Datasets:

Infi-MM
/

InfiMM-WebMath-40B

Dataset card Data Studio Files Files and versions Community

Split (1)

train · ~22.8M rows (showing the first 514k)

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.

URL stringlengths 15 1.68k	text_list listlengths 1 199	image_list listlengths 1 199	metadata stringlengths 1.19k 3.08k
https://de.maplesoft.com/support/help/maple/view.aspx?path=DEtools%2FXchange	[ "", null, "Xchange - Maple Help\n\nDEtools\n\n Xchange\n change variables in an ODE point symmetry generator", null, "Calling Sequence Xchange(tr, X, y(x), [t, u(t)])", null, "Parameters\n\n tr - set of transformation equations of the form $\\left\\{x=..,y\\left(x\\right)=..\\right\\}$ from the old variables on the left hand side to the new variables on the right hand side X - list of coefficients of a point symmetry generator (infinitesimals) as in [xi(x, y), eta(x, y)] y(x) - 'dependent variable' of the problem (it can be any indeterminate function of one variable) [t, u(t)] - (optional) new independent and dependent variables (required when they cannot be inferred from the transformation)", null, "Description\n\n • The Xchange command receives a change of variables transformation set and a pair of infinitesimals $\\left[\\mathrm{\\xi },\\mathrm{\\eta }\\right]$, representing the coefficients of a point symmetry generator of an ODE, and the dependent variable y(x), and returns the symmetry in the new variables. The change of variables is performed by making calls to dchange, and hence the same extra arguments accepted by dchange are accepted by Xchange as well. This change of variables takes into account that $\\left[\\mathrm{\\xi }\\left(x,y\\right),\\mathrm{\\eta }\\left(x,y\\right)\\right]$ are the coefficients of a differential operator $\\mathrm{expr}$. Thus, if the new variables are $\\left\\{t,u\\left(t\\right)\\right\\}$, then the returned list is constituted by the coefficients of the differential operator $\\mathrm{expr}$, where $\\mathrm{ξ2}\\left(t,u\\left(t\\right)\\right)$ and $\\mathrm{η2}\\left(t,u\\left(t\\right)\\right)$ also incorporate any factor that appears when changing variables in $\\frac{d}{\\mathrm{dx}}$ and $\\frac{d}{\\mathrm{dy}}$.\n • This function is part of the DEtools package, and so it can be used in the form Xchange(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[Xchange](..).", null, "Examples\n\n > $\\mathrm{with}\\left(\\mathrm{DEtools}\\right):$\n\nNote that the infinitesimals $\\left[\\mathrm{\\xi },\\mathrm{\\eta }\\right]$ can be expressed in terms of $y\\left(x\\right)$, but also directly in terms of $y$:\n\n > $X≔\\left[-y,x\\right]$\n ${X}{≔}\\left[{-}{y}{,}{x}\\right]$ (1)\n > $\\mathrm{tr}≔\\left\\{x=u\\left(t\\right)\\mathrm{cos}\\left(t\\right),y\\left(x\\right)=u\\left(t\\right)\\mathrm{sin}\\left(t\\right)\\right\\}$\n ${\\mathrm{tr}}{≔}\\left\\{{x}{=}{u}{}\\left({t}\\right){}{\\mathrm{cos}}{}\\left({t}\\right){,}{y}{}\\left({x}\\right){=}{u}{}\\left({t}\\right){}{\\mathrm{sin}}{}\\left({t}\\right)\\right\\}$ (2)\n > $\\mathrm{Xchange}\\left(\\mathrm{tr},X,y\\left(x\\right)\\right)$\n $\\left[{1}{,}{0}\\right]$ (3)\n > $X≔\\left[\\frac{1}{x+y},1\\right]$\n ${X}{≔}\\left[\\frac{{1}}{{x}{+}{y}}{,}{1}\\right]$ (4)\n > $\\mathrm{tr}≔\\left\\{x=-\\mathrm{LambertW}\\left(-r\\mathrm{exp}\\left(-s\\left(r\\right)-1\\right)\\right)-s\\left(r\\right)-1,y\\left(x\\right)=s\\left(r\\right)\\right\\}$\n ${\\mathrm{tr}}{≔}\\left\\{{x}{=}{-}{\\mathrm{LambertW}}{}\\left({-}{r}{}{{ⅇ}}^{{-}{s}{}\\left({r}\\right){-}{1}}\\right){-}{s}{}\\left({r}\\right){-}{1}{,}{y}{}\\left({x}\\right){=}{s}{}\\left({r}\\right)\\right\\}$ (5)\n > $\\mathrm{Xchange}\\left(\\mathrm{tr},X,y\\left(x\\right)\\right)$\n $\\left[{0}{,}{1}\\right]$ (6)\n\nThis example illustrates a more general change of variables where the new variables are $\\left[v,u\\left(v\\right)\\right]$:\n\n > $X≔\\left[x,y\\right]$\n ${X}{≔}\\left[{x}{,}{y}\\right]$ (7)\n > $\\mathrm{tr}≔\\left\\{x=\\mathrm{\\phi }\\left(u,v\\left(u\\right)\\right),y\\left(x\\right)=\\mathrm{\\psi }\\left(u,v\\left(u\\right)\\right)\\right\\}$\n ${\\mathrm{tr}}{≔}\\left\\{{x}{=}{\\mathrm{\\phi }}{}\\left({u}{,}{v}{}\\left({u}\\right)\\right){,}{y}{}\\left({x}\\right){=}{\\mathrm{\\psi }}{}\\left({u}{,}{v}{}\\left({u}\\right)\\right)\\right\\}$ (8)\n > $\\mathrm{Xchange}\\left(\\mathrm{tr},X,y\\left(x\\right),\\left[u,v\\left(u\\right)\\right]\\right)$\n $\\left[\\frac{{\\mathrm{\\phi }}{}\\left({u}{,}{v}\\right){}\\left(\\frac{{\\partial }}{{\\partial }{v}}\\phantom{\\rule[-0.0ex]{0.4em}{0.0ex}}{\\mathrm{\\psi }}{}\\left({u}{,}{v}\\right)\\right){-}{\\mathrm{\\psi }}{}\\left({u}{,}{v}\\right){}\\left(\\frac{{\\partial }}{{\\partial }{v}}\\phantom{\\rule[-0.0ex]{0.4em}{0.0ex}}{\\mathrm{\\phi }}{}\\left({u}{,}{v}\\right)\\right)}{\\left(\\frac{{\\partial }}{{\\partial }{u}}\\phantom{\\rule[-0.0ex]{0.4em}{0.0ex}}{\\mathrm{\\phi }}{}\\left({u}{,}{v}\\right)\\right){}\\left(\\frac{{\\partial }}{{\\partial }{v}}\\phantom{\\rule[-0.0ex]{0.4em}{0.0ex}}{\\mathrm{\\psi }}{}\\left({u}{,}{v}\\right)\\right){-}\\left(\\frac{{\\partial }}{{\\partial }{v}}\\phantom{\\rule[-0.0ex]{0.4em}{0.0ex}}{\\mathrm{\\phi }}{}\\left({u}{,}{v}\\right)\\right){}\\left(\\frac{{\\partial }}{{\\partial }{u}}\\phantom{\\rule[-0.0ex]{0.4em}{0.0ex}}{\\mathrm{\\psi }}{}\\left({u}{,}{v}\\right)\\right)}{,}{-}\\frac{{\\mathrm{\\phi }}{}\\left({u}{,}{v}\\right){}\\left(\\frac{{\\partial }}{{\\partial }{u}}\\phantom{\\rule[-0.0ex]{0.4em}{0.0ex}}{\\mathrm{\\psi }}{}\\left({u}{,}{v}\\right)\\right){-}{\\mathrm{\\psi }}{}\\left({u}{,}{v}\\right){}\\left(\\frac{{\\partial }}{{\\partial }{u}}\\phantom{\\rule[-0.0ex]{0.4em}{0.0ex}}{\\mathrm{\\phi }}{}\\left({u}{,}{v}\\right)\\right)}{\\left(\\frac{{\\partial }}{{\\partial }{u}}\\phantom{\\rule[-0.0ex]{0.4em}{0.0ex}}{\\mathrm{\\phi }}{}\\left({u}{,}{v}\\right)\\right){}\\left(\\frac{{\\partial }}{{\\partial }{v}}\\phantom{\\rule[-0.0ex]{0.4em}{0.0ex}}{\\mathrm{\\psi }}{}\\left({u}{,}{v}\\right)\\right){-}\\left(\\frac{{\\partial }}{{\\partial }{v}}\\phantom{\\rule[-0.0ex]{0.4em}{0.0ex}}{\\mathrm{\\phi }}{}\\left({u}{,}{v}\\right)\\right){}\\left(\\frac{{\\partial }}{{\\partial }{u}}\\phantom{\\rule[-0.0ex]{0.4em}{0.0ex}}{\\mathrm{\\psi }}{}\\left({u}{,}{v}\\right)\\right)}\\right]$ (9)" ]	[ null, "https://bat.bing.com/action/0", null, "https://de.maplesoft.com/support/help/maple/arrow_down.gif", null, "https://de.maplesoft.com/support/help/maple/arrow_down.gif", null, "https://de.maplesoft.com/support/help/maple/arrow_down.gif", null, "https://de.maplesoft.com/support/help/maple/arrow_down.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7532291,"math_prob":1.0000036,"size":2305,"snap":"2022-40-2023-06","text_gpt3_token_len":849,"char_repetition_ratio":0.14254671,"word_repetition_ratio":0.0125,"special_character_ratio":0.2121475,"punctuation_ratio":0.14774776,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9998996,"pos_list":[0,1,2,3,4,5,6,7,8,9,10],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-10-06T14:36:13Z\",\"WARC-Record-ID\":\"<urn:uuid:9fe64a0f-f103-464c-82f9-f5336a2a60c2>\",\"Content-Length\":\"310976\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7ca8f927-3011-40d8-9b89-0653f72bf6eb>\",\"WARC-Concurrent-To\":\"<urn:uuid:fa62e75c-379e-407a-b38c-3089aa69f470>\",\"WARC-IP-Address\":\"199.71.183.28\",\"WARC-Target-URI\":\"https://de.maplesoft.com/support/help/maple/view.aspx?path=DEtools%2FXchange\",\"WARC-Payload-Digest\":\"sha1:QFX5ARL4K5ZZMACWSD3ANBJGNCZ2C6GZ\",\"WARC-Block-Digest\":\"sha1:QDUHS3QYR2WUEIXZRS432IZFX6VFZL3S\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-40/CC-MAIN-2022-40_segments_1664030337836.93_warc_CC-MAIN-20221006124156-20221006154156-00059.warc.gz\"}"}
https://alexmennen.com/index.php/2016/03/	[ "## Ordered algebraic geometry\n\nEdit: Shortly after posting this, I found where the machinery I develop here was discussed in the literature. Real Algebraic Geometry by Bochnak, Coste, and Roy covers at least most of this material. I may eventually edit this to clean it up and adopt more standard notation, but don't hold your breath.\n\n### Introduction\n\nIn algebraic geometry, an affine algebraic set is a subset of $\\mathbb{C}^{n}$ which is the set of solutions to some finite set of polynomials. Since all ideals of $\\mathbb{C}\\left[x_{1},...,x_{n}\\right]$ are finitely generated, this is equivalent to saying that an affine algebraic set is a subset of $\\mathbb{C}^{n}$ which is the set of solutions to some arbitrary set of polynomials.\n\nIn semialgebraic geometry, a closed semialgebraic set is a subset of $\\mathbb{R}^{n}$ of the form $\\left\\{ \\bar{x}\\in\\mathbb{R}^{n}\\mid f\\left(\\bar{x}\\right)\\geq0\\,\\forall f\\in F\\right\\}$ for some finite set of polynomials $F\\subseteq\\mathbb{R}\\left[x_{1},...,x_{n}\\right]$. Unlike in the case of affine algebraic sets, if $F\\subseteq\\mathbb{R}\\left[x_{1},...,x_{n}\\right]$ is an arbitrary set of polynomials, $\\left\\{ \\bar{x}\\in\\mathbb{R}^{n}\\mid f\\left(\\bar{x}\\right)\\geq0\\,\\forall f\\in F\\right\\}$ is not necessarily a closed semialgebraic set. As a result of this, the collection of closed semialgebraic sets are not the closed sets of a topology on $\\mathbb{R}^{n}$. In the topology on $\\mathbb{R}^{n}$ generated by closed semialgebraic sets being closed, the closed sets are the sets of the form $\\left\\{ \\bar{x}\\in\\mathbb{R}^{n}\\mid f\\left(\\bar{x}\\right)\\geq0\\,\\forall f\\in F\\right\\}$ for arbitrary $F\\subseteq\\mathbb{R}\\left[x_{1},...,x_{n}\\right]$. Semialgebraic geometry usually restricts itself to the study of semialgebraic sets, but here I wish to consider all the closed sets of this topology. Notice that closed semialgebraic sets are also closed in the standard topology, so the standard topology is a refinement of this one. Notice also that the open ball $B_{r}\\left(\\bar{p}\\right)$ of radius $r$ centered at $\\bar{p}$ is the complement of the closed semialgebraic set $\\left\\{ \\bar{x}\\in\\mathbb{R}^{n}\\mid\\left\|\\bar{x}-\\bar{p}\\right\|^{2}-r^{2}\\geq0\\right\\}$, and these open balls are a basis for the standard topology, so this topology is a refinement of the standard one. Thus, the topology I have defined is exactly the standard topology on $\\mathbb{R}^{n}$.\n\nIn algebra, instead of referring to a set of polynomials, it is often nicer to talk about the ideal generated by that set instead. What is the analog of an ideal in ordered algebra? It's this thing:\n\nDefinition: If $A$ is a partially ordered commutative ring, a cone $C$ in $A$ is a subsemiring of $A$ which contains all positive elements, and such that $C\\cap-C$ is an ideal of $A$. By \"subsemiring\", I mean a subset that contains $0$ and $1$, and is closed under addition and multiplication (but not necessarily negation). If $F\\subseteq A$, the cone generated by $F$, denoted $\\left\\langle F\\right\\rangle$, is the smallest cone containing $F$. Given a cone $C$, the ideal $C\\cap-C$ will be called the interior ideal of $C$, and denoted $C^{\\circ}$.\n\n$\\mathbb{R}\\left[x_{1},...,x_{n}\\right]$ is partially ordered by $f\\geq g\\iff f\\left(\\bar{x}\\right)\\geq g\\left(\\bar{x}\\right)\\,\\forall\\bar{x}\\in\\mathbb{R}^{n}$. If $F\\subseteq\\mathbb{R}\\left[x_{1},...,x_{n}\\right]$ is a set of polynomials and $\\bar{x}\\in\\mathbb{R}^{n}$, then $f\\left(\\bar{x}\\right)\\geq0\\,\\forall f\\in F\\iff f\\left(\\bar{x}\\right)\\geq0\\,\\forall f\\in\\left\\langle F\\right\\rangle$. Thus I can consider closed sets to be defined by cones. We now have a Galois connection between cones of $\\mathbb{R}\\left[x_{1},...,x_{n}\\right]$ and subsets of $\\mathbb{R}^{n}$, given by, for a cone $C$, its positive-set is $P_{\\mathbb{R}}\\left(C\\right):=\\left\\{ \\bar{x}\\in\\mathbb{R}^{n}\\mid f\\left(\\bar{x}\\right)\\geq0\\,\\forall f\\in C\\right\\}$ (I'm calling it the \"positive-set\" even though it is where the polynomials are all non-negative, because \"non-negative-set\" is kind of a mouthful), and for $X\\subseteq\\mathbb{R}^{n}$, its cone is $C_{\\mathbb{R}}\\left(X\\right):=\\left\\{ f\\in\\mathbb{R}\\left[x_{1},...,x_{n}\\right]\\mid f\\left(\\bar{x}\\right)\\geq0\\,\\forall\\bar{x}\\in X\\right\\}$$P_{\\mathbb{R}}\\circ C_{\\mathbb{R}}$ is closure in the standard topology on $\\mathbb{R}^{n}$ (the analog in algebraic geometry is closure in the Zariski topology on $\\mathbb{C}^{n}$). A closed set $X$ is semialgebraic if and only if it is the positive-set of a finitely-generated cone.\n\n### Quotients by cones, and coordinate rings\n\nAn affine algebraic set $V$ is associated with its coordinate ring $\\mathbb{C}\\left[V\\right]:=\\mathbb{C}\\left[x_{1},...,x_{n}\\right]/I\\left(V\\right)$. We can do something analogous for closed subsets of $\\mathbb{R}^{n}$.\n\nDefinition: If $A$ is a partially ordered commutative ring and $C\\subseteq A$ is a cone, $A/C$ is the ring $A/C^{\\circ}$, equipped with the partial order given by $f+C^{\\circ}\\geq g+C^{\\circ}$ if and only if $f-g\\in C$, for $f,g\\in A$.\n\nDefinition: If $X\\subseteq\\mathbb{R}^{n}$ is closed, the coordinate ring of $X$ is $\\mathbb{R}\\left[X\\right]:=\\mathbb{R}\\left[x_{1},...,x_{n}\\right]/C\\left(X\\right)$. This is the ring of functions $X\\rightarrow\\mathbb{R}$ that are restrictions of polynomials, ordered by $f\\geq g$ if and only if $f\\left(\\bar{x}\\right)\\geq g\\left(\\bar{x}\\right)\\,\\forall\\bar{x}\\in X$. For arbitrary $X\\subseteq\\mathbb{R}^{n}$, the ring of regular functions on $X$, denoted $\\mathcal{O}\\left(X\\right)$, consists of functions on $X$ that are locally ratios of polynomials, again ordered by $f\\geq g$ if and only if $f\\left(\\bar{x}\\right)\\geq g\\left(\\bar{x}\\right)\\,\\forall\\bar{x}\\in X$. Assigning its ring of regular functions to each open subset of $X$ endows $X$ with a sheaf of partially ordered commutative rings.\n\nFor closed $X\\subseteq\\mathbb{R}^{n}$, $\\mathbb{R}\\left[X\\right]\\subseteq\\mathcal{O}\\left(X\\right)$, and this inclusion is generally proper, both because it is possible to divide by polynomials that do not have roots in $X$, and because $X$ may be disconnected, making it possible to have functions given by different polynomials on different connected components.\n\n### Positivstellensätze\n\nWhat is $C_{\\mathbb{R}}\\circ P_{\\mathbb{R}}$? The Nullstellensatz says that its analog in algebraic geometry is the radical of an ideal. As such, we could say that the radical of a cone $C$, denoted $\\text{Rad}_{\\mathbb{R}}\\left(C\\right)$, is $C_{\\mathbb{R}}\\left(P_{\\mathbb{R}}\\left(C\\right)\\right)$, and that a cone $C$ is radical if $C=\\text{Rad}_{\\mathbb{R}}\\left(C\\right)$. In algebraic geometry, the Nullstellensatz shows that a notion of radical ideal defined without reference to algebraic sets in fact characterizes the ideals which are closed in the corresponding Galois connection. It would be nice to have a description of the radical of a cone that does not refer to the Galois connection. There is a semialgebraic analog of the Nullstellensatz, but it does not quite characterize radical cones.\n\nPositivstellensatz 1: If $C\\subseteq\\mathbb{R}\\left[x_{1},...,x_{n}\\right]$ is a finitely-generated cone and $p\\in\\mathbb{R}\\left[x_{1},...,x_{n}\\right]$ is a polynomial, then $p\\left(\\bar{x}\\right)>0\\,\\forall\\bar{x}\\in P_{\\mathbb{R}}\\left(C\\right)$ if and only if $\\exists f\\in C$ such that $pf-1\\in C$.\n\nThere are two ways in which this is unsatisfactory: first, it applies only to finitely-generated cones, and second, it tells us exactly which polynomials are strictly positive everywhere on a closed semialgebraic set, whereas we want to know which polynomials are non-negative everywhere on a set.\n\nThe second problem is easier to handle: a polynomial $p$ is non-negative everywhere on a set $S$ if and only if there is a decreasing sequence of polynomials $\\left(p_{i}\\mid i\\in\\mathbb{N}\\right)$ converging to $p$ such that each $p_{i}$ is strictly positive everywhere on $S$. Thus, to find $\\text{Rad}_{\\mathbb{R}}\\left(C\\right)$, it is enough to first find all the polynomials that are strictly positive everywhere on $P_{\\mathbb{R}}\\left(C\\right)$, and then take the closure under lower limits. Thus we have a characterization of radicals of finitely-generated cones.\n\nPositivstellensatz 2: If $C\\subseteq\\mathbb{R}\\left[x_{1},...,x_{n}\\right]$ is a finitely-generated cone, $\\text{Rad}_{\\mathbb{R}}\\left(C\\right)$ is the closure of $\\left\\{ p\\in\\mathbb{R}\\left[x_{1},...,x_{n}\\right]\\mid\\exists f\\in C\\, pf-1\\in C\\right\\}$, where the closure of a subset $X\\subseteq\\mathbb{R}\\left[x_{1},...,x_{n}\\right]$ is defined to be the set of all polynomials in $\\mathbb{R}\\left[x_{1},...,x_{n}\\right]$ which are infima of chains contained in $X$.\n\nThis still doesn't even tell us what's going on for cones which are not finitely-generated. However, we can generalize the Positivstellensatz to some other cones.\n\nPositivstellensatz 3: Let $C\\subseteq\\mathbb{R}\\left[x_{1},...,x_{n}\\right]$ be a cone containing a finitely-generated subcone $D\\subseteq C$ such that $P_{\\mathbb{R}}\\left(D\\right)$ is compact. If $p\\in\\mathbb{R}\\left[x_{1},...,x_{n}\\right]$ is a polynomial, then $p\\left(\\bar{x}\\right)>0\\,\\forall\\bar{x}\\in P_{\\mathbb{R}}\\left(C\\right)$ if and only if $\\exists f\\in C$ such that $pf-1\\in C$. As before, it follows that $\\text{Rad}_{\\mathbb{R}}\\left(C\\right)$ is the closure of $\\left\\{ p\\in\\mathbb{R}\\left[x_{1},...,x_{n}\\right]\\mid\\exists f\\in C\\, pf-1\\in C\\right\\}$.\n\nproof: For a given $p\\in\\mathbb{R}\\left[x_{1},...,x_{n}\\right]$$\\left\\{ \\bar{x}\\in\\mathbb{R}^{n}\\mid p\\left(\\bar{x}\\right)\\leq0\\right\\} \\cap P_{\\mathbb{R}}\\left(C\\right)=\\left\\{ \\bar{x}\\in\\mathbb{R}^{n}\\mid p\\left(\\bar{x}\\right)\\leq0\\right\\} \\cap\\bigcap\\left\\{ P_{\\mathbb{R}}\\left(\\left\\langle f\\right\\rangle \\right)\\mid f\\in C\\right\\}$, an intersection of closed sets contained in the compact set $P_{\\mathbb{R}}\\left(D\\right)$, which is thus empty if and only if some finite subcollection of them has empty intersection within $P_{\\mathbb{R}}\\left(D\\right)$. Thus if $p$ is strictly positive everywhere on $P_{\\mathbb{R}}\\left(C\\right)$, then there is some finitely generated subcone $E\\subseteq C$ such that $p$ is strictly positive everywhere on $P_{\\mathbb{R}}\\left(E\\right)\\cap P_{\\mathbb{R}}\\left(D\\right)=P_{\\mathbb{R}}\\left(\\left\\langle E\\cup D\\right\\rangle \\right)$, and $\\left\\langle E\\cup D\\right\\rangle$ is finitely-generated, so by Positivstellensatz 1, there is $f\\in\\left\\langle E\\cup D\\right\\rangle \\subseteq C$ such that $pf-1\\in\\left\\langle E\\cup D\\right\\rangle \\subseteq C$$\\square$\n\nFor cones that are not finitely-generated and do not contain any finitely-generated subcones with compact positive-sets, the Positivstellensatz will usually fail. Thus, it seems likely that if there is a satisfactory general definition of radical for cones in arbitrary partially ordered commutative rings that agrees with this one in $\\mathbb{R}\\left[x_{1},...,x_{n}\\right]$, then there is also an abstract notion of \"having a compact positive-set\" for such cones, even though they don't even have positive-sets associated with them.\n\n### Beyond $\\mathbb{R}^{n}$\n\nAn example of cone for which the Positivstellensatz fails is $C_{\\infty}:=\\left\\{ f\\in\\mathbb{R}\\left[x\\right]\\mid\\exists x\\in\\mathbb{R}\\,\\forall y\\geq x\\, f\\left(y\\right)\\geq0\\right\\}$, the cone of polynomials that are non-negative on sufficiently large inputs (equivalently, the cone of polynomials that are either $0$ or have positive leading coefficient). $P_{\\mathbb{R}}\\left(C\\right)=\\emptyset$, and $-1$ is strictly positive on $\\emptyset$, but for $f\\in C_{\\infty}$$-f-1\\notin C_{\\infty}$.\n\nHowever, it doesn't really look $C_{\\infty}$ is trying to point to the empty set; instead, $C_{\\infty}$ is trying to describe the set of all infinitely large reals, which only looks like the empty set because there are no infinitely large reals. Similar phenomena can occur even for cones that do contain finitely-generated subcones with compact positive-sets. For example, let $C_{\\varepsilon}:=\\left\\{ f\\in\\mathbb{R}\\left[x\\right]\\mid\\exists x>0\\,\\forall y\\in\\left[0,x\\right]\\, f\\left(y\\right)\\geq0\\right\\}$$P_{\\mathbb{R}}\\left(C_{\\varepsilon}\\right)=\\left\\{ 0\\right\\}$, but $C_{\\varepsilon}$ is trying to point out the set containing $0$ and all positive infinitesimals. Since $\\mathbb{R}$ has no infinitesimals, this looks like $\\left\\{ 0\\right\\}$.\n\nTo formalize this intuition, we can change the Galois connection. We could say that for a cone $C\\subseteq\\mathbb{R}\\left[x_{1},...,x_{n}\\right]$$P_{\\text{}\\mathbb{R}}\\left(C\\right):=\\left\\{ \\bar{x}\\in\\left(\\text{}\\mathbb{R}\\right)^{n}\\mid f\\left(\\bar{x}\\right)\\geq0\\,\\forall f\\in C\\right\\}$, where $\\text{}\\mathbb{R}$ is the field of hyperreals. All you really need to know about $\\text{}\\mathbb{R}$ is that it is a big ordered field extension of $\\mathbb{R}$. $P_{\\text{}\\mathbb{R}}\\left(C_{\\infty}\\right)$ is the set of hyperreals that are bigger than any real number, and $P_{\\text{}\\mathbb{R}}\\left(C_{\\varepsilon}\\right)$ is the set of hyperreals that are non-negative and smaller than any positive real. The cone of a subset $X\\subseteq\\left(\\text{}\\mathbb{R}\\right)^{n}$, denoted $C_{\\text{}\\mathbb{R}}$$\\left(X\\right)$ will be defined as before, still consisting only of polynomials with real coefficients. This defines a topology on $\\left(\\text{}\\mathbb{R}\\right)^{n}$ by saying that the closed sets are the fixed points of $P_{\\text{}\\mathbb{R}}\\circ C_{\\text{}\\mathbb{R}}$. This topology is not $T_{0}$ because, for example, there are many hyperreals that are larger than all reals, and they cannot be distinguished by polynomials with real coefficients. There is no use keeping track of the difference between points that are in the same closed sets. If you have a topology that is not $T_{0}$, you can make it $T_{0}$ by identifying any pair of points that have the same closure. If we do this to $\\left(\\text{}\\mathbb{R}\\right)^{n}$ , we get what I'm calling ordered affine $n$-space over $\\mathbb{R}$.\n\nDefinition: An $n$-type over $\\mathbb{R}$ is a set $\\Phi$ of inequalities, consisting of, for each polynomial $f\\in\\mathbb{R}\\left[x_{1},..,x_{n}\\right]$, one of the inequalities $f\\left(\\bar{x}\\right)\\geq0$ or $f\\left(\\bar{x}\\right)<0$, such that there is some totally ordered field extension $\\mathcal{R}\\supseteq\\mathbb{R}$ and $\\bar{x}\\in\\mathcal{R}^{n}$ such that all inequalities in $\\Phi$ are true about $\\bar{x}$. $\\Phi$ is called the type of $\\bar{x}$. Ordered affine $n$-space over $\\mathbb{R}$, denoted $\\mathbb{OA}_{\\mathbb{R}}^{n}$ is the set of $n$-types over $\\mathbb{R}$.\n\nCompactness Theorem: Let $\\Phi$ be a set of inequalities consisting of, for each polynomial $f\\in\\mathbb{R}\\left[x_{1},..,x_{n}\\right]$, one of the inequalities $f\\left(\\bar{x}\\right)\\geq0$ or $f\\left(\\bar{x}\\right)<0$. Then $\\Phi$ is an $n$-type if and only if for any finite subset $\\Delta\\subseteq\\Phi$, there is $\\bar{x}\\in\\mathbb{R}$ such that all inequalities in $\\Delta$ are true about $\\bar{x}$.\n\nproof: Follows from the compactness theorem of first-order logic and the fact that ordered field extensions of $\\mathbb{R}$ embed into elementary extensions of $\\mathbb{R}$. The theorem is not obvious if you do not know what those mean. $\\square$\n\nAn $n$-type represents an $n$-tuple of elements of an ordered field extension of $\\mathbb{R}$, up to the equivalence relation that identifies two such tuples that relate to $\\mathbb{R}$ by polynomials in the same way. One way that a tuple of elements of an extension of $\\mathbb{R}$ can relate to elements of $\\mathbb{R}$ is to equal a tuple of elements of $\\mathbb{R}$, so there is a natural inclusion $\\mathbb{R}^{n}\\subseteq\\mathbb{OA}_{\\mathbb{R}}^{n}$ that associates an $n$-tuple of reals with the set of polynomial inequalities that are true at that $n$-tuple.\n\nA tuple of polynomials $\\left(f_{1},...,f_{m}\\right)\\in\\left(\\mathbb{R}\\left[x_{1},...,x_{n}\\right]\\right)^{m}$ describes a function $f:\\mathbb{R}^{n}\\rightarrow\\mathbb{R}^{m}$, which extends naturally to a function $f:\\mathbb{OA}_{\\mathbb{R}}^{n}\\rightarrow\\mathbb{OA}_{\\mathbb{R}}^{m}$ by $f\\left(\\Phi\\right)$ is the type of $\\left(f_{1}\\left(\\bar{x}\\right),...,f_{m}\\left(\\bar{x}\\right)\\right)$, where $\\bar{x}$ is an $n$-tuple of elements of type $\\Phi$ in an extension of $\\mathbb{R}$. In particular, a polynomial $f\\in\\mathbb{R}\\left[x_{1},...,x_{n}\\right]$ extends to a function $f:\\mathbb{OA}_{\\mathbb{R}}^{n}\\rightarrow\\mathbb{OA}_{\\mathbb{R}}^{1}$, and $\\mathbb{OA}_{\\mathbb{R}}^{1}$ is totally ordered by $\\Phi\\geq\\Psi$ if and only if $x\\geq y$, where $x$ and $y$ are elements of type $\\Phi$ and $\\Psi$, respectively, in an extension of $\\mathbb{R}$$f\\left(\\Phi\\right)\\geq0$ if and only if $\\text{\"}f\\left(\\bar{x}\\right)\\geq0\\text{\"}\\in\\Phi$, so we can talk about inequalities satisfied by types in place of talking about inequalities contained in types.\n\nI will now change the Galois connection that we are talking about yet again (last time, I promise). It will now be a Galois connection between the set of cones in $\\mathbb{R}\\left[x_{1},...,x_{n}\\right]$ and the set of subsets of $\\mathbb{OA}_{\\mathbb{R}}^{n}$. For a cone $C\\subseteq\\mathbb{R}\\left[x_{1},...,x_{n}\\right]$, $P\\left(C\\right):=\\left\\{ \\Phi\\in\\mathbb{OA}_{\\mathbb{R}}^{n}\\mid f\\left(\\Phi\\right)\\geq0\\,\\forall f\\in C\\right\\}$. For a set $X\\subseteq\\mathbb{OA}_{\\mathbb{R}}^{n}$, $C\\left(X\\right):=\\left\\{ f\\in\\mathbb{R}\\left[x_{1},...,x_{n}\\right]\\mid f\\left(\\Phi\\right)\\geq0\\,\\forall\\Phi\\in X\\right\\}$. Again, this defines a topology on $\\mathbb{OA}_{\\mathbb{R}}^{n}$ by saying that fixed points of $P\\circ C$ are closed. $\\mathbb{OA}_{\\mathbb{R}}^{n}$ is $T_{0}$; in fact, it is the $T_{0}$ topological space obtained from $\\left(\\text{}\\mathbb{R}\\right)^{n}$ by identifying points with the same closure as mentioned earlier. $\\mathbb{OA}_{\\mathbb{R}}^{n}$ is also compact, as can be seen from the compactness theorem. $\\mathbb{OA}_{\\mathbb{R}}^{n}$ is not $T_{1}$ (unless $n=0$). Note that model theorists have their own topology on $\\mathbb{OA}_{\\mathbb{R}}^{n}$, which is distinct from the one I use here, and is a refinement of it.\n\nThe new Galois connection is compatible with the old one via the inclusion $\\mathbb{R}^{n}\\subseteq\\mathbb{OA}_{\\mathbb{R}}^{n}$, in the sense that if $X\\subseteq\\mathbb{R}^{n}$, then $C_{\\mathbb{R}}\\left(X\\right)=C\\left(X\\right)$ (where we identify $X$ with its image in $\\mathbb{OA}_{\\mathbb{R}}^{n}$), and for a cone $C\\subseteq\\mathbb{R}\\left[x_{1},...,x_{n}\\right]$$P_{\\mathbb{R}}=P\\left(C\\right)\\cap\\mathbb{R}^{n}$.\n\nLike our intermediate Galois connection $\\left(P_{\\text{}\\mathbb{R}},C_{\\text{*}\\mathbb{R}}\\right)$, our final Galois connection $\\left(P,C\\right)$ succeeds in distinguishing $P\\left(C_{\\infty}\\right)$ and $P\\left(C_{\\varepsilon}\\right)$ from $\\emptyset$ and $\\left\\{ 0\\right\\}$, respectively, in the desirable manner. $P\\left(C_{\\infty}\\right)$ consists of the type of numbers larger than any real, and $P\\left(C_{\\varepsilon}\\right)$ consists of the types of $0$ and of positive numbers smaller than any positive real.\n\nJust like for subsets of $\\mathbb{R}^{n}$, a closed subset $X\\subseteq\\mathbb{OA}_{\\mathbb{R}}^{n}$ has a coordinate ring $\\mathbb{R}\\left[X\\right]:=\\mathbb{R}\\left[x_{1},...,x_{n}\\right]/C\\left(X\\right)$, and an arbitrary $X\\subseteq\\mathbb{OA}_{\\mathbb{R}}^{n}$ has a ring of regular functions $\\mathcal{O}\\left(X\\right)$ consisting of functions on $X$ that are locally ratios of polynomials, ordered by $f\\geq0$ if and only if $\\forall\\Phi\\in X$, where $f=\\frac{p}{q}$ is a representation of $f$ as a ratio of polynomials in a neighborhood of $\\Phi$, either $p\\left(\\Phi\\right)\\geq0$ and $q\\left(\\Phi\\right)>0$, or $p\\left(\\Phi\\right)\\leq0$ and $q\\left(\\Phi\\right)<0$, and $f\\geq g$ if and only if $f-g\\geq0$. As before, $\\mathbb{R}\\left[X\\right]\\subseteq\\mathcal{O}\\left(X\\right)$ for closed $X\\subseteq\\mathbb{OA}_{\\mathbb{R}}^{n}$.\n\n$\\mathbb{OA}_{\\mathbb{R}}^{n}$ is analogous to $\\mathbb{A}_{\\mathbb{C}}^{n}$ from algebraic geometry because if, in the above definitions, you replace \"$\\geq$\" and \"$<$\" with \"$=$\" and \"$\\neq$\", replace totally ordered field extensions with field extensions, and replace cones with ideals, then you recover a description of $\\mathbb{A}_{\\mathbb{C}}^{n}$, in the sense of $\\text{Spec}\\left(\\mathbb{C}\\left[x_{1},...,x_{n}\\right]\\right)$.\n\nWhat about an analog of projective space? Since we're paying attention to order, we should look at spheres, not real projective space. The $n$-sphere over $\\mathbb{R}$, denoted $\\mathbb{S}_{\\mathbb{R}}^{n}$, can be described as the locus of $\\left\|\\bar{x}\\right\|^{2}=1$ in $\\mathbb{OA}_{\\mathbb{R}}^{n}$.\n\nFor any totally ordered field $k$, we can define $\\mathbb{OA}_{k}^{n}$ similarly to $\\mathbb{OA}_{\\mathbb{R}}^{n}$, as the space of $n$-types over $k$, defined as above, replacing $\\mathbb{R}$ with $k$ (although a model theorist would no longer call it the space of $n$-types over $k$). The compactness theorem is not true for arbitrary $k$, but its corollary that $\\mathbb{OA}_{k}^{n}$ is compact still is true.\n\n### Visualizing $\\mathbb{OA}_{\\mathbb{R}}^{n}$ and $\\mathbb{S}_{\\mathbb{R}}^{n}$\n\n$\\mathbb{S}_{\\mathbb{R}}^{n}$ should be thought of as the $n$-sphere with infinitesimals in all directions around each point. Specifically, $\\mathbb{S}_{\\mathbb{R}}^{0}$ is just $\\mathbb{S}^{0}$, a pair of points. The closed points of $\\mathbb{S}_{\\mathbb{R}}^{n+1}$ are the points of $\\mathbb{S}^{n+1}$, and for each closed point $p$, there is an $n$-sphere of infinitesimals around $p$, meaning a copy of $\\mathbb{S}_{\\mathbb{R}}^{n}$, each point of which has $p$ in its closure.\n\n$\\mathbb{OA}_{\\mathbb{R}}^{n}$ should be thought of as $n$-space with infinitesimals in all directions around each point, and infinities in all directions. Specifically, $\\mathbb{OA}_{\\mathbb{R}}^{n}$ contains $\\mathbb{R}^{n}$, and for each point $p\\in\\mathbb{R}^{n}$, there is an $n-1$-sphere of infinitesimals around $p$, and there is also a copy of $\\mathbb{S}_{\\mathbb{R}}^{n-1}$ around the whole thing, the closed points of which are limits of rays in $\\mathbb{R}^{n}$.\n\n$\\mathbb{OA}_{\\mathbb{R}}^{n}$ and $\\mathbb{S}_{\\mathbb{R}}^{n}$ relate to each other the same way that $\\mathbb{R}^{n}$ and $\\mathbb{S}^{n}$ do. If you remove a closed point from $\\mathbb{S}_{\\mathbb{R}}^{n}$, you get $\\mathbb{OA}_{\\mathbb{R}}^{n}$, where the sphere of infinitesimals around the removed closed point becomes the sphere of infinities of $\\mathbb{OA}_{\\mathbb{R}}^{n}$.\n\nMore generally, if $k$ is a totally ordered field, let $k^{r}$ be its real closure. $\\mathbb{OA}_{k}^{n}$ consists of the Cauchy completion of $\\left(k^{r}\\right)^{n}$ (as a metric space with distances valued in $k^{r}$), and for each point $p\\in\\left(k^{r}\\right)^{n}$ (though not for points that are limits of Cauchy sequences that do not converge in $\\left(k^{r}\\right)^{n}$), an $n-1$-sphere $\\mathbb{S}_{k}^{n-1}$ of infinitesimals around $p$, and an $n-1$-sphere $\\mathbb{S}_{k}^{n-1}$ around the whole thing, where $\\mathbb{S}_{k}^{n}$ is the locus of $\\left\|\\bar{x}\\right\|^{2}=1$ in $\\mathbb{OA}_{k}^{n}$. $\\mathbb{OA}$ does not distinguish between fields with the same real closure.\n\n### More Positivstellensätze\n\nThis Galois connection gives us a new notion of what it means for a cone to be radical, which is distinct from the old one and is better, so I will define $\\text{Rad}\\left(C\\right)$ to be $C\\left(P\\left(C\\right)\\right)$. A cone $C$ will be called radical if $C=\\text{Rad}\\left(C\\right)$. Again, it would be nice to be able to characterize radical cones without referring to the Galois connection. And this time, I can do it. Note that since $\\mathbb{OA}_{\\mathbb{R}}^{n}$ is compact, the proof of Positivstellensatz 3 shows that in our new context, the Positivstellensatz holds for all cones, since even the subcone generated by $\\emptyset$ has a compact positive-set.\n\nPositivstellensatz 4: If $C\\subseteq\\mathbb{R}\\left[x_{1},...,x_{n}\\right]$ is a cone and $p\\in\\mathbb{R}\\left[x_{1},...,x_{n}\\right]$ is a polynomial, then $p\\left(\\Phi\\right)>0\\,\\forall\\Phi\\in P\\left(C\\right)$ if and only if $\\exists f\\in C$ such that $pf-1\\in C$.\n\nHowever, we can no longer add in lower limits of sequences of polynomials. For example, $-x+\\varepsilon\\in C_{\\varepsilon}$ for all real $\\varepsilon>0$, but $-x\\notin C_{\\varepsilon}$, even though $C_{\\varepsilon}$ is radical. This happens because, where $\\Sigma$ is the type of positive infinitesimals, $-\\Sigma+\\varepsilon>0$ for real $\\varepsilon>0$, but $-\\Sigma<0$. However, we can add in lower limits of sequences contained in finitely-generated subcones, and this is all we need to add, so this characterizes radical cones.\n\nPositivstellensatz 5: If $C\\subseteq\\mathbb{R}\\left[x_{1},...,x_{n}\\right]$ is a cone, $\\text{Rad}\\left(C\\right)$ is the union over all finitely-generated subcones $D\\subseteq C$ of the closure of $\\left\\{ p\\in\\mathbb{R}\\left[x_{1},...,x_{n}\\right]\\mid\\exists f\\in D\\, pf-1\\in D\\right\\}$ (again the closure of a subset $X\\subseteq\\mathbb{R}\\left[x_{1},...,x_{n}\\right]$ is defined to be the set of all polynomials in $\\mathbb{R}\\left[x_{1},...,x_{n}\\right]$ which are infima of chains contained in $X$).\n\nProof: Suppose $D\\subseteq C$ is a subcone generated by a finite set $\\left\\{ f_{1},...,f_{m}\\right\\}$, and $q$ is the infimum of a chain $\\left\\{ q_{\\alpha}\\right\\} _{\\alpha\\in A}\\subseteq\\left\\{ p\\in\\mathbb{R}\\left[x_{1},...,x_{n}\\right]\\mid\\exists f\\in D\\, pf-1\\in D\\right\\}$. For any $\\bar{x}\\in\\mathbb{R}^{n}$, if $f_{i}\\left(\\bar{x}\\right)\\geq0$ for each $i$, then $q_{\\alpha}\\left(\\bar{x}\\right)>0$ for each $\\alpha$, and hence $q\\left(\\bar{x}\\right)\\geq0$. That is, the finite set of inequalities $\\left\\{ f_{i}\\left(\\bar{x}\\right)\\geq0\\mid1\\leq i\\leq m\\right\\} \\cup\\left\\{ q\\left(\\bar{x}\\right)<0\\right\\}$ does not hold anywhere in $\\mathbb{R}^{n}$. By the compactness theorem, there are no $n$-types satisfying all those inequalities. Given $\\Phi\\in P\\left(C\\right)$$f_{i}\\left(\\Phi\\right)\\geq0$, so $q\\left(\\Phi\\right)\\nless0$; that is, $q\\left(\\Phi\\right)\\geq0$.\n\nConversely, suppose $q\\in\\text{Rad}\\left(C\\right)$. Then by the compactness theorem, there are some $f_{1},...,f_{m}\\in C$ such that $q\\in\\text{Rad}\\left(\\left\\langle f_{1},...,f_{m}\\right\\rangle \\right)$. Then $\\forall\\varepsilon>0$, $q+\\varepsilon$ is strictly positive on $P\\left(\\left\\langle f_{1},...,f_{m}\\right\\rangle \\right)$, and hence by Positivstellensatz 4, $\\exists f\\in\\left\\langle f_{1},...,f_{m}\\right\\rangle$ such that $pf-1\\in\\left\\langle f_{1},...,f_{m}\\right\\rangle$. That is, $\\left\\{ q+\\varepsilon\\mid\\varepsilon>0\\right\\}$ is a chain contained in $\\left\\langle f_{1},...,f_{m}\\right\\rangle$, a finitely-generated subcone of $C$, whose infimum is $q$. $\\square$\n\n### Ordered commutative algebra\n\nEven though they are technically not isomorphic, $\\mathbb{C}^{n}$ and $\\text{Spec}\\left(\\mathbb{C}\\left[x_{1},...,x_{n}\\right]\\right)$ are closely related, and can often be used interchangeably. Of the two, $\\text{Spec}\\left(\\mathbb{C}\\left[x_{1},...,x_{n}\\right]\\right)$ is of a form that can be more easily generalized to more abstruse situations in algebraic geometry, which may indicate that it is the better thing to talk about, whereas $\\mathbb{C}^{n}$ is merely the simpler thing that is easier to think about and just as good in practice in many contexts. In contrast, $\\mathbb{R}^{n}$ and $\\mathbb{OA}_{\\mathbb{R}}^{n}$ are different in important ways. The situation in algebraic geometry provides further reason to pay more attention to $\\mathbb{OA}_{\\mathbb{R}}^{n}$ than to $\\mathbb{R}^{n}$.\n\nThe next thing to look for would be an analog of the spectrum of a ring for a partially ordered commutative ring (I will henceforth abbreviate \"partially ordered commutative ring\" as \"ordered ring\" in order to cut down on the profusion of adjectives) in a way that makes use of the order, and gives us $\\mathbb{OA}_{\\mathbb{R}}^{n}$ when applied to $\\mathbb{R}\\left[x_{1},...,x_{n}\\right]$. I will call it the order spectrum of an ordered ring $A$, denoted $\\text{OrdSpec}\\left(A\\right)$. Then of course $\\mathbb{OA}_{A}^{n}$ can be defined as $\\text{OrdSpec}\\left(A\\left[x_{1},...,x_{n}\\right]\\right)$$\\text{OrdSpec}\\left(A\\right)$ should be, of course, the set of prime cones. But what even is a prime cone?\n\nDefinition: A cone $\\mathfrak{p}\\subseteq A$ is prime if $A/\\mathfrak{p}$ is a totally ordered integral domain.\n\nDefinition: $\\text{OrdSpec}\\left(A\\right)$ is the set of prime cones in $A$, equipped with the topology whose closed sets are the sets of prime cones containing a given cone.\n\nAn $n$-type $\\Phi\\in\\mathbb{OA}_{\\mathbb{R}}^{n}$ can be seen as a cone, by identifying it with $\\left\\{ f\\in\\mathbb{R}\\left[x_{1},...,x_{n}\\right]\\mid f\\left(\\Phi\\right)\\geq0\\right\\}$, aka $C\\left(\\left\\{ \\Phi\\right\\} \\right)$. Under this identification, $\\mathbb{OA}_{\\mathbb{R}}^{n}=\\text{OrdSpec}\\left(\\mathbb{R}\\left[x_{1},...,x_{n}\\right]\\right)$, as desired. The prime cones in $\\mathbb{R}\\left[x_{1},...,x_{n}\\right]$ are also the radical cones $C$ such that $P\\left(C\\right)$ is irreducible. Notice that irreducible subsets of $\\mathbb{OA}_{\\mathbb{R}}^{n}$ are much smaller than irreducible subsets of $\\mathbb{A}_{\\mathbb{C}}^{n}$; in particular, none of them contain more than one element of $\\mathbb{R}^{n}$.\n\nThere is also a natural notion of maximal cone.\n\nDefinition: A cone $\\mathfrak{m}\\subseteq A$ is maximal if $\\mathfrak{m}\\neq A$ and there are no strictly intermediate cones between $\\mathfrak{m}$ and $A$. Equivalently, if $\\mathfrak{m}$ is prime and closed in $\\text{OrdSpec}\\left(A\\right)$.\n\nMaximal ideals of $\\mathbb{C}\\left[x_{1},...,x_{n}\\right]$ correspond to elements of $\\mathbb{C}^{n}$. And the cones of elements of $\\mathbb{R}^{n}$ are maximal cones in $\\mathbb{R}\\left[x_{1},...,x_{n}\\right]$, but unlike in the complex case, these are not all the maximal cones, since there are closed points in $\\mathbb{OA}_{\\mathbb{R}}^{n}$ outside of $\\mathbb{R}^{n}$. For example, $C_{\\infty}$ is a maximal cone, and the type of numbers greater than all reals is closed. To characterize the cones of elements of $\\mathbb{R}^{n}$, we need something slightly different.\n\nDefinition: A cone $\\mathfrak{m}\\subseteq A$ is ideally maximal if $A/\\mathfrak{m}$ is a totally ordered field. Equivalently, if $\\mathfrak{m}$ is maximal and $\\mathfrak{m}^{\\circ}$ is a maximal ideal.\n\nElements of $\\mathbb{R}^{n}$ correspond to ideally maximal cones of $\\mathbb{R}\\left[x_{1},...,x_{n}\\right]$.\n\n$\\text{OrdSpec}$ also allows us to define the radical of a cone in an arbitrary partially ordered commutative ring.\n\nDefinition: For a cone $C\\subseteq A$, $\\text{Rad}\\left(C\\right)$ is the intersection of all prime cones containing $C$. $C$ is radical if $C=\\text{Rad}\\left(C\\right)$.\n\nConjecture: $\\text{Rad}\\left(C\\right)$ is the union over all finitely-generated subcones $C\\subseteq D$ of the closure of $\\left\\{ p\\in A\\mid\\exists f\\in D\\, pf-1\\in D\\right\\}$ (as before, the closure of a subset $X\\subseteq A$ is defined to be the set of all elements of $A$ which are infima of chains contained in $X$).\n\n### Order schemes\n\nDefinition: An ordered ringed space is a topological space equipped with a sheaf of ordered rings. An ordered ring is local if it has a unique ideally maximal cone, and a locally ordered ringed space is an ordered ringed space whose stalks are local.\n\n$\\text{OrdSpec}\\left(A\\right)$ can be equipped with a sheaf of ordered rings $\\mathcal{O}_{A}$, making it a locally ordered ringed space.\n\nDefinition: For a prime cone $\\mathfrak{p}\\subseteq A$, the localization of $A$ at $\\mathfrak{p}$, denoted $A_{\\mathfrak{p}}$, is the ring $A_{\\mathfrak{p}^{\\circ}}$ equipped with an ordering that makes it a local ordered ring. This will be the stalk at $\\mathfrak{p}$ of $\\mathcal{O}_{A}$. A fraction $\\frac{a}{b}\\in A_{\\mathfrak{p}}$ ($b\\notin\\mathfrak{p}^{\\circ}$) is also an element of $A_{\\mathfrak{q}}$ for any prime cone $\\mathfrak{q}\\subseteq A$ whose interior ideal does not contain $b$. This is an open neighborhood of $\\mathfrak{p}$ (its complement is the set of prime cones containing $\\left\\langle b,-b\\right\\rangle$). There is a natural map $A_{\\mathfrak{p}}\\rightarrow\\text{Frac}\\left(A/\\mathfrak{p}\\right)$ given by $\\frac{a}{b}\\mapsto\\frac{a+\\mathfrak{p}^{\\circ}}{b+\\mathfrak{p}^{\\circ}}$, and the total order on $A/\\mathfrak{p}$ extends uniquely to a total order on the fraction field, so for $a,b\\in A_{\\mathfrak{p}}$, we can say that $a\\geq b$ at $\\mathfrak{p}$ if this is true of their images in $\\text{Frac}\\left(A/\\mathfrak{p}\\right)$. We can then say that $a\\geq b$ near $\\mathfrak{p}$ if $a\\geq b$ at every point in some neighborhood of $\\mathfrak{p}$, which defines the ordering on $A_{\\mathfrak{p}}$.\n\nDefinition: For open $U\\subseteq\\text{OrdSpec}\\left(A\\right)$, $\\mathcal{O}_{A}\\left(U\\right)$ consists of elements of $\\prod_{\\mathfrak{p}\\in U}A_{\\mathfrak{p}}$ that are locally ratios of elements of $A$. $\\mathcal{O}_{A}\\left(U\\right)$ is ordered by $a\\geq b$ if and only if $\\forall\\mathfrak{p}\\in\\text{OrdSpec}\\left(A\\right)$ $a\\geq b$ near $\\mathfrak{p}$ (equivalently, if $\\forall\\mathfrak{p}\\in\\text{OrdSpec}\\left(A\\right)$ $a\\geq b$ at $\\mathfrak{p}$).\n\n$A\\subseteq\\mathcal{O}_{A}\\left(\\text{OrdSpec}\\left(A\\right)\\right)$, and this inclusion can be proper. Conjecture: $\\text{OrdSpec}\\left(\\mathcal{O}_{A}\\left(U\\right)\\right)\\cong U$ as locally ordered ringed spaces for open $U\\subseteq\\text{OrdSpec}\\left(A\\right)$. This conjecture says that it makes sense to talk about whether or not a locally ordered ringed space looks locally like an order spectrum near a given point. Thus, if this conjecture is false, it would make the following definition look highly suspect.\n\nDefinition: An order scheme is a topological space $X$ equipped with a sheaf of ordered commutative rings $\\mathcal{O}_{X}$ such that for some open cover of $X$, the restrictions of $\\mathcal{O}_{X}$ to the open sets in the cover are all isomorphic to order spectra of ordered commutative rings.\n\nI don't have any uses in mind for order schemes, but then again, I don't know what ordinary schemes are for either and they are apparently useful, and order schemes seem like a natural analog of them." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.95483637,"math_prob":0.99996793,"size":20315,"snap":"2021-31-2021-39","text_gpt3_token_len":4415,"char_repetition_ratio":0.14849097,"word_repetition_ratio":0.08025027,"special_character_ratio":0.21023874,"punctuation_ratio":0.11828499,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000049,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-09-24T02:41:15Z\",\"WARC-Record-ID\":\"<urn:uuid:9be149a0-fef5-45e7-a6fa-77eb65bd8174>\",\"Content-Length\":\"189442\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7958ef5e-cd6a-4ddc-bae2-d423380c3081>\",\"WARC-Concurrent-To\":\"<urn:uuid:38c64fdd-44f1-46c6-bb67-f431fc9ac802>\",\"WARC-IP-Address\":\"209.182.203.193\",\"WARC-Target-URI\":\"https://alexmennen.com/index.php/2016/03/\",\"WARC-Payload-Digest\":\"sha1:IJN3H6MW7Y2DOFVTSVBDQDXLRG3XMEEW\",\"WARC-Block-Digest\":\"sha1:6RM2Q5YIEKY45F2QGW4UCGBWRAMDWDVZ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-39/CC-MAIN-2021-39_segments_1631780057496.18_warc_CC-MAIN-20210924020020-20210924050020-00688.warc.gz\"}"}
https://brainmass.com/computer-science/c/few-class-c-subnetting-questions-186719	[ "Explore BrainMass\n\n# Few Class C subnetting questions\n\nNot what you're looking for? Search our solutions OR ask your own Custom question.\n\nThis content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!\n\nAnswer the following questions for each of the configuration in the table below.\n\n1. How many subnets?\n2. How many hosts per subnet?\n3. What are the valid subnets?\n4. What are the valid hosts per subnet?\n\nCONFIGURATION A 192.168.10.0 255.255.255.192\nCONFIGURATION B 220.110.16.0 255.255.255.224\nCONFIGURATION C 219.220.250.0 255.255.255.248\n\nhttps://brainmass.com/computer-science/c/few-class-c-subnetting-questions-186719\n\n#### Solution Preview\n\nAll the given configurations belong to Class C network, so subnetting involves only the last octet. This answer looks at the Class C specific subnetting computations in different ways.\n\nConfiguration A:\n\nFirst, we find out how many bits of host part are used for subnetting in this case.\n\n192 = (128 + 64 + 0 + 0 + 0 + 0 + 0 + 0) OR 11000000 in binary.\n\nNumber of bits used for denoting subnetwork (Ns) = 2\nNumber of bits for denoting hosts in subnetwork (Nh) = 8 - 2 = 6.\n\n1. No. of subnets = 2^Ns - 2 = 2^2 - 2 = 2\n\n2. No. of hosts per subnet = 2^Nh - 2 = 2^6 - 2 = 62\n\n3. To find the first subnet, we compute (256 - subnetmask in fourth octet), and to find further subnets we add this computed value to previous subnet id and continue till we get to the above mentioned subnetmask value.\n\nFirst subnet id = 256 - 192 = 64\nSecond subnet id = 64 + 64 = 128\n\nSo, the valid subnets are 192.168.10.64 and 192.168.10.128 .\n\n4. Since the bits in host id can not be all zeros or all ones. We compute the valid host ids in each of the above subnets as following.\n\nSubnet ...\n\n#### Solution Summary\n\nSolution not only gives detailed explanation along with computations, but also illustrates different ways of subnetting computations.\n\n\\$2.49" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8442882,"math_prob":0.886675,"size":1860,"snap":"2021-31-2021-39","text_gpt3_token_len":526,"char_repetition_ratio":0.14331897,"word_repetition_ratio":0.02173913,"special_character_ratio":0.3236559,"punctuation_ratio":0.16458853,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.996266,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-08-01T13:19:21Z\",\"WARC-Record-ID\":\"<urn:uuid:a9fbe629-3e5d-4b7e-881e-1e97813b857f>\",\"Content-Length\":\"332022\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:13404d39-a01f-4fc0-9acc-8d8034eca2eb>\",\"WARC-Concurrent-To\":\"<urn:uuid:394e3363-5c93-4dfb-973e-9d82c24c8044>\",\"WARC-IP-Address\":\"172.67.75.38\",\"WARC-Target-URI\":\"https://brainmass.com/computer-science/c/few-class-c-subnetting-questions-186719\",\"WARC-Payload-Digest\":\"sha1:Y3J6EIQEQT6QN5P33HPBUPI5FSVVWF53\",\"WARC-Block-Digest\":\"sha1:O2OYHM35PHTANLERAHCIHBX6CGVZTNXE\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-31/CC-MAIN-2021-31_segments_1627046154214.36_warc_CC-MAIN-20210801123745-20210801153745-00535.warc.gz\"}"}
