"word_repetition_ratio":0.08,"special_character_ratio":0.23277737,"punctuation_ratio":0.089147285,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9549091,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-07-02T14:34:17Z\",\"WARC-Record-ID\":\"<urn:uuid:40d7f849-513f-4bad-8d87-985b7b120925>\",\"Content-Length\":\"50744\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a1d13da7-e51c-453e-b1fc-92a9a53e8fc2>\",\"WARC-Concurrent-To\":\"<urn:uuid:132b3733-a1b9-46f6-aee9-589c34f0f758>\",\"WARC-IP-Address\":\"162.0.208.14\",\"WARC-Target-URI\":\"https://ace-writers.com/help-with-homework-online-class-work-and-assignments-17350/\",\"WARC-Payload-Digest\":\"sha1:XCUB5RAGBYO7GLQQM2RQU2YIOCVQRX5T\",\"WARC-Block-Digest\":\"sha1:JSUMZEBFJ5KY7S6CCA46C7SU5TJUBWMW\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656104141372.60_warc_CC-MAIN-20220702131941-20220702161941-00029.warc.gz\"}"}
https://vsbattles.fandom.com/wiki/User_blog:Morning_Star_TM/RWBY_-_Weiss_creates_an_ice_construct_that_knock_off_a_mecha	[ "# VS Battles Wiki\n\nWe have moved to a new external forum hosted at https://vsbattles.com\n\nWe have a new automated signup system for our wiki members, with a procedure that must be exactly followed in order to register.\n\n## Introduction\n\nPage that holds feat in question: Scan.\n\nBasically, Weiss creates an ice construct that knocks off this giant robot. I will calculate the amount of ice that she created.\n\n## Calculation\n\nSince the whole ice construct is not definite in shape, i will try to measure each icicle 1 by 1.\n\n1st, let's bring forth the height of each icicle. ( I swear this job is going to kill me with shortened age).\n\nRuby (The small black dot in front of the ice) = 10 px = 1.62 m.\n\nPixel scaling = 1.62/10 = 0.162 m/px.\n\n### Icicles' Heights\n\nHeight of 1st icicle = 43 px = 6.966 meters.\n\nHeight of 2nd icicle = Height of 1st icicle = 6.966 meters.\n\nHeight of 3rd icicle = 56 px = 9.072 meters.\n\nHeight of 4th icicle = 70 px = 11.34 meters.\n\nHeight of 5th icicle = 77 px = 12.474 meters.\n\nHeight of 6th icicle = 82 px = 13.284 meters.\n\nHeight of 7th icicle = 82 px = 13.284 meters.\n\nHeight of 8th icicle = 99 px = 16.038 meters.\n\nHeight of 9th icicle = 117 px = 18.954 meters.\n\nHeight of 10th icicle = 202 px = 32.724 meters.\n\nHeight of 11th icicle = 223 px = 36.126 meters.\n\nWith that out of the way, let's determine the radius of each icicle. (I really want to die right now)\n\nDiameter of 1st icicle = 6 px = 0.972 meters.\n\nRadius of 1st icicle = 0.486 meters.\n\nDiameter of 2nd icicle = 8 px = 1.296 meters.\n\nRadius of 2nd icicle = 0.648 meters.\n\nDiameter of 3rd icicle = 5 px = 0.81 meters.\n\nRadius of 3rd icicle = 0.405 meters.\n\nDiameter of 4th icicle = 6 px = 0.972 meters.\n\nRadius of 4th icicle = 0.486 meters.\n\nDiameter of 5th icicle = 16 px = 2.592 meters.\n\nRadius of 5th icicle = 1.296 meters.\n\nDiameter of 6th icicle = 37 px = 5.994 meters.\n\nRadius of 6th icicle = 2.997 meters.\n\nDiameter of 7th icicle = 7 px = 1.134 meters.\n\nRadius of 7th icicle = 0.567 meters.\n\nDiameter of 8th icicle = 10 px = 1.62 meters.\n\nRadius of 8th icicle = 0.81 meters.\n\nDiameter of 9th icicle = 18 px = 2.916 meters.\n\nRadius of 9th icicle = 1.458 meters.\n\nDiameter of 10th icicle = 75 px = 12.15 meters.\n\nRadius of 10th icicle = 6.075 meters.\n\nDiameter of 11th icicle = 38 px = 6.156 meters.\n\nRadius of 11th icicle = 3.078 meters.\n\nOkay. With that out of the way, each and every single 1 of those icicles resemble a cylinder to me, so i will gather each and every single 1 of them and apply the volume of a cylinder to each icicle.\n\n### Icicles' Volumes\n\nVolume of 1st icicle = 5.1689922538252 m^3.\n\nVolume of 2nd icicle = 9.1893195623559 m^3.\n\nVolume of 3rd icicle = 4.674799195966 m^3.\n\nVolume of 4th icicle = 8.4146385527387 m^3.\n\nVolume of 5th icicle = 65.821172679201 m^3.\n\nVolume of 6th icicle = 374.84543438625 m^3.\n\nVolume of 7th icicle = 13.416673692422 m^3.\n\nVolume of 8th icicle = 33.057508600045 m^3.\n\nVolume of 9th icicle = 126.58020565763 m^3.\n\nVolume of 10th icicle = 3794.1004188688 m^3.\n\nVolume of 11th icicle = 1075.2438847796 m^3.\n\nAnyways, with that out of the way, taking into account that the density of water is 1000 kg/m^3, let's determine the mass of each icicle.\n\n### Icicles' Masses\n\nMass of 1st icicle = 5.1689922538252 * 1000 = 5169 kg.\n\nMass of 2nd icicle = 9.1893195623559 * 1000 = 9189.32 kg.\n\nMass of 3rd icicle = 4.674799195966 * 1000 = 4674.8 kg.\n\nMass of 4th icicle = 8.4146385527387 * 1000 = 8414.64 kg.\n\nMass of 5th icicle = 65.821172679201 * 1000 = 65821.2 kg.\n\nMass of 6th icicle = 374.84543438625 * 1000 = 374845.4344 kg.\n\nMass of 7th icicle = 13.416673692422 * 1000 = 13416.7 kg.\n\nMass of 8th icicle = 33.057508600045 * 1000 = 33057.51 kg.\n\nMass of 9th icicle = 126.58020565763 * 1000 = 126580.206 kg.\n\nMass of 10th icicle = 3794.1004188688 * 1000 = 3794100.42 kg.\n\nMass of 11th icicle = 1075.2438847796 * 1000 = 1075244 kg.\n\nLastly, we find the E of all of this.\n\n### Icicles' (E)\n\nI am using the ocean water method due to the fact that Weiss created the ice on the surface of what looks like an ocean. This doesn't look like a lake at all.\n\nSpecific Heat Capacity of water = 4181 J/kg.\n\nTemperature = 15 C. (Let's go with 15 degrees for temperature difference. 20 degrees would have meant that the water would be almost room temperature anyways).\n\nNow, to find the E:\n\n#### (E) through Total Mass\n\nTotal Mass of all icicles = 5510513.2304 kg.\n\nQ1 = MC(Delta T) = (5510513.2304)(4181)(15) = 345591837244.5 Joules.\n\nQ2 = (5510513.2304)(334000) = 1.840511418954e12 Joules.\n\nE = Q1 + Q2 = 2.186103256199e12 Joules = 522.5 Tons of TNT (Multi-City Block level).\n\n#### Revised Calculation by Mr. TheRustyOne\n\nQuoting from said Mister:\n\n “ This is amazing but those aren't cylinders, cylinders don't have a point at the top. I think using a cone would be more accurate. „ ~ TheRustyOne\n\nVolume 1 = 1.7229974179417 m^3\n\nVolume 2 = 3.0631065207853 m^3\n\nVolume 3 = 1.5582663986553 m^3\n\nVolume 4 = 2.8048795175796 m^3\n\nVolume 5 = 21.940390893067 m^3\n\nVolume 6 = 124.94847812875 m^3\n\nVolume 7 = 4.4722245641408 m^3\n\nVolume 8 = 11.019169533348 m^3\n\nVolume 9 = 42.193401885876 m^3\n\nVolume 10 = 1264.7001396229 m^3\n\nVolume 11 = 358.41462825985 m^3\n\nTotal Volume = 1836.83768274 m^3\n\nTotal Mass of all icicles = 1836.83768274 x 1000 = 1836837.68274 kg.\n\nQ1 = MC(Delta T) = (1836837.68274)(4181)(15) = 115197275273 Joules.\n\nQ2 = (1836837.68274)(334000) = 613503786035.2 Joules.\n\nE = Q1 + Q2 = 728701061308.2 Joules = 174.2 Tons of TNT (Multi-City Block level).\n\n## Final Conclusion\n\nWeiss creates an ice construct that knock off a mecha (Total Mass Method) = 522.5 Tons of TNT (Multi-City Block level).\n\nWeiss creates an ice construct that knock off a mecha (Mr. TheRustyOne Method) = 174.2 Tons of TNT (Multi-City Block level).\n\nCredits to Mr. TheRustyOne for giving me the more accurate scan and helping me with the calculation, and for Mr. Dargoo Faust for giving me the accurate idea of how to calculate this." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8081274,"math_prob":0.9907117,"size":5641,"snap":"2021-43-2021-49","text_gpt3_token_len":2059,"char_repetition_ratio":0.2130566,"word_repetition_ratio":0.10174418,"special_character_ratio":0.4602021,"punctuation_ratio":0.16639611,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99978405,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-12-01T21:18:54Z\",\"WARC-Record-ID\":\"<urn:uuid:7844d569-e7d7-42fa-bf01-914bc21e0630>\",\"Content-Length\":\"385652\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:5bf859ef-325b-4cda-996d-059034dc217c>\",\"WARC-Concurrent-To\":\"<urn:uuid:609847df-f485-4ef6-84ea-cda9c6472353>\",\"WARC-IP-Address\":\"151.101.192.194\",\"WARC-Target-URI\":\"https://vsbattles.fandom.com/wiki/User_blog:Morning_Star_TM/RWBY_-_Weiss_creates_an_ice_construct_that_knock_off_a_mecha\",\"WARC-Payload-Digest\":\"sha1:GTZWNW2XOJACLTBTMWUWTM25TAB6CP47\",\"WARC-Block-Digest\":\"sha1:2DHTWNS4GFCG5YMPQVKP7P25W6F3CZSM\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964360951.9_warc_CC-MAIN-20211201203843-20211201233843-00346.warc.gz\"}"}
https://www.colorhexa.com/14ff7f	[ "# #14ff7f Color Information\n\nIn a RGB color space, hex #14ff7f is composed of 7.8% red, 100% green and 49.8% blue. Whereas in a CMYK color space, it is composed of 92.2% cyan, 0% magenta, 50.2% yellow and 0% black. It has a hue angle of 147.3 degrees, a saturation of 100% and a lightness of 53.9%. #14ff7f color hex could be obtained by blending #28fffe with #00ff00. Closest websafe color is: #00ff66.\n\n• R 8\n• G 100\n• B 50\nRGB color chart\n• C 92\n• M 0\n• Y 50\n• K 0\nCMYK color chart\n\n#14ff7f color description : Vivid cyan - lime green.\n\n# #14ff7f Color Conversion\n\nThe hexadecimal color #14ff7f has RGB values of R:20, G:255, B:127 and CMYK values of C:0.92, M:0, Y:0.5, K:0. Its decimal value is 1376127.\n\nHex triplet RGB Decimal 14ff7f `#14ff7f` 20, 255, 127 `rgb(20,255,127)` 7.8, 100, 49.8 `rgb(7.8%,100%,49.8%)` 92, 0, 50, 0 147.3°, 100, 53.9 `hsl(147.3,100%,53.9%)` 147.3°, 92.2, 100 00ff66 `#00ff66`\nCIE-LAB 88.541, -76.301, 47.128 39.876, 73.197, 32.104 0.275, 0.504, 73.197 88.541, 89.682, 148.298 88.541, -78.956, 75.339 85.555, -66.524, 37.64 00010100, 11111111, 01111111\n\n# Color Schemes with #14ff7f\n\n• #14ff7f\n``#14ff7f` `rgb(20,255,127)``\n• #ff1494\n``#ff1494` `rgb(255,20,148)``\nComplementary Color\n• #1fff14\n``#1fff14` `rgb(31,255,20)``\n• #14ff7f\n``#14ff7f` `rgb(20,255,127)``\n• #14fff5\n``#14fff5` `rgb(20,255,245)``\nAnalogous Color\n• #ff141f\n``#ff141f` `rgb(255,20,31)``\n• #14ff7f\n``#14ff7f` `rgb(20,255,127)``\n• #f514ff\n``#f514ff` `rgb(245,20,255)``\nSplit Complementary Color\n• #ff7f14\n``#ff7f14` `rgb(255,127,20)``\n• #14ff7f\n``#14ff7f` `rgb(20,255,127)``\n• #7f14ff\n``#7f14ff` `rgb(127,20,255)``\n• #94ff14\n``#94ff14` `rgb(148,255,20)``\n• #14ff7f\n``#14ff7f` `rgb(20,255,127)``\n• #7f14ff\n``#7f14ff` `rgb(127,20,255)``\n• #ff1494\n``#ff1494` `rgb(255,20,148)``\n• #00c75a\n``#00c75a` `rgb(0,199,90)``\n• #00e066\n``#00e066` `rgb(0,224,102)``\n• #00fa72\n``#00fa72` `rgb(0,250,114)``\n• #14ff7f\n``#14ff7f` `rgb(20,255,127)``\n• #2eff8d\n``#2eff8d` `rgb(46,255,141)``\n• #47ff9b\n``#47ff9b` `rgb(71,255,155)``\n• #61ffa9\n``#61ffa9` `rgb(97,255,169)``\nMonochromatic Color\n\n# Alternatives to #14ff7f\n\nBelow, you can see some colors close to #14ff7f. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #14ff44\n``#14ff44` `rgb(20,255,68)``\n• #14ff58\n``#14ff58` `rgb(20,255,88)``\n• #14ff6b\n``#14ff6b` `rgb(20,255,107)``\n• #14ff7f\n``#14ff7f` `rgb(20,255,127)``\n• #14ff93\n``#14ff93` `rgb(20,255,147)``\n• #14ffa6\n``#14ffa6` `rgb(20,255,166)``\n• #14ffba\n``#14ffba` `rgb(20,255,186)``\nSimilar Colors\n\n# #14ff7f Preview\n\nThis text has a font color of #14ff7f.\n\n``<span style=\"color:#14ff7f;\">Text here</span>``\n#14ff7f background color\n\nThis paragraph has a background color of #14ff7f.\n\n``<p style=\"background-color:#14ff7f;\">Content here</p>``\n#14ff7f border color\n\nThis element has a border color of #14ff7f.\n\n``<div style=\"border:1px solid #14ff7f;\">Content here</div>``\nCSS codes\n``.text {color:#14ff7f;}``\n``.background {background-color:#14ff7f;}``\n``.border {border:1px solid #14ff7f;}``\n\n# Shades and Tints of #14ff7f\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, #000000 is the darkest color, while #ecfff5 is the lightest one.\n\n• #000000\n``#000000` `rgb(0,0,0)``\n• #001409\n``#001409` `rgb(0,20,9)``\n• #002812\n``#002812` `rgb(0,40,18)``\n• #003b1b\n``#003b1b` `rgb(0,59,27)``\n• #004f24\n``#004f24` `rgb(0,79,36)``\n• #00622d\n``#00622d` `rgb(0,98,45)``\n• #007636\n``#007636` `rgb(0,118,54)``\n• #008a3f\n``#008a3f` `rgb(0,138,63)``\n• #009d48\n``#009d48` `rgb(0,157,72)``\n• #00b151\n``#00b151` `rgb(0,177,81)``\n• #00c559\n``#00c559` `rgb(0,197,89)``\n• #00d862\n``#00d862` `rgb(0,216,98)``\n• #00ec6b\n``#00ec6b` `rgb(0,236,107)``\n• #00ff74\n``#00ff74` `rgb(0,255,116)``\n• #14ff7f\n``#14ff7f` `rgb(20,255,127)``\n• #28ff8a\n``#28ff8a` `rgb(40,255,138)``\n• #3bff94\n``#3bff94` `rgb(59,255,148)``\n• #4fff9f\n``#4fff9f` `rgb(79,255,159)``\n• #62ffaa\n``#62ffaa` `rgb(98,255,170)``\n• #76ffb4\n``#76ffb4` `rgb(118,255,180)``\n• #8affbf\n``#8affbf` `rgb(138,255,191)``\n• #9dffca\n``#9dffca` `rgb(157,255,202)``\n• #b1ffd4\n``#b1ffd4` `rgb(177,255,212)``\n• #c5ffdf\n``#c5ffdf` `rgb(197,255,223)``\n• #d8ffea\n``#d8ffea` `rgb(216,255,234)``\n• #ecfff5\n``#ecfff5` `rgb(236,255,245)``\nTint Color Variation\n\n# Tones of #14ff7f\n\nA tone is produced by adding gray to any pure hue. In this case, #809389 is the less saturated color, while #14ff7f is the most saturated one.\n\n• #809389\n``#809389` `rgb(128,147,137)``\n• #779c88\n``#779c88` `rgb(119,156,136)``\n• #6ea587\n``#6ea587` `rgb(110,165,135)``\n• #65ae86\n``#65ae86` `rgb(101,174,134)``\n• #5cb785\n``#5cb785` `rgb(92,183,133)``\n• #53c085\n``#53c085` `rgb(83,192,133)``\n• #4ac984\n``#4ac984` `rgb(74,201,132)``\n• #41d283\n``#41d283` `rgb(65,210,131)``\n• #38db82\n``#38db82` `rgb(56,219,130)``\n• #2fe481\n``#2fe481` `rgb(47,228,129)``\n• #26ed81\n``#26ed81` `rgb(38,237,129)``\n• #1df680\n``#1df680` `rgb(29,246,128)``\n• #14ff7f\n``#14ff7f` `rgb(20,255,127)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #14ff7f 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":0.5635354,"math_prob":0.8246377,"size":3691,"snap":"2021-43-2021-49","text_gpt3_token_len":1613,"char_repetition_ratio":0.13642527,"word_repetition_ratio":0.011049724,"special_character_ratio":0.54700625,"punctuation_ratio":0.23224352,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98765594,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-12-07T18:32:08Z\",\"WARC-Record-ID\":\"<urn:uuid:96649fa4-944f-411c-b624-546747d7c465>\",\"Content-Length\":\"36173\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:12f7c2b1-8272-40e2-8b67-c42967b6bd1c>\",\"WARC-Concurrent-To\":\"<urn:uuid:abe577ae-125f-46c5-a2bb-38adf637f3d6>\",\"WARC-IP-Address\":\"178.32.117.56\",\"WARC-Target-URI\":\"https://www.colorhexa.com/14ff7f\",\"WARC-Payload-Digest\":\"sha1:TQPNUCVN52NUCKHCPU4VXW43Y2GXIJMD\",\"WARC-Block-Digest\":\"sha1:YTKBGTNE3MH7RE75QOD2VHTQ5DFN67FS\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964363405.77_warc_CC-MAIN-20211207170825-20211207200825-00396.warc.gz\"}"}
