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https://en.wikipedia.org/wiki/Iterated_elimination_of_dominated_strategies
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[
"# Strategic dominance\n\nIn game theory, strategic dominance (commonly called simply dominance) occurs when one strategy is better than another strategy for one player, no matter how that player's opponents may play. Many simple games can be solved using dominance. The opposite, intransitivity, occurs in games where one strategy may be better or worse than another strategy for one player, depending on how the player's opponents may play.\n\n## Terminology\n\nWhen a player tries to choose the \"best\" strategy among a multitude of options, that player may compare two strategies A and B to see which one is better. The result of the comparison is one of:\n\n• B is equivalent to A: choosing B always gives the same outcome as choosing A, no matter what the other players do.\n• B strictly dominates A: choosing B always gives a better outcome than choosing A, no matter what the other players do.\n• B weakly dominates A: choosing B always gives at least as good an outcome as choosing A, no matter what the other players do, and there is at least one set of opponents' action for which B gives a better outcome than A. (Notice that if B strictly dominates A, then B weakly dominates A. Therefore, we can say \"B dominates A\" as synonymous of \"B weakly dominates A\".)\n• B and A are intransitive: B and A are not equivalent, and B neither dominates, nor is dominated by, A. Choosing A is better in some cases, while choosing B is better in other cases, depending on exactly how the opponent chooses to play. For example, B is \"throw rock\" while A is \"throw scissors\" in Rock, Paper, Scissors.\n• B is weakly dominated by A: There is at least one set of opponents' actions for which B gives a worse outcome than A, while all other sets of opponents' actions give A the same payoff as B. (Strategy A weakly dominates B).\n• B is strictly dominated by A: choosing B always gives a worse outcome than choosing A, no matter what the other player(s) do. (Strategy A strictly dominates B).\n\nThis notion can be generalized beyond the comparison of two strategies.\n\n• Strategy B is strictly dominant if strategy B strictly dominates every other possible strategy.\n• Strategy B is weakly dominant if strategy B dominates all other strategies, but some (or all) strategies are only weakly dominated by B.\n• Strategy B is strictly dominated if some other strategy exists that strictly dominates B.\n• Strategy B is weakly dominated if some other strategy exists that weakly dominates B.\n\nStrategy: A complete contingent plan for a player in the game. A complete contingent plan is a full specification of a player's behavior, describing each action a player would take at every possible decision point. Because information sets represent points in a game where a player must make a decision, a player's strategy describes what that player will do at each information set.\n\nRationality: The assumption that each player acts to in a way that is designed to bring about what he or she most prefers given probabilities of various outcomes; von Neumann and Morgenstern showed that if these preferences satisfy certain conditions, this is mathematically equivalent to maximizing a payoff. A straightforward example of maximizing payoff is that of monetary gain, but for the purpose of a game theory analysis, this payoff can take any form. Be it a cash reward, minimization of exertion or discomfort, promoting justice, spread of ones genes, or amassing overall “utility” - the assumption of rationality states that players will always act in the way that best satisfies their ordering from best to worst of various possible outcomes.\n\nCommon Knowledge: The assumption that each player has knowledge of the game, knows the rules and payoffs associated with each course of action, and realizes that every other player has this same level of understanding. This is the premise that allows a player to make a value judgment on the actions of another player, backed by the assumption of rationality, into consideration when selecting an action.\n\n## Dominance and Nash equilibria\n\nC D 1, 1 0, 0 0, 0 0, 0\n\nIf a strictly dominant strategy exists for one player in a game, that player will play that strategy in each of the game's Nash equilibria. If both players have a strictly dominant strategy, the game has only one unique Nash equilibrium. However, that Nash equilibrium is not necessarily \"efficient\", meaning that there may be non-equilibrium outcomes of the game that would be better for both players. The classic game used to illustrate this is the Prisoner's Dilemma.\n\nStrictly dominated strategies cannot be a part of a Nash equilibrium, and as such, it is irrational for any player to play them. On the other hand, weakly dominated strategies may be part of Nash equilibria. For instance, consider the payoff matrix pictured at the right.\n\nStrategy C weakly dominates strategy D. Consider playing C: If one's opponent plays C, one gets 1; if one's opponent plays D, one gets 0. Compare this to D, where one gets 0 regardless. Since in one case, one does better by playing C instead of D and never does worse, C weakly dominates D. Despite this, $(D,D)$",
null,
"is a Nash equilibrium. Suppose both players choose D. Neither player will do any better by unilaterally deviating—if a player switches to playing C, they will still get 0. This satisfies the requirements of a Nash equilibrium. Suppose both players choose C. Neither player will do better by unilaterally deviating—if a player switches to playing D, they will get 0. This also satisfies the requirements of a Nash equilibrium.\n\n## Iterated elimination of strictly dominated strategies (IESDS)\n\nThe iterated elimination (or deletion) of dominated strategies (also denominated as IESDS or IDSDS) is one common technique for solving games that involves iteratively removing dominated strategies. In the first step, at most one dominated strategy is removed from the strategy space of each of the players since no rational player would ever play these strategies. This results in a new, smaller game. Some strategies—that were not dominated before—may be dominated in the smaller game. The first step is repeated, creating a new even smaller game, and so on. The process stops when no dominated strategy is found for any player. This process is valid since it is assumed that rationality among players is common knowledge, that is, each player knows that the rest of the players are rational, and each player knows that the rest of the players know that he knows that the rest of the players are rational, and so on ad infinitum (see Aumann, 1976).\n\nThere are two versions of this process. One version involves only eliminating strictly dominated strategies. If, after completing this process, there is only one strategy for each player remaining, that strategy set is the unique Nash equilibrium.\n\nStrict Dominance Deletion Step-by-Step Example:\n\n1. C is strictly dominated by A for Player 1. Therefore, Player 1 will never play strategy C. Player 2 knows this. (see IESDS Figure 1)\n2. Of the remaining strategies (see IESDS Figure 2), Z is strictly dominated by Y and X for Player 2. Therefore, Player 2 will never play strategy Z. Player 1 knows this.\n3. Of the remaining strategies (see IESDS Figure 3), B is strictly dominated by A for Player 1. Therefore, Player 1 will never play B. Player 2 knows this.\n4. Of the remaining strategies (see IESDS Figure 4), Y is strictly dominated by X for Player 2. Therefore, Player 2 will never play Y. Player 1 knows this.\n5. Only one rationalizable strategy is left {A,X} which results in a payoff of (10,4). This is the single Nash Equilibrium for this game.\n\nAnother version involves eliminating both strictly and weakly dominated strategies. If, at the end of the process, there is a single strategy for each player, this strategy set is also a Nash equilibrium. However, unlike the first process, elimination of weakly dominated strategies may eliminate some Nash equilibria. As a result, the Nash equilibrium found by eliminating weakly dominated strategies may not be the only Nash equilibrium. (In some games, if we remove weakly dominated strategies in a different order, we may end up with a different Nash equilibrium.)\n\nWeak Dominance Deletion Step-by-Step Example:\n\n1. O is strictly dominated by N for Player 1. Therefore, Player 1 will never play strategy O. Player 2 knows this. (see IESDS Figure 5)\n2. U is weakly dominated by T for Player 2. If Player 2 chooses T, then the final equilibrium is (N,T)\n1. O is strictly dominated by N for Player 1. Therefore, Player 1 will never play strategy O. Player 2 knows this. (see IESDS Figure 6)\n2. T is weakly dominated by U for Player 2. If Player 2 chooses U, then the final equilibrium is (N,U)\n\nIn any case, if by iterated elimination of dominated strategies there is only one strategy left for each player, the game is called a dominance-solvable game."
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"https://wikimedia.org/api/rest_v1/media/math/render/svg/0f14de04ee675be80958307e46b611ae80a5f43e",
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https://lists.w3.org/Archives/Public/w3c-rdfcore-wg/2001Nov/0011.html
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[
"# datatypes and MT\n\nFrom: Pat Hayes <[email protected]>\nDate: Thu, 1 Nov 2001 18:31:11 -0600\nMessage-Id: <p0510101eb8074f2b836b@[205.160.76.193]>\n\n```I'm sorry I dropped the ball on this issue just as it heated up. Ive\nbeen (literally) laid low with flu since Friday and unable to do\nanything much except whimper and cough.\n\nLet me try to first summarize the MT changes that I managed to\nextract from the pfps/ph interchange - I think these are all pretty\nmuch the same as what Peter got out of it, in all but stylistic\ndetails - and then use this extended MT to respond to a lot of the\nrecent comments, some of which I think are based on\nmisunderstandings. I will also try to show how *all* the various\nproposals for encoding datatyping information in RDF syntax can be\nincorporated into this MT extension in a uniform way, and so can a\nfew others that havn't been made yet.\n\nOK. The basic idea of the MT extension arises from the fact that\nwhile the same literal label might mean different things in different\ncontexts, datatyping information seems to remove what would otherwise\nbe an 'ambiguity' when one looks at particular occurrences of literal\nlabels. For example, it seems perfectly clear, just speaking\nintuitively, that the following graph\n\naaa rdfs:range xsd:integer .\nbbb rdfs:range xsd:string .\nfoo aaa \"101001\" .\nbaz bbb \"101001\" .\n\ncan be unambiguously interpreted as saying that the aaa-value of foo\nis the integer (89456 + 11545) and the bbb-value of baz is the\ncharacter string \"101001\". (The double quotes in the RDF triples are\nnot to be interpreted as actual quotation marks, of course, but only\nas literal-markers, a convention which I abhor but will stick with\nthroughout this message.) The fact that there is no ambiguity in this\ngraph, even though the same syntactic label is used with two\ndifferent meanings, suggests that the datatyping information should\nnot be thought of as a mapping from the literal labels - which is\nwhat one would get by extending the MT in a very straightforward way\nby simply including literal labels into the vocabulary of an\ninterpretation - but rather as attached to the node itself. The\ndatatyping extension to the MT therefore introduces a new kind of\nmapping, which is very similar to an interpretation mapping, but\nwhich applies to the nodes of the graph rather than to the literals\nwhich are used to label those nodes. We might call such a thing a\n'datatyping interpretation', but I think that might be confusing, so\nI will call it a 'typing'. An interpretation is a mapping from a URI\nvocabulary to entities; a typing is a mapping from nodes of the graph\nto datatypes. (I'll tell you what a datatype is in a minute.)\n\nThe current MT does need to be altered in a tiny degree to make this\npossible. Currently it says that XL is a global mapping from literals\nto LV, and then it uses that mapping once, and never mentions it\nagain. We need to change that so we define XL to be a mapping from\n*occurrences* (tokens, inscriptions, whatever) of literals to LV; or,\nless mysteriously, from *literal nodes* to LV. Then the model theory\nworks exactly as before; this mapping is still 'global' (though that\nis now perhaps an unfortunate word to use) in the sense that it is\nindependent of the interpretation. (When we introduce a datatyping\nscheme, however, there is some connection between them, as we will\nsee.)\n\nTechnicalities.\nA datatype is a mapping from a lexical domain (a subset of literals)\nto a range of values (a set). A datatype scheme is a set DT plus a\nfixed mapping DTS from DT to datatypes. It is convenient to define\nDTC(x) to be the range (not rdfs:Range, just range) of the datatype\nDTS(x). If nnn is the URI of a type, we will write XD(nnn) for the\nmember of DT that the uri identifies. This mapping XD is similar in\nsome ways to an interpretation mapping I, but unlike I, it is assumed\nthat XD is computable (in some way), ie given the uri of a datatyping\nscheme, the machine can somehow access the actual DTS and DTC\nmappings associated with that datatyping scheme and apply them.\n\n(DTS and DT aren't *strictly* needed, in fact; one could have a\nsingle global mapping directly from urirefs to datatypes; but it is\nin the same spirit as the rest of the model theory so I will go on\ndoing it this way. Think of the member of DT as the abstract\ndatatype-thingie and DTS as the datatype version of the IEXT mapping\nin the MT. DTC is analogous to ICEXT.)