https://socratic.org/questions/what-is-the-derivative-of-e-3x-2	[ "# What is the derivative of -e^(3x^2)?\n\n##### 1 Answer\nApr 19, 2018\n\n$\\frac{\\mathrm{dy}}{\\mathrm{dx}} = - 6 x {e}^{3 {x}^{2}}$\n\n#### Explanation:\n\nBy using chain rule for the function of function concept\n\ny=-e^(3x)^2)\n\n$y = - {e}^{t}$\n\n$\\frac{\\mathrm{dy}}{\\mathrm{dx}} = - {e}^{t} \\frac{\\mathrm{dt}}{\\mathrm{dx}}$\n\n$t = 3 {x}^{2}$\n\n$t = 3 u$\n\n$u = {x}^{2}$\n\n$\\frac{\\mathrm{du}}{\\mathrm{dx}} = 2 x$\n\n(dt)/(dx)=3)du)/(dx)\n\n3(du/(dx)=3xx2x\n\n$\\frac{\\mathrm{dt}}{\\mathrm{dx}} = 3 \\times 2 x$\n\n$3 \\times 2 x = 6 x$\n\n$\\frac{\\mathrm{dt}}{\\mathrm{dx}} = 6 x$\n\n$\\frac{\\mathrm{dy}}{\\mathrm{dx}} = - {e}^{t} \\frac{\\mathrm{dt}}{\\mathrm{dx}}$\n\n$\\frac{\\mathrm{dy}}{\\mathrm{dx}} = - {e}^{3 {x}^{2}} \\left(6 x\\right)$\n\n$\\frac{\\mathrm{dy}}{\\mathrm{dx}} = - 6 x {e}^{3 {x}^{2}}$" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.60022837,"math_prob":1.00001,"size":259,"snap":"2021-21-2021-25","text_gpt3_token_len":64,"char_repetition_ratio":0.12156863,"word_repetition_ratio":0.0,"special_character_ratio":0.25096524,"punctuation_ratio":0.06382979,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000077,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-13T21:32:01Z\",\"WARC-Record-ID\":\"<urn:uuid:f838dda3-6ea0-4c90-9ce3-11c350e28c03>\",\"Content-Length\":\"33143\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f5bf1c5e-795d-43b7-a46b-d0f12f5288a5>\",\"WARC-Concurrent-To\":\"<urn:uuid:693f0141-6478-46fd-a98b-242d98024d3b>\",\"WARC-IP-Address\":\"216.239.36.21\",\"WARC-Target-URI\":\"https://socratic.org/questions/what-is-the-derivative-of-e-3x-2\",\"WARC-Payload-Digest\":\"sha1:QWUQGJEAPN5XJ2STVCDKR74KAYMYEN42\",\"WARC-Block-Digest\":\"sha1:GMRJ2AKJN2DALC4JTFX5CBAAI3QIXCUG\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623487610841.7_warc_CC-MAIN-20210613192529-20210613222529-00139.warc.gz\"}"}
https://listserv.uni-heidelberg.de/cgi-bin/wa?A3=0705&L=LATEX-L&E=7bit&P=2221&B=--&T=text%2Fplain;%20charset=ISO-8859-1&header=1	[ "On 5/5/07, Andreas Matthias <[log in to unmask]> wrote:\n> 1) I suppose there's a bug in \\prop_map_function_aux:NNn.\n\nYes, thanks. Will fix in source as soon as I get access from my laptop\n(tonight I hope).\n\n> 2) I have no idea how to use \\prop_map_inline:NN. Can someone\n> show me? There seems to be a missing \\prop_map_function:N.\n\nHmm, the code does look odd. I wonder why it wasn't written in a more\nstraightforward manner. Here's a combined test+fix document.\n\n\\documentclass{article}\n\n\\usepackage{l3prop,l3quark}\n\\CodeStart\n\n% also missing:\n\\def:Npn \\prop_map_function:Nc {\\exp_args:NNc \\prop_map_function:NN}\n% fix\n\\def:Npn \\prop_map_function_aux:NNn #1#2#3{\n\\if_meaning:NN \\q_nil #2\n\\exp_after:NN \\prop_map_break:w\n\\fi:\n#1#2{#3}\n\\prop_map_function_aux:NNn #1\n}\n% fix\n\\def_long:Npn \\prop_map_inline:Nn #1#2 {\n\\num_incr:N \\l_prop_inline_level_num\n\\def_long:cpn {prop_map_inline_ \\num_use:N \\l_prop_inline_level_num :n}\n##1##2{#2}\n\\prop_map_function:Nc #1\n{prop_map_inline_ \\num_use:N \\l_prop_inline_level_num :n}\n\\num_decr:N \\l_prop_inline_level_num\n}\n\n\\prop_new:N \\l_testa_plist\n\\prop_put:NNn \\l_testa_plist \\keya {info1}\n\\prop_put:NNn \\l_testa_plist \\keyb {info2}\n\\prop_put:NNn \\l_testa_plist \\keyc {info3}\n\n\\prop_map_inline:Nn \\l_testa_plist{\n\\io_put_term:x{\\token_to_string:N #1\\space =\\space #2}\n}\n\n\\stop\n\n--\nMorten" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7490299,"math_prob":0.43843728,"size":1315,"snap":"2023-40-2023-50","text_gpt3_token_len":445,"char_repetition_ratio":0.1884058,"word_repetition_ratio":0.0,"special_character_ratio":0.3163498,"punctuation_ratio":0.2,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9633838,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-02T22:58:42Z\",\"WARC-Record-ID\":\"<urn:uuid:a7a3285f-43d3-498f-b647-f4b288f13234>\",\"Content-Length\":\"44873\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:5d4a4966-89cc-4068-bc73-4bbdfac5e32a>\",\"WARC-Concurrent-To\":\"<urn:uuid:b9f76a25-005f-4770-ba82-e27a43bf5f95>\",\"WARC-IP-Address\":\"129.206.100.94\",\"WARC-Target-URI\":\"https://listserv.uni-heidelberg.de/cgi-bin/wa?A3=0705&L=LATEX-L&E=7bit&P=2221&B=--&T=text%2Fplain;%20charset=ISO-8859-1&header=1\",\"WARC-Payload-Digest\":\"sha1:7SIZU5VL4RX7XMVYYFHLEPJVKWGAG5O7\",\"WARC-Block-Digest\":\"sha1:RV6BWY425VR5HTXASLA27BXSEPMS2SLT\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100452.79_warc_CC-MAIN-20231202203800-20231202233800-00286.warc.gz\"}"}
https://numbermatics.com/n/6648538482576/	[ "# 6648538482576\n\n## 6,648,538,482,576 is an even composite number composed of four prime numbers multiplied together.\n\nWhat does the number 6648538482576 look like?\n\nThis visualization shows the relationship between its 4 prime factors (large circles) and 135 divisors.\n\n6648538482576 is an even composite number. It is composed of four distinct prime numbers multiplied together. It has a total of one hundred thirty-five divisors.\n\n## Prime factorization of 6648538482576:\n\n### 24 × 32 × 1512 × 14232\n\n(2 × 2 × 2 × 2 × 3 × 3 × 151 × 151 × 1423 × 1423)\n\nSee below for interesting mathematical facts about the number 6648538482576 from the Numbermatics database.\n\n### Names of 6648538482576\n\n• Cardinal: 6648538482576 can be written as Six trillion, six hundred forty-eight billion, five hundred thirty-eight million, four hundred eighty-two thousand, five hundred seventy-six.\n\n### Scientific notation\n\n• Scientific notation: 6.648538482576 × 1012\n\n### Factors of 6648538482576\n\n• Number of distinct prime factors ω(n): 4\n• Total number of prime factors Ω(n): 10\n• Sum of prime factors: 1579\n\n### Divisors of 6648538482576\n\n• Number of divisors d(n): 135\n• Complete list of divisors:\n• Sum of all divisors σ(n): 18743884804827\n• Sum of proper divisors (its aliquot sum) s(n): 12095346322251\n• 6648538482576 is an abundant number, because the sum of its proper divisors (12095346322251) is greater than itself. Its abundance is 5446807839675\n\n### Bases of 6648538482576\n\n• Binary: 11000001011111110111100011001000111100100002\n• Base-36: 2CUAO5HG0\n\n### Squares and roots of 6648538482576\n\n• 6648538482576 squared (66485384825762) is 44203063954293980655595776\n• 6648538482576 cubed (66485384825763) is 293885771747891584367365438230275198976\n• 6648538482576 is a perfect square number. Its square root is 2578476\n• The cube root of 6648538482576 is 18803.6458171681\n\n### Scales and comparisons\n\nHow big is 6648538482576?\n• 6,648,538,482,576 seconds is equal to 211,402 years, 49 weeks, 5 days, 21 hours, 9 minutes, 36 seconds.\n• To count from 1 to 6,648,538,482,576 would take you about five hundred twenty-eight thousand, five hundred seven years!\n\nThis is a very rough estimate, based on a speaking rate of half a second every third order of magnitude. If you speak quickly, you could probably say any randomly-chosen number between one and a thousand in around half a second. Very big numbers obviously take longer to say, so we add half a second for every extra x1000. (We do not count involuntary pauses, bathroom breaks or the necessity of sleep in our calculation!)\n\n• A cube with a volume of 6648538482576 cubic inches would be around 1567 feet tall.\n\n### Recreational maths with 6648538482576\n\n• 6648538482576 backwards is 6752848358466\n• 6648538482576 is a Harshad number.\n• The number of decimal digits it has is: 13\n• The sum of 6648538482576's digits is 72\n• More coming soon!" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6864196,"math_prob":0.9528132,"size":4964,"snap":"2021-31-2021-39","text_gpt3_token_len":1587,"char_repetition_ratio":0.14758064,"word_repetition_ratio":0.052333806,"special_character_ratio":0.52699435,"punctuation_ratio":0.13801453,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9903613,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-07-27T08:55:46Z\",\"WARC-Record-ID\":\"<urn:uuid:cb322c8f-775f-4076-a2ea-1e3c340a6e83>\",\"Content-Length\":\"28666\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:c45e2bbe-e3f9-47d8-bc57-2ab71eb2a217>\",\"WARC-Concurrent-To\":\"<urn:uuid:27cf0011-5082-47a7-900b-b9aac68b314c>\",\"WARC-IP-Address\":\"72.44.94.106\",\"WARC-Target-URI\":\"https://numbermatics.com/n/6648538482576/\",\"WARC-Payload-Digest\":\"sha1:Z5P3NBFUPVSOBNMC5TLHOKH72L3O6ZNG\",\"WARC-Block-Digest\":\"sha1:ZONHASAZGXCJ5DBILAZ2HDUUYD2K5WFV\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-31/CC-MAIN-2021-31_segments_1627046153223.30_warc_CC-MAIN-20210727072531-20210727102531-00062.warc.gz\"}"}
http://ilarioaltobelli.it/parent-functions-algebra-2.html	[ "# Parent Functions Algebra 2\n\nThis topic covers: - Evaluating functions - Domain & range of functions - Graphical features of functions - Average rate of change of functions - Function combination and composition - Function transformations (shift, reflect, stretch) - Piecewise functions - Inverse functions - Two-variable functions. Algebra 2 -27 - Functions, Equations, and Graphs SECTION 2. You may use your graphing calculator to compare & sketch. Algebra, functions, & patterns cover 20-30% of the GED Math test. From parent functions in algebra to value, we have got all the pieces included. x is not a function of y, because the input y = 3 has multiple outputs: x = 1 and x = 2. Merely said, the answers to parent function packet algebra 2 is universally compatible following any devices to read. 2 worksheet Name _____ Parent Functions and Transformations Per ____ Date _____ Graph the parent function (black), the transformation (another color), and the domain and range for the. As discussed in the previous section, quadratic functions have y = x 2 as their parent function. g (x) = 2x³ – 5 Part A State the…. They are simple, yet powerful in their ability to model real world situations. The quadratic formula and its uses. Jul 10, 2017 - Explore Marla Barkman's board \"parent functions\" on Pinterest. 2X-2 _ 1 2 d. = -2(x - 1/2) 2 + 7/2. By the end of. Introduction to Parent Functions. Can you identify different functions and their translations? 1. IF Interpreting Functions Understand the concept of a function and use function notation. The graphs of these functions are drawn on the next page. f (x) = −2x+1 f (x) = - 2 x + 1 The parent function is the simplest form of the type of function given. Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. Last Name . Square Root Function f(x) = x 7. A set of basic functions used as building blocks for more complicated functions. This function can be transformed in the same way as other functions. logs 2 (Oft) 7. For each function, graph the indicated change and write a function statement. (Do not use any of the examples from #1 -12!). We could alternatively write these functions as (x +2)2,(x 2)2, (2x) 2,(x 2) ,and(x). Algebra 1 Honors – Ch 3. If a parabola opens downward, it has a highest point. Each poster gives the parent function, graph, and the following characteristics: domain, range, x-intercept, y-intercept, maximum number of roots, end behavior, increasing interval (s), decreasing interval (s) and asymptotes. 4 Exercises - Page 203 28 including work step by step written by community members like you. Pre-AP Algebra 2 Function Transformations ©a x2b0U1\\8s mKEuatXa` DSgoxfYtvwAarr[eG FLCLaCt. Should you seek advice with math and in particular with parent function of algebra 2 or dividing rational expressions come pay a visit to us at Solve-variable. Algebra 2: Unit 4 Instructional Focus –– Modeling with Functions Topic Instructional Foci le In this unit students synthesize and generalize what they have learned about a variety of function families. Similar graph for all odd roots. Just as with other parent functions, we can apply the four types of transformations—shifts, reflections, stretches, and compressions—to the parent function $f\\left(x\\right)={b}^{x}\\\\$ without loss of shape. BUSH ALGEBRA 2. Mathematics: Parent functions \"give birth\" to child functions. Planning a Parent Workshop: Toolkit for Parent Engagement Maine-Endwell Educator Named 2020 New York State Teacher of the Year Released 2019 3-8 ELA and Mathematics State Test Questions. horizontal stretching of functions common core algebra 2 homework answer key, Some students in my introductory algebra class struggle with basic concepts. A point of special interest is the point (0,0) on the parent function. Linear; translation down 6. In mathematics, you see certain graphs over and over again. Graphs of Common Parent Functions; Get full access to over 1,300 online videos and slideshows from multiple courses ranging from Algebra 1 to Calculus. Graph is a horizontal line. Horizontal Translation. Quadratic; translation left 5. Algebra 2 Rational Functions Test Answers algebra 2 rational functions test review sheet. Graphing parent functions with transformations by Dan Tating - September 10, 2012. Algebra 3/MAT 150 Adv. In order to translate any function to the right or left, place an addition or subtraction \"inside\" of the Parent function. Common Parent Functions Linear Function: f(x) = x Domain: All real numbers. d: Perform operations on functions, including function composition, and determine domain and range for each of the given functions. 2-7 Parent Functions & Transformations •identify and use parent functions •describe transformations of functions •F. What is a parent function in Algebra 2? A parent function is the simplest function of a family of functions. U2 C6 Parent Functions. Algebra 2 – Unit 4: Working with Polynomials; Algebra 2 – Unit 5: Polynomial Equations & Graphs; Algebra 2 – Unit 6: Rational Functions; Algebra 2 – Unit 7: Exponential and Logarithmic Functions; Algebra 2 – Unit 8: Inequalities; Algebra 2 – Unit 9: Sequences and Series; Algebra 2 – Unit 10: Cumulative Review; Algebra 2 – Unit. and has specific properties and key points to assist in graphing. Browse other questions tagged algebra-precalculus functions transformation absolute-value or ask your own question. Parent Function: y = x. quadratic. f(x) = -2/+3 Parent Function: List all transformations: Parent Function: List all transformations: 7. (Do not use any of the examples from #1 -12!). 00 Parent Functions Wall Posters $3. The parent function of a parabola is where are the vertex. The functions in Items 2 and 3 are examples of horizontal translations. For example, if we begin by graphing the parent function $$f(x)=2^x$$, we can then graph the two reflections alongside it. s of Symmetry: Plug back in to find the vertex! EXAMPLE: VERTEX FORM: Axis of Symmetry. Inverse of a Function 79 4. In addition to watching the pre-recorded lessons or viewing the online slides, you may alsopurchase the PowerPoint (PPT) or Keynote file for this lesson for$3. 3: LINEAR FUNCTIONS AND SLOPE-INTERCEPT FORM MACC. }y 5 1 3 Ï} x is a vertical shrink of y 5 Ï} x. Introduction to Parent Functions Identify the parent function for from its equation. Range of a Function, and Onto Functions 78 4. Featured on Meta State of the Stack Q1 2021 Blog Post. Be sure to graph the squaring function using a dashed curve because it will be used as a guide and is not the answer. See answer below Given: y = sqrt(x) - 2 The parent function is g(x) = sqrt(x) The domain is limited because of the radical: x >= 0 The -2 outside of the radical determines the vertical shift, 2 units down. The functions in Items 2 and 3 are examples of horizontal translations. = -2(x - 1/2) 2 + 7/2. Anchor Points for Parent Function Graphs f(x) = x2 Anchor Points: (-1, 1), (0, 0), (1, 1), (-2, 4), (2, 4) D = { x\| x ∈ R } or (-∞, ∞) R = { x\| x ∈ R , x ≥0. Worksheet 3 Graphing exponential functions g(x) =- Hour Identify each transformation from the parent function of Tell if the function is a decay or growth function. Teacher Resources Based on extensive research, SmartGraphs: Algebra is designed for students studying Algebra 1 or Algebra 2. 1) x y-8-6-4-22468-8-6-4-2 2 4 6 8 2) x y-8-6. Algebra 3/MAT 150 Adv. But you can see where this could get really hard, right?. A set of basic functions used as building blocks for more complicated functions. 1 vertex; 1 line of symmetry; The highest degree (the greatest exponent) of the function is 2; The graph is a parabola; Parent and Offspring. See back of book. That was a great start with my students in Algebra 2, as we were able to use it right away starting with linears. By the end of. d: Perform operations on functions, including function composition, and determine domain and range for each of the given functions. 2 WS: Evaluate Parent Functions Name _____ Pre-AP Algebra II Date _____ Per _____ Assessment: Match the function (write the number of the graph on the blank) with. Transformations of exponential graphs behave similarly to those of other functions. IF Interpreting Functions Understand the concept of a function and use function notation. and has specific properties and key points to assist in graphing. Study Guide and Intervention Parent Functions and Transformations Name Characteristics Parent Function. Identified parent functions and transformations. Our focus on in-depth instruction is also ideal for homeschool parents looking to offer their child the equivalent of a $30,000-a-year private school math education for a tiny fraction of the cost. Algebra 1 Parent Functions and Transformations. The word logarithm, abbreviated log, is introduced to satisfy this need. 1-5 Guided Notes SE - Parent Functions and Transformations. absolute value B. Functions in the same family are transformations of their parent function. Parabola Parent Function - MathBitsNotebook (A2 - CCSS Math) A parent function is the simplest function of a family of functions. Parent Functions 1. ) • Polar graphs (r=cos2θ) • Parametric. Given f (x) = –x2 + 5x + 2, find the expression, in terms of f , for a leftward shift of five units. g(x) = x 2 – 6 Parent: _____. share to google. Before exploring the family of related functions, let’s clarify some of the. Scaling means shrinking or magnifying the function. 2 Transformations of Linear & Absolute Value Functions 1. 2 Represent Functions and Relations A2. Learn about the properties and graphs of linear parent functions quadratic parent functions absolute value parent functions and reciprocal parent functions. This exercise differs from the previous one in that I not only have to do the operations with the functions, but I also have to evaluate at a particular x-value. Tell students that the parent function, y = x 2, can be modified to create other functions by adding or subtracting values from x 2, or by multiplying x 2 by a coefficient, or both. One to one functions: is a function from to. This is a set of nine parent function posters to display in your classroom. [Algebra 2] What is this parent function. Parent Function Posters for Algebra 2This is a set of nine parent function posters to display in your classroom. Linear Function (Identity) f(x) = x 3. Then describe the transformations. Algebra 2 Team Tutorials – If I am not available, try one of the other great Algebra 2 teachers. This kind of circumstance is so stressful and through the help of some. Constant Linear Quadratic Absolute Value Cubic Square Root. THEOREM 2. Function Family Fun. The foldable is a great guided practice, the interactive notebook is a great way for students to collaborate and create and manipulate, the practice sheet can IA-2. f(x) = 2x 2. Chapter 2 43 Glencoe Algebra 2 2-7 Parent Graphs The parent graph, which is the graph of the parent function, is the simplest of the graphs in a family. 2 Represent Functions and Relations A2. This is the graph that is transformed to create other members in a family of graphs. Luckily, this is not because function problems are inherently more difficult to solve than any other math problem, but because most students have simply not dealt with functions as much as they have other SAT math topics. Vertical Expansions and Compressions. Bookmark File PDF Algebra 2 Parent Function Project With Graphing Parent Functions Project - Algebra 2 2018-19 Create a “Family Album” for the 7 function families listed below. Algebra 1 Honors – Ch 3. The other leg (the region in Quadrant 1) points upwards. An absolute value function has an x-intercept. Merely said, the answers to parent function packet algebra 2 is universally compatible with any devices to read. g(x) = 3 9. Learn how to shift graphs up, down, left, and right by looking at their equations. Unit 2 – Linear Functions & Systems Unit 3 – Intro to Parent Functions and Transformations Unit 4 – Solving Quadratics and Complex Numbers Unit 5 – Polynomial Functions Unit 6 – Radical Functions Unit 7 – Exponential and Logarithmic Functions Unit 8 – Rational Functions Unit 9 – Conic Sections Unit 10 – Sequences and Series. We can use a handy trick to find out whether this data set matches a linear or quadratic function. PARENT FUNCTIONS Author: Pete Falzone Created Date: 8/2/2001 12:36:31 AM. The Vertex Formula. Lesson 2-6 NAME DATE PERIOD PDF Pass Chapter 2 37 Glencoe Algebra 2 Exercises Graph each function. translation parent function 1–8. What is the line of reflection? A. Square Root Function f(x) = x 7. y= constant f(x) = constant. For the family of quadratic functions, y = ax2 + bx + c, the simplest function. Algebra 2 TAP. 1) 2 x>10 3. b: Use transformation to draw the graph of a relation and determine a relation that fits a graph. When we multiply the parent function f (x) = b x f (x) = b x by −1, −1, we get a reflection about the x-axis. View ALG2 HW Parent Functions. The graph below represents the function f (x) = - x2. IF Interpreting Functions Understand the concept of a function and use function notation. Vertical Expansions and Compressions. Come to Algebra-net. 𝑓𝑥 𝑥2 v 𝑓 𝑥 2 v 𝑥 3 𝑥 u2. Provide at least 2 examples of a function Provide 1 example of a non-function 3. h(x) = −2 ⋅ f(x) Multiply the output by −2. (link on my webpage) An explanation of what a function is. Name the parent function. Parent Function: Domain: X. Transform rational functions by changing parameters. ex: f(x) = (1/2 x — 4)2 becomes (1/2 (x — 8))2 is stretched horizontally by a factor of 2 5. 2 - Parent Functions 2 4 6 8 ©v f2D0c1f4Y QK]untXao uStomfMtHwxavrEev XLaLBCd. The other leg (the region in Quadrant 1) points upwards. Monday, 4/20/15- 7. Quadratic; translation 2 units up 8. Analyze functions using different representations. Range of a Function, and Onto Functions 78 4. Use the video above instead of going on a field trip. Chapter 1: Linear Functions 1. Then describe which transformation of the parent function it represents. This Custom Polygraph is designed to spark vocabulary-rich conversations about graphs of parent functions. A scientist notes the bacteria count in a petrie dish is 50. Functions are equal if they have the same domain and rule of correspondence. The vertex is the point (1/2, 7/2). 1 5 Parent Functions And Transformations April 18th, 2019 - Function Of Time Is Given By F T At2 V 0 T X 0 Where A Is The Acceleration V 0 Is The Initial Velocity And X0 Is The Initial Position Of The Object Describe The Transformation S Of The Parent 2 / 7. One leg of this graph (the region in Quadrant 3) points downwards. PDF Lessons 9. Parabola Parent Function - MathBitsNotebook (A2 - CCSS Math) A parent function is the simplest function of a family of functions. Parent functions included: linear, absolute value, quadratic, cubic, square root, cube root, reciprocal, exponential, and logarithmic. General Form for Describing the TRANSOFMRATIONS for a function f(x): F(x) ( a•f(x – h. This lowest or highest point is the vertex of the parabola. com contains practical advice on Glencoe Algebra 2 Answer Key, synthetic division and equations in two variables and other math topics. Luckily, this is not because function problems are inherently more difficult to solve than any other math problem, but because most students have simply not dealt with functions as much as they have other SAT math topics. Algebra 2: Parent Functions Name _____ Parent function: A parent function is the simplest function in a family of function – the simplest form of the given function. 3 2 hx x= 11. Planning a Parent Workshop: Toolkit for Parent Engagement Maine-Endwell Educator Named 2020 New York State Teacher of the Year Released 2019 3-8 ELA and Mathematics State Test Questions. h(x) = - x2 + 1 10. Unit 2 – Linear Functions & Systems Unit 3 – Intro to Parent Functions and Transformations Unit 4 – Solving Quadratics and Complex Numbers Unit 5 – Polynomial Functions Unit 6 – Radical Functions Unit 7 – Exponential and Logarithmic Functions Unit 8 – Rational Functions Unit 9 – Conic Sections Unit 10 – Sequences and Series. What are parent functions? The parent functions are a base of functions you should be able to recognize the graph of given the function and the other way around. Algebra 2 - Parent Functions. The equation is y= 5(2)^x+1 +3 The parent function here is y= 2^x The parent function is just where the graph y= 5(2)^x+1 +3 originates from. Function Family Fun. Finally, they learned how to write equations given the graph with dilations. After reviewing the Warm Up, I hand students a list of Graphing Vocabulary, and a Graphic Organizer for students to take notes on the Parent Functions. 1{1 Functions 78 4. 1 Linear and Quadratic Functions; 2. Reflection through the x-axis. They are simple, yet powerful in their ability to model real world situations. ( )2 4 9 hx x=− +5 14. Then describe the transformation. function for the graph at the right. Examples of doing distributive property in algebra workshhet that i can print; how to use my casio in algebra; free geometry basic transformation worksheets; 7th grade pre algebra test; algebra 2 online book; how changes in one or two dimensional of an object affect perimeter, area, surface area, and volume. All notes and homework should be done in student binder. You must understand that there's different parent functions. Algebra, functions, & patterns cover 20-30% of the GED Math test. (Limited to linear and quadratic functions only. 3 Polynomial Functions of Higher degree; 2. 3 5 f xx= 10. In cases where you have to have assistance with algebra and in particular with essay on parent functions in algebra 2 or math come pay a visit to us at Emaths. When in the function , a horizontal shrinking of the graph of will occur. If n is an odd whole number (1,3,5, etc) the graph will generally start in the third quadrant and end in the first quadrant. Algebra 2 Parent Functions Worksheet Answers - It is tiring when your kids request you in assisting these algebra residence operates, and you also are not able to do that home works, or you may not find out about them where you have not. To shift the graph side to side, I need to add or subtract inside the argument of the function (that is, inside the parentheses). translation parent function 1–8. In order to translate any function to the right or left, place an addition or subtraction \"inside\" of the Parent function. Transformations Of Parent Functions. Lesson #1: Parent Functions and Transformations: Vertical & Horizontal Shifts and Reflections September 13, 2019 September 13, 2019 mshallmaa Leave a comment Homework – Due Monday, September 16. Big Idea To develop an understanding for the transformations of parent functions through repeated reasoning (MP8) and the structure (MP7) of the equation. The vertex of the parent function y = x 2 lies on the origin. Common Parent Functions Linear Function: f(x) = x Domain: All real numbers. The Concept of a Function 72 4. Teacher Resources Based on extensive research, SmartGraphs: Algebra is designed for students studying Algebra 1 or Algebra 2. 9A The student is expected to use the parent function to investigate, describe, and predict the effects of parameter changes on the graphs of square root functions and describe limitations on the domains and ranges. TYPES OF TRANSLATIONS. See back of book. Solution for State the parent function of g (x), and describe how the graph of g (x) is related to to its parent function. All for only$14. 9) x y 10) x y 11) x y 12) x y Identify the parent function f (x) and write an equation for the function given. Parent Organizations; Paw Print; This website is for all Unit 5 students taking Algebra 1. Algebra 2 Parent Function Toolkit 5 Name: Sine Parent Equation: yx sin Description of the Locator: y-intercept (h, a+k) General Equation: y a b x h k sin ( ) a = amplitude b = angular frequency pb 2S h = horizontal shift k = vertical shift Properties: starts in the middle and ends in the middle Domain: , f f all real numbers f fx. Algebra 2: Parent Functions Name _____ Parent function: A parent function is the simplest function in a family of function – the simplest form of the given function. Pascal's Triangle. Gradesheet – This is to help the students keep track of their grades and daily homework checks. Algebra 1 - Unit 5 Vocabulary: Domain Range Translation Function Linear Function Quadratic Function Absolute Value Function Transformation Relation Parent Function Maximum Minimum Increasing Decreasing x-intercept y-intercept Function Notation F. Home Functions Logarithms Exponents Compounding Interest Probability Parent Funtions. In a quadratic function, the variable is always squared. This graph is known as the \"Parent Function\" for parabolas, or quadratic functions. For the following functions, identify the parent function and the transformations, then sketch the graph. Step Functions 83 4. The quadratic formula and its uses. What is a parent function and how do you do them? My teacher is no help and i'm stuck not knowing how to do my homework. The other leg (the region in Quadrant 1) points upwards. The restricted (natural) domain of the parent function is also of special interest as it is the non-negative real numbers, ie the interval. The \"Parent\" Graph: The simplest parabola is y = x 2, whose graph is shown at the right. g x 3x Quadratic Cubic Linear Identify the parent function for each graph. They are simple, yet powerful in their ability to model real world situations. Glenn’s convinced me to use the “transformation form” (AKA h,k form) last year in Geometry. These worksheets are written so that you do not have to be a mathematician to help your child. > Algebra 2A: Unit 2 - Introduction to Functions. Learn about algebra 2 parent functions with free interactive flashcards. YES! Now is the time to redefine your true self using Slader’s Algebra 2: A Common Core Curriculum answers. Common Parent Functions Linear Function: f(x) = x Domain: All real numbers. Quadratic 2. Algebra 2: Unit 1. Partial Fractions. This Page 11/28. Given f (x) = –x2 + 5x + 2, find the expression, in terms of f , for a leftward shift of five units. Quadratic Function f(x) = x 2 5. Graphing quadratic functions Quadratic functions are functions in which the 2nd power, or square, is the highest to which […]. Parent Functions and Transformations Worksheet, Word Docs, & PowerPoints. Start studying Parent Functions Algebra 2. When we multiply the input by $$−1$$,we get a reflection about the y-axis. Parent Functions DRAFT. Describe the transformation from the parent graph. 1) x y-8-6-4-22468-8-6-4-2 2 4 6 8 2) x y-8-6. the graph of the function changes, compared to the parent function. Algebra 2 Team Tutorials – If I am not available, try one of the other great Algebra 2 teachers. Algebra, Functions, and Data Analysis 2014 AFDA Vocabulary Cards Page 2 Parent Functions Linear, Quadratic. , 46) but is having difficulty with more advanced exponent problems. Solution for State the parent function of g (x), and describe how the graph of g (x) is related to to its parent function. The parent function f(x) = x2 has its vertex at the origin. 0 mathematicsvisionproject. Transformations of exponential graphs behave similarly to those of other functions. ti 89 help videos; Pre algebra final. Algebra 1 - Unit 5 Vocabulary: Domain Range Translation Function Linear Function Quadratic Function Absolute Value Function Transformation Relation Parent Function Maximum Minimum Increasing Decreasing x-intercept y-intercept Function Notation F. answers to parent function packet algebra 2 is available in our digital library an online access to it is set as public so you can get it instantly. Algebra 2 TAP. The vertex is the point (1/2, 7/2). 00 Algebra 2 End of Year Review Escape Room Activity $8. Title: HONORS Algebra 2: Intro Activity to Graphing Absolute Value Functions Author: lausd_user. Transformations of exponential graphs behave similarly to those of other functions. 2) Algebra II. Here you will learn to identify primary function families by their equations and graphs. Let us start with a function, in this case it is f(x) = x 2, but it could be anything: f(x) = x 2. The graph of the function g(x) = is a reflection of the parent function f(x) =. f(x) = -x - 11 + 2 6b. What effect do the variables a and k have on the. ) Click to print the worksheet 2. Identify the domain and range. 7 Contrast to Parent Function: 2-1 0 1 2. The purpose of this lesson is to introduce students to some of the Parent Functions, and the vocabulary used to explain graphs. This is the graph that is transformed to create other members in a family of graphs. share to facebook share to twitter Questions. A quiz over the main 8 parent functions. Identified parent functions and transformations. September 22, 2019 0 Comment. Parent Function: y = x. 3: LINEAR FUNCTIONS AND SLOPE-INTERCEPT FORM MACC. Algebra 2 provides further development of the concept of functions, systems of functions, and more advanced nonlinear functions. Name _ Date _ Algebra 2, Parent Function Worksheet # 1- 6 Give the name of the parent function. But you can see where this could get really hard, right?. Algebra, functions, & patterns cover 20-30% of the GED Math test. Name the parent function. The restricted (natural) domain of the parent function is also of special interest as it is the non-negative real numbers, ie the interval. ( f ∘ g) ( x) = f [ g ( x)] = f [ 2 x − 1] ( f ∘ g) ( x) = f [ g ( x)] = f [ 2 x − 1] Now, notice that since we’ve got a formula for g ( x) g ( x) we went ahead and plugged that in first. Algebra 1 Parent Functions and Transformations. g(x) 5 x 3 2 4 Parent function: Transformation: 6. This equation is rewritten as y = log 2 x. Exponential functions increase faster than either linear or quadratic functions, while square root functions increase slower than either linear or quadratic functions. y x= +2 2 y x= −2 2 The graph of the function shifts up 2. Identified parent functions and transformations. Algebra Il: Unit 4 Group Competency Part 2 Name. Jul 10, 2017 - Explore Marla Barkman's board \"parent functions\" on Pinterest. 9A The student is expected to use the parent function to investigate, describe, and predict the effects of parameter changes on the graphs of square root functions and describe limitations on the domains and ranges. Specifically, we use th. 0 mathematicsvisionproject. Parent Letter; Unit 2 - Exponents; Parent Letter; Unit 3 - Linear Equations; Expanding Space Station; Parent Letter; Unit 4 - Functions Relations; Parent Letter; Unit 5 - Slippery Slope; Walk the Graph; Parent Letter; Unit 6 - Parallel Lines & Congruence; Parent Letter; Unit 7 - Systems of Equations and Inequalities; Cara’s Candles & DVD Club. A scientist notes the bacteria count in a petrie dish is 50. In cases where you have to have assistance with algebra and in particular with essay on parent functions in algebra 2 or math come pay a visit to us at Emaths. absolute value B. Algebra ‒ Parent Functions and Transformations THE PARENT FUNCTION GRAPHS AND TRANSFORMATIONS!. All notes and homework should be done in student binder. Algebra 1 - Unit 5 Vocabulary: Domain Range Translation Function Linear Function Quadratic Function Absolute Value Function Transformation Relation Parent Function Maximum Minimum Increasing Decreasing x-intercept y-intercept Function Notation F. Lesson 2-6 NAME DATE PERIOD PDF Pass Chapter 2 37 Glencoe Algebra 2 Exercises Graph each function. Then describe the transformations. represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. Can you identify different functions and their translations? 1. Example: y = 2x The blue line shows the graph of the stretch of the parent function, y = 2x, by a scale factor of 2. y x= +2 2 y x= −2 2 The graph of the function shifts up 2. Study Guide and Intervention Parent Functions and Transformations Name Characteristics Parent Function. Get full access to over 1,300 online videos and slideshows from multiple courses ranging from Algebra 1 to Calculus. A shrink is also referred to as a widening of the graph of the function closer to the x-axis. 4 Combining Functions; 1. Functions in the same family are transformations of their parent function. 1 5 Parent Functions And Transformations April 18th, 2019 - Function Of Time Is Given By F T At2 V 0 T X 0 Where A Is The Acceleration V 0 Is The Initial Velocity And X0 Is The Initial Position Of The Object Describe The Transformation S Of The Parent 2 / 7. When the parent function f (x) = x2 has an a -value that is less than 0, the graph reflects across the x -axis before it is transformed. We can use a handy trick to find out whether this data set matches a linear or quadratic function. Write “none” for transformations that do not exist. h(x) = −2 ⋅ f(x) Multiply the output by −2. Let's look at the effect of the addition or subtraction. It's never been late to practice some questions before the test. 1{1 Functions 78 4. PARENT FUNCTIONS Pre -Ap Algebra 2 Complete your Parent Function Packet!!!! • There are two slides per Parent Function. We can use a handy trick to find out whether this data set matches a linear or quadratic function. 1 5 Parent Functions And Transformations April 18th, 2019 - Function Of Time Is Given By F T At2 V 0 T X 0 Where A Is The Acceleration V 0 Is The Initial Velocity And X0 Is The Initial Position Of The Object Describe The Transformation S Of The Parent 2 / 7. 1 3 fx x=− + −2. g(x) 5 x 3 2 4 Parent function: Transformation: 6. Algebra 1 Module 1 Unit 2 Scatterplots and Lines of Best Fit Student Edition; Algebra 1 Module 2 Unit 3 Solving Equations & Inequalities Student Edition; Algebra 1 Module 2 Unit 4 Plus Honors Matrices Student Edition; Algebra 1 Module 2 Unit 4 Systems of Equations & Inequalities Student Ed; Algebra 1 Module 3 Functions; Algebra 1 Module 4. Additition resources: PDF 1, PDF 2, PDF 3, PDF 4. Provide at least 2 examples of a function Provide 1 example of a non-function 3. Our practice tests include the answers to all math problems. [Algebra 2] What is this parent function. The graph of the function shifts down 2. 1 vertex; 1 line of symmetry; The highest degree (the greatest exponent) of the function is 2; The graph is a parabola; Parent and Offspring. 2X-2 _ 1 2 d. Last Name . 03/17/21 3-4: Parent Functions 2 What is a parent function? The parent function is the simplest function with the defining characteristics of the family. PAP-Algebra II Parent Functions/Transformations Test 1 Name:_____ II. y = x +k y = a x: y = a. Algebra 3/MAT 150. When we multiply the parent function f (x) = b x f (x) = b x by −1, −1, we get a reflection about the x-axis. Algebra 1 Parent Functions and Transformations. and has specific properties and key points to assist in graphing. If x = 2 y, then y = (the power on base 2) to equal x. Introduction: Identifying and graphing parent functions and their translations has been a primary focus of. a) F(x)= x2-6x+1 b) F(x)= x 3 2 c) F(x)= 1 4 3 x d) F(x)= \|x-6\|+2 Sketch with graphing calculator and identify the requested point. This is the graph that is transformed to create other members in a family of graphs. The concept of parent function is less clear for polynomials of higher power because of the extra turning points, but for the family of n-degree polynomial functions for any given n, the parent function is sometimes taken as x n, or, to simplify further, x 2 when n is even and x 3 for odd n. Algebra 2 Pre-AP Overview 2020 - 2021 This document is designed provide parents/guardians/community an overview of the curriculum taught in the FBISD classroom. What effect do the variables a and k have on the. THEOREM 2. Algebra 2 - Parent Functions DRAFT. Unit 3 - Linear Functions. Homework Help. Jul 10, 2017 - Explore Marla Barkman's board \"parent functions\" on Pinterest. Duration : 25 min Activity 5. Learn about algebra 2 parent functions with free interactive flashcards. High School Math. Y Range: Class Notes special. horizontal stretching of functions common core algebra 2 homework answer key, Some students in my introductory algebra class struggle with basic concepts. 2X-2 _ 1 2 d. Square root; translation 4 units left 2. This document supports families in understanding the learning goals for the course, and how students will demonstrate what they know and are able to do. On this page you can read or download gina wilson all things algebra 2015 parent functions and transformations in PDF format. Jun 29, 2015 - Parent Functions and Transformations (Algebra 2 Curriculum - Unit 3) DISTANCE LEARNING UPDATE: This unit now contains a Google document with: (1) Links to instructional videos. Algebra 1 - Unit 5 Vocabulary: Domain Range Translation Function Linear Function Quadratic Function Absolute Value Function Transformation Relation Parent Function Maximum Minimum Increasing Decreasing x-intercept y-intercept Function Notation F. exponential. 2 Power Functions; 2. The restricted (natural) domain of the parent function is also of special interest as it is the non-negative real numbers, ie the interval. Use the video above instead of going on a field trip. Algebra, functions, & patterns cover 20-30% of the GED Math test. To find the answers, I can either work symbolically (like in the previous example) and then evaluate, or else I can find the values of the functions at x = 2 and then work from there. Browse other questions tagged algebra-precalculus functions transformation absolute-value or ask your own question. To shift the graph side to side, I need to add or subtract inside the argument of the function (that is, inside the parentheses). Luckily, this is not because function problems are inherently more difficult to solve than any other math problem, but because most students have simply not dealt with functions as much as they have other SAT math topics. Similar graph for all odd powers. image/svg+xml. SAT functions have the dubious honor of being one of the trickiest topics on the SAT math section. Y Quadratic; Linear; Cubic; reflection translation left translation down across the y-axis Graph the data from the table. Those being Linear (y=x), Absolute Value (y=IxI), Quadratic (y=x (power of 2)), Radical (y= (square root of x)), Cubic (y=x (power of 3)), Cube Root (y=3 (square root of x)), Exponential (y=b), Log (y=logb (x. Algebra 2 - Parent Functions DRAFT. pdf from MATH 2410 at Fremont High School. 2: Tesing Cause and Effect with Experiments and Conclusions From Observational Studies PDF Lesson 9. Algebra 2 Parent Functions Worksheet Answers - It is tiring when your kids request you in assisting these algebra residence operates, and you also are not able to do that home works, or you may not find out about them where you have not. Observe that this function increases when x is positive and decreases while x is negative. This document supports families in understanding the learning goals for the course, and how students will demonstrate what they know and are able to do. We are talking about transformations of exponential functions here. com/ Lesson 5. In a quadratic function, the variable is always squared. Teacher Resources Based on extensive research, SmartGraphs: Algebra is designed for students studying Algebra 1 or Algebra 2. This will be a major grade project and should demonstrate thorough and thoughtful work. Similar graph for all odd roots. x +k : Math 2 Unit 6 Lesson 1 Absolute Value Functions Page 5. High School Math. For example, if we begin by graphing the parent function $$f(x)=2^x$$, we can then graph the two reflections alongside it. Algebra 2 TAP. Y Quadratic; Linear; Cubic; reflection translation left translation down across the y-axis Graph the data from the table. 1 - Algebra 2 - parent functions and transformations. translation parent function 1–8. 2: Parent Functions and Transformations Name: Algebra II, Unit 1: Functions Block: In this course, you will learn shortcuts that allow you to sketch many different types of graphs quickly and accurately. How many zeros of the function are there in this graph? 6. Algebra and Trigonometry 10th Edition answers to Chapter 2 - 2. The table shows the linear and quadratic parent functions. In cases where you have to have assistance with algebra and in particular with essay on parent functions in algebra 2 or math come pay a visit to us at Emaths. (link on my webpage) An explanation of what a function is. Learn how to shift graphs up, down, left, and right by looking at their equations. Jump to navigation Jump to search. 