https://googology.fandom.com/wiki/Subcubic_graph_number	[ "# Googology Wiki\n\nThis wiki's URL has been migrated to the primary fandom.com domain.Read more here\n\nThe subcubic graph numbers are the outputs of a fast-growing combinatorial function. They were devised by Harvey Friedman, who showed that it eventually dominates every recursive function provably total in the theory of $$\\Pi^1_1$$-$$\\text{CA}_0$$, and is itself provably total in the theory of $$\\Pi_1^1-\\textrm{CA}+\\textrm{BI}$$[citation needed].\n\nOne output of the sequence, SCG(13), is a subject of extensive research. It is known to surpass TREE(3), a number that arises from a related sequence.\n\n## Definition\n\nA subcubic graph is a finite graph in which each vertex has a valence of at most three, i.e. no vertex is connected to more than three edges. (For the sake of this article, subcubic graphs are allowed to be multigraphs, and are not required to be connected.) We also define the graph minor relation as follows: A is said to be a graph minor of B if we can derive A from the following process: start with B, remove vertices and edges, and contract edges. An example of a graph minor derivation is shown in the infobox of this article.\n\nGiven an integer k, suppose we have a sequence of subcubic graphs G1, G2, ... such that each graph Gi has at most i + k vertices and for no i < j is Gi homeomorphically embeddable into Gj (i.e. is a graph minor).\n\nThe Robertson-Seymour theorem proves that subcubic graphs are well-quasi-ordered by homeomorphic embeddability, implying such a sequence cannot be infinite. So, for each value of k, there is a sequence with maximal length. We denote this maximal length using SCG(k).\n\n## Specific values\n\nIt is possible to show that SCG(0) = 6. The first graph is one vertex with a loop,\n\nthe second is two vertices connected by a single edge, and the next four graphs consist of 3, 2, 1, and 0 unconnected vertices. All maximal sequences will peak and decline this way.\n\nThe following bounds have been claimed by Googology Wiki user Hyp cos.\n\n• $$\\text{SCG}(1) > f_{\\varepsilon_22}(f_{\\varepsilon_02}(f_{\\varepsilon_0+1}(f_{\\varepsilon_0}(f_{\\omega^\\omega+1}(f_{\\omega^5+\\omega^2+\\omega}(\\\\f_{\\omega^23+1}(f_{\\omega^22+1}(f_{\\omega^2+\\omega3+1}(f_{\\omega^2+1}(f_{\\omega^2}(3\\times2^{3\\times2^{95}})))))))))))$$.\n\n• $$\\text{SCG}(2) > f_{\\vartheta(\\Omega^\\omega)}(f_{\\varepsilon_22}(f_{\\varepsilon_02}(f_{\\varepsilon_0+1}(f_{\\varepsilon_0}(f_{\\omega^\\omega+1}(\\\\f_{\\omega^5+\\omega^2+\\omega}(f_{\\omega^23+1}(f_{\\omega^22+1}(f_{\\omega^2+\\omega3+1}(f_{\\omega^2+1}(f_{\\omega^2}(3\\times2^{3\\times2^{95}}))))))))))))$$\n\nThese bounds use a non-standard choice of fundamental sequences for ordinals — by using a particular, highly complex bijection between ordinals and small graphs, which we will denote here by $$f$$, we define $$\\alpha[n]=\\max\\{\\beta: \\beta<\\alpha\\text{ and } f(\\beta)\\text{ is a graph with }\\leq n\\text{ vertices}\\}$$.\n\nSince the graph indices start at one, it is also valid to say that SCG(-1) = 1, consisting only of the empty graph.\n\nFriedman stated that SCG(13) is greater than the halting time of any Turing machine such that it can be proven to halt in at most 22,000 symbols in $$\\Pi^1_1$$-$$\\text{CA}_0$$. It is therefore far larger than TREE(3).\n\nSCG(n) is computable,therefore it is naturally surpassed by $$\\Sigma(n)$$ for some n.\n\n## Matrix interpretation\n\nAn alternate way of describing the SCG function is as follows. Define an incidence matrix as a matrix with entries in {0, 1, 2} where each column sums to exactly 2 and each row sums to at most 3. Given a nonnegative integer k, we construct a sequence of n incidence matrices M1, M2, ..., Mn such that each matrix Mi has at most i + k rows, and no matrix can be changed into an earlier one by repeated applications of any of the following processes:\n\n• Reordering rows or columns.\n• Deleting columns.\n• Deleting rows, then deleting all columns that do not sum to 2.\n• Take two rows A and B such that A + B contains a 2 in position i for some i. Remove A and B, append A + B to the matrix, and finally remove column i.\n\nSCG(k), then, is the largest possible value of n.\n\n## Simple subcubic graph numbers\n\nIf we require the subcubic graphs to be simple (i.e. no loops or multiple edges), we get the simple subcubic graph numbers, denoted SSCG. Although this community believed that Adam P. Goucher has shown that SSCG(2) << TREE(3) << SSCG(3) in his article, it just contains a rough estimation without a proof. Moreover, the community believed that he has shown that even TREEn(3) for even very large n (for example n=TREE(3)) does not compete at all with SSCG(3). Later, he proved that TREE(3) < SSCG(3) in a different blog post.\n\nGoucher claimed that he had proved that $$\\text{SSCG}(4n+3) \\geq \\text{SCG}(n)$$ in his comment and hence SCG(n) and SSCG(n) have comparable growth rates. He later proved it in a later blog post.\n\nSimilar to the fact that there are many wrong informations on \"the actual results on TREE with proofs\", there are many statements on SCG which are said to be proved but do not have actual proofs. See also issues on analysis of TREE.\n\n### Values and bounds\n\n• SSCG(0) = 2\n• SSCG(1) = 5\n• SSCG(2) $$\\geq 3 \\cdot 2^{3 \\cdot 2^{95}}-8 \\approx 10^{3.5775 \\cdot 10^{28}}$$ (it is possible, that sequence of subcubic graphs that Adam P. Goucher has shown is really optimal, but it remains unproven.)\n\n## Sources\n\n1. Harvey Friedman, FOM 279:Subcubic Graph Numbers/restated\n2. Technically a topological minor, but topological minors and graph minors are equivalent for subcubic graphs.\n3. User blog:Hyp cos/SCG(n) and some related" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8843949,"math_prob":0.99720067,"size":5765,"snap":"2021-43-2021-49","text_gpt3_token_len":1669,"char_repetition_ratio":0.1258462,"word_repetition_ratio":0.01343785,"special_character_ratio":0.2891587,"punctuation_ratio":0.1021645,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99962914,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-11-29T09:06:50Z\",\"WARC-Record-ID\":\"<urn:uuid:10fc680b-c66e-44b5-8db0-6500fce91c96>\",\"Content-Length\":\"316477\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:bf0d7f15-81eb-4f21-9816-9fb03dec0627>\",\"WARC-Concurrent-To\":\"<urn:uuid:603634c5-e3ff-4c2a-9c26-8bfaac8a4d96>\",\"WARC-IP-Address\":\"151.101.192.194\",\"WARC-Target-URI\":\"https://googology.fandom.com/wiki/Subcubic_graph_number\",\"WARC-Payload-Digest\":\"sha1:IPYB2HWDEZ5S4IP2LFWMT5KG5L3CRIGQ\",\"WARC-Block-Digest\":\"sha1:X77UJRE3RRNUFMYA62ZQEE7G2QQR776I\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964358702.43_warc_CC-MAIN-20211129074202-20211129104202-00291.warc.gz\"}"}
https://www.studyadda.com/notes/10th-class/mathematics/triangles/triangles/15131	[ "10th Class Mathematics Triangles\n\nTriangles\n\nCategory : 10th Class\n\nTRIANGLES\n\nFUNDAMENTALS\n\nSimilar figures:", null, "• Figures having the same shape (not necessarily the same size) are called similar figures. Same shapes ensure that the corresponding angles are equal and their corresponding sides are proportional.\n\nCongruent figures:", null, "• Figures having the same shape and the same size are called congruent figures. Here, apart from angles, corresponding sides are also equal\n\nSimilar Triangles:\n\n• Two triangles are said to be similar, if their corresponding angles are equal and corresponding sides are proportional.", null, "e.g., If in $\\Delta \\,ABC$and $\\Delta \\,PQR$\n\n$\\angle A=\\angle P,\\angle B=\\angle Q,\\angle C=\\angle R$\n\nand $\\frac{AB}{PQ}=\\frac{BC}{QR}=\\frac{AC}{PR},$ then, $\\Delta \\text{ }ABC\\sim \\Delta \\,PQR;$ where symbol $\\sim$ is read as ‘is similar to’.\n\n• When two triangles ($\\Delta \\text{ }ABC$and $\\Delta \\text{ }DEF$as below) are similar, then all above results about angles and ratio of sides hold good. However, in questions, when you are asked to prove similarly, you can either prove:", null, "(i) $\\angle A=\\angle D,\\angle B=\\angle E$ (called AA similarly)\n\nOr\n\n(ii) $\\frac{AB}{DE}=\\frac{BC}{EF}=\\frac{AC}{DF}$ (called SSS similarity)\n\nOr\n\n(iii) $\\frac{AB}{PQ}=\\frac{BC}{QR}$ and $\\angle B=\\angle Q$ (called SAS similarity)\n\nAny one of the above three, would be sufficient for proving similarity.\n\nConversely: If $\\Delta \\text{ }ABC$is similar to$\\Delta \\text{ }PQR$, then\n\n$\\angle A=\\angle D;\\angle B=\\angle E;\\angle C=\\angle Q$ and $\\frac{AB}{DE}=\\frac{BC}{EF}=\\frac{AC}{DF}$\n\n• When two triangles are congruent (notation for congruent is $\\cong$)\n\nThen, $\\angle A=\\angle D;\\text{ }\\angle B=\\angle E;\\text{ }\\angle C=\\angle F$\n\nand, $AB=DE,BC=EF$ and $CA=FD$\n\nHowever, to prove congruency, we need to prove any one of the following only:\n\n(i) $\\text{ }AB=DE,\\text{ }BC=EF\\And CA=FD$(SSS congruency)\n\nor\n\n(ii) $AB=DE;\\text{ }BC=EF\\And \\angle B=\\angle E$ (SAS congruency)\n\nor\n\n(iii) $\\angle A=\\angle D;\\angle B=\\angle E$ and $AB=DE$(ASA congruency)\n\nHow to look for similarity and congruency of angles\n\nWhile looking for similarity and congruency, you should not only see external appearance but also which of the corresponding angles are equal (or, which of corresponding sides are in the same ratio).", null, "", null, "In the above figure, $\\angle B=\\angle F;\\text{ }\\angle C=\\angle E$and $\\angle A\\cong \\angle D;$ thus $\\Delta \\,ACB\\cong \\Delta \\,DEF$(and not$\\Delta \\,ABC\\cong \\Delta \\,DEF$)", null, "Mathematical statement of the theorem$\\frac{AD}{DB}=\\frac{AE}{EC}$ (where$DE\\parallel ~BC$)\n\ni.e., if in $\\Delta \\text{ }ABC$as shown above, $DE\\parallel BC\\Rightarrow \\frac{AD}{DB}=\\frac{AE}{EC}$\n\nConverse of Basic Proportionality Theorem:", null, "Mathematical statement of the theorem if in $\\Delta \\,ABC$(as shown above),\n\n$\\frac{AD}{DB}=\\frac{AE}{EC}\\Rightarrow DE\\parallel BC)$\n\nOther Topics\n\nNotes - Triangles\n\nYou need to login to perform this action.\nYou will be redirected in 3 sec", null, "" ]	[ null, "https://www.studyadda.com/upload/html_folder/8_Triangles_NZM_GOF-10/8_Triangles_NZM_GOF-10_files/image001.jpg", null, "https://www.studyadda.com/upload/html_folder/8_Triangles_NZM_GOF-10/8_Triangles_NZM_GOF-10_files/image002.jpg", null, "https://www.studyadda.com/upload/html_folder/8_Triangles_NZM_GOF-10/8_Triangles_NZM_GOF-10_files/image003.jpg", null, "https://www.studyadda.com/upload/html_folder/8_Triangles_NZM_GOF-10/8_Triangles_NZM_GOF-10_files/image004.jpg", null, "https://www.studyadda.com/upload/html_folder/8_Triangles_NZM_GOF-10/8_Triangles_NZM_GOF-10_files/image005.jpg", null, "https://www.studyadda.com/upload/html_folder/8_Triangles_NZM_GOF-10/8_Triangles_NZM_GOF-10_files/image006.jpg", null, "https://www.studyadda.com/upload/html_folder/8_Triangles_NZM_GOF-10/8_Triangles_NZM_GOF-10_files/image007.png", null, "https://www.studyadda.com/upload/html_folder/8_Triangles_NZM_GOF-10/8_Triangles_NZM_GOF-10_files/image008.jpg", null, "https://www.studyadda.com/assets/frontend/images/msg-gif.GIF", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5572174,"math_prob":0.99949825,"size":2157,"snap":"2022-05-2022-21","text_gpt3_token_len":750,"char_repetition_ratio":0.19647004,"word_repetition_ratio":0.0,"special_character_ratio":0.32591563,"punctuation_ratio":0.12177986,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000097,"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,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-01-17T17:15:13Z\",\"WARC-Record-ID\":\"<urn:uuid:338a9459-8aba-4b30-a6fd-f69a6db75a32>\",\"Content-Length\":\"134867\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:bca05626-4cac-435a-8d36-50352957078c>\",\"WARC-Concurrent-To\":\"<urn:uuid:58371296-1cab-40cc-a1e9-8d8a85befbef>\",\"WARC-IP-Address\":\"151.106.35.148\",\"WARC-Target-URI\":\"https://www.studyadda.com/notes/10th-class/mathematics/triangles/triangles/15131\",\"WARC-Payload-Digest\":\"sha1:MD6LW3JGWUDNXGZRX72GSEBI5YPE4NHU\",\"WARC-Block-Digest\":\"sha1:LBDW5NDXUFYBTJ5P3TWUCWQUBNSA6BYB\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-05/CC-MAIN-2022-05_segments_1642320300574.19_warc_CC-MAIN-20220117151834-20220117181834-00305.warc.gz\"}"}