\n\nNow, a datatyping of a graph is simply a mapping from the literal\nnodes of the graph to a datatype scheme, ie an assignment of a type\nto each literal node of the graph. Of course, a graph by itself might\nhave any number of datatypings, just as a vocabulary can have any\nnumber of interpretations, but we want to make sure that if we have\none of each than that fixes the interpretation of every part of a\nlabelled graph unambiguously, by placing mutual constraints on how\nthey act together. So, in this spirit, a typed interpretation is a\npair <I,D> of an interpretation (of the uri vocabulary) and a\ndatyping D (of the literal nodes of the graph) which together satisfy\nthe following constraints:\n\n1. XL(n) = DTS(D(n))(label(n))\n2. ICEXT(d) is a subset of DTC(d) for any d in (DT intersect IR)\n\n(In 1 we are using two rather different kinds of function; label(n)\nis just a kind of syntactic selection function which refers to the\nlabel on the node n, while the other mappings are semantic.\nMathematically these are all just functions, but intuitively they\nhave different roles.)\n\nWhat these mean, intuitively, is exactly what one would expect; that\nthe value of a literal (in the interpretation) is understood to be\ndetermined by the datatyping of the node on which it occurs; and that\ndatatypes are treated appropriately as rdfs classes, in the sense\nthat the RDFS class of a given datatype in the interpretation is a\nsubset of set of things which actually have that datatype according\nto D.\n\nNow, the only other semantic condition we need on an interpretation\nis that it 'understands' that the urirefs that denote datatypes are\nin fact interpreted to refer to those datatypes, which can be stated\nas the condition\n\n3. I(nnn)= XD(nnn)\n\nIf I satisfies condition 3 for the datatype named nnn then we will\nsay that I 'recognizes' that datatype, and it is natural to require\nthat I recognizes every datatype which is mentioned in the graph (ie\nwhere nnn occurs in the graph as a node label), so we will add this\nas a third condition on a typed interpretation <I,D>.\n\nOK, that is all we require. We could sum all this up by saying that I\nhas to 'respect' D by agreeing to use the datatype URIs in the\nappropriate ways, and by agreeing to interpret the datatype mappings\nas rdf:properties in a consistent way (consistent with D, that is.)\nIn return, D will undertake to guarantee that all literal occurrences\nare interpreted in a way that will be consistent with anything that\nis said in the RDF triples. As long as I and D promise to keep to\ntheir respective vows concerning each other, we can be sure that\ntheir marriage will be happy.\n\n(A less anthropomorphic analogy might be to think of D as a kind of\ndatatyping network between nodes along which datatyping information\ncan flow, and which can carry different information to different\nnodes.)\n\nThis isn't really very different from extending the interpretation\nidea from urirefs to literals, but it acknowledges explicitly the\nways that literal evaluation differs from uriref interpretations; the\nfact that once the datatyping has been determined, the value of the\nliteral is *fixed* and is not subject to variation from one\ninterpretation to another; but that also, in an odd way, the precise\nmeaning of any given literal is hostage to the datatyping that is\nused to interpret it, and different occurrences might be interpreted\ndifferently. It also makes it clear how the rest of the RDF graph,\nwhile it may not change the datatyping mappings themselves (they are\nstill 'global' to the graph) , can provide information (ie constrain\nthe satisfying interpretations) which forces a particular literal\nlabel to be interpreted according to one or another datatyping scheme.\n\n[Aside. Heres how to say this without using the DT /DTS/DTC stuff.\nJust assume a global mapping XD from urirefs to datatypes, and\ndefine a datatyping as a mapping D from nodes to datatypes. Then the\nfirst two conditions above can be stated:\n1. LV(n)=D(n)(label(n))\n2. ICEXT(I(x)) is a subset of {y: XD(x)(lit)=y & lit is a literal}\nThis shows how very simply it really is, but I still prefer the\nearlier way of stating it.]\n\nHow this works.\n\nTo illustrate this idea I will use the example graph given earlier.\nSince I want to refer to the actual nodes of the graph, however, I\nwill use Ntriples++ notation to give the literal nodes labels, OK?\nRemember, this is the *same* graph, Ive just given two of the nodes\nunique labels in the Ntriples, is all.\n\naaa rdfs:range xsd:integer .\nbbb rdfs:range xsd:string .\nfoo aaa _:1:\"101001\" .\nbaz bbb _:2:\"101001\" .\n\nHere is one datatyping of this graph:\n\nDTS(D(_:1)) = (lambda (?x)(eval ?x))\nDTS(D(_:2)) = (lambda (?x) 17)\n\nThis says that literals on node#1 are interpreted by LISP and\nanything on node#2 is interpreted to be the number 17. Fortunately,\nthat datatyping scheme has no URI, but it is a scheme. There are\ninfinitely many damn silly schemes, but most of them wouldn't make\nthe graph come out to be true no matter what interpretation you were\nto combine them with.\n\nObviously the scheme we want is this:\nDTS(D(_:1)) = XD(xsd:integer)\nDTS(D(_:2)) = XD(xsd:string))\n\nAnd the cute thing is that this is guaranteed by the semantic\nconditions, since the graph says enough to lock down this as the only\npossible datatyping that would yield a satisfying typed interpetation\n<I,D>. Let me just do this for the integer case. If <I,D> is a typed\ninterpretation satisfying this graph, then I must recognize\nxsd:integer, so from 3:\nI(xsd:integer) = XD(xsd:integer)\nand by 2,\nICEXT(I(xsd:integer)) must be a subset of DTC(I(xsd:integer)), ie of\nDTC(XD(xsd:integer)), ie the value space of xsd: integers. It follows\nfrom the ordinary rdfs:range semantic conditions that the only way\nthat I could make the fourth triple true is if XL(_:1) is in the\nclass DTC(XD(xsd:integer)), which requires (by the first semantic\ncondition) that the datatyping D(_:1) on _:1 be the lexical-value\nmapping specified by xsd:integer, since\nXL(_:1)= DTS(D(_:1))(\"101001\").\n(Actually what I said above isn't exactly true. The conditions do not\nguarantee that this datatyping is specified uniquely; but they do\nimpose that whatever datatyping is used, it must *agree* with\nxsd:integer and xsd:string on all the literal nodes in the graph. I\ncould invent a silly datatyping which was like xsd on all numerals\nwork. The point being that you wouldn't know the difference, so it\ndoesn't matter.)\n\nAlternative syntaxes\n\nNow let me illustrate how a variety of alternative ideas for\nspecifying datatyping information in RDF graphs can be seen as\nextensions to this, by adding different kinds of constraints on\ndatatyped interpretations.\n\n1. Explicit schema datatyping.\n\nAssume that every occurrence of a literal is somehow explicitly\nlabelled with a datatype in the graph itself, eg by redefining\n'literal labels' to be pairs of a literal and a uri indicating a\ndatatype. Let me use label(n) and dtype(n) respectively for these two\nparts of the literal label at a node; then simply define a datatyping\nscheme in the obvious way, by requiring it to be defined by the dtype\nlabels:\n4.1 D(n)=I(dtype(n))\n\nSubstituting 4.1 into 1 gives:\n\nLV(n) = DTS(I(dtype(n)))(label(n))\n\nand assuming that dtype(n) is indeed a datatype uriref, then 3 in turn gives:\n\nLV(n)= DTS(XD(dtype(n)))(label(n))\n\nshowing that in this case we can eliminate D from the equations\naltogether, and simply use the single 'global fixed' mapping XD to\ndetermine the literal value of a literal independently of the\ninterpretation I and hence independently of the rest of the RDF\ngraph. (This is in fact what I had in the back of my mind when\ndefining the original MT, by declaring XL to be a global, fixed\nmapping.) So if we were to impose this very strict local datatype\nlabelling scheme, in effect incorporating the datatype into the\nliteral label itself, then there really is no need for this extension\nto the model theory. However, what this does show is that this kind\nof strict labelling scheme does not contradict the more flexible\nscheme, so they could be used together without any risk of being\nmutually incompatible.\n\n2. Bnodes as literal values.\n\nLet me put together here under one heading a variety of variations of\nthe following theme: that occurrences of literals in value position\nin a triple should be understood as an abbreviation of a pair of\ntriples with an 'intermediate' bnode, where the bnode denotes the\ntriple value and the literal label itself is relegated to a minor\nrole of somehow 'illustrating' how that value could be written in\nsome notation. For example: rewrite\n\naaa bbb lit .\n\nas\naaa bbb _:1 .\n_:1 rdf:value lit .\n\nBefore analyzing this, I note that it has the advantage of putting a\nnode that denotes the literal value in subject position, where other\nproperties (such as rdf:type) can be asserted of it. This is indeed a\nmajor advantage - I think the *only* advantage - of this proposal,\nbut we could render this moot simply by declaring that RDF shall\nallow literals in subject position. I will return to this point later.\n\nThe first of these triples seems easy to interpret: it means exactly\nwhat the original triple meant, in fact. It's the second triple that\nseems rather odd. Since _:1 is supposed to denote the literal value,\nit would seem to have the same value at both ends. We have provided\nnow two nodes to refer to the same thing: one blank, but in subject\nposition; the other with a literal label to say what it is. The\nfirst, blank, node can now safely be asserted to have an rdf:type\nwhich is a datatype, and if we make reasonable assumptions about\ninterpretations being in accordance with the global XD mappings from\ndatatype URIs, (similar to the conditions listed here) then we will\nbe able to infer the datatypes of those blank nodes by normal rdfs\ninference. But the only way to know what the first node actually\n*means* is to look at the label on the second node. So we now have an\nodd juxtaposition, where rdf:value has a literal label at one end\nwhich has no assigned datatype, and a datatype at the other but no\nliteral there to use it on.\n\nThere are several ways to get around this. The simplest, in the\ncurrent framework, is simply to declare that rdf:value is equality,\nie that <x,y> is in IEXT(I(rdf:value)) iff x=y. This effectively\nforces I(_:1) to be the same as XL(lit), and then the <I,D> semantic\nmachinery works in this case in exactly the same way that it works in\nthe 'plain' case. This however is not how the proponents of this\nkind of scheme usually think of it.\n\nAnother way is to insist that the literal label in the second triple\nis treated in a nonstandard way: rather than denoting a literal\nvalue, it is being mentioned rather than used; it simply indicates\nthe literal itself. (Or, equivalently, all literals are treated as\nstrings.) The intuitive meaning of rdf:value is then a kind of\ninversion of the denotation mapping itself: it assigns a literal\nlabel to the literal value that is the semantic value of that label\nin the given interpretation. This seems to be what Sergey has in\nmind, and it is also I think what Dan C. suggested a while back as\nthe best way to handle literals.\n\nNow, this seems to me to have a fatal flaw, which arises from the\nfact that the value spaces of two different datatypes might overlap.\nFor example, suppose that there are datatypes xxd:octal and\nxxd:decimal, then the following would seem to be perfectly true:\n\n_:1 rdf:type xxd:octal\n_:1 rdf:type xxd:decimal\n_:1 rdf:value \"32\"\n_:1 rdf:value \"26\"\n\nsince indeed the number twenty-six is the value of both a decimal and\nan octal numeral, so that number, which is what I(_:1) should be in\nany satisfying interpretation, is in both class extensions, and <26,\n\"26\"> is in IEXT(I(rdf:value)) when 26 is of type xxd:decimal, and\n<26,\"32\"> is in it when 26 is of type xxd:octal, so *both* of those\npairs have to be in it. The point being that in cases like this, it\nisn't enough to just attach a datatype to the *interpretation* of the\nliteral, ie the literal value denoted by the blank node. You have to\nsomehow get it attached *to the literal itself*, and by making the\nseparation a syntactic separation, there is no way to do that. The\nonly way to do that in this kind of scheme would be to impose a\nsyntactic constraint on RDF graphs that required any node to be the\nsubject of at most one rdf:value arc; and since literals only appear\nat the object ends of those arcs, the only function of the rdf:value\nedge in the graph is to attach a unique literal label to the blank\nnode. It seems less trouble, and much clearer in meaning, to simply\nattach it directly to the 'blank' node and throw that edge away, and\nthen we are back where we started. (Notice that if we did that, then\nthis kind of pathological example becomes impossible, since we would\nhave to attach two different literal labels to the 'blank' node. The\nsyntactic separation of lexical and value spaces in the RDF graph\nsimply creates new opportunities for confusion, which is what usually\nhappens when one gets use and mention mixed up in this kind of way.)\n\nOn balance, therefore, I would prefer to adopt the first\ninterpretation of rdf:value as simply meaning equality. However, if\nwe are going to introduce equality into RDF, let us do so properly.\nThis is quite a significant change to the language, making it much\nmore expressive. We ought to be able to make inferences which follow\nfrom the transitivity and substutivity of equality, for example, and\nto be able to assert equalities between urirefs as well as literals\nand blank nodes.\n\nOn the other hand, we could also gain all the expressive advantages\nof this device for literals without going this far, by making one\nsimple change.\n\n3. Literals as subjects.\n\nThis proposal modifies the RDF syntax in a small way, by allowing\nliterals to be subjects. The current MT goes through in exactly the\nsame way, except of course the artificial restrictions in the closure\nrules for RDFS closures are removed. This would have several notable\n\nFirst, information about literal nodes - in particular, their data\ntype - can be expressed directly, instead of resorting to subterfuges\nlike the introduction of blank nodes. In fact, as pointed out above,\nyou can think of this as what you would get just by taking the\nnotational devices used in the 'blank-node' syntax proposals and\nconflating the blank node with the literal node that it is linked to.\nFor example, the following graph written in bnode-style:\n\naaa bbb _:1 .\n_:1 rdf:value \"345\" .\n_:1 rdf:type xsd:integer .\n\nwould be boiled down into:\n\naaa bbb _:1:\"345\" .\n_:1 rdf:type xsd:integer .