9th - University grade. The Parent Functions are numbered in the bottom right corner of each slide. To do this, we simply add a constant term to the function. The 'Parent' Graph: The simplest parabola is y = x2, whose graph is shown at the right. Parent Functions And Transformations. Range of a Function, and Onto Functions 78 4. Just as with other parent functions, we can apply the four types of transformations—shifts, reflections, stretches, and compressions—to the parent function $f\\left(x\\right)={b}^{x}\\\\$ without loss of shape. From Wikiversity. exponential. Introduction to Parent Functions. This is illustrated below in the graph. Transformers Parts 2-5 - Algebra 2 Parent Functions For Students 10th - 12th Standards. the previous day. Scribd is the world's largest social reading and publishing site. Choose from 500 different sets of flashcards about algebra 2 parent functions on Quizlet. Learn about the properties and graphs of linear parent functions quadratic parent functions absolute value parent functions and reciprocal parent functions. parent function - the most basic function in a family; strip away all coefficients, negative signs, constants, etc you get the parent. Parent Graph: the graph of the parent function - it's the ; Function. pdf), Text File (. The parent graph, which is the graph of the parent function, is the simplest of the graphs in a family. 2 units up of the graph of the parent linear function. Changing variable names does not change the function. Tip #1: To Connect The Math Vocabulary To Prior Learning Okay, so you've probably heard this before, but as the teacher, we must connect new information to. IF Interpreting Functions Understand the concept of a function and use function notation. The graphs of these functions are drawn on the next page. Take a look. Transform rational functions by changing parameters. Range of a Function, and Onto Functions 78 4. When we multiply the parent function f (x) = b x f (x) = b x by −1, −1, we get a reflection about the x-axis. Graph is a horizontal line. Linear; translation down 6. Algebra 1 - Unit 5 Vocabulary: Domain Range Translation Function Linear Function Quadratic Function Absolute Value Function Transformation Relation Parent Function Maximum Minimum Increasing Decreasing x-intercept y-intercept Function Notation F. Functions are equal if they have the same domain and rule of correspondence. add 2 to each y-coordinate. Identified parent functions and transformations. Range of a Function, and Onto Functions 78 4. Here are a few of the types of functions we’ll learn about this year, and what their basic equations are, what their basic graphs look like, and what their domain and range are. log, c 4 16 9. Please watch through fi. y x= +2 2 y x= −2 2 The graph of the function shifts up 2. Algebra 1 Unit 7 – Quadratic Functions Monday Tuesday Wednesday Thursday Friday Mar 2 A Day 3 B Day 4 A Day 5 B Day 6 A Day Quadratic Parent Function Characteristics − Find the AOS, vertex, roots/zeros/ x-intercepts/solutions. Transform rational functions by changing parameters. For example, the function A = s² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a. 5 Inverses; 1. b: Use transformation to draw the graph of a relation and determine a relation that fits a graph. Describe what happened to the parent a. shifted up two units b. Day 5/6 - Parent Graphs and Transformations - Notes Day 6 - Parent. PARENT FUNCTIONS Author: Pete Falzone Created Date: 8/2/2001 12:36:31 AM. If you need a review on polynomials in general, feel free to go to Tutorial 6: Polynomials. Vertical Expansions and Compressions. Click on the lesson that. Discover (and save!) your own Pins on Pinterest. 0 mathematicsvisionproject. 2X-2 _ 1 2 d. View ALG2 HW Parent Functions. Parent Functions And Transformation - Displaying top 8 worksheets found for this concept. A point of special interest is the point (0,0) on the parent function. notebook 2 September 27, 2013 Sep 276:48 AM Vertex Axis of Symmetry Parent Function Sep 276:50 AM ex. 3 5 f xx= 10. 1-5 Assignment - Parent Functions and Transformations. Parent Functions In Algebra 2, we are going to learn about a lot of different types of functions. Plus each one comes with an answer key. Restrictions on Domain Most of the functions we have studied in Algebra I are defined for all real numbers. Get full access to over 1,300 online videos and slideshows from multiple courses ranging from Algebra 1 to Calculus. Basically, the graph of a polynomial function is a smooth continuous curve. The module culminates with the fundamental theorem of algebra as the ultimate result in factoring. f(x) = - 2 # 7- 12 Draw the basic graph for the parent function. Then write a function g that represents the translation of h. If you need a review on functions, feel free to go to Tutorial 30: Introduction to Functions. the graph of (x) = 2x, reflected across the x axis. 1) x y-8-6-4-22468-8-6-4-2 2 4 6 8 2) x y-8-6. To graph the function, plot the points on the graph with x-values 2, QUADRATIC FUNCTIONS PARENT FUNCTION: Creates a U-shape curve called a The is a vertical X line that divides the parabola into two equal parts. Similar graph for all even roots. IF Interpreting Functions Understand the concept of a function and use function notation. Jun 29, 2015 - Parent Functions and Transformations (Algebra 2 Curriculum - Unit 3) DISTANCE LEARNING UPDATE: This unit now contains a Google document with: (1) Links to instructional videos. In these examples, the k value is what is changing. If a parabola opens downward, it has a highest point. x is not a function of y, because the input y = 3 has multiple outputs: x = 1 and x = 2. That was a great start with my students in Algebra 2, as we were able to use it right away starting with linears. < Algebra II. Before exploring the family of related functions, let’s clarify some of the. The graph of the function shifts down 2. In Topic C, students solve rational and radical equations, identifying extraneous solutions, then modeling and solving equations in situations where rational and radical functions are necessary. Functions are equal if they have the same domain and rule of correspondence. Algebra 2 Team Tutorials – If I am not available, try one of the other great Algebra 2 teachers. Given no information regarding the specific equation of. The parent function is the simplest function with the defining characteristics of the family. The turning point is called the STANDARD FORM: Ax. A point of special interest is the point (0,0) on the parent function. It results in a graph of the same shape and size but possibly in a different position. Each poster gives the parent function, graph, and the following characteristics: domain, range, x-intercept, y-intercept, maximum number of roots, end behavior, increasing interval (s), decreasing interval (s) and asymptotes. Algebra 2 provides further development of the concept of functions, systems of functions, and more advanced nonlinear functions. By the end of. Unit 3 will include the following subtopics: Exponential Functions including Growth and Decay (compound interest) Direct and Inverse Variation ( Note: Please see Unit 2 for the above topics) Parent. Algebra and Trigonometry 10th Edition answers to Chapter 2 - 2. 2 Shift and Stretch A Solidify Understanding Task In 4. The math involved in the calculation is easy as long as you are careful in every step of … How to Tell if a Function is Even, Odd or Neither Read More ». Linear Absolute Value Quadratic Square Root Exponential Logarithmic. = -2(x - 1/2) 2 + 7/2. Algebra Il: Unit 4 Group Competency Part 2 Name. Yay Math in Studio returns, with the help of baby daughter, to share some knowledge about parent functions and their transformations. ) • Polar graphs (r=cos2θ) • Parametric. The parent graph of this family of graphs is shown below. Then graph the function. Graphing parent functions with transformations by Dan Tating - September 10, 2012. of this form is y = x2. Parent Function: y = x. They are simple, yet powerful in their ability to model real world situations. 1 Parent Functions & Transformations 1. owvc parent transformations intercept asymptote Simplify each logarithm. Common Parent Functions Linear Function: f(x) = x Domain: All real numbers. Parent functions included: linear, absolute value, quadratic, cubic, square root, cube root, reciprocal, exponential, and logarithmic. General Form for Describing the TRANSOFMRATIONS for a function f(x): F(x) ( a•f(x – h. 7 Statistics and Statistical Graphs. That was a great start with my students in Algebra 2, as we were able to use it right away starting with linears. On this page you can read or download gina wilson all things algebra 2015 parent functions and transformations in PDF format. Algebra 2 Syllabus – General information about Algebra 2 as well as materials needed for the class. Gina Wilson All Things Algebra Test. translation parent function 1–8. Mathematics: Parent functions \"give birth\" to child functions. g(x) = x 2 – 1 4. IF Interpreting Functions Understand the concept of a function and use function notation. The Parent and Student Study Guide Workbook includes: •A 1-page worksheet for every lesson in the. Grieser Page 2 3) Sketch the graphs using transformations. Parent Functions and Transformations. f(x) = -2/+3 Parent Function: List all transformations: Parent Function: List all transformations: 7. Graph is a horizontal line. Gina Wilson All Things Algebra Test. Unit# 4 Algebraic Expressions and Algebraic Formulas Exercise 4. Analyze functions using different representations. 1-0 0, describe how. Duration : 40 min Activity 5. Click on the lesson that. The phrase \"y is a function of x\" means that the value of y depends upon the value of. Transformers Parts 2-5 - Algebra 2 Parent Functions For Students 10th - 12th Standards. Identified parent functions and transformations. Functions in the same family are transformations of their parent function. Similar graph for all odd powers. High School Math. Relations The first section deals with the difference between relations and functions. Home Functions Logarithms Exponents Compounding Interest Probability Parent Funtions. 1 Algebra 2 Project With Free PDF ebook Download: Algebra 2 Project With Download or Read Online ebook algebra 2 parent function project with graphing in PDF Format From The Best User Guide Database 4th 9 Weeks Pre-AP Algebra II Project. From Wikiversity. Observe that this function increases when x is positive and decreases while x is negative. We add and subtract 1/4, because (-1/2) 2 = 1/4, and -1 is the coefficient of x. 00 Algebra 1 Review Flip Book$4. 4 Finding Real Zeros of Polynomials of Higher. Parent Functions. Unit 2: Functions & Their Graphs Date: Homework 6: Parent Functions & Transformations * This is a 2-page document ** Directions: Given each function, identify both the parent function and the transformations from the parent function. 4 years ago by. Each graph in a family of graphs has similar characteristics. • Function notation provides an efficient way to define and communicate functions. Lesson 2-6 NAME DATE PERIOD PDF Pass Chapter 2 37 Glencoe Algebra 2 Exercises Graph each function." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.85394484,"math_prob":0.9694573,"size":48179,"snap":"2021-21-2021-25","text_gpt3_token_len":11420,"char_repetition_ratio":0.23487286,"word_repetition_ratio":0.38511446,"special_character_ratio":0.23767616,"punctuation_ratio":0.12067665,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9887865,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-05-13T12:22:31Z\",\"WARC-Record-ID\":\"<urn:uuid:ffe64b0c-697f-43f8-ae66-0582bab2186f>\",\"Content-Length\":\"52979\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:c09b3335-c0d3-4c0d-bd26-c553582798fb>\",\"WARC-Concurrent-To\":\"<urn:uuid:fc03b614-a08d-4622-a33a-b596ea2fad06>\",\"WARC-IP-Address\":\"172.67.218.234\",\"WARC-Target-URI\":\"http://ilarioaltobelli.it/parent-functions-algebra-2.html\",\"WARC-Payload-Digest\":\"sha1:NPFKJWAV43T5RIE2FEWGIQGXSAZROWAL\",\"WARC-Block-Digest\":\"sha1:53CLGZNRG3PSJE4LJYVSYW3YVBVQJ6TQ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-21/CC-MAIN-2021-21_segments_1620243989916.34_warc_CC-MAIN-20210513111525-20210513141525-00502.warc.gz\"}"}
https://www.colorhexa.com/ebcb4f	[ "# #ebcb4f Color Information\n\nIn a RGB color space, hex #ebcb4f is composed of 92.2% red, 79.6% green and 31% blue. Whereas in a CMYK color space, it is composed of 0% cyan, 13.6% magenta, 66.4% yellow and 7.8% black. It has a hue angle of 47.7 degrees, a saturation of 79.6% and a lightness of 61.6%. #ebcb4f color hex could be obtained by blending #ffff9e with #d79700. Closest websafe color is: #ffcc66.\n\n• R 92\n• G 80\n• B 31\nRGB color chart\n• C 0\n• M 14\n• Y 66\n• K 8\nCMYK color chart\n\n#ebcb4f color description : Soft yellow.\n\n# #ebcb4f Color Conversion\n\nThe hexadecimal color #ebcb4f has RGB values of R:235, G:203, B:79 and CMYK values of C:0, M:0.14, Y:0.66, K:0.08. Its decimal value is 15453007.\n\nHex triplet RGB Decimal ebcb4f `#ebcb4f` 235, 203, 79 `rgb(235,203,79)` 92.2, 79.6, 31 `rgb(92.2%,79.6%,31%)` 0, 14, 66, 8 47.7°, 79.6, 61.6 `hsl(47.7,79.6%,61.6%)` 47.7°, 66.4, 92.2 ffcc66 `#ffcc66`\nCIE-LAB 82.347, -2.191, 63.683 57.029, 60.941, 16.156 0.425, 0.454, 60.941 82.347, 63.72, 91.971 82.347, 27.714, 74.49 78.064, -6.213, 42.375 11101011, 11001011, 01001111\n\n# Color Schemes with #ebcb4f\n\n• #ebcb4f\n``#ebcb4f` `rgb(235,203,79)``\n• #4f6feb\n``#4f6feb` `rgb(79,111,235)``\nComplementary Color\n• #eb7d4f\n``#eb7d4f` `rgb(235,125,79)``\n• #ebcb4f\n``#ebcb4f` `rgb(235,203,79)``\n• #bdeb4f\n``#bdeb4f` `rgb(189,235,79)``\nAnalogous Color\n• #7d4feb\n``#7d4feb` `rgb(125,79,235)``\n• #ebcb4f\n``#ebcb4f` `rgb(235,203,79)``\n• #4fbdeb\n``#4fbdeb` `rgb(79,189,235)``\nSplit Complementary Color\n• #cb4feb\n``#cb4feb` `rgb(203,79,235)``\n• #ebcb4f\n``#ebcb4f` `rgb(235,203,79)``\n• #4febcb\n``#4febcb` `rgb(79,235,203)``\n• #eb4f6f\n``#eb4f6f` `rgb(235,79,111)``\n• #ebcb4f\n``#ebcb4f` `rgb(235,203,79)``\n• #4febcb\n``#4febcb` `rgb(79,235,203)``\n• #4f6feb\n``#4f6feb` `rgb(79,111,235)``\n• #d5ae18\n``#d5ae18` `rgb(213,174,24)``\n• #e6bd21\n``#e6bd21` `rgb(230,189,33)``\n• #e8c438\n``#e8c438` `rgb(232,196,56)``\n• #ebcb4f\n``#ebcb4f` `rgb(235,203,79)``\n• #eed266\n``#eed266` `rgb(238,210,102)``\n• #f0d97d\n``#f0d97d` `rgb(240,217,125)``\n• #f3df94\n``#f3df94` `rgb(243,223,148)``\nMonochromatic Color\n\n# Alternatives to #ebcb4f\n\nBelow, you can see some colors close to #ebcb4f. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #eba44f\n``#eba44f` `rgb(235,164,79)``\n• #ebb14f\n``#ebb14f` `rgb(235,177,79)``\n• #ebbe4f\n``#ebbe4f` `rgb(235,190,79)``\n• #ebcb4f\n``#ebcb4f` `rgb(235,203,79)``\n• #ebd84f\n``#ebd84f` `rgb(235,216,79)``\n• #ebe54f\n``#ebe54f` `rgb(235,229,79)``\n• #e4eb4f\n``#e4eb4f` `rgb(228,235,79)``\nSimilar Colors\n\n# #ebcb4f Preview\n\nThis text has a font color of #ebcb4f.\n\n``<span style=\"color:#ebcb4f;\">Text here</span>``\n#ebcb4f background color\n\nThis paragraph has a background color of #ebcb4f.\n\n``<p style=\"background-color:#ebcb4f;\">Content here</p>``\n#ebcb4f border color\n\nThis element has a border color of #ebcb4f.\n\n``<div style=\"border:1px solid #ebcb4f;\">Content here</div>``\nCSS codes\n``.text {color:#ebcb4f;}``\n``.background {background-color:#ebcb4f;}``\n``.border {border:1px solid #ebcb4f;}``\n\n# Shades and Tints of #ebcb4f\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 #fdfaee is the lightest one.\n\n• #000000\n``#000000` `rgb(0,0,0)``\n• #120f02\n``#120f02` `rgb(18,15,2)``\n• #231d04\n``#231d04` `rgb(35,29,4)``\n• #352b06\n``#352b06` `rgb(53,43,6)``\n• #473a08\n``#473a08` `rgb(71,58,8)``\n• #58480a\n``#58480a` `rgb(88,72,10)``\n• #6a570c\n``#6a570c` `rgb(106,87,12)``\n• #7b650e\n``#7b650e` `rgb(123,101,14)``\n• #8d7310\n``#8d7310` `rgb(141,115,16)``\n• #9f8212\n``#9f8212` `rgb(159,130,18)``\n• #b09014\n``#b09014` `rgb(176,144,20)``\n• #c29f16\n``#c29f16` `rgb(194,159,22)``\n``#d4ad18` `rgb(212,173,24)``\n• #e5bb1a\n``#e5bb1a` `rgb(229,187,26)``\n• #e7c12c\n``#e7c12c` `rgb(231,193,44)``\n• #e9c63d\n``#e9c63d` `rgb(233,198,61)``\n• #ebcb4f\n``#ebcb4f` `rgb(235,203,79)``\n• #edd061\n``#edd061` `rgb(237,208,97)``\n• #efd572\n``#efd572` `rgb(239,213,114)``\n• #f1db84\n``#f1db84` `rgb(241,219,132)``\n• #f3e095\n``#f3e095` `rgb(243,224,149)``\n• #f5e5a7\n``#f5e5a7` `rgb(245,229,167)``\n• #f7eab9\n``#f7eab9` `rgb(247,234,185)``\n• #f9efca\n``#f9efca` `rgb(249,239,202)``\n• #fbf5dc\n``#fbf5dc` `rgb(251,245,220)``\n• #fdfaee\n``#fdfaee` `rgb(253,250,238)``\nTint Color Variation\n\n# Tones of #ebcb4f\n\nA tone is produced by adding gray to any pure hue. In this case, #a09f9a is the less saturated color, while #fad440 is the most saturated one.\n\n• #a09f9a\n``#a09f9a` `rgb(160,159,154)``\n• #a7a393\n``#a7a393` `rgb(167,163,147)``\n• #afa78b\n``#afa78b` `rgb(175,167,139)``\n• #b6ac84\n``#b6ac84` `rgb(182,172,132)``\n• #beb07c\n``#beb07c` `rgb(190,176,124)``\n• #c5b575\n``#c5b575` `rgb(197,181,117)``\n• #cdb96d\n``#cdb96d` `rgb(205,185,109)``\n• #d4be66\n``#d4be66` `rgb(212,190,102)``\n• #dcc25e\n``#dcc25e` `rgb(220,194,94)``\n• #e3c757\n``#e3c757` `rgb(227,199,87)``\n• #ebcb4f\n``#ebcb4f` `rgb(235,203,79)``\n• #f3cf47\n``#f3cf47` `rgb(243,207,71)``\n``#fad440` `rgb(250,212,64)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #ebcb4f 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":0.5145055,"math_prob":0.5319821,"size":3701,"snap":"2020-10-2020-16","text_gpt3_token_len":1660,"char_repetition_ratio":0.12902354,"word_repetition_ratio":0.011111111,"special_character_ratio":0.5228317,"punctuation_ratio":0.23634337,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9714125,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-02-24T19:10:29Z\",\"WARC-Record-ID\":\"<urn:uuid:15b57d11-a874-42d4-bec3-45cb7accaace>\",\"Content-Length\":\"36309\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b857bb5b-9902-4837-b9db-e3e2a2e92947>\",\"WARC-Concurrent-To\":\"<urn:uuid:937cc2aa-ef6f-4b9a-9104-32c14502cf25>\",\"WARC-IP-Address\":\"178.32.117.56\",\"WARC-Target-URI\":\"https://www.colorhexa.com/ebcb4f\",\"WARC-Payload-Digest\":\"sha1:YJF2AICG72JSXHPMPSRFKVXDCRAJEDZZ\",\"WARC-Block-Digest\":\"sha1:NKJ4J66TFJSN5BVOPAGZMHUFONH66LHH\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-10/CC-MAIN-2020-10_segments_1581875145966.48_warc_CC-MAIN-20200224163216-20200224193216-00504.warc.gz\"}"}
https://stats.stackexchange.com/questions/tagged/skellam	[ "# Questions tagged [skellam]\n\nThe Skellam distribution is a discrete distribution that describes the difference between two independent Poisson distributions with possibly different parameters.\n\n19 questions\nFilter by\nSorted by\nTagged with\n363 views\n\n### What probability distribution would be suitable for modelling scores in a basketball match?\n\nIn football (not American football but what Americans call soccer), it is pretty clear: we may consider the difference of two independent Poisson variables. In basketball, in theory, we could increase ...\n554 views\n\n### What is the expectation of the absolute value of the Skellam distribution?\n\nIn particular, for a Skellam distribution obtained by substracting two iid Poisson Processes. Thank you!\n293 views\n\n184 views\n\n### Estimator of Bessel function?\n\nI am trying to estimate the parameters of the modified Bessel function of the first kind for integer order case. $I_n(wt) = \\sum\\limits_{m=0}^\\infty \\frac{1}{m!(m+n)!}(\\frac{wt}{2})^{2m+n}$ In ...\n1k views\n\n### Is the absolute value of the difference between two Poisson distributions a Poisson distribution?\n\nWhat is the distribution of the absolute value of the Skellam distribution?\n218 views\n\n### How to compare rates of occurence in consecutive time series count data?\n\nMy data consists of occurrences of words in time windows. E.g.: Day; Word; Frequency 1; \"dog\"; 45 1; \"cat\"; 2 ... 2; \"dog\"; 90 2; \"cat\"; 4 ... I would like to ...\n3k views\n\n### The difference between discrete and continuous variables\n\nIs the number of hydrogen bonds or the number of rings in a molecule a discrete or a continuous variable ? Can I say that the number of rings: 1, 3, 4, $\\dots$ is a continuous variable because in ...\n401 views\n\n### Poisson processes\n\nI have two realizations of a poisson stochastic process, they are over the same space with rate $\\lambda_{1}$ and $\\lambda_{2}$. What is the probability that N elements in both sequences are the same, ...\n7k views\n\n### How to calculate cumulative Poisson probabilities without adding each one if no. of outcomes is large and hence, adding (without excel) is difficult?\n\nOne can calculate the probability of a correct score of a football match by Poisson(Number of Team1 goals, Mean average Team1 goals) x Poisson(Number of Team2 goals, Mean average Team2 goals) So for ...\nI want to estimate the parameter $\\mu$ using the difference between two Poisson distributions with the same parameter, i.e. a Skellam distribution with $\\mu_1=\\mu_2 = \\mu$. I can calculate the ..." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8817341,"math_prob":0.9805266,"size":5631,"snap":"2019-26-2019-30","text_gpt3_token_len":1337,"char_repetition_ratio":0.18073574,"word_repetition_ratio":0.038331453,"special_character_ratio":0.23228556,"punctuation_ratio":0.13828786,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9936652,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-07-23T09:51:33Z\",\"WARC-Record-ID\":\"<urn:uuid:4cc661d1-693c-482a-8de3-9e228b62c91b>\",\"Content-Length\":\"161581\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e6b01d10-afbb-442c-8490-792fff6d7066>\",\"WARC-Concurrent-To\":\"<urn:uuid:ef8aea08-f4c7-4a43-8bde-e311bed3feb2>\",\"WARC-IP-Address\":\"151.101.1.69\",\"WARC-Target-URI\":\"https://stats.stackexchange.com/questions/tagged/skellam\",\"WARC-Payload-Digest\":\"sha1:LMEYTDLKNRWN4ZUJLTWGTBERAS3KIP5C\",\"WARC-Block-Digest\":\"sha1:YMRVIKPCF3EUUJFHP4B6IYFOFQLZR2SL\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-30/CC-MAIN-2019-30_segments_1563195529175.83_warc_CC-MAIN-20190723085031-20190723111031-00056.warc.gz\"}"}
https://number.rocks/add/3367/plus/18	[ "What is the sum of 3367 plus 18 and How much is 3367 + 18 percent?\n\nAddition of 3367 and 18 is 3385 and 3367 plus 18 percent is 3973.06\n\n3367 + 18\n=\n3385\n3367 + 18 percent\n=\n3973.06\n\nStep by Step Calculation for what is 3367 plus 18%\n\n= 3367 + 18% of 3367\n\n= 3367 + 18 * 3367 / 100\n\n= 3367 + 18 * 3367/100\n\n= 3367 + 30303/50\n\n= 63569/16\n\n63569/16 or fraction as a decimal is 3973.06" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8697019,"math_prob":0.9997507,"size":328,"snap":"2020-24-2020-29","text_gpt3_token_len":140,"char_repetition_ratio":0.20987654,"word_repetition_ratio":0.028571429,"special_character_ratio":0.61585367,"punctuation_ratio":0.042857144,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99995875,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-09T15:09:52Z\",\"WARC-Record-ID\":\"<urn:uuid:ac456dbf-4175-4a76-b91e-5e312cefb64b>\",\"Content-Length\":\"5483\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:41b7e8db-7586-4b61-8f1e-7b5fd90e5649>\",\"WARC-Concurrent-To\":\"<urn:uuid:ceb97c19-2e77-4343-a19e-be74854f1683>\",\"WARC-IP-Address\":\"166.62.10.36\",\"WARC-Target-URI\":\"https://number.rocks/add/3367/plus/18\",\"WARC-Payload-Digest\":\"sha1:QXIDY7TIBFTSXSZ26IO42X5WKITXADVY\",\"WARC-Block-Digest\":\"sha1:PAVUWNWG6QTBAIRPDKM5AZGV6EVRGS42\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-29/CC-MAIN-2020-29_segments_1593655900335.76_warc_CC-MAIN-20200709131554-20200709161554-00194.warc.gz\"}"}
https://westcottcourses.com/subsite-publishing/algebra1-standards.html	[ "### California State Standard for Algebra l\n\nSymbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences. In addition, algebraic skills and concepts are developed and used in a wide variety of problem-solving situations.\n\nCalifornia Standards for Algebra l Omega Math's Alignment to California Standards\n1.0 Students identify and use the arithmetic properties of subsets of integers and rational, irrational, and real numbers, including closure properties for the four basic arithmetic operations where applicable: Algebra l - e1.1\n1.1 Students use properties of numbers to demonstrate whether assertions are true or false. Algebra l - e1.1, e1.5, e1.7, e3.1, e3.2, e4.1, e5.5, e6.5, e7.2, e7.3, e7.4, e8.5, e10.3\n2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents. Algebra l - e1.4, chapter 8, e2.1, e2.2, e6.4\n3.0 Students solve equations and inequalities involving absolute values. Algebra l - Chapter 3, e1.7, e6.5\n4.0 Students simplify expressions before solving linear equations and inequalities in one variable, such as 3(2x-5) + 4(x-2) = 12. Algebra l - e3.2\n5.0 Students solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step. Algebra l - Chapter 10, e3.5\n6.0 Students graph a linear equation and compute the x-and y-intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the region defined by 2x + 6y < 4). Algebra l - e6.2, e6.3, e6.5\n7.0 Students verify that a point lies on a line, given an equation of the line. Students are able to derive linear equations by using the point-slope formula. Algebra l - e6.4\n8.0 Students understand the concepts of parallel lines and perpendicular lines and how those slopes are related. Students are able to find the equation of a line perpendicular to a given line that passes through a given point. Algebra l - e6.2, e6.4\n9.0 Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets. Algebra 1 - e7.4\n10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques. Algebra l - e2.4, e2.5, e2.7, e3.5, e4.6, e6.4, e7.3, e8.6, e10.1, e10.2, e10.3\n11.0 Students apply basic factoring techniques to second-and simple third-degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials. Algebra l - Chapter 4\n12.0 Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms. Algebra l - e5.4\n13.0 Students add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques. Algebra l - Chapter 5\n14.0 Students solve a quadratic equation by factoring or completing the square. Algebra l - e4.6, e9.2\n15.0 Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems. Algebra l - Chapter 10\n16.0 Students understand the concepts of a relation and a function, determine whether a given relation defines a function, and give pertinent information about given relations and functions. Algebra ll - Chapter 10\n17.0 Students determine the domain of independent variables and the range of dependent variables defined by a graph, a set of ordered pairs, or a symbolic expression. Algebra ll - i10.2\n18.0 Students determine whether a relation defined by a graph, a set of ordered pairs, or a symbolic expression is a function and justify the conclusion. Algebra ll - i10.1\n19.0 Students know the quadratic formula and are familiar with its proof by completing the square. Algebra l - e9.3\n20.0 Students use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations. Algebra l - e9.3\n21.0 Students graph quadratic functions and know that their roots are the x-intercepts. Algebra ll - i8.5\n22.0 Students use the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will intersect the x-axis in zero, one, or two points. Algebra l-e9.3\n23.0 Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity. Algebra l - e4.6\n24.0 Students use and know simple aspects of a logical argument. Geometry - g2.1\n24.1 Students explain the difference between inductive and deductive reasoning and identify and provide examples of each. N/A\n24.2 Students identify the hypothesis and conclusion in logical deduction. 24.3 Students use counterexamples to show that an assertion is false and recognize that a single counterexample is sufficient to refute an assertion. Geometry - g2.1\n25.0 Students use properties of the number system to judge the validity of results, to justify each step of a procedure, and to prove or disprove statements: Algebra l - e1.1\n25.1 Students use properties of numbers to construct simple, valid arguments (direct and indirect) for, or formulate counterexamples to, claimed assertions. Geometry - g2.1, g2.2, g2.3\n25.2 Students judge the validity of an argument according to whether the properties of the real number system and the order of operations have been applied correctly at each step. Algebra l - e1.1, e1.5, e1.6, e1.7, e3.1, e3.2, e3.4, e5.5, e6.5, e7.2, e7.3, e7.4, e8.2, e8.5, e10.3\n25.3 Given a specific algebraic statement involving linear, quadratic, or absolute value expressions or equations or inequalities, students determine whether the statement is true sometimes, always, or never. Algebra 1 - e3.2" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.83725655,"math_prob":0.9907529,"size":6222,"snap":"2021-43-2021-49","text_gpt3_token_len":1570,"char_repetition_ratio":0.16806047,"word_repetition_ratio":0.057494868,"special_character_ratio":0.2582771,"punctuation_ratio":0.18716578,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9989885,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-22T15:14:28Z\",\"WARC-Record-ID\":\"<urn:uuid:a0e019fe-e8b3-421a-a8ad-9d8544ebf9d5>\",\"Content-Length\":\"17285\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:76188c31-28f6-41a2-824e-090c17ad3384>\",\"WARC-Concurrent-To\":\"<urn:uuid:381c16dc-a474-4035-936d-65477f4302a0>\",\"WARC-IP-Address\":\"54.68.116.64\",\"WARC-Target-URI\":\"https://westcottcourses.com/subsite-publishing/algebra1-standards.html\",\"WARC-Payload-Digest\":\"sha1:PTYTO7H4FNKHKD3PXVFIFBIQMFAGUBSU\",\"WARC-Block-Digest\":\"sha1:QK2R2GB7MBRBCYFJCXGDR2I6C3ZLQHPZ\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323585516.51_warc_CC-MAIN-20211022145907-20211022175907-00496.warc.gz\"}"}
https://www.arxiv-vanity.com/papers/0809.1501/	[ "# Quantum Semi-Markov Processes\n\nHeinz-Peter Breuer Physikalisches Institut, Universität Freiburg, Hermann-Herder-Strasse 3, D-79104 Freiburg, Germany Hanse-Wissenschaftskolleg, Institute for Advanced Study, D-27753 Delmenhorst, Germany Bassano Vacchini Dipartimento di Fisica dell’Università di Milano and INFN Sezione di Milano, Via Celoria 16, I-20133 Milano, Italy\nMarch 19, 2023\n###### Abstract\n\nWe construct a large class of non-Markovian master equations that describe the dynamics of open quantum systems featuring strong memory effects, which relies on a quantum generalization of the concept of classical semi-Markov processes. General conditions for the complete positivity of the corresponding quantum dynamical maps are formulated. The resulting non-Markovian quantum processes allow the treatment of a variety of physical systems, as is illustrated by means of various examples and applications, including quantum optical systems and models of quantum transport.\n\n###### pacs:\n03.65.Yz,42.50.Lc,02.50.Ga,03.65.Ta\n\nThe analysis of the time evolution of open systems plays a central role in many applications of modern quantum theory, including quantum information science, quantum transport theory, quantum thermodynamics, and quantum process tomography and control (see e.g. general ). The state of an open quantum system that is coupled to the degrees of freedom of its surroundings is represented by a time-dependent density matrix . In the Markovian regime the dynamics is governed by a master equation of the relatively simple form\n\n ddtρ(t)=Lρ(t), (1)\n\n Lρ=−i[H,ρ]+∑α[AαρA†α−12{A†αAα,ρ}]. (2)\n\nThe Hamiltonian describes the coherent part of the time evolution and the are certain operators representing the various decay modes. The solution of Eq. (1) can be written in terms of a linear map that transforms the initial state into the state at time . The physical interpretation of this map requires that it preserves the trace and the positivity of the density matrix . According to general physical principles must be a completely positive (CP) map KRAUS ; TheWork . Hence, represents a CP dynamical semigroup known as quantum Markov process, whose generator has been proven GORINI-LINDBLAD to be of the form (2).\n\nThe quantum dynamics given by Eq. (2) has a clearcut connection to a classical Markov process for the case in which one has a closed system of equations for the populations in a fixed orthonormal basis of the open system’s Hilbert space, typically the energy eigenbasis. In fact, in this case one recovers the Pauli master equation,\n\n ddtPn(t)=∑m[ΓnmPm(t)−ΓmnPn(t)], (3)\n\nwhich describes a classical Markovian jump process with transition rates , justifying the notion of a quantum Markov process.\n\nThe most important physical assumption which underlies the master equation (1) is the validity of the Markov approximation of short environmental correlation times. If this approximation is violated non-Markovian dynamics emerges which is characterized by pronounced memory effects, finite revival times and non-exponential relaxation and decoherence. These effects can result from long-range correlation functions, from correlations and entanglement in the initial state, as well as from the neglection of extra degrees of freedom affecting the dynamics BUDINI ; nmgroup . As a consequence the theoretical treatment of non-Markovian quantum dynamics is generally extremely demanding. A widely used non-Markovian generalization of Eq. (1) is given by the integrodifferential equation\n\n ddtρ(t)=∫t0dτK(τ)ρ(t−τ). (4)\n\nIn this equation one takes into account quantum memory effects through the introduction of the memory kernel which means that the rate of change of the state at time depends on the states at previous times . Equations of the form (4) arise, for instance, by employing the standard Nakajima-Zwanzig projection operator technique NAKAJIMA-ZWANZIG . Obviously, the Markovian master equation (1) is obtained if the memory kernel is taken to be proportional to a -function, .\n\nIn order to be physically acceptable the superoperator appearing in Eq. (4) must grant the CP of the resulting quantum dynamical map . This is a very stringent requirement and, in fact, the general structural characterization of physically admissible memory kernels is an unsolved problem of central importance in the field of non-Markovian quantum dynamics BUDINI ; Daffer-Lidar . It has been realized recently that even the most simple and natural choices for the memory kernel can lead to unphysical results BARNETT ; BUDINI . To improve this situation we will construct a class of non-Markovian quantum master equations that arises naturally as a quantum mechanical generalization of classical semi-Markov processes Feller . The approach proposed here leads to important physical insights guiding the phenomenological determination of the memory kernel, and, at the same time, enables a compact formulation of sufficient conditions that guarantee the existence and the CP of the quantum dynamical map. Moreover, for a specific class of processes one can formulate CP conditions which are not only sufficient but also necessary.\n\nWe consider memory kernels with the general structure\n\n K(τ)ρ = −i[H(τ),ρ]−12∑α{A†α(τ)Aα(τ),ρ} (5) +∑αAα(τ)ρA†α(τ),\n\nthat is to say of the form given by Eq. (2) apart from the time dependence of the considered operators. As previously done in the Markovian case let us consider the situation in which the populations obey a closed system of equations of motion, which then takes the form\n\n ddtPn(t) = ∫t0dτ∑m[Wnm(τ)Pm(t−τ) (6) −Wmn(τ)Pn(t−τ)],\n\nwhere . This is the master equation for a general type of classical non-Markovian processes known as semi-Markov processes Feller . Thus, whenever the populations obey closed equations, Eq. (2) yields the classical Markovian master equation (3), while Eq. (4) with the kernel (5) leads under the same conditions to the generalized master equation (6) for a classical semi-Markov process. This justifies on the same footing as before the name quantum semi-Markov process.\n\nTo clarify the physical content of Eq. (6) let us consider as an example the situation in which the kernel functions factorize as , where and . The corresponding process can then be interpreted as describing a particle moving on a lattice with sites labelled by , where the are the probabilities for jumps from site to site . Jumps out of a given site take place after a certain waiting time that follows the waiting time distribution . The characteristic feature of semi-Markov processes is the fact that, by contrast to the Markovian case, need not be an exponential function, but can be any probability distribution, thus giving rise to memory effects. These waiting time distributions are uniquely determined by the functions according to the relation GILLESPIE\n\n fn(t)=∫t0dτkn(τ)gn(t−τ)≡(kn∗gn)(t), (7)\n\nwhere the function\n\n gn(t)=1−∫t0dτfn(τ) (8)\n\ndenotes the probability not to have left site by time , the so-called survival probability, and is the usual convolution product. The generalized master equation (6) therefore provides a physically acceptable time evolution for the populations , granting in particular their positivity, provided the functions allow an interpretation in terms of waiting time distributions GILLESPIE ; Esposito2008a .\n\nHowever, these classical conditions are clearly not enough to ensure the existence of a well-defined dynamics in the quantum case, and a general characterization at the quantum level can hardly be achieved. Therefore our next goal is the formulation of sufficient conditions that guarantee the CP of the dynamical map corresponding to the non-Markovian master equation defined by Eqs. (4) and (5), no longer assuming that closed equations for the populations exist. This map is defined by\n\n ddtV(t)=∫t0dτK(τ)V(t−τ), (9)\n\ntogether with the initial condition , with the identity map. We now employ ideas recently formulated in Ref. KOSSAKOWSKI , decomposing the memory kernel as , where is the CP map defined by\n\n B(τ)ρ=∑αAα(τ)ρA†α(τ), (10)\n\nand is given by the first line of (5). We further introduce the map as the solution of the equation\n\n ddtV0(t)=∫t0dτC(τ)V0(t−τ), (11)\n\nwith the initial condition . Considering the Laplace transforms of Eqs. (9) and (11) one obtains a resolvent-like identity for the dynamical map leading in the time domain to the equation\n\n V(t)=V0(t)+(V0∗B∗V)(t). (12)\n\nRegarding formally the superoperator as a perturbation and iterating Eq. (12) one finds that the full dynamical map can be represented as a series,\n\n V(t) = V0(t)+(V0∗B∗V0)(t) (13) +(V0∗B∗V0∗B∗V0)(t)+….\n\nDue to the fact that the set of CP maps is closed under addition and convolution, we can conclude from Eq. (13) that is CP if is CP. To bring this condition into an explicit form let us assume that the Hermitian operators and are diagonal in the time-independent orthonormal basis , that is and\n\n ∑αA†α(τ)Aα(τ)=∑nkn(τ)\|n⟩⟨n\|. (14)\n\nThen we can solve Eq. (11) to obtain\n\n V0(t)ρ(0)=∑nmgnm(t)\|n⟩⟨n\|ρ(0)\|m⟩⟨m\|, (15)\n\nwhere the functions are the solutions of\n\n ˙gnm(t)=−∫t0dτ[zn(τ)+z∗m(τ)]gnm(t−τ), (16)\n\ncorresponding to the initial conditions , and . To prove Eq. (15) one first shows that . Using this relation one easily demonstrates that the expression (15) indeed represents the desired solution of Eq. (11). It is important to notice that the functions do actually coincide with the survival probabilities introduced by Eq. (8).\n\nEmploying the Kraus representation KRAUS we see that the map given by Eq. (15) is CP if and only if the matrix with elements is positive,\n\n G(t)=(gnm(t))≥0. (17)\n\nHence, we arrive at a sufficient condition for CP: The quantum dynamical map corresponding to the non-Markovian master equation (4) with the memory kernel (5) is CP if the condition (17) is fulfilled. A necessary condition for (17) to hold is the positivity of the diagonal elements of , which are given by the survival probabilities . This necessary condition in turn implies the positivity of the functions according to Eq. (7), which can then be interpreted as true waiting time distributions. The positivity of the matrix therefore represents a natural quantum generalization of the classical conditions.\n\nWe illustrate the result (17) with the help of several examples, which all fall into the class of quantum semi-Markov processes introduced by means of Eqs. (4) and (5). A prototypical system showing strong non-Markovian behavior is a two-level atom interacting with a damped field mode described by the memory kernel\n\n K(τ)ρ = −iε(τ)[σ+σ−,ρ] (18) +k(τ)[σ−ρσ+−12{σ+σ−,ρ}].\n\nExcited and ground state are denoted by and , respectively, and are the corresponding raising and lowering operators. The index thus takes on the two values . For a positive function the memory kernel (18) is of the form specified above with , , , and . Hence, the matrix takes the form\n\n G(t)=(g++(t)g+−(t)g∗+−(t)1), (19)\n\nwith and determined by Eq. (16). Thus we see that the condition (17) for CP is equivalent to . The master equation corresponding to the memory kernel (18) can be solved analytically. One then finds that for this case the condition (17) is not only sufficient but also necessary for CP.\n\nA further very instructive example involving an infinite dimensional Hilbert space is the model of a quantum oscillator with non-Markovian damping studied in Ref. BARNETT . The memory kernel for this model reads\n\n K(τ)ρ=k(τ)[aρa†−12{a†a,ρ}], (20)\n\nwhere and , are the raising and lowering operators of the oscillator. This kernel is again of the form (5) with a single Lindblad operator . Here, the basis states are the number states of the oscillator, and . Solving Eq. (16) by means of a Laplace transformation, we find\n\n gnn(t)=e−γt/2[cosh(dnt/2)+γdnsinh(dnt/2)],\n\nwhere . For the necessary condition to hold must be real. This shows that condition (17) is certainly violated if . Because can be arbitrary large we conclude that condition (17) is never fulfilled. The interesting aspect of this example is the fact that the non-Markovian master equation indeed violates not only CP but also positivity. This fact has been demonstrated in BARNETT and clearly shows again the relevance of our CP conditions.\n\nMany further physical systems lead to a generalized master equation of the form introduced here if one applies the Nakajima-Zwanzig projection operator technique, such as the tight-binding quantum diffusion model studied in GASPARD , and the quantum transport model introduced in DIFBAL , which leads to a memory kernel of the form\n\n K(τ)ρ=k(τ)[12TρT†+12T†ρT−ρ]. (21)\n\nThis kernel describes the motion of an excitation in a modular system consisting of weakly coupled subunits labelled by the index , where represents the corresponding translation operator. The model features strong non-Markovian behavior and a transition from diffusive to ballistic quantum transport. The memory kernel is obviously of the form introduced above. The Hamiltonian contribution vanishes, , and all kernel functions are equal to each other, , which corresponds to the special case treated in Refs. KOSSAKOWSKI ; BUDINI with a loss term proportional to the identity operator. Equation (16) shows that also all matrix elements of are equal, , and, hence, condition (17) reduces to the condition . Clearly this condition leads to important restrictions on the form of the kernel function which is determined by the correlation function of the microscopic model.\n\nAs our final example we discuss memory kernels of the following general structure,\n\n K(τ)ρ = −i[H(τ),ρ]−12∑nkn(τ){\|n⟩⟨n\|,ρ}, (22) +∑nmπnmkm(τ)\|n⟩⟨m\|ρ\|m⟩⟨n\|\n\nof which (18) provides an example. For this memory kernel the coherences of the density matrix, i. e. the off-diagonal elements , , are simply given by . On the other hand, the diagonals of the density matrix, i. e. the populations obey a closed transport equation as in (6). It is remarkable that in this case one can go one step further to derive a condition for the CP which is not only sufficient but also necessary. To this end one writes the quantum dynamical map corresponding to the non-Markovian quantum master equation (4) with the memory kernel (22) in terms of the functions and of the conditional transition probabilities obeying the classical master equation (6). The quantity represents the probability that the particle is at site at time given that it started at site at time . With the help of the resulting expression for the map we then find the following result. Given a classical semi-Markov process, the quantum dynamical map is CP if and only if the condition\n\n ~G(t)=(~gnm(t))≥0 (23)\n\nis satisfied. Here, the off-diagonal elements of the matrix coincide with those of , while the diagonals of are given by the conditional transition probabilities, . Note that the probabilities are in fact in general greater than the corresponding survival probabilities , since the system can be in state at time both because it has not left it and because it has come back to the initial state. Eq. (23) thus provides a complete characterization of the CP of the class of quantum semi-Markov processes given by (22).\n\nBuilding on an analogy with classical semi-Markov processes we have constructed a large class of non-Markovian master equations with memory kernel and formulated sufficient conditions for the CP of the resulting quantum dynamical map. The latter impose strong restrictions on the structure of physically acceptable non-Markovian quantum master equations, which are particularly useful in phenomenological approaches. For a specific class of quantum semi-Markov processes necessary and sufficient conditions for CP have also been formulated. Important further developments of the theory should include the case of temporarily negative kernel functions and effects from correlations and entanglement in the initial state.\n\n###### Acknowledgements.\nOne of us (HPB) gratefully acknowledges a Fellowship of the Hanse-Wissenschaftskolleg, Delmenhorst.\n\n## References\n\n• (1) J. Gemmer, M. Michel, and G. Mahler, Quantum Thermodynamics, Lecture Notes in Physics, vol 657 (Springer, Berlin, 2004); S. Mancini, V. I. Man’ko, and H. M. Wiseman (eds.), J. Opt. B: Quantum Semiclass. Opt. 7 (2005); R. Alicki, D. A. Lidar, and P. Zanardi Phys. Rev. A 73, 052311 (2006); H. P. Breuer, J. Gemmer, M. Michel, and U. Schollwöck (eds.), Eur. Phys. J. Special Topics 151 (2007).\n• (2) V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, J. Math. Phys. 17, 821 (1976); G. Lindblad, Commun. Math. Phys. 48, 119 (1976).\n• (3) E. C. G. Sudarshan, P. M. Mathews, and J. Rau, Phys. Rev. 121, 920 (1961); K. Kraus, States, Effects, and Operations, Lecture Notes in Physics, vol 190 (Springer, Berlin, 1983).\n• (4) H. P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (OUP, Oxford, 2007).\n• (5) A. A. Budini, Phys. Rev. A 69, 042107 (2004).\n• (6) H. P. Breuer, J. Gemmer, and M. Michel, Phys. Rev. E 73, 016139 (2006); H. P. Breuer, Phys. Rev. A 75, 022103 (2007); A. A. Budini, Phys. Rev. E 72, 056106 (2005); Phys. Rev. A 74, 053815 (2006); A. A. Budini and H. Schomerus, J. Phys. A: Math. Gen. 38, 9251 (2005); B. Vacchini, Phys. Rev. A 78, 022112 (2008).\n• (7) S. Nakajima, Progr. Theor. Phys. 20, 948 (1958); R. Zwanzig, J. Chem. Phys. 33, 1338 (1960).\n• (8) S. Daffer et al., Phys. Rev. A 70, 010304 (R) (2004); A. Shabani and D. A. Lidar, Phys. Rev. A 71, 020101 (R) (2005).\n• (9) S. M. Barnett and S. Stenholm, Phys. Rev. A 64, 033808 (2001).\n• (10) W. Feller, An Introduction to Probability Theory and its Applications. Vol. II, John Wiley & Sons Inc., New York, 1971; Proc. Natl. Acad. Sci. U.S.A. 51, 653 (1964); B. D. Hughes, Random Walks and Random Environments (OUP, New York, 1995).\n• (11) D. T. Gillespie, Phys. Lett. A 64, 22 (1977).\n• (12) M. Esposito and K. Lindenberg, Phys. Rev. E 77, 051119 (2008).\n• (13) A. Kossakowski and R. Rebolledo, eprint arXiv:0801.1057.\n• (14) M. Esposito and P. Gaspard, Phys. Rev. B 71, 214302 (2005).\n• (15) R. Steinigeweg, H. P. Breuer, and J. Gemmer, Phys. Rev. Lett. 99, 150601 (2007)." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.87949073,"math_prob":0.97787726,"size":16958,"snap":"2023-14-2023-23","text_gpt3_token_len":3942,"char_repetition_ratio":0.15447682,"word_repetition_ratio":0.029829029,"special_character_ratio":0.23682038,"punctuation_ratio":0.14981389,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99365085,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-03-25T01:49:12Z\",\"WARC-Record-ID\":\"<urn:uuid:b67f39a1-a1eb-4fba-8bc3-7cc0e0a037cf>\",\"Content-Length\":\"388937\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7cbc0f44-4bd2-41d2-9e05-3ce1a7b9fe1e>\",\"WARC-Concurrent-To\":\"<urn:uuid:7081e9b1-f217-4ab2-9b06-619e8743a929>\",\"WARC-IP-Address\":\"104.21.14.110\",\"WARC-Target-URI\":\"https://www.arxiv-vanity.com/papers/0809.1501/\",\"WARC-Payload-Digest\":\"sha1:WXJNFLU3JWYNKZJWGSDPWP36PAOLEON7\",\"WARC-Block-Digest\":\"sha1:7Y5NFOKKVBPRVKNG2U4I3SVJKJAXWWR6\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-14/CC-MAIN-2023-14_segments_1679296945292.83_warc_CC-MAIN-20230325002113-20230325032113-00068.warc.gz\"}"}