https://mathlesstraveled.com/2015/11/13/mablowrimo-12-groups-and-order/?shared=email&msg=fail	[ "## MaBloWriMo 12: Groups and Order\n\nContinuing our discussion of groups (see here and here), today I want to discuss the concept of order, which is defined both for groups themselves and for the elements of a group.\n\nThe order of a group simply means the size of the set", null, "$G$. So instead of “a group with 8 elements” you will often hear “a group of order 8” instead.\n\nElements of a group also have an order. Recall the example", null, "$\\mathbb{Z}_8$, the group with elements", null, "$\\{0, 1, \\dots, 7\\}$, and with binary operation addition", null, "$\\pmod 8$. Consider what happens if we start with", null, "$1$ and keep combining it with itself according to the group operation.", null, "$1 +_8 1 = 2$, then", null, "$1 +_8 1 +_8 1 = 3$, and so on, until we get to adding eight copies of", null, "$1$, giving", null, "$1 +_8 \\dots +_8 1 = 0$. If we add up copies of", null, "$2$, on the other hand, it takes only", null, "$4$ copies to reach", null, "$0$. What about", null, "$3$? Well,", null, "$3 +_8 3 = 6$; three copies yield", null, "$1$; four copies yield", null, "$4$; and so on. As you can verify, we actually need eight copies of", null, "$3$ before we get to", null, "$0$.\n\nIn general, if", null, "$G$ is a group and", null, "$g \\in G$ is some element of the group, the order of", null, "$g$ is defined as the smallest number of copies of", null, "$g$ which combine to yield the identity element. So in", null, "$\\mathbb{Z}_8$,", null, "$1$ has order eight,", null, "$2$ has order four,", null, "$3$ also has order eight, and so on.", null, "$0$ itself has order 1, because one copy of", null, "$0$ is already", null, "$0$.\n\nA few questions immediately suggest themselves:\n\n1. Does every group element have a well-defined order?\n2. How does the order of group elements relate to the order of the group?\n\nTo answer the first question, recall that the integers form a group under addition, and obviously if you start with some nonzero integer", null, "$n$, you can add", null, "$n$ to itself as many times as you like but you will never get zero! In such cases we say that the order is infinite.\n\nOK, but the group", null, "$\\mathbb{Z}$ itself has infinitely many elements. Let’s refine our question a bit:\n\n1. (revised) Does every element of a finite group have a well-defined, finite order?\n\nEven in a finite group, you could imagine having some element", null, "$g \\in G$ such that no matter how much you combine it with itself, you will never get the identity element. Can this happen? (Also, as a tangential challenge, can you come up with an example of an infinite group where some elements other than the identity do have a finite order?)\n\nFor now, I’ll let you think about these questions! Tomorrow, we’ll answer question 1; the next day, we’ll talk about a simple answer to question 2 which will be sufficient for our purposes (there is also a more nuanced answer which is harder to prove).\n\nAs an aside, remember that we’re trying to prove the Lucas-Lehmer test, and currently we’re trying to understand the related number", null, "$\\omega = 2 + \\sqrt{3}$. The point of all this stuff about groups is that (1) we’re going to construct a group containing", null, "$\\omega$ as an element, and then (2) we’re going to figure out something about the order of", null, "$\\omega$ in that group. We’ll see that if the Lucas-Lehmer test didn’t work, then the order of", null, "$\\omega$ would be in contradiction to the answer to question 2.", null, "" ]	[ null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://s0.wp.com/latex.php", null, "https://0.gravatar.com/avatar/cc113924265dbeb535c8b2fefe4e33ee", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9518181,"math_prob":0.9959022,"size":2881,"snap":"2022-40-2023-06","text_gpt3_token_len":640,"char_repetition_ratio":0.15015641,"word_repetition_ratio":0.0,"special_character_ratio":0.22735162,"punctuation_ratio":0.11774461,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.998789,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76],"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,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,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\":\"2022-10-06T00:23:37Z\",\"WARC-Record-ID\":\"<urn:uuid:d051995f-258f-4d9e-a097-247fb4687828>\",\"Content-Length\":\"95346\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b73676bf-8a44-4f80-95ed-79a7943a855b>\",\"WARC-Concurrent-To\":\"<urn:uuid:2e359079-25a4-44b1-907e-8ef9f4b2ecfc>\",\"WARC-IP-Address\":\"192.0.78.24\",\"WARC-Target-URI\":\"https://mathlesstraveled.com/2015/11/13/mablowrimo-12-groups-and-order/?shared=email&msg=fail\",\"WARC-Payload-Digest\":\"sha1:JM7A5RSMSZT26VRY2DXF62XDLYWGHVI7\",\"WARC-Block-Digest\":\"sha1:J6AKCCENFPI4DJSHI5FHIAXK6J22GNAA\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-40/CC-MAIN-2022-40_segments_1664030337680.35_warc_CC-MAIN-20221005234659-20221006024659-00150.warc.gz\"}"}
https://e-zaitaku.info/essay-beowulf/5-1-homework-system-of-equations-76233.php	[ "# 5. 1 homework system of equations\n\n61%\n545 1 month ago\nA system of linear equations is a set of linear equations that have common variables. Common systems consist of two variables, x and y, and two linear equations. The solution to the system is the value of x and y that satisfy both equations. There are two ways to solve systems: substitution and elimination. Step 1: Solve one equation in terms of one of the variables whichever is easier. Step 2: Substitute one equation into the other in terms of the variable found in step one.", null, "", null, "## Unit 5: Linear Systems of Equations", null, "", null, "", null, "", null, "## Homework: Systems of Equations Word Problems Quiz - Quizizz\n\nTeachers Pay Teachers is an online marketplace where teachers buy and sell original educational materials. Are you getting the free resources, updates, and special offers we send out every week in our teacher newsletter? Grade Level. Resource Type. Log In Join Us. View Wish List View Cart. You Selected: Keyword systems of equations graphing worksheet.", null, "", null, "", null, "### 5.1: Solve Systems of Equations by Graphing\n\nIf you missed this problem, review Example 1. If you missed this problem, review Example 2. We have solved systems of linear equations by graphing and by substitution. Graphing works well when the variable coefficients are small and the solution has integer values.", null, "", null, "", null, "", null, "", null, "There are two methods that will be used in this lesson to solve a system of linear equations algebraically. They are 1 substitution , and 2 elimination. They are both aimed at eliminating one variable so that normal algebraic means can be used to solve for the other variable. Once one variable is solved, then substitution will be used in both above methods to find the second variable. Special Circumstances.", null, "Category: essay beowulf\nReport this post:\nCause:\n\n### Your comments\n\nRicky S. 23.04.2021\nIm not good at making calculations so I decided to turn to professionals with my excel exercises." ]	[ null, "https://e-zaitaku.info/img/3097579425faaa6a4734880637558506.jpg", null, "https://e-zaitaku.info/img/100652.jpg", null, "https://e-zaitaku.info/img/5-1-homework-system-of-equations.jpg", null, "https://e-zaitaku.info/img/5-1-homework-system-of-equations-2.jpg", null, "https://e-zaitaku.info/img/5-1-homework-system-of-equations-3.jpg", null, "https://e-zaitaku.info/img/5-1-homework-system-of-equations-4.jpg", null, "https://e-zaitaku.info/img/b714a044d5eb2752ed4c59c035cb6f22.jpg", null, "https://e-zaitaku.info/img/544737.jpg", null, "https://e-zaitaku.info/img/bd2cf76751d75fa231312c758c407822.jpg", null, "https://e-zaitaku.info/img/5-1-homework-system-of-equations-5.png", null, "https://e-zaitaku.info/img/2211797714f564b207a880e2777777b3.jpg", null, "https://e-zaitaku.info/img/5-1-homework-system-of-equations-6.jpg", null, "https://e-zaitaku.info/img/5-1-homework-system-of-equations-7.jpg", null, "https://e-zaitaku.info/img/643642.jpg", null, "https://e-zaitaku.info/img/797530.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9265103,"math_prob":0.96014786,"size":1571,"snap":"2021-21-2021-25","text_gpt3_token_len":314,"char_repetition_ratio":0.14613912,"word_repetition_ratio":0.023166023,"special_character_ratio":0.19350731,"punctuation_ratio":0.11824324,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99469984,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30],"im_url_duplicate_count":[null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-16T16:53:37Z\",\"WARC-Record-ID\":\"<urn:uuid:3c795702-b2c3-4331-a163-9488abd32fcf>\",\"Content-Length\":\"32349\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:9c7670e6-7410-492d-a6ca-125d6ea2b8c3>\",\"WARC-Concurrent-To\":\"<urn:uuid:8782fd81-c092-4c29-96d0-703f031f000b>\",\"WARC-IP-Address\":\"172.67.199.67\",\"WARC-Target-URI\":\"https://e-zaitaku.info/essay-beowulf/5-1-homework-system-of-equations-76233.php\",\"WARC-Payload-Digest\":\"sha1:D5N4QO6RDY2NE7S25GP2BBN5KCE2BB2R\",\"WARC-Block-Digest\":\"sha1:W3XMHK35QHVBLLX5G4AY4DUL54YS7YCX\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623487625967.33_warc_CC-MAIN-20210616155529-20210616185529-00165.warc.gz\"}"}
http://oalevelsolutions.com/past-papers-solutions/cambridge-international-examinations/as-a-level-mathematics-9709/pure-mathematics-p2-9709-02/year-2003-october-november-p2-9709-02/cie_03_on_9709_02_q_7/	[ "# Past Papers’ Solutions \| Cambridge International Examinations (CIE) \| AS & A level \| Mathematics 9709 \| Pure Mathematics 2 (P2-9709/02) \| Year 2003 \| Oct-Nov \| (P2-9709/02) \| Q#7\n\nHits: 210\n\nQuestion\n\ni. By differentiating", null, ", show that if y = cot x then", null, "ii. Hence, show that", null, "By using appropriate trigonometrical identities, find the exact value of\n\niii.", null, "iv.", null, "Solution\n\ni.\n\nWe are given;", null, "Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve", null, "with respect to", null, "is:", null, "", null, "If", null, "and", null, "are functions of", null, ", and if", null, ", then;", null, "Let", null, "and", null, ", then;", null, "", null, "Rule for differentiation of", null, "is;", null, "Rule for differentiation of", null, "is;", null, "", null, "", null, "", null, "We have the trigonometric identity;", null, "", null, "", null, "provided that", null, "", null, "Therefore, if;", null, "", null, "provided that", null, "Hence;", null, "For;", null, "As demonstrated above;", null, "Hence;", null, "ii.", null, "As we have demonstrated in (i) that;", null, "Therefore, the inverse of differentiate ie integral must be as;", null, "Hence;", null, "", null, "", null, "", null, "provided that", null, "", null, "", null, "", null, "iii.", null, "Utilizing the identity;", null, "", null, "We can write;", null, "Rule for integration of", null, "is:", null, "", null, "As demonstrated in (ii);", null, "Hence;", null, "Rule for integration of", null, "is:", null, "", null, "", null, "", null, "", null, "iv.", null, "", null, "", null, "", null, "We have the trigonometric identity;", null, "", null, "", null, "", null, "", null, "Rule for integration of", null, "is:", null, "", null, "", null, "provided that", null, "", null, "We have shown in (ii) that;", null, "Therefore;", null, "" ]	[ null, "http://oalevelsolutions.com/CIE/GCE/Maths/P2/CIE_GCE_AS_Maths_P2_03_Nov_02_Q_7_files/image001.png", null, "http://oalevelsolutions.com/CIE/GCE/Maths/P2/CIE_GCE_AS_Maths_P2_03_Nov_02_Q_7_files/image002.png", null, "http://oalevelsolutions.com/CIE/GCE/Maths/P2/CIE_GCE_AS_Maths_P2_03_Nov_02_Q_7_files/image003.png", null, "http://oalevelsolutions.com/CIE/GCE/Maths/P2/CIE_GCE_AS_Maths_P2_03_Nov_02_Q_7_files/image004.png", null, "http://oalevelsolutions.com/CIE/GCE/Maths/P2/CIE_GCE_AS_Maths_P2_03_Nov_02_Q_7_files/image005.png", null, 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https://ambrevar.xyz/homogeneous/index.html	[ "# Homogeneous Coordinates\n\nA Concrete Explanation\n\nComputer graphics programs usually work with homogeneous coordinates, whose main purpose is to allow for some operations like translation to be linear and thus vectorizable. Vectorized operations can be heavily parallelized on the processing unit which results in a tremendous performance boost. Video games make heavy use of this technique.\n\nHomogeneous coordinates are often introduced as a trick. I think this is a misconception and it paves the way for a lot of misunderstandings and errors when working with computer graphics.\n\nIn the following article I will detail a more natural approach to the concept.\n\n## Notations\n\nIn the following we will consider canonical Euclidean planes and spaces. $$x$$, $$y$$ and $$z$$ are the canonical Euclidean dimensions or the vector components depending on the context.\n\nThe initial space is called the projective space.\n\nAll the transformations are supposed to be endomorphisms.\n\n## Usual transformations\n\nLinear transformations in a Euclidean space can be written as a matrix $$M$$ so that the transformed point $$P'$$ of $$P$$ is:\n\n$P' = M \\cdot P$\n\nIn 2D, the matrix is of size $$(2,2)$$. A few examples follow:\n\n• 2D scaling $$(x,y) \\to (sx, sy)$$:\n\\begin{pmatrix} s & 0 \\\\ 0 & s \\\\ \\end{pmatrix}\n• Rotation of angle $$\\theta$$:\n\nIn 2D:\n\n\\begin{pmatrix} \\cos(\\theta) & -\\sin(\\theta) \\\\ \\sin(\\theta) & \\cos(\\theta) \\\\ \\end{pmatrix}\n\nIn 3D, around the $$z$$-axis:\n\n\\begin{pmatrix} \\cos(\\theta) & -\\sin(\\theta) & 0 \\\\ \\sin(\\theta) & \\cos(\\theta) & 0 \\\\ 0 & 0 & 1 \\\\ \\end{pmatrix}\n\nOther linear transformations include shearing and mirroring.