\n\nwhere I have been obliged to use Ntriples++ to indicate that the\nsubject node of the second triple is the object node of the first one.\n\n(Notice that the use of nodeIDs in this style of Ntriples++ notation\nis *exactly* parallel to its use in the 'bnode' graph; the only\ndifference is that some of the nodes being identified are no longer\nblank. In fact, one could get the Ntriples++ for the second graph\nfrom the Ntriples for the first one simply by making sure that the\nrdf:value triples occur after the triples that generated them, and\nthen replacing all strings of the form\nA<white>. <white>?<newline><white>?A<white>rdf:value<white>\nwhere A is a nodeID, with\nA:\nand leaving the rest alone. )\n\nSecond, this now gives us a very sweet way to characterize how to\ndetermine the datatype of any given literal node: Generate the rdfs\nclosure of the graph, and see if it contains\n\n_:node rdf:type <datatype> .\n\nwhere _:node is the ID of the node and <datatype> is a datatype URI.\nIf it does, then the literal on that node has to be interpreted in\naccordance with that datatype; if not, it doesn't. This works no\nmatter how the datatyping information is provided; it could be said\ndirectly, or inferred from range information or even by\nsubclass-transitivity inference; all those variations are absorbed in\nthe details of the rdfs closure rules. (If it is provided by explicit\nnode labelling then we would need to incorporate an extra closure\nrule to get it from the label into the graph explicitly.) The key\npoint is, that if the graph somehow establishes membership in a class\nknown to be a datatype, then that fixes it; if not, it is not fixed.\n\nOK, that's all for now. I have to go to bed, I'm bushed.\n\nPat\n\nPS. How about having an rdfs class called rdfs:Datatype? Then the\nsemantic rule on recognition could be relativised to membership in\nthat class, and an rdfs graph would have an explicit internal note of\nthe datatype/simple-class distinction.\n\n--\n---------------------------------------------------------------------\nIHMC\t\t\t\t\t(850)434 8903 home\n40 South Alcaniz St.\t\t\t(850)202 4416 office\nPensacola, FL 32501\t\t\t(850)202 4440 fax\[email protected]\nhttp://www.coginst.uwf.edu/~phayes\n```\nReceived on Thursday, 1 November 2001 19:31:15 UTC\n\nThis archive was generated by hypermail 2.4.0 : Friday, 17 January 2020 20:24:06 UTC"
] |
[
null
] |
{"ft_lang_label":"__label__en","ft_lang_prob":0.93359005,"math_prob":0.8840547,"size":21234,"snap":"2022-40-2023-06","text_gpt3_token_len":4879,"char_repetition_ratio":0.1642487,"word_repetition_ratio":0.008705114,"special_character_ratio":0.22784214,"punctuation_ratio":0.10902778,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9659858,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-02-04T07:27:01Z\",\"WARC-Record-ID\":\"<urn:uuid:1ccde6d2-e3b7-475e-926b-e8e41a5df59e>\",\"Content-Length\":\"34537\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:108ada1b-f96f-480e-86ae-de975b04e6cc>\",\"WARC-Concurrent-To\":\"<urn:uuid:d10bebe2-a59b-4def-80ed-33b464b04ba0>\",\"WARC-IP-Address\":\"104.18.22.19\",\"WARC-Target-URI\":\"https://lists.w3.org/Archives/Public/w3c-rdfcore-wg/2001Nov/0011.html\",\"WARC-Payload-Digest\":\"sha1:O4IQAGXRHOZA4RX6H4EW3KUGM4E2ZRMJ\",\"WARC-Block-Digest\":\"sha1:ZDFJC7MXSY7EXHVFBV62GNCO7WZMOHPF\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764500094.26_warc_CC-MAIN-20230204044030-20230204074030-00573.warc.gz\"}"}
|
https://www.gasspringsshop.co.uk/product/replacement-for-suspa-liftline-01625001-30-450n/
|
[
"# Replacement for Suspa Liftline 01625001 30-450N\n\n£28.13 (excl. VAT)\n\nReplacement gas spring for the Suspa Liftline 01625001 30-450 Newton. This fits a (possibly already present) ball with a diameter of 10mm. A corresponding ball is included. Brand: Hahn.\n Force Choose an option30 Newton40 Newton50 Newton60 Newton80 Newton100 Newton120 Newton140 Newton150 Newton160 Newton180 Newton200 Newton220 Newton240 Newton250 Newton260 Newton280 Newton300 Newton320 Newton340 Newton350 Newton360 Newton380 Newton400 Newton420 Newton440 Newton450 NewtonClear\nThis gas spring is also known as 16-1 016 25001, 16-1-72,5-45-A246-B246.",
null,
"",
null,
"",
null,
"Category:"
] |
[
null,
"https://www.gasspringsshop.co.uk/wp-content/featherlight/gasspring-replacements/images/UK/replaces-suspa-liftline-01625001-30-450N.png",
null,
"https://www.gasspringsshop.co.uk/wp-content/plugins/gasveer_tools_serverside//assets/img/caddrawings/WX22.png",
null,
"https://www.gasspringsshop.co.uk/wp-content/plugins/gasspring_replacements/assets//img/fl_elbow_ball_removal.png",
null
] |
{"ft_lang_label":"__label__en","ft_lang_prob":0.82986325,"math_prob":0.5577546,"size":325,"snap":"2021-43-2021-49","text_gpt3_token_len":105,"char_repetition_ratio":0.10280374,"word_repetition_ratio":0.0,"special_character_ratio":0.39384615,"punctuation_ratio":0.14492753,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9820209,"pos_list":[0,1,2,3,4,5,6],"im_url_duplicate_count":[null,1,null,2,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-18T15:13:32Z\",\"WARC-Record-ID\":\"<urn:uuid:aba8c194-b862-4389-8694-66be7750918f>\",\"Content-Length\":\"103346\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:38fb8b46-5bcc-440b-8eec-2ebc84ac047c>\",\"WARC-Concurrent-To\":\"<urn:uuid:9bd95c1c-dc3a-4b5c-8d32-d85f9a70f347>\",\"WARC-IP-Address\":\"104.21.40.179\",\"WARC-Target-URI\":\"https://www.gasspringsshop.co.uk/product/replacement-for-suspa-liftline-01625001-30-450n/\",\"WARC-Payload-Digest\":\"sha1:UCNU4LHEPKDYLXFIHRMGSKW6BWKOLTXH\",\"WARC-Block-Digest\":\"sha1:BHXPZTZIQLFM5EA2NVKPYUIX4DN2Q5KU\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323585203.61_warc_CC-MAIN-20211018124412-20211018154412-00107.warc.gz\"}"}
|
https://eprint.iacr.org/2021/1215
|
[
"## Cryptology ePrint Archive: Report 2021/1215\n\nOptimization of Homomorphic Comparison Algorithm on RNS-CKKS Scheme\n\nEunsang Lee and Joon-Woo Lee and Young-Sik Kim and Jong-Seon No\n\nAbstract: Since the sign function can be used to implement the comparison operation, max function, and rectified linear unit (ReLU) function, several studies have been conducted to efficiently evaluate the sign function in the Cheon-Kim-Kim-Song (CKKS) scheme, one of the most promising fully homomorphic encryption schemes. Recently, Lee et al. (IEEE Trans. Depend. Sec. Comp.) proposed a practically optimal approximation method of sign function on the CKKS scheme using a composition of minimax approximate polynomials. However, homomorphic comparison, max function, and ReLU function algorithms that use this approximation method have not yet been successfully implemented on the residue number system variant CKKS (RNS-CKKS) scheme, and the sets of degrees of the component polynomials used by the algorithms are not optimized for the RNS-CKKS scheme. In this paper, we propose the optimized homomorphic comparison, max function, and ReLU function algorithms on the RNS-CKKS scheme using a composition of minimax approximate polynomials for the first time. We propose a fast algorithm for inverse minimax approximation error, a subroutine required to find the optimal set of degrees of component polynomials. This proposed algorithm makes it possible to find the optimal set of degrees of component polynomials with higher degrees than the previous study. In addition, we propose a method to find the degrees of component polynomials optimized for the RNS-CKKS scheme using the proposed algorithm for inverse minimax approximation error. We successfully implement the homomorphic comparison, max function, and ReLU function algorithms on the RNS-CKKS scheme with a low comparison failure rate ($< 2^{-15}$) and provide the various parameter sets according to the precision parameter $\\alpha$. We reduce the depth consumption of the homomorphic comparison, max function, and ReLU function algorithms by one depth for several $\\alpha$. In addition, the numerical analysis demonstrates that the proposed homomorphic comparison, max function, and ReLU function algorithms reduce running time by 6%, 7%, and 6% on average compared with the previous best-performing algorithms, respectively.\n\nCategory / Keywords: Cheon-Kim-Kim-Song (CKKS) scheme, fully homomorphic encryption (FHE), homomorphic comparison operation, minimax approximate polynomial, Remez algorithm, residue number system variant CKKS (RNS-CKKS) scheme."
] |
[
null
] |
{"ft_lang_label":"__label__en","ft_lang_prob":0.81418204,"math_prob":0.9579879,"size":2844,"snap":"2021-43-2021-49","text_gpt3_token_len":617,"char_repetition_ratio":0.15387323,"word_repetition_ratio":0.12935324,"special_character_ratio":0.19936709,"punctuation_ratio":0.11642411,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9920547,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-27T01:32:39Z\",\"WARC-Record-ID\":\"<urn:uuid:bcf5a136-e436-4d14-9a5f-492ca8e239ea>\",\"Content-Length\":\"5336\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:81292d6b-afa4-4e9d-9fed-6e2b5bead2cd>\",\"WARC-Concurrent-To\":\"<urn:uuid:172920bc-351b-47a5-959c-bca375fd39e8>\",\"WARC-IP-Address\":\"165.232.149.190\",\"WARC-Target-URI\":\"https://eprint.iacr.org/2021/1215\",\"WARC-Payload-Digest\":\"sha1:DDU5SSOEB77UFMR7PSKSNLAHSLELCVZ5\",\"WARC-Block-Digest\":\"sha1:Z467JLN3XLXUCZIB2GJJQ4APSRBRXMAD\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323587963.12_warc_CC-MAIN-20211026231833-20211027021833-00619.warc.gz\"}"}
|
https://gis.stackexchange.com/questions/313482/strange-output-from-qgis-raster-terrain-analysis-plugin
|
[
"# Strange output from QGIS Raster Terrain Analysis plugin\n\nThis is closely related to several other questions on this site: here, here, here etc. However, my issue is not resolved by following the answers provided.\n\nI am using the Raster Terrain Analysis plugin in QGIS ver 2.14.3. I want to create a Slope layer based on SRTM imagery. I understand that confusion can arise because the lat-lon units are degrees, and the vertical units are in metres. I calculated the z-factor with the formula `z = 1/(111320*cos(latitude*pi/180))` provided in this answer. Which gives `0.00000899928` (I'm using imagery which spans the equator so used `latitude = 0`).",
null,
"However, I get a layer of either `0` or `90`. It is very similar to the question here. Re-saving the layer with WGS84 before using the plugin does not help.",
null,
"I am able to produce a sensible looking Slope layer if I convert the CRS to World Mollweide, but I would like to know why it isn't working with CRS in degrees? What am I doing wrong?\n\nSince slope is determined by `rise` over `run`, If you elevation units are in metres and your distance units are in degree-decimals, even small changes in your elevations will easily calculate your slope at 90 degree inclinations.\nSince your data is located at the equator, see if you can find where your region is along the Mercator system and `warp` your raster to an EPSG CRS that is appropriate for that area."
] |
[
null,
"https://i.stack.imgur.com/ekUsI.png",
null,
"https://i.stack.imgur.com/G8Dz4.png",
null
] |
{"ft_lang_label":"__label__en","ft_lang_prob":0.9323342,"math_prob":0.663331,"size":922,"snap":"2022-27-2022-33","text_gpt3_token_len":228,"char_repetition_ratio":0.09259259,"word_repetition_ratio":0.0,"special_character_ratio":0.2537961,"punctuation_ratio":0.11282051,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95291233,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,9,null,9,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-08-09T20:01:05Z\",\"WARC-Record-ID\":\"<urn:uuid:de71fe24-d18c-4400-a327-6cd25526c8a8>\",\"Content-Length\":\"225162\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:8ad85e0c-7f6e-4920-8d6e-fa8b56513edc>\",\"WARC-Concurrent-To\":\"<urn:uuid:730db39e-dd28-4445-a3d0-d11f170fb250>\",\"WARC-IP-Address\":\"151.101.129.69\",\"WARC-Target-URI\":\"https://gis.stackexchange.com/questions/313482/strange-output-from-qgis-raster-terrain-analysis-plugin\",\"WARC-Payload-Digest\":\"sha1:VVI4NCX22OFLGPHTH5PTJ3DDMCI5EOON\",\"WARC-Block-Digest\":\"sha1:PHJDER7GJOQBQ4WMXFUF6J5DJDMPD5OD\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-33/CC-MAIN-2022-33_segments_1659882571086.77_warc_CC-MAIN-20220809185452-20220809215452-00203.warc.gz\"}"}
|
https://www.onlinemathlearning.com/statistics-lecture-23.html
|
[
"# Statistics Lectures - 23: Introduction To ANOVA & One-Way ANOVA\n\nA series of free Statistics Lectures with lessons, examples & solutions in videos.\n\nThis is page twenty-three of the series of free video lessons, “Statistics Lectures”. These lectures covers an introduction to ANOVA - analysis of variance, one-way anova, effect size for one-way anova, post-hoc tests for one-way ANOVA.\n\nShare this page to Google Classroom\n\n### Statistics - Lecture 70: Introduction to Analysis of Variance (ANOVA)\n\nANOVA is a statistical method used to compare the means of two or more groups.\nTypes of ANOVA:\n\n• Repeated-Measures ANOVA - One factor with at least two levels, levels are dependent.\n• Factorial ANOVA - Two or more factors (each of which with at least two levels), levels can be either independent, dependent, or both (mixed). Assumptions in ANOVA:\n1. Normality of Sampling Distribution of Mass - The distribution of sample means is normally distributed.\n2. Independence of Errors - Errors between cases are independent of one another\n3. Absence of Outliers - Outlying scores have been removed from the data set.\n\n### Statistics - Lecture 71: One-Way ANOVA\n\nOne factor with at least two levels, levels are independent.\n\n1. Define Null and Alternative Hypotheses\n2. State Alpha\n3. Calculate Degrees of Freedom\n4. State Decision Rule\n5. Calculate Test Statistic\n6. State Results\n7. State Conclusion\n\n### Statistics - Lecture 72: Effect Size For One-Way ANOVA\n\nThe most common measure of effect size for a One-Way ANOVA is Eta-squared.\n\n### Statistics - Lecture 73: Post-Hoc Tests for One-Way ANOVA\n\nExample:\nResearchers want to test a new anti-anxiety medication. They split participants into three conditions (0mg, 50mg, and 100mg), then ask them to rate their anxiety level on a scale of 1-10. Are there any differences between the three conditions using alpha = 0.05?\n\n### Statistics Lecture Series - Table Of Contents\n\nTry the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.",
null,
"We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page."