https://softmath.com/math-com-calculator/distance-of-points/poems-of-equation-that-are.html	[ "English \| Español\n\n# Try our Free Online Math Solver!", null, "Online Math Solver\n\n Depdendent Variable\n\n Number of equations to solve: 23456789\n Equ. #1:\n Equ. #2:\n\n Equ. #3:\n\n Equ. #4:\n\n Equ. #5:\n\n Equ. #6:\n\n Equ. #7:\n\n Equ. #8:\n\n Equ. #9:\n\n Solve for:\n\n Dependent Variable\n\n Number of inequalities to solve: 23456789\n Ineq. #1:\n Ineq. #2:\n\n Ineq. #3:\n\n Ineq. #4:\n\n Ineq. #5:\n\n Ineq. #6:\n\n Ineq. #7:\n\n Ineq. #8:\n\n Ineq. #9:\n\n Solve for:\n\n Please use this form if you would like to have this math solver on your website, free of charge. Name: Email: Your Website: Msg:\n\nYahoo users came to this page today by entering these algebra terms:\n\n• how to solve eigenvectors on ti 89\n• math word problem quizes\n• Math Trivias\n• online factor solver\n• trapezoidal rule program for TI-84\n• free 8th grade math worksheet on variables\n• Iowa Practice test 6th grade math\n• free worksheet area circle\n• nth term\n• easy to teach multiplying dividing integers\n• a online convert to mix number calculator\n• lesson plan cube and square roots in Maths\n• percent of savings math worksheet\n• enter a standard form and get a vertex form\n• multiples and factors worksheet grade 4\n• algebra 2 homework helo\n• Prentice Hall algebraic concepts answers cheat\n• square root conversion\n• rational expressions video\n• ti 83 modular\n• ti83 plus rom and emulator\n• show how to solve quadratic equations in a flow chart\n• multiple choice questions to do with simplify expressions using rational exponents\n• cube roots on TI-83 Plus\n• least common multiple technique\n• algebra help grouping\n• Free Algebra Lesson Worksheet\n• for kids how to add a fraction to a other fraction\n• free worksheets on multiplying integers\n• express decimals in fractions in matlab\n• sample of math investigatory problem\n• maths homework help-year 10 percentages\n• online factorization\n• when and why to use linear equations\n• gallian solutions chapter 10\n• free printable elementary algebra worksheets\n• graphing functions\n• powerpoint presentation on graphing linear equations and functions\n• GCF variables calculator\n• balancing chemical animations\n• vertical intercept for radical functions\n• math eqations\n• grade seven calculating percents practice\n• christmas worksheets for simplifying fractions\n• dividing decimals worksheet\n• algebra exercises pdf\n• online graphing calculator binomial\n• converting fractions to decimals worksheets\n• 5th grade iowa practice tests\n• prentice hall geometry practice workbook answer sheet\n• general aptitude questions\n• 8th grade math worksheets exponents\n• laws of logarithm expand calculator\n• free pre-algebra worksheet for beginners\n• free equation workshet\n• handheld calculator factor polynomial equations\n• solving a system of differential equations matlab\n• why do you have to divide by the coefficient of x^2\"\n• Online Algebra Solver\n• introductory algebra eight edition help\n• describe differences between multiplication and addition\n• life in the UK free sample test papers\n• simplifying complex rational expressions help\n• factoring worksheets\n• \"Fluid Mechanics\" \"Problems with solutions\" exams\n• algebra worksheets using a calculator\n• pre algebra fractions worksheets\n• \"math problems\" + \"ratio\" + \"scale\"\n• exponential fractions worksheets\n• \"cubed root\", geometric interpretation\n• excel math exercise\n• free prealgebra calculator\n• balancing chemical equations using algebra\n• integer exponents worksheet\n• Intermediate Algebra Exam Questions\n• multiplying and dividing intergers interactive activities\n• reduce fractions power unknown\n• simultaneous equation solver step by step\n• matlab+numerically solving non-linear equations\n• maths percentages assignment year 6-8\n• free online ti 83 calculator\n• step by step answers for Algebra and Trigonometry, Book 2\n• 1998 free ks3 exam papers\n• algebra worksheets distributive\n• how to solve algebra equations\n• Free Algebra Solver\n• list 4th root\n• converting whole numbers and fractions to decimals calculator\n• calculas\n• algebra practice beginners\n• aptitude solved questions in c\n• TEACH YOURSELF MATHEMATICS FREE\n• radical expressions program ti 84\n• multiplication with the addition or subtraction method\n• area polynomial worksheets\n• Quadratic equation solver graphic calculator\n• quadratic equation in 2 variables\n• complex numbers solver\n• what is the definition multiplying fraction property\n• multiplying rational expressions on a ti-89\n• mix number\n• TI 89 laplace transform\n• COMPOUND INTEREST LOG PROBLEMS WORKSHEET\n• vector exercises for GCSE\n• fraction subtractor\n• help me solve my rational number equations\n• online factorer\n• algebra problems yr 8\n• online TI-84\n• grammer chart\n• merrill math books\n• printable compound interest worksheets\n• how do you multiply the square root of 2 times the square roote of 2\n• FORMULA FOR A square of a trinomial\n• prentice hall conceptual physics workbook answers\n• Subtracting mixed Functions\n• ti 82 lesson sixth grade resources\n• math grade 7, fractions, practice questions\n• implicit differentiation solver\n• mixed number as a decimal\n• Modulo mit Texas TI-84\n• ti 84 calculator online\n• Worksheet on adding and subtracting signed numbers\n• math cheats\n• Physic at work 1A heat exercise answer\n• how to solve properties of exponents\n• free math worksheets using compound interest\n• factor third order\n• answers to Mcdougal littell 6th grade mathematical concepts and skills book\n• math help check the solution\n• Algebra Equation Calculator\n• roots and radicals calculator online\n• trigonomic\n• Simplify 3512\n• english aptitude\n• deviding polynomials\n• free online electrician practice tests\n• distributive property applet\n• how to do systems of linear equations with three varibles on a calculator\n• Free \"Algebra Worksheet\" college\n• how to store answers on a ti 89\n• linear equation graphical solutions problem solver\n• solving equations worksheets\n• MATHS SUMS FOR 8TH STANDARD\n• Arabic Exam grade 8 GCSE level\n• proportion worksheets for 5th grade\n• \"Logarithmic Equation Solver\"\n• binary notation on ti 83\n• evaluating expressions+worksheets\n• adding and subtracting integers free worksheets\n• FIRST SEMESTER REVIEW TEST FOR FOURTH GRADE MATH\n• how to solve pamela equation\n• 6th grade algebra lesson plans\n• practice solving logarithms\n• trig for dummies online\n• grade nine math programs online\n• math equation poem\n• lowest common denominator chart\n• double summation notation + matlab\n• Dividing polynomials on TI 84\n• Pre-Algebra Tutors, San Francisco\n• \"equation of a cube\"\n• how to do simplified roots\n• how to teach geometry nets\n• worksheets adding positive and negative integers\n• solve slope y-intercept\n• Maths Online Tests for 6th Grade-How Our Numbers Work\n• nonlinear difference equation with matlab\n• glencoe mcgraw hill worksheets algebra 2\n• find the lcd of each fraction calculator\n• simplifying combinations math\n• solutions Rudin \"chapter 8\"\n• exponents on calculator\n• factoring calculator\n• TI84 Study Notes\n• college algerbra\n• HCL Aptitude model question paper with reasonable answer\n• how to teach slope algebra\n• solving problem using lagrange multipliers in maple\n• decimal order of operations worksheet\n• algebra-diamond foil method\n• multiply and simplify square roots\n• how to solve through synthetic division using a scientific calculator\n• ti-89 laplace\n• mathematics+formulae and questions+trigonometry\n• kumon exercises online\n• how to program the quadratic in a ti 84\n• online synthetic division calculators\n• McDougal Littell Pre-Algebra mathbook\n• FREE ASSESSMENT TEST + 6TH GRADE\n• Free Math Trivia\n• worksheet exponets\n• pythagorean theory worksheet\n• two-step equations worksheet\n• simultaneous nonlinear equations\n• FRACTIONS FROM GREATEST TO LEAST\n• algebra 1 texas book\n• lesson 11 vocabulary answers McDougal littell\n• \"How to calculate lcm\"\n• percent worksheets\n• how to order fractions with a ti-83 plus\n• COMPLEX NUMBER EQUATION SOLVER\n• answers for chapter 11 algebra 2 test form A\n• intermediate algebra cheat sheet\n• demical into a mixed number\n• solving problems+ti89\n• free antiderivative programs\n• free printable 8th grade math problems\n• multiplying in standard form\n• graphing one step linear equalities worksheets\n• Hyperbola Grapher\n• foiling equation\n• formula to convert decimal to mixed number using a calculator\n• laplace from TI\n• online antiderivative calculator\n• are parentheses necessary in signed numbers\n• blitzer college algebra 4th edition practice exercises answers\n• nonlinear simultaneous\n• Sixth Grade Printable Fraction Tests\n• solve systems ti 83\n• printable free homework for 5th graders\n• laplace transform ti 89\n• calculating entropy from a chemical equation\n• online graphing calculator find range\n• polynomial.ppt\n• subsitition method on TI calculator\n• polynomials.ppt\n• intermediate algebra work sheets\n• polynomial equations \"graphing in matlab\"\n• find the lowest common denominator for (12,26,65)\n• saxon help with abstract fractional equations\n• multiplying fractions sixth grade worksheets\n• kumon algebra tutoring\n• least common multiple of 147\n• Gr8 Arabic exam paper\n• equation analysis test #2 answers\n• how to solve Quadratic Equations and FUnctions\n• worksheet order of operations\n• math test ks3 in arabic\n• saxon algebra 2 answer key\n• free algebra help\n• two step equation worksheets\n• GET THE ANSWERS TO QUOTIENT BINOMIALS AND POLYNOMIAL PROBLEMS\n• java emulator ti 84\n• +caculator linear equation\n• math statistics C# .NET freeware\n• standard form polynomial\n• algebraic problem solver\n• c programs to find roots of the quadratic equation\n• Algebra Equations Solver\n• trigonometric chart\n• learning concepts worksheet UOP\n• fun variables worksheet grade 6\n• year 11 chemistry online practise printouts\n• free algebra equation calculator\n• easy way to simplifying radicals\n• algebra 2 polynomial problems\n• two step algebra problems\n• calculator that rationalizes the denominator\n• least common multiple powerpoint\n• identifying decimal numbers worksheet\n• alegebra help\n• multiplying integers worksheet\n• square root algebra\n• ti 83 finding gcf\n• aptitude exam papers\n• ti 89 rom\n• free online calculator for integers and albebraic expressions\n• free algebra solver\n• boolean algebra calculator\n• least to greatest fractions\n• Algebra 2 Problems\n• when to use factoring to solve quadratic equations\n• hyperbola maker\n• free pdf chemistry notes for gcse\n• temperature conversion worksheets\n• free answer to algebra question\n• simplify cube root\n• write a system of equations +ordered pairs\n• GCF of negative numbers\n• ti-83 plus linear equations and slope\n• math b advanced homework help\n• Free circuit equation solver\n• mcdougal littell teacher access code algorithm\n• Abstract Algebra Problems\n• Permutations Combinations Problem Set Answers\n• vertex formula graph\n• solve multiply square roots\n• question and answer for c language objectives\n• how to solve standard form equations\n• Virginia Algebra 1 Prentice Hall Mathematics\n• rotations in maths worksheets\n• 9th grade algebra practice worksheets\n• nonlinear equations, ti-89\n• 7th grade impact mathmatics book pg 100\n• negative exponenets\n• PERMUTATIONS AND COMBINATIONS 3RD GRADE\n• jeeves vertex maths\n• Third order equations\n• factorization solver\n• get answers to algebra charts\n• how to solve for probability equation\n• cheating on algebra\n• ALGEBRA WITH PIZAZZ\n• solving nonlinear equations with maple\n• prentice hall algebra 1 answers\n• quardratic projects\n• Distributive property puzzlers for fifth grade\n• geometry pizzazz\n• multivariable word problem\n• TI-83 inverse log\n• free worksheets/probability\n• samples of quadratic equations with irrational numbers\n• Write Addition and subtraction expressions\n• graphing equation review\n• permutation explain combinations 5th grade\n• problem solving worksheets on slopes\n• Subtracting whole numbers and Functions\n• calculator solver order casio\n• how to teach cubic polynomial\n• logarithms test for dummies\n• subtract rational fractions calculator online\n• free online ti-83 plus calculator\n• finding standard form of circles worksheets\n• how to solve polynominals\n• Linear systems in TI-89\n• prentice hall mathematics course 2 answer book for teachers\n• grade 12 activites on random number generator using graphing calculator\n• LINEAR ALGEBRA WITH APPLICATION\n• Equivalent Decimals worksheet\n• Pre algerbra\n• factoring using quadratic formula calculator\n• free online math tutor\n• square roots, cube roots, n roots and radicals differences\n• integers/games\n• printable algebra equations\n• TI calculator worksheets for 6th grade\n• Difference of two square\n• year 10+australia+maths+syllabus\n• worksheets for working with psoitive and negative\n• square root of a difference\n• finding the inverse of radical functions\n• given points quadratic equation online\n• Algebra SOl review sheet\n• math exercises positive and negative numbers\n• algebra 2 McDougal Littell answers\n• FRACTIONS WITH A +COMMOM DENOMINATOR\n• algebraic formula for balancing equations\n• worksheets of adding and subtracting mixed fractions and fractions\n• FACTOR TREE WORKSHEETS\n• TI-83 square roots\n• free math practice sheets mixed adding subtracting\n• lesson plans percents to sixth grader\n• plotting pictures worksheets\n• math sheet i can print gcse free\n• TI-84 EMULATOR\n• how do u write a square equation in C#\n• year 9 printable free worksheets\n• multiply and divide fractions worksheets\n• solving inequalities equations worksheets\n• decimals lesson plan second grade\n• linear equations graphing programs\n• 11+ maths test free\n• basic percent proportion\n• alegbra rules\n• examples of fractions from least to greatest\n• Algebra 2 math book lessons, quizzes, games\n• matlab solve second order differential equation\n• Visual basic exponentiation calculator projects\n• program algebraic equations TI 83\n• math help with algebra 2 and trig\n• algerbra 1\n• ks2 maths worksheets printouts\n• ti-83 factor program\n• algebra problem solvers\n• symbolic summation in matlab\n• basic math skills fo gmat\n• cubed root calculator online\n• Maath Highschool trivia\n• algerbra equations\n• physics practice worksheets\n• division printouts\n• Completing the Square Activity\n• find factors on ti-83\n• adding, subtracting, dividing, multiplying integers worksheet\n• Calculator for Greatest Common Factor\n• how to solve cube roots on a ti 83\n• texas integrated physics and chemistry worksheet answers\n• ti 83 laplace transform program\n• getting cubed root of a number in excel\n• solveing second order differential equations\n• TI89 solver\n• introduction to algebra free worksheets negative numbers\n• How to find exponential relationships y-intercept made easy\n• trig Identities solver\n• free +matric calculator\n• solve the third degree\n• cheating for maths homework\n• ellipse, graphing\n• Linear Extrapolation Method 8th grade\n• rational expression calculator fractions\n• how to do tricks on a ti-84 calculator\n• elementary algebra substitutions\n• what is simplified radical form\n• negatives and positives calculator (adding,subtracting,multiplying,and dividing)\n• expressions worksheets\n• fluids vocabulary and lessons grade 8\n• math poems\n• ucsmp functions, statistics and trigonometry answers\n• graphing hyperbola worksheet\n• vertex form algerbra 2a\n• merrill Integrated Mathematics course 1 answer key\n• LCM problems for elementary school\n• Kumon printables\n• square root of negative fraction\n• calculate eigenvalues vb code\n• exponent as variable\n• algrabra software\n• free binomial worksheets\n• algebra software\n• positive & negative numbers + word problems\n• antiderivative solver\n• 5 math trivias\n• entering parabolas on algebrator\n• what is the greatest common factor of 63 and 81\n• graph an ellipse calculator\n• algebraic pyramid solver\n• Online Algebra Problem Solver\n• order fractions from least to greatest calculator\n• online maths exam revision sheets for grade 9\n• drawing pictures ti-89\n• converting among decimals\n• working with fractions and least common denominator\n• different math trivia\n• real life exponential expressions\n• solving second-order systems with matlab\n• finding absolute value on a TI-89 calculator\n• square root of (3x+y) squared\n• nonlinear equation solver\n• online calculator using \"square root\"\n• english aptitude?\n• polynomials test online\n• balancing equations problems\n• ti89 complex numbers\n• write the equation of the parabola in the vertex form\n• show me a math matics cheats\n• inverse logarithm ti-89\n• world history modern times glencoe worksheet answers\n• algebra 2 hot math for workbooks test problems for pretice hall mathematics\n• permutations and combinations free practice questions for gmat\n• merrill math worksheets algebra 1b\n• numbers equations trivia\n• algebra homework cheats\n• video on pre-calculus family of graphs- quadratic, cubic, logarithm\n• Application problems Differential Equations using MATLAB\n• binary fractions calculator\n• how to solve polynomials\n• algebra for beginners\n• matlab newton raphson with interpolation\n• java aptitude questions\n• KS3 cHRISTMAS PRINTABLES\n• partial fraction solver\n• home work with graphing linear equations using my own problems\n• rudin chapter 7 solutions\n• how to do log4 on TI-83 PLUS\n• calculate square maximum and minimum\n• tutorial of modern algebra\n• answer to lesson 7-5 glencoe/McGraw-Hill\n• how to write a standard form function in vertex form\n• free best maths word problems for 3 rd grade online\n• equations with variables on each side printable worksheets\n• pie values\n• hard grade seven math problems\n• balancing math equations\n• yr 6 mental sats quizzes\n• Greatest Common Factor List\n• Simplifying Square Root Calculator\n• pre algebra lessons polynomials worksheet answer sheet\n• fluid mechanics in pdf\n• rectangle ratio of 5:7\n• christmas maths\n• cubic factor calculator\n• Free Worksheets Algebraic Expressions\n• math kids least common denominator\n• ti trig solver\n• java count number of times a loop is executed\n• using algebra to balance chemical equations\n• simplifying calculator\n• math scale factor\n• simplifying exponent equations with variables worksheets\n• algebra solver division\n• answers for practice workbook McDougal littell Algebra 1 pg. 74\n• +sample printout of lcm\n• help slove elimination\n• simplifying square root variable\n• merrill algebra 2\n• PPT on solved example of mathematics\n• easy way to calculate greatest common factor\n• tic tac toe method for factoring quadratics\n• prentice hall algebra 2 book online\n• square root for math cad\n• Matlab solving equations\n• hyperbola algebra\n• SOLVE equations by completing the square\n• PRINTABLE FRACTIONS TEST\n• adding and subtracting integer equations\n• Algebra 2/Trigonometry Worksheets\n• exponential+probabilities+ti83\n• expressions of square roots problem solver\n• TI86 log base 2\n• mathlab solve equation\n• power point on writing linear equations in standard form\n• addition and subtraction problems up to 18 worksheets\n• greatest common factor table\n• answers for algerbra 2 homework\n• usable Online TI 83 Calculator\n• square root exponent\n• combination worksheets\n• List of Real Numbers Square Roots\n• log + Texas TI 82 + bas\n• q & a homework solver\n• elimination method powerpoint algebra I\n• daily uses of linear equation\n• factoring using graphs to find roots\n• divide and conquer lectures.ppt\n• second order differential equation nonlinear missing y\n• mcgraw hill, investments, 7th eduition, teacher edition, quiz\n• Samples of Math Trivia\n• linear standard form calculator\n• third order elipse\n• lineal metre\n• solve basic algebra on TI-83\n• free video lessons on 2-step equations\n• design\n• Least common denominator calculator\n• calculators math free\n• answers for algebra 1 homework\n• formulae for converting decimals back to fractions\n• Free year 9 SAT Papers online for maths and science\n• dividing decimals-gcse\n• 1st grade line plot worksheet\n• simplify square root\n• least common multiples tutorial for fifth graders\n• Glencoe Algebra 1 / Workbook/ Answers\n• glencoe enrichment algebra 2\n• cube root of variables with exponents\n• aptitude questions + solutions\n• Free Online TI 83 calculator\n• multivariable algebra\n• whats the hardest equation of algebRA?\n• who invented fraction equations\n• addition and subtraction of fractions with like denominators worksheets\n• dividing decimals by decimals worksheets\n• 7th grade math combining like terms\n• multivariable linear word problem\n• solve nonlinear polynomial inequalities\n• a calculator for fractions\n• Three Value Least Common Multiple Calculator\n• area of circle equation+ practical maths examples\n• mathsimplification\n• 72375901685815\n• solve by substitution method online calculator\n• tenth grade free math work\n• algebra tile worksheets\n• trigonometry games ks3\n• nonlinear inequality solve matlab\n• FREE cost accounting reviewer questions\n• computers and the approximation of the roots of algebraic equations\n• reverse foiling solver\n• \"Grade 9 Practice Math Exam\"\n• how do i convert a mixed number to a decimal\n• basic algebra fraction equations\n• algebraic equations with square roots\n• combine like terms and solve worksheet\n• mcdougal littell algebra 1 online answer key\n• TI 83 plus log calculations\n• maple numerical extrema surface\n• KS3 SATs questions science\n• simultaneous equations in excel\n• least common multiple solver\n• free beginning algebra worksheets\n• cheats for maths exams\n• florida prentice hall mathematics algebra 1 pearson\n• free singapore math test paper\n• mathmatical tenths thousands\n• converting mixed numbers to decimals\n• logarithms for dummy\n• integrated 3 mcdougal littell high school mathematics answers\n• Converting a Mixed Number to a Decimal\n• graphing ellipses\n• math-dilation\n• balancing chemical equation calculator\n• graphing polynomial equations in matlab\n• holt algebra Lesson 2-4 Practice A print out\n• math trivia division\n• fraction caculator\n• elementary algebra help\n• mcdougal littell algebra\n• Simple Algebra Problems\n• solution of diffrential equation of second order when the function is sin\n• kumon mathematics work sheets\n• pre-algebra scale drawing length how\n• seventh grade algebra angle printables\n• square root of 120 simplified\n• how to convert fractions with odd denominators to percent\n• math equation for burning calories\n• system of linear equations pdf files grade 10\n• approximate the radical expression using calculator\n• logarithmic equation calculator\n• free rhombus theorem worksheets\n• fun math activities gr10\n• evaluate indefinite integral calculator\n• how to do standard deviation with 2 variables on ti 83 plus\n• common multiples calculator\n• 2-step math equations worksheets\n• free online guide for ti-82\n• Free Online Math Tutor\n• online aptitude test for middle school kids\n• symbolic TI-84 tutorial\n• investigatory project cosine problems with explanation and abstraction\n• maths work sheets on factors and products\n• prime number is difference of two square\n• rectangular to polar ti-84 plus guide steps help\n• grade 9 tutorial algebra rules and applications\n• learning how to use permutations and combinations\n• algebra magic X factoring\n• simplifying square roots fractions\n• algebra 2 connections answers to book\n• simultaneous equations maths free worksheets\n• printable worksheets finding fractions with least common denominator\n• HOW TO SOLVE ALGABRA\n• sum or difference of logarithms calculator\n• complex trinomials\n• relations and functions worksheets\n• Prentice Hall Pre-algebra Math Book\n• print out 3rd grade saxon problem set paper\n• free worksheet for solving two step equations\n• teaching yourself how to do a hyperbola\n• how to solve graphing square roots and cube root functions\n• maths test expanding brackets\n• free online quiz test maths coordinate geometry formula slope\n• matlab algebraic solver\n• free homeschool printouts for 7th graders\n• simplifying algebraic equations\n• Simplification Algebra Calculator\n• calculating slope on a ti-83 plus\n• sample motion problems with solution(algebra)\n• how to factorize complexe equation\n• invented the algebra\n• math problems using divide & multiply\n• solving inequalities using addition and subtraction - activities\n• algebra worksheet domain range free\n• how to solve Quadratic Equations finding Square roots\n• ordered triple solver\n• square root property solver\n• SQUARE ROOTS 1-60\n• ti 84 logarithm base\n• glencoe algebra books\n• linear equation worksheets\n• cool math phrases\n• turn decimals into fractions\n• Order of Operations Homework Sheets\n• solving fractions equations for multiplications and division\n• square route finder math\n• finding the slope of a table\n• free maths trivia + answers\n• money math assessment worksheet test free\n• wibro aptitude question paper\n• TI-85 Calculator ROM\n• TI-89 ROM image\n• algegra factoring reducing\n• plato math cheat\n• boolean algebra software\n• \"ratio formula\"\n• simultaneous equations solver\n• free printable maths works sheets .com\n• problems for substitution method in algebra\n• ks3 maths calculator help\n• sample problem equation of a square\n• glencoe advanced mathematical concepts lecture notes\n• order of operations cheat sheet\n• cool math for kids.com\n• Sample Pre- Algebra Tests\n• Liner Equations\n• inputting logarithmic into ti-83\n• Answer to Prentice Hall Algebra Book\n• How to rewrite square roots into exponent form\n• quadratic formula in everyday life\n• ks2 how to work fraction sums\n• complex numbers in matrix + TI-83\n• adding fractions with square roots\n• how to subtact thousanths\n• free printable year 6 sats math questions\n• PLUS FACTORING\n• factors+greatest common factor activities\n• geometric sequence online calculator\n• \"completing the square with two variables\"\n• using log button ti-83\n• example clep question from physical science\n• maths hardest question\n• how to solve nonlinear differential equations\n• college algebra tutoring program\n• generator for solving quadratic equations\n• finding common denominator\n• free printable coordinated pairs math\n• Factoring Polynomials Tests\n• Free Online Algebra Solver\n• simplifying rational expression mixed types\n• solving set theory problems\n• square root of an exponent\n• how to balance chemical equations using diatomic elements\n• resolving math\n• fractional exponent calculator\n• free printable sat test\n• calculate lineal metre\n• usable free calculators\n• find least common monomials answers\n• teachers guide of algebra 1 glencoe mathematics florida edition\n• graphing linear equations worksheets\n• problems and questions on solving two linear equations in two variables\n• convert linear Riemann sum\n• Long Division of Polynomials Solver\n• problems with circles in mathcad\n• pre-algebra lesson distributive property worksheet\n• tutoring in conceptual physics\n• ti89 howto\n• homeschool 8th grade algebra lesson plan\n• simplify fraction solver\n• simplifying expression worksheets\n• math caculator finding GCF of big numbers\n• divisibility practice worksheets\n• algebra percent\n• mixed number calculator\n• arithematic\n• number factor calculator\n• ti 86 online calculator\n• world history fifth addition Mcdougal littell\n• differential equations ti89 integral\n• balancing chemical equations calculator\n• what is the name of the remainder in dividing\n• how to solve multiplying and dividing radical expressions\n• free Study Guide for GCE O level Accounting\n• Permutations +activities\n\nBing visitors found our website yesterday by entering these math terms :\n\nOrder fractions from least to greatest, math lesson on scale factor, online algebra 2 solver, An Investegatory Project about Math, algebra lessons 6th gra, fraction, least common denominator.\n\nAlgebra homework, first grade math sheets, factoring polynomials online.\n\nSymbolic math solver online, Free Maths Worksheets For Grade 9, algebra help scales, how to decimal to fraction, worksheets on factoring trinomials, maths problem solver, free probability worksheets KS3.\n\nFree prealgebra worksheets, Dynamics Lesson Plan, input/output table activities - 6th grade math, factorise by grouping logarythms explained.\n\nSection Tests Answer Sheet World History Copyright© by the McGraw-Hill Companies, Inc., subtract rational fractions calculator, solved papers of Aptitude, mathematical translation worksheets, free online Ti84 calculator, variable in the exponent.\n\n\"permutations worksheet\", cube root worksheets, solving nonlinear equations algebraically, charts to solve algebra problems, homework cheat and answers, convert fraction to mix number to a percent, free math answers to the algebra 2 prentice hall.\n\nHelp with the sums of radicals, free basic algebra solvers, extra numeracy classes for 9yr olds.\n\nGeometry online answers from Glencoe, sums of permutations and combinations, ratio worksheets, nonhomogeneous differential equation sin.\n\nSimplifying radicals on a graphing calculator, answers for prentice hall engineering graphics, how to write a fraction or mixed number as a decimal, simplify radicals TI-89.\n\nHow to calculate standard grade physics equations, hp 48g+ \"complex numbers\" \"solve equation\", College Algebra Calculators, calculator / find the least common multiple.\n\nAsymptotes quiz for high school, free past SATs papers, McDougal Littell Ebook, MATH TRIVIAS.\n\nSquare root property formula, quadratic equation word problems, gmat ppt free, prentice hall alegebra 1 unit 8, extrapolation calculator online, simplifying radicals calculator, gallian algebra solution.\n\nCalculator cu radical, LCM calculation, worksheet on decomposition for partial fractions.\n\n\"greatest common factor\"+\"factoring polynomials\"+\"activities\", math + Christmas + graphs, fraction formula.\n\nGlencoe science 8 answers to Chapter 3 test, factorer online, GED algebra II sample tests.\n\nMultiplacation sites, math exercise for 7 years old, objective Questions Fluid Mechanics.\n\nHow to factor trinomials with a table, algebra help and answers, \"linear equations\" + \"used for\", adding/subtracting unlike fraction worksheet, balancing equations online, aptitude test paper format.\n\n\"Algebra 2\" +Quadratic Equations and Functions, calculator to find the focus on a hyperbola, solving \"mathematical series\" answers, sats papers for year 6 for free on line, physics formula and test questions answers, Roots of Real Numbers Simplify Radicals, procedure to find square root of a algebraic expression.\n\nComplete the square and write in vertex form, solving cubed equations, divide square root calculator, factoring cubed power.\n\nFinding range of an equation, the worlds hardest easy geometry problem, intermediate algebra + Games.\n\nStandard form linear equations and worksheet, quadratic equations solving by square root, math \"grade ten\" ratio of triangles.\n\nSolve third order quadratic equation using determinants, Probability exercises for 11th graders, free CHALLENGING QUADRATIC EQUATION QUESTIONS, minute degree on ti83plus, holt algebra 1 texas answer key, samples of comparison problem in elementary algebra, how to multiply a linear expression by an integer.\n\n\"boolean reduction\" in excel, generating subtraction worksheets for kids, canadian math questions and quizes for grade 5, printable word problems worksheets for adults.\n\nFinding eigenvalue using ti-83, bar graph worksheets, comprehention worksheets with answersfor 6th grade, holt introductory algebra 1/homework, College Algebra Tutor, powerpoint pre algebra lessons.\n\nBooks on cost accounting, factoring complex numbers, math probloms, finding x and y intercepts on a TI84 Plus, maths online for class 5, solving second order differential equations in matlab.\n\nPrintable science tests with answers (, mathmatical conversion quizz, free mixed numbers and decimals.\n\nAddition subtraction equations worksheets, ti-83 square root GCF, Algebra Quadrant Solver, show me how to find the lcd for each pair of fractions, java test Question & Answer aptitued, vocabulary answers by McDougal littell, log math problems formulas.\n\nFinding the vertex on a graphing calculator, free math worksheets on dividing equations, formula rational expression, do my math factoring online, free Algebra practices, free online algebra 2 games, EXAMPLES OF MATH TRIVIA.\n\nCognitive tutor cheats, fft square java code, large roots on calculator, glencoe mathematics answers, banking and finance problems.ppt.\n\nMaths past sat paper, permutations and combinations on the GRE, how to calculate the square root of an equation, Descartes Rule of Signs chart, printable pythagorean theorem word problems, solving radical expressions solver.\n\nAlgebra and Trigonometry Structure and Method Book 2, Solution Manual method book2, yr 8 maths, TI-84 plus silver edition game cheats.\n\n\"numerical methods\" temperature matlab, pocket pc algebra solver, simpliying boolean polynomials, absolute value equations for 9th grade lesson plan, online slope calculators.\n\nTI-83 Plus \"cube root\", fraction worksheet adding with like denominators, prentice hall answers, Equation Checker.\n\nDistributive property algebra variables, write quadratic calculator program test it, holt pre algebra answers, pdf Algebraic Expressions, Equations, and Inequalities problems and answers, fraction worksheets for kids.\n\nAlgebra sums, least common multiples worksheets, cost accounting books, explanation of simple interest for grade 11, Glenco algebra word search puzzle, quadratic equations for dummies.\n\nSaxon Math Algebra 1/2 Textbook, problem set 24,23, pre-algebra least common multiple sample test, 9th grade algebraic symbols online, glencoe algebra 1 answers, college algebra calculator.\n\nMerrill Algebra One homework help, cheats for the ged in texas, 2 grade math free worksheets greater and least than, ti84 simplify square root, Ineed a study guide for Arithmetic Reasoning.\n\nConverting decimals to mixed numbers, \"online math answers\", diamond foil-math, Scale Factor in Algebra, tests on maths free online to do at home ks3 year 8, graphing calculator ti-89 online, convert standard form to vertex form.\n\nFactoring Quadratic Expressions, Simplifying Radical Expressions Solver, order of operations with square roots worksheet, finding slope using a calculator, algebra 1 basic concepts.\n\nMcdougal Littell Inc./chapter 6, online equation solvers, simplifying fractions online calculator, border word problems algebra, Evaluating square roots, +\"FUN MATHS\" +printables+grade 6, scientific notation worksheets and adding and subtracting.\n\nWorksheets on Multiplying and dividing Decimal, grade 8 patterning and algebra worksheets, equation for the product of the roots of a quadratic equation.\n\nReal roots of multiple variables maple, solving algebraic square root problems, algerbra problem solver, polynomial solver c++.\n\nEquations with variables worksheets, algebra I two step equations free printable handouts, high school worksheet printouts, solution to the world's hardest geometry problem, finding coordinates on a graph worksheets for kids.\n\nFreeware Algebra Calculator, Math Permutation and Combination word problems, square roots and multiple variables, math worksheet on permutation and combination.\n\nFree sample GMAT questions & answers in pdf file, formula to find square root of polynomial, boolean algebra practice sheets, hard GMAT math tutorial free.\n\nCollege algebra-matrices examples, how to teach square roots, solving radical expressions equation solver, PreAlgebra Math Software, online calculator that will do negative numbers.\n\nGraphic equation +ppt, answers to prentice hall, free worksheets of graphing pictures on a coordinate plane, intermidiate algebra solutions, ninth grade algebra.\n\nAmple algebra word problems about football, solving nonlinear equations algebraically exponential, solved problems by Hungerford, algebra calculator Radicals.\n\nHow to do log base on a TI-83, factoring expressions on a ti-83 calculator, fraction simplifier machine, absolute value equation minimum point, java calculator square root.\n\nIntegers worksheets, square root of 17 fraction, mcdougal littell inc worksheet answers, finding the fourth root of a number, partial sum algorithm worksheet, solve algebra problems.\n\nRadical form calculator, decimal as a fraction or mixed number in simplest form., simplifying angles.\n\nTrig cheats, permutations for 5th graders, algebra practice quiz for clep exams.\n\nSample dividing problems 3rd graders, simplify cubed root, THIRD ROOT, factor tree - printable, free algebra online calculator, printable gr.4 area & perimeter worksheets.\n\nAlgebric tiles worksheet, 8th grade worksheet pdf, solving nonlinear equations apps ti 89, faction trees-math, simple maths grade 5 printerble sheets, equation analysis test answers.\n\nGcse algebra expansion and factorization, Algebra 2 calculators, Algebra logarithm teacher worksheets.\n\nConverting decimal to fraction lessons, Algabraic symbols, abstract algebra sheets, addition and subtraction equations cheat sheet, problem solver to mixed numbers into simplest form, world history prentice hall quiz answers.\n\nTrigonometry calculation of unknowns, equation calculator with fractions and square roots, Chemistry year 10 revision games, NINETH GRADE ALGEBRA DOWNLOADABLE STUDY AND WORK SHEETS, adding subtracting multiplying and dividing integers questions really Hard, ks2 sats questions online.\n\nGive the value of the least common multiple of 9 and 12., Free evaluating expressions worksheets, math fifth grade word problems videos, fraction simplifier solver, using log base ti-89, Glencoe/McGraw-Hill practice workbook for algebra 1 answers.\n\nMath poems on graphing, free step-by-step fractions instructions, ti-84 plus distance formula free download, McDougal Littell teacher keys book, fundamentals of college algebra CLEP, synthetic division on ti-89 titanium, common factors math quadratic equations multiple variables.\n\nFree worksheets Greatest common factor, two variable equation solver, algebra 2 worksheet, matlab differential equation \"second order\", dividing radicals calculator.\n\nConceptual physics third edition, free printable worksheets for multiplication and division of integers, algebra workbook answer, input/output worksheets for fourth grade.\n\nOnline math tests on interest, differential equation programs for ti-89, mcdougal littell answers, factoring polinomial worksheet, integrated 3 mcdougal littell answers.\n\nFree online algeber for dummies, Differential Equations Edwards Penny Solution download, reduce factor math algebra, algebra 1 cheats.\n\nApplications of systems of equations tests, intermediate algebra project, solving radical fractions, Second order Polynomial in Excel.\n\nPolynomials solving rational exponents, exponents + hands-on, beginning algebra problem solvers, completing the sq.\n\nHow to subtract negative decimals, ti 83 accounting programs, solving cubed functions.\n\nConvolution laplace ti-89, how to solve four nonlinear algebric equations, fractions poetry activity a fraction of me.\n\n6th Grade Logic Matrices, \"The practice of statistics\" \"chapter 2 test\", Vocabulary from Classical Roots A–Teacher`s Guide/Answer Key, pictograph samples printable, sum of numbers in java, square root of 24 solve by factoring, excel simultaneous quadratic.\n\n+java +convert +time +to +int, dividing and rationalizing radicals worksheet, ks3 math resources ppt, complex equation solver.\n\nOrdering fractions from least to greatest worksheet, Prentice hall algebra 1 test, Algebrater, convert pointslope form into slope intercept form, ks2 long division worksheet.\n\nGrade level test/quiz, mcdougal biology study guide answer, solve multiple variable equation, ellipses in real life, how to find the lcm of a monomial, free prealgebra worksheets to print.\n\nChristmas pictographs worksheets, convert pH values to hydronium ion concentration with a TI-83 plus, \"math pattern worksheets\", greatest common factor tables.\n\nOnline cost accounting book, 6th grade decimal equations, \"crc32 calculator\", Algebrator, quadratric equation, section a: adding and subtracting fractions prentice-hall cheats, algebra two factor.\n\nWorksheets Describe and represent relations and functions, using tables, graphs, and rules., Free Online Algebra for Dummies, Functions, Statistics, and Trigonometry answers, challenging algebra problems, GED work sheet math.\n\nFree slope equation solvers, Prentice Hall mathematics Algebra 1 help, algebra 1 cpm chapter ca, adding positive and negative numbers calculator, chemical equation dissolution of NaCl, on line calculator for algebra, binomial factorial simplification.\n\nAlgebra help, Glencoe Algebra 2 McGraw-Hill answers, word problem for least common multiples, free algebra problems, free beginning algebra printable worksheets, writing linear inequalities worksheets.\n\nSolving systems of quadratic equations using matrices, 8 th and 9th grade math work sheets, ti 83 FACTOR DIFFERENCE OF SQUARES.\n\nHow to do ti 89 log, free printable absolute value worksheets, quadratic calculator intercepts fraction form.\n\nMath problems for a sixth grader worksheets, merrill mathematics books, applications of quadratic equations, Glencoe Math Even Answers, Algebrator version.\n\nHow to solve radicals, Algebra 2 how to convert standard form into vertex form, \"learning radicals\", quadratic formula application for TI-84, least common denominator+calculator, algebrea 1.\n\nSaxon math 76 help pg 245 answer key, ellipse equation solver, \"scientific notation powerpoints\", third grade worksheets on expressions, equations, inequalities, brain teaser equation analysis test #2 answers, slope intercept formula picture, multiplying and dividing rational expressions math solver.\n\nAlgebra expression calculator online, \"downloadable algebraic calculator\", substitution method with multiple variables.\n\nEquation with a fraction as an exponent, sample 1st grade aptitude test, solve simultaneous equations on excell cramer's method, square root solver.\n\nPre algebra homework, simplifing expressions several variables worksheets, free 4th grade probability worksheets, online grade 8 algebra quiz, Where is the log base button on TI-83 plus, simplify radicals by factoring.\n\nOnline graph differentiation, Convert Whole Fraction, general aptitude question, solve equation on texas Ti-83, numeracy christmas worksheets printable.\n\nMath transformations worksheets, Free Cost Accounting Presentations, fun square root activity 7th grade, algebra 2 chapter 6 tutor, solver integers for kids, symbols algebraic for fractions, answers for math books.\n\nOnline algebra solver, how to check ordered pairs in linear and quadratic equations, dictionary for kids plus calculator, problem solving algebra.\n\nContinuous compound help, writing balanced equations calculator, GCF CALCULATOR literal symbols, graphic exponential equations, TI-83 roots, calculating absolute value inequalities.\n\nPre-algebra calculator, masteringphysics answers, ti-89 trigonometry programs, binomial coefficients dice, Free GMAT question and answer downloads, 3rd order equation solver, practiccal use of differential equations.\n\nPrintable worksheet balancing simple algebra, negative and positive integers chart, Factoring Difference of Squares program TI 83 plus, Factoring Cubed Equations, nc test line math test, TAKING DECIMALS FROM MIXED NUMBERS.\n\nAlgebrator & precalculus, alegbra practice worksheets inequalities, fractions into decimal calculator, learn algebra free online, decimal as a fraction in simplest form, solving equations word problems worksheet, 72363292565055.\n\nConvert \\$12.50 into a fraction, factoring third order polynomials, ks3 christmas maths, adding & subtracting integars, gcse maths basics og graph x y, explain the math expressions f(x-5) and f(x)-f(5), chicago advanced algebra lesson answers.\n\nEquations ks2, printable math factor trees, glencoe algebra 1 teachers edition, online algebra 2 tutoring, advanced algebra answers, permutations and combinations concepts for gmat.\n\nConverting fractions into decimal the definition, ti89+roots, tasks dividing integers, algebra 2 chapter 6 resource book answers.