\n\n## Non-linear operations\n\nIn the following section we will consider the projective space to be a 2D plane. This is only to make the explanation more visual, since this can be easily generalized to any dimension.\n\nSome operations are not linear in the projective plane. But what if we consider the projective plane as being a plane in a 3D space? Then we can apply any 3D transformation on the points of this plane.\n\nWe say that we promote the plane to the next dimension.\n\nThe fundamental principle here is that the higher dimension allows for more complex transformations, e.g. some specific shearing operations in 3D can be seen as a translation on the plane.\n\nAfter the desired transformations have been applied, we project the resulting points back to the plane.\n\nThere are some parameters we need to specify:\n\n• How do we promote the points? I.e. where does the projective plane lie in 3D?\n• How do we project the transformation result back to the plane?\n\nIn order to answer this we need to analyze how some non-linear operations behave.\n\n### Translation operation\n\n#### Promotion\n\nTranslations are not linear in the 2D-Euclidean space. Now if we promote the input points to a 3D space, they will all lie in a plane. It is obviously simpler to keep the $$x$$ and $$y$$ vectors identical. Then the only parameter for defining the plane is $$z$$.\n\nLet’s have a look at the following shearing of some point $$P=(i,j,k)$$ by the scalars $$a$$ and $$b$$:\n\n$\\begin{pmatrix} 1 & 0 & a \\\\ 0 & 1 & b \\\\ 0 & 0 & 1 \\\\ \\end{pmatrix} \\cdot (i, j, k) = (i+ka, j+kb, k)$\n\nIf $$k=1$$, then the transformed point is $$(i+a, j+b, 1)$$, which is a translation on the plane $$z=1$$.\n\nConsidering the projective plane at $$z=1$$ makes it easy to write a translation matrix.\n\nConclusion: the points from a projective space of any dimension can be promoted to the next dimension by adding a coordinate with the scalar value 1.\n\n#### Projection\n\nIf the transformed points lie in the plane, the projection back to the projective space is trivial: we simply cut off the last dimension.\n\nHowever, the transformed points might end up somewhere outside the plane. In that case we need some way to map the 3D points to the 2D points. There are several ways of doing this.\n\n• One would be to simply cut off the last dimension.\n• Another solution would be to divide all dimensions by the last one when it is non-zero. The last dimension would then always be 1, which matches the position of the plane $$z=1$$ as we defined it previously.\n\nFirst let’s consider some point $$P=(i,j,k)$$ and let’s see what happens if we apply the same transformation as before:\n\n$\\begin{pmatrix} 1 & 0 & a \\\\ 0 & 1 & b \\\\ 0 & 0 & 1 \\\\ \\end{pmatrix} \\cdot (i,j,k) = (i+ka, j+ka, k) = P'$\n\n• If we project by cutting off the last coordinate, the projection of $$P$$ is $$(i, j)$$, the projection of $$P'$$ is $$(i+ka, j+kb)$$. The transformation is generally not the translation by $$(a, b)$$.\n• Using the division by the last coordinate, the projection of $$P$$ is $$(i/k, j/k)$$, the projection of $$P'$$ is $$(i/k+a, j/k+b)$$. This time the transformation is the translation by $$(a, b)$$.\n\nConclusion: the projection back to the projective space is done by dividing all dimensions by the last one when non-zero. This allows for translations in the projective space.\n\n### Perspective operation\n\nWe have seen one example that covers it all. But what if the above analysis was good enough for translations only? Let us have a look at a very common operation in computer graphics: perspective correction.\n\nIn this section we consider a 3D projective space. The higher dimension is 4D and the 4th coordinates we be written $$w$$.\n\nWhen the camera and the screen are centered on the $$z$$-axis, perspective projection results from simple triangular relations, i.e. by dividing $$x$$ and $$y$$ by $$\\alpha+\\beta\\cdot z$$, where $$\\alpha$$ and $$\\beta$$ are scalars depending on the distance from the camera and the screen to the origin.\n\nIn the previous scheme, $$i' = i\\frac{z_s-z_c}{k-z_c}$$ and $$j' = j\\frac{z_s-z_c}{k-z_c}$$.\n\nIn the simplest case the camera is at the origin and the screen is at $$z=1$$, as above. We get the simple transformation where $$\\alpha=1$$ and $$\\beta=1$$, that is, in our example, $$i' = i/k$$ and $$j' = j/k$$.\n\nBut there is no way a matrix multiplication can induce a division by one of the dimension. After all, matrices are a notation for linear transformations.\n\nThere is one way out though: a division by one dimension is performed during the projection back to the projective space. As such, our lead is to “inject” $$z$$ into $$w$$:\n\n$\\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & \\beta & \\alpha \\\\ \\end{pmatrix} \\cdot (i, j, k, 1) = (i, j, k, \\alpha+\\beta k)$\n\nIf we project the result back to the projective space, we get the 3D point $$(\\frac{i}{\\alpha+\\beta k}, \\frac{j}{\\alpha+\\beta k}, \\frac{k}{\\alpha+\\beta k})$$. If we project the result on the plane $$z=z_s$$, we obtain the perspective projection of $$(i, j, k)$$.\n\nConclusion: one more time, the division by the last coordinate is a projection technique that allows for operations in the next dimension to be properly mapped to the projective space.\n\n## Homogeneous coordinates\n\nThe above analysis issued a relation between points in the projective space and points in the next dimension.\n\nMaking use of the above reasoning, homogeneous coordinates offer an alternative notation to cartesian coordinates. We insist on the word ’notation’ as there are no new points nor new spaces. It means that $$(i,j,k)$$ and $$(i,j,k,1)_h$$ specify the same point in 3D.\n\nWe use the $$_h$$ index to avoid confusion between homogeneous coordinates and cartesian coordinates in a higher dimension.\n\nThe mapping between the two coordinate systems is central: the cartesian coordinates are obtained by dividing all dimensions by the homogeneous component if not null, and discarding this last component. This has several consequences:\n\n• Infinity can be represented using finite coordinates, e.g. $$(x, y, 0)_h$$ in 2D.\n• The $$0_h$$ vector is ignored.\n• The origin is the vector where all components are 0 but the last one is 1, e.g. $$(0,0,1)_h$$ in 2D.\n• All points on a line going through $$0_h$$ represent the same point. A non-null scalar multiplication over homogeneous coordinates does not change the point. For instance $$(i, j) = (i, j, 1)_h = (2i, 2j, 2)_h = (\\alpha i, \\alpha j, \\alpha)_h$$ for all scalars $$\\alpha$$.\n\n## Examples\n\n### Rendering pipeline\n\nIn computer graphics, homogeneous coordinates play an important role as they allow for perspective transformations in 3D.\n\nThe 3D rendering pipeline can be summarized as follows (C stands for cartesian, H stands for homogeneous):\n\nVertex data (3D C) → Object data (3D H) → Eye coordinates (3D H) → Clip coordinates (3D H) → Normalized device coordinates (3D C) → Window coordinates (2D C)\n\n• Vertex data are 3D vectors. The graphic pipeline uses homogeneous coordinates for object data, with the homogeneous component being 1, for the reason detailed above.\n• Object data is then translated and rotated as needed via $$(4,4)_h$$-matrices. This yields the homogeneous eye coordinates.\n• A $$(4,4)_h$$-perspective matrix is applied on the eye coordinates, which leads to the homogeneous clip coordinates.\n• The homogeneous clip coordinates are converted to cartesian coordinates, a.k.a. normalized device coordinates (NDC), by dividing every coordinate by the last one, as usual.\n• The NDC are converted to the 2D window coordinates by removing the third dimension and applying some scaling.\n\n### 2D perspective correction\n\nLet us consider the following problem: we want to rectify the perspective of a picture, e.g. we want a tilted building facade picture to appear as if it had been taken orthogonally.\n\nWe proceed with control points: we select pixels in the input picture and tell the program where we would like it to be on the transformed picture.\n\nA $$(2,2)$$-matrix will not do the trick as it is not able to handle translations for instance.\n\nUsing homogeneous coordinates, we can transform the entire plane to the orthogonal picture we would like. In 3D, this can be seen as tilting the plane to make it match the screen.\n\nThis transformation can be embodied within a $$(3,3)_h$$-matrix $$M$$. The answer to our problem is $$M$$.\n\nLet us consider a set of known point pairs $$\\langle \\text{ control point } (i,j), \\text{ target point } (I,J)\\rangle$$, so that\n\n$\\begin{pmatrix} m_{11} & m_{12} & m_{13} \\\\ m_{21} & m_{22} & m_{23} \\\\ m_{31} & m_{32} & m_{33} \\\\ \\end{pmatrix} \\cdot (i, j, 1)_h = (\\alpha I, \\alpha J, \\alpha)_h$\n\nWe could be tempted to think we only need the first 2 rows of the matrix: this is wrong! The result of the matrix multiplication yields the homogeneous coordinates of a point, which still needs to be projected to the 2D plane to be significant. Therefore we need to compute the homogeneous component of every point.\n\nThe previous matrix operation yields the following equations:\n\n$\\left\\{ \\begin{array}{l l} m_{11}i + m_{12}j + m_{13} &= \\alpha I \\\\ m_{21}i + m_{22}j + m_{23} &= \\alpha J \\\\ m_{31}i + m_{32}j + m_{33} &= \\alpha \\\\ \\end{array} \\right.$\n\nor, cancelling the scale factor:\n\n$\\left\\{\\begin{array}{l l} \\frac{m_{11}i + m_{12}j + m_{13}}{m_{31}i + m_{32}j + m_{33}} &= I \\\\ \\frac{m_{21}i + m_{22}j + m_{23}}{m_{31}i + m_{32}j + m_{33}} &= J \\\\ \\end{array} \\right.$\n\nThe matrix has 9 unknown coefficients, but since it is homogeneous, it is defined up to a scale factor. For instance, we have $$M=M/m_{11}$$ if $$m_{11}≠0$$. As such, only 8 coefficient determines a homogeneous matrix uniquely.\n\nWe can solve the system of 8 linearly independent equations constructed as above from 4 well chosen points. The above equation pair yields\n\n$\\left\\{\\begin{array}{l l} m_{11}i + m_{12}j + m_{13} - m_{31}iI - m_{32}jI - m_{33}I &= 0 \\\\ m_{21}i + m_{22}j + m_{23} - m_{31}iJ - m_{32}jJ - m_{33}J &= 0 \\\\ \\end{array} \\right.$\n\nwhich can be written as a multiplication\n\n$\\left\\{\\begin{array}{l l} (i, j, 1, 0, 0, 0, -iI, -jI, -I) \\cdot m^T &= 0 \\\\ (0, 0, 0, i, j, 1, -iJ, -jJ, -J) \\cdot m^T &= 0 \\\\ \\end{array} \\right.$\n\nwith $$m=(m_{11}, m_{12}, m_{13}, m_{21}, m_{22}, m_{23}, m_{31}, m_{32}, m_{33})$$.\n\nWe can build the following matrix from 4 point sets:\n\n$S = \\begin{pmatrix} i_1 & j_1 & 1 & 0 & 0 & 0 & -i_1I_1 & -j_1I_1 & -I_1 \\\\ 0 & 0 & 0 & i_1 & j_1 & 1 & -i_1J_1 & -j_1J_1 & -J_1 \\\\ ... \\\\ i_4 & j_4 & 1 & 0 & 0 & 0 & -i_4I_4 & -j_4I_4 & -I_4 \\\\ 0 & 0 & 0 & i_4 & j_4 & 1 & -i_4J_4 & -j_4J_4 & -J_4 \\\\ \\end{pmatrix}$\n\nIf $$det(S) ≠ 0$$, then the result can be found by solving\n\n$S\\cdot m = 0$\n\nwhich is equivalent to finding the eigenvalues of $$S$$. A singular-value decomposition will numerically find $$m$$.\n\n## Final note\n\nThe present article does not claim to provide any new results when it comes to homogeneous coordinates. It only aims at explaining the theory more visually, and thus making it easier to understand and remember. It all makes perfect sense, it is not just a trick. Mathematically speaking, the key argument that I introduced above is to consider the projective space as an affine hyperplane in the next dimension. Then everything falls into place:\n\n• the link between cartesian and homogeneous coordinates;\n• why this actually works;\n• when we should manipulate homogeneous coordinates, when we should project back to the projective space.\n\nThe central idea can be seen as a hook in the process: we momentarily promote our data to the next dimension, manipulate it in there, then project the result back to the original dimension. It generally does not make sense to exploit a result in homogeneous coordinates." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8433144,"math_prob":0.9995235,"size":13212,"snap":"2020-45-2020-50","text_gpt3_token_len":3644,"char_repetition_ratio":0.1580103,"word_repetition_ratio":0.07155477,"special_character_ratio":0.30146837,"punctuation_ratio":0.114450864,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9998186,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-26T03:29:41Z\",\"WARC-Record-ID\":\"<urn:uuid:517c586d-a1da-464f-9b40-df9eda20ddb8>\",\"Content-Length\":\"20315\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:5c5c3bba-1025-4443-8a52-d9e728d0801b>\",\"WARC-Concurrent-To\":\"<urn:uuid:108b6f35-8b2a-430e-b209-fb2e57d67cec>\",\"WARC-IP-Address\":\"35.185.44.232\",\"WARC-Target-URI\":\"https://ambrevar.xyz/homogeneous/index.html\",\"WARC-Payload-Digest\":\"sha1:XPEW6INBPJLNQ2ZGWVAR2FZ463P5UCGZ\",\"WARC-Block-Digest\":\"sha1:B55T6PMPYV22D5QD4CGYVFKMUT454TY3\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-45/CC-MAIN-2020-45_segments_1603107890273.42_warc_CC-MAIN-20201026031408-20201026061408-00125.warc.gz\"}"}