] |
[
null,
"https://www.onlinemathlearning.com/objects/default_image.gif",
null
] |
{"ft_lang_label":"__label__en","ft_lang_prob":0.833435,"math_prob":0.86586195,"size":1326,"snap":"2021-21-2021-25","text_gpt3_token_len":310,"char_repetition_ratio":0.090771556,"word_repetition_ratio":0.0,"special_character_ratio":0.21644042,"punctuation_ratio":0.12062257,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98732185,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-20T21:51:07Z\",\"WARC-Record-ID\":\"<urn:uuid:8d642a91-e237-4ad9-b4ce-c8fb2ef51ac1>\",\"Content-Length\":\"53057\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ebde0b78-09d5-4f72-a07b-4897eb9b62be>\",\"WARC-Concurrent-To\":\"<urn:uuid:f9cacb0a-94b4-4e3e-b51b-ca89b1050a84>\",\"WARC-IP-Address\":\"173.247.219.45\",\"WARC-Target-URI\":\"https://www.onlinemathlearning.com/statistics-lecture-23.html\",\"WARC-Payload-Digest\":\"sha1:LBEWWUJ5FR2RJCZI3Z6ILAAATEQ5QDRF\",\"WARC-Block-Digest\":\"sha1:7B5L23NC7EYVFJKYVJ7J3YTADHIBNCWC\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623488257796.77_warc_CC-MAIN-20210620205203-20210620235203-00059.warc.gz\"}"}
|
https://diyi0t.com/arduino-pwm-tutorial/
|
[
"# PWM Tutorial for Arduino, ESP8266 and ESP32\n\nIn this tutorial you learn what Pulse Width Modulation (PWM) is and how to create different PWM signals from your Arduino, ESP8266 or ESP32 microcontroller.\n\nIn two detailed example you change the brightness of an LED and control the speed of a DC motor by using the PWM signal from the microcontroller.",
null,
"## How does PWM for Microcontroller Work?\n\nThe digital inputs / outputs on your microcontroller have a constant voltage of 3.3V (for ESP8266 and ESP32 boards) or 5V (for Arduino boards). But in some cases you want to control the voltage to a specific value between 0V and the maximum voltage.\n\nIn case of PWM, a signal is pulsing between HIGH (3.3V or 5V) and LOW (0V). How often the signal is changing between HIGH and LOW is defined by the PWM frequency. The PWM frequency on Arduino pins are 976 cycles per seconds (Herz), for the ESP8266 up to 1 kHz and for the ESP32 up to 40 MHz.\n\nTo generate a PWM signal you use the function analogWrite(pin, value). This function create a square wave PWM signal. You can control the shape of the PWM signal with the duty cycle of (value/255). A 0% to 100% duty cycle corresponds to a value between 0 and 255.\n\nThe following table shows broadly the relation between the duty cycle and the average output voltage if the maximum voltage is 5V for Arduino microcontroller and 3.3V for ESP8266 and ESP32 microcontroller.\n\nDuty CycleOutput voltage (Arduino)Output voltage (ESP8266 and ESP32)analogWrite(X)\n0%0V0V0 = 0%*255\n0.25%1.25V0.825V63.75 = 25%*191.25\n0.5%2.5V1.65V127.5 = 50%*255\n0.75%3.75V2.475V191.25 = 75%*255\n100%5V3.3V255 = 100%*255\n\nI used my oscilloscope to measure the duty cycles of the table above for the Arduino output voltage. Therefore I connected digital pin 11 of the Arduino Uno to my PicoScope and set up a measurement for the duty cycle. You can do the exactly same measurement for the ESP8266 or the ESP32\n\nThe code for the quick measurement is the following. For each measurement I changed the value for the analogWrite function.\n\nDuty Cycle 0%\nNote that the 0% duty cycle is measured as 100% because there is no square wave at 0% and also at 100%. Therefore be careful how the duty cycle is measured. In my case I have to see if the voltage is 0 at 100% duty cycle.\n\nDuty Cycle 25%\n\nDuty Cycle 50%\n\nDuty Cycle 75%\n\nDuty Cycle 100%\n\nThe following table gives you an overview of all components and parts that I used for this tutorial. I get commissions for purchases made through links in this table.\n\nArduino Nano AmazonAliExpress\nArduino Pro Mini AmazonAliExpress\nArduino Uno AmazonAliExpress\nArduino Mega AmazonAliExpress\nESP32 ESP-WROOM-32AmazonAliExpress\nESP8266 NodeMCU AmazonAliExpress\nESP8266 WeMos D1 Mini AmazonAliExpress\nResistor and LED in Package AmazonAliExpress\nDC Motor -AliExpress\nL298N Motor Driver Module -AliExpress\nDC Motor with L298N Motor Driver Module Amazon-\n\n## How to Change the Brightness of an LED by PWM\n\nIn the following example we want to change the brightness of an LED by changing the duty cycle of the regarding PWM signal.\n\nThe following fritzing sketches show the circuit done with different Arduino, ESP8266 and ESP32 microcontroller boards. We have to make sure that the LED is connected to a microcontroller pin that is PWM able.\n\nI recommend to get the Microcontroller Datasheet eBook where you can find the pinouts of the different microcontroller boards, that include the information which pins are able to use PWM. You get the eBook for free if you join the DIYI0T newsletter.\n\n### Wiring between LED and Arduino Microcontroller for PWM\n\nThe following pictures show the wiring between different Arduino boards, the LED that brightness we want to control and a 220Ω resistor to prevent the LED from too high voltages.\n\nFor this example I use the digital pin 11 that is able to create PWM signals but you can use any other PWM able pin as well.\n\n### Wiring between LED and ESP8266 Microcontroller for PWM\n\nThe following pictures show the connection between different ESP8266 boards like the ESP8266 NodeMCU or the ESP8266 WeMos D1 Mini and the LED as well as the 220Ω resistor.\n\nWe need a resistor in series to the LED to prevent the LED for too high voltages.\n\nYou can choose any digital I/O pin of the ESP8266 for this example, because all digital I/O pins of the ESP8266 are able to create PWM signals, but make sure that you also change the pin in the program script. I select pin D4 for this example.\n\n### Wiring between LED and ESP32 Microcontroller for PWM\n\nThe following picture shows the wiring between the ESP32 ESP-WROOM-32 microcontroller board, the LED and the 220Ω resistance to prevent the LED for too high voltages.\n\nBecause any digital I/O pin of the ESP32 is able to create PWM signals, you can choose any other digital pin but make sure to change the program script. In my case, I connect pin 4 of the ESP32 to the LED.\n\n### Program Code to Change the Brightness of an LED by PWM\n\nThe program code is pretty straight forward. We have to define the pins and some variables for the time and the height to increment the brightness of the LED. If you want to know how to change the color for multicolor LEDs, using different PWM signals, you find this in my LED tutorial.\n\n```int LEDpin = 11; // for Arduino microcontroller\n//int LEDpin = D4; // for ESP8266 microcontroller\n//int LEDpin = 4; // for ESP32 microcontroller\n\nint bright = 0; // initial value of LED brightness\nint incremt = 5; // incremental change in PWM frequency\nint time = 100; // time period the PWM frequency is changing\n\nvoid setup()\n{\npinMode(LEDpin, OUTPUT); // define the LEDpin as output pin\n}\n\nvoid loop()\n{\nanalogWrite(LEDpin, bright); // set LED brightness as PWM signal\ndelay(time); // wait for a time period\nbright = bright + incremt; // increment LED brightness\n// if the brightness is out of range, reduce brightness\nif (bright <=0 || bright >=255) incremt = - incremt;\n}```\n\nIn the first part of the Arduino code, we define the pin that connects the LED to the microcontroller. Because this script is for Arduino, ESP32 and ESP8266 microcontrollers, you have to comment out two of the first three lines that defines the pin.\n\nAlso we have to define three additional variables:\n\n• bright: initial value of the LED brightness and therefore 0 to shut down the LED.\n• increment: the incremental change in the PWM frequency. Each increment the LED increases and decreases the brightness.\n• time: the time period in milliseconds for each PWM cycle.\n\nIn the setup function, we fine the pin, that we defines as LED pin at the beginning of the script, as output pin to use the pin with PWM.\n\nWe start the loop function, with the analogWrite(pin, value) function we set the analog value (PWM wave) for the brightness to the LED pin. After the delay of 0.1 seconds, the brightness in incremented. If the brightness reaches the minimum (0) or maximum (255) value, the increment is changed from a positive value to a negative value.\n\nThe following video shows the change of the brightness of the LED due to the PWM function.",
null,
"## How to Control the Speed of a DC Motor by PWM\n\nThe second example shows how we can change the speed of an DC motor with the help of a PWM signal. We use the L298N motor driver module to connect the Arduino, ESP8266 or ESP32 to the DC motor. It is also recommend to power the DC motor with an external power supply. Therefore I use 4 AA batteries, each with 1.5V so in total 6V.\n\nI want to focus on the PWM signal in this example. Therefore I do not further describe the DC motor example. But if you are interested in DC motors, I will write an extra article about DC motors, that explains everything in detail.",
null,
"Microcontroller Datasheet eBook\n\nThe 35 pages Microcontroller Datasheet Playbook contains the most useful information of 14 Arduino, ESP8266 and ESP32 microcontroller boards.\n\n### Wiring between DC Motor and Arduino Microcontroller for PWM\n\nThe following pictures show the connections for the speed control of a DC motor via PWM signal from the Arduino microcontroller. The DC motor is connected to the L298N motor driver.\n\nWe use 4 AA batteries or a laboratory power supply for the power supply of the L298N motor driver. The Arduino microcontroller is powered via USB or also via the laboratory power supply\n\nTo control the speed and rotation direction of the DC motor, we connect the pins ENA, INT1 and INT2 to the digital pins of the Arduino board.\n\n### Wiring between DC Motor and ESP8266 Microcontroller for PWM\n\nThe wiring between the ESP8266 board, the power supply, the L298N motor driver and the DC motor is shown in the following picture.\n\nFour AA batteries of a laboratory power supply can be used as power supply for the motor driver. The ESP8266 board is powered from the USB connection and ground is connection to each other.\n\nThe speed and rotation direction of the DC motor, are controlled via the pins ENA, INT1 and INT2 pins of the L298N motor driver and connected to the digital pins of the ESP8266 board. The DC motor itself is connected directly to the motor driver.\n\n### Wiring between DC Motor and ESP32 Microcontroller for PWM\n\nThe following picture shows how to connect a DC motor to the ESP32 ESP-WROOM-32. For the power supply of 6V, you can use 4 AA batteries or a laboratory power supply. The ESP32 board is powered from the USB connection or also from the laboratory power supply.\n\nTo control the DC motor, we can not connect the DC motor directly to the ESP32, but we need the L298N motor driver. Pins ENA, INT1 and INT2 to control the speed and rotation direction of the DC motor are connected to three digital pins of the ESP32. The DC motor is then connected to the L298N motor driver.\n\n### Program Code to Control the Speed of a DC Motor by PWM\n\nFor the program code we want to increase the motor speed each second. At the start, the voltage provided through the PWM signal is to low to start the motor. This is why it takes some intervals in the for loop until the motor is turning.\n\n```// Arduino connection to L298N\nint enA = 10; // PWM for Motor A\nint in1 = 9; // Control Rotation of Motor A\nint in2 = 8; // Control Rotation of Motor A\n\n// ESP8266 connection to L298N\n//int enA = D4; // PWM for Motor A\n//int in1 = D5; // Control Rotation of Motor A\n//int in2 = D6; // Control Rotation of Motor A\n\n// ESP32 connection to L298N\n//int enA = 4; // PWM for Motor A\n//int in1 = 0; // Control Rotation of Motor A\n//int in2 = 2; // Control Rotation of Motor A\n\nint motor_speed = 0;\n\nvoid setup()\n{\n// set all the motor control pins to outputs\npinMode(enA, OUTPUT);\npinMode(in1, OUTPUT);\npinMode(in2, OUTPUT);\n}\n\nvoid loop()\n{\n// run the motor between 0 and 250 in increments of 10\ndigitalWrite(in1, LOW); // Input1 LOW = move forward\ndigitalWrite(in2, HIGH); // Input2 HIGH = move forward\nfor(motor_speed = 0; motor_speed < 250; motor_speed += 10)\n{\nanalogWrite(enA, motor_speed); // PWM output\ndelay(1000);\n}\n}```\n\nIn the first part of the script we define the pins that connect the microcontroller to the L298N module, that is connected to the DC motor. In total we have to define 3 pins:\n\n• enA: transfers the PWM signal to the L298N module and must be a pin that is able to produce a PWM signal.\n• int1 and int2: control the rotation direction of the motor. The following table shows how to control int1 and int2 to stop the motor, move forward and move backwards.\nint1\nLOWHIGH\nint2LOWMotor stopsMotor moves backwards\nHIGHMotor moves forwardMotor stops\n\nBecause this script is for Arduino, EPS8266 and ESP32 microcontroller boards, you have to select only the lines for your microcontrolle and comment the other lines. The current script is commented in a way, that is for Arduino boards.\n\nTo increase the motor speed, the speed has to be saved to a variable that we call motor_speed and has an initial value of 0.\n\nIn the setup function we define all digital pins, that connects the board to the L298N as outputs.\n\nWe start the loop function by setting int1 LOW and int2 HIGH. Therefore the DC motor rotates in the forward direction. In the for loop we increase the motor speed each second by 10 between 0 and 250. The motor speed in written as analog output (PWM signal) to the enA connection.\n\nThe following video shows the Arduino script in action for the Arduino Uno as example.",
null,
"Conclusion\n\nI hope you enjoined this article about the PWM signal. The PWM signal is a very handy tool which is used a lot in practical examples. It is recommended to know how PWM is working. Therefore if you have any further questions, use the comment section below to ask.\n\n### 2 thoughts on “PWM Tutorial for Arduino, ESP8266 and ESP32”\n\n1. I will right awɑy clսtch ｙour rѕs feed as I can’t to find your e-mail subscription link or e-newsletter service.\n\nDo you have any? Please let me know іn ordеr that I could ѕubscribe.\nThanks."
] |
[
null,
"https://diyi0t.com/wp-content/uploads/2019/11/PWM-thumbnail.jpg",
null,
"https://diyi0t.com/wp-content/plugins/borlabs-cookie/assets/images/cb-no-thumbnail.png",
null,
"https://diyi0t.com/wp-content/uploads/2021/01/Datasheet-Playbook_Cover_Landscape_v03.png",
null,
"https://diyi0t.com/wp-content/plugins/borlabs-cookie/assets/images/cb-no-thumbnail.png",
null
] |
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https://liqas.org/code/
|
[
"# Code\n\nLi, J. C.-H. (in press). Effect size measures in a two independent-samples case with non-normal and non-homogeneous data. Behavior Research Methods. (Supplementary materials: https://osf.io/msy3h/)\n\nSupplementary Materials\n\nThe following sections include the supplemental materials for the study entitled as “Effect Size Measures in a Two Independent-Samples Case with Non-Normal and Non-Homogeneous Data”. The first section presents a Mathematica code that can be used to estimate the six effect sizes given a real-world database (named “data.csv”). The second section includes the Monte Carlo simulation code used in the study. The third section presents a URL that links to the full report of the percentage biases that were used to create Figures 1 and 2 in the study.\n\n1. A Mathematica Code Used to Obtain the Six ES Estimates Based on the Hypothetical Example in the Conclusion and Discussion Section\n\n• First, save the data as “data.csv”, where the first column contains the vale labels for the two groups (i.e., 0 and 1), and the second includes the observations for each participant.\n• For example, enter the hypothetical data shown in the conclusion and discussion section of the study and save it as “data.csv”. Alternatively, download the data file in excel format (data) and save it as “data.csv”. Note that the data file should be placed to the location that can be retrieved in the Mathematica code below (i.e., line 1: data = Import[“C:/data.csv”], where C:/ means that the data is saved to the C:/ drive of your computer).\n• Second, run the following Mathematica code.\n\ncode1\n\n• Third, obtain the 6 ES estimates below.\n\n—————————————-Output——————————————-\n\nd = -0.135157, dr* = 0.611744, dr = 0.39274, rpb = -0.0674249, CL = 0.446244, Aw = 0.6416\n\n——————————————————————————————\n\n2. The Monte Carlo Simulation Code Used in the Study\n\ncode2\n\n3. The Full Values for the Percentage Biases in Figures 1 and 2.\n\nTable"
] |
[
null
] |
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|
https://markupuk.org/webhelp/ar07s03s02.html
|
[
"### Representing XML Attribute Change in diff3\n\nXML attributes present a particular challenge for diff3 format. Here is an example of a change of value for an attribute.\n\nTable 3. XML attribute value change\n\nA.txtO.txtB.txt\n```<span\nclass=\"two\"\ndir=\"rtr\"\nid=\"23\">```\n```<span\nclass=\"one\"\nid=\"23\"\ndir=\"TBA\">```\n```<span\nid=\"23\"\nclass=\"three\"\ndir=\"ltr\">```\n\nThis could be represented as shown below. Note here that we are not showing the result of running 'diff3 -m' but rather we have run an XML-aware comparison yielding results that we want to express in the diff3 format.\n\n```<span id=\"23\"\n<<<<<<< A.txt\nclass=\"two\" dir=\"rtr\"\n||||||| O.txt\nclass=\"one\" dir=\"TBA\"\n=======\nclass=\"three\" dir=\"ltr\"\n>>>>>>> B.txt\n>```\n Figure 1. Attribute example in Visual Studio code",
null,
"In Figure 1, “Attribute example in Visual Studio code”, we see how this can be displayed and managed in Microsoft Visual Studio code.\n\nThe above will produce syntactically correct results, though it is not ideal because it would be more natural to choose the attributes separately rather than as a pair. This separation can be achieved by inserting additional white space so that we get two choices as shown below.\n\n```<span id=\"23\"\n<<<<<<< A.txt\nclass=\"two\"\n||||||| O.txt\nclass=\"one\"\n=======\nclass=\"three\"\n>>>>>>> B.txt\n\n<<<<<<< A.txt\ndir=\"rtr\"\n||||||| O.txt\ndir=\"TBA\"\n=======\ndir=\"ltr\"\n>>>>>>> B.txt\n>```\n\nThere is another representation that takes the common attribute name out of the choice, but it may be less easy for a user to see what is happening. This representation is shown below.\n\n```<span class=\n<<<<<<< A.txt\n\"two\"\n||||||| O.txt\n\"one\"\n=======\n\"three\"\n>>>>>>> B.txt\ndir=\n<<<<<<< A.txt\n\"rtr\"\n||||||| O.txt\n\"TBA\"\n=======\n\"ltr\"\n>>>>>>> B.txt\n>```"
] |
[
null,
"https://markupuk.org/webhelp/papers-2019/robin/MarkupUK2019-submitted2/vscode-screen.png",
null
] |
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|
https://www.convertunits.com/from/kilopond/square+centimetre/to/decipascal
|
[
"## ››Convert kilopond/square centimetre to decipascal\n\n kilopond/square centimetre decipascal\n\nHow many kilopond/square centimetre in 1 decipascal? The answer is 1.0197162129779E-6.\nWe assume you are converting between kilopond/square centimetre and decipascal.\nYou can view more details on each measurement unit:\nkilopond/square centimetre or decipascal\nThe SI derived unit for pressure is the pascal.\n1 pascal is equal to 1.0197162129779E-5 kilopond/square centimetre, or 10 decipascal.\nNote that rounding errors may occur, so always check the results.\nUse this page to learn how to convert between kiloponds/square centimeter and decipascals.\nType in your own numbers in the form to convert the units!\n\n## ››Quick conversion chart of kilopond/square centimetre to decipascal\n\n1 kilopond/square centimetre to decipascal = 980665 decipascal\n\n2 kilopond/square centimetre to decipascal = 1961330 decipascal\n\n3 kilopond/square centimetre to decipascal = 2941995 decipascal\n\n4 kilopond/square centimetre to decipascal = 3922660 decipascal\n\n5 kilopond/square centimetre to decipascal = 4903325 decipascal\n\n6 kilopond/square centimetre to decipascal = 5883990 decipascal\n\n7 kilopond/square centimetre to decipascal = 6864655 decipascal\n\n8 kilopond/square centimetre to decipascal = 7845320 decipascal\n\n9 kilopond/square centimetre to decipascal = 8825985 decipascal\n\n10 kilopond/square centimetre to decipascal = 9806650 decipascal\n\n## ››Want other units?\n\nYou can do the reverse unit conversion from decipascal to kilopond/square centimetre, or enter any two units below:\n\n## Enter two units to convert\n\n From: To:\n\n## ››Definition: Decipascal\n\nThe SI prefix \"deci\" represents a factor of 10-1, or in exponential notation, 1E-1.\n\nSo 1 decipascal = 10-1 pascals.\n\nThe definition of a pascal is as follows:\n\nThe pascal (symbol Pa) is the SI unit of pressure.It is equivalent to one newton per square metre. The unit is named after Blaise Pascal, the eminent French mathematician, physicist and philosopher.\n\n## ››Metric conversions and more\n\nConvertUnits.com provides an online conversion calculator for all types of measurement units. You can find metric conversion tables for SI units, as well as English units, currency, and other data. Type in unit symbols, abbreviations, or full names for units of length, area, mass, pressure, and other types. Examples include mm, inch, 100 kg, US fluid ounce, 6'3\", 10 stone 4, cubic cm, metres squared, grams, moles, feet per second, and many more!"