\n\nHow to do permutations basics 5th grade free math help, algebra print-out graphs, rules for college algerba, holt middle school math course 3 workbook help, maths exercise for 11 plus for free, Ti-84 PLUS ROM.\n\nAlgebra 2 math book answers McDougal Littell, math practice percentage fractions, surds worksheet, practice sheet for algebra 2.\n\nAlgebra rules, mixed number written as a decimal, dividing and subtracting fractions, adding variables, ti83 lcm.\n\nHow to solve hyperbolas, Least common multiple of two monomials, how to do permutations basics 5th grade, how to calculate divisions with remainder in excel, +cube root of the square root of 8 times the fourth root of 4.\n\nOperating with radical expressions, how do I solve radicals, biology question paper 8th std.\n\nAplication linear algebra, examples of standard form to vertex form, converting functions standard to vertex, Multiply Divide Rational Expressions, rudin chapter 7 problem 12, solutions, used glencoe algebra textbooks.\n\nEquations ks3 examples, simplifying radical expressions practice problems, ti83 emulator online.\n\n(weekly Math challenge) sixth grade, boolean algebra calculator online, properties of exponents for idiots, converting functions to vertex form, free high school AIMS prep worksheets, solve equations multiple variables, equation by using the square root property.\n\nCalculating sum of difference ti89, factor third polynomials, trinomial root calculator, write percent as fraction.\n\nAlgebra 1 texas book answers, quatric equation calculator, equations with variables third grade, How to scale in math.\n\nAnswers to math problems in the algebra 2 book, exercices des matric, problem solving in hyperbola with equation, fraction lcd worksheet, solving equations with complex numbers using ti 89, adding and subtracting multiple negative numbers, circle math trivia questions.\n\nDistributive property arithmetic, solving problem with square roots and radicals, holt mathematics answers, nys seventh grade math topics, simplifying exponents from square roots, slopes in Algebra, easy way to learn standard form equations.\n\nMath stat trivia, multiply and divide decimals by 10 worksheets, factorization by completing square.\n\nAlgebra caculator java, clep \"college algebra\" \"practice\" free, cost Accounting book, Matlab+second order ODE, k12 answers cheats, mcdougal littell geometry answers.\n\nPdf algebra, ti calculator emulator, simplify fractions online calculator.\n\nHow do i turn a decimal into a fraction on my T I 86, free worksheets on systems of inequalities by graphing, examples of scale factor for 7th grade.\n\nWebsites where i can do a easy multistep equations, cube roots+ti89, glencoe accounting worksheets, grade 6 maths worksheets on fractions, quiz on combining like terms, live maths video simultaneous equations substitution.\n\nAlgebra expression calculator, implicit differentiation calculator, GCF find calculate, simplifying radicals tutorial, maple worksheet electrical downloads, \"real life algebra\", convert decimal mixed.\n\nRationalizing denominator of radical expressions using TI 83 calculator, solve my algebra problem, writing quadratic functions in vertex form help, Fourth Root Calculator, least common multiple football games, Solving Algebraic Equations.\n\nPre algebra combine like terms, Applications of Parabolas, algebrator software, ti89+algebra operations, online partial fractions calculator.\n\nFree simplifying algebraic fractions worksheets, answers to math problems in algebra 1 book colorado, factorise calculator, simple graph worksheets, 9th grade world history section 15 test answer sheet, simplifying fractions + lesson plans + 4th grade, partial fraction solving software.\n\nFraction converter and ti84, solving quadratic roots with complex coefficients, factoring square roots radicals, ti 86 help cramer, fit data to quadratic equation calculator.\n\nPizzazz math answers, softmath, i need maths/algebra websites, grade 10 math algebra tutorial variables, quadratic equation in football, integers multiplying dividing word problems, math book test, algebra 1 structure and method,.\n\nJava Mixed Number examples, simultaneous equation solving program, free accounting lesson plans, middle school math with pizzazz book D ANSWERS, solving rational expressions online free, mcdougal littell inc answer sheet.\n\nEXCEL FORMULA DIVIDE, add rational expressions, exponents square roots free worksheets, im a 7th grader and i have a prentice hall math book is there an answer book, \"dependant system\" definition.\n\nBalancing equations using algebra practice problems, ti89+radicals, Intermediate Algebra/Factoring Polynomials, Solving Fraction Equations :Multiplication and Division, solve third order equations.\n\nHolt, Rinehart, and Winston Algebra 1 solutions manual, ti 83 emulator rom, how do you multiply roots and radicals, polynomial factoring calculator, worksheets on converting units for grade 4, logarithmic using ti-83 calculator.\n\nMath trivias, multiplying variables with fractional exponents, chemistry mcdougal littell answers.\n\nSubtract and divide questions, Change Mixed Number into a Decimal, algebra tile project, free 6th grade math online tutoring, Rudin Chapter 7 solution, yr 4 english homework sheet, Radical Calculator, multiply square roots.\n\n\"simultaneous equations\" in maple, how to teach synthetic division, Pearson College Algebra Blitzer Chapter Test Prep Video free download, algebra help programs, writing solving equations free printable, Step-by-step answers for math textbook problems, online.\n\nHandheld calculator quadratic equation solver, saxon math venn diagram 4th grade, Pre-algebra for beginners, implicit differentiation calculator online, prealgebra software for beginners.\n\nAlgebra 2 an integrated approach 1998, quadratic equations by graphing worksheet, equation solver for finding mixing equations, Roots and Rational Exponents, California Algebra 1 Concepts And Skills Answers, set theory worksheet 8th grade, find perimeter with missing width.\n\nTrig equtions, The use of the prime no. 1 in math, adding subtracting decimals worksheet, algebra tiles solving equations, How do I calculate money word problems in a quadratic equation?.\n\nThree given points quadratic equation online, how to find the LCM of Monomials, free online help precalculus GCF 7th grade homework, algebra 1 guides, find the small of 10 number using loops.\n\nPh pre algebra ca edition/ answers, quadradic equations made easy, algebraic expressions/algebra 1, online unlike fraction calculator type in, Pizzazz! Book D online book.\n\nSum cube of digits java, mathCAD simultaneous differential equations, Rudin chapter 7 solutions.\n\nAlgebraic lessons level e, free ti-82 guide online, divide polynomials in life, algebra with pizzazz.\n\n5th grade algebraic equation worksheets, rational number online calculator, what does int stand for on a graphing calulator?, polynomial functions solver online, answers to pre-alg textbooks, negative fraction calculator simplify, eigenvalues in ti 83.\n\nLetter grade on line calculator, online factoring, compound inequalities solver, \"programming tutorial\" for ti 84 silver edition, gcf exponents calculator.\n\nGraph system of equation word problems, calculator for dividing polynomials, convert decimal to fraction formula, how to put second in a graphing calculator, monomial calculator online, Instant Algebra Answers, ti-83 rom download.\n\nCompleting the square worksheet, free algebra tutor, General Aptitude Solved Questions, Glencoe Mathematics Answers.\n\nTi-89 solve polynomials, algebraic expressio, factors multiples exercises.\n\nMultiplying integers calculator, boolean algebra samples, free integer worksheet for 6th graders, HCF of 3 number java, Resolving factoring equations online.\n\nThe worlds hardest easy geometry problem answer, grade 7 exam online free, slope math games, rules for adding factions with variables.\n\nFactoring equation calculator, glencoe mathematics algebra 1 9th grade, online graphing calculator TI-89, solving multiple equations on matlab, how to cube root on ti 83, Worksheets for equations with one variable for 8th grade math.\n\nFree trial on the eleven plus exam, permutations and combinations worksheets, online algebraic expression calculator..\n\nFactor radical square roots, system of equations, explain how to do inequalities, algebraic graphing, what are compound inequalities, quadratic calculator, solving rational equations why is it necessary to check them.\n\nCompare Algebra printable worksheet softwares, what is a polynomial, Graphing Linear Equations, simplify the expression a3b2 over a2b.\n\nPolynomial Calculator, ti-84 free online calculator, algebra grade 5, clearing fracions worksheets, simplifying radicals.\n\nWww.algebra-help.com, Simple Order of Operations Worksheet, simultaneous equations solver, How two solve two step equation's?, math poetry high school., algebra problem solving help, Linear Equations in two variables.\n\nSimplify variables with exponents, variables in algebra, Free 9th Grade Geometry Worksheets, solve algebra problems online, online prealgbra help, solving linear equations.\n\nMultiplying polynomials, free finite math for dummies, 9th grade math problems examples.\n\nPractice exams for algebra II trig, linear equations with a fraction, 9th grade worksheets, collecting like terms in algebra calculator.\n\nRationalizing numerator, Combinations Calculator, vertex form of a parabola, algebrator, 8-4 factor trinomials pg 28.\n\nAlgebra inequality, how to solve inequalities, \"Chapter 9 Test, Form 1B\", How do i simplify problem in chapter 9.2 of algebra 1, answers of problems of california pre algebra prentice hall math free.\n\nEvaluating Summation Notation with ti83 plus, free rationalizing denominators worksheet, How do u do one-step equations with fractions, get algebra 2 answers, GGmain.\n\nEquations with variables on both sides, solve the following equation-7/5m = 3/8, POLYNOMIAL DIVISION.\n\nMultipying and dividing radical expressions, answers to algebra problems, do my algebra for me, finite math for dummies, parabola equation.\n\nFactoring quadratic equations calculator, linear inequalities, The perfect square trinomial that results from completing the square for x2 - 16x is, Algebrator, examples on how to do linear equations, free maths Tiles Worksheet, prentice hall connections to today powerpoints.\n\nWriting Equations for Parabolas, algebra rule, rule method algebra, how to factor a greater than 1, how to factor a binomial, rationalizing denominators worksheet.\n\nStep by step algebra solver, introduction algrebra, Vocabulary Power Plus for the new SAT Book, factoring polynomials calculator online free, college algebra help, quadratic formula, what is the answer to radical 18 divided by radical 6.\n\nAlgebra Tiles Worksheet, algebra v - 4 (4 - v) = -2 (2v-1), 9th Grade Math Problems Worksheets.\n\nFormulas and literal equations, what are linear equations, directrix of a parabola is:D: y = 2 and the vertex is V(2, 4), what is the focus?.\n\nFree online prealgbra help, Examples of Linear Equations, search prentice hall mathematics.\n\nRational expressions, scholastic math algebra, algebra problem solver, solving compound inequality with a fraction, College Algebra for Dummies, a polynomial equation with rational coefficients has the roots 5+ sqrt 1 , 4 - sqrt 7. Find two additional roots.\n\nMath grade 10 system equations speed, solving compound inequalities, the square root of the difference of two numbers equals 5 the sum of the numbers is 31, chapter 9 page 442 radical expressions algebra 2 answers, hard math equations, adding and subtracting rational expressions.\n\nSamples on synthetic division with answers, simplify the expression, math calculations, videos on how to do equations with fractions using variables, solve one step equations worksheet, mathematics a level worksheets.\n\n9th grade math workbooks, what is rational numbers, simplify expressions in positive terms, solving college algebra math problems, 9th Grade Math Worksheets, free algebra step by step solver.\n\nAlgebra software, roots and radicals, my teacher rate .com, algebra, Solving Systems of Linear Equations by Graphing.\n\nWhat is a easy way to do equations with variables on both sides, given equation of a parabola y=-3/7(x-14/14)what is axis of symmetry?, quadratic formula, what is a linear equation, math question.\n\nSolve useing an equation, trigonometry, what is the algebraic expression for 2x 3, for x = -3, 0, 1, and 4, vocabulary book power plus, problems and answers for algebra 1.\n\nFree math help for mcdougal littel algebra 2, dividing by monomials, examples of math trivia with answers mathematics, rational equations, algebra 2 calculator, rational expressons calculatorhttp://www.1728.com/triang.htm.\n\nFun online 9th grade algebra activities, List of All Math Formulas LINEAR, rational expression with one solution,two solutions and no real solution, whats the answer to this algebra problem {1/3 6 - 4/5 -4}{-1/5 3}, solving algebra problems, how to solve rational equations.\n\nFree Algebra step by step, solvin compond inequality with a fraction, Free Algebrator, Simplifying Rational Expressions Step by Step, how do u put negative and positive numbers in algebraic expression?, algebra worksheets for 6th grade.\n\nWhat is an algebraic expression for \"three less than five times a number x\"?, Search Solving Linear Equations, solve system of equation matlab, nelson math grade 4 workbook answers.\n\nQuadratic formula, graph each pair of inequalities, math ratio problem solver, rational equations showing answer, operations of radicals .\n\nHow do you solve compound inequalities with fractions, Why do you factor a quadratic equation before you solve, Graph of a linear equation in two variables, rational expressions math.\n\nGraphing linear equations, what are some good math websites to help solve percent, how to graph an inequality, free multiplying and simplifying with radical expressions solver, compound inequality, how do you find the coefficient of a polynomial.\n\nStandard form formula of parabola, free algebra II/trig homework help, what is the system of equation by graphing for x+y=4, algebrator software, free algebra solver step by step, 9th grade mathematics homework.\n\nSleeping parabola examples, equation, how do you do cubed root on calculator.\n\nSolving inequalities as algebriac equations, How do you graph linear equations, radical function calculator, how to solve a matched-pair t test on a ti89, Algebra homework solutions, algebra use the function find the outpoint y,for each input x, free step by step algebra solver.\n\nOnline inequality calculator, Free Algebra Answers step by step, grade 9 math worksheets, algebra rational expressions, rational, solve algebra equations, free online algebraic equation calculator.\n\nRationaliza the denominator, Algebrator, grade 10 factoring simplifiying, c# arithmetic composition rules class, algebraic inequalities graph.\n\nFree online algebra programs, learn algebra 1, create equation, study guide, algebra, structure method, book 1, good algebra I and II books, independent and dependent variables maths.\n\nLinear equation graphing calculator ti - 89, simplifying complex fractions calculator, solutions of questions of abstract algebra, complex rational expressions calculator, variable E, simplify math terms calculator, mcdougal littell algebra 2 answer keys.\n\nCollege algebra problem solver, radical expressions calculator online, absolute value equations solver, hrw algebra 2, divide and simplifying radicals calculator.\n\nIs college algebra the same as intermediate algebra, how to learn algebra, finite math vrs algebra.\n\nHow is algebra used in jobs, coding simplest radical form, rudin answers chapter 3, Change the subject of a formula calculator, algebra facts, Respuestas De Problem Sets De Saxon Algebra 2, integration algebra.\n\nAdding radical expressions calculator, free enter your math problem algebra, online simplifying decimal calculator, While graphing an equation or an inequality, what are the basic rules?.\n\nAlgebraic fractions calculator, what are the six steps of algebra, 8th grade math pre algebra, how to solve algebra for 6th grade.\n\nSolve square feet, algebra expression calculator with division, 8th grade algebra, graphing an equation process, step by step directions to algebra.\n\nBest algebra software, how to find a quadratic equation from a table, cliffnotes algebra 1 test samples, cognitive tutor algebra 1.\n\nAlegbra for beginners, where to buy college algebra programs, Hungerford Algebra, pizzazz middle school math worksheets.\n\nHow to factor math problems, nj algebra test, how to solvecollege algebra math problems, meage math.com.\n\nCalculator computer science, Abstract algebra question, poem about algebra, simplifying a product of radical expressions calculator.\n\nMathematics the book of ninth grade in lebanon, solve online differential, algebra picture, algebra problem solver step by step free, simplifying exponents worksheet.\n\nCollege algebra university of arizona old exams, why is algebra important, equation operating system, i want to learn percentage, free college algebra book, cheat on the sat, algerbra real-life application of radicals solutions.\n\nIowa math testing sample questions, maths easy learning\"integration\", transformation graphing circles, www.algebra-answer.com algebra-practise-questions.\n\nFactoring and expanding polynomials, exponential math worksheet i can complete now, addition principle fractions, math dictionary algebra, least common denominator finder, how to factor equations in algebra 1, Why should we clear fractions and decimals when solving linear equations.\n\nAnswers for college algebra problems, matematica algebra, simplifying variable expressions worskheets, algebra factorization problems, Thinkwell college algebra answers.\n\nRadical expressions calculator, hungerford solution manual, factor exponents.\n\nOnline complex equation solver, solve differential equations online, how to solve variables in exponents.\n\nIs there a basic difference between solving a system of equations by the algebraic method and the graphical method?, equivelant, using diamonds to factor polynomials, combining unlike terms, answers to my math problems for free.\n\nFree math test adults, beginning and intermediate algebra 4th edition, poems about polynomials.\n\nStep by step word problem solver \"free\", number line graph inequalities, differences geometry algebra, how to get your teacher to kiss you, i need help on algebra, kumon worksheets download.\n\nFinding square root of algebric expressions, free 7th algebra problems, pre algebra cheat sheet, ALGERBRA2, linear equations distributive property, algebaic proofs.\n\nOnline interpolation, mathcad tutorials, boolean algebra simplification calculator, best way to study linear algebra.\n\nDecimal to mixed number calculator, examples ofalgebra, algebrator online, mantissa calculation, algebra 5, algebraic division calculator.\n\nAlgebra age problem examples with solution, is there any free math tutors gr 12 canada, transformation equation, solve intermediate algebra problems online, Help with Abstract Algebra, what is decomposition in maths.\n\nExplaining algebra poem, How Is Algebra Used in Everyday Life, maths (set).\n\nAlgebraforkids, HELP learning transforming formulas, exponents calculator factored to exponential, how to solve complex fraction, exponential fractions, root mean square on matlab.\n\nCollege math problems, substituting values math problems, fraction simplify calculator, mcdougal littell algebra 2 answers online, prentice hall algebra 1 workbook answers, basic principle that can be used to simplify a polynomial, boca raton math tutoring.\n\nList of algebra formulas, larsen's fifth edition intermediate algebra book, downward division and radical expressions, houghton mifflin algebra 2 answer book, answer finite math questions for free, dxdy, collecting like terms.\n\nCalculator shows work, sample algebra and functions, how to do quadratics, 2nd grade algebra in and out sat 10, how to solve brackets, online word problem solver, how to work out improper fractions.\n\nAlgebra equations on expanding and simplifying, solve your numericals, free online matrix solver, solving mathematical induction inequalities, college algebra cheat sheet, glencoe math algebra 2, algebraic fractions in real life.\n\nYear 7 algebra worksheets, algerbra symbols, solve your math problems online, solving fraction radicals, extrapolate math.\n\nPowers worksheets, adding algebraic fractions with same denominators, how do you simplify?, teaching factoring algebraic expressions, solutions to linear algebra and its applications, sat ti83.\n\nTelecharger algebrator, free 5th grade equations patterns and expressions worksheets, rudin solution, college algebra free tests.\n\nAlgebra with division solver, intermediate algebra problem solver, pre algebra powers and exponents worksheets\\, solving sequences, algebra solving single equation two variables, what is a factor in mathematics, solving exponential fractions.\n\nSolve variable calculator, understanding rational expressions, solving compound inequalities.\n\nWhat math do you learn 11th grade, Polynomial Calculator c#, principles of mathematical analysis problem 6, Finding quadratic functions in real life, radical of fraction, factoring formula, linear algebra and its applications answer.\n\nFree algebra 2 problem solver step-by-step, Where Can I Go To Get The Answers To Algebra Problems In My Math Class, basic algebra abbreviation.\n\nAlgebra calculator, solving radical expressions with fractions, linear algebra simplification, how much algebra is on the asvab, what is algebra used for, how to solve algebra equations.\n\nFactor problems, algebra for 1st grade, math investigatory word problems.\n\nHomework help type in problem for math, sample math poems, algebra workout, algebra formulas calculator, linear equations problem solving variable on both sides, 2 equations in 2 unknown division.\n\nFind quadratic equation by looking a a data table, algebra 1 mcdougal littell answers, algebra 2 factorial equation, 9th grade textbooks, basic algebra tutorial.\n\nAlgebra 2 solver, factorize exponents, math solver.\n\nLeast common multiple lesson plan, algebra Structure and method book 1 solution key, simple way of clearing fractions, online differential equation solver.\n\nEven answers for dugopolski, free rational expression solver, how to do simplification of algebra, order of operations hands on activities, 7th grade pre algebra problems, how to do mixed fractions.\n\nWhy does ti 89 calculator gives me a fraction as a answer to linear equations, simplify quadraTIC FUNCTIONS USING ALGEBRA TILES, factorising algebra, algebra what does x - mean, practical use of simplifying algebraic expression, math concepts explained, algebraic pyramids.\n\nIs Algebra 2 easier than Geometry, developing skills in algebra book c, free college algebra, how do you solve equations with three variables, linear algebra tutorial software.\n\nInterest rate problems, learn cancelling fraction, solving system of equations in three variables, how do u solve an algebra expression, l'algeber avence, math calculator shows work, geometry radicals.\n\nFree algebra course, Algebraic Proofs calculator, make a program that simplify frActions, investment problems with solutions, solve for the variable calculator, great analysis proofs, complex fractions solver.\n\nGetting rid of exponents, ratio conversions to fractions quiz, explain step by step basic algebra functions, algebra words into symbols, i need a program to do geometry on.\n\nSamples of pre algebra problems, ordering fractions with a common denominator calculator, algebra 2 solver software, ESL algebra 1 inequalities, written algebra problems, how to find the lcd of trinomials, binomial solver.\n\nSimplify equations online, rudin's chapter 3 #17 solution, abstract algebra solutions manual, college algebra for dummies, complex fractions calculator, sat math ti-89.\n\nNumber system tricks cat, solving fractional algebraic formulas, algebrator program, algebra formula list, how to do college algebra study guide, download Algebra For Dummies.\n\nAlgebraic fractions worksheets, alebra help with factoring, algebraic trivias, Algebra Factoring, describe reciprocal equation.\n\nWww.the scien citife of algebra, fraction to a faction exponent, multiplaying and dividing.\n\nMath tiles, how do i work out algebra, factoring negative exponents, intermediate algebra math solver, intermediate algebra solver.\n\nMath factoring program, free online intermediate algebra tutorials, algebra compound inequalities, Algebra for Dummies, algebra fraction calculator online, formulas percents, Math problems using factors.\n\nAlgebra open ended questions, algebraic simiplication program, long subtraction+algebra, factoring list, rule of order for division, multiplication, addition and subtraction.\n\nAbstract algebra fraleigh, basic algebra jacobson, Solving linear programming ti-89, algebra real life applications, algebra identities, radical expression fraction degrees.\n\nMath poem, ti-89 solving systems of equations, algebra division calculator, factoring perfect square trinomials, how to do matrices in ti-89.\n\nLinear programming ti89, algebraic sequences, Basic algerbra teaching software, how math is used in cryptography, math radicals.\n\nWhat do you learn in pre algebra, what are some hard math promblems for i graders, college algebra answers free.\n\nFind the rule, how to do simple algerbra problems, some equation for rational numbers, algebra structure and method book 1, What is one basic principle that can be used to simplify a polynomial, mark dugopolski.\n\nMultiplying brackets, abstract algebra with power point presentation, how to solve a complex fraction.\n\nAlgebra factoring calculator, all free algebra math answers, visual algebra workbook, Solving Equations fractions, Foerster calculus.\n\nReal life quadratic function, Solution to #17 chapter 3 of Rudin's Principles of Mathematical Analysis, teachers edition introductory algebra 10th editionby m. l. bittinger, modern abstract algebra proofs.\n\nSolving inequalities lesson plans, wHAT IS BINOMIALS, mantissa exponent.\n\nTopics in Algebra by I. N. Herstein, making a math text, algebra 1 review CD, exponential inequalities tutorial, algebra sequence problem, algebra help what does ! means, expanding quadratics calculator.\n\nTest questions on alebraic expressions, solve fraction variable equations calculator, high school linear programming, solving continuous functions, what is an example of college algebra equation, algebra activities for elementary, teaching slope algebra.\n\nHow do independent functions look?, solve inequality calculator, Prentice Hall Mathematics Algebra 2 book on line, what is a variable in algebra?, Solve matrices, algebra worksheets for 8th grade, basic algebra objective.\n\nIDIOTS GUIDE TO ALGEBRA, algebra hungerford all solutions, get prepared for 9th grade, software for working out algebra equations, complex polynomials, inserting algebra symbols in computer.\n\nC program to solve two unknowns, cube root of 405, lesson plans solving inequalities, free step-by-step calculator for Algebra 2, year 6 algebra, learning to percents, how to solve equations with fractional coefficients.\n\nIs there a basic difference between solving a system of equations by the algebraic method and the graphical method? Why?, ti-84 binary converter, Pre Algebra problem solver for free, answers to beginning Algebra 6th edition, algebraic calculator algebra help, algebra series problem interval, algebra on the asvab.\n\nAlgebra pythagorean theorem, hungerford abstract algebra solutions, congruence theory, ALGEBRA AND RANGE, words for math expressions, work a math problem of me.\n\nTi-89 trigonometry Programs, elgebra caculator, hb stat calculator, learn mechanic fluid, systems and equations within the context of everyday lives, prentice hall algebra 2 solution manual.\n\nArtin algebra solutions, algebraic symbols, real life functions.\n\nAlice Kaseburg, examples of ordering of operations for 8th graders, finite math, tutor, Differential equation online.\n\nSolutions intermediate, scientific calculator online fractions, elementary abstract algebra proofs, worksheets on functions.\n\nAlgebra homework solver, parent help algebra and fractions, step by step for factoring, answer to homework algebra 1, cliff notes college algebra, free college algebra solver.\n\nThe newest saxon algebra 2 series, printable pre-algebra equations, algebra book answers, How do I use my TI 83?.\n\nBello math, how to solve how long it takes to do something algebra problems, download free algebrator.\n\nMcdougal littell pre-algebra practice workbook teacher's edition, step by step- algebra, answer my math problem for free.\n\nIntermediate algebra tutorial, word problems two equations two vriables, answers to my two step equations homework, what i need to know to learn college algebra.\n\nPoems about algebra, algebra 2 parent functions, pre algebra formulas, probability tutoring, Distributive Property Free Worksheet, Algebra Cheat software, writing math expressions.\n\nAlgerbra formulas, distributive property math worksheets with answers, finding roots of polynomials calculator, Algebra 1 study guides pdf, mcdougal littell pre algebra.\n\nI need to master algebra 2, maths inequality solver, algebra practice problems, Advance Algebra question, binomial algebra, GCF on my T1-84 calculator.\n\nGlencoe algebra 1 answers, beginning and intermediate algebra teacher's edition, system of equations three variables, ti 83 plus sat.\n\nOnline inequalities algebrator, algebra 2 honors workbook, lcd calculator for rational numbers, pre algebra solver, passing college algebra, ks3 algebra worksheets.\n\nActivities algebraic expressions equations, Intermediate Algebra answer cheats, ONLINE EQUATION SOLVERS WITH ABSOLUTE VALUE, free college algebra practice test.\n\nHow is algebra used in everyday life, basic concepts of algebra, myalgebra, honors algebra 1.\n\nDifferential equation solver online, how to solve matrix problems, cas solve trigonometry ti-89.\n\nAlgebra sixth grade, Cliffs Quick Review algebra torrent, algebra 2 review, radical inequalities, VIDEOS LECTURES ON ANGLES BETWEEN TWO LINES CALCULAS, algebra for kids, product rule algebra.\n\nSolve this math problem for me, linear equations practice, why should we clear fractions and decimals when solving linear equations, TI-83, real life application algebra, factorial division algebra.\n\nSolve two linear equations with two unknowns using c, Algebrator, factorization tutorial.\n\nSolve math problems, show step by step in algebra for free, algebra pyramids, contemporary abstract algebra solutions, inequalities calculator.\n\nMaths algebraic identities, Iowa Algebra Aptitude Test, hard algebraic equations, USING EXCEL IN ALGEBRA, painless algebra, math help verbal expressions.\n\n1 penny doubled everyday for 30 days, solution for solving algebra problems for gre, ucsmp algebra, math trivias with solutions.\n\nFree algebra helper, trinomial solver, solve by factoring radicals.\n\nAlgebra solution finder software, level e maths test answers, intermediate algebra review, help with rational expressions, algerbra 2, algebra factoring.\n\nMath solve by comparison, compound inequality solver, trig helper, algebraic expression worksheets.\n\nWhen solving a rational equation, why is it necessary to perform a check?, free online algebra tutor, algebra for dummies college, free algebra equations with solutions, college algebra refresher.\n\nSimplification of boolean algebra software, free algebra solver with steps, answers to algebra problems, equivelant.\n\nAbstract algebra dummit solutions manual, alegbra rulkes, exponent math calculator, algebra tricky answers, topics in algebra herstein solutions.\n\nFactor polynomial calculator, 1/square root x divided by x, grade 11 math radical inequality, beginning and intermediate algebra third edition, tic tac math, inequalities, math answers to rational expression.\n\nDomain of difference of cubes, online squaring calculator, solve my math problems for me, algebra equations list, algebraic expressions brain teasers, formula solver.\n\nMymathtutor.com, algebra in everyday life, ti 89 sum, Algebra answer that equals to 11, free step by step algebra solver, solving matrix steps software.\n\nFunctions AND independent and dependent variables, answers to question in page 1 in algebra 2 hrw book, division games for fifth grade, hard math investigatory PROBLEM.\n\nAlgebra practice worksheets, rudin chapter 5 solutions, Free tutorials on Year 8 Algebra.\n\nSimplifying calculator, pre algebra worksheets for 8th graders, calculate fractions, a poem about algebra, algebra for college examples, what is binomial.\n\nHow to solve math functions, simplification calculator, abstract algebra hungerford, algebra ii problems 10th grade, solve a math word problem, algebra multiple choice.\n\nHb calculator, 6th grade math algebra, online solve differential equations, algebra order of perations free worksheets, Simplify Calculator, solving rational expressions calculator online, \"algebra helper\" download.\n\nMultiplication calculator that shows work, Roots and Rational Exponents بحث في, free Hands On Equations Worksheet and answer key, rudin chapter 3 solutions.\n\nAlgebra explanation, ANSWERS FOR MATHS QUESTIONS WHEN STUCK ON ALGEBRA, how to learn algebra quickly, howdo algebra to, factoring complex polynomials, sample algebra problems, free worksheets on square roots.\n\nPolynomial poem, holt pre algebra chapter 2 answer key, algebra rules radicals, simplifying indices, rewriting equations with positive exponents, examples on arithmatic progression for finding perimeter of triangle.\n\nDifferential online, teach me linear algebra, balancing equations calculator online, When solving a rational equation, why is it necessary to perform a check?, Algebra I textbooks, PACE.\n\nAlgebrator free, free tutoring printables elementary and college algebra, simplifying fractions with variables, Factoring Calculator.\n\nWhere is algebra used in everyday life?, collect like terms, algebra math basketball, figure out two unknowns, cramer's law, tutor \"business card\" samples, advanced algebra questions.\n\nAlgebra expanding calculator, rational expressions real life, compass test algebra cheat sheets, discrete mathematics homework answers, EASY STEPS FOR ALGEBRA 2, Dr Lee Carlson, motion problems in algebra.\n\nHungerford algebra solution, college algebra word problems, algebra steps.\n\nInequalities solving, alegebraic, algebraic inequalities line graph, make program to simplify fractions.\n\nPrintable worksheet on exponents, What is the difference between evaluation and simplification of an expression?, 7th grade lesson inequalities, how to graph interval notation on a number line.\n\nFree algebra answers, solve complex equations exponential, decimal to fraction baking, solve my algebra 2 problem for free, basic algebra prep.\n\nHow to cheat on the sats, What does it mean to have an “extraneous solution” to a radical equation? Give an example., math 105 help, learning algebra easily, free online cheater math tutor, mcdougal littell algebra 1 answer keys.\n\nBeginners prealgebra equations, solving radical inequality math questions, 6th grade math word problems printable, learn algebra 1.\n\nOne step llinear Equations, I don't understand linear equations, where can I find help?, order in algebra brackets, \"Advance algebra\", enduring understanding on linear equations, factor my polynomial, dummit and foote solutions.\n\nSimplifying fractional expressions calculator, elementary and intermediate algebra answers, mcdougal littell algebra 1 structure and method teacher's edition, answers for prentice hall california pre-algebra, interpolation algebra.\n\nSolve algebramath equations for me, algebrator free, math trivias algebra, 8th grade pre algebra, calculator casio tutorial, 89 monte carlo ss.\n\nRational expressions+definition, examples of algebra in real life, algebraic fractions worksheets generators, the best ways to work on word problems.\n\nDifferential equations calculator online, solve for variables, poems about math division, learn y7 algebra, fucking algebra, subject Algebra 1 - California Edition 2009.\n\n\"linear and nonlinear\" + \"practice\" + \"STEP\", what is 10th grade math like, solve my math.\n\nUnit 1 test for algebra in 9th grade, absolute value equations+worksheets, do my math problems for me online, Easy steps to algebra, solving quadratic formulas programs, Free SAT II Math.\n\nLogarithmic equation solver, the best way to learn college math, take algebra 1 online, factorising help, mcdougal littell algebra 1.\n\nNature of discriminate, problems for the rational algebraic expression, show me how to do algebra and concepts with equations, i need help with algebra, abstract algebra exercises, algebra I for dummies download, use a ti 89 to cheet.\n\nRational expressions applications, college algebra, program simplify fractions, principles of mathematical analysis solutions manual, Boolean algebra programs, ti 84 convert decimal to radical, abstract algebra problems.\n\nSaxon Algebra 2 Answers, basic principle used to simplify a polynomial, What are the parent functions algebra 2 definitions.\n\nExample of a Verbal Model, algebra practice sheets for inequalities, Solution manual abstract algebra by i.n herstein, College Level Algebra Help.\n\nFactor in mathematics, algebra 1 brown dolciani chapter 3, different problems for math investigatory, multiplyin fractions out of brackets, solving binomials by factoring, learning basic math in college, applications with quadratic.\n\nEasiest way to learn algebra 7th grade\\, congruence algebra, algebraic fractions simplifier online, what is one basic principle that can be used to simplify a polynomial, ti-89 arithmetic series, parent functions algebra 2, properties of equations.\n\nAlgebra for dummies online, simplifying equations, contemporary abstract algebra solution manual, step by step help for algebra rational expressions, \"free college algebra help\", distributive property activities.\n\nWrite a program for ti-84 cubic equation, simplifying algebraic expressions powerpoint, failing 7th grade math, basic algebra rules.\n\nInequalities and their graphs, how to unfoil, algebra nth term, solve math equations for me for free, f x math, continuous function grade 11, algebra worksheets for 8th graders.\n\nYahoo visitors found our website yesterday by typing in these math terms :" ]	[ null, "https://softmath.com/images/video-pages/solver-top.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.79384154,"math_prob":0.9927415,"size":96093,"snap":"2020-10-2020-16","text_gpt3_token_len":20837,"char_repetition_ratio":0.2471667,"word_repetition_ratio":0.009319401,"special_character_ratio":0.1938955,"punctuation_ratio":0.121602766,"nsfw_num_words":1,"has_unicode_error":false,"math_prob_llama3":0.9999081,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-04-03T18:20:06Z\",\"WARC-Record-ID\":\"<urn:uuid:2a508265-eb1c-49c5-bc8d-e84bb73cd8a5>\",\"Content-Length\":\"220387\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:9df867d2-1ca4-4510-88b2-e9bf094ff7ef>\",\"WARC-Concurrent-To\":\"<urn:uuid:ebd58309-30eb-40af-8107-e2ab271105b6>\",\"WARC-IP-Address\":\"52.43.142.96\",\"WARC-Target-URI\":\"https://softmath.com/math-com-calculator/distance-of-points/poems-of-equation-that-are.html\",\"WARC-Payload-Digest\":\"sha1:5HEWA4767NT7ZLSBDXVSWKYIOOOI7BQJ\",\"WARC-Block-Digest\":\"sha1:PW2N5EGQLOMOQNLQKHM3YP6DT2SSMFM7\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-16/CC-MAIN-2020-16_segments_1585370515113.54_warc_CC-MAIN-20200403154746-20200403184746-00215.warc.gz\"}"}
https://stackoverflow.com/questions/2461667/centering-strings-with-printf	[ "# Centering strings with printf()\n\nBy default, `printf()` seems to align strings to the right.\n\n``````printf(\"%10s %20s %20s\\n\", \"col1\", \"col2\", \"col3\");\n/* col1 col2 col3 /\n``````\n\nI can also align text to the left like this:\n\n``````printf(\"%-10s %-20s %-20s\", \"col1\", \"col2\", \"col3\");\n``````\n\nIs there a quick way to center text? Or do I have to write a function that turns a string like `test` into `(space)(space)test(space)(space)` if the text width for that column is 8?\n\nprintf by itself can't do the trick, but you could play with the \"indirect\" width, which specifies the width by reading it from an argument. Lets' try this (ok, not perfect)\n\n``````void f(char s)\n{\nprintf(\"---%s%s---\\n\",10+strlen(s)/2,s,10-strlen(s)/2,\"\");\n}\nint main(int argc, char *argv)\n{\nf(\"uno\");\nf(\"quattro\");\nreturn 0;\n}\n``````\n• I tested this in some C++ code I wrote to create table, and it was truncating valid text, and I had to figure out why. I've provided an alternative answer based on how I resolved it. Nov 25, 2018 at 15:57\n• Correct, indeed I warned that the solution is \"not perfect\", due to hard coded constants it contains. My goal was just to give a hint, your solution completes the idea. Nov 26, 2018 at 18:21\n• The `` expects an `int`, `strlen()` will return `size_t`. This function causes UB when `SIZE_MAX>INT_MAX`. Possible solution: Change it to `(int) ((10+strlen(s)/2)&INT_MAX)` Aug 13, 2021 at 20:36\n• Correct. In most cases casting strlen directly to int is enough. It's unlikely that a string that was meant to be centered exceeds INT_MAX in length. Aug 23, 2021 at 7:24\n\n@GiuseppeGuerrini's was helpful, by suggesting how to use print format specifiers and dividing the whitespace. Unfortunately, it can truncate text.\n\nThe following solves the problem of truncation (assuming the field specified is actually large enough to hold the text).\n\n``````void centerText(char text, int fieldWidth) {\nint padlen = (fieldWidth - strlen(text)) / 2;\n}\n``````\n• note that the resulting total width will be different depending on if `fieldWidth - strlen(text)` is even or odd Feb 21, 2019 at 16:38\n• added a modified version that always prints `fieldWidth` characters (e.g. for use in tables etc.) Dec 4, 2019 at 14:29\n• The behavior is implementations defined when the result of `(fieldWidth - strlen(text)) / 2;` is out of the range of an `int`. Which can be the case when `SIZE_MAX>INT_MAX` and `strlen(text)>fieldWidth`, causing `(fieldWidth - strlen(text)) == SIZE_MAX-(strlen(text)-fieldWidth-1)`. Aug 13, 2021 at 20:43\n• @12431234123412341234123 Reminds me of the old adage \"Just because you can do something, doesn't mean you should.\" Aug 16 at 0:28\n\nThere is no `printf()` format specifier to centre text.\n\nYou will need to write your own function or locate a library which provides the functionality that you're looking for.\n\n• Your opportunity to go the extra 9 yards to provide a solution, eh? Nov 25, 2018 at 15:14\n\nYou may try write own function for this problem.\n\n``````/\n Returns a sting \"str\" centered in string of a length width \"new_length\".\n* Padding is done using the specified fill character \"placeholder\".\n/\nchar \nstr_center(char str[], unsigned int new_length, char placeholder)\n{\nsize_t str_length = strlen(str);\n\n// if a new length is less or equal length of the original string, returns the original string\nif (new_length <= str_length)\nreturn str;\n\nchar buffer;\nunsigned int i, total_rest_length;\n\nbuffer = malloc(sizeof(char) new_length);\n\n// length of a wrapper of the original string\ntotal_rest_length = new_length - str_length;\n\n// write a prefix to buffer\ni = 0;\nwhile (i < (total_rest_length / 2)) {\nbuffer[i] = placeholder;\n++i;\n}\nbuffer[i + 1] = '\\0';\n\n// write the original string\nstrcat(buffer, str);\n\n// write a postfix to the buffer\ni += str_length;\nwhile (i < new_length) {\nbuffer[i] = placeholder;\n++i;\n}\nbuffer[i + 1] = '\\0';\n\nreturn buffer;\n}\n``````\n\nResults:\n\n``````puts(str_center(\"A\", 0, '-')); // A\nputs(str_center(\"A\", 1, '-')); // A\nputs(str_center(\"A\", 10, '-')); // ----A-----\nputs(str_center(\"text\", 10, '')); // text\nputs(str_center(\"The C programming language\", 26, '!')); // The C programming language\nputs(str_center(\"The C programming language\", 27, '!')); // The C programming language!\nputs(str_center(\"The C programming language\", 28, '!')); // !The C programming language!\nputs(str_center(\"The C programming language\", 29, '!')); // !The C programming language!!\nputs(str_center(\"The C programming language\", 30, '!')); // !!The C programming language!!\nputs(str_center(\"The C programming language\", 31, '!')); // !!The C programming language!!!\n``````\n• Cool that it is c/stdlib only implementation for the standpoint of educating about C but otherwise at least replace the prefix/postfix loops with `memcpy()` or `bcopy()` Nov 27, 2018 at 13:55\n• And replace malloc with alloca.. ewww Apr 22, 2019 at 15:33\n• alloca wouldn't work in this instance, as the string being returned is in that memory - you don't want it be free'd at the point of the 'str_center' returns! But as Martin points out, as written this leaks memory. Apr 19, 2021 at 10:06\n\nThere are two solutions, the first is similar to the above, by placing macros in printf, and the second is a custom macro, which calculates the length of the formatted string in advance through snprintf, and then calls the printf function to output.\n\n``````#include <stdio.h>\n#include <string.h>\n\n#define LEFT(str, w) \\\n({int m = w + strlen(str); m % 2 ? (m + 1) / 2 : m / 2;})\n\n#define RIGHT(str, w) \\\n({ int m = w - strlen(str); m % 2 ? (m - 1) / 2 : m / 2; })\n\n#define STR_CENTER(str, width) \\\nLEFT(str, width), str, RIGHT(str, width), \"\"\n\n#define PRINTF_CENTER(width, start, fmt, end, ...) ({ \\\nint n = snprintf(NULL, 0, fmt, __VA_ARGS__); \\\nint m = width - n; \\\nint left = m % 2 ? (m + 1) / 2 : m / 2; \\\nint right = m % 2 ? (m - 1) / 2 : m / 2; \\\nprintf(start \"%s\" fmt \"%s\" end, left, \"\", \\\n__VA_ARGS__, right, \"\"); \\\n})\n\n#define MYFORMAT_CENTER(width, fmt, ...) \\\nPRINTF_CENTER(40, \"[\", fmt , \"]\\n\", __VA_ARGS__)\n\nint main(int argc, char const argv[])\n{\nprintf(\"%s%s\\n\\n\", STR_CENTER(\"--- Hello World ---\", 40));\nprintf(\"[%s%s]\\n\", STR_CENTER(\"I am okay today\", 40));\n\nMYFORMAT_CENTER(40, \"%d, e is %f\", 1, 2.71828);\nMYFORMAT_CENTER(40, \"%d, pi is %f\", 2, 3.1415926);\nMYFORMAT_CENTER(40, \"%s %d.\", \"This is such a long string that it exceeds the given size:\", 40);\n\nreturn 0;\n}\n``````\n\nOutput:\n\n`````` --- Hello World ---\n\n[ I am okay today ]\n[ 1, e is 2.718280 ]\n[ 2, pi is 3.141593 ]\n[ This is such a long string that it exceeds the given size: 40. ]\n``````\n\nIll drop my 2 cents after dealing with similar issue of trying to center a table headers in a row with printf.\n\nThe following macros will need to be printed before/after the text and will align regardless of the length of the text itself. Notice that if we have odd length strings, we will not align as should(because the normal devision will result in missing space). Therefor a round up is needed, and I think this is the elegant way to solve that issue:\n\n``````#define CALC_CENTER_POSITION_PREV(WIDTH, STR) (((WIDTH + ((int)strlen(STR))) % 2) \\\n? ((WIDTH + ((int)strlen(STR)) + 1)/2) : ((WIDTH + ((int)strlen(STR)))/2))\n#define CALC_CENTER_POSITION_POST(WIDTH, STR) (((WIDTH - ((int)strlen(STR))) % 2) \\\n? ((WIDTH - ((int)strlen(STR)) - 1)/2) : ((WIDTH - ((int)strlen(STR)))/2))\n``````\n\nUsage example:\n\n``````printf(\"%s%s\" , CALC_CENTER_POSITION_PREV(MY_COLUMN_WIDTH, \"Header\")\n``````\n\nYes, you will either have to write your own function that returns \" test \" etc, e.g.\n\n``````printf(\"%s %s %s\", center(\"col1\", 10), center(\"col2\", 20), center(\"col3\", 20));\n``````\n\nOr you have a center_print function, something like the following:\n\n``````void center_print(const char s, int width)\n{\nint length = strlen(s);\nint i;\nfor (i=0; i<=(width-length)/2; i++) {\nfputs(\" \", stdout);\n}\nfputs(s, stdout);\ni += length;\nfor (; i<=width; i++) {\nfputs(\" \", stdout);\n}\n}\n``````\n• The first suggestion: How can this be impl'd without leaking memory? Mar 31, 2017 at 13:57\n• If you preallocate some buffers based on some criteria that does not seem unreasonable (like for instance no more than 20 arguments will be centred for one printf, and none of the centred results will be longer than 200 bytes), you could let the center function just rotate buffers on each invocation. Apr 2, 2017 at 15:18\n\nA more compact version of PADYMKO's function above (which still leaks memory):\n\n``````char str_center(char str[], unsigned int new_length, char placeholder)\n{\nsize_t str_length = strlen(str);\nchar buffer;\n/------------------------------------------------------------------\n* If a new length is less or equal length of the original string,\n* returns the original string\n------------------------------------------------------------------/\nif (new_length <= str_length)\n{\nreturn(str);\n}\nbuffer = malloc(sizeof(char) * (new_length + 1));\nmemset(buffer, placeholder, new_length);\nbuffer[new_length] = '\\0';\nbcopy(str, buffer + (( new_length - str_length) / 2), str_length);\nreturn(buffer);\n}\n``````\n\nThis sets the whole of newly allocated buffer to the padding character, null terminates that, and then drops the string to be centred into the middle of the buffer - no loops, or keeping track of where to copy to..\n\nIf you want to be able to use a `printf()` format string for that and you accept to be limited to the GNU clib, you can extend `printf()` with your own conversion specifier for centering a string with. Add the conversion specifier with `register_printf_function()`. See here for the documentation: https://www.gnu.org/software/libc/manual/html_node/Customizing-Printf.html The other answers already provide you with a solution on how to manually print a string in the center, which you still need when using your own conversion specifier.\n\nYou can use either of the following two options:\n\n``````char name[] = \"Name1\";\n\n//Option One\nprintf(\"%s\", 40+strlen(name)/2, name, 40-strlen(name)/2, \"\");\nputs(\"\");//skip one line\n``````\n``````//Option two\nprintf(\"%s\", 40+strlen(\"Name2\")/2, \"Name2\", 40-strlen(\"Name2\")/2, \"\");\n``````\n\nThe output is:\n\nName1(center)\nName2(center)\n\n• First of all, this looks like a ripoff of Giuseppe's answer, and, secondly, you have more printf() arguments than you have format specifiers to accommodate them. Did you even test this? Nov 25, 2018 at 15:13" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5075467,"math_prob":0.9026396,"size":1433,"snap":"2023-40-2023-50","text_gpt3_token_len":480,"char_repetition_ratio":0.11336599,"word_repetition_ratio":0.19485295,"special_character_ratio":0.448709,"punctuation_ratio":0.27439025,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9632326,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-30T14:23:00Z\",\"WARC-Record-ID\":\"<urn:uuid:f9d77788-39ec-4afb-8b43-66d0b5152db0>\",\"Content-Length\":\"281371\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:8bc52ebf-ab12-4a8d-8366-0bbfaf8d1471>\",\"WARC-Concurrent-To\":\"<urn:uuid:5964cefc-d8e7-4577-83f3-933bae3c599c>\",\"WARC-IP-Address\":\"104.18.23.201\",\"WARC-Target-URI\":\"https://stackoverflow.com/questions/2461667/centering-strings-with-printf\",\"WARC-Payload-Digest\":\"sha1:F62WA2KMSZN33OV7NVTLRC7K7YSVJUAF\",\"WARC-Block-Digest\":\"sha1:YGOYEEOF3Z5T3NWAAO7KKNLEGJUFHKJ3\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233510676.40_warc_CC-MAIN-20230930113949-20230930143949-00424.warc.gz\"}"}