https://curriculum.illustrativemathematics.org/k5/teachers/grade-4/unit-4/lesson-8/preparation.html	[ "# Lesson 8\n\nBeyond 100,000\n\n### Lesson Purpose\n\nThe purpose of this lesson is to read, write, and represent numbers within 1,000,000 using base-ten blocks, diagrams, and expanded form.\n\n### Lesson Narrative\n\nIn previous grades, students used base-ten blocks, diagrams, and expanded form, a specific way of writing a number as a sum of hundreds, tens, and ones, to represent numbers within 1,000. In this lesson, they extend their understanding of place value to write and represent numbers within 1,000,000.\n\nThroughout the lesson, students determine the value represented by given sets of blocks and consider how to use blocks to represent given numbers. The reasoning students use helps to develop conceptual understanding of expanded form, allows them to practice reading and writing large numbers, and prompts them to think about the relative value of each place. In the next lesson, students generalize observations in terms of the relationship between any two adjacent digits in a multi-digit number.\n\nThe emphasis in this lesson and subsequent ones is not on how to write a number in expanded form. However, this notation may be helpful for students to notice a relationship between the same digits in adjacent places in large numbers. For example, when students expand 23,450 as $$20,\\!000 +3,\\!000+400+50$$ and 2,345 as $$2,\\!000+300+40+5$$, they see that the digit 2 in 23,450 has ten times the value of the 2 in 2,345.\n\nWhen students use strategies that are based on place value and our number system, they are looking for and making use of structure (MP7).\n\n• Representation\n\n### Learning Goals\n\nTeacher Facing\n\n• Represent, read, and write multi-digit whole numbers within 1,000,000, including in expanded form.\n\n### Student Facing\n\n• Let’s read, write, and represent numbers beyond 100,000.\n\n### Required Materials\n\nMaterials to Gather\n\n### Lesson Timeline\n\n Warm-up 10 min Activity 1 15 min Activity 2 10 min Activity 3 10 min Lesson Synthesis 10 min Cool-down 5 min\n\n### Teacher Reflection Questions\n\nReflect on your experience with “How Many Do You See?” in the curriculum. What moves or questions have improved the learning for each of your students during this routine? What improvements would you make next time?\n\n### Suggested Centers\n\n• Greatest of Them All (1–5), Stage 2: Three-digit Numbers (Supporting)\n• Mystery Number (1–4), Stage 4: Fractions with Denominators 5, 8, 10, 12, 100 (Supporting)\n\n### Print Formatted Materials\n\nFor access, consult one of our IM Certified Partners." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.87155825,"math_prob":0.8927485,"size":2290,"snap":"2023-40-2023-50","text_gpt3_token_len":549,"char_repetition_ratio":0.11373578,"word_repetition_ratio":0.053370785,"special_character_ratio":0.25502184,"punctuation_ratio":0.1418919,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9500142,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-10-01T22:43:21Z\",\"WARC-Record-ID\":\"<urn:uuid:89261f74-d6fd-4c73-911e-141c6e626b85>\",\"Content-Length\":\"71656\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:6f596254-6182-4181-9494-a1a823bdc70a>\",\"WARC-Concurrent-To\":\"<urn:uuid:974e3b2d-9c71-4b5f-b008-d38a2b10eede>\",\"WARC-IP-Address\":\"54.196.16.164\",\"WARC-Target-URI\":\"https://curriculum.illustrativemathematics.org/k5/teachers/grade-4/unit-4/lesson-8/preparation.html\",\"WARC-Payload-Digest\":\"sha1:KQEKDWCXM6DS3CUS4LMURD52CCJVZMVD\",\"WARC-Block-Digest\":\"sha1:OFWWBKOX5UMT4LZJBFNEHUC7IOVA4B7W\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233510941.58_warc_CC-MAIN-20231001205332-20231001235332-00494.warc.gz\"}"}
http://ixtrieve.fh-koeln.de/birds/litie/document/31516	[ "# Document (#31516)\n\nAuthor\nHudson, J.\nTitle\nOn-the-job training for cataloging and classification\nSource\nCataloging and classification quarterly. 7(1987) no.4, S.69-78\nYear\n1987\nAbstract\nTraining for cataloging and classification within the Cataloging Department is discussed for two levels of staff, copy catalogers and original catalogers. A general pattern which moves from learning to catalog straightforward materials which require minimal editing to processing materials which are progressively more complex is described for copy catalogers. A survey of heads of cataloging departments reveals that there is a general feeling that cataloging and classification are being shortchanged in library school training and recommendations for such training are made. Training programs for original catalogers also follow a pattern, from working with LC copy to preparing original cataloging records. Some variations to the training patterns for each group are noted. The article concludes with a discussion of continuing education for both groups.\nFootnote\nSimultaneously published as Education and Training for Catalogers and Classifiers\nTheme\nAusbildung\nFormalerschließung\nLocation\nUSA\n\n## Similar documents (content)\n\n1. Gomez, J.; LaGrange, J.: ¬A Chinese challenge : utilizing students for special cataloging projects (1990) 0.31\n```0.3051187 = sum of:\n0.3051187 = product of:\n0.953496 = sum of:\n0.06604984 = weight(abstract_txt:department in 1625) [ClassicSimilarity], result of:\n0.06604984 = score(doc=1625,freq=2.0), product of:\n0.09656785 = queryWeight, product of:\n6.19062 = idf(docFreq=237, maxDocs=42740)\n0.015599059 = queryNorm\n0.6839734 = fieldWeight in 1625, product of:\n1.4142135 = tf(freq=2.0), with freq of:\n2.0 = termFreq=2.0\n6.19062 = idf(docFreq=237, maxDocs=42740)\n0.078125 = fieldNorm(doc=1625)\n0.014908394 = weight(abstract_txt:which in 1625) [ClassicSimilarity], result of:\n0.014908394 = score(doc=1625,freq=1.0), product of:\n0.06505126 = queryWeight, product of:\n1.4215829 = boost\n2.9334934 = idf(docFreq=6181, maxDocs=42740)\n0.015599059 = queryNorm\n0.22917917 = fieldWeight in 1625, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n2.9334934 = idf(docFreq=6181, maxDocs=42740)\n0.078125 = fieldNorm(doc=1625)\n0.046423588 = weight(abstract_txt:materials in 1625) [ClassicSimilarity], result of:\n0.046423588 = score(doc=1625,freq=1.0), product of:\n0.121179886 = queryWeight, product of:\n1.5842146 = boost\n4.903635 = idf(docFreq=861, maxDocs=42740)\n0.015599059 = queryNorm\n0.3830965 = fieldWeight in 1625, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n4.903635 = idf(docFreq=861, maxDocs=42740)\n0.078125 = fieldNorm(doc=1625)\n0.037795503 = weight(abstract_txt:classification in 1625) [ClassicSimilarity], result of:\n0.037795503 = score(doc=1625,freq=1.0), product of:\n0.12094704 = queryWeight, product of:\n1.9383937 = boost\n3.9999528 = idf(docFreq=2127, maxDocs=42740)\n0.015599059 = queryNorm\n0.3124963 = fieldWeight in 1625, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n3.9999528 = idf(docFreq=2127, maxDocs=42740)\n0.078125 = fieldNorm(doc=1625)\n0.16240333 = weight(abstract_txt:original in 1625) [ClassicSimilarity], result of:\n0.16240333 = score(doc=1625,freq=3.0), product of:\n0.22164503 = queryWeight, product of:\n2.6240575 = boost\n5.414848 = idf(docFreq=516, maxDocs=42740)\n0.015599059 = queryNorm\n0.7327181 = fieldWeight in 1625, product of:\n1.7320508 = tf(freq=3.0), with freq of:\n3.0 = termFreq=3.0\n5.414848 = idf(docFreq=516, maxDocs=42740)\n0.078125 = fieldNorm(doc=1625)\n0.1919416 = weight(abstract_txt:copy in 1625) [ClassicSimilarity], result of:\n0.1919416 = score(doc=1625,freq=1.0), product of:\n0.35733968 = queryWeight, product of:\n3.3318465 = boost\n6.8753986 = idf(docFreq=119, maxDocs=42740)\n0.015599059 = queryNorm\n0.5371405 = fieldWeight in 1625, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n6.8753986 = idf(docFreq=119, maxDocs=42740)\n0.078125 = fieldNorm(doc=1625)\n0.2884648 = weight(abstract_txt:catalogers in 1625) [ClassicSimilarity], result of:\n0.2884648 = score(doc=1625,freq=2.0), product of:\n0.40957487 = queryWeight, product of:\n4.1188917 = boost\n6.3746233 = idf(docFreq=197, maxDocs=42740)\n0.015599059 = queryNorm\n0.704303 = fieldWeight in 1625, product of:\n1.4142135 = tf(freq=2.0), with freq of:\n2.0 = termFreq=2.0\n6.3746233 = idf(docFreq=197, maxDocs=42740)\n0.078125 = fieldNorm(doc=1625)\n0.1455089 = weight(abstract_txt:cataloging in 1625) [ClassicSimilarity], result of:\n0.1455089 = score(doc=1625,freq=1.0), product of:\n0.37431583 = queryWeight, product of:\n4.822569 = boost\n4.975782 = idf(docFreq=801, maxDocs=42740)\n0.015599059 = queryNorm\n0.38873297 = fieldWeight in 1625, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n4.975782 = idf(docFreq=801, maxDocs=42740)\n0.078125 = fieldNorm(doc=1625)\n0.32 = coord(8/25)\n```\n2. Kao, M.L.: Cataloging and classification for library technicians (2001) 0.25\n```0.24878125 = sum of:\n0.24878125 = product of:\n1.2439063 = sum of:\n0.092847176 = weight(abstract_txt:materials in 6364) [ClassicSimilarity], result of:\n0.092847176 = score(doc=6364,freq=1.0), product of:\n0.121179886 = queryWeight, product of:\n1.5842146 = boost\n4.903635 = idf(docFreq=861, maxDocs=42740)\n0.015599059 = queryNorm\n0.766193 = fieldWeight in 6364, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n4.903635 = idf(docFreq=861, maxDocs=42740)\n0.15625 = fieldNorm(doc=6364)\n0.075591005 = weight(abstract_txt:classification in 6364) [ClassicSimilarity], result of:\n0.075591005 = score(doc=6364,freq=1.0), product of:\n0.12094704 = queryWeight, product of:\n1.9383937 = boost\n3.9999528 = idf(docFreq=2127, maxDocs=42740)\n0.015599059 = queryNorm\n0.6249926 = fieldWeight in 6364, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n3.9999528 = idf(docFreq=2127, maxDocs=42740)\n0.15625 = fieldNorm(doc=6364)\n0.18752721 = weight(abstract_txt:original in 6364) [ClassicSimilarity], result of:\n0.18752721 = score(doc=6364,freq=1.0), product of:\n0.22164503 = queryWeight, product of:\n2.6240575 = boost\n5.414848 = idf(docFreq=516, maxDocs=42740)\n0.015599059 = queryNorm\n0.84607 = fieldWeight in 6364, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n5.414848 = idf(docFreq=516, maxDocs=42740)\n0.15625 = fieldNorm(doc=6364)\n0.3838832 = weight(abstract_txt:copy in 6364) [ClassicSimilarity], result of:\n0.3838832 = score(doc=6364,freq=1.0), product of:\n0.35733968 = queryWeight, product of:\n3.3318465 = boost\n6.8753986 = idf(docFreq=119, maxDocs=42740)\n0.015599059 = queryNorm\n1.074281 = fieldWeight in 6364, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n6.8753986 = idf(docFreq=119, maxDocs=42740)\n0.15625 = fieldNorm(doc=6364)\n0.50405765 = weight(abstract_txt:cataloging in 6364) [ClassicSimilarity], result of:\n0.50405765 = score(doc=6364,freq=3.0), product of:\n0.37431583 = queryWeight, product of:\n4.822569 = boost\n4.975782 = idf(docFreq=801, maxDocs=42740)\n0.015599059 = queryNorm\n1.3466105 = fieldWeight in 6364, product of:\n1.7320508 = tf(freq=3.0), with freq of:\n3.0 = termFreq=3.0\n4.975782 = idf(docFreq=801, maxDocs=42740)\n0.15625 = fieldNorm(doc=6364)\n0.2 = coord(5/25)\n```\n3. 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Howarth, L.C.: (Re)making the serials cataloger : the SCCTP within an educational framework (2000) 0.24\n```0.24051598 = sum of:\n0.24051598 = product of:\n1.0021499 = sum of:\n0.058125988 = weight(abstract_txt:continuing in 370) [ClassicSimilarity], result of:\n0.058125988 = score(doc=370,freq=1.0), product of:\n0.11173133 = queryWeight, product of:\n1.0756506 = boost\n6.658944 = idf(docFreq=148, maxDocs=42740)\n0.015599059 = queryNorm\n0.52023 = fieldWeight in 370, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n6.658944 = idf(docFreq=148, maxDocs=42740)\n0.078125 = fieldNorm(doc=370)\n0.093763605 = weight(abstract_txt:original in 370) [ClassicSimilarity], result of:\n0.093763605 = score(doc=370,freq=1.0), product of:\n0.22164503 = queryWeight, product of:\n2.6240575 = boost\n5.414848 = idf(docFreq=516, maxDocs=42740)\n0.015599059 = queryNorm\n0.423035 = fieldWeight in 370, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n5.414848 = idf(docFreq=516, maxDocs=42740)\n0.078125 = fieldNorm(doc=370)\n0.1919416 = weight(abstract_txt:copy in 370) [ClassicSimilarity], result of:\n0.1919416 = score(doc=370,freq=1.0), product of:\n0.35733968 = queryWeight, product of:\n3.3318465 = boost\n6.8753986 = idf(docFreq=119, maxDocs=42740)\n0.015599059 = queryNorm\n0.5371405 = fieldWeight in 370, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n6.8753986 = idf(docFreq=119, maxDocs=42740)\n0.078125 = fieldNorm(doc=370)\n0.3532958 = weight(abstract_txt:catalogers in 370) [ClassicSimilarity], result of:\n0.3532958 = score(doc=370,freq=3.0), product of:\n0.40957487 = queryWeight, product of:\n4.1188917 = boost\n6.3746233 = idf(docFreq=197, maxDocs=42740)\n0.015599059 = queryNorm\n0.8625915 = fieldWeight in 370, product of:\n1.7320508 = tf(freq=3.0), with freq of:\n3.0 = termFreq=3.0\n6.3746233 = idf(docFreq=197, maxDocs=42740)\n0.078125 = fieldNorm(doc=370)\n0.1455089 = weight(abstract_txt:cataloging in 370) [ClassicSimilarity], result of:\n0.1455089 = score(doc=370,freq=1.0), product of:\n0.37431583 = queryWeight, product of:\n4.822569 = boost\n4.975782 = idf(docFreq=801, maxDocs=42740)\n0.015599059 = queryNorm\n0.38873297 = fieldWeight in 370, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n4.975782 = idf(docFreq=801, maxDocs=42740)\n0.078125 = fieldNorm(doc=370)\n0.15951402 = weight(abstract_txt:training in 370) [ClassicSimilarity], result of:\n0.15951402 = score(doc=370,freq=1.0), product of:\n0.39796457 = queryWeight, product of:\n4.9725776 = boost\n5.130556 = idf(docFreq=686, maxDocs=42740)\n0.015599059 = queryNorm\n0.4008247 = fieldWeight in 370, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n5.130556 = idf(docFreq=686, maxDocs=42740)\n0.078125 = fieldNorm(doc=370)\n0.24 = coord(6/25)\n```\n5. Ferris, A.M.: Results of an expanded survey on the use of Classification Web : they will use it, if you buy it! 