] |
[
null
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|
https://encyclopedia2.thefreedictionary.com/Expected+value
|
[
"# expected value\n\nAlso found in: Dictionary, Thesaurus, Medical, Legal, Financial, Acronyms, Wikipedia.\n\n## expected value\n\n[ek′spek·təd ′val·yü]\n(mathematics)\nFor a random variable x with probability density function ƒ(x), this is the integral from -∞ to ∞ of (x) dx. Also known as expectation.\nFor a random variable x on a probability space (Ω, P), the integral of x with respect to the probability measure P.\n(systems engineering)\nIn decision theory, a measure of the value or utility expected to result from a given strategy, equal to the sum over states of nature of the product of the probability of the state times the consequence or outcome of the strategy in terms of some value or utility parameter. Abbreviated EV. Also known as expected utility (EU).\nReferences in periodicals archive ?\n(181) Thus, using expected value is consistent with \"the essence of existing standing doctrine,\" (182) and its use as a standard for fear-based injury finds support in case law and legal scholarship.\nClaims of comparatively moderate scope have an expected value above\nFor the expected value [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the corresponding lower and upper integrals are called lower and upper expected values, and denoted by [E.bar][a] and [bar.E][a].\nLike the simple cost-benefit model (Equation 1), the expected value\nThe reformulated objective is equivalent to the maximization of the expected value because it results in the same optimal decisions.\n, [x.sub.n]).sup.t] are said to be statistically consistent if they reasonably fit the normal consistency model which postulates that the joint n-variate sampling pdf of x is normal N(1[micro], D) with unknown expected value 1[micro] and variance-covariance matrix D = [[[sigma].sub.ij]].\nFor a given value of P*, equilibrium entry occurs such that zero expected value is attained by the lowest valued entrant.\nThe expected value of taking 100 cartons to market, therefore, is calculated as follows: (0.3 x 250) + (0.5 x 250) + (0.2 x 250) = 250 [pounds sterling].\nthat has high expected value in the current period but adds little to\nHe expected value buying in pivotals including Bank Muscat and National Bank of Oman.\nThe first inequality is a comparison of the expected value of a ratio to the ratio of the expected value, a problem that arises in pricing foreign exchange rates.\nThe expected value of the contract is approximately \\$140 million dollars.\n\nSite: Follow: Share:\nOpen / Close"
] |
[
null
] |
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|
https://www.maplesoft.com/support/help/maplesim/view.aspx?path=componentLibrary/electrical/spice3/sources/current/I_pulse&L=E
|
[
"",
null,
"I_pulse - MapleSim Help\n\nI_pulse\n\nPulse current source",
null,
"Description\n\nThe I_pulse component models a periodic pulse source with unlimited number of periods. A single pulse is described by the following table:\n\n time value 0 I1 TD I1 TD+TR I2 TD+TR+PW I2 TD+TR+PW+TF I1 TSTOP I1\n\nIntermediate points are determined by linear interpolation. A pulse it looks like a saw tooth, use this parameters e.g.:\n\n Parameter Value I1 0 I2 1 TD 0 TR 1 TF 1 PW 2 PER 1\n\nNote:\n\n • All parameters of sources should be set explicitly.\n • since TSTEP and TSTOP are not available for modeling in Modelica, differences to SPICE may occur if not all parameters are set.\n Equations $0={i}_{p}+{i}_{n}$ $i={i}_{p}={I}_{1}+\\left\\{\\begin{array}{cc}0& t<{T}_{\\mathrm{D}}\\vee \\mathrm{counter2}=0\\vee {T}_{0}+\\mathrm{Tfalling}\\le t\\\\ \\left(t-{T}_{0}\\right)\\frac{{I}_{2}-{I}_{1}}{\\mathrm{Trising}}& t<{T}_{0}+\\mathrm{Trising}\\\\ {I}_{2}-{I}_{1}& t<{T}_{0}+\\mathrm{Twidth}\\\\ \\left({T}_{0}+\\mathrm{Tfalling}-t\\right)\\frac{{I}_{2}-{I}_{1}}{\\mathrm{Tfalling}-\\mathrm{Twidth}}& \\mathrm{otherwise}\\end{array}$ $v={v}_{p}-{v}_{n}$\n\nVariables\n\n Name Units Description Modelica ID $v$ $V$ Voltage drop between the two pins $\\left({v}_{p}-{v}_{n}\\right)$ v $i$ $A$ Current flowing from pin p to pin n i\n\nConnections\n\n Name Description Modelica ID $p$ Positive pin p $n$ Negative pin n\n\nParameters\n\n Name Default Units Description Modelica ID ${I}_{1}$ $0$ $A$ Initial value I1 ${I}_{2}$ $0$ $A$ Pulsed value I2 ${T}_{\\mathrm{D}}$ $0$ $s$ Delay time TD ${T}_{R}$ $1$ $s$ Rise time TR ${T}_{F}$ ${T}_{R}$ $s$ Fall time TF $\\mathrm{PW}$ $\\mathrm{\\infty }$ $s$ Pulse width PW $\\mathrm{PER}$ $\\mathrm{\\infty }$ $s$ Period PER\n\n Modelica Standard Library The component described in this topic is from the Modelica Standard Library. To view the original documentation, which includes author and copyright information, click here."
] |
[
null,
"https://bat.bing.com/action/0",
null,
"https://www.maplesoft.com/support/help/content/11797/image7.png",
null
] |
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|
http://ccspp.org.br/index.php/books/commutative-algebra-and-its-applications-de-gruyter-proceedings-in-mathematics
|
[
"",
null,
"By Marco Fontana, Salah-Eddine Kabbaj, Bruce Olberding, Irena Swanson\n\nISBN-10: 311020746X\n\nISBN-13: 9783110207460\n\nThis quantity comprises chosen refereed papers in line with lectures awarded on the ??;Fifth foreign Fez convention on Commutative Algebra and Applications?? that used to be held in Fez, Morocco in June 2008. the amount represents new traits and components of classical study in the box, with contributions from many alternative international locations. furthermore, the amount has as a different concentration the learn and effect of Alain Bouvier on commutative algebra during the last thirty years.\n\nSimilar abstract books\n\nNew PDF release: Cohomology of finite groups\n\nAdem A. , Milgram R. J. Cohomology of finite teams (Springer, 1994)(ISBN 354057025X)\n\nSyzygies and Homotopy Theory by F.E.A. Johnson PDF\n\nCrucial invariant of a topological area is its basic staff. whilst this can be trivial, the ensuing homotopy thought is easily researched and common. within the normal case, even if, homotopy idea over nontrivial basic teams is far extra difficult and much much less good understood. Syzygies and Homotopy thought explores the matter of nonsimply attached homotopy within the first nontrivial situations and offers, for the 1st time, a scientific rehabilitation of Hilbert's approach to syzygies within the context of non-simply hooked up homotopy idea.\n\nAdditional resources for Commutative Algebra and its Applications (De Gruyter Proceedings in Mathematics)\n\nExample text\n\nOn -divisorial ideals. The following is a characterization of -Mori rings in terms of Mori rings in the sense of . 34 A. 2]). Let R 2 H . R/ is a Mori ring. The following is a characterization of -Mori rings in terms of Mori domains. 5]). Let R 2 H . R/ is a Mori domain. 7]). Let R 2 H be a -Mori ring. c. on nonnil divisorial ideals of R. In particular, R is a Mori ring. 3 is not valid as it can be seen by the following example. 8]). L=D/, the idealization of L=D over D. Then R 2 H is a Mori ring which is not a -Mori ring.\n\nLet R 2 H be a -Mori ring and I be a nonzero -divisorial ideal of R. Then I contains a power of its radical. We recall a few definitions regarding special types of ideals in integral domains. II 1 /v D D. We will extend these concepts to the rings in H . Let I be a nonnil ideal of a ring R 2 H . II 1 /v D R 36 A. Badawi and -v-invertible if . R/. I / is, respectively, strong, strongly divisorial or v-invertible. In [51, Proposition 1], J. Querré proved that if P is a prime ideal of a Mori domain D, then P is divisorial when it is height one.\n\nLet R 2 H . R/ is ring-isomorphic to a ring A obtained from the following pullback diagram: A # T ! A=M # ! T =M where T is a zero-dimensional quasilocal ring with maximal ideal M , A=M is a ZPUI ring that is a subring of T =M , the vertical arrows are the usual inclusion maps, and the horizontal arrows are the usual surjective maps. 7 -Krull rings We say that a ring R 2 H is a discrete -chained ring if R is a -chained ring with at most one nonnil prime ideal and every nonnil ideal of R is principal."