https://www.numbersaplenty.com/232909	[ "Search a number\nBaseRepresentation\nbin111000110111001101\n3102211111021\n4320313031\n524423114\n64554141\n71660015\noct706715\n9384437\n10232909\n11149a96\n12b2951\n1382021\n1460c45\n1549024\nhex38dcd\n\n232909 has 4 divisors (see below), whose sum is σ = 234016. Its totient is φ = 231804.\n\nThe previous prime is 232907. The next prime is 232919. The reversal of 232909 is 909232.\n\nIt is a semiprime because it is the product of two primes, and also a Blum integer, because the two primes are equal to 3 mod 4, and also a brilliant number, because the two primes have the same length.\n\nIt is a cyclic number.\n\nIt is not a de Polignac number, because 232909 - 21 = 232907 is a prime.\n\nIt is an alternating number because its digits alternate between even and odd.\n\nIt is a Duffinian number.\n\nIt is a congruent number.\n\nIt is not an unprimeable number, because it can be changed into a prime (232901) by changing a digit.\n\nIt is a pernicious number, because its binary representation contains a prime number (11) of ones.\n\nIt is a polite number, since it can be written in 3 ways as a sum of consecutive naturals, for example, 129 + ... + 694.\n\nIt is an arithmetic number, because the mean of its divisors is an integer number (58504).\n\n2232909 is an apocalyptic number.\n\nIt is an amenable number.\n\n232909 is a deficient number, since it is larger than the sum of its proper divisors (1107).\n\n232909 is an equidigital number, since it uses as much as digits as its factorization.\n\n232909 is an odious number, because the sum of its binary digits is odd.\n\nThe sum of its prime factors is 1106.\n\nThe product of its (nonzero) digits is 972, while the sum is 25.\n\nThe square root of 232909 is about 482.6064649380. The cubic root of 232909 is about 61.5264829650.\n\nThe spelling of 232909 in words is \"two hundred thirty-two thousand, nine hundred nine\".\n\nDivisors: 1 283 823 232909" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9253408,"math_prob":0.9935881,"size":1719,"snap":"2022-40-2023-06","text_gpt3_token_len":463,"char_repetition_ratio":0.19183673,"word_repetition_ratio":0.0063492064,"special_character_ratio":0.32693428,"punctuation_ratio":0.1401099,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99638176,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-01-28T14:06:20Z\",\"WARC-Record-ID\":\"<urn:uuid:75d7c18e-b3c3-44c0-9ede-a475eadac951>\",\"Content-Length\":\"9226\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:9d35f2cb-3a71-4d55-939b-db205e028f75>\",\"WARC-Concurrent-To\":\"<urn:uuid:6b0341f8-4764-4399-b27c-dc51f77b9e70>\",\"WARC-IP-Address\":\"89.46.108.74\",\"WARC-Target-URI\":\"https://www.numbersaplenty.com/232909\",\"WARC-Payload-Digest\":\"sha1:4O2SI6T7NGDN2KP7NL7SONQ2XEVLEVJW\",\"WARC-Block-Digest\":\"sha1:K47IBNUI64X24WXTFJIECJC3MLKS7IHC\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764499634.11_warc_CC-MAIN-20230128121809-20230128151809-00021.warc.gz\"}"}
https://forum.uipath.com/t/filter-relative-comparison/564085	[ "# Filter relative comparison\n\nHello all,\n\nI use filter to comparison but I want relative comparison ± 0.05. Thank all\nAmount column: 10602.38\nTong Tien column: 10602.39\n\[email protected]\n\n\"[Amount] >= \" + (10602.38 - 0.05).ToString + \" And [Amount] <= \" + (10602.39 + 0.05).ToString\n\nHope this works\n\n1 Like\n\nHi,\n\nUnfortunately, FilterDataTable activity doesn’t support such calculation.\nCan you try as the following?\n\n``````arrDr = dt.AsEnumerable.Where(Function(r) Math.Abs(CDbl(r(\"Amount\"))-CDbl(r(\"Tong Tien\")))<=0.05).ToArray()\n``````\n\nNote: arrDr is DataRow array type.\n\nRegards,\n\n1 Like\n\nHi @Yoichi\n\nBecause i need to compare 2sheet. How can I use your code ?\nAssign: Column ‘Tổng Tiền’ does not belong to table DataTable.\n\n“Tổng Tiền” column in dt2\n“AMOUNT” column in DTfilter\n\nHi,\n\nDo you want to compare same row index data in both datatable?\n\nIf so,can you try the following expression. it’s unnecesary to use ForEachRow activity.\n\n``````dtFilter = dt.AsEnumerable.Where(Function(r,i) Math.Abs(CDbl(r(\"Amount\"))-CDbl(dt2.Rows(i)(\"Tong Tien\")))<=0.05).ToArray()\n``````\n\nRegards,\n\nhi, when I assign have some issues, and the condition for true? Thank you\n\nHi,\n\nSorry, It’s my mistake. Can you try to use arrDR (DataRow array type) as the following? (Because if there is no row in filtered datatable, it will throws error)\n\nRegards," ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5808557,"math_prob":0.6772615,"size":1120,"snap":"2023-40-2023-50","text_gpt3_token_len":336,"char_repetition_ratio":0.12007169,"word_repetition_ratio":0.24675325,"special_character_ratio":0.31339285,"punctuation_ratio":0.20588236,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9883428,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-11-30T17:10:44Z\",\"WARC-Record-ID\":\"<urn:uuid:2b66b875-30c6-4eb9-a6c9-a0030b296ee5>\",\"Content-Length\":\"63319\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:58f65765-b611-4e7c-aaac-07232a2b5876>\",\"WARC-Concurrent-To\":\"<urn:uuid:e7af8940-9a91-4535-9d81-9d2560105672>\",\"WARC-IP-Address\":\"184.105.99.75\",\"WARC-Target-URI\":\"https://forum.uipath.com/t/filter-relative-comparison/564085\",\"WARC-Payload-Digest\":\"sha1:P2FW64XOWZYYDUHGEU7OQ3LD2DHZOZIP\",\"WARC-Block-Digest\":\"sha1:CL5R4DS6XLAZTSGE75LOV74GQD6YVRGN\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100229.44_warc_CC-MAIN-20231130161920-20231130191920-00123.warc.gz\"}"}
https://studyrankersonline.com/85511/prism-of-glass-is-shown-in-figure-ray-incident-normally-on-one-face-is-totally-reflected	[ "# A prism of glass is shown in figure A ray incident normally on one face is totally reflected.\n\n592 views\nin Science\n\nA prism of glass is shown in figure A, ray incident normally on one face is totally reflected. If θ is 45°, the index of refraction of glass is _______ .\n\n(A) < √2 (B) > √2 (C) = 2 (D) None of these", null, "by Expert (37.9k points)\n\nThe correct option is (B) > √2.\n\nExplanation:\n\nFor total internal reflection i > θc ......(1)\n\nwhen θc is critical angle\n\nsin θc = (1/η) i.e. η = (1 / sin θc)\n\ni = 45°\n\nhence 45° > sin –1(1/η) From (1)\n\nsin 45° > (1/η)\n\n(1 / √2) > (1/η) hence η > √2" ]	[ null, "https://www.studyrankersonline.com/", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.77055854,"math_prob":0.9813811,"size":439,"snap":"2021-43-2021-49","text_gpt3_token_len":178,"char_repetition_ratio":0.10344828,"word_repetition_ratio":0.0,"special_character_ratio":0.44646925,"punctuation_ratio":0.13461539,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97577935,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-19T22:17:21Z\",\"WARC-Record-ID\":\"<urn:uuid:537ba703-465e-4cad-b34c-a4973309ac04>\",\"Content-Length\":\"77870\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ab4acd87-06d7-417c-962b-a01d50dc4ca8>\",\"WARC-Concurrent-To\":\"<urn:uuid:47f4fc61-af37-4c8a-a773-8e913b3f59b7>\",\"WARC-IP-Address\":\"139.59.83.9\",\"WARC-Target-URI\":\"https://studyrankersonline.com/85511/prism-of-glass-is-shown-in-figure-ray-incident-normally-on-one-face-is-totally-reflected\",\"WARC-Payload-Digest\":\"sha1:OO5JYDFCB6PIAY4T6VAWPDTMZOQ4DI3H\",\"WARC-Block-Digest\":\"sha1:ST4SV7IT6NQQCVOH7GRGQDWC7U2A7MEK\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323585281.35_warc_CC-MAIN-20211019202148-20211019232148-00333.warc.gz\"}"}
https://naumathstat.github.io/seminars/acgtSpring2020/	[ "# ACGT Seminar\n\nThe Algebra, Combinatorics, Geometry, and Topology (ACGT) Seminar meets on Tuesdays at 1:00-1:50PM in Room 221 of the Adel Mathematics Building.\n\n# Schedule Spring 2020\n\nNote that talks are listed in reverse chronological order.\n\nNo meeting March 3 or March 10.\n\n### Hypergraphic polymatroids and Dilworth truncation\n\nDate: January 28, February 4, February 11, February 18, February 25\n\nSpeaker: Michael Falk (NAU)\n\nAbstract: Matroids capture the combinatorial structure of vector or line configurations (or, equivalently, linear subspaces). Graphs (allowing loops and multiple edges) naturally give rise to line configurations and matroids, which reflect the graph structure to a large degree. For instance, planarity of a graph is detected by realizability (over the field of two elements) of the dual matroid. Whitney’s 2-isomorphism theorem states that graphs with isomorphic matroids are related by splitting, joining, or twisting.\n\nPolymatroids capture the combinatorial structure of configurations of subspaces, and hypergraphs naturally give rise to such. The purpose of these talks is to examine the generalization of Whitney’s Theorem to hypergraphs, due to Vertigan and Whittle, which involves a very interesting operation on subspace arrangements or polymatroids known as Dilworth truncation.\n\n### Binary Linear Codes from Graphs\n\nDate: January 14, January 21\n\nSpeaker: Sudipta Mallik (NAU)\n\nAbstract: A binary linear code of length $n$ is a subspace of $\\mathbb{F}_2^n$. I will start with a brief introduction to binary linear codes. Then I will present a construction of binary linear codes from graphs, in particular, by the generator matrix $[I_n\\mid A]$ where A is the adjacency matrix of a graph on n vertices. Finally a graph theoretic interpretation of the minimum distance of such codes will be given." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.92014754,"math_prob":0.8586755,"size":1697,"snap":"2022-27-2022-33","text_gpt3_token_len":391,"char_repetition_ratio":0.10277614,"word_repetition_ratio":0.007905139,"special_character_ratio":0.20860341,"punctuation_ratio":0.13029316,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96542716,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-07-05T00:24:04Z\",\"WARC-Record-ID\":\"<urn:uuid:1e02d77e-c0bb-4ccc-b56e-ec6e3dd18df8>\",\"Content-Length\":\"5898\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:591ae764-3eb9-40d2-a5fa-ebef507bf78f>\",\"WARC-Concurrent-To\":\"<urn:uuid:c2c0fdf3-13c2-45be-9e97-d6806cc42bfa>\",\"WARC-IP-Address\":\"185.199.111.153\",\"WARC-Target-URI\":\"https://naumathstat.github.io/seminars/acgtSpring2020/\",\"WARC-Payload-Digest\":\"sha1:XUXHOKP6PQLRVI27733SIX6QVH53PSKS\",\"WARC-Block-Digest\":\"sha1:N6KZI3YMC6ET5SX7U2FSSWA7L7ADBNXV\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656104506762.79_warc_CC-MAIN-20220704232527-20220705022527-00264.warc.gz\"}"}
https://www.crio.do/projects/python-preprocessor-cli/	[ "#### Objective\n\nYou will implement a Command Line Interface (CLI) which will preprocess your dataset and save your time.\n\n#### Project Context\n\nMachine Learning is a subset of the larger field of artificial intelligence (AI) that focuses on teaching computers how to learn without the need to be programmed for specific tasks. In fact, the key idea behind ML is that it is possible to create algorithms that learn from and make predictions on the data.\n\nExamples of Machine Learning are present everywhere including the spam filter that flags messages in your email, the recommendation engine Netflix uses to suggest content you might like, and the self-driving cars being developed by Google and other companies.\n\nBut before applying Machine Learning on any dataset, you need to convert it in such a way that the algorithms could understand the dataset. These steps are preprocessing steps.\n\nData preprocessing is an integral step in Machine Learning as the quality of data and the useful information that can be derived from it directly affects the ability of your model to learn; therefore, it is extremely crucial that you preprocess your data before feeding it into your model.\n\nOne more advantage of preprocessing is that it is considered time consuming for many machine learning developers. This simple CLI tool will save your time so that you can utilize it in applying different machine learning algorithms.\n\nYou will apply the following preprocessing steps:\n\n• Handling NULL values\n• Encoding Categorical Data\n• Feature Scaling\n\n#### Product Architecture\n\nThe Product Architecture consists of 6 parts which are as follows:", null, "#### High-level approach\n\nThis project consists of the following milestones:\n\n• Input the Dataset\n• Data Description\n• Handling NULL Values\n• Encoding Categorical Data\n• Feature Scaling\n\nPandas and scikit learn will be used throughout the project to perform the preprocessig steps.\n\nThe desired end result of this project is like this:\n\n#### Objective\n\nYou will implement a Command Line Interface (CLI) which will preprocess your dataset and save your time.\n\n#### Project Context\n\nMachine Learning is a subset of the larger field of artificial intelligence (AI) that focuses on teaching computers how to learn without the need to be programmed for specific tasks. In fact, the key idea behind ML is that it is possible to create algorithms that learn from and make predictions on the data.\n\nExamples of Machine Learning are present everywhere including the spam filter that flags messages in your email, the recommendation engine Netflix uses to suggest content you might like, and the self-driving cars being developed by Google and other companies.\n\nBut before applying Machine Learning on any dataset, you need to convert it in such a way that the algorithms could understand the dataset. These steps are preprocessing steps.\n\nData preprocessing is an integral step in Machine Learning as the quality of data and the useful information that can be derived from it directly affects the ability of your model to learn; therefore, it is extremely crucial that you preprocess your data before feeding it into your model.\n\nOne more advantage of preprocessing is that it is considered time consuming for many machine learning developers. This simple CLI tool will save your time so that you can utilize it in applying different machine learning algorithms.\n\nYou will apply the following preprocessing steps:\n\n• Handling NULL values\n• Encoding Categorical Data\n• Feature Scaling\n\n#### Product Architecture\n\nThe Product Architecture consists of 6 parts which are as follows:", null, "#### High-level approach\n\nThis project consists of the following milestones:\n\n• Input the Dataset\n• Data Description\n• Handling NULL Values\n• Encoding Categorical Data\n• Feature Scaling\n\nPandas and scikit learn will be used throughout the project to perform the preprocessig steps.\n\nThe desired end result of this project is like this:\n\n#### Input the Dataset\n\nThere are several types of Machine Learning such as Supervised learning, Unsupervised learning etc. Here, you are writing python scripts to make preprocessed dataset for performing supervised learning.\n\nSupervised learning consists of mapping input data (independent variables) to known targets (dependent variable), which humans have provided. Predicting house prices is a good example.\n\nImportant: For simultaneously testing out our application you will be performing the preprocessing on a very popular ML dataset - Titanic survival Dataset. You need to download train.csv dataset from the mentioned website.\n\nThe idea of this milestone is that you are correctly taking the input of the dataset.\n\n#### Requirements\n\n• This is a python application and the project structure is shown below.", null, "• Make a main class for the Project where all below functions will be defined -\n• A function that only takes a ‘.csv’ input from the command line.\n• A function that chooses a target (dependent) variable and removes it from the dataset, so that you can start preprocessing on all the independent variables.\n• Each task will have a separate class and will be called by their respective object.\n• Handle all the exceptions in the input.", null, "### References\n\n#### Expected Outcome\n\nAt the end of this milestone, the project should work like this.\n\n#### Data Description\n\nNow that you are done with the initial step, you can implement various components of your project.\n\nIn this milestone, you need to implement the functionality that will enable users to describe the dataset’s properties like mean, max, standard deviation etc.\n\nThe idea of this milestone is that you can correctly show some basic statistical details (mean, standard deviation, percentiles, total number of values, maximum, minimum), datatype of columns of the dataset using methods provided by pandas library.\n\n#### Requirements\n\n• Make a separate class for Data Description where all below functions will be defined -\n• A function that shows properties (mean, standard deviation, percentiles, total number of values, maximum, minimum) of each numeric column. This function should also show the datatypes along with the null value count of each column.\n• A function that shows the property of any specific column.\n• A numeric column should show properties like mean, standard deviation, percentiles, total number of values, maximum and minimum.\n• A string column should show properties like total number of values and number of distinct values.\n• A function that takes a number of rows ‘n’ as input and prints the dataset.\n• Handle all the exceptions in the input.", null, "### References\n\n#### Expected Outcome\n\nAt the end of this milestone, the project should work like this.\n\n#### Handling NULL Values\n\nThe next step of data preprocessing is to handle missing data in the datasets. If your dataset contains some missing data, then it may create a huge problem for your machine learning model. Hence it is necessary to handle missing values present in the dataset.\n\nThe handling of missing values is also called Data Imputation.\n\nThe idea of this milestone is to remove all the NULL values from the dataset.\n\n#### Requirements\n\n• Make a separate class for Data Imputation where all below functions will be defined -\n• A function that removes the specified column.\n• Three respective functions that fill the null values in a column with mean, median and mode.\n• Handle all the exceptions in the input.", null, "### References\n\n#### Expected Outcome\n\nAt the end of this milestone, the project should work like this.\n\n#### Encoding Categorical Data\n\nCategorical data is data which has some categories. Machine learning models completely works on mathematics and numbers, but if your dataset would have a categorical variable, then it may create trouble while building the model. So, it is necessary to encode these categorical variables into numbers.\n\nThe idea of this milestone is to map all the categorical columns into numbers.\n\n#### Requirements\n\n• Make a separate class for Encoding Categorical data where all below functions will be defined -\n• A function that shows all the Categorical columns present in the dataset\n• A function that performs one hot encoding for a categorical column.\n• Handle all the exceptions in the input.", null, "### References\n\n#### Expected Outcome\n\nAt the end of this milestone, the project should work like this.\n\n#### Feature Scaling\n\nFeature scaling is a method used to normalize the range of independent variables or columns of data. It is done to handle highly varying magnitudes among different columns.\n\nIf feature scaling is not done, then a machine learning algorithm tends to weigh greater values, higher and consider smaller values as the lower values, regardless of the unit of the values. To avoid this, feature scaling is done.\n\nThere are 2 main ways of doing feature scaling:\n\n• Normalization\n• Standardization\n\nThe idea of this milestone is that you are able to correctly scale the dataset.\n\n#### Requirements\n\n• Make a separate class for Feature Scaling where all below functions will be defined -\n• A function that performs normalization of any specific column or group of columns.\n• A function that performs standardization of any specific column or group of columns.\n• Handle all the exceptions in the input.", null, "### References\n\n#### Expected Outcome\n\nAt the end of this milestone, the project should work like this.\n\nAs all the preprocessing is done, you can implement the functionality to download the preprocessed dataset.\n\nThe idea of this milestone is that you can correctly download the dataset in the correct file format.\n\n#### Requirements", null, "" ]	[ null, "https://storage.googleapis.com/crio-content-container-assets/ME_ME_PROJECT_PREP_CLI_MODULE_ME_PROJECT_PREP_CLI_MODULE_PREP_CLI_product-architecture.png", null, "https://storage.googleapis.com/crio-content-container-assets/ME_ME_PROJECT_PREP_CLI_MODULE_ME_PROJECT_PREP_CLI_MODULE_PREP_CLI_product-architecture.png", null, "https://storage.googleapis.com/crio-content-container-assets/ME_ME_PROJECT_PREP_CLI_MODULE_ME_PROJECT_PREP_CLI_MODULE_PREP_CLI_project-structure.png", null, "https://storage.googleapis.com/crio-content-container-assets/ME_ME_PROJECT_PREP_CLI_MODULE_ME_PROJECT_PREP_CLI_MODULE_PREP_CLI_step1-main.png", null, "https://storage.googleapis.com/crio-content-container-assets/ME_ME_PROJECT_PREP_CLI_MODULE_ME_PROJECT_PREP_CLI_MODULE_PREP_CLI_step2-description.png", null, "https://storage.googleapis.com/crio-content-container-assets/ME_ME_PROJECT_PREP_CLI_MODULE_ME_PROJECT_PREP_CLI_MODULE_PREP_CLI_step3-imputation.png", null, "https://storage.googleapis.com/crio-content-container-assets/ME_ME_PROJECT_PREP_CLI_MODULE_ME_PROJECT_PREP_CLI_MODULE_PREP_CLI_step4-encoding_categorical.png", null, "https://storage.googleapis.com/crio-content-container-assets/ME_ME_PROJECT_PREP_CLI_MODULE_ME_PROJECT_PREP_CLI_MODULE_PREP_CLI_step5-feature_scaling.png", null, "https://storage.googleapis.com/crio-content-container-assets/ME_ME_PROJECT_PREP_CLI_MODULE_ME_PROJECT_PREP_CLI_MODULE_PREP_CLI_step6-download.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.914605,"math_prob":0.7936518,"size":6991,"snap":"2023-40-2023-50","text_gpt3_token_len":1288,"char_repetition_ratio":0.1509947,"word_repetition_ratio":0.61613476,"special_character_ratio":0.18166214,"punctuation_ratio":0.09272582,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9587907,"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,8,null,8,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-03T08:12:38Z\",\"WARC-Record-ID\":\"<urn:uuid:4852a44e-bd12-452f-895c-241de5b4852f>\",\"Content-Length\":\"124000\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:42711804-5099-447b-8015-db3189fc07c0>\",\"WARC-Concurrent-To\":\"<urn:uuid:95ad2c6d-4971-4399-9ad3-aa5fe46bf119>\",\"WARC-IP-Address\":\"52.85.151.114\",\"WARC-Target-URI\":\"https://www.crio.do/projects/python-preprocessor-cli/\",\"WARC-Payload-Digest\":\"sha1:3CE7KEIYMG2QOKKKRZRVNDFDKNWYZUSL\",\"WARC-Block-Digest\":\"sha1:EXA6JUETBM2TRYEFJQICSIDZKFEUM7WO\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100489.16_warc_CC-MAIN-20231203062445-20231203092445-00819.warc.gz\"}"}
http://kali.kaysen.me/kalitools/powerfuzzer_zh.html	[ "## Powerfuzzer包装说明\n\nPowerfuzzer是基于从众多安全资源和网站上聚集了许多可供其他开放源代码模糊器和信息的高度自动化，完全可定制的web模糊器（HTTP协议的应用程序的fuzzer）。它被设计成用户友好的，现代化的，有效的工作。\n\n• 跨站点脚本（XSS）\n• 注射剂（SQL中，LDAP，代码，命令和XPATH）\n• CRLF\n• HTTP 500状态（通常是指示是否有误/安全漏洞（包括图表）缓冲区溢出）\n\nPowerfuzzer首页 \| 卡利Powerfuzzer回购\n\n• 作者：马辛·科兹洛夫斯基\n• 许可：GPLv3的\n\nWeb应用程序漏洞扫描。\n\n### Powerfuzzer用法示例\n\n`[email protected]/* <![CDATA[ /!function(t,e,r,n,c,a,p){try{t=document.currentScript\|\|function(){for(t=document.getElementsByTagName('script'),e=t.length;e--;)if(t[e].getAttribute('data-cfhash'))return t[e]}();if(t&&(c=t.previousSibling)){p=t.parentNode;if(a=c.getAttribute('data-cfemail')){for(e='',r='0x'+a.substr(0,2)\|0,n=2;a.length-n;n+=2)e+='%'+('0'+('0x'+a.substr(n,2)^r).toString(16)).slice(-2);p.replaceChild(document.createTextNode(decodeURIComponent(e)),c)}p.removeChild(t)}}catch(u){}}()/ ]]> */:~# powerfuzzer`" ]	[ null ]	{"ft_lang_label":"__label__zh","ft_lang_prob":0.8746025,"math_prob":0.88898563,"size":460,"snap":"2020-10-2020-16","text_gpt3_token_len":334,"char_repetition_ratio":0.18421052,"word_repetition_ratio":0.0,"special_character_ratio":0.17608696,"punctuation_ratio":0.08163265,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9537122,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-04-03T05:26:55Z\",\"WARC-Record-ID\":\"<urn:uuid:ae2e1d21-c342-4d8d-a5b5-a8b96a191aad>\",\"Content-Length\":\"3306\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:dc27a0b3-42f1-4e73-af0e-e2bcec9da71e>\",\"WARC-Concurrent-To\":\"<urn:uuid:e3b0b172-743e-42c0-81c9-87dcbad07029>\",\"WARC-IP-Address\":\"101.200.135.227\",\"WARC-Target-URI\":\"http://kali.kaysen.me/kalitools/powerfuzzer_zh.html\",\"WARC-Payload-Digest\":\"sha1:53RFWRIBTHU3OCQXB2CZ6BLPZONLGKWZ\",\"WARC-Block-Digest\":\"sha1:CDSJXF6H4D6C5ZXP6X2WH6BTIQUIMLA2\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-16/CC-MAIN-2020-16_segments_1585370510287.30_warc_CC-MAIN-20200403030659-20200403060659-00106.warc.gz\"}"}
https://socratic.org/questions/a-line-segment-with-endpoints-at-1-2-and-6-7-is-rotated-clockwise-by-pi-2-what-a	[ "# A line segment with endpoints at (1 , -2 ) and (6, -7 ) is rotated clockwise by pi/2. What are the new endpoints of the line segment?\n\nFeb 22, 2018\n\nNew coordinates of the end points are\n\nA’((-2),(-1)) and B’((-7),(-6))\n\n#### Explanation:\n\n$A \\left(1 , - 2\\right) , B \\left(6 , - 7\\right)$. Rotted clockwise by $\\frac{\\pi}{2}$", null, "A((1),(-2)) => A’((-2),(-1)) IV to III quadrant.\n\nB((6),(-7)) => B’((-7),(-6)) IV to III quadrant." ]	[ null, "https://useruploads.socratic.org/oZYbQRzaTjCwH9zctC06_ECD0D721-0026-4B6C-8948-AA966ED22DEA.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7582403,"math_prob":0.99960166,"size":340,"snap":"2023-40-2023-50","text_gpt3_token_len":86,"char_repetition_ratio":0.116071425,"word_repetition_ratio":0.0,"special_character_ratio":0.27941176,"punctuation_ratio":0.092307694,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9573386,"pos_list":[0,1,2],"im_url_duplicate_count":[null,3,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-25T09:46:49Z\",\"WARC-Record-ID\":\"<urn:uuid:f05a7043-a11a-49a3-8e7c-06ea8d2d3020>\",\"Content-Length\":\"33772\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:38fbfebf-d5bb-4ab9-ba43-dee5b592853a>\",\"WARC-Concurrent-To\":\"<urn:uuid:97ddfbe2-4343-4e8a-af0f-5a153ca217b9>\",\"WARC-IP-Address\":\"216.239.38.21\",\"WARC-Target-URI\":\"https://socratic.org/questions/a-line-segment-with-endpoints-at-1-2-and-6-7-is-rotated-clockwise-by-pi-2-what-a\",\"WARC-Payload-Digest\":\"sha1:COSITLJA75VYWRLK6ZIJA6CAOL6QSUGN\",\"WARC-Block-Digest\":\"sha1:HOKWVXVKUX6DVEDDMUNIKVX3UIEKJIO6\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233508959.20_warc_CC-MAIN-20230925083430-20230925113430-00869.warc.gz\"}"}
https://socratic.org/questions/if-you-roll-two-dice-what-is-the-probability-of-rolling-a-number-less-than-5-and	[ "# If you roll two dice, what is the probability of rolling a number less than 5 and another odd number?\n\nNov 6, 2015\n\nBecause we see the word AND we will have to multiply, but see the \"but\" later on.\n\n#### Explanation:\n\nThere are 6 possible outcomes:\n\n$P \\left(< 5\\right) = \\frac{4}{6} = \\frac{2}{3}$\n\n$P \\left(o \\mathrm{dd}\\right) = \\frac{3}{6} = \\frac{1}{2}$\n\nMultiplying we get $\\frac{2}{3} \\cdot \\frac{1}{2} = \\frac{2}{6} = \\frac{1}{3}$\n\nBUT!\nThis only counts for the first die giving the under 5, and the second die giving the odd. The other way around has the same probability.\nIt's either..or , so we add .\n$\\frac{1}{3} + \\frac{1}{3} = \\frac{2}{3}$" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7985205,"math_prob":0.99995816,"size":567,"snap":"2022-27-2022-33","text_gpt3_token_len":137,"char_repetition_ratio":0.09058615,"word_repetition_ratio":0.0,"special_character_ratio":0.23809524,"punctuation_ratio":0.12605043,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9999794,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-07-05T16:05:35Z\",\"WARC-Record-ID\":\"<urn:uuid:ca7ba46e-5ee7-4fc2-beb6-30a1f5ba6117>\",\"Content-Length\":\"33755\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e57dc174-c6fc-4d99-b265-33a0b8e0b73a>\",\"WARC-Concurrent-To\":\"<urn:uuid:366de50c-f162-4c45-98d6-5042ff230291>\",\"WARC-IP-Address\":\"216.239.32.21\",\"WARC-Target-URI\":\"https://socratic.org/questions/if-you-roll-two-dice-what-is-the-probability-of-rolling-a-number-less-than-5-and\",\"WARC-Payload-Digest\":\"sha1:334BQC5G6GSQBKUM4CDFFN6JEXHS4G3J\",\"WARC-Block-Digest\":\"sha1:R3FHLC2GFOUQKWOB6V4SBC3NBGDKDNS2\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656104585887.84_warc_CC-MAIN-20220705144321-20220705174321-00159.warc.gz\"}"}
http://mechdesigner.support/md-qst13_3-rocker-rotating-rocker.htm	[ " Getting Started Tutorials - MechDesigner > Tutorial 13: Forces: Introduction > Step 13.3: Rotating Rocker\n\n# Step 13.3: A Rocker\n\n### A Rotating Rocker – a Crank\n\n We connect a Linear-Motion FB to rotate the Rocker with a constant angular-velocity. This Step helps to understand: 1.Centripetal Acceleration and Force from the circular motion of the Center-of-Mass 2.Superposition of Gravitational Force and Centripetal Force 3.Why the Moment of the Centripetal-Force equals zero. STEP 1: Delete the Spring FB or Open the Spring FB dialog-box and make the Spring-Rate = 0 N/mm STEP 2: Add a Linear-Motion FB STEP 3: Connect a wire from the output-connector of the Linear-Motion FB to the input-connector of the Motion-Dimension FB. The Machine Speed Setting is 60RPM, or 1 Cycle/second = 2π radians/second. The mechanism below shows the forces that act at the Pin-Joint - the rotation axis of the Rocker.", null, "Addition of Vertical Forces acting on the Rocker (Point 2) : (↑+ve). ∑FV=0 : R2V(N) - 1(kg)9.807(m/s/s) = 0; R2V = 9.807N (upwards) Addition of Horizontal Forces acting on the Rocker (Point 2) : (→ +ve). ∑FV=0 : +R2H(N) + 1(kg)0.1(m)(2π)2(1/s/s) = 0; R2H = -3.948N (to the left) Moments are: •An Anti-clockwise Moment that acts on the Rocker is 0.98Nm - as before. •The Centripetal Force acts through the center of rotation. It does not apply a moment to the Arm. Hence, the Moment is the same. The image to the left is the same as the image above. I have added to the image the Horizontal and Vertical Force Vectors that ACT-ON the Rocker The two components are: 1.Reaction to the Gravitational Force - it acts on the Rocker. Equal to 9.81N 2.Centripetal Force acts on the Rocker to the left. Equal to 3.948N The forces are perpendicular(⊥) when the Rocker is horizontal. Hence, we can use Pythagoras, to give 10.571N Ftotal = √(Fg2 + Fc2) (Total Force = SQRT(SQR(Gravitation Force) + SQR(Centripetal Force)) = SQRT((m.g)2 + (m.r.ω2)2) Total Force = √((1g)2 + (10.1(2π)2)2) = √(9.812 + 3.9482) = 10.571N", null, "" ]	[ null, "http://mechdesigner.support/gst-rocker-forces-10.png", null, "http://mechdesigner.support/gst-rocker-forces-11.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7709826,"math_prob":0.99272776,"size":1941,"snap":"2022-27-2022-33","text_gpt3_token_len":623,"char_repetition_ratio":0.15229736,"word_repetition_ratio":0.025,"special_character_ratio":0.30602783,"punctuation_ratio":0.13625304,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9986768,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,1,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-07-01T04:12:18Z\",\"WARC-Record-ID\":\"<urn:uuid:914a318e-a5f7-4850-8e7f-ccb46e6cdb6e>\",\"Content-Length\":\"18855\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:3061195d-afb3-4a67-8ba2-c2cd38a3b676>\",\"WARC-Concurrent-To\":\"<urn:uuid:7d026eac-f0b3-49fe-ad99-56db1002d68c>\",\"WARC-IP-Address\":\"212.67.221.235\",\"WARC-Target-URI\":\"http://mechdesigner.support/md-qst13_3-rocker-rotating-rocker.htm\",\"WARC-Payload-Digest\":\"sha1:3Q5RZG54FWCSCHDUV6OFVI374X7RAEQL\",\"WARC-Block-Digest\":\"sha1:SDHUBTD4XAHU7RXUCHWN3MQS5JT67OM7\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656103920118.49_warc_CC-MAIN-20220701034437-20220701064437-00645.warc.gz\"}"}
https://developers.google.com/earth-engine/guides/resample	[ "# Resampling and Reducing Resolution\n\nAs noted in the Projections doc, Earth Engine performs nearest neighbor resampling by default during reprojection. You can change this behavior with the `resample()` or `reduceResolution()` methods. Specifically, when one of these methods is applied to an input image, any required reprojection of the input will be done using the indicated resampling or aggregation method.\n\n## Resampling\n\n`resample()` causes the indicated resampling method (`'bilinear'` or `'bicubic'`) to be used at the next reprojection. Since inputs are requested in the output projection, an implicit reprojection may happen before any other operation on the input. For this reason, call `resample()` directly on the input image. Consider the following simple example:\n\n```// Load a Landsat image over San Francisco, California, UAS.\nvar landsat = ee.Image('LANDSAT/LC08/C01/T1_TOA/LC08_044034_20160323');\n\n// Set display and visualization parameters.\nMap.setCenter(-122.37383, 37.6193, 15);\nvar visParams = {bands: ['B4', 'B3', 'B2'], max: 0.3};\n\n// Display the Landsat image using the default nearest neighbor resampling.\n// when reprojecting to Mercator for the Code Editor map.\n\n// Force the next reprojection on this image to use bicubic resampling.\nvar resampled = landsat.resample('bicubic');\n\n// Display the Landsat image using bicubic resampling.\n```\n\nNote that the `'bicubic'` resampling results in the output pixels appearing smooth relative to the original image (Figure 1).\n\nThe order of operations for this code sample is diagrammed in Figure 2. Specifically, the implicit reprojection to the maps mercator projection takes place with the resampling method specified on the input image.", null, "Figure 2. Flow chart of operations when resample() is called on the input image prior to display in the Code Editor. Curved lines indicate the flow of information to the reprojection: specifically, the output projection, scale and resampling method to use.\n\n## Reduce Resolution\n\nSuppose that instead of resampling during reprojection, your goal is to aggregate pixels to larger pixels in a different projection. This is useful when comparing image datasets at different scales, for example 30-meter pixels from a Landsat-based product to coarse pixels (higher scale) from a MODIS-based product. You can control this aggregation process with the `reduceResolution()` method. As with `resample()`, call `reduceResolution()` on the input, in order to affect the next reprojection of the image. The following example uses `reduceResolution()` to compare forest cover data at 30-meters resolution to a vegetation index at 500-meters resolution:\n\n```// Load a MODIS EVI image.\nvar modis = ee.Image(ee.ImageCollection('MODIS/006/MOD13A1').first())\n.select('EVI');\n\n// Display the EVI image near La Honda, California.\nMap.setCenter(-122.3616, 37.5331, 12);\nMap.addLayer(modis, {min: 2000, max: 5000}, 'MODIS EVI');\n\n// Get information about the MODIS projection.\nvar modisProjection = modis.projection();\nprint('MODIS projection:', modisProjection);\n\n// Load and display forest cover data at 30 meters resolution.\nvar forest = ee.Image('UMD/hansen/global_forest_change_2015')\n.select('treecover2000');\nMap.addLayer(forest, {max: 80}, 'forest cover 30 m');\n\n// Get the forest cover data at MODIS scale and projection.\nvar forestMean = forest\n// Force the next reprojection to aggregate instead of resampling.\n.reduceResolution({\nreducer: ee.Reducer.mean(),\nmaxPixels: 1024\n})\n// Request the data at the scale and projection of the MODIS image.\n.reproject({\ncrs: modisProjection\n});\n\n// Display the aggregated, reprojected forest cover data.\nMap.addLayer(forestMean, {max: 80}, 'forest cover at MODIS scale');\n```\n\nIn this example, note that the output projection is explicitly set with `reproject()`. During the reprojection to the MODIS sinusoidal projection, rather than resampling, the smaller pixels are aggregated with the specified reducer (`ee.Reducer.mean()` in the example). This sequence of operations is illustrated in Figure 3. Although this example uses `reproject()` to help visualize the effect of `reduceResolution()`, most scripts don't need to explicitly reproject; see the warning here.", null, "Figure 3. Flow chart of operations when reduceResolution() is called on an input image prior to reproject(). Curved lines indicate the flow of information to the reprojection: specifically, the output projection, scale and pixel aggregation method to use.\n\nNote that a second reprojection occurs (implicitly) to display the data on the Code Editor map. Visually inspect the results and observe the correspondence between the pixels from the MODIS layer and the forest cover data reprojected to MODIS scale and projection. In general, you should rarely need to explicitly `reproject()` in Earth Engine.\n\n### Pixel weights for `ReduceResolution`\n\nThe weights of pixels used during the `reduceResolution()` aggregation process are based on the overlap between the smaller pixels being aggregated and the larger pixels specified by the output projection. This is illustrated in Figure 4.", null, "Figure 4. Input pixels (black) and output pixel (blue) for reduceResolution().\n\nThe default behavior is that input pixel weights are computed as the fraction of the output pixel area covered by the input pixel. In the diagram, the output pixel has area a, the weight of the input pixel with intersection area b is computed as b/a and the weight of the input pixel with intersection area c is computed as c/a. This behavior can result in unexpected results when using a reducer other than the mean reducer. For example, to compute forested area per pixel, use the mean reducer to compute the fraction of a pixel covered, then multiply by area (instead of computing areas in the smaller pixels then adding them up with the sum reducer):\n\n```// Compute forest area per MODIS pixel.\nvar forestArea = forest.gt(0)\n// Force the next reprojection to aggregate instead of resampling.\n.reduceResolution({\nreducer: ee.Reducer.mean(),\nmaxPixels: 1024\n})\n// The reduce resolution returns the fraction of the MODIS pixel\n// that's covered by 30 meter forest pixels. Convert to area\n// after the reduceResolution() call.\n.multiply(ee.Image.pixelArea())\n// Request the data at the scale and projection of the MODIS image.\n.reproject({\ncrs: modisProjection\n});\nMap.addLayer(forestArea, {max: 500 * 500}, 'forested area at MODIS scale');\n```" ]	[ null, "https://developers.google.com/earth-engine/images/Resample.png", null, "https://developers.google.com/earth-engine/images/ReduceResolution.png", null, "https://developers.google.com/earth-engine/images/ReduceResolution_weights.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.76309055,"math_prob":0.85707366,"size":5804,"snap":"2020-34-2020-40","text_gpt3_token_len":1341,"char_repetition_ratio":0.16896552,"word_repetition_ratio":0.08043218,"special_character_ratio":0.23104756,"punctuation_ratio":0.15804878,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96205556,"pos_list":[0,1,2,3,4,5,6],"im_url_duplicate_count":[null,7,null,7,null,4,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-09-28T20:13:20Z\",\"WARC-Record-ID\":\"<urn:uuid:2764b8e0-52f6-4225-b669-67eeb357a310>\",\"Content-Length\":\"86098\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:73caa1fb-7420-4c5b-a250-368474448699>\",\"WARC-Concurrent-To\":\"<urn:uuid:fca8d959-f8d2-487f-8815-f5b9aa34075c>\",\"WARC-IP-Address\":\"172.217.13.238\",\"WARC-Target-URI\":\"https://developers.google.com/earth-engine/guides/resample\",\"WARC-Payload-Digest\":\"sha1:ZVK2ZGZBFUF2YKQBVN6BO66V6M7CE3RE\",\"WARC-Block-Digest\":\"sha1:CZ7RKFUPLZX5D6WXSUCEM6SPSKXU6WS4\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-40/CC-MAIN-2020-40_segments_1600401604940.65_warc_CC-MAIN-20200928171446-20200928201446-00313.warc.gz\"}"}
https://www.reference.com/science/molar-mass-kci03-3a3f2b6015d6328b	[ "# What Is the Molar Mass of KCI03?\n\nThe molar mass of KCI03 is 431.82 grams per mole. KCI03 a compound containing one atom of potassium, one atom of chlorine and three atoms of iodine. The molar mass of the similarly written compound KClO3, which is the common chemical potassium chlorate, is 122.54 grams per mole.\n\nThe molar mass of either compound can be calculated by adding together the atomic mass of each atom of each element comprising the compound. For example, a compound containing one atom of potassium and one atom of chlorine includes at least the mass of 39.09 grams per mole for a potassium atom and 35.45 grams per mole for a chlorine atom.\n\nSimilar Articles" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.87479424,"math_prob":0.9764706,"size":620,"snap":"2019-35-2019-39","text_gpt3_token_len":142,"char_repetition_ratio":0.16720779,"word_repetition_ratio":0.057142857,"special_character_ratio":0.23225807,"punctuation_ratio":0.103174604,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99507266,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-08-22T11:32:50Z\",\"WARC-Record-ID\":\"<urn:uuid:b9d17177-1e65-401f-a7c3-4af3a5062470>\",\"Content-Length\":\"171073\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2c104b28-4dd3-40e9-ac70-8caa429fdac2>\",\"WARC-Concurrent-To\":\"<urn:uuid:7c1991ee-8359-4f1f-8f27-3fb67cc17ab3>\",\"WARC-IP-Address\":\"151.101.250.114\",\"WARC-Target-URI\":\"https://www.reference.com/science/molar-mass-kci03-3a3f2b6015d6328b\",\"WARC-Payload-Digest\":\"sha1:GNIMKGNXIGHGXIWKUHUAZR4DETLRKRVT\",\"WARC-Block-Digest\":\"sha1:A23YCTEOU2EZTZWBAHMJCIC45MMRQVPA\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-35/CC-MAIN-2019-35_segments_1566027317113.27_warc_CC-MAIN-20190822110215-20190822132215-00295.warc.gz\"}"}