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https://mathematica.stackexchange.com/questions/278807/how-to-plot-a-point-with-two-colors	[ "# How to plot a point with two colors?\n\nHow can I use this type of marker for this?", null, "Plot[{x + 3, x^2 + 1}, {x, -5, 5}, PlotStyle -> Thickness[0.01],\nEpilog -> {Red, PointSize[0.03], Point[{2, 5}]}]\n\n\nI would be nice if the colors are added automatically from the two curves.", null, "f1[x_] := x + 3\nf2[x_] := 1 + x^2\n\nmeshStyle = (# /. Point[x_] :>\nMap[Inset[PieChart[{1, 1}, SectorOrigin -> Bottom],\n#, Center, Scaled[.1]] &, x] &);\n\nPlot[{f1[x], f2[x]}, {x, -5, 5},\nPlotStyle -> Thickness[0.01],\nMeshFunctions -> {f1[#] - f2[#] &},\nMesh -> {{0}},\nMeshStyle -> meshStyle]", null, "• We can define PlotMarkers by any Graphics object and using ListPlot to add such markers of points.\n\n• We using MeshFunctions to get the intersection points of two curves.\n\n• The AspectRatio can be any ratio,here we set AspectRatio -> 1/2.\n\nClear[f, g, hemipoint, plot, indexes, meshs];\nhemipoint =\nGraphics[{{ColorData,\nDisk[{0, 0}, 1, {π/2, 2 π - π/2}]}, {ColorData[\n2], Disk[{0, 0}, 1, {-π/2, π/2}]}}, ImageSize -> 20];\nf[x_] = x + 3;\ng[x_] = x^2 + 1;\nplot = Plot[{f[x], g[x]}, {x, -5, 5}, PlotStyle -> Thickness[0.01],\nMesh -> {{0}}, MeshFunctions -> {f[#] - g[#] &}, MeshStyle -> None];\npts = Cases[plot,\nGraphicsComplex[pts_, rest__] :> pts, ∞][];\nindexes = Cases[plot, Point[index_] :> index, ∞];\nmeshs = pts[[#]] & /@ indexes;\nShow[plot, ListPlot[{meshs[[1, 2]]}, PlotMarkers -> hemipoint],\nAspectRatio -> 1/2]", null, "• Define two types of PlotMarkers.\nClear[hemipoint1,hemipoint2,plot,indexes,meshs];\nhemipoint1 =\nGraphics[{{ColorData,\nDisk[{0, 0}, 1, {π/2, 2 π - π/2}]}, {ColorData[\n2], Disk[{0, 0}, 1, {-π/2, π/2}]}}, ImageSize -> 20];\nhemipoint2 =\nGraphics[{{ColorData,\nDisk[{0, 0}, 1, {π/2, 2 π - π/2}]}, {ColorData[\n1], Disk[{0, 0}, 1, {-π/2, π/2}]}}, ImageSize -> 20];\nf[x_] = x + 3;\ng[x_] = x^2 + 1;\nplot = Plot[{f[x], g[x]}, {x, -5, 5}, PlotStyle -> Thickness[0.01],\nMesh -> {{0}}, MeshFunctions -> {f[#] - g[#] &}, MeshStyle -> None];\npts = Cases[plot,\nGraphicsComplex[pts_, rest__] :> pts, ∞][];\nindexes = Cases[plot, Point[index_] :> index, ∞];\nmeshs = pts[[#]] & /@ indexes;\nShow[plot,\nListPlot[{{meshs[[1, 2]]}, {meshs[[1, 1]]}},\nPlotMarkers -> {hemipoint1, hemipoint2}], AspectRatio -> 1/2]", null, "It's not so easy to do this automatically, and you have to account for the aspect ratio of your plot, without which you end up with a squashed marker. Here's my attempt for manual points and colour selection:\n\nmarker[pos_, radius_, col1_, col2_] :=\nWith[{d = Disk[pos, radius, {-Pi/2, Pi/2}]},\n{EdgeForm[Black], col1, d, col2, Rotate[d, Pi, pos]}]\n\nxrange = {-5, 5};\nyrange = {-5, 30};\naspect = Subtract @@ yrange/Subtract @@ xrange;\nPlot[{x + 3, x^2 + 1}, {x, xrange[], xrange[]},\nPlotRange -> {xrange, yrange}, PlotStyle -> Thickness[0.01],\nEpilog -> {\nRGBColor[0.368417, 0.506779, 0.709798],\nRGBColor[0.880722, 0.611041, 0.142051]]}, AspectRatio -> 1]", null, "• Thanks, it is much more complex than I though even without automatically adding colors. It works great but does not work when I use it like this Show[plot1, plot_with_marker]. How can I fix this?" ]	[ null, "https://i.stack.imgur.com/oSCzT.png", null, "https://i.stack.imgur.com/ley0O.png", null, "https://i.stack.imgur.com/2Cpjg.png", null, "https://i.stack.imgur.com/wCNZ6.png", null, "https://i.stack.imgur.com/GppCC.png", null, "https://i.stack.imgur.com/4aj4r.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5981787,"math_prob":0.99399143,"size":3531,"snap":"2023-40-2023-50","text_gpt3_token_len":1259,"char_repetition_ratio":0.09781684,"word_repetition_ratio":0.4317757,"special_character_ratio":0.4276409,"punctuation_ratio":0.27468354,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9977684,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12],"im_url_duplicate_count":[null,1,null,1,null,1,null,1,null,1,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-11-30T05:46:50Z\",\"WARC-Record-ID\":\"<urn:uuid:9525da5a-04ac-4beb-b6d7-0670b36cc117>\",\"Content-Length\":\"186811\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:35e5b7f5-1ef9-4e22-932c-56797cd0b214>\",\"WARC-Concurrent-To\":\"<urn:uuid:3462ca0a-f7e5-48f0-b2f4-e5d3de271fa9>\",\"WARC-IP-Address\":\"104.18.43.226\",\"WARC-Target-URI\":\"https://mathematica.stackexchange.com/questions/278807/how-to-plot-a-point-with-two-colors\",\"WARC-Payload-Digest\":\"sha1:QPIHSVCXSEXKOOZVSRTFRGVMJK5UOYUL\",\"WARC-Block-Digest\":\"sha1:RNLOONZSZ7DEXU5XUVDAU5KEZPWH6OWB\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100164.87_warc_CC-MAIN-20231130031610-20231130061610-00546.warc.gz\"}"}
https://metacpan.org/release/Statistics-ROC/source/lib/Statistics/ROC.pm	[ "#LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL\n#\n# ROC.pm - A Perl module implementing receiver-operator-characteristic (ROC)\n# curves with nonparametric confidence bounds\n#\n# This program is free software; you may redistribute it and/or\n# modify it under the same terms as Perl itself.\n#\n# This code implements a method for constructing nonparametric confidence\n# for ROC curves described in\n# R.A. Hilgers, Distribution-Free Confidence Bounds for ROC Curves,\n# Meth Inform Med 1991; 30:96-101\n# Additionally some auxilliary functions were ported (and corrected) from\n# Fortran (Applied Statistics, ACM).\n#\n# Written in Perl by Hans A. Kestler.\n# Hans A. Kestler <[email protected]>\n# <[email protected]>\n#\n#\n#LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL\n\npackage Statistics::ROC;\nrequire 5;\nuse Carp;\nuse strict;\nuse vars qw($VERSION @ISA @EXPORT @EXPORT_OK); require Exporter; @ISA = ('Exporter'); @EXPORT = qw( roc rank loggamma betain Betain xinbta Xinbta );$VERSION = '0.01';\n\n# Algorithm 291, Logarithm of the gamma function.\n# in Collected Algorithms of the ACM, Vol II, 1980\n# M.C. Pike and I.D. Hill with remark by M.R. Hoare\n# see also Pike, M.C. and Hill, I.D. (1966). Algorithm 291. Logarithm of the\n# gamma function. Commun. Ass. Comput. Mach., 9,684.\n\nsub loggamma($){ # This procedure evaluates the natural logarithm of gamma(x) for all # x>0, accurate to 10 decimal places. Stirlings formula is used for the # central polynomial part of the procedure my$x= shift; # default arg is @_\nmy ($f,$z);\n\nif($x==0){return 743.746924740801} # this is: loggamma(9.9999999999E-324) if($x < 7)\n{\nfor($z=$x,$f=1;$z<7;$z++) {$x=$z;$f=$z; }$x++;\n$f= -log($f); # log returns the natural logarithm\n}\nelse{ $f=0;}$z=1/($x$x);\nreturn $f+($x-.5)log($x)-$x+.918938533204673+\n(((-.000595238095238$z+.000793650793651)$z-\n.002777777777778)$z+.083333333333333)/$x;\n}\n\n# Algorithm AS 63 with remark AS R19,\n# Computes incomplete beta function ratio\n# K.L. Majumder and G.P. Bhattacharjee (1973). The Incomplete Beta Integral,\n# Appl. Statist.,22:409:411 and\n# G.W. Cran, K.J. Martin and G.E. Thomas (1977).Remark AS R19 and\n# Algorithm AS109, A Remark on Algorithms AS 63: The Incomplete Beta Integral\n# AS 64: Inverse of the Incomplete Beta Function Ratio,\n# Appl. Statist., 26:111-114.\n#\n# Remarks:\n# Complete beta function: B(p,q)=gamma(p)gamma(q)/gamma(p+q)\n# log(B(p,q))=ln(gamma(p))+ln(gamma(q))-ln(gamma(p+q))\n#\n# Incomplete beta function ratio:\n# I_x(p,q)=1/B(p,q) \\int_0^x t^{p-1}(1-t)^{q-1} dt\n#\n# --> log(B(p,q)) has to be supplied to calculate I_x(p,q)\n# log denotes the natural logarithm\n# $beta = log(B(p,q)) #$x = x\n# $p = p #$q = q\n# The subroutine returns I_x(p,q). If an error occurs a negative value\n# {-1,-2} is returned.\n\nsub betain(){\n# Computes incomplete beta function ratio for arguments\n# $x between zero and one,$p and $q positive. # Log of complete beta function,$beta, is assumed to be known.\n\nmy ($x,$p, $q,$beta) = @_;\nmy ($xx,$psq, $cx,$pp, $qq,$index, $betain,$ns, $term,$ai, $rx,$temp);\nmy $ACU=1.0E-14; # accuracy # tests for admissibility of arguments if($p<=0 \|\| $q<=0){ return -1;} if($x<0 \|\| $x>1) { return -2;} # change tail if necessary and determine s$psq=$p+$q; $cx=1-$x;\nif($p<$psq$x){$xx=$cx;$cx=$x;$pp=$q;$qq=$p;$index=1;}\nelse{ $xx=$x; $pp=$p; $qq=$q; $index=0;}$term=1; $ai=1;$betain=1;\n$ns=$qq+$cx$psq;\n\n# use Soper's reduction formulae\n$rx=$xx/$cx; do{ if($ns>=0){$temp=$qq-$ai; if($ns==0){$rx=$xx;}}\nelse{ $temp=$psq; $psq++;}$term = $temp$rx/($pp+$ai);\n$betain+=$term;\n$temp=abs($term); $ai++;$ns--;}\nuntil($temp<=$ACU && $temp<=$ACU$betain); # calculate result$betain = exp($pplog($xx)+($qq-1)log($cx)-$beta)/$pp;\nif($index){ return 1-$betain;}\nelse{ return $betain;} } sub Betain($$){ # Computes the incomplete beta function # by calling loggamma() and betain() my (x, p, q) = @_; if(x==1){return 1;} elsif(x==0){return 0;} else{ return betain(x, p, q,loggamma(p)+loggamma(q)-loggamma(p+q));} } sub max($$){ # computes the maximum of two numbers my ($a, $b) = @_; if($a>$b){ return$a;}\nelse{ return $b;} } # Algorithm AS 109, # Computes inverse of incomplete beta function ratio # G.W. Cran, K.J. Martin and G.E. Thomas (1977).Remark AS R19 and # Algorithm AS109, A Remark on Algorithms AS 63: The Incomplete Beta Integral # AS 64: Inverse of the Incomplete Beta Function Ratio, # Appl. Statist., 26:111-114. # # Remark AS R83 and the correction in vol40(1) of Appl. Statist.(1991), p.236 # have been incorporated in this version. # K.J. Berry, P.W. Mielke, Jr and G.W. Cran (1990) Algorithm AS R83, A Remark # on Algorithm AS 109: Inverse of the Incomplete Beta Function Ratio, # Appl. Statist., 39:309-310. # # Remarks: # # Complete beta function: B(p,q)=gamma(p)gamma(q)/gamma(p+q) # log(B(p,q))=ln(gamma(p))+ln(gamma(q))-ln(gamma(p+q)) # # Incomplete beta function ratio: # alpha = I_x(p,q) = 1/B(p,q) * \\int_0^x t^{p-1}(1-t)^{q-1} dt # # --> log(B(p,q)) has to be supplied to calculate I_x(p,q) # log denotes the natural logarithm #$beta = log(B(p,q))\n# $alpha= I_x(p,q) #$p = p\n# $q = q # The subroutine returns x. If an error occurs a negative value {-1,-2,-3} # is returned. sub xinbta(){ # Computes inverse of incomplete beta function ratio # for given positive values of the arguments$p and $q, #$alpha between zero and one.\n# Log of complete beta function, $beta is assumed to be known. # # copyright by H.A. Kestler, 1998 my ($p, $q,$beta, $alpha) = @_; my ($a, $y,$pp, $qq,$index, $r,$h, $t,$w, $xinbta,$yprev, $prev,$s, $sq,$tx, $adj,$g);\nmy $SAE=-37; my$FPU=10$SAE; my$ACU;\n\n# test for admissibility of parameters\nif($p<=0 \|\|$q<=0){ return -1;}\nif($alpha<0 \|\|$alpha>1) { return -2;}\nif($alpha==0 \|\|$alpha==1){ return $alpha;} # change tail if necessary if($alpha>.5){ $a=1-$alpha; $pp=$q; $qq=$p; $index=1;} else{$a=$alpha;$pp=$p;$qq=$q;$index=0;}\n\n# calculate the initial approximation\n$r=sqrt(-log($a$a));$y=$r-(2.30753+.27061$r)/(1+(.99229+.04481$r)$r);\nif($pp>1 &&$qq > 1)\n{\n$r=($y$y-3)/6;$s=1/($pp+$pp-1); $t=1/($qq+$qq-1);$h=2/($s+$t);\n$w=$ysqrt($h+$r)/$h-($t-$s)($r+5/6-2/(3$h));$xinbta=$pp/($pp+$qqexp($w+$w)); } else {$r=$qq+$qq; $t=1/(9$qq);\n$t=$r(1-$t+$ysqrt($t))3; if($t<=0){\n$xinbta=1-exp((log((1-$a)$qq)+$beta)/$qq); } else{$t=(4$pp+$r-2)/$t; if($t<=1){ $xinbta=exp((log($a$pp)+$beta)/$pp);} else{$xinbta=1-2/($t+1);} } } # solve for$x by a modified newton-raphson method\n# using subroutine betain()\n$r=1-$pp; $t=1-$qq; $yprev=0;$sq=1; $prev=1; if($xinbta<.0001){ $xinbta=.0001;} if($xinbta>.9999){ $xinbta=.9999;}$ACU=10(max(-5/$pp2-1/$a*.2-13,$SAE)); do{$y=betain($xinbta,$pp,$qq,$beta);\nif($y==-1 \|\|$y==-2){ return -3;} # betain returns an exception\n$y=($y-$a)exp($beta+$rlog($xinbta)+$tlog(1-$xinbta));\nif($y$y<=0){ $prev=max($sq,$FPU);}$g=1;\nLabel10: do{\ndo{ $adj=$g$y;$sq=$adj$adj; $g/=3;} while($sq>=$prev);$tx=$xinbta-$adj;}\nuntil($tx>=0 &&$tx<=1);\nif($prev<=$ACU \|\| $y$y<=$ACU){ goto Label12;} if($tx==0 \|\| $tx==1){ goto Label10;}$xinbta=$tx;$yprev=$y;} until($adj==0);\n\nLabel12:\nif($index){ return 1-$xinbta;}\nelse{ return $xinbta;} } sub Xinbta($$){ # Computes the inverse of the incomplete beta function # by calling loggamma() and xinbta() # # copyright by H.A. Kestler, 1998 my (p, q, alpha) = @_; if(alpha==1){return 1;} elsif(alpha==0){return 0;} else{ return xinbta(p, q,loggamma(p)+loggamma(q)-loggamma(p+q), alpha);} } sub rank(\\@){ # Computes the ranks of the values specified as the second # argument (an array). Returns a vector of ranks # corresponding to the input vector. # Different types of ranking are possible ('high', 'low', 'mean'), # and are specified as first argument. # These differ in the way ties of the input vector, i.e. identical # values, are treated: # 'high' --> replace ranks of identical values with their # highest rank # 'low' --> replace ranks of identical values with their # lowest rank # 'mean' --> replace ranks of identical values with the mean # of their rank # # copyright by H.A. Kestler, 1998 my (type, r) = @_; # type: type of ranking 'high', 'low' or 'mean' # r: reference to array of values to be ranked my (@s, s, i, @e, @rk, rk_m); # calculate initial rank's @s=sort{$$r[$a]<=>$$r[b]} 0..