] |
[
null,
"https://images-na.ssl-images-amazon.com/images/I/41trmU%2BnoKL._SX354_BO1,204,203,200_.jpg",
null
] |
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|
http://raganwald.com/2018/09/08/practical-use-for-the-y-combinator.html
|
[
"This is an unpublished work. It contains errors and/or omissions. Please do not share it with others.\n\nThis essay has been incorporated into Why Y? Deriving the Y Combinator in JavaScript\n\nThe Y Combinator is an important result in theoretical computer science. A famous technology investment firm and startup incubator takes its name from the Y Combinator, likely because the Y Combinator acts as a kind of “bootstrap” to allow a function to build upon itself.\n\nIn this essay, after a brief review of the work we’ve already done on the Mockingbird, Why Bird, M Combinator, and Y Combinator, we’ll derive the “Decoupled Trampoline,” a/k/a “Long-Tailed Widowbird.” The decoupled trampoline builds on the why bird and Y Combinator to allow us to write tail-recursive functions that execute in constant stack space, while hewing closely to idomatic JavaScript.\n\nWhile this use case is admittedly rare in production code, it does arise from time to time and it is pleasing to contemplate a direct connection between one of programming’s most cerebrally theoretical constructs, and a tool for overcoming the limitations of today’s JavaScript implementations.",
null,
"### revisiting the why bird\n\nThis review of the why bird recapitulates the material from To Grok a Mockingbird and Deriving the Y Combinator and Why Bird from the Mockingbird. Readers familiar with these two essays can skim it quickly.\n\nIn To Grok a Mockingbird, we explored the mockingbird, a recursive combinator that decouples recursive functions from themselves. We explored how writing recursive functions “in mockingbird form” decreases couplingand helps us increase reuse and composition.1\n\nWe then moved on to Deriving the Y Combinator and Why Bird from the Mockingbird, where we derived the Why Bird. Although it has tremendous application to combinatory logic as a fixed point combinator, our interest in the why bird was how it helps us obtain all of the benefits of the mockingbird, but we saw that functions written”in why bird form” were much closer to idiomatic JavaScript.\n\nThis is the compact expression of the why bird:\n\n``````const why =\nfn =>\n(x => x(x))(\nmaker =>\n(...args) =>\nfn(maker(maker), ...args)\n);\n``````\n\nWith the why bird, instead of writing a recursive function like this:\n\n``````const isEven =\nn =>\n(n === 0) || !isEven(n - 1);\n``````\n\nIn why bird form, we write it like this:\n\n``````const _isEven =\n(myself, n) =>\n(n === 0) || !myself(n - 1);\n``````\n\nWe use the why bird in conjunction with a function written “in why bird form” like this:\n\n``````why(_isEven)(42)\n//=> true\n``````\n\nThis arrangement decouples the recursive function from itself, allowing us to use an anonymous function if we wish, like this:\n\n``````why(\n(myself, n) =>\n(n === 0) || !myself(n - 1)\n)(42)\n//=> true\n``````\n\nIt also allows us to decorate the recursive function easily, whether anonymous or not.\n\nThe why bird is an idiomatic JavaScript version of combinatorial logic’s Y Combinator. The Y Combinator works with functions in curried form, i.e. functions that take only one argument.\n\nThe full expression of the Y Combinator looks like this:\n\n``````const Y =\nfn =>\n(m => a => fn(m(m))(a))(\nm => a => fn(m(m))(a)\n);\n``````\n\nIts compact form, like the compact why bird, makes use of the M Combinator, reduced to `x => x(x)`:\n\n``````const Y =\nfn =>\n(x => x(x))(m => a => fn(m(m))(a));\n``````\n\nEither expression of the Y Combinator is used with functions in curried form:\n\n``````Y(\nmyself =>\nn =>\n(n === 0) || !myself(n - 1)\n)(1962)\n``````\n\nWe will work with the why bird in this essay, however everything we do can also be done with the Y Combinator, albeit by writing functions in a form that is not usual for JavaScript.",
null,
"### tail recursion\n\nThis function for determining whether a number is even is extremely slow, and it has another problem:\n\n``````const isEven =\nn =>\n(n === 0) || !isEven(n - 1);\n\nisEven(1000042)\n//=> Maximum call stack exceeded\n``````\n\nRevising it to work with the why bird does not fix the issue:\n\n``````why(\n(myself, n) =>\n(n === 0) || !myself(n - 1)\n)(1000042)\n//=> Maximum call stack exceeded\n``````\n\nOur function consumes stack space equal to the magnitude of the argument `n`. Naturally, this is a contrived example, but recursive functions that consume the entire stack to occur from time to time, and it is not always appropriate to rewrite them in iterative form.\n\nOne solution to this problem is to rewrite the function in tail-recursive form. If the JavaScript engine supports tail-call optimization, the function will execute in constant stack space:\n\n``````// Safari Browser, c. 2018\n\nwhy(\n(myself, n) => {\nif (n === 0)\nreturn true;\nelse if (n === 1)\nreturn false;\nelse return myself(n - 2);\n}\n)(1000042)\n//=> true\n``````\n\nHowever, not all engines support tail-call optimization, despite it being part of the JavaScript specification. If we wish to execute such a function in constant stack space, one of our options is to “greenspun” tail-call optimization ourselves by implementing a trampoline:2\n\nA trampoline is a loop that iteratively invokes thunk-returning functions (continuation-passing style). A single trampoline is sufficient to express all control transfers of a program; a program so expressed is trampolined, or in trampolined style; converting a program to trampolined style is trampolining. Trampolined functions can be used to implement tail-recursive function calls in stack-oriented programming languages.–Wikipedia\n\nAs we saw in To Grok a Mockingbird, this necessitates having our recursive function become tightly coupled to its execution strategy. In other words, above and beyond being rewritten in tail-recursive form, it must explicitly return thunks rather than call `myself`:\n\n``````class Thunk {\nconstructor (delayed) {\nthis.delayed = delayed;\n}\n\nevaluate () {\nreturn this.delayed();\n}\n}\n\nconst trampoline =\nfn =>\n(...initialArgs) => {\nlet value = fn(...initialArgs);\n\nwhile (value instanceof Thunk) {\nvalue = value.evaluate();\n}\n\nreturn value;\n};\n\nconst isEven =\ntrampoline(\nfunction myself (n, parity = 0) {\nif (n === 0) {\nreturn parity === 0;\n} else {\nreturn new Thunk(() => myself(n - 1, 1 - parity));\n}\n}\n);\n\nisEven(1000001)\n//=> false\n``````\n\nIn To Grok a Mockingbird, we solved this problem for functions written “in mockingbird form” with the Jackson’s Widowbird function. We created a function with the same contract as the mockingbird, but its implementation used a trampoline to execute recursive functions in constant stack space.\n\nFunctions written “in why bird form” are more idiomatically JavaScript than functions written in mockingbird form. If we can create a similar function that has the same contract as the why bird, but uses a trampoline to evaluate the recursive function, we could execute tail-recursive functions in constant stack space.\n\nWe will call this function the “Long-tailed Widowbird.” Let’s derive it.",
null,
"### deriving the long-tailed widowbird from the why bird\n\nOur goal is to create a trampolining function. So let’s start with the basic outline of a trampoline, and call it `longtailed`:\n\n``````class Thunk {\nconstructor (delayed) {\nthis.delayed = delayed;\n}\n\nevaluate () {\nreturn this.delayed();\n}\n}\n\nconst longtailed =\nfn =>\n(...initialArgs) => {\nlet value = fn(...initialArgs);\n\nwhile (value instanceof Thunk) {\nvalue = value.evaluate();\n}\n\nreturn value;\n};\n``````\n\nWe won’t even bother trying this, we know that `fn(...initialArgs)` is not going to work without injecting a function for `myself`. But we do know a function that we can call with `...initialArgs`:\n\n``````class Thunk {\nconstructor (delayed) {\nthis.delayed = delayed;\n}\n\nevaluate () {\nreturn this.delayed();\n}\n}\n\nconst longtailed =\nfn =>\n(...initialArgs) => {\nlet value = why(fn)(...initialArgs);\n\nwhile (value instanceof Thunk) {\nvalue = value.evaluate();\n}\n\nreturn value;\n};\n``````\n\nThis works, but never actually creates any thunks. To do that, let’s reduce `why(fn)`:\n\n``````class Thunk {\nconstructor (delayed) {\nthis.delayed = delayed;\n}\n\nevaluate () {\nreturn this.delayed();\n}\n}\n\nconst longtailed =\nfn =>\n(...initialArgs) => {\nlet value =\n(x => x(x))(\nmaker =>\n(...args) =>\nfn(maker(maker), ...args)\n)(...initialArgs);\n\nwhile (value instanceof Thunk) {\nvalue = value.evaluate();\n}\n\nreturn value;\n};\n``````\n\nNow we see where the value for `myself` comes from, it’s `maker(maker)`. Let’s replace that with a function that, given some arguments, returns a new thunk that—when evaluated—returns `maker(maker)` invoked with those arguments:\n\n``````class Thunk {\nconstructor (delayed) {\nthis.delayed = delayed;\n}\n\nevaluate () {\nreturn this.delayed();\n}\n}\n\nconst longtailed =\nfn =>\n(...initialArgs) => {\nlet value =\n(x => x(x))(\nmaker =>\n(...args) =>\nfn((...argsmm) => new Thunk(() => maker(maker)(...argsmm)), ...args)\n)(...initialArgs);\n\nwhile (value instanceof Thunk) {\nvalue = value.evaluate();\n}\n\nreturn value;\n};\n\nlongtailed(\n(myself, n) => {\nif (n === 0)\nreturn true;\nelse if (n === 1)\nreturn false;\nelse return myself(n - 2);\n}\n)(1000042)\n//=> true\n``````\n\nIt works! And it executes in constant stack space! But this Is code that only its author could love.",
null,
"### from long-tailed widowbird to decoupled trampoline\n\nLet’s begin our cleanup by moving `Thunk` inside our function. This has certain technical advantages if we ever create a recursive program that itself returns thunks. Since it is now a special-purpose class that only ever invokes a single function, we’ll give it a more specific implementation:\n\n``````const longtailed =\nfn => {\nclass Thunk {\nconstructor (fn, ...args) {\nthis.fn = fn;\nthis.args = args;\n}\n\nevaluate () {\nreturn this.fn(...this.args);\n}\n}\n\nreturn (...initialArgs) => {\nlet value =\n(x => x(x))(\nmaker =>\n(...args) =>\nfn((...argsmm) => new Thunk(maker(maker), ...argsmm), ...args)\n)(...initialArgs);\n\nwhile (value instanceof Thunk) {\nvalue = value.evaluate();\n}\n\nreturn value;\n};\n};\n``````\n\nNext, let’s extract the creation of a function that delays the invocation of `maker(maker)`:\n\n``````const longtailed =\nfn => {\nclass Thunk {\nconstructor (fn, ...args) {\nthis.fn = fn;\nthis.args = args;\n}\n\nevaluate () {\nreturn this.fn(...this.args);\n}\n}\n\nconst thunkify =\nfn =>\n(...args) =>\nnew Thunk(fn, ...args);\n\nreturn (...initialArgs) => {\nlet value =\n(x => x(x))(\nmaker =>\n(...args) =>\nfn(thunkify(maker(maker)), ...args)\n)(...initialArgs);\n\nwhile (value instanceof Thunk) {\nvalue = value.evaluate();\n}\n\nreturn value;\n};\n};\n``````\n\nAnd now we have a considerably less ugly long-tailed widowbird. Well, actually, we are ignoring “the elephant in the room,” the name of the function. “Long-tailed Widowbird” is a touching tribute to the genius of Raymond Smullyan, and there is an amusing correlation between its long tail and the business of optimizing tail-recursive functions.\n\nNevertheless, if we are to work with others, we might want to consider the possibility that they would prefer a less poetic approach:\n\n``````const decoupledTrampoline =\nfn => {\nclass Thunk {\nconstructor (fn, ...args) {\nthis.fn = fn;\nthis.args = args;\n}\n\nevaluate () {\nreturn this.fn(...this.args);\n}\n}\n\nconst thunkify =\nfn =>\n(...args) =>\nnew Thunk(fn, ...args);\n\nreturn (...initialArgs) => {\nlet value =\n(x => x(x))(\nmaker =>\n(...args) =>\nfn(thunkify(maker(maker)), ...args)\n)(...initialArgs);\n\nwhile (value instanceof Thunk) {\nvalue = value.evaluate();\n}\n\nreturn value;\n};\n};\n``````\n\nAnd there we have our decoupled trampoline in its final form.",
null,
"### summarizing the use case for the decoupled trampoline\n\nTo recapitulate the use case for the decoupled trampoline, in the rare but nevertheless valid case where we wish to refactor a singly recursive function into a trampolined function to ensure that it does not consume the stack, we previously had to:\n\n1. Refactor the function into tail-recursive form;\n2. Refactor the tail-recursive version to explicitly invoke a trampoline;\n3. Wrap the result in a trampoline function.\n\nWith the decoupled trampoline, we can:\n\n1. Refactor the function into tail-recursive form;\n2. Refactor the function into “why bird form,” then;\n3. Wrap the result in the decoupled trampoline.\n\nWhy is this superior? We’re going to refactor into tail-recursive form either way, and we’re going to wrap the function either way, however:\n\n1. Refactoring into “why bird form” is less intrusive than rewriting the code to explicitly return thunks, and;\n2. The refactored code is decoupled from trampolining, so it is easier to reverse the procedure if need be, or even just used with the why bird;\n\nIf we compare and contrast:\n\n``````const isEven =\ntrampoline(\nfunction myself (n) {\nif (n === 0)\nreturn true;\nelse if (n === 1)\nreturn false;\nelse return new Thunk(() => myself(n - 2));\n}\n);\n``````\n\nWith:\n\n``````const isEven =\ndecoupledTrampoline(\n(myself, n) => {\nif (n === 0)\nreturn true;\nelse if (n === 1)\nreturn false;\nelse return myself(n - 2);\n}\n);\n``````\n\nThe latter has clearer separation of concerns and is thus easier to grok at first sight. And thus, we have articulated a practical (albeit infrequently needed) use for the Y Combinator.\n\nThat’s all!\n\nThe essays in this series on recursive combinators are: To Grok a Mockingbord, Deriving the Y Combinator and Why Bird from the Mockingbird, and A practical (albeit infrequently needed) use for the Y Combinator. Enjoy them all!\n\n1. The mockingbird is more formally known as the M Combinator. Our naming convention is that when discussing formal combinators from combinatory logic, or direct implementations in JavaScript, we will use the formal name. But when using variations designed to work more idiomatically in JavaScript–such as versions that work with functions taking more than one argument), we will use Raymond Smullyan’s ornithological nicknames.\n\nFor a formalist, the M Combinator’s direct translation is `const M = fn => fn(fn)`. This is only useful if `fn` is implemented in “curried” form, e.g. `const isEven = myself => n => n === 0 || !myself(n - 1)`. If we wish to use a function written in idiomatic JavaScript form, such as `const isEven = (myself, n) => n === 0 || !myself(n - 1)`, we use the mockingbird, which is given later as `const mockingbird = fn => (...args) => fn(fn, ...args)`. This is far more practical for programming purposes.\n\n2. A more complete exploration of ways to convert recursive functions to non-recusrives functions can be found in Recursion? We don’t need no stinking recursion!, and its follow-up, A Trick of the Tail"
] |
[
null,
"http://raganwald.com/assets/images/y2.jpg",
null,
"http://raganwald.com/assets/images/spiral.jpg",
null,
"http://raganwald.com/assets/images/long-tailed-widowbird.jpg",
null,
"http://raganwald.com/assets/images/ink-and-water.jpg",
null,
"http://raganwald.com/assets/images/schickard.jpg",
null
] |
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|
https://convertoctopus.com/21-3-knots-to-miles-per-hour
|
[
"Conversion formula\n\nThe conversion factor from knots to miles per hour is 1.1507794480225, which means that 1 knot is equal to 1.1507794480225 miles per hour:\n\n1 kt = 1.1507794480225 mph\n\nTo convert 21.3 knots into miles per hour we have to multiply 21.3 by the conversion factor in order to get the velocity amount from knots to miles per hour. We can also form a simple proportion to calculate the result:\n\n1 kt → 1.1507794480225 mph\n\n21.3 kt → V(mph)\n\nSolve the above proportion to obtain the velocity V in miles per hour:\n\nV(mph) = 21.3 kt × 1.1507794480225 mph\n\nV(mph) = 24.51160224288 mph\n\nThe final result is:\n\n21.3 kt → 24.51160224288 mph\n\nWe conclude that 21.3 knots is equivalent to 24.51160224288 miles per hour:\n\n21.3 knots = 24.51160224288 miles per hour\n\nAlternative conversion\n\nWe can also convert by utilizing the inverse value of the conversion factor. In this case 1 mile per hour is equal to 0.040797006662037 × 21.3 knots.\n\nAnother way is saying that 21.3 knots is equal to 1 ÷ 0.040797006662037 miles per hour.\n\nApproximate result\n\nFor practical purposes we can round our final result to an approximate numerical value. We can say that twenty-one point three knots is approximately twenty-four point five one two miles per hour:\n\n21.3 kt ≅ 24.512 mph\n\nAn alternative is also that one mile per hour is approximately zero point zero four one times twenty-one point three knots.\n\nConversion table\n\nknots to miles per hour chart\n\nFor quick reference purposes, below is the conversion table you can use to convert from knots to miles per hour\n\nknots (kt) miles per hour (mph)\n22.3 knots 25.662 miles per hour\n23.3 knots 26.813 miles per hour\n24.3 knots 27.964 miles per hour\n25.3 knots 29.115 miles per hour\n26.3 knots 30.265 miles per hour\n27.3 knots 31.416 miles per hour\n28.3 knots 32.567 miles per hour\n29.3 knots 33.718 miles per hour\n30.3 knots 34.869 miles per hour\n31.3 knots 36.019 miles per hour"
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[
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http://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/chapter13/section05.html
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[
"## 13.5 Solving Two General Equations in Two Variables\n\nOne nice feature of Newton's Method (and of Poor Man's Newton as well) is that it can easily be generalized to two or even three dimensions.\n\nThat is, suppose we have two standard functions, f and g of two variables, x and y. Each of the equations, f(x, y) = 0, and g(x, y) = 0 will typically be satisfied on curves, just as similar linear equations are satisfied on straight lines. And suppose we seek simultaneous solutions to both equations, which will then be at the intersections, if any, of these curves.\n\nIf we could solve either of the equations for x say, in terms of y, we could find a parametric representations of the curve solutions to it (with y as parameter), and use the divide and conquer method of the last section on the other function, cutting the parameter interval in half at each step as in one dimension.\n\nThis is a slow and steady method that can be implemented fairly easily. But it assumes we can obtain a parametric representation of one or the other curve.\n\nWe can always try Newton's method, which is fairly easy to implement in general.\n\nTo use Newton's method, we compute the gradients of f and g at some initial point, and find a new point at which the linear approximations to f and g defined by the gradients at the initial point are both 0. We then iterate this step.\n\nTo implement this method takes roughly three times the work of using Newton's method in one dimension. On the other hand three times almost nothing is still small.\n\nThis method suffers from the same problems as Newton's method suffers from in one dimension.\n\nWe may wander far from where we want to be in the xy plane, especially if we come to a point at which the gradients are small.\n\nAnd of course two random curves need not intersect at all, so there may not even be a solution, or there could be many of them.\n\nBut again if we can start near a solution, under the right circumstances the method does very well indeed.\n\nHow do we set it up?\n\nFirst you pick initial guesses of x0 and y0; on the spreadsheet, enter them in the first two columns (say). Then you need a column to enter f(x, y) and one for g(x, y); and one each for the x and y derivatives of f and g (four derivatives all together).\n\nYou now have all the information you need to compute x1 and y1. Once you have done this you need only copy everything down say 30 rows, and you can see what happens. If f and g go to zero, which means that xi and yi both converge, you will have a solution.\n\nSo how do we iterate?\n\nWe must solve the two linear equations in x and y that state that the linear approximations to f and to g defined at (x0, y0) both are 0.\n\nWhat are these equations? They are",
null,
"And what are the solutions?\n\nWe can use Cramer's (ratio of determinant) rule to tell us that the solutions are",
null,
"and",
null,
"Of course all further iterations are identical to this one with the new guesses. Thus after entering these formulae once, copying them down is all that is necessary to apply the method.\n\nExercise 13.10 Try this method on a spreadsheet with the following function: f(x, y) = exp(x * y) - y2 , g(x, y) = cos(x + y).\nFind three solutions for which both variables are positive. How many such solutions are there?\n\nThe same approach can be implemented in three dimensions, though the amount of work required on a spreadsheet starts to be slightly tedious.\n\nYou have to enter three variables, three functions, their nine partial derivatives and of course Cramer's rule now involves the ratio of two three by three determinants each, now three times.\n\nYou could do it, though if you ever really wanted to, and actually find the solution to three arbitrary non-linear equations in three variables, with reasonable luck."