http://wellness.bcbsla.com/Conditions/Heart/90,P01598	[ "# Heart Disease\n\nHealth Library Explorer\n\n# Determining Body Mass Index for Teens\n\n## What is body mass index?\n\nDetermining how much your teen should weigh is not a simple matter of looking at an insurance height-weight chart. It includes considering the amount of bone, muscle, and fat in his or her body. The amount of fat is the critical measurement.\n\nA good indicator of how much fat your teen carries is the body mass index (BMI). Although it's not a perfect measure, it gives a fairly accurate assessment of how much of your teen's body is composed of fat.\n\nThe formulas below apply to adults only. For children and teens ages 2 to 19 years, the BMI varies by age and sex. An additional step must be done after the BMI has been determined using one of the formulas below. The BMI-for-age percentile is determined by comparing your teen's weight to that of other teens of the same age and sex.\n\nIn other words, by plotting your child's BMI value into the CDC's BMI-for-age growth chart, you can determine if your child is underweight, within normal range, overweight, or obese. Or you can easily calculate it from CDC's online tool\n\n## To calculate an adult's BMI using the English formula\n\nFor example, a person who weighs 165 pounds and is 5 feet 4 inches tall has a BMI of 28.165 lbs x 703=28(64 inches) x (64 inches)\n\n## To calculate an adult's BMI using the metric formula\n\nA BMI between 25 to 29.9 is considered overweight. Anything over 30 is considered obese. Normal BMI is between 18.5 and 24.9. BMI cut-offs for people of South Asian descent are different.\n\nBMI can be calculated using pounds and inches.BMI=Weight in Pounds x 703(Height in Inches) x (Height in Inches)\n\nMultiply your weight in pounds by 703.\n\nBMI can be calculated using kilograms and meters.BMI=Weight in Kilograms(Height in Meters) x (Height in Meters)\n\nDivided by your height in meters." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.90081084,"math_prob":0.9439957,"size":4852,"snap":"2019-26-2019-30","text_gpt3_token_len":1080,"char_repetition_ratio":0.100247525,"word_repetition_ratio":0.02122347,"special_character_ratio":0.20836769,"punctuation_ratio":0.09021739,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9514196,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-07-24T03:47:08Z\",\"WARC-Record-ID\":\"<urn:uuid:3c2cb384-0b9d-44cf-8db0-cf7c3b64cbe1>\",\"Content-Length\":\"101218\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:33e918b9-3e0f-4812-a366-6d59e811d8eb>\",\"WARC-Concurrent-To\":\"<urn:uuid:e597c1ce-50f6-4f42-ab5a-872e8a3ea82d>\",\"WARC-IP-Address\":\"216.115.94.237\",\"WARC-Target-URI\":\"http://wellness.bcbsla.com/Conditions/Heart/90,P01598\",\"WARC-Payload-Digest\":\"sha1:XBHXOTMSCDOXULNKYLW6SOJFMNUMXWRN\",\"WARC-Block-Digest\":\"sha1:IL4JUWLND4ZRCATUGVKJPU6ZGBYT6LBQ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-30/CC-MAIN-2019-30_segments_1563195530250.98_warc_CC-MAIN-20190724020454-20190724042454-00117.warc.gz\"}"}
http://mathandmultimedia.com/tag/problem-solving-in-mathematics/	[ "## Introduction to Math Word Problem Solving\n\nAs I have mentioned in the previous post, I will be starting a Math Word Problem Solving Series for elementary school and high school students. If you can recall, I have already started this series before, but it was discontinued for a while.\n\nIn this series, we are going to learn some of the basics of solving word problems and learn some strategies and tricks to make problem solving easier. This series will include tutorials on how to solve number problems, age problems, number digit problems, distance-rate-time problems, base-rate-percentage problems, mixture problems, etc.\n\nProblem solving ability is developed over time. If you want to be good at it, then you have to practice solving a lot of problems. Reading this series will probably help you a little, but solving the problems yourself will make you a better problem solver. » Read more" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9578097,"math_prob":0.7676203,"size":892,"snap":"2023-14-2023-23","text_gpt3_token_len":175,"char_repetition_ratio":0.18018018,"word_repetition_ratio":0.0,"special_character_ratio":0.19170403,"punctuation_ratio":0.11176471,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97315294,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-03-27T01:43:34Z\",\"WARC-Record-ID\":\"<urn:uuid:8c02d0ff-af52-4616-af42-3fe33773de7d>\",\"Content-Length\":\"47252\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b7c2ec22-5165-4ed5-bb32-2bd610f645e5>\",\"WARC-Concurrent-To\":\"<urn:uuid:94ae20de-35f3-47ac-8739-329db4d1acce>\",\"WARC-IP-Address\":\"166.62.28.131\",\"WARC-Target-URI\":\"http://mathandmultimedia.com/tag/problem-solving-in-mathematics/\",\"WARC-Payload-Digest\":\"sha1:UAPPCCCN6WRLMIIIWUF2L32GYO4AY5SK\",\"WARC-Block-Digest\":\"sha1:ZYRMESXHY2S2RHOT7JIYM4EI52GJQFAH\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-14/CC-MAIN-2023-14_segments_1679296946584.94_warc_CC-MAIN-20230326235016-20230327025016-00794.warc.gz\"}"}
https://projecteuclid.org/journals/electronic-journal-of-probability/volume-21/issue-none/Optimal-binomial-Poisson-and-normal-left-tail-domination-for-sums/10.1214/16-EJP4474.full	[ "Translator Disclaimer\n2016 Optimal binomial, Poisson, and normal left-tail domination for sums of nonnegative random variables\nIosif Pinelis\nElectron. J. Probab. 21: 1-19 (2016). DOI: 10.1214/16-EJP4474\n\n## Abstract\n\nExact upper bounds on the generalized moments $\\operatorname{\\mathsf {E}} f(S_n)$ of sums $S_n$ of independent nonnegative random variables $X_i$ for certain classes $\\mathcal{F}$ of nonincreasing functions $f$ are given in terms of (the sums of) the first two moments of the $X_i$’s. These bounds are of the form $\\operatorname{\\mathsf {E}} f(\\eta )$, where the random variable $\\eta$ is either binomial or Poisson depending on whether $n$ is fixed or not. The classes $\\mathcal{F}$ contain, and are much wider than, the class of all decreasing exponential functions. As corollaries of these results, optimal in a certain sense upper bounds on the left-tail probabilities $\\operatorname{\\mathsf {P}} (S_n\\le x)$ are presented, for any real $x$. In fact, more general settings than the ones described above are considered. Exact upper bounds on the exponential moments $\\operatorname{\\mathsf {E}} \\exp \\{hS_n\\}$ for $h<0$, as well as the corresponding exponential bounds on the left-tail probabilities, were previously obtained by Pinelis and Utev. It is shown that the new bounds on the tails are substantially better.\n\n## Citation\n\nIosif Pinelis. \"Optimal binomial, Poisson, and normal left-tail domination for sums of nonnegative random variables.\" Electron. J. Probab. 21 1 - 19, 2016. https://doi.org/10.1214/16-EJP4474\n\n## Information\n\nReceived: 11 August 2015; Accepted: 25 February 2016; Published: 2016\nFirst available in Project Euclid: 11 March 2016\n\nzbMATH: 1338.60061\nMathSciNet: MR3485362\nDigital Object Identifier: 10.1214/16-EJP4474\n\nSubjects:\nPrimary: 60E15\nSecondary: 60G42, 60G48\n\nJOURNAL ARTICLE\n19 PAGES", null, "SHARE\nVol.21 • 2016", null, "" ]	[ null, "https://projecteuclid.org/Content/themes/SPIEImages/Share_black_icon.png", null, "https://projecteuclid.org/images/journals/cover_ejp.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7949791,"math_prob":0.95443517,"size":1343,"snap":"2021-31-2021-39","text_gpt3_token_len":349,"char_repetition_ratio":0.11426438,"word_repetition_ratio":0.010309278,"special_character_ratio":0.25241995,"punctuation_ratio":0.1125,"nsfw_num_words":1,"has_unicode_error":false,"math_prob_llama3":0.9967948,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-08-04T01:05:21Z\",\"WARC-Record-ID\":\"<urn:uuid:7bbfeacf-6304-4879-84f8-4a5425cf2406>\",\"Content-Length\":\"141912\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e6f5d01a-136a-4a79-bbd8-717610085743>\",\"WARC-Concurrent-To\":\"<urn:uuid:34b7e2cd-2fb5-496c-809a-5342c30646cb>\",\"WARC-IP-Address\":\"45.60.100.145\",\"WARC-Target-URI\":\"https://projecteuclid.org/journals/electronic-journal-of-probability/volume-21/issue-none/Optimal-binomial-Poisson-and-normal-left-tail-domination-for-sums/10.1214/16-EJP4474.full\",\"WARC-Payload-Digest\":\"sha1:KTKYCTCO5OZA62ZHJMZGRXYGSESBXXEQ\",\"WARC-Block-Digest\":\"sha1:WTMAL3OO6ZRRL6MCSMYSC3ZDKBTF2YCM\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-31/CC-MAIN-2021-31_segments_1627046154486.47_warc_CC-MAIN-20210803222541-20210804012541-00423.warc.gz\"}"}
https://www.encyclopediaofmath.org/index.php/Weil_algebra	[ "# Weil algebra\n\nMotivated by algebraic geometry, A. Weil [a3] suggested the treatment of infinitesimal objects as homomorphisms from algebras of smooth functions", null, "into some real finite-dimensional commutative algebra", null, "with unit. The points in", null, "correspond to the choice", null, ", while the algebra", null, ",", null, ", of dual numbers (also called Study numbers) leads to the tangent vectors at points in", null, "(viewed as derivations on functions). At the same time, Ch. Ehresmann established similar objects, jets (cf. also Jet), in the realm of differential geometry, cf. [a1].\n\nSince", null, "is formally real (i.e.", null, "is invertible for all", null, "), the values of the homomorphisms in", null, "are in formally real subalgebras. Now, for each finite-dimensional real commutative unital algebra", null, "which is formally real, there is a decomposition of the unit", null, "into all minimal idempotent elements. Thus,", null, ", where", null, ", and", null, "are nilpotent ideals in", null, ". A real unital finite-dimensional commutative algebra", null, "is called a Weil algebra if it is of the form", null, "where", null, "is the ideal of all nilpotent elements in", null, ". The smallest", null, "with the property", null, "is called the depth, or order, of", null, ".\n\nIn other words, one may also characterize the Weil algebras as the formally real and local (i.e. the ring structure is local, cf. also Local ring) finite-dimensional commutative real unital algebras. See [a2], 35.1, for details.\n\nAs a consequence of the Nakayama lemma, the Weil algebras can be also characterized as the local finite-dimensional quotients of the algebras of real polynomials", null, ". Consequently, the Weil algebras", null, "correspond to choices of ideals", null, "in", null, "of finite codimension. The algebra of Study numbers", null, "is given by", null, ", for example. Equivalently, one may consider the algebras of formal power series or the algebras of germs of smooth functions at the origin", null, "(cf. also Germ) instead of the polynomials.\n\nThe width of a Weil algebra", null, "is defined as the dimension of the vector space", null, ". If", null, "is an ideal of finite codimension in", null, ",", null, ", then the width of", null, "equals", null, ". For example, the Weil algebra", null, "has width", null, "and order", null, ", and it coincides with the algebra", null, "of", null, "-jets of smooth functions at the origin in", null, ". Moreover, each Weil algebra of width", null, "and order", null, "is a quotient of", null, ".\n\nTensor products of Weil algebras are Weil algebras again. For instance,", null, ".\n\nThe infinitesimal objects of type", null, "attached to points in", null, "are simply", null, ". All smooth functions", null, "extend to", null, "by the evaluation of the Taylor series (cf. also Whitney extension theorem)", null, "where", null, ",", null, ",", null, "are multi-indices,", null, ". Applying this formula to all components of a mapping", null, ", one obtains an assignment functorial in both", null, "and", null, ". Of course, this definition extends to a functor on all locally defined smooth mappings", null, "and so each Weil algebra gives rise to a Weil functor", null, ". (See Weil bundle for more details.)\n\nThe automorphism group", null, "of a Weil algebra is a Lie subgroup (cf. also Lie group) in", null, "and its Lie algebra coincides with the space of all derivations (cf. also Derivation in a ring) on", null, ",", null, ", i.e. all mappings", null, "satisfying", null, ", cf. [a2], 42.9." ]	[ null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w1200501.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w1200502.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w1200503.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w1200504.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w1200505.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w1200506.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w1200507.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w1200508.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w1200509.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005010.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005011.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005012.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005013.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005014.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005015.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005016.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005017.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005018.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005019.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005020.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005021.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005022.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005023.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005024.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005025.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005026.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005027.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005028.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005029.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005030.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005031.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005032.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005033.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005034.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005035.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005036.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005037.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005038.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005039.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005040.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005041.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005042.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005043.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005044.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005045.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005046.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005047.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005048.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005049.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005050.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005051.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005052.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005053.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005054.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005055.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005056.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005057.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005058.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005059.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005060.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005061.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005062.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005063.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005064.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005065.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005066.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005067.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005068.png", null, "https://www.encyclopediaofmath.org/legacyimages/w/w120/w120050/w12005069.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.83798206,"math_prob":0.9744947,"size":3643,"snap":"2020-10-2020-16","text_gpt3_token_len":926,"char_repetition_ratio":0.1415224,"word_repetition_ratio":0.010169491,"special_character_ratio":0.24430415,"punctuation_ratio":0.1764706,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9962325,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138],"im_url_duplicate_count":[null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,2,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,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,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,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,2,null,1,null,1,null,2,null,1,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-03-31T16:51:55Z\",\"WARC-Record-ID\":\"<urn:uuid:4acb0111-a97c-4d9b-85fe-273c5cf9ea4e>\",\"Content-Length\":\"25862\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a000154e-c3d3-4c65-832b-920cd68fb807>\",\"WARC-Concurrent-To\":\"<urn:uuid:98ed99c9-5fba-45dd-8a79-e42d6940e822>\",\"WARC-IP-Address\":\"80.242.138.72\",\"WARC-Target-URI\":\"https://www.encyclopediaofmath.org/index.php/Weil_algebra\",\"WARC-Payload-Digest\":\"sha1:4U7LGUMOT2IDGYF6ZAIRIMEIRIXXBM46\",\"WARC-Block-Digest\":\"sha1:LOJUURCRKH2E6O7D6TURRKCEVQDKJMYU\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-16/CC-MAIN-2020-16_segments_1585370502513.35_warc_CC-MAIN-20200331150854-20200331180854-00432.warc.gz\"}"}
https://slideplayer.com/slide/6381895/	[ "", null, "Graphing in Science. Types of Charts  Most scientific graphs are made as line graphs.  However, occasionally bar graphs, pie charts, or scatter plots.\n\nPresentation on theme: \"Graphing in Science. Types of Charts  Most scientific graphs are made as line graphs.  However, occasionally bar graphs, pie charts, or scatter plots.\"— Presentation transcript:\n\nGraphing in Science\n\nTypes of Charts  Most scientific graphs are made as line graphs.  However, occasionally bar graphs, pie charts, or scatter plots are used.\n\nWhy do we use graphs?  All types of graphs and charts make trends in data easier to see.  It is a way for scientists to analyze their data.  It can also be used to predict data that is not measured on the graph.\n\nSteps to constructing a line graph  Identify your variables  Independent variable goes on the x-axis  Dependent variable goes on the y-axis  Determine the variable range  Subtract the lowest data value from the highest data value for each variable  Determine the scale of the graph  This is the numerical value for each square that best fits the range of each variable  Use most of the graphing space available.\n\nSteps to constructing a line graph (continued)  Number and Label each axis  Do not forget to include units!!  Plot the data points.  Draw the graph  Connect the dots  Best Fit  Title the graph  Titles should clearly show what the graph is about  If you have more than one set of data, you need to include a key to identify the different lines\n\nExample Using the data table, answer the questions below:  What is the independent variable?  pH of the water  What is the dependent variable?  Number of tadpoles  What goes on the x-axis? What is the range?  pH of water (2.5)  What goes on the y-axis? What is the range?  Number of tadpoles (65) pH of water# of tadpoles 8.045 7.569 7.078 6.588 6.043 5.523\n\nExample pH of water# of tadpoles 8.045 7.569 7.078 6.588 6.043 5.523\n\nDownload ppt \"Graphing in Science. Types of Charts  Most scientific graphs are made as line graphs.  However, occasionally bar graphs, pie charts, or scatter plots.\"\n\nSimilar presentations" ]	[ null, "https://slideplayer.com/static/blue_design/img/slide-loader4.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.87451947,"math_prob":0.9070901,"size":1718,"snap":"2022-05-2022-21","text_gpt3_token_len":411,"char_repetition_ratio":0.13710618,"word_repetition_ratio":0.06020067,"special_character_ratio":0.24388824,"punctuation_ratio":0.11111111,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9852575,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-01-28T17:40:02Z\",\"WARC-Record-ID\":\"<urn:uuid:9f82a5ab-b35e-4f8f-b909-03e38d5837fd>\",\"Content-Length\":\"148752\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:dca8cd9f-56e2-49d3-8a92-5a772e57a7d1>\",\"WARC-Concurrent-To\":\"<urn:uuid:1efc3d9c-5e67-406a-892c-360afa2b4f90>\",\"WARC-IP-Address\":\"138.201.58.10\",\"WARC-Target-URI\":\"https://slideplayer.com/slide/6381895/\",\"WARC-Payload-Digest\":\"sha1:XF23X2SIDLE5FJOIGPD6ESQJ6BNFVHJH\",\"WARC-Block-Digest\":\"sha1:5WMAPSKUPORZT7QRTJX5SE3TPJRXKLYE\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-05/CC-MAIN-2022-05_segments_1642320306301.52_warc_CC-MAIN-20220128152530-20220128182530-00057.warc.gz\"}"}
https://newpathworksheets.com/math/grade-8/linear-equations/virgin-islands-common-core-standards	[ "## ◂Math Worksheets and Study Guides Eighth Grade. Linear equations\n\n### The resources above correspond to the standards listed below:\n\n#### Virgin Islands Common Core Standards\n\nVI.CC.EE.8. Expressions and Equations\nUnderstand the connections between proportional relationships, lines, and linear equations.\nEE.8.5. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.\nEE.8.6. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.\nVI.CC.F.8. Functions\nDefine, evaluate, and compare functions.\nF.8.3. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s^2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.\nUse functions to model relationships between quantities.\nF.8.4. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8998012,"math_prob":0.9979954,"size":1648,"snap":"2020-10-2020-16","text_gpt3_token_len":358,"char_repetition_ratio":0.14902677,"word_repetition_ratio":0.055555556,"special_character_ratio":0.21723302,"punctuation_ratio":0.14749263,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9993462,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-04-02T03:38:36Z\",\"WARC-Record-ID\":\"<urn:uuid:4c99a900-2da5-42d1-bee0-40a31bde7b86>\",\"Content-Length\":\"30664\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:191d9078-b772-44e4-b470-af655ddbd15d>\",\"WARC-Concurrent-To\":\"<urn:uuid:e14259d9-a65a-431c-914a-75a728c39df0>\",\"WARC-IP-Address\":\"13.58.13.153\",\"WARC-Target-URI\":\"https://newpathworksheets.com/math/grade-8/linear-equations/virgin-islands-common-core-standards\",\"WARC-Payload-Digest\":\"sha1:YU5PS74EOAOUFPFSHGCD457EXOZXYLQE\",\"WARC-Block-Digest\":\"sha1:A72P64BTBWJ7SCLAR4WRR6VLNU7U46ZY\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-16/CC-MAIN-2020-16_segments_1585370506580.20_warc_CC-MAIN-20200402014600-20200402044600-00281.warc.gz\"}"}
http://sigmapedia.com/includes/term.cfm?word_id=2179&lang=eng	[ "", null, "# Normality Plot\n\nGo Back\n\nDefinition\n\nThis is the probability plot produced by statistical software when a normality test is performed. Perfectly normal data would form a straight line on the probability plot. Typically, a normality test such as the Anderson-Darling Normality Test statistic is shown on the graph to aid in its interpretation.\n\nApplication\n\nThe image shows normal probability plots for three different datasets. The one on the left comes from an Uniform distribution, the one in the center comes from a Normal distribution, and the one on the right comes from an Exponential distribution.\n\nAlthough there are more sophisticated ways of telling whether or not a sample comes from a normal population, a common informal test is the ‘Fat Pencil Test’. Take the normal probability plot and lay an imaginary fat pencil across the plotted data. If the pencil covers all the data points, the data are most likely from a normal distribution.", null, "" ]	[ null, "https://media.moresteam.com/main/pics/sigmapedia.png", null, "http://sigmapedia.com/sigmapedia/pics/norm_2.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9006504,"math_prob":0.9419583,"size":920,"snap":"2020-24-2020-29","text_gpt3_token_len":175,"char_repetition_ratio":0.14737992,"word_repetition_ratio":0.0,"special_character_ratio":0.17934783,"punctuation_ratio":0.07926829,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99067926,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,null,null,5,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-06-05T14:32:52Z\",\"WARC-Record-ID\":\"<urn:uuid:dfab41cd-5caa-4e8c-a240-f9f472f4148d>\",\"Content-Length\":\"19043\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:3ff214f2-b922-40b1-ba8d-65f3a04be992>\",\"WARC-Concurrent-To\":\"<urn:uuid:4c7e9368-b1a1-400e-919e-be11b28014fd>\",\"WARC-IP-Address\":\"52.176.89.191\",\"WARC-Target-URI\":\"http://sigmapedia.com/includes/term.cfm?word_id=2179&lang=eng\",\"WARC-Payload-Digest\":\"sha1:XQS67B7WNUNZYKYO2FRQTYEIRCEOWX6K\",\"WARC-Block-Digest\":\"sha1:53TNGMTVVP6CK4GU7DH2YGECEL7BXBEN\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-24/CC-MAIN-2020-24_segments_1590348502097.77_warc_CC-MAIN-20200605143036-20200605173036-00051.warc.gz\"}"}
https://www.physicsforums.com/threads/billiard-balls-elastic-collision.311193/	[ "# Billiard Balls-Elastic Collision\n\nBilliard Balls--Elastic Collision\n\nI'm going through my old homework problems to prepare for the AP exam, and I got this right back in December, but now I have no idea how to solve it\n\n## Homework Statement\n\nA cue ball traveling at 7.0 m/s makes a glancing, elastic collision with a target ball of equal mass that is initially at rest. The cue ball is deflected so that it makes an angle of 30° with its original direction of travel. Find:\na) the angle between the velocity vectors of the two balls after the collision\nb) the speed of each ball after the collision\n\nKE = 1/2mv2\np = mv\n\n## The Attempt at a Solution\n\nThe collision is elastic, so both kinetic energy and momentum are conserved, so\n\n1/2mv02 = 1/2mv12 + 1/2 mv22\nv02 = v12 + v22\n\nand\n\nmv0 = mv1 + mv2\nv0 = v1 + v2\n\nThe x component of the cue ball's final velocity is v1cos30\nThe x component of the target ball's final velocity is v2cosΘ\n\nthis gives me two equations, and three unknowns:\n\n49 = (v1cos30)2 + (v2cosΘ)2\n\n7 = v1cos30 + v2cosΘ\n\nAm I missing something that will tell me the angle of the velocity of the second ball? Thanks\n\n## The Attempt at a Solution\n\nRelated Introductory Physics Homework Help News on Phys.org\n\nFor KE, you use the total magnitude of the velocity and don't break it up into x components. Also, the third equation you might be looking for is the y-component of momentum. Three equations and three unknowns is solvable.\n\nOkay, so then my equations should be:\n\n49 = v12 + v22\n7 = v1cos30 + v2cosΘ\n\nand, since i feel like the y components of momentum ought to cancel out for some reason,\n\n0 = v1sin30 - v2sinΘ\n\nThe algebra is getting pretty thorny...\n\nThe y-components of the momentum are correct, since there is no initial y-momentum. The equations do look correct, although it is pretty late and I might be forgetting something, but lets hope not. Now you need to wade through some messy algebra, and definitely check your final answers when you are done.\n\nAndrew Mason\nHomework Helper\n\nI\nThe collision is elastic, so both kinetic energy and momentum are conserved, so\n\n1/2mv02 = 1/2mv12 + 1/2 mv22\nv02 = v12 + v22\nOk.\n\nand\n\nmv0 = mv1 + mv2\nv0 = v1 + v2\nThis is correct only if these are vectors.\n\n$$\\vec v_0 = \\vec v_1 + \\vec v_2$$\n\nDraw the vector triangle made by these velocity vectors.\n\nSince\n\n$$v_0^2 = v_1^2 + v_2^2$$\n\nwhat kind of triangle is it?\n\nAM\n\nOk.\n\nThis is correct only if these are vectors.\n\n$$\\vec v_0 = \\vec v_1 + \\vec v_2$$\n\nDraw the vector triangle made by these velocity vectors.\n\nSince\n\n$$v_0^2 = v_1^2 + v_2^2$$\n\nwhat kind of triangle is it?\n\nAM\n\ni think the answer you wanted me to get for the triangle question is that it's a right triangle, probably because I'm supposed to recognize the Pythagorean theorem? Didn't see that, maybe next time.\n\nAnyway, I slogged through the algebra and got the correct answer. I'll remember to look for that in the future though.\n\nThanks!\n\nAndrew Mason\nHomework Helper\n\ni think the answer you wanted me to get for the triangle question is that it's a right triangle, probably because I'm supposed to recognize the Pythagorean theorem? Didn't see that, maybe next time.\n\nAnyway, I slogged through the algebra and got the correct answer. I'll remember to look for that in the future though.\n\nThanks!\nIn an elastic collision between two objects of equal mass at an oblique angle, the angle between the directions after the collision is always 90 degrees. That makes it very easy to solve this problem.\n\nYou can observe this all the time in curling and billiards (except that in billiards the cue ball spin can dramatically change how the cue ball moves after collision).\n\nAM" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9012801,"math_prob":0.9860921,"size":2110,"snap":"2020-10-2020-16","text_gpt3_token_len":621,"char_repetition_ratio":0.11728395,"word_repetition_ratio":0.8856448,"special_character_ratio":0.28625593,"punctuation_ratio":0.056872036,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9988631,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-02-26T05:05:31Z\",\"WARC-Record-ID\":\"<urn:uuid:c3eb5453-b7f4-40c0-ab56-c2e3ec0b3e5c>\",\"Content-Length\":\"88664\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7de71a1f-73e7-4209-b22e-2dfc0403a950>\",\"WARC-Concurrent-To\":\"<urn:uuid:e92e778f-bb8b-4e6c-a801-24c8054c3dd7>\",\"WARC-IP-Address\":\"23.111.143.85\",\"WARC-Target-URI\":\"https://www.physicsforums.com/threads/billiard-balls-elastic-collision.311193/\",\"WARC-Payload-Digest\":\"sha1:YDRCRHC6ZTKUQEL4IQUS45JXPUF57HXW\",\"WARC-Block-Digest\":\"sha1:YYXUFKTOYOMI3M2A5HJLXH3VTYPWC5TF\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-10/CC-MAIN-2020-10_segments_1581875146186.62_warc_CC-MAIN-20200226023658-20200226053658-00046.warc.gz\"}"}
https://www.tutorialfor.com/questions-324080.htm	[ "Home>\n\n### python - meaning of dlib face detection function arguments\n\nReference site\n\nI have a face detection&stamp program with reference to the above site, but I had one question.\nI didn't understand it even after checking it, so I asked a question.\n\nI don't understand the meaning of the number in the second argument of dlib.get_frontal_face_detector ().\nCould you tell me?\n\n(Hereafter detector.py in the site)\n\n``````#coding: utf-8\nimport json\nimport math\nimport os\nimport cv2\nimport dlib\n\ndef main ():\n## input path and read image\nimg_path = \"../data/img/\" + os.listdir (\"../ data/img /\") \nimg_filename = os.path.basename (img_path)\nimg_ext = img_filename.split (\".\") [-1]\nimg = cv2.imread (img_path, cv2.IMREAD_COLOR)\nif img is None:\nprint (\"Could not read input image\")\nexit ()\n\n## instance detector/predictor\ndetector = dlib.get_frontal_face_detector ()\npredictor = dlib.shape_predictor (\"predictor/shape_predictor_68_face_landmarks.dat\")\n\n## detect faces\n## I don't understand the meaning of the second argument below! !! !! !!\nfaces = detector (img, 1)\nif len (faces) == 0:\nprint (\"Could not detect any faces\")\nexit ()\n\n## compute face properties\nface_properties = {}\nfor i, f in enumerate (faces):\n## get 68 points\npoints = predictor (img, f) .parts ()\n## Left/Right temples\nt_L = points \nt_R = points \n## calc properties\nface_width = math.sqrt ((t_L.x --t_R.x) 2 + (t_L.y --t_R.y) 2)\nface_center = [(t_L.x + t_R.x)/2, (t_L.y + t_L.y)/2]\ndegree_acw = -1 * math.degrees (math.atan2 ((t_R.y --t_L.y), (t_R.x --t_L.x)))\n## add result\nface_properties [i] = {\n\"stamp\": \"01.png\",\n\"center\": face_center,\n\"width\": face_width,\n\"angle\": degree_acw\n}\n## save\nwith open (\"../ data/json /\" + img_filename.replace (img_ext, \"json\"), \"w\") as f:\njson.dump (face_properties, f, ensure_ascii = False, indent = 4, separators = (',',':'))\n\nif __name__ == \"__main__\":\nmain ()``````\n• Answer # 1\n\nIt is a parameter called upsample_num_times.\nIt seems that the original image is upsampled that number of times before it is applied to the detector.\nhttp://dlib.net/python/index.html\n\nThe higher the number, the smaller the area that can be detected, but the larger the processing time and memory required.\nThe value of the default parameter is 0, and the smallest area detected at that time is 80x80. It seems that 1 is 40x40 and 2 is 20x20.\nhttps://qiita.com/nonbiri15/items/9561c8194ba0b2041bd0 # About the minimum detection size of face detection with dlib" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7421677,"math_prob":0.92407346,"size":4641,"snap":"2021-21-2021-25","text_gpt3_token_len":1175,"char_repetition_ratio":0.13219754,"word_repetition_ratio":0.03307888,"special_character_ratio":0.2783883,"punctuation_ratio":0.12244898,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98589295,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-12T20:11:51Z\",\"WARC-Record-ID\":\"<urn:uuid:3f6106e4-264e-4a75-8489-39a28d898751>\",\"Content-Length\":\"15777\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:340a3a58-b372-435f-9e2b-1aa551c83c65>\",\"WARC-Concurrent-To\":\"<urn:uuid:1207ef14-7cd8-4199-9883-d696d2378d99>\",\"WARC-IP-Address\":\"172.67.222.192\",\"WARC-Target-URI\":\"https://www.tutorialfor.com/questions-324080.htm\",\"WARC-Payload-Digest\":\"sha1:B2E4ZDQEZTDIJUBUXZ3MRJ3JNVZY5AU7\",\"WARC-Block-Digest\":\"sha1:A4YBFBK7N5WROHEMOM6TM5BBSOPFH434\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623487586390.4_warc_CC-MAIN-20210612193058-20210612223058-00190.warc.gz\"}"}
https://www.homeandlearn.co.uk/php/php6p8.html	[ "Home and Learn: PHP Programming Course\n\n# The PHP count function\n\nThe count( ) function is useful when you want to return how many elements are in your array. You can then use this in a for loop. Here's an example we used earlier, only this time with the count function:\n\n\\$seasons = array(\"Autumn\", \"Winter\", \"Spring\", \"Summer\");\n\n\\$array_count = count(\\$seasons);\n\nfor (\\$key_Number = 0; \\$key_Number < \\$array_count; \\$key_Number++) {\n\nprint \\$seasons[\\$key_Number];\n\n}\n\nTo get how many elements are in the array, we used this:\n\n\\$array_count = count(\\$seasons);\n\nSo you type the word count and then the round brackets. In between the round brackets, you type the name of your array. The function then counts how many elements are in the array, which we then assign to a variable called \\$array_count. You can then use this value as the end condition in you loop:\n\nfor (\\$key_Number = 0; \\$key_Number < \\$array_count; \\$key_Number++)\n\nHere, we're saying, \"keep looping round as long as the value in \\$key_Number is less than the value in \\$array_count.\n\nTo round off this section on arrays, there are some script for you to try out in the next part.\n\nBack to the PHP Contents Page\n\nEmail us: enquiry at homeandlearn.co.uk" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8411462,"math_prob":0.93119437,"size":1185,"snap":"2023-40-2023-50","text_gpt3_token_len":290,"char_repetition_ratio":0.16257408,"word_repetition_ratio":0.085427135,"special_character_ratio":0.26413503,"punctuation_ratio":0.14035088,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9876578,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-02T17:40:51Z\",\"WARC-Record-ID\":\"<urn:uuid:c7e6e6c3-9d81-4ebf-a0a0-e5e47c79707a>\",\"Content-Length\":\"8564\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ed8d6915-8216-4a84-9f18-28c13d392a64>\",\"WARC-Concurrent-To\":\"<urn:uuid:2f35a077-0aa5-4b99-954d-7dee0528fac2>\",\"WARC-IP-Address\":\"62.182.20.50\",\"WARC-Target-URI\":\"https://www.homeandlearn.co.uk/php/php6p8.html\",\"WARC-Payload-Digest\":\"sha1:N3KIHDKAKN4UGF6B5JRINOPT7CQKZ43J\",\"WARC-Block-Digest\":\"sha1:3PESOKPWBEY5SH5TNWOQGNWXPA4AWXXF\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100448.65_warc_CC-MAIN-20231202172159-20231202202159-00106.warc.gz\"}"}
https://nforum.ncatlab.org/discussion/15134/a-noneditable-entry-double-derivation/	[ "# Start a new discussion\n\n## Not signed in\n\nWant to take part in these discussions? Sign in if you have an account, or apply for one below\n\n## Site Tag Cloud\n\nVanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.\n\n• CommentRowNumber1.\n• CommentAuthorzskoda\n• CommentTime7 days ago\n\nI tried to edit the entry double derivation and even if I write a single letter or none (!) it sends back an error\n\n500 Internal Server Error\n\nYour edit was blocked by spam filtering\n\nPlease report this on the nForum in the nLab Technical Matters category. Please give as precise details as you can as to what triggered the error.\n\nHere is the error:\n\nNo such file or directory @ rb_sysopen - page_content/submitted_edits/nlab/double_derivation\n\netc. If I try to click on “discuss” button it does NOT open the latest changes page but just links to front page of $n$Forum, not linked to a page.\n\n• CommentRowNumber2.\n• CommentAuthorzskoda\n• CommentTime7 days ago\nHere is the edited page which I did not succeed to send:\n\nGiven a _commutative_ ring $k$ and an associative $k$-algebra $A$ over $k$, the tensor product $A\\otimes_k A$ is equipped with two bimodule structures, \"outer\" and \"inner\". For the outer structure $a\\cdot_o(b\\otimes c)\\cdot_o d = a b\\otimes c d$ and for the inner $a\\cdot_i(b\\otimes c)\\cdot_i d = b d\\otimes a c$. The two bimodule structures mutually commute. A $k$-linear map $\\alpha\\in Hom_k(A,A\\otimes A)$ is called a __double derivation__ if it is also a map of $A$-bimodules with respect to the _outer_ bimodule structure ($\\alpha\\in A Mod A({}_A A_A,{}_A A\\otimes_k A_A)$); thus the $k$-module $Der(A,A\\otimes A)$ of all double derivations becomes an $A$-bimodule with respect to the _inner_ $A$-bimodule structure.\n\nThe tensor algebra $T_A Der(A,A\\otimes A)$ of the $A$-bimodule $Der(A,A\\otimes A)$ (which is the free monoid on $Der(A,A\\otimes A)$ in the monoidal category of $A$-bimodules) is a step in the definition of the deformed preprojective algebras of Bill Crawley-Boevey. A theorem of Van den Bergh says that for any associative $A$ the tensor algebra $T_A Der(A,A\\otimes A)$ has a canonical double Poisson bracket.\n\n* Michel Van den Bergh, _Double Poisson algebras_, Trans. Amer. Math. Soc. __360__ (2008) 5711--5769, [arXiv:math.AG/0410528](https://arXiv.org/abs/math/0410528)\n* Anne Pichereau, Geert Van de Weyer, _Double Poisson cohomology of path algebras of quivers_, J. Alg. __319__, 5 (2008) 2166--2208 [doi](https://doi.org/10.1016/j.jalgebra.2007.09.021)\n* Jorge A. Guccione, Juan J. Guccione, _A characterization of quiver algebras based on double derivations_, [arXiv:0807.1148](https://arxiv.org/abs/0807.1148)\n\ncategory: algebra\n• CommentRowNumber3.\n• CommentAuthorzskoda\n• CommentTime7 days ago\nCopying the content of double derivation to another pagename does not work: spam message appears instead of creating a page, including at my personal web.\n\nI created William Crawley-Boevey with redirect Bill Crawley-Boevey wanted at double derivation which still shows that Bill Crawley-Boevey does not exist.\n• CommentRowNumber4.\n• CommentAuthorUrs\n• CommentTime6 days ago\n\nHave been busy all day. Am forwarding this to the technical team now…\n\n1. The problem was that \"gucci\" was a pattern in the spam filter. I've removed it. You should now be able to submit your edit.\n• CommentRowNumber6.\n• CommentAuthorzskoda\n• CommentTime5 days ago\n\nUnfortunately, it still seem to persist if I leave “Guccione” so I temporarily wrote G uccione with a space. Then it accepts.\n\n• V. Ginzburg, T. Schedler, Differential operators and BV structures in noncommutative geometry, Sel. Math. New Ser. 16, 673–730 (2010) doi\n• CommentRowNumber7.\n• CommentAuthorUrs\n• CommentTime5 days ago\n\nOne can work around this problem by escaping some html characters. For example:\n\n Guccione\n\n\ngives\n\nGuccione\n\n2. Unfortunately, it still seem to persist if I leave “Guccione”\n\nRight, I forgot to actually update the deployment. Should be fixed now.\n\n• CommentRowNumber9.\n• CommentAuthorzskoda\n• CommentTime5 days ago\n\n7 thanks for the tip" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.89591026,"math_prob":0.6313333,"size":1265,"snap":"2022-40-2023-06","text_gpt3_token_len":307,"char_repetition_ratio":0.07930214,"word_repetition_ratio":0.06306306,"special_character_ratio":0.2229249,"punctuation_ratio":0.09881423,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96456826,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-09-26T00:03:15Z\",\"WARC-Record-ID\":\"<urn:uuid:f370b4d1-e9fb-493f-8e95-5e0ed0dcefa4>\",\"Content-Length\":\"55486\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f7d95606-3acf-41fe-a55a-0eca52a3c882>\",\"WARC-Concurrent-To\":\"<urn:uuid:4be271b0-eedd-4507-988e-ae9e8033ebd1>\",\"WARC-IP-Address\":\"172.67.177.13\",\"WARC-Target-URI\":\"https://nforum.ncatlab.org/discussion/15134/a-noneditable-entry-double-derivation/\",\"WARC-Payload-Digest\":\"sha1:EBWWKL5JJR5WMMHM3564T3HP3EHCLVI3\",\"WARC-Block-Digest\":\"sha1:RLY3RAX5L364FB5ZBYZ264PPZHMBCASS\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-40/CC-MAIN-2022-40_segments_1664030334620.49_warc_CC-MAIN-20220925225000-20220926015000-00424.warc.gz\"}"}
https://mruanova.medium.com/softmax-function-dfe8f223208e	[ "In mathematics, the softmax function, also known as softargmax or normalized exponential function, is a function that takes as input a vector z of K real numbers, and normalizes it into a probability distribution consisting of K probabilities proportional to the exponentials of the input numbers.\n\nThat is, prior to applying softmax, some vector components could be negative, or greater than one; and might not sum to 1; but after applying softmax, each component will be in the interval (0,1), and the components will add up to 1, so that they can be interpreted as probabilities.\n\nFurthermore, the larger input components will correspond to larger probabilities.\n\nSoftmax is often used in neural networks, to map the non-normalized output of a network to a probability distribution over predicted output classes.\n\nWritten by" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.90274817,"math_prob":0.98121697,"size":829,"snap":"2021-04-2021-17","text_gpt3_token_len":163,"char_repetition_ratio":0.13333334,"word_repetition_ratio":0.0,"special_character_ratio":0.18817852,"punctuation_ratio":0.125,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9975628,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-01-21T04:57:22Z\",\"WARC-Record-ID\":\"<urn:uuid:85746dd4-533b-433e-b6ac-0ee45b2fa67a>\",\"Content-Length\":\"118457\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:4dcf413b-0ae1-4788-beb9-ac6c39080396>\",\"WARC-Concurrent-To\":\"<urn:uuid:367caa78-d197-4996-b517-07a49663eecc>\",\"WARC-IP-Address\":\"104.17.31.52\",\"WARC-Target-URI\":\"https://mruanova.medium.com/softmax-function-dfe8f223208e\",\"WARC-Payload-Digest\":\"sha1:F6XTNWP75VBDJQIQJUP6FGWETO7ZATNG\",\"WARC-Block-Digest\":\"sha1:XP7E5QGQSZ6SYH54WT23G7G4U7XIGGVU\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-04/CC-MAIN-2021-04_segments_1610703522242.73_warc_CC-MAIN-20210121035242-20210121065242-00425.warc.gz\"}"}