#{r}; # sort idx num. by values of @r for(i=0,@rk=@s;i<@rk;i++){ rk[s[i]]=i+1;} # set rank's # treat ties for(i=1,@e=(); i<@s; i++){ if($$r[$s[$i]]==$$r[s[i-1]]){ # test if there are ties push @e,i-1;} # save index numbers of tied values (minus 1) elsif(@e){ # ties have occured and are now being treated if(type eq'mean'){ # calculate mean value of tied ranks rk_m=0; for(@e,e[-1]+1){ rk_m+=rk[s[_]];} rk_m/=@e+1; } for(@e,e[-1]+1){ if(type eq 'high'){ rk[s[_]]=rk[s[e[-1]+1]];} elsif(type eq 'low' ){ rk[s[_]]=rk[s[e]];} elsif(type eq 'mean'){ rk[s[_]]=rk_m;} else{ croak \"Wrong type of ranking (high\|low\|mean).\\n\";} } @e=(); # reinitialize @e } } return @rk; } sub locate(\\@){ # Routine to find the index for table lookup which is below # the value to be interpolated. # Given a reference to an array xx and a value x a value j # is returned such that x is between xx[j] and xx[j+1]. # xx must be monotonic, either increasing or decreasing. # # This routine is adapted from \"Numerical Recipes in C\", # second edition, by Press, Teukolsky, Vetterling and Flannery, # Cambridge University Press, 1992. # It uses bisection to find the right place, which has a # comutational complexity of O(log_2(n)). # # copyright by H.A. Kestler, 1998 my (xx,x)=@_; my (jl,ju)=(0,#{xx}); # initialize lower and upper limits my (jm,ascend); ascend=$$xx[$ju] > $$xx; # test if x is inside of the array if((x>$$xx[$ju] \|\| $x<$$xx[jl]) && ascend) { croak \"Value out of range for table lookup (1): x.\\n\";} if((x<$$xx[$ju] \|\| $x>$$xx[jl]) && !ascend) { croak \"Value out of range for table lookup (2): x.\\n\";} while((ju-jl)>1) { # If we are not yet done jm=int((ju+jl)/2); # compute a midpoint, if(x > xx->[jm] == ascend) { jl=jm;} # and replace either the lower limit else { ju=jm;} # or the upper limit, as appropriate. } return jl; } sub linlocate(\\@$$){ # Routine to find the index for table lookup which is below # the value to be interpolated. # Given a reference to an array$xx and a value $x a value$j\n# is returned such that $x is between$xx[$j] and$xx[$j+1]. #$xx must be monotonic, either increasing or decreasing.\n#\n# Starts searching linearly from an initial index value\n# provided as the third argument.\n# If no index value can be found a negative value is\n# returned, i.e. -1.\n#\n# copyright by H.A. Kestler, 1998\n\nmy ($xx,$x,$index)=@_; my ($jl,$ju)=(0,$#{$xx}); # initialize lower and upper limits my$ascend;\n\n$ascend=$$xx[ju] >$$xx; # test if$x is inside of the array\nif(($x>$$xx[ju] \|\| x<$$xx[$jl]) && $ascend) { croak \"Value out of range for table lookup.\\n\";} if(($x<$$xx[ju] \|\| x>$$xx[$jl]) && !$ascend)\n{ croak \"Value out of range for table lookup.\\n\";}\n\n# step through the table sequentially\nif($ascend &&$xx->[$index]<$x){ # ascending\nwhile($x>$xx->[$index] and$index<=$ju) {$index++;}}\nelsif(!$ascend &&$xx->[$index]>$x){ # descending\nwhile($x<$xx->[$index] and$index<=$ju) {$index++;}}\nelse{ return -1;} # starting index is too high\n\nreturn $index-1; } sub interp(\\@\\@\\@){ # Interpolates (table lookup) piecewise linearly an # array (third argument). Returns # The table is represented by the first two arguments, i.e. @xx and @yy. # Assumes the @xx values to be monotonically increasing. # # copyright by H.A. Kestler, 1998 use vars ('@xx', '@yy', '@x'); local (xx, yy, x)=@_; my ($i, $index, @y); # make checks if(@xx != @yy) {croak \"Sizes of xx and yy arrays are not equal.\\n\";} for($i=0; $i<@x;$i++)\n{\n$index=locate(@xx,$x[$i]);$y[$i]=($yy[$index+1]-$yy[$index])/($xx[$index+1]-$xx[$index]) ($x[$i]-$xx[$index]) +$yy[$index]; } return @y; } sub roc($$\\@){ # ROC (receiver operator characteristic) curves with confidence bounds # # Determines the ROC curve and its nonparametric confidence bounds. # The ROC curve shows the relationship of \"probability of false # alarm\" (x-axis) to \"probability of detection\" (y-axis) for a # certain test. # Or in medical terms: the \"probability of a positive test, given no # disease\" to the \"probability of a positive test, given disease\". # The ROC curve may be used to determine an \"optimal\" cutoff # point for the test. # # The routine takes three arguments: # (1) type of model: 'decrease' or 'increase', this states the assumption # that a higher ('increase') value of the data tends to be an # indicator of a positive test result or for the model 'decrease' # a lower value. # (2) two-sided confidence interval (usually 0.95 is chosen). # (3) the data stored as a list-of-lists: # each entry in this list consits of an \"value / true group\" pair, # i.e. value / disease present. Group values are from {0,1}. # 0 stands for disease (or signal) not present (prior knowledge) and # 1 for disease (or signal) present (prior knowledge). # Example: @s=([2, 0], [12.5, 1], [3, 0], [10, 1], [9.5, 0], [9, 1]); # Notice the small overlap of the groups. The # optimal cutoff point to separate the two groups would be between # 9 and 9.5 if the criterion of optimality is to maximize the # probability of detection and simultaneously minimize the # probability of false alarm. # # Returns a list-of-lists with the three curves: # @ROC=([@lower_b], [@roc], [@upper_b]) each of the curves is # again a list-of-lists with each entry consisting of one (x,y) pair. # The routine impelements the method described in: # R.A. Hilgers, Distribution-Free Confidence Bounds for ROC Curves, # Meth Inform Med 1991; 30:96-101 # # copyright by H.A. Kestler, 1998 my$model_type = shift; # assign\nmy $conf = shift; use vars '@val_grp'; local (val_grp)=@_; my ($cu, $cl,$elem, $n1,$n0, $i,$j);\nmy @grp1=();my @grp0=();\nmy (@f_l_1,@f_m_1,@f_h_1,@f_l_0,@f_m_0,@f_h_0,@mat,@xx,@yy,@x,@y,@index);\nmy (@lower_b ,@roc ,@upper_b, @ROC);\n\n# make checks\nif($conf>=1 \|\|$conf<=0){ croak\n\"The nominal 2-sided confidence limit must be a number of [0,1].\\n\";}\nif($model_type ne 'increase' &&$model_type ne 'decrease'){ croak\n\"Wrong model type specified!\\n\";}\n\n$cu=(sqrt($conf)+1)/2; # calculate the one-sided upper\n$cl=1-$cu; # and lower confidence limits\n\n# extract values\nfor($i=0;$i<@val_grp;$i++){ if($val_grp[$i]==1) { push @grp1,$val_grp[$i];} else { push @grp0,$val_grp[$i];} } # compute ranks and values of inverse incomplete beta function @f_l_1=rank('low' ,@grp1); @f_m_1=rank('mean',@grp1); @f_h_1=rank('high',@grp1); @f_l_0=rank('low' ,@grp0); @f_m_0=rank('mean',@grp0); @f_h_0=rank('high',@grp0);$n1=@grp1; $n0=@grp0; # number of elements in both arrays for$elem (@f_l_1){ $elem=Xinbta($elem,$n1+1-$elem,$cl);} for$elem (@f_m_1){ $elem=Xinbta($elem,$n1+1-$elem,0.5);}\nfor $elem (@f_h_1){$elem=Xinbta($elem,$n1+1-$elem,$cu);}\nfor $elem (@f_l_0){$elem=Xinbta($elem,$n0+1-$elem,$cl);}\nfor $elem (@f_m_0){$elem=Xinbta($elem,$n0+1-$elem,0.5);} for$elem (@f_h_0){ $elem=Xinbta($elem,$n0+1-$elem,$cu);} # merge and sort @mat=(); for($i=0;$i<$n1;$i++){ push @mat, [($grp1[$i], -1, -1, -1,$f_l_1[$i],$f_m_1[$i],$f_h_1[$i])];} for($i=0;$i<$n0;$i++){ push @mat, [($grp0[$i],$f_l_0[$i],$f_m_0[$i],$f_h_0[$i], -1, -1, -1)];} # sort numerically according to value in first column @mat=@mat[sort{$mat[$a] <=>$mat[$b]} 0..$#mat];\n\n# for practical purposes augment @mat and fill missing data (-1)\n# at the beginning and end of the matrix\nunshift @mat,[-1, 0, 0, Xinbta(1,$n0,$cu), 0, 0, Xinbta(1,$n1,$cu)];\npush @mat,[-1, Xinbta($n0,1,$cl), 1, 1, Xinbta($n1,1,$cl), 1, 1];\nfor($i=1;$mat[$i]==-1;$i++){\n$mat[$i]=0; $mat[$i]=0; $mat[$i]=$mat;} for($i=1;$mat[$i]==-1; $i++){$mat[$i]=0;$mat[$i]=0;$mat[$i]=$mat;}\nfor($i=$#mat-1;$mat[$i]==-1; $i--){$mat[$i]=$mat[$#mat];$mat[$i]=1;$mat[$i]=1;} for($i=$#mat-1;$mat[$i]==-1;$i--){\n$mat[$i]=$mat[$#mat]; $mat[$i]=1; $mat[$i]=1;}\n\n# replace missing data (-1) with a piecewise linear interpolation\nfor($j=1;$j<7;$j++) # iterate thru columns { for($i=1,@xx=(),@yy=(),@x=(),@index=();$i<$#mat;$i++){ push @xx,$mat[$i] if$mat[$i][$j] !=-1;\npush @yy, $mat[$i][$j] if$mat[$i][$j] !=-1;\npush @x, $mat[$i] if $mat[$i][$j] ==-1; push @index,$i if $mat[$i][$j] ==-1; } @y=interp(@xx,@yy,@x); for($i=0;$i<@index;$i++){ $mat[$index[$i]][$j]=$y[$i];}\n}\n\n# calculate (x,y) pairs of ROC curve and its limit curves\n# (lower, ROC, upper) according to specified model\nfor($i=0,@lower_b=(),@roc=(),@upper_b=(),;$i<@mat;$i++){ if($model_type eq 'decrease'){\npush @lower_b, [($mat[$i], $mat[$i])];\npush @roc, [($mat[$i], $mat[$i])];\npush @upper_b, [($mat[$i], $mat[$i])];\n}\nelse{\npush @lower_b, [(1-$mat[$i], 1-$mat[$i])];\npush @roc, [(1-$mat[$i], 1-$mat[$i])];\npush @upper_b, [(1-$mat[$i], 1-$mat[$i])];\n}\n}\n\n@ROC=([@lower_b], [@roc], [@upper_b]);\nreturn @ROC;\n}\n\n# Autoload methods go after =cut, and are processed by the autosplit program.\n\n1;\n__END__\n\nStatistics::ROC - receiver-operator-characteristic (ROC) curves with nonparametric confidence bounds\n\nuse Statistics::ROC;\n\nmy ($y) = loggamma($x);\nmy ($y) = betain($x, $p,$q, $beta); my ($y) = Betain($x,$p, $q); my ($y) = xinbta($p,$q, $beta,$alpha);\nmy ($y) = Xinbta($p, $q,$alpha);\nmy (@rk) = rank($type, \\@r); my (@ROC) = roc($model_type,$conf,\\@val_grp); =head1 DESCRIPTION This program determines the ROC curve and its nonparametric confidence bounds for data categorized into two groups. A ROC curve shows the relationship of B<probability of false alarm> (x-axis) to B<probability of detection> (y-axis) for a certain test. Expressed in medical terms: the B<probability of a positive test, given no disease> to the B<probability of a positive test, given disease>. The ROC curve may be used to determine an I<optimal> cutoff point for the test. The main function is B<roc()>. The other exported functions are used by B<roc()>, but might be useful for other nonparametric statistical procedures. =over 4 =item B<loggamma> This procedure evaluates the natural logarithm of gamma(x) for all x>0, accurate to 10 decimal places. Stirlings formula is used for the central polynomial part of the procedure. For C<x=0> a value of 743.746924740801 will be returned: this is loggamma(9.9999999999E-324). =item B<betain> Computes incomplete beta function ratio Remarks: Complete beta function: B(p,q)=gamma(p)gamma(q)/gamma(p+q) log(B(p,q))=ln(gamma(p))+ln(gamma(q))-ln(gamma(p+q)) Incomplete beta function ratio: I_x(p,q)=1/B(p,q) * \\int_0^x t^{p-1}(1-t)^{q-1} dt --> log(B(p,q)) has to be supplied to calculate I_x(p,q) log denotes the natural logarithm$beta = log(B(p,q))\n$x = x$p = p\n$q = q The subroutine returns I_x(p,q). If an error occurs a negative value {-1,-2} is returned. =item B<Betain> Computes the incomplete beta function by calling B<loggamma()> and B<betain()>. =item B<xinbta> Computes inverse of incomplete beta function ratio Remarks: Complete beta function: B(p,q)=gamma(p)gamma(q)/gamma(p+q) log(B(p,q))=ln(gamma(p))+ln(gamma(q))-ln(gamma(p+q)) Incomplete beta function ratio: alpha = I_x(p,q) = 1/B(p,q) * \\int_0^x t^{p-1}(1-t)^{q-1} dt --> log(B(p,q)) has to be supplied to calculate I_x(p,q) log denotes the natural logarithm$beta = log(B(p,q))\n$alpha= I_x(p,q)$p = p\n$q = q The subroutine returns x. If an error occurs a negative value {-1,-2,-3} is returned. =item B<Xinbta> Computes the inverse of the incomplete beta function by calling B<loggamma()> and B<xinbta()>. =item B<rank> Computes the ranks of the values specified as the second argument (an array). Returns a vector of ranks corresponding to the input vector. Different types of ranking are possible ('high', 'low', 'mean'), and are specified as first argument. These differ in the way ties of the input vector, i.e. identical values, are treated: =over 10 =item B<high>: replace ranks of identical values with their highest rank =item * B<low>: replace ranks of identical values with their lowest rank =item * B<mean>: replace ranks of identical values with the mean of their ranks =back =item B<roc> Determines the ROC curve and its nonparametric confidence bounds. The ROC curve shows the relationship of \"probability of false alarm\" (x-axis) to \"probability of detection\" (y-axis) for a certain test. Or in medical terms: the \"probability of a positive test, given no disease\" to the \"probability of a positive test, given disease\". The ROC curve may be used to determine an \"optimal\" cutoff point for the test. The routine takes three arguments: (1) type of model: 'decrease' or 'increase', this states the assumption that a higher ('increase') value of the data tends to be an indicator of a positive test result or for the model 'decrease' a lower value. (2) two-sided confidence interval (usually 0.95 is chosen). (3) the data stored as a list-of-lists: each entry in this list consits of an \"value / true group\" pair, i.e. value / disease present. Group values are from {0,1}. 0 stands for disease (or signal) not present (prior knowledge) and 1 for disease (or signal) present (prior knowledge). Example: @s=([2, 0], [12.5, 1], [3, 0], [10, 1], [9.5, 0], [9, 1]); Notice the small overlap of the groups. The optimal cutoff point to separate the two groups would be between 9 and 9.5 if the criterion of optimality is to maximize the probability of detection and simultaneously minimize the probability of false alarm. Returns a list-of-lists with the three curves: @ROC=([@lower_b], [@roc], [@upper_b]) each of the curves is again a list-of-lists with each entry consisting of one (x,y) pair. =back =over 4 =head2 Examples$,=\" \";\nprint loggamma(10), \"\\n\";\nprint Xinbta(3,4,Betain(.6,3,4)),\"\\n\";\n\n@e=(0.7, 0.7, 0.9, 0.6, 1.0, 1.1, 1,.7,.6);\nprint rank('low',@e),\"\\n\";\nprint rank('high',@e),\"\\n\";\nprint rank('mean',@e),\"\\n\";\n\n@var_grp=([1.5,0],[1.4,0],[1.4,0],[1.3,0],[1.2,0],[1,0],[0.8,0],\n[1.1,1],[1,1],[1,1],[0.9,1],[0.7,1],[0.7,1],[0.6,1]);\n@curves=roc('decrease',0.95,@var_grp);\nprint \"$curves$curves \\n\";\n\nHans A. Kestler, I<[email protected]> B<or>\nI<[email protected]>\n\nPerl/Tk userinterface for drawing ROC curves (requires installed Tk and X11 on MacOS X).\n\nR.A. Hilgers, Distribution-Free Confidence Bounds for ROC Curves (1991),\nI<Meth Inform Med>, 30:96-101\n\nAlgorithm 291, Logarithm of the gamma function.\nI<Collected Algorithms of the ACM>, Vol II, 1980\n\nI<Numerical Recipes in C>, second edition, by Press, Teukolsky, Vetterling and Flannery,\nCambridge University Press, 1992.\n\nG.W. Cran, K.J. Martin and G.E. Thomas (1977).Remark AS R19 and\nAlgorithm AS109, A Remark on Algorithms AS 63: The Incomplete Beta Integral\nAS 64: Inverse of the Incomplete Beta Function Ratio,\nI<Appl Statist>, 26:111-114.\n\nK.J. Berry, P.W. Mielke, Jr and G.W. Cran (1990) Algorithm AS R83, A Remark\non Algorithm AS 109: Inverse of the Incomplete Beta Function Ratio,\nI<Appl Statist>, 39:309-310.\n\n=cut" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5065602,"math_prob":0.99644506,"size":23263,"snap":"2020-10-2020-16","text_gpt3_token_len":8153,"char_repetition_ratio":0.11556817,"word_repetition_ratio":0.31495583,"special_character_ratio":0.39036238,"punctuation_ratio":0.19945674,"nsfw_num_words":1,"has_unicode_error":false,"math_prob_llama3":0.9999745,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-04-01T02:14:47Z\",\"WARC-Record-ID\":\"<urn:uuid:79f9fe68-bbb2-4d4b-91ae-7fd278dc1c5a>\",\"Content-Length\":\"42519\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:9e3d6eed-586a-4911-8485-b1b5c36771cc>\",\"WARC-Concurrent-To\":\"<urn:uuid:193a7c61-7920-42e6-984e-367e9d1c18b0>\",\"WARC-IP-Address\":\"151.101.194.217\",\"WARC-Target-URI\":\"https://metacpan.org/release/Statistics-ROC/source/lib/Statistics/ROC.pm\",\"WARC-Payload-Digest\":\"sha1:3YVKBPUST3AOVWM7BREHHKKHYHZFFUVY\",\"WARC-Block-Digest\":\"sha1:A6ALH3XQVDGOPIO3TKWCYRATF55ZSZHU\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-16/CC-MAIN-2020-16_segments_1585370505359.23_warc_CC-MAIN-20200401003422-20200401033422-00164.warc.gz\"}"}