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[
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"http://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/chapter13/equations/sections_eqn15.gif",
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"http://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/chapter13/equations/sections_eqn16.gif",
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"http://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/chapter13/equations/sections_eqn17.gif",
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https://supercustomessay.com/essay-stock-price-valuation/2832/
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[
"# Essay: Stock Price Valuation\n\n###### Essay: Stock Price Valuation\n\nSample Essay\n\nThe expected value of stock can be determined through several techniques which include Price to Earnings Ratio – P/E ratio, discounting of expected future dividends and discounted expected future free cash flows. The common factor in the last two methods is dependence on expected future values instead of current financial statement of market values. Expected future values of dividends or cash flows are estimated by implementing an appropriate growth rate. The P/E ratio on the other hand estimates the stock price based on earnings per share. The main question is whether two investors should pay the same price for a stock or not. The answer to this question is in the affirmative when the two investors expect the pattern of future dividends to be similar based on a similar growth rate.\n\nThe dividends discounts model approach uses expected dividends, a discount rate such as cost of common stock and a growth rate to arrive at a current price of the stock. If there is a difference in the growth rate, expected future dividends or discount rates the dividend discount model would yield different stock prices for the two investors. The difference in time periods for estimation of stock prices would not be considered substantial as both investors would discount the future dividends to present values which would be similar.\n\nPlease go to the order form to order essays, research papers, term papers, thesis, dissertation, case study, assignments on this essay topic.\n\n#### Related Essays, Research Papers, Term Papers, Thesis, Dissertation, Case Study, Assignments entries.",
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"Tags"
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[
null,
"https://supercustomessay.com/wp-content/uploads/2011/10/optn_01.png",
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{"ft_lang_label":"__label__en","ft_lang_prob":0.932664,"math_prob":0.84534866,"size":1626,"snap":"2022-40-2023-06","text_gpt3_token_len":299,"char_repetition_ratio":0.14488286,"word_repetition_ratio":0.0,"special_character_ratio":0.18081181,"punctuation_ratio":0.072916664,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97634786,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-02-03T14:16:56Z\",\"WARC-Record-ID\":\"<urn:uuid:35893bd5-5f18-4d0e-acdf-a2893a05a0fa>\",\"Content-Length\":\"70021\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:99aa2114-abcb-49bd-8170-31d06db4fd30>\",\"WARC-Concurrent-To\":\"<urn:uuid:c54e665d-1175-4732-8bf1-99b4510df0e0>\",\"WARC-IP-Address\":\"5.189.163.89\",\"WARC-Target-URI\":\"https://supercustomessay.com/essay-stock-price-valuation/2832/\",\"WARC-Payload-Digest\":\"sha1:MAGT6NZ737ISPYRE6P5W6AH4MPIK47HV\",\"WARC-Block-Digest\":\"sha1:V6ANRZ6T42ED47JV6P7EWNEXCTJBHP2R\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764500056.55_warc_CC-MAIN-20230203122526-20230203152526-00822.warc.gz\"}"}
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https://math.stackexchange.com/questions/2392148/how-do-you-write-the-following-statements-in-predicate-logic
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[
"# How do you write the following statements in predicate logic?\n\nWhich symbolization is right for the following statements ?\n\nThe statements:\n\n1. Some computer science students are not regular.\n2. Every user in this website has some rank.\n3. All the questions related to programming has an answer on stackoverflow.\n\nSybolization:\n\nFor 1st statement:\n\na. $\\exists$xP(x)$\\rightarrow\\neg$Q(x) (P(x) = cs students, Q(x) = are regular)\n\nb. is it $\\exists$xP(x)$\\land\\neg$Q(x)\n\nwhich one is it \"a\" or \"b\" ?\n\nFor 2nd statement:\n\na.$\\forall$P(x)$\\rightarrow$Q(x) (P(x) = user in this website, Q(x) = has some rank)\n\nb.$\\forall$P(x)$\\land$Q(x)\n\nwhich one is it \"a\" or \"b\" ?\n\nFor 3rd statement:\n\na.$\\forall$P(x)$\\rightarrow$Q(x) (P(x) = questions related to programming, Q(x) = has an answer on stackoverflow)\n\nb.$\\forall$P(x)$\\land$Q(x)\n\nwhich one is it \"a\" or \"b\" ?\n\nthanks.\n\n## 2 Answers\n\nThe wording of your statements isn't quite correct. For example, '$P(x)=$ cs students' and '$Q(x)=$ are regular'. What on earth do these statements mean? They're not statements - they don't say anything. They don't mention $x$ in them and we don't even know what $x$ is.\n\nInstead, you should have something like: let $X$ be the set of all students and let $P(x)=$ '$x$ is a cs student', $Q(x)=$ '$x$ is regular'.\n\nYou'll then see both\n\n$$(\\exists x\\in X)\\quad P(x)\\implies\\lnot Q(x)$$\n\nand\n\n$$(\\exists x\\in X)\\quad P(x)\\land\\lnot Q(x)$$\n\nhold true for the first one.\n\nNote that\n\n\n\n$$P(x)\\land\\lnot Q(x)$$ $\\implies$ $$\\lnot Q(x)$$ $\\implies$ $$\\lnot P(x)\\lor\\lnot Q(x)$$ $\\iff$ $$(P(x)\\implies\\lnot Q(x))$$\n\nso the first is just a weaker statement than the second.\n\nSimilar problems with wording of your statements apply for the other two, and yet again you'll see each time that both are true with one of the statements being implied by the other.\n\nEdit: In conclusion, only the b statements are equivalent to the original statement. So although a is true in each case, it isn't what you're after.\n\n• @Ahtisham In fact, taking away the context, you have exactly the same pairs of statements in all three cases (with different quantifiers and substituting $Q(x)$ for $\\lnot Q(x)$ in the first), so just apply the above again to see one implies the other in each case. – Shuri2060 Aug 13 '17 at 12:25\n• really is there no difference between $\\exists$P(x)$\\rightarrow\\neg$Q(x) and $\\exists$P(x)$\\land\\neg$Q(x) ? – Ahtisham Aug 13 '17 at 12:30\n• @Ahtisham No - as shown above, one is a weaker statement than the other. The second implies the first. Eg. $(\\forall x\\in X)\\quad \\lnot P(x)\\Rightarrow (P(x)\\implies\\lnot Q(x))$ but $(\\forall x\\in X)\\quad \\lnot P(x)\\not\\Rightarrow (P(x)\\land\\lnot Q(x))$. – Shuri2060 Aug 13 '17 at 12:56\n• @Ahtisham Ah I see - yes, in which case $∃P(x)→¬Q(x)$ isn't what you're after, although it is true. Only the b statements are equivalent to the original statement. This is because the a statements hold true when $P(x)$ is false regardless of the original statements. – Shuri2060 Aug 13 '17 at 13:03\n\nTechnically, in all 3 cases, it is neither a) nor b), since all statements contain free variables due to missing parentheses.\n\nFor example, the statement $\\exists x P(x) \\rightarrow Q(x)$ will be interpreted as $(\\exists x P(x)) \\rightarrow Q(x)$, meaning that the $x$ in $Q(x)$ is free, meaning that this is a formula, but not a sentence.\n\nSo, to be a sentence, you have to add parentheses, and make it $\\exists x (P(x) \\rightarrow Q(x))$\n\nSimilar for all the other expressions.\n\nNow, in general, you typically want to use a $\\land$ in combination with a $\\exists$, and a $\\rightarrow$ in combination with a $\\forall$. There are always exceptions of course, but if you ever see a $\\rightarrow$ in combination with a $\\exists$, then most likely something is wrong.\n\nAnd so it is in this case. Do yourself a favor and remember the following 4 'Aristotelean Categorical sentence forms':\n\n1. 'All P are Q'. This translates as $\\forall x (P(x) \\rightarrow Q(x))$\n\n2. 'Some p are Q'. This is $\\exists x ( P(x) \\land Q(x))$\n\n3. 'No P are Q' (or, what is the same thing: 'All P are not q'): $\\forall x (P(x) \\rightarrow \\neg Q(x))$\n\n4. 'Some P are not Q': $\\exists x (P(x) \\land \\neg Q(x))$\n\nNotice how for these 4 very common sentence patterns, you indeed get a $\\land$ whenever you have a $\\exists$, and a $\\rightarrow$ whenever you have a $\\forall$"
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https://cs.stackexchange.com/questions/63083/space-complexity-of-string-indices-o1-or-ologs/63106
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[
"# Space complexity of string indices: O(1) or O(log|S|)?\n\nSay you have a string S and wish to store indices of it, e.g. letter at index 3 of \"toast\" is 'a'. Seems that people generally consider an index as taking O(1) space to store*. But doesn't it take O(log(|S|)) space?\n\nIf we use binary bits...\n\n• length 4 string => index must be at least 2 bits\n• length 8 string => index must be at least 3 bits\n• length n string => index must be at least log(n) bits\n\n*An example of where I'm seeing O(1) suggested: educational sources state that a suffix tree has O(|S|) space complexity, where S is the input string, because there are O(|S|) nodes in the tree, and each node is just an index in the input string. This implies that each index takes O(1) space, but this seems untrue to me. Seems like the tree takes O(|S| log(|S|)) space.\n\n*More examples: Many basic data structures (hash tables, BSTs, etc) use memory pointers (equivalent to string indices), and everyone considers these pointers fixed-size.\n\n• Look up the phrase \"transdichotomous model\". – Pseudonym Sep 2 '16 at 3:16\n• @Pseudonym Thanks, that answers it. I hadn't heard that term. So I assume these educational sources are all using the transdichotomous model. (Edited because I misunderstood at first what this means.) I'm still unsure why they use this model, but that's a different discussion to have... I'll go ask a professor. Thanks again! – sudo Sep 2 '16 at 6:58\n• I'm going to suggest that once you've worked out the details, write an answer for your own question. Explaining it will be very educational for you. – Pseudonym Sep 2 '16 at 7:53\n• Just as another comment, one of the things that complicates the way we talk about this is that complexity classes like \"P\" and \"NP\" are defined in terms of Turing machines; a language is in \"P\" if its language can be accepted in $O(p(n))$ time on a deterministic Turing machine where $p$ is a polynomial and $n$ is the number of starting symbols on the tape. This is fine for broad classes like \"P\", because \"polynomial\" on a TM is almost always \"polynomial\" on a word RAM machine. But finer distinctions (e.g. \"linear\")... not so much. – Pseudonym Sep 4 '16 at 23:24\n• The answer I got: 2^64 is far beyond any input we'd ever receive, and we'll just assume 64-bit integers are always used. This is reasonable in the real world, but I don't see the harm in including the log term, and I do see harm in leaving it out since big-O is supposed to consider arbitrarily large input. Also, I've looked around and have seen things similar to what you say about TMs. Makes some sense, but since there are ways to abuse it, it doesn't sit well with me. – sudo Sep 11 '16 at 17:53\n\nThese are correct (unless you explicitly specify a non-standard model of computing):\n\n• $O(1)$ space,\n• $O(1)$ words of space,\n• $O(\\log|S|)$ bits of space.\n• I don't understand how bullets 1 and 3 are not contradictory. – Raphael Sep 2 '16 at 18:40\n• @Raphael: \"$O(1)$ space\" without any further details in the context of algorithms and data structures almost always means \"$O(1)$ words of space in the word-RAM model with $\\Theta(\\log n)$-bit words\". And bits are always bits, without any ambiguity. – Jukka Suomela Sep 2 '16 at 18:46\n• I see. Maybe you want to add that to your answer. – Raphael Sep 2 '16 at 20:16\n• And are words the same as string indexes if you're using the model where words are O(1)? In the data structure I described, the index of the string, not a RAM address, is being stored. You can change the index size without changing the machine's word size. – sudo Sep 4 '16 at 21:11\n\nDepends on which model you are interested in.\n\nOn RAMs with the uniform or unit cost model, (not too large) numbers take constant space to store, and constant time to work with.\n\nOn RAMs with the logarithmic cost model and TMs, they take logarithmic space."