https://advancesincontinuousanddiscretemodels.springeropen.com/articles/10.1186/s13662-019-2362-3	[ "Theory and Modern Applications\n\n# Some new inequalities for generalized fractional conformable integral operators\n\n## Abstract\n\nThe present paper aims to establish certain new classes of integral inequalities for a class of n ($$n\\in \\mathbb{N}$$) positive continuous and decreasing functions by utilizing the generalized fractional conformable integral operators (FCIO) recently defined by Khan and Khan. From these results, we also derive several particular cases.\n\n## 1 Introduction\n\nFractional calculus earned more recognition due to its applications in diverse domains. Recent research focuses on developing a large number of the fractional integral operators (FIO) and their applications in multiple disciplines of sciences (see [13, 14, 20, 26]). In , Liu et al. introduced interesting integral inequalities for continuous functions on $$[a,b]$$. Later on, Dahmani generalized the work of involving the Riemann–Liouville fractional integral operators. In , Dahmani and Tabharit introduced weighted Grüss type inequalities involving fractional integral operators. Dahmani established some new inequalities for fractional integrals. Polya–Szego and Chebyshev type inequalities involving the Riemann–Liouville fractional integral operators are found in . Nisar et al. established some inequalities involving extended gamma and the Kummar confluent hypergeometric k-functions. In , Set et al. established generalized Grüss type inequalities for k-fractional integrals and applications. Certain Gronwall inequalities associated with Riemann–Liouville k and Hadamard k-fractional derivatives and their applications are found in the work of Nisar et al. . The $$(k,s)$$-fractional integrals and their applications are found in . Rahman et al. presented certain inequalities via $$(k,\\rho )$$-fractional integral operators. In , the authors introduced the idea of fractional conformable derivative operators with a shortcoming that the new derivative operator does not tend to the original function when the order $$\\rho \\rightarrow 0$$. In , the author studied certain various properties of the fractional conformable derivative operators and raised the problem of how to use conformable derivative operators to generate more general types of nonlocal fractional derivative operators, after that the method was demonstrated in .\n\nThe generalized FCIO defined in is given by\n\n\\begin{aligned} {}_{\\alpha }^{\\mu }\\mathfrak{I}_{r^{+}}^{\\beta }f(x)= \\frac{1}{\\varGamma ( \\beta )} \\int _{r}^{x} \\biggl(\\frac{x^{\\alpha +\\mu }-\\tau ^{\\alpha + \\mu }}{\\alpha +\\mu } \\biggr)^{\\beta -1}\\frac{f(\\tau )}{ \\tau ^{1-\\alpha -\\mu }}\\,d\\tau ,\\quad x>r, \\end{aligned}\n(1)\n\nand\n\n\\begin{aligned} {}_{\\alpha }^{\\mu }\\mathfrak{I}_{s^{-}}^{\\beta }f(x)= \\frac{1}{\\varGamma ( \\beta )} \\int _{x}^{s} \\biggl(\\frac{\\tau ^{\\alpha +\\mu }-x^{\\alpha + \\mu }}{\\alpha +\\mu } \\biggr)^{\\beta -1}\\frac{f(\\tau )}{ \\tau ^{1-\\alpha -\\mu }}\\,d\\tau ,\\quad x< s, \\end{aligned}\n(2)\n\nwhere $$\\beta \\in \\mathbb{C}$$, $$\\Re (\\beta )>0$$, $$\\alpha \\in (0,1]$$, $$\\mu \\in \\mathbb{R}$$, $$\\alpha +\\mu \\neq 0$$, and Γ is the gamma function .\n\n### Remark 1\n\n(i) If we set $$\\mu =0$$ in (1) and (2), then we have the following Riemann–Liouville (R-L) type FCIO:\n\n\\begin{aligned} {}_{\\alpha }\\mathfrak{I}_{r^{+}}^{\\beta }f(x)= \\frac{1}{\\varGamma (\\beta )} \\int _{r}^{x} \\biggl(\\frac{x^{\\alpha }-\\tau ^{\\alpha }}{\\alpha } \\biggr) ^{\\beta -1}\\frac{f(\\tau )}{\\tau ^{1-\\alpha }}\\,d\\tau ,\\quad x>r, \\end{aligned}\n(3)\n\nand\n\n\\begin{aligned} {}_{\\alpha }\\mathfrak{I}_{s^{-}}^{\\beta }f(x)= \\frac{1}{\\varGamma (\\beta )} \\int _{x}^{s} \\biggl(\\frac{\\tau ^{\\alpha }-x^{\\alpha }}{\\alpha } \\biggr) ^{\\beta -1}\\frac{f(\\tau )}{\\tau ^{1-\\alpha }}\\,d\\tau ,\\quad x< s, \\end{aligned}\n(4)\n\nwhere $$\\beta \\in \\mathbb{C}$$, $$\\Re (\\beta )>0$$, $$\\alpha \\in (0,1]$$.\n\n(ii) If $$\\alpha =1$$ in 3 and 4, then we obtain the following R-L FIO:\n\n\\begin{aligned} \\mathfrak{I}_{r^{+}}^{\\beta }f(x)=\\frac{1}{\\varGamma (\\beta )} \\int _{r} ^{x} (x-\\tau )^{\\beta -1}f(\\tau )\\,d \\tau ,\\quad x>r, \\end{aligned}\n(5)\n\nand\n\n\\begin{aligned} \\mathfrak{I}_{s^{-}}^{\\beta }f(x)=\\frac{1}{\\varGamma (\\beta )} \\int _{x} ^{s} (\\tau -x )^{\\beta -1}f(\\tau )\\,d \\tau ,\\quad x< s, \\end{aligned}\n(6)\n\nwhere $$\\beta \\in \\mathbb{C}$$, $$\\Re (\\beta )>0$$.\n\nRecently, the researchers [21, 24] established inequalities of Grüss type and Čebyšev type by utilizing fractional conformable integral operators. Rahman et al. established certain Chebyshev type inequalities involving fractional conformable integral operators. In , the authors introduced the Minkowski inequalities via generalized proportional fractional integral operators. Some new inequalities involving fractional conformable integrals are found in the work of Nisar et al. . Adjabi et al. presented generalized fractional integral operators and Gronwall type inequalities with applications. In , Abdeljawad established a Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel. Abdeljawad et al. introduced Lyapunov type inequalities for mixed nonlinear forced differential equations within conformable derivatives. Fractional operators with exponential kernels and a Lyapunov type inequality are found in . Abdeljawad et al. presented a generalized Lyapunov type inequality in the frame of conformable derivatives.\n\nOur aim in this paper is to generalize the inequalities obtained earlier by [8, 15] by employing the left generalized fractional conformable integral operator (1).\n\n## 2 Main results\n\nIn this section, we employ the left generalized FCIO to establish the generalization of some classical inequalities.\n\n### Theorem 1\n\nLet $$(g_{i})_{i=1,2,3,\\ldots ,n}$$ be n positive continuous and decreasing functions on the interval $$[r,s]$$. Let $$r< x\\leq s$$, $$\\vartheta >0$$, $$\\sigma \\geq \\gamma _{p}>0$$ for any fixed $$p\\in \\{1,2,3, \\ldots ,n\\}$$. Then, for generalized fractional conformable integral (1), we have\n\n\\begin{aligned} \\frac{{}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\beta } [\\prod_{i \\neq p}^{n}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) ]}{{}_{\\alpha } ^{\\mu }\\mathfrak{I}_{r}^{\\beta } [\\prod_{i=1}^{n}g_{i}^{\\gamma_{i}}(x) ]}\\geq \\frac{{}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{ \\beta } [(x-r)^{\\vartheta }\\prod_{i\\neq p}^{n}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) ]}{{}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{ \\beta } [(x-r)^{\\vartheta }\\prod_{i=1}^{n}g_{i}^{\\gamma _{i}}(x) ]}, \\end{aligned}\n(7)\n\nwhere $$\\beta \\in \\mathbb{C}$$, $$\\alpha \\in (0,1]$$, $$\\mu \\in \\mathbb{R}$$, $$\\alpha +\\mu \\neq 0$$, and $$\\Re (\\beta )>0$$.\n\n### Proof\n\nSince $$(g_{i})_{i=1,2,3,\\ldots ,n}$$ are n positive continuous and decreasing functions on the interval $$[r,s]$$. Therefore, we have\n\n\\begin{aligned} \\bigl((\\rho -r)^{\\vartheta }-(t-r)^{\\vartheta } \\bigr) \\bigl(g_{p} ^{\\sigma -\\gamma _{p}}(t)-g_{p}^{\\sigma -\\gamma _{p}}(\\rho ) \\bigr) \\geq 0, \\end{aligned}\n(8)\n\nwhere $$r< x\\leq s$$, $$\\vartheta >0$$, $$\\sigma \\geq \\gamma _{p}>0$$, $$t, \\rho \\in [r,x]$$ and for any fixed $$p\\in \\{1,2,3,\\ldots ,n\\}$$.\n\nDefine a function\n\n\\begin{aligned} {}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\beta }(x,\\rho ,t) &=\\frac{1}{ \\varGamma (\\beta )} \\biggl(\\frac{x^{\\alpha +\\mu }-t^{\\alpha +\\mu }}{ \\alpha +\\mu } \\biggr)^{\\beta -1} \\\\ &\\quad {}\\times \\frac{\\prod_{i=1}^{n}g_{i}^{\\gamma _{i}}(t)}{t^{1-\\alpha - \\mu }} \\bigl((\\rho -r)^{\\vartheta }-(t-r)^{\\vartheta } \\bigr) \\bigl(g_{p}^{\\sigma -\\gamma _{p}}(t)-g_{p}^{\\sigma -\\gamma _{p}}( \\rho ) \\bigr). \\end{aligned}\n(9)\n\nWe observe that the above function satisfies all the assumptions stated in Theorem 1, and hence the function $${}_{\\alpha }^{\\mu } \\mathfrak{I}_{r}^{\\beta }(x,\\rho ,t)$$ is positive for all $$t\\in (r,x)$$ ($$x>r$$). Integrating both sides of (9) with respect to t over $$(r,x)$$, we have\n\n\\begin{aligned} 0&\\leq \\int _{r}^{x}{}_{\\alpha }^{\\mu } \\mathfrak{I}_{r}^{\\beta }(x, \\rho ,t)\\,dt \\\\ &=\\frac{1}{\\varGamma (\\beta )} \\int _{r}^{x} \\biggl(\\frac{x^{ \\alpha +\\mu }-t^{\\alpha +\\mu }}{\\alpha +\\mu } \\biggr)^{\\beta -1} \\\\ &\\quad {}\\times \\prod_{i=1}^{n}g_{i}^{\\gamma _{i}}(t) \\bigl((\\rho -r)^{\\vartheta }-(t-r)^{\\vartheta } \\bigr) \\bigl(g_{p}^{\\sigma -\\gamma _{p}}(t)-g_{p}^{\\sigma -\\gamma _{p}}( \\rho ) \\bigr)\\frac{dt}{t^{1-\\alpha -\\mu }} \\\\ &= \\Biggl[(\\rho -r)^{\\vartheta }{}_{\\alpha }^{\\mu } \\mathfrak{I}_{r} ^{\\beta }\\prod_{i\\neq p}^{n}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) \\Biggr]+g_{p}^{\\sigma -\\gamma _{p}}(\\rho ){}_{\\alpha }^{\\mu } \\mathfrak{I}_{r} ^{\\beta } \\Biggl[(x-r)^{\\vartheta }\\prod _{i=1}^{n}g_{i}^{\\gamma _{i}}(x) \\Biggr] \\\\ &\\quad {}- (\\rho -r)^{\\vartheta }g_{p}^{\\sigma -\\gamma _{p}}(\\rho ) \\Biggl[{}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\beta } \\prod_{i=1 }^{n}g_{i}^{ \\gamma _{i}}(x) \\Biggr]-{}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\beta } \\Biggl[(x-r)^{\\vartheta }\\prod_{i=1}^{n}g_{i}^{\\gamma _{i}}(x) \\Biggr]. \\end{aligned}\n(10)\n\nMultiplying (10) by $$\\frac{1}{\\varGamma (\\beta )} (\\frac{x ^{\\alpha +\\mu }-\\rho ^{\\alpha +\\mu }}{\\alpha +\\mu } )^{\\beta -1}\\frac{ \\prod_{i=1}^{n}g_{i}^{\\gamma _{i}}(\\rho )}{\\rho ^{1-\\alpha -\\mu }}$$ and integrating the resultant identity with respect to ρ over $$(r,x)$$, we have\n\n\\begin{aligned} 0&\\leq \\Biggl[{}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\beta } \\prod_{i \\neq p}^{n}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) \\Biggr]{}_{\\alpha } ^{\\mu }\\mathfrak{I}_{r}^{\\beta } \\Biggl[(x-r)^{\\vartheta }\\prod_{i=1} ^{n}g_{i}^{\\gamma _{i}}(x) \\Biggr] \\\\ &\\quad {}-{}_{\\alpha }^{\\mu } \\mathfrak{I}_{r} ^{\\beta } \\Biggl[(x-r)^{\\vartheta }\\prod_{i\\neq p}^{n}g_{i}^{\\gamma_{i}}g_{p}^{\\sigma }(x) \\Biggr] \\Biggl[{}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\beta } \\prod_{i=1 }^{n}g_{i}^{\\gamma _{i}}(x) \\Biggr], \\end{aligned}\n\nwhich completes the desired inequality (7). □\n\n### Corollary 1\n\nLet $$(g_{i})_{i=1,2,3,\\ldots ,n}$$ be n positive continuous and decreasing on $$[r,s]$$. Let $$r< x\\leq s$$, $$\\vartheta >0$$, $$\\sigma \\geq \\gamma _{p}>0$$ for any fixed $$p\\in \\{1,2,3,\\ldots ,n\\}$$. Then, for R-L type fractional conformable integral (3), we have\n\n\\begin{aligned} \\frac{{}_{\\alpha }\\mathfrak{I}_{r}^{\\beta } [\\prod_{i\\neq p}^{n}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) ]}{{}_{\\alpha }\\mathfrak{I}_{r}^{\\beta } [\\prod_{i=1}^{n}g_{i}^{\\gamma _{i}}(x) ]} \\geq \\frac{{}_{\\alpha }\\mathfrak{I}_{r}^{\\beta } [(x-r)^{\\vartheta }\\prod_{i\\neq p}^{n}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) ]}{{}_{\\alpha }\\mathfrak{I}_{r}^{\\beta } [(x-r)^{\\vartheta }\\prod_{i=1} ^{n}g_{i}^{\\gamma _{i}}(x) ]}, \\end{aligned}\n(11)\n\nwhere $$\\beta \\in \\mathbb{C}$$, $$\\alpha \\in (0,1]$$, and $$\\Re (\\beta )>0$$.\n\n### Corollary 2\n\nLet $$(g_{i})_{i=1,2,3,\\ldots ,n}$$ be n positive continuous and decreasing on $$[r,s]$$. Let $$r< x\\leq s$$, $$\\vartheta >0$$, $$\\sigma \\geq \\gamma _{p}>0$$ for any fixed $$p\\in \\{1,2,3,\\ldots ,n\\}$$. Then, for R-L fractional integral (5), we have\n\n\\begin{aligned} \\frac{\\mathfrak{I}_{r}^{\\beta } [\\prod_{i\\neq p}^{n}g_{i}^{\\gamma_{i}}g_{p}^{\\sigma }(x) ]}{\\mathfrak{I}_{r}^{\\beta } [\\prod_{i=1} ^{n}g_{i}^{\\gamma _{i}}(x) ]}\\geq \\frac{\\mathfrak{I}_{r}^{\\beta } [(x-r)^{\\vartheta }\\prod_{i\\neq p}^{n}g_{i}^{\\gamma _{i}}g_{p} ^{\\sigma }(x) ]}{\\mathfrak{I}_{r}^{\\beta } [(x-r)^{\\vartheta }\\prod_{i=1}^{n}g_{i}^{\\gamma _{i}}(x) ]}, \\end{aligned}\n(12)\n\nwhere $$\\beta \\in \\mathbb{C}$$ and $$\\Re (\\beta )>0$$.\n\n### Remark 2\n\nThe inequality in Theorem 1 will reverse if $$(g_{i})_{i=1,2,3, \\ldots ,n}$$ are increasing on the interval $$[r,s]$$. If we let $$\\alpha =1$$, $$\\mu =0$$, then Theorem 1 will lead to Theorem 3.1 . Moreover, setting $$\\mu =0$$, $$\\alpha =\\beta =n=1$$, $$x=s$$, then Theorem 1 reduces to the well-known Theorem 3 .\n\n### Theorem 2\n\nLet $$(g_{i})_{i=1,2,3,\\ldots ,n}$$ be n positive continuous and decreasing on $$[r,s]$$. Let $$r< x\\leq s$$, $$\\vartheta >0$$, $$\\sigma \\geq \\gamma _{p}>0$$ for any fixed $$p\\in \\{1,2,3,\\ldots ,n\\}$$. Then, for generalized fractional conformable integral (1), we have\n\n\\begin{aligned} &\\frac{{}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\beta } [{ \\prod_{i\\neq p}^{n}}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) ]{}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\lambda } [(x-r)^{\\vartheta } { \\prod_{i=1}^{n}} g_{i}^{\\gamma _{i}}(x) ] +{}_{\\alpha }^{\\mu } \\mathfrak{I}_{r}^{\\beta } [{ \\prod_{i\\neq p}^{n}}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) ]{}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\lambda } [(x-r)^{\\vartheta } { \\prod_{i=1}^{n}}g_{i}^{\\gamma _{i}}(x) ]}{{}_{\\alpha }^{\\mu } \\mathfrak{I}_{r}^{\\lambda } [(x-r)^{\\vartheta }{ \\prod_{i\\neq p}^{n}}g_{i}^{\\gamma _{i}}(x) ]{}_{\\alpha }^{\\mu } \\mathfrak{I}_{r}^{\\beta } [{ \\prod_{i=1}^{n}}g_{i}^{\\gamma _{i}}(x) ]+ {}_{\\alpha }^{\\mu } \\mathfrak{I}_{r}^{\\beta } [(x-r)^{\\vartheta }{ \\prod_{i\\neq p}^{n}}g_{i}^{\\gamma _{i}}(x) ] {}_{\\alpha }^{ \\mu }\\mathfrak{I}_{r}^{\\lambda } [{ \\prod_{i=1}^{n}}g_{i}^{\\gamma _{i}}(x) ]} \\\\ &\\quad \\geq 1, \\end{aligned}\n(13)\n\nwhere $$\\beta ,\\lambda \\in \\mathbb{C}$$, $$\\alpha \\in (0,1]$$, $$\\mu \\in \\mathbb{R}$$, $$\\alpha +\\mu \\neq 0$$, $$\\Re (\\beta )>0$$, and $$\\Re (\\lambda )>0$$.\n\n### Proof\n\nFirstly, multiplying both sides of equation (10) by $$\\frac{1}{\\varGamma (\\lambda )} (\\frac{x^{\\alpha +\\mu }- \\rho ^{\\alpha +\\mu }}{\\alpha +\\mu } )^{\\lambda -1}\\frac{\\prod_{i=1}^{n}g_{i}^{\\gamma _{i}}(\\rho )}{\\rho ^{1-\\alpha -\\mu }}$$ and integrating the resultant identity with respect to ρ over $$(r,x)$$, we have\n\n\\begin{aligned} 0&\\leq \\int _{r}^{x} \\int _{r}^{x}\\frac{1}{\\varGamma (\\lambda )} \\biggl( \\frac{x ^{\\alpha +\\mu }-\\rho ^{\\alpha +\\mu }}{\\alpha +\\mu } \\biggr)^{\\lambda -1}\\frac{ \\prod_{i=1}^{n}g_{i}^{\\gamma _{i}}(\\rho )}{\\rho ^{1-\\alpha -\\mu }} {}_{\\alpha }^{\\mu } \\mathfrak{I}_{r}^{\\beta }(x,\\rho ,t)\\,dt\\,d\\rho \\\\ &= {}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\beta } \\Biggl[\\prod_{i\\neq p} ^{n}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) \\Biggr]{}_{\\alpha }^{\\mu } \\mathfrak{I}_{r}^{\\lambda } \\Biggl[(x-r)^{\\vartheta }\\prod_{i=1}^{n}g_{i}^{\\gamma _{i}}(x) \\Biggr] \\\\ &\\quad {}+{}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{ \\lambda } \\Biggl[\\prod_{i\\neq p}^{n}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) \\Biggr] {}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\beta } \\Biggl[(x-r)^{\\vartheta } \\prod_{i=1}^{n}g_{i}^{\\gamma _{i}}(x) \\Biggr] \\\\ &\\quad {}- {}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\beta } \\Biggl[(x-r)^{\\vartheta }\\prod_{i\\neq p}^{n}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) \\Biggr]{}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\lambda } \\Biggl[ \\prod_{i=1}^{n}g_{i}^{\\gamma _{i}}(x) \\Biggr] \\\\ &\\quad {}-{}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{ \\lambda } \\Biggl[(x-r)^{\\vartheta }\\prod_{i\\neq p}^{n}g_{i}^{\\gamma_{i}}g_{p}^{\\sigma }(x) \\Biggr]{}_{\\alpha }^{\\mu }\\mathfrak{I}_{r} ^{\\beta } \\Biggl[ \\prod_{i=1}^{n}g_{i}^{\\gamma _{i}}(x) \\Biggr]. \\end{aligned}\n(14)\n\nHence, dividing both sides of (14) by\n\n$${}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\beta } \\Biggl[(x-r)^{\\vartheta } \\prod_{i\\neq p}^{n}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) \\Biggr]{}_{ \\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\lambda } \\Biggl[ \\prod_{i=1}^{n}g_{i}^{\\gamma _{i}}(x) \\Biggr]+{}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{ \\lambda } \\Biggl[(x-r)^{\\vartheta }\\prod_{i\\neq p}^{n}g_{i}^{\\gamma_{i}}g_{p}^{\\sigma }(x) \\Biggr]{}_{\\alpha }^{\\mu }\\mathfrak{I}_{r} ^{\\beta } \\Biggl[ \\prod_{i=1}^{n}g_{i}^{\\gamma _{i}}(x) \\Biggr],$$\n\nwe get the desired proof. □\n\n### Corollary 3\n\nLet $$(g_{i})_{i=1,2,3,\\ldots ,n}$$ be n positive continuous and decreasing on $$[r,s]$$. Let $$r< x\\leq s$$, $$\\vartheta >0$$, $$\\sigma \\geq \\gamma _{p}>0$$ for any fixed $$p\\in \\{1,2,3,\\ldots ,n\\}$$. Then, for R-L type fractional conformable integral (3), we have\n\n\\begin{aligned} &\\frac{{}_{\\alpha }\\mathfrak{I}_{r}^{\\beta } [{ \\prod_{i\\neq p}^{n}}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) ]{}_{\\alpha }\\mathfrak{I}_{r}^{\\lambda } [(x-r)^{\\vartheta }{ \\prod_{i=1}^{n}}g_{i}^{\\gamma _{i}}(x) ] +{}_{\\alpha } \\mathfrak{I}_{r}^{\\beta } [{ \\prod_{i\\neq p}^{n}}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) ]{}_{\\alpha }\\mathfrak{I}_{r}^{\\lambda } [(x-r)^{\\vartheta }{ \\prod_{i=1}^{n}}g_{i}^{\\gamma _{i}}(x) ]}{{}_{\\alpha } \\mathfrak{I}_{r}^{\\lambda } [(x-r)^{\\vartheta }{ \\prod_{i\\neq p}^{n}}g_{i}^{\\gamma _{i}}(x) ]{}_{\\alpha } \\mathfrak{I}_{r}^{\\beta } [{ \\prod_{i=1}^{n}}g_{i}^{\\gamma _{i}}(x) ]+ {}_{\\alpha } \\mathfrak{I}_{r}^{\\beta } [(x-r)^{\\vartheta }{ \\prod_{i\\neq p}^{n}}g_{i}^{\\gamma _{i}}(x) ] {}_{\\alpha } \\mathfrak{I}_{r}^{\\lambda } [{ \\prod_{i=1}^{n}}g_{i}^{\\gamma _{i}}(x) ]} \\\\ &\\quad \\geq 1, \\end{aligned}\n(15)\n\nwhere $$\\beta ,\\lambda \\in \\mathbb{C}$$, $$\\alpha \\in (0,1]$$, $$\\Re ( \\beta )>0$$, and $$\\Re (\\lambda )>0$$.\n\n### Corollary 4\n\nLet $$(g_{i})_{i=1,2,3,\\ldots ,n}$$ be n positive continuous and decreasing on $$[r,s]$$. Let $$r< x\\leq s$$, $$\\vartheta >0$$, $$\\sigma \\geq \\gamma _{p}>0$$ for any fixed $$p\\in \\{1,2,3,\\ldots ,n\\}$$. Then, for R-L fractional integral (5), we have\n\n\\begin{aligned} &\\frac{\\mathfrak{I}_{r}^{\\beta } [{ \\prod_{i\\neq p}^{n}}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) ] \\mathfrak{I}_{r}^{\\lambda } [(x-r)^{\\vartheta }{ \\prod_{i=1}^{n}}g_{i}^{\\gamma _{i}}(x) ] +\\mathfrak{I}_{r}^{ \\beta } [{ \\prod_{i\\neq p}^{n}}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) ] \\mathfrak{I}_{r}^{\\lambda } [(x-r)^{\\vartheta }{ \\prod_{i=1}^{n}}g_{i}^{\\gamma _{i}}(x) ]}{\\mathfrak{I}_{r}^{ \\lambda } [(x-r)^{\\vartheta }{ \\prod_{i\\neq p}^{n}}g_{i}^{\\gamma _{i}}(x) ]\\mathfrak{I}_{r}^{ \\beta } [{ \\prod_{i=1}^{n}}g_{i}^{\\gamma _{p}}(x) ]+ \\mathfrak{I}_{r}^{ \\beta } [(x-r)^{\\vartheta }{ \\prod_{i\\neq p}^{n}}g_{i}^{\\gamma _{i}}(x) ] \\mathfrak{I}_{r} ^{\\lambda } [{ \\prod_{i=1}^{n}}g_{i}^{\\gamma _{i}}(x) ]} \\\\ &\\quad \\geq 1, \\end{aligned}\n(16)\n\nwhere $$\\beta ,\\lambda \\in \\mathbb{C}$$, $$\\Re (\\beta )>0$$, and $$\\Re (\\lambda )>0$$.\n\n### Remark 3\n\nApplying Theorem 2 for $$\\beta =\\lambda$$, we get Theorem 1. Again, the inequality will reverse if $$(g_{i})_{i=1,2,3, \\ldots ,n}$$ are increasing functions on the interval $$[r,s]$$. If we let $$\\alpha =1$$, $$\\mu =0$$, then Theorem 1 will lead to Theorem 3.4 . Moreover, setting $$\\mu =0$$, $$\\alpha =\\beta = \\lambda =n=1$$, $$x=s$$, then again Theorem 1 reduces to the well-known Theorem 3 .\n\n### Theorem 3\n\nLet $$(g_{i})_{i=1,2,3,\\ldots ,n}$$ and h be positive continuous on the interval $$[r,s]$$ such that h is increasing and $$(g_{i})_{i=1,2,3, \\ldots ,n}$$ are decreasing functions on $$[r,s]$$. Let $$r< x\\leq s$$, $$\\vartheta >0$$, $$\\sigma \\geq \\gamma _{p}>0$$ for any fixed $$p\\in \\{1,2,3, \\ldots ,n\\}$$. Then, for generalized fractional conformable integral (1), we have\n\n\\begin{aligned} \\frac{{}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\beta } [\\prod_{i \\neq p}^{n}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) ]{}_{\\alpha } ^{\\mu }\\mathfrak{I}_{r}^{\\beta } [h^{\\vartheta }(x) \\prod_{i=1} ^{n}g_{i}^{\\gamma _{i}}(x) ]}{{}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\beta } [h^{\\vartheta }(x)\\prod_{i\\neq p}^{n}g_{i}^{\\gamma_{i}}g_{p}^{\\sigma }(x) ]{}_{\\alpha }^{\\mu }\\mathfrak{I}_{r} ^{\\beta } [ \\prod_{i=1}^{n}g_{i}^{\\gamma _{i}}(x) ]}\\geq 1, \\end{aligned}\n(17)\n\nwhere $$\\beta \\in \\mathbb{C}$$, $$\\alpha \\in (0,1]$$, $$\\mu \\in \\mathbb{R}$$, $$\\alpha +\\mu \\neq 0$$, and $$\\Re (\\beta )>0$$.\n\n### Proof\n\nUnder the conditions stated in Theorem 3, we can write\n\n\\begin{aligned} \\bigl(h^{\\vartheta }(\\rho )-h^{\\vartheta }(t) \\bigr) \\bigl(g_{p} ^{\\sigma -\\gamma _{p}}(t)-g_{p}^{\\sigma -\\gamma _{p}}(\\rho ) \\bigr) \\geq 0, \\end{aligned}\n(18)\n\nwhere $$r< x\\leq s$$, $$\\vartheta >0$$, $$\\sigma \\geq \\gamma _{p}>0$$, $$t, \\rho \\in [r,x]$$ and for any fixed $$p\\in \\{1,2,3,\\ldots ,n\\}$$.\n\nDefine a function\n\n\\begin{aligned} {}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\beta }(x,\\rho ,t) &=\\frac{1}{ \\varGamma (\\beta )} \\biggl(\\frac{x^{\\alpha +\\mu }-t^{\\alpha +\\mu }}{ \\alpha +\\mu } \\biggr)^{\\beta -1} \\\\ &\\quad {}\\times \\frac{\\prod_{i=1}^{n}g_{i}^{\\gamma _{i}}(t)}{t^{1-\\alpha - \\mu }} \\bigl(h^{\\vartheta }(\\rho )-h^{\\vartheta }(t) \\bigr) \\bigl(g_{p}^{\\sigma -\\gamma _{p}}(t)-g_{p}^{\\sigma -\\gamma _{p}}( \\rho ) \\bigr). \\end{aligned}\n(19)\n\nWe observe that the above function satisfies all the assumptions stated in Theorem 3, and hence the function $${}_{\\alpha }^{\\mu } \\mathfrak{I}_{r}^{\\beta }(x,\\rho ,t)$$ is positive for all $$t\\in (r,x)$$ ($$x>r$$). Therefore, integrating both sides of (19) with respect to t over $$(r,x)$$, we have\n\n\\begin{aligned} 0&\\leq \\int _{r}^{x}{}_{\\alpha }^{\\mu } \\mathfrak{I}_{r}^{\\beta }(x, \\rho ,t)\\,dt \\\\ &=\\frac{1}{\\varGamma (\\beta )} \\int _{r}^{x} \\biggl(\\frac{x^{ \\alpha +\\mu }-t^{\\alpha +\\mu }}{\\alpha +\\mu } \\biggr)^{\\beta -1} \\\\ &\\quad {}\\times \\prod_{i=1}^{n}g_{i}^{\\gamma _{i}}(t) \\bigl(h^{\\vartheta }( \\rho )-h^{\\vartheta }(t) \\bigr) \\bigl(g_{p}^{\\sigma -\\gamma _{p}}(t)-g_{p}^{\\sigma -\\gamma _{p}}( \\rho ) \\bigr)\\frac{dt}{t^{1-\\alpha -\\mu }} \\\\ &= \\Biggl[h(\\rho )^{\\vartheta }{}_{\\alpha }^{\\mu } \\mathfrak{I}_{r} ^{\\beta }\\prod_{i\\neq p}^{n}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) \\Biggr]+g_{p}^{\\sigma -\\gamma _{p}}(\\rho ){}_{\\alpha }^{\\mu } \\mathfrak{I}_{r} ^{\\beta } \\Biggl[h^{\\vartheta }(x) \\prod _{i=1}^{n}g_{i}^{\\gamma _{i}}(x) \\Biggr] \\\\ &\\quad {}- h^{\\vartheta }(\\rho ) g_{p}^{\\sigma -\\gamma _{p}}(\\rho ) \\Biggl[{}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\beta } \\prod_{i=1 }^{n}g_{i}^{ \\gamma _{i}}(x) \\Biggr]-{}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\beta } \\Biggl[h^{\\vartheta }(x) \\prod_{i=1}^{n}g_{i}^{\\gamma _{i}}(x) \\Biggr]. \\end{aligned}\n(20)\n\nMultiplying (20) by $$\\frac{1}{\\varGamma (\\beta )} (\\frac{x ^{\\alpha +\\mu }-\\rho ^{\\alpha +\\mu }}{\\alpha +\\mu } )^{\\beta -1}\\frac{ \\prod_{i=1}^{n}g_{i}^{\\gamma _{i}}(\\rho )}{\\rho ^{1-\\alpha -\\mu }}$$ and integrating the resultant identity with respect to ρ over $$(r,x)$$, we have\n\n\\begin{aligned} 0&\\leq \\Biggl[{}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\beta } \\prod_{i \\neq p}^{n}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) \\Biggr]{}_{\\alpha } ^{\\mu }\\mathfrak{I}_{r}^{\\beta } \\Biggl[h^{\\vartheta }(x) \\prod_{i=1} ^{n}g_{i}^{\\gamma _{i}}(x) \\Biggr] \\\\ &\\quad {}-{}_{\\alpha }^{\\mu } \\mathfrak{I}_{r} ^{\\beta } \\Biggl[h^{\\vartheta }(x)\\prod_{i\\neq p}^{n}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) \\Biggr] \\Biggl[{}_{\\alpha }^{\\mu }\\mathfrak{I}_{r} ^{\\beta }\\prod_{i=1 }^{n}g_{i}^{\\gamma _{i}}(x) \\Biggr], \\end{aligned}\n\nwhich completes the desired inequality (17) of Theorem 3. □\n\n### Corollary 5\n\nLet $$(g_{i})_{i=1,2,3,\\ldots ,n}$$ and h be positive continuous on $$[r,s]$$ such that h is increasing and $$(g_{i})_{i=1,2,3,\\ldots ,n}$$ are decreasing functions on the interval $$[r,s]$$. Let $$r< x\\leq s$$, $$\\vartheta >0$$, $$\\sigma \\geq \\gamma _{p}>0$$ for any fixed $$p\\in \\{1,2,3, \\ldots ,n\\}$$. Then, for R-L type fractional conformable integral (3), we have\n\n\\begin{aligned} \\frac{{}_{\\alpha }\\mathfrak{I}_{r}^{\\beta } [\\prod_{i\\neq p}^{n}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) ]{}_{\\alpha }\\mathfrak{I}_{r}^{\\beta } [h^{\\vartheta }(x) \\prod_{i=1}^{n}g_{i}^{\\gamma_{i}}(x) ]}{{}_{\\alpha }\\mathfrak{I}_{r}^{\\beta } [h^{ \\vartheta }(x)\\prod_{i\\neq p}^{n}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) ] {}_{\\alpha }\\mathfrak{I}_{r}^{\\beta } [ \\prod_{i=1}^{n}g_{i}^{ \\gamma _{i}}(x) ]}\\geq 1, \\end{aligned}\n(21)\n\nwhere $$\\beta \\in \\mathbb{C}$$, $$\\alpha \\in (0,1]$$, and $$\\Re (\\beta )>0$$.\n\n### Corollary 6\n\nLet $$(g_{i})_{i=1,2,3,\\ldots ,n}$$ and h be positive continuous on $$[r,s]$$ such that h is increasing and $$(g_{i})_{i=1,2,3,\\ldots ,n}$$ are decreasing functions on the interval $$[r,s]$$. Let $$r< x\\leq s$$, $$\\vartheta >0$$, $$\\sigma \\geq \\gamma _{p}>0$$ for any fixed $$p\\in \\{1,2,3, \\ldots ,n\\}$$. Then, for R-L fractional integral (5), we have\n\n\\begin{aligned} \\frac{\\mathfrak{I}_{r}^{\\beta } [\\prod_{i\\neq p}^{n}g_{i}^{\\gamma_{i}}g_{p}^{\\sigma }(x) ]\\mathfrak{I}_{r}^{\\beta } [h^{ \\vartheta }(x) \\prod_{i=1}^{n}g_{i}^{\\gamma _{i}}(x) ]}{ \\mathfrak{I}_{r}^{\\beta } [h^{\\vartheta }(x)\\prod_{i\\neq p}^{n}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) ]\\mathfrak{I}_{r}^{\\beta } [ \\prod_{i=1}^{n}g_{i}^{\\gamma _{i}}(x) ]}\\geq 1, \\end{aligned}\n(22)\n\nwhere $$\\beta \\in \\mathbb{C}$$ and $$\\Re (\\beta )>0$$.\n\n### Theorem 4\n\nLet $$(g_{i})_{i=1,2,3,\\ldots ,n}$$ and h be positive continuous functions on $$[r,s]$$ such that h is increasing and $$(g_{i})_{i=1,2,3, \\ldots ,n}$$ are decreasing functions on the interval $$[r,s]$$. Let $$r< x\\leq s$$, $$\\vartheta >0$$, $$\\sigma \\geq \\gamma _{p}>0$$ for any fixed $$p\\in \\{1,2,3,\\ldots ,n\\}$$. Then, for generalized fractional conformable integral (1), we have\n\n\\begin{aligned} &\\frac{{}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\beta } [{ \\prod_{i\\neq p}^{n}}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) ]{}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\lambda } [h^{\\vartheta }(x) { \\prod_{i=1}^{n}}g_{i}^{\\gamma _{i}}(x) ]+{}_{\\alpha }^{\\mu } \\mathfrak{I}_{r}^{\\lambda } [{ \\prod_{i\\neq p}^{n}}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) ]{}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\beta } [h^{\\vartheta }(x) { \\prod_{i=1}^{n}}g_{i}^{\\gamma _{i}}(x) ]}{{}_{\\alpha }^{\\mu } \\mathfrak{I}_{r}^{\\beta } [h^{\\vartheta }(x){ \\prod_{i\\neq p}^{n}}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) ]{}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\lambda } [ { \\prod_{i=1}^{n}}g_{i}^{\\gamma _{i}}(x) ]+{}_{\\alpha }^{\\mu } \\mathfrak{I}_{r}^{\\lambda } [h^{\\vartheta }(x){ \\prod_{i\\neq p}^{n}}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) ]{}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\beta } [ \\prod_{i=1}^{n}g_{i}^{\\gamma _{i}}(x) ]} \\\\ &\\quad \\geq 1, \\end{aligned}\n(23)\n\nwhere $$\\beta ,\\lambda \\in \\mathbb{C}$$, $$\\alpha \\in (0,1]$$, $$\\mu \\in \\mathbb{R}$$, $$\\alpha +\\mu \\neq 0$$, $$\\Re (\\beta )>0$$, and $$\\Re (\\lambda )>0$$.\n\n### Proof\n\nMultiplying (20) by $$\\frac{1}{\\varGamma (\\lambda )} (\\frac{x ^{\\alpha +\\mu }-\\rho ^{\\alpha +\\mu }}{\\alpha +\\mu } )^{\\lambda -1}\\frac{ \\prod_{i=1}^{n}g_{i}^{\\gamma _{i}}(\\rho )}{\\rho ^{1-\\alpha -\\mu }}$$ then integrating with respect to ρ over $$(r,x)$$, we have\n\n\\begin{aligned} 0&\\leq \\Biggl[{}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\beta } \\prod_{i \\neq p}^{n}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) \\Biggr]{}_{\\alpha } ^{\\mu }\\mathfrak{I}_{r}^{\\lambda } \\Biggl[h^{\\vartheta }(x) \\prod_{i=1} ^{n}g_{i}^{\\gamma _{i}}(x) \\Biggr] \\\\ &\\quad {}+ \\Biggl[{}_{\\alpha }^{\\mu } \\mathfrak{I}_{r}^{\\lambda } \\prod_{i\\neq p}^{n}g_{i}^{\\gamma _{i}}g_{p} ^{\\sigma }(x) \\Biggr]{}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\beta } \\Biggl[h^{\\vartheta }(x) \\prod_{i=1}^{n}g_{i}^{\\gamma _{i}}(x) \\Biggr] \\\\ &\\quad {}- {}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\beta } \\Biggl[h^{\\vartheta }(x) \\prod_{i\\neq p}^{n}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) \\Biggr] \\Biggl[{}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\lambda } \\prod_{i=1 }^{n}g_{i}^{ \\gamma _{i}}(x) \\Biggr] \\\\ &\\quad {} -{}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\lambda } \\Biggl[h^{\\vartheta }(x)\\prod_{i\\neq p}^{n}g_{i}^{\\gamma _{i}}g_{p} ^{\\sigma }(x) \\Biggr] \\Biggl[{}_{\\alpha }^{\\mu } \\mathfrak{I}_{r}^{ \\beta }\\prod_{i=1 }^{n}g_{i}^{\\gamma _{i}}(x) \\Biggr]. \\end{aligned}\n\nAfter simplification, we get the desired result. □\n\n### Remark 4\n\nApplying Theorem 4 for $$\\beta =\\lambda$$, we get Theorem 3.\n\n### Corollary 7\n\nLet $$(g_{i})_{i=1,2,3,\\ldots ,n}$$ and h be positive continuous on $$[r,s]$$ such that h is increasing and $$(g_{i})_{i=1,2,3,\\ldots ,n}$$ are decreasing functions on the interval $$[r,s]$$. Let $$r< x\\leq s$$, $$\\vartheta >0$$, $$\\sigma \\geq \\gamma _{p}>0$$ for any fixed $$p\\in \\{1,2,3, \\ldots ,n\\}$$. Then, for R-L type fractional conformable integral (3), we have\n\n\\begin{aligned} &\\frac{{}_{\\alpha }\\mathfrak{I}_{r}^{\\beta } [{ \\prod_{i\\neq p}^{n}}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) ]{}_{\\alpha }\\mathfrak{I}_{r}^{\\lambda } [h^{\\vartheta }(x) { \\prod_{i=1}^{n}}g_{i}^{\\gamma _{i}}(x) ]+{}_{\\alpha } \\mathfrak{I}_{r}^{\\lambda } [{ \\prod_{i\\neq p}^{n}}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) ]{}_{\\alpha }\\mathfrak{I}_{r}^{\\beta } [h^{\\vartheta }(x) { \\prod_{i=1}^{n}}g_{i}^{\\gamma _{i}}(x) ]}{{}_{\\alpha } \\mathfrak{I}_{r}^{\\beta } [h^{\\vartheta }(x){ \\prod_{i\\neq p}^{n}}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) ]{}_{\\alpha }\\mathfrak{I}_{r}^{\\lambda } [ { \\prod_{i=1}^{n}}g_{i}^{\\gamma _{i}}(x) ]+{}_{\\alpha } \\mathfrak{I}_{r}^{\\lambda } [h^{\\vartheta }(x){ \\prod_{i\\neq p}^{n}}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) ]{}_{\\alpha }\\mathfrak{I}_{r}^{\\beta } [ { \\prod_{i=1}^{n}}g_{i}^{\\gamma _{i}}(x) ]} \\\\ &\\quad \\geq 1, \\end{aligned}\n(24)\n\nwhere $$\\beta ,\\lambda \\in \\mathbb{C}$$, $$\\alpha \\in (0,1]$$, $$\\Re ( \\beta )>0$$, and $$\\Re (\\lambda )>0$$.\n\n### Corollary 8\n\nLet $$(g_{i})_{i=1,2,3,\\ldots ,n}$$ and h be positive continuous on $$[r,s]$$ such that h is increasing and $$(g_{i})_{i=1,2,3,\\ldots ,n}$$ are decreasing on the interval $$[r,s]$$. Let $$r< x\\leq s$$, $$\\vartheta >0$$, $$\\sigma \\geq \\gamma _{p}>0$$ for any fixed $$p\\in \\{1,2,3,\\ldots ,n\\}$$. Then, for R-L fractional integral (5), we have\n\n\\begin{aligned} &\\frac{\\mathfrak{I}_{r}^{\\beta } [{ \\prod_{i\\neq p}^{n}}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) ] \\mathfrak{I}_{r}^{\\lambda } [h^{\\vartheta }(x) \\prod_{i=1}^{n}g_{i}^{\\gamma _{i}}(x) ]+\\mathfrak{I}_{r}^{\\lambda } [{ \\prod_{i\\neq p}^{n}}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) ] \\mathfrak{I}_{r}^{\\beta } [h^{\\vartheta }(x) { \\prod_{i=1}^{n}}g_{i}^{\\gamma _{i}}(x) ]}{\\mathfrak{I}_{r}^{ \\beta } [h^{\\vartheta }(x){ \\prod_{i\\neq p}^{n}}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) ] \\mathfrak{I}_{r}^{\\lambda } [ { \\prod_{i=1}^{n}}g_{i}^{\\gamma _{i}}(x) ]+\\mathfrak{I}_{r}^{ \\lambda } [h^{\\vartheta }(x){ \\prod_{i\\neq p}^{n}}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) ] \\mathfrak{I}_{r}^{\\beta } [ { \\prod_{i=1}^{n}}g_{i}^{\\gamma _{i}}(x) ]} \\\\ &\\quad \\geq 1, \\end{aligned}\n(25)\n\nwhere $$\\beta ,\\lambda \\in \\mathbb{C}$$, $$\\Re (\\beta )>0$$, and $$\\Re (\\lambda )>0$$.\n\n### Theorem 5\n\nLet $$(g_{i})_{i=1,2,3,\\ldots ,n}$$ and h be positive continuous on $$[r,s]$$, and let for any fixed $$p\\in \\{1,2,3,\\ldots ,n\\}$$,\n\n$$\\bigl(g_{p}^{\\vartheta }(t)h^{\\vartheta }(\\rho )-g_{p}^{\\vartheta }( \\rho )h^{\\vartheta }(t) \\bigr) \\bigl(g_{p}^{\\sigma -\\gamma _{p}}(t) -g_{p}^{\\sigma -\\gamma _{p}}(\\rho ) \\bigr)\\geq 0,$$\n\n$$r< x\\leq s$$, $$\\vartheta >0$$, $$\\sigma \\geq \\gamma _{p}>0$$, then we have\n\n\\begin{aligned} \\frac{{}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\beta } [{ \\prod_{i\\neq p}^{n}}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma +\\vartheta }(x) ] {}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\beta } [h^{\\vartheta }(x) { \\prod_{i=1}^{n}}g_{i}^{\\gamma _{i}}(x) ]}{{}_{\\alpha }^{\\mu } \\mathfrak{I}_{r}^{\\beta } [h^{\\vartheta }(x){ \\prod_{i\\neq p}^{n}}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma }(x) ]{}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\beta } [g_{p}^{\\vartheta } { \\prod_{i=1}^{n}}g_{i}^{\\gamma _{i}}(x) ]}\\geq 1, \\end{aligned}\n(26)\n\nwhere $$\\beta \\in \\mathbb{C}$$, $$\\alpha \\in (0,1]$$, $$\\mu \\in \\mathbb{R}$$, $$\\alpha +\\mu \\neq 0$$, and $$\\Re (\\beta )>0$$.\n\n### Proof\n\nThe proof of Theorem 5 is similar to the proof of Theorem 3 if we replace $$h^{\\vartheta }(\\rho )-h^{\\vartheta }(t)$$ by $$(g_{p}^{\\vartheta }(t)h^{\\vartheta }(\\rho )-g_{p}^{\\vartheta }( \\rho )h^{\\vartheta }(t) )$$. □\n\n### Theorem 6\n\nLet $$(g_{i})_{i=1,2,3,\\ldots ,n}$$ and h be positive continuous functions on $$[r,s]$$, and let for any fixed $$p\\in \\{1,2,3,\\ldots ,n\\}$$,\n\n$$\\bigl(g_{p}^{\\vartheta }(t)h^{\\vartheta }(\\rho )-g_{p}^{\\vartheta }( \\rho )h^{\\vartheta }(t) \\bigr) \\bigl(g_{p}^{\\sigma -\\gamma _{p}}(t) -g_{p}^{\\sigma -\\gamma _{p}}(\\rho ) \\bigr)\\geq 0,$$\n\n$$r< x\\leq s$$, $$\\vartheta >0$$, $$\\sigma \\geq \\gamma _{p}>0$$, then we have\n\n\\begin{aligned} &\\Biggl({}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\beta } \\Biggl[{ \\prod_{i\\neq p}^{n}}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma +\\vartheta }(x) \\Biggr] {}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\lambda } \\Biggl[h^{\\vartheta }(x) { \\prod_{i=1}^{n}}g_{i}^{\\gamma _{i}}(x) \\Biggr] \\\\ &\\qquad {}+{}_{\\alpha }^{\\mu } \\mathfrak{I}_{r}^{\\lambda } \\Biggl[{ \\prod_{i\\neq p}^{n}}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma +\\vartheta }(x) \\Biggr] {}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\beta } \\Biggl[h^{\\vartheta }(x) { \\prod_{i=1}^{n}}g_{i}^{\\gamma _{i}}(x) \\Biggr]\\Biggr) \\\\ &\\qquad {}\\Big/\\Biggl({}_{\\alpha }^{\\mu } \\mathfrak{I}_{r}^{\\beta } \\Biggl[h^{\\vartheta }(x){ \\prod_{i\\neq p}^{n}}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma +\\vartheta }(x) \\Biggr] {}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\lambda } \\Biggl[g_{p}^{\\vartheta }{ \\prod_{i=1}^{n}}g_{i}^{\\gamma _{i}}(x) \\Biggr] \\\\ &\\qquad {}+{}_{\\alpha }^{\\mu } \\mathfrak{I}_{r}^{\\lambda } \\Biggl[h^{\\vartheta }(x){ \\prod_{i\\neq p}^{n}}g_{i}^{\\gamma _{i}}g_{p}^{\\sigma +\\vartheta }(x) \\Biggr] {}_{\\alpha }^{\\mu }\\mathfrak{I}_{r}^{\\beta } \\Biggl[g_{p}^{\\vartheta } { \\prod_{i=1}^{n}}g_{i}^{\\gamma _{i}}(x) \\Biggr]\\Biggr) \\\\ &\\quad \\geq 1, \\end{aligned}\n(27)\n\nwhere $$\\beta ,\\lambda \\in \\mathbb{C}$$, $$\\alpha \\in (0,1]$$, $$\\mu \\in \\mathbb{R}$$, $$\\alpha +\\mu \\neq 0$$, $$\\Re (\\beta )>0$$, and $$\\Re (\\lambda )>0$$.\n\n### Proof\n\nThe proof of Theorem 6 runs parallel as to the proof of Theorem 4 if we replace $$h^{\\vartheta }(\\rho )-h^{\\vartheta }(t)$$ by $$(g_{p}^{\\vartheta }(t)h^{\\vartheta }(\\rho )-g_{p}^{\\vartheta }( \\rho )h^{\\vartheta }(t) )$$. □\n\n### Remark 5\n\nIn a similar way, we can get the inequalities for the generalized right FCIO (2) and special cases for integrals (4) and (6).\n\n## References\n\n1. Abdeljawad, T.: On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015). https://doi.org/10.1016/j.cam.2014.10.016\n\n2. Abdeljawad, T.: A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel. J. Inequal. Appl. 2017, 130 (2017). https://doi.org/10.1186/s13660-017-1400-5\n\n3. Abdeljawad, T.: Fractional operators with exponential kernels and a Lyapunov type inequality. Adv. Differ. Equ. 2017, 313 (2017). https://doi.org/10.1186/s13662-017-1285-0\n\n4. Abdeljawad, T., Agarwal, R.P., Alzabut, J., Jarad, F., ÖZbekler, A.: Lyapunov-type inequalities for mixed non-linear forced differential equations within conformable derivatives. J. Inequal. Appl. 2018, 143 (2018). https://doi.org/10.1186/s13660-018-1731-x\n\n5. Abdeljawad, T., Alzabut, J., Jarad, F.: A generalized Lyapunov-type inequality in the frame of conformable derivatives. A generalized Lyapunov-type inequality in the frame of conformable derivatives. Adv. Differ. Equ. 2017, 321 (2017). https://doi.org/10.1186/s13662-017-1383-z\n\n6. Adjabi, Y., Jarad, F., Abdeljawad, T.: On generalized fractional operators and a Gronwall type inequality with applications. Filomat 31(17), 5457–5473 (2017). https://doi.org/10.2298/FIL1717457A\n\n7. Dahmani, Z.: New inequalities in fractional integrals. Int. J. Nonlinear Sci. 9(4), 493–497 (2010)\n\n8. Dahmani, Z.: New classes of integral inequalities of fractional order. Matematiche 69(1), 237–247 (2014). https://doi.org/10.4418/2014.69.1.18\n\n9. Dahmani, Z., Tabharit, L.: On weighted Gruss type inequalities via fractional integration. J. Adv. Res. Pure Math. 2, 31–38 (2010)\n\n10. Jarad, F., Ugurlu, E., Abdeljawad, T., Baleanu, D.: On a new class of fractional operators. Adv. Differ. Equ. 2017, 247 (2017)\n\n11. Khalil, R., Horani, M.A., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264(65), 65–70 (2014)\n\n12. Khan, T.U., Khan, M.A.: Generalized conformable fractional integral operators. J. Comput. Appl. Math. 346, 378–389 (2019). https://doi.org/10.1016/j.cam.2018.07.018\n\n13. Kilbas, A.A.: Hadamard-type fractional calculus. J. Korean Math. Soc. 38, 1191–1204 (2001)\n\n14. Kiryakova, V.: Generalized Fractional Calculus and Applications. Pitman Res. Notes Math. Ser., vol. 301. Longman, New York (1994)\n\n15. Liu, W., Ngǒ, Q.A., Huy, V.N.: Several interesting integral inequalities. J. Math. Inequal. 3(2), 201–212 (2009)\n\n16. Nisar, K.S., Qi, F., Rahman, G., Mubeen, S., Arshad, M.: Some inequalities involving the extended gamma function and the Kummer confluent hypergeometric k-function. J. Inequal. Appl. 2018, 135 (2018)\n\n17. Nisar, K.S., Rahman, G., Choi, V., Mubeen, S., Arshad, M.: Certain Gronwall type inequalities associated with Riemann–Liouville k- and Hadamard k-fractional derivatives and their applications. East Asian Math. J. 34(3), 249–263 (2018)\n\n18. Nisar, K.S., Tassaddiq, A., Rahman, G., Khan, A.: Some inequalities via fractional conformable integral operators. J. Inequal. Appl. 2019, 217 (2019)\n\n19. Ntouyas, K.S., Agarwal, P., Tariboon, J.: On Polya–Szego and Chebyshev types inequalities involving the Riemann–Liouville fractional integral operators. J. Math. Inequal. 10(2), 491–504 (2016)\n\n20. Podlubny, I.: Fractional Differential Equations. Academic Press, London (1999)\n\n21. Qi, F., Rahman, G., Hussain, S.M., Du, W.S., Nisar, K.S.: Some inequalities of Čebyšev type for conformable k-fractional integral operators. Symmetry 10, 614 (2018). https://doi.org/10.3390/sym10110614\n\n22. Rahman, G., Khan, A., Abdeljawad, T., Nisar, K.S.: The Minkowski inequalities via generalized proportional fractional integral operators. Adv. Differ. Equ. 2019, 287 (2019). https://doi.org/10.1186/s13662-019-2229-7\n\n23. Rahman, G., Nisar, K.S., Mubeen, S., Choi, J.: Certain inequalities involving the $$(k,\\rho )$$-fractional integral operator. Far East J. Math. Sci.: FJMS 103(11), 1879–1888 (2018)\n\n24. Rahman, G., Nisar, K.S., Qi, F.: Some new inequalities of the Gruss type for conformable fractional integrals. AIMS Mathematics 3(4), 575–583 (2018)\n\n25. Rahman, G., Ullah, Z., Khan, A., Set, E., Nisar, K.S.: Certain Chebyshev type inequalities involving fractional conformable integral operators. Mathematics 7, 364 (2019). https://doi.org/10.3390/math7040364\n\n26. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, Reading (1993)\n\n27. Sarikaya, M.Z., Dahmani, Z., Kiris, M.E., Ahmad, F.: $$(k, s)$$-Riemann–Liouville fractional integral and applications. Hacet. J. Math. Stat. 45(1), 77–89 (2016)\n\n28. Set, E., Tomar, M., Sarikaya, M.Z.: On generalized Grüss type inequalities for k-fractional integrals. Appl. Math. Comput. 269, 29–34 (2015)\n\n29. Srivastava, H.M., Choi, J.: Zeta and q-Zeta Functions and Associated Series and Integrals. Elsevier, Amsterdam (2012)\n\n### Acknowledgements\n\nThe authors G. Rahman and A. Khan thanks to the Higher Education Commission of Pakistan for the support under the Start-Up Research Grant Project.\n\nNot applicable.\n\nNot applicable.\n\n## Author information\n\nAuthors\n\n### Contributions\n\nThe authors have contributed equally to this manuscript. They read and approved the final manuscript.\n\n### Corresponding author\n\nCorrespondence to Kottakkaran Sooppy Nisar.\n\n## Ethics declarations\n\n### Competing interests\n\nThe authors declare that they have no competing interests.", null, "" ]	[ null, "https://advancesincontinuousanddiscretemodels.springeropen.com/track/article/10.1186/s13662-019-2362-3", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.67107934,"math_prob":1.0000006,"size":20230,"snap":"2023-40-2023-50","text_gpt3_token_len":6763,"char_repetition_ratio":0.16103035,"word_repetition_ratio":0.4166368,"special_character_ratio":0.36307463,"punctuation_ratio":0.22624227,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000095,"pos_list":[0,1,2],"im_url_duplicate_count":[null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-29T17:26:55Z\",\"WARC-Record-ID\":\"<urn:uuid:ea5af691-f49a-40ca-8c9b-d8546d6a69b3>\",\"Content-Length\":\"354589\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:92942dcc-45e2-44e2-b1c1-0624487f9526>\",\"WARC-Concurrent-To\":\"<urn:uuid:4a8942d9-ce7a-4e13-a27f-b4d64d149739>\",\"WARC-IP-Address\":\"146.75.32.95\",\"WARC-Target-URI\":\"https://advancesincontinuousanddiscretemodels.springeropen.com/articles/10.1186/s13662-019-2362-3\",\"WARC-Payload-Digest\":\"sha1:LF7DR4E2LZULB4MPOTZR3Y32QMDFDQWG\",\"WARC-Block-Digest\":\"sha1:WRMQN6ACVWIAUFAEFZUFPY4WHNAJVQAZ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233510520.98_warc_CC-MAIN-20230929154432-20230929184432-00407.warc.gz\"}"}
https://codebun.com/category/problem-solving/page/13/	[ "Category: Problem Solving\n\nWrite a java program to find Sum of Squares of Even Digits.\n\nWrite a program to read a number, calculate the sum of squares of even digits (values) present in the given number. Input and Output Format: Input consists of a positive integer n. Output is a single integer . Refer sample output for formatting specifications. Sample Input 1: 56895 Sample Output 1: 100 Sum of Squares\n\nWrite a java program check Number Validation.\n\nWrite a program to read a string of 10 digit number, check whether the string contains a 10 digit number in the format XXX-XXX-XXXX where ‘X’ is a digit(Number Validation). Input and Output Format: Input consists of a string. The output is a string specifying the given string is valid or not. Refer sample output for\n\nWrite a java program to check sum of odd digits.\n\nWrite a program to read a number, calculate the sum of odd digits (values) present in the given number. Check Sum of Odd Digits in java Input and Output Format: Input consists of a positive integer n. Refer sample output for formatting specifications. Sample Input 1: 56895 Sample Output 1: Sum of odd digits is" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.697768,"math_prob":0.99293154,"size":948,"snap":"2019-26-2019-30","text_gpt3_token_len":211,"char_repetition_ratio":0.14724576,"word_repetition_ratio":0.3580247,"special_character_ratio":0.2278481,"punctuation_ratio":0.10638298,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9971382,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-06-19T18:05:57Z\",\"WARC-Record-ID\":\"<urn:uuid:1196fdde-468b-40e5-ab41-1d14140d6aa4>\",\"Content-Length\":\"42192\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:31dde19d-537f-4f08-9ecc-a07c8a94dee4>\",\"WARC-Concurrent-To\":\"<urn:uuid:d39d5ac0-b945-4955-a093-287a004c3940>\",\"WARC-IP-Address\":\"198.54.115.144\",\"WARC-Target-URI\":\"https://codebun.com/category/problem-solving/page/13/\",\"WARC-Payload-Digest\":\"sha1:KKN5THSY6L5WBODRS2AITDBF4JYFTWAP\",\"WARC-Block-Digest\":\"sha1:JPQKSSZ3QIGVGHRM4LJXIUNAUSLJXXQP\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-26/CC-MAIN-2019-26_segments_1560627999003.64_warc_CC-MAIN-20190619163847-20190619185847-00145.warc.gz\"}"}
https://www.mbatious.com/topic/1022/quant-boosters-sibanand-pattnaik-set-2/?page=2%E2%8C%A9=en-US	[ "Quant Boosters - Sibanand Pattnaik - Set 2\n\n• 2310 = 2 * 3 * 5 * 7 * 11\nThese 5 primes to be distributed into 3 places.\nTotal : 3^5 ways , but this is ordered ( arranged) and we need unordered.\nWe know that any triplet with distinct elements (a,b,c) is arranged in 3! ways, so all the distinct triplets in 3^5 = 243 are arranged in 3! Ways.\nExcept for one triplet (1, 1, 2310) which is arranged in 3!/2= 3 ways\nSo remove this from total and we are left with only distinct, is divide by 3! ,\nSo (3^5 - 3)/3! , and just add that one case of 1, 1, 2310\nSo (3^5-3)/3! + 1 = 41\n\n• Q11) How many pairs of positive integers (m, n) satisfy 1/m + 4/n = 1/12. Where n is an odd integer less than 60.\n\n• (m -12) * (n-48) = 12 * 48 = 2^6 * 3^2\nnow for \"n\" to be odd , (n-48) has to be odd\nso n - 48 can be either 1 or 3 or 3^2 (because we need ODD)\n(m -12) * (n - 48) = 2^6 * 3^2 * 1\nOR (m -12) * (n - 48) = 2^6 * 3 * 3\nor (m -12) * (n - 48) = 2^6 * 3^2\nwhen n - 48 = 1, n = 49 (acceptable)\nwhen n - 48 = 3, n = 51 (acceptable)\nwhen n - 48 = 3^2, n = 57 (acceptable)\nonly these 3 values are possible for n < 60 and odd\n\n• Q12) Find number of whole number solutions of a + b + c + d = 20 where a, b, c and d ≤ 8\n\n• Cofficient of x^20 in (1 + x + x^2 + x^3 + ... + x^8)^4\n= coff of x^20 in (1 - x^9)^4 * (1 - x)^-4\n= 23c3 - 4 * 14c3 + 4c2 * 5c3\n= 1771 - 1456 + 60\n= 375\n\nFor more detailed explanation of the concept, refer the video.\n\n• Q13) If k is a factor of (n+5) and (6n+54) where n is a natural number, how many possible values of k are there?\na) 6\nb) 7\nc) 8\nd) 9\n\n• As given k must be a factor of 6n+30 and is also a factor of 6n+54.\nSo it must be a factor of (6n+54) – (6n+30) = 24 = 2^3 x 3\nTherefore k can take 8 values\nHence (c)\n\n• Q14) The sum of all the factors of 480 including 1 and the number itself is 1512. What is the sum of the reciprocals of all the factors?\na) 3.15\nb) 2.85\nc) 2.55\nd) 3.45\n\n• Sum of reciprocals = (sum of factors)/480\n1512/480= 3.15\n\n• Q15) Find the last two digits of 89^102\na) 23\nb) 21\nc) 01\nd) 69\n\n• (89^2)^51\n89^2 same as 11^2 -> 21^51 = 21\n\n• Q16) The highest power of 5 in N! is 34. Find the highest power of 7 in N!.\na) 23\nb) 26\nc) 22\nd) Cannot be determined\n\n• Highest power of 5 in 125!= 31.\nHighest power of 5 in 140!=34\nBut highest power of 5 in 141!, 142!, 143! And 144! Is also 34.\nSo N can be 140 to 144, any of these values\nHowever the highest power of 7 in any of these values is same i.e. 22\n\n• Q17) A! + B! = N^2, Where A, B and N are whole numbers. How many possible combinations are there?\na) 1\nb) 2\nc) 3\nd) 4\n\n• 0! + 4! = 5^2\n1! + 4! = 5^2\n0! + 5! = 11^2\n1! + 5! = 11^2\nHence 4 combinations are possible\n\n• Q18) x, y and z are natural numbers such that x > y > z > 1 and PRODUCT= 3003. Let A and B be the maximum and minimum values of x + y + z. Find A-B.\na) 152\nb) 45\nc) 107\nd) None of the above\n\n• 3003= 3 x 7 x 11 x 13\nA= 143+7+3= 153\nB= 21 + 11 + 13= 45\nA-B= 108\nHence (d)\n\n• Q19) How many natural numbers less than 4444 are \"not co primes\" to 247?\na) 585\nb) 341\nc) 244\nd) 557\n\n• 247= 13 x 19\nSo all multiples of 13 and 19 has to be excluded till 4444\nMultiples of 13 : 341\nMultiples of 19: 233\nMultiple of both 13 and 19 i.e. of 247: 17\nTherefore number of numbers less than 4444 which are co-prime to 247: 341+233-17= 557\n\n• Q20) N , a natural number has 2 prime factors. N^3 can have which of the following number of factors?\na) 18\nb) 90\nc) 70\nd) 84\n\n61\n\n61\n\n61\n\n64\n\n58\n\n61\n\n61\n\n62" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7658342,"math_prob":0.99922,"size":3380,"snap":"2019-43-2019-47","text_gpt3_token_len":1421,"char_repetition_ratio":0.097748816,"word_repetition_ratio":0.039653037,"special_character_ratio":0.48994082,"punctuation_ratio":0.1056338,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9996685,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-10-23T10:09:45Z\",\"WARC-Record-ID\":\"<urn:uuid:256829e6-0813-4586-b8ee-079e41cd0619>\",\"Content-Length\":\"184838\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:03bfe22d-296c-4214-a396-35d0e45ce28b>\",\"WARC-Concurrent-To\":\"<urn:uuid:d3b11995-4f5d-4e73-9fb4-dc72d7680b93>\",\"WARC-IP-Address\":\"104.28.9.51\",\"WARC-Target-URI\":\"https://www.mbatious.com/topic/1022/quant-boosters-sibanand-pattnaik-set-2/?page=2%E2%8C%A9=en-US\",\"WARC-Payload-Digest\":\"sha1:OMAFMA3SBF54OJ4UG76F5P5GOR6MZOI6\",\"WARC-Block-Digest\":\"sha1:2XW3GW7TVMN5AEQ5ZSN5572NUBRBNUN2\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-43/CC-MAIN-2019-43_segments_1570987833089.90_warc_CC-MAIN-20191023094558-20191023122058-00122.warc.gz\"}"}