https://www.emworks.com/application/inductance-capacitance-and-resistance-of-a-pcb-structure	[ "HOME / Applications / Inductance Capacitance and Resistance of a PCB structure\n\n# Inductance Capacitance and Resistance of a PCB structure\n\nUsed Tools:", null, "", null, "## Parasitic parameters of a PCB\n\nEMI, or electromagnetic interference, is an undesirable electromagnetic noise from a device or a system that interferes with the normal operation of neighboring devices or systems.\nThe basic process of EMI modeling and prediction requires the extraction of parasitic parameters of a PCB and circuit components to build high frequency circuit models.\n\n## Parasitic capacitance, parasitic resistance, and parasitic inductance\n\nIn electronic design automation (EDA), parasitic extraction is the calculation of the parasitic effects in both the designed devices and the required wiring interconnects of an electronic circuit: parasitic capacitance, parasitic resistance, and parasitic inductance. In this article, we illustrate the use of EMS to calculate these circuit parameters and compare the results to published data .\n\n## Soildworks PCB model\n\nA PCB structure shown in Figure 1, contains two copper traces on a 4-oz FR4 PCB square board and a 5 mils thickness copper Ground. Some parameters used in simulation are given below where all dimensions are in mils.\nConductivity of Copper = 5.8 107S/m\nRelative Permittivity of FR4 = 4.4", null, "Figure 1 - A PCB structure used for simulation where all dimensions are in mils\n\n## Capacitance calculation\n\nTo compute the parasitic capacitance of the PCB structure, shown in Figure 1, the Electrostatic module is invoked. Figure 2 shows the model and the mesh for the PCB structure.", null, "Figure 2 - Model and mesh of the PCB structure\n\n## Three floating conductors\n\nTo account for the capacitance of a given conductor, it is assigned a floating boundary condition, including the ground plane, in EMS. As a result, this PCB structure has three floating conductors, i.e. the left and right traces as well as the ground plane.\nThe EMS capacitance results and that of the Reference are shown in Figure 3 and Table 1.", null, "Figure 3 - Parasitic capacitance computed by EMS\n\n Reference EMS The capacitance between the left trace and the ground plane -4.3047 pF -4.3563 pF The capacitance between the right trace and the ground plane -4.3046 pF -4.3552 pF The capacitance between the two copper traces -0.1673 pF -0.1825 pF\nTable 1 - Capacitance results of EMS compared to Reference \n\n## DC Inductance & DC Resistance calculation\n\nTo compute the DC resistance and inductance of the PCB structure, shown in Figure 1, the EMS Magnetostatic module is invoked. To compute the DC inductance and DC resistance of the copper traces, they are modeled as coils. The DC inductance and DC resistance results obtained by EMS and compared to Reference , are shown below:", null, "Figure 4 - DC Resistance computed by EMS", null, "Figure 5 - DC Inductance computed by EMS\n\nDC loop Inductance is obtained by the following formula:\nL Loop = L11+L22- 2*M12 ; where L11, L22: self inductances; M12: mutual inductance\n\n Reference EMS DC resistance 5.4304 m Ohm 5.4304 m Ohm DC Loop inductance 50.742 n Henry 54.775 n Henry\nTable 2 - DC resistance and DC inductance results produced by EMS and compared to Reference \n\n## AC Inductance & AC Resistance calculation\n\nIn addition to DC inductance and resistance calculation, EMS is equipped with AC Magnetic and eddy current capabilities which are used to compute the AC resistance and AC loop inductance for the PCB structure at hand at the frequencies: 1Khz, 2Khz, 5Khz, 10 KHz, 20 KHz, 50 KHz, 100 KHz, 200 KHz, 500 KHz, 1 Mhz.\n\n### Reduced model\n\nDue to the small size of the skin depth of the field for the conducting regions, i.e. order of 1e-005 to 1e-004 mm, important computer resources, both in terms of CPU and RAM, are needed. Therefore, only 1/20 of the model is simulated as shown in Figure 6. In turn, the obtained inductance and resistance by the reduced model are multiplied by 20 to recover the full model results.", null, "Figure 6 - 1/20 of the structure is modeled for AC Magnetic analysis\n\nSimilar to the above Magnetostatic analysis, the two traces are modeled as coils. In order to account for the skin depth in the AC resistance calculation, the coils are modeled as solids, i.e. wound coils do not support eddy current. Figure 7 shows the AC resistance for frequencies from 1 KHz to 1 MHz computed by EMS and compared to Reference . Whereas Figure 8 shows the AC inductance results and comparison for the same frequency range.", null, "Figure 7 - AC resistance computed by EMS and compared to Reference", null, "Figure 8 - AC inductance computed by EMS and compared to Reference \n\n## Conclusion\n\nClearly, EMS compares well with the capacitance, AC and DC inductance and resistance published results for PCB structures. Hence EMS can readily be used to easily and quickly extract parasitic parameters for PCB and electronic structures which, in turn, can be used to build high frequency circuit models.\n\n## References\n\n Jingen Qian, “RF Models for Active IPEMs”, MS Thesis in Electrical Engineering, Virginia Polytechnic Institute and State University, January 31, 2003." ]	[ null, "https://www.emworks.com/media/images/product/large/EMS-logo.png", null, "https://www.emworks.com/assets/img/cad/solidworks-icon.png", null, "https://www.emworks.com/ckfinder/userfiles/images/PCB-structure-used-for-simulation.jpg", null, "https://www.emworks.com/ckfinder/userfiles/images/model-and-mesh-of-the-PCB-structure.jpg", null, "https://www.emworks.com/ckfinder/userfiles/images/Parasitic-capacitance-computed-by-EMS.jpg", null, "https://www.emworks.com/ckfinder/userfiles/images/DC-Resistance-computed-by-EMS.jpg", null, "https://www.emworks.com/ckfinder/userfiles/images/DC-Inductance-computed-by-EMS.jpg", null, "https://www.emworks.com/ckfinder/userfiles/images/simulate-a-portion-of-the-model.jpg", null, "https://www.emworks.com/ckfinder/userfiles/images/AC-resistance-comparison.png", null, "https://www.emworks.com/ckfinder/userfiles/images/AC-inductance-comparison.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8653579,"math_prob":0.9625016,"size":3021,"snap":"2022-40-2023-06","text_gpt3_token_len":772,"char_repetition_ratio":0.13523367,"word_repetition_ratio":0.011904762,"special_character_ratio":0.24793115,"punctuation_ratio":0.116949156,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9900506,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],"im_url_duplicate_count":[null,null,null,null,null,5,null,5,null,5,null,5,null,5,null,5,null,5,null,5,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-02-04T11:21:28Z\",\"WARC-Record-ID\":\"<urn:uuid:7ceb3729-ec06-4ced-8cc9-18d0873f0050>\",\"Content-Length\":\"92948\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a5141c4b-331a-4b01-8c49-a1e1ce6c3775>\",\"WARC-Concurrent-To\":\"<urn:uuid:e38d6900-1fa2-4eaa-af5a-30fd521c8c06>\",\"WARC-IP-Address\":\"192.124.249.7\",\"WARC-Target-URI\":\"https://www.emworks.com/application/inductance-capacitance-and-resistance-of-a-pcb-structure\",\"WARC-Payload-Digest\":\"sha1:3WGSOBT2NFPRO4TCLWTYW7FIBDSE3W6S\",\"WARC-Block-Digest\":\"sha1:CWKAT643YUTT7MC54LPLRRSIMAQWPAQ3\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764500126.0_warc_CC-MAIN-20230204110651-20230204140651-00563.warc.gz\"}"}
http://neuralensemble.blogspot.com/2015/04/elephant-010-released.html	[ "## Wednesday, April 8, 2015\n\n### Elephant 0.1.0 released\n\nWe are pleased to announce the first release of the Elephant toolbox for the analysis of neurophysiology data.\n\nElephant builds on the Python scientific stack (NumPy, SciPy) to provide a set of well-tested analysis functions for spike train data and time series recordings from electrodes, such as spike train statistics, power spectrum analysis, filtering, cross-correlation and spike-triggered averaging. The toolbox also provides tools to generate artificial spike trains, either from stochastic processes or by controlled perturbations of real spike trains. Elephant is built on the Neo data model, and takes advantage of the broad file-format support provided by the Neo library. A bridge to the Pandas data analysis library is also provided.\n\nElephant is a community-based effort, aimed at providing a common platform to test and distribute code from different laboratories, with the goal of improving the reproducibility of modern neuroscience research. If you are a neuroscience researcher or student using Python for data analysis, please consider joining us, either to contribute your own code or to help with code review and testing.\n\nElephant is the direct successor to NeuroTools and maintains ties to complementary projects such as OpenElectrophy and SpykeViewer. It is also the default tool for electrophysiology data analysis in the Human Brain Project.\n\nAs a simple example, let's generate some artificial spike train data using a homogeneous Poisson process:\n\nfrom elephant.spike_train_generation import homogeneous_poisson_process\nfrom quantities import Hz, s, ms\nspiketrains = [\nhomogeneous_poisson_process(rate=10.0Hz, t_start=0.0s, t_stop=100.0s)\nfor i in range(100)]\n\nand visualize it in Matplotlib:\n\nimport matplotlib.pyplot as plt\nimport numpy as np\nfor i, spiketrain in enumerate(spiketrains):\nt = spiketrain.rescale(ms)\nplt.plot(t, i np.ones_like(t), 'k.', markersize=2)\nplt.axis('tight')\nplt.xlim(0, 1000)\nplt.xlabel('Time (ms)', fontsize=16)\nplt.ylabel('Spike Train Index', fontsize=16)\nplt.gca().tick_params(axis='both', which='major', labelsize=14)\nplt.show()", null, "Now we calculate the coefficient of variation of the inter-spike interval for each of the 100 spike trains.\n\nfrom elephant.statistics import isi, cv\ncv_list = [cv(isi(spiketrain)) for spiketrain in spiketrains]\n\nAs expected for a Poisson process, the values cluster around 1:\n\nplt.hist(cv_list)\nplt.xlabel('CV', fontsize=16)\nplt.ylabel('count', fontsize=16)\nplt.gca().tick_params(axis='both', which='major', labelsize=14)\nplt.show()", null, "" ]	[ null, "http://elephant.readthedocs.org/en/latest/_images/tutorial_1_figure_1.png", null, "http://elephant.readthedocs.org/en/latest/_images/tutorial_1_figure_2.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7161352,"math_prob":0.82869303,"size":4954,"snap":"2019-13-2019-22","text_gpt3_token_len":1119,"char_repetition_ratio":0.12020202,"word_repetition_ratio":0.9583975,"special_character_ratio":0.21760194,"punctuation_ratio":0.14994487,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9686778,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,2,null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-03-23T11:16:35Z\",\"WARC-Record-ID\":\"<urn:uuid:5a7404c6-4df0-4727-ba4b-f255e7d0d6eb>\",\"Content-Length\":\"64393\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:52858443-612c-4a12-b4a9-df86cb743ad1>\",\"WARC-Concurrent-To\":\"<urn:uuid:15d646ef-bb16-410b-be3e-c24f96fcdced>\",\"WARC-IP-Address\":\"172.217.7.193\",\"WARC-Target-URI\":\"http://neuralensemble.blogspot.com/2015/04/elephant-010-released.html\",\"WARC-Payload-Digest\":\"sha1:EWPQC3V7NX75KGF6ACMTE7RWPPWY2XQA\",\"WARC-Block-Digest\":\"sha1:LZNJ4WYB7XFC5SQQKYNAXEVZXNP5CC4B\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-13/CC-MAIN-2019-13_segments_1552912202781.83_warc_CC-MAIN-20190323101107-20190323123107-00211.warc.gz\"}"}