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https://ixtrieve.fh-koeln.de/birds/litie/document/8409
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[
"# Document (#8409)\n\nAuthor\nHayashi, K\nSekilima, A.\nTitle\nMediating interface between hypertext and structured documents\nSource\nElectronic publishing. 6(1993) no.4, S.423-434\nYear\n1993\nAbstract\nDescribes a unified document model for an authoring system that focuses on the stage of drafting and revising and takes advantage of both hypertext and structured document models. Discusses the underlying structure and the surface of the document models and its key features. Describes Nelumbo, a prototype system currently being developed which integrates different types of editors that ahndle features of hypertext and structured documents. Users can use any of the tools at will, and editing with the tools affects the underlying structure consistently\nTheme\nElektronisches Publizieren\nHypertext\n\n## Similar documents (content)\n\n1. Richy, H.: ¬A hypertext electronic index based on the Grif structured document editor (1994) 0.31\n```0.31072018 = sum of:\n0.31072018 = product of:\n0.97100055 = sum of:\n0.09928713 = weight(abstract_txt:advantage in 2703) [ClassicSimilarity], result of:\n0.09928713 = score(doc=2703,freq=1.0), product of:\n0.13189219 = queryWeight, product of:\n6.0223207 = idf(docFreq=286, maxDocs=43556)\n0.021900559 = queryNorm\n0.7527901 = fieldWeight in 2703, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n6.0223207 = idf(docFreq=286, maxDocs=43556)\n0.125 = fieldNorm(doc=2703)\n0.04904334 = weight(abstract_txt:system in 2703) [ClassicSimilarity], result of:\n0.04904334 = score(doc=2703,freq=2.0), product of:\n0.0824158 = queryWeight, product of:\n1.1179199 = boost\n3.3662362 = idf(docFreq=4086, maxDocs=43556)\n0.021900559 = queryNorm\n0.5950721 = fieldWeight in 2703, product of:\n1.4142135 = tf(freq=2.0), with freq of:\n2.0 = termFreq=2.0\n3.3662362 = idf(docFreq=4086, maxDocs=43556)\n0.125 = fieldNorm(doc=2703)\n0.050452266 = weight(abstract_txt:describes in 2703) [ClassicSimilarity], result of:\n0.050452266 = score(doc=2703,freq=1.0), product of:\n0.1058167 = queryWeight, product of:\n1.2667257 = boost\n3.8143141 = idf(docFreq=2610, maxDocs=43556)\n0.021900559 = queryNorm\n0.47678927 = fieldWeight in 2703, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n3.8143141 = idf(docFreq=2610, maxDocs=43556)\n0.125 = fieldNorm(doc=2703)\n0.07572783 = weight(abstract_txt:structure in 2703) [ClassicSimilarity], result of:\n0.07572783 = score(doc=2703,freq=1.0), product of:\n0.13871947 = queryWeight, product of:\n1.4503545 = boost\n4.36725 = idf(docFreq=1501, maxDocs=43556)\n0.021900559 = queryNorm\n0.54590625 = fieldWeight in 2703, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n4.36725 = idf(docFreq=1501, maxDocs=43556)\n0.125 = fieldNorm(doc=2703)\n0.121118456 = weight(abstract_txt:features in 2703) [ClassicSimilarity], result of:\n0.121118456 = score(doc=2703,freq=2.0), product of:\n0.15057886 = queryWeight, product of:\n1.51108 = boost\n4.550104 = idf(docFreq=1250, maxDocs=43556)\n0.021900559 = queryNorm\n0.80435234 = fieldWeight in 2703, product of:\n1.4142135 = tf(freq=2.0), with freq of:\n2.0 = termFreq=2.0\n4.550104 = idf(docFreq=1250, maxDocs=43556)\n0.125 = fieldNorm(doc=2703)\n0.10756004 = weight(abstract_txt:document in 2703) [ClassicSimilarity], result of:\n0.10756004 = score(doc=2703,freq=1.0), product of:\n0.20064645 = queryWeight, product of:\n2.1363225 = boost\n4.28854 = idf(docFreq=1624, maxDocs=43556)\n0.021900559 = queryNorm\n0.5360675 = fieldWeight in 2703, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n4.28854 = idf(docFreq=1624, maxDocs=43556)\n0.125 = fieldNorm(doc=2703)\n0.22043903 = weight(abstract_txt:structured in 2703) [ClassicSimilarity], result of:\n0.22043903 = score(doc=2703,freq=1.0), product of:\n0.32373515 = queryWeight, product of:\n2.713601 = boost\n5.447392 = idf(docFreq=509, maxDocs=43556)\n0.021900559 = queryNorm\n0.680924 = fieldWeight in 2703, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n5.447392 = idf(docFreq=509, maxDocs=43556)\n0.125 = fieldNorm(doc=2703)\n0.24737243 = weight(abstract_txt:hypertext in 2703) [ClassicSimilarity], result of:\n0.24737243 = score(doc=2703,freq=1.0), product of:\n0.34959486 = queryWeight, product of:\n2.8198993 = boost\n5.6607795 = idf(docFreq=411, maxDocs=43556)\n0.021900559 = queryNorm\n0.70759743 = fieldWeight in 2703, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n5.6607795 = idf(docFreq=411, maxDocs=43556)\n0.125 = fieldNorm(doc=2703)\n0.32 = coord(8/25)\n```\n2. Zhang, Z.; Heuer, A.; Engel, T.; Meinel, C.: DAPHNE - a tool for distributed Web authoring and publishing (1999) 0.29\n```0.2861222 = sum of:\n0.2861222 = product of:\n0.7153055 = sum of:\n0.033925645 = weight(abstract_txt:system in 691) [ClassicSimilarity], result of:\n0.033925645 = score(doc=691,freq=5.0), product of:\n0.0824158 = queryWeight, product of:\n1.1179199 = boost\n3.3662362 = idf(docFreq=4086, maxDocs=43556)\n0.021900559 = queryNorm\n0.41164008 = fieldWeight in 691, product of:\n2.236068 = tf(freq=5.0), with freq of:\n5.0 = termFreq=5.0\n3.3662362 = idf(docFreq=4086, maxDocs=43556)\n0.0546875 = fieldNorm(doc=691)\n0.07018206 = weight(abstract_txt:editing in 691) [ClassicSimilarity], result of:\n0.07018206 = score(doc=691,freq=1.0), product of:\n0.18160278 = queryWeight, product of:\n1.1734152 = boost\n7.0666823 = idf(docFreq=100, maxDocs=43556)\n0.021900559 = queryNorm\n0.3864592 = fieldWeight in 691, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n7.0666823 = idf(docFreq=100, maxDocs=43556)\n0.0546875 = fieldNorm(doc=691)\n0.17037496 = weight(abstract_txt:authoring in 691) [ClassicSimilarity], result of:\n0.17037496 = score(doc=691,freq=5.0), product of:\n0.19183104 = queryWeight, product of:\n1.2060072 = boost\n7.2629623 = idf(docFreq=82, maxDocs=43556)\n0.021900559 = queryNorm\n0.88815117 = fieldWeight in 691, product of:\n2.236068 = tf(freq=5.0), with freq of:\n5.0 = termFreq=5.0\n7.2629623 = idf(docFreq=82, maxDocs=43556)\n0.0546875 = fieldNorm(doc=691)\n0.02783021 = weight(abstract_txt:documents in 691) [ClassicSimilarity], result of:\n0.02783021 = score(doc=691,freq=1.0), product of:\n0.123497844 = queryWeight, product of:\n1.3684696 = boost\n4.1206813 = idf(docFreq=1921, maxDocs=43556)\n0.021900559 = queryNorm\n0.22534975 = fieldWeight in 691, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n4.1206813 = idf(docFreq=1921, maxDocs=43556)\n0.0546875 = fieldNorm(doc=691)\n0.03313092 = weight(abstract_txt:structure in 691) [ClassicSimilarity], result of:\n0.03313092 = score(doc=691,freq=1.0), product of:\n0.13871947 = queryWeight, product of:\n1.4503545 = boost\n4.36725 = idf(docFreq=1501, maxDocs=43556)\n0.021900559 = queryNorm\n0.23883398 = fieldWeight in 691, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n4.36725 = idf(docFreq=1501, maxDocs=43556)\n0.0546875 = fieldNorm(doc=691)\n0.03519969 = weight(abstract_txt:tools in 691) [ClassicSimilarity], result of:\n0.03519969 = score(doc=691,freq=1.0), product of:\n0.14443561 = queryWeight, product of:\n1.4799348 = boost\n4.4563212 = idf(docFreq=1373, maxDocs=43556)\n0.021900559 = queryNorm\n0.24370506 = fieldWeight in 691, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n4.4563212 = idf(docFreq=1373, maxDocs=43556)\n0.0546875 = fieldNorm(doc=691)\n0.052989326 = weight(abstract_txt:features in 691) [ClassicSimilarity], result of:\n0.052989326 = score(doc=691,freq=2.0), product of:\n0.15057886 = queryWeight, product of:\n1.51108 = boost\n4.550104 = idf(docFreq=1250, maxDocs=43556)\n0.021900559 = queryNorm\n0.35190415 = fieldWeight in 691, product of:\n1.4142135 = tf(freq=2.0), with freq of:\n2.0 = termFreq=2.0\n4.550104 = idf(docFreq=1250, maxDocs=43556)\n0.0546875 = fieldNorm(doc=691)\n0.047057517 = weight(abstract_txt:document in 691) [ClassicSimilarity], result of:\n0.047057517 = score(doc=691,freq=1.0), product of:\n0.20064645 = queryWeight, product of:\n2.1363225 = boost\n4.28854 = idf(docFreq=1624, maxDocs=43556)\n0.021900559 = queryNorm\n0.23452953 = fieldWeight in 691, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n4.28854 = idf(docFreq=1624, maxDocs=43556)\n0.0546875 = fieldNorm(doc=691)\n0.13638969 = weight(abstract_txt:structured in 691) [ClassicSimilarity], result of:\n0.13638969 = score(doc=691,freq=2.0), product of:\n0.32373515 = queryWeight, product of:\n2.713601 = boost\n5.447392 = idf(docFreq=509, maxDocs=43556)\n0.021900559 = queryNorm\n0.42130023 = fieldWeight in 691, product of:\n1.4142135 = tf(freq=2.0), with freq of:\n2.0 = termFreq=2.0\n5.447392 = idf(docFreq=509, maxDocs=43556)\n0.0546875 = fieldNorm(doc=691)\n0.10822544 = weight(abstract_txt:hypertext in 691) [ClassicSimilarity], result of:\n0.10822544 = score(doc=691,freq=1.0), product of:\n0.34959486 = queryWeight, product of:\n2.8198993 = boost\n5.6607795 = idf(docFreq=411, maxDocs=43556)\n0.021900559 = queryNorm\n0.3095739 = fieldWeight in 691, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n5.6607795 = idf(docFreq=411, maxDocs=43556)\n0.0546875 = fieldNorm(doc=691)\n0.4 = coord(10/25)\n```\n3. Amann, B.; Scholl, M.: Schema-based authoring and querying of large hypertexts (1995) 0.28\n```0.2796116 = sum of:\n0.2796116 = product of:\n0.8737862 = sum of:\n0.0433486 = weight(abstract_txt:system in 4568) [ClassicSimilarity], result of:\n0.0433486 = score(doc=4568,freq=4.0), product of:\n0.0824158 = queryWeight, product of:\n1.1179199 = boost\n3.3662362 = idf(docFreq=4086, maxDocs=43556)\n0.021900559 = queryNorm\n0.5259744 = fieldWeight in 4568, product of:\n2.0 = tf(freq=4.0), with freq of:\n4.0 = termFreq=4.0\n3.3662362 = idf(docFreq=4086, maxDocs=43556)\n0.078125 = fieldNorm(doc=4568)\n0.1539351 = weight(abstract_txt:authoring in 4568) [ClassicSimilarity], result of:\n0.1539351 = score(doc=4568,freq=2.0), product of:\n0.19183104 = queryWeight, product of:\n1.2060072 = boost\n7.2629623 = idf(docFreq=82, maxDocs=43556)\n0.021900559 = queryNorm\n0.8024515 = fieldWeight in 4568, product of:\n1.4142135 = tf(freq=2.0), with freq of:\n2.0 = termFreq=2.0\n7.2629623 = idf(docFreq=82, maxDocs=43556)\n0.078125 = fieldNorm(doc=4568)\n0.031532668 = weight(abstract_txt:describes in 4568) [ClassicSimilarity], result of:\n0.031532668 = score(doc=4568,freq=1.0), product of:\n0.1058167 = queryWeight, product of:\n1.2667257 = boost\n3.8143141 = idf(docFreq=2610, maxDocs=43556)\n0.021900559 = queryNorm\n0.2979933 = fieldWeight in 4568, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n3.8143141 = idf(docFreq=2610, maxDocs=43556)\n0.078125 = fieldNorm(doc=4568)\n0.05622551 = weight(abstract_txt:documents in 4568) [ClassicSimilarity], result of:\n0.05622551 = score(doc=4568,freq=2.0), product of:\n0.123497844 = queryWeight, product of:\n1.3684696 = boost\n4.1206813 = idf(docFreq=1921, maxDocs=43556)\n0.021900559 = queryNorm\n0.45527524 = fieldWeight in 4568, product of:\n1.4142135 = tf(freq=2.0), with freq of:\n2.0 = termFreq=2.0\n4.1206813 = idf(docFreq=1921, maxDocs=43556)\n0.078125 = fieldNorm(doc=4568)\n0.04732989 = weight(abstract_txt:structure in 4568) [ClassicSimilarity], result of:\n0.04732989 = score(doc=4568,freq=1.0), product of:\n0.13871947 = queryWeight, product of:\n1.4503545 = boost\n4.36725 = idf(docFreq=1501, maxDocs=43556)\n0.021900559 = queryNorm\n0.3411914 = fieldWeight in 4568, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n4.36725 = idf(docFreq=1501, maxDocs=43556)\n0.078125 = fieldNorm(doc=4568)\n0.057926565 = weight(abstract_txt:models in 4568) [ClassicSimilarity], result of:\n0.057926565 = score(doc=4568,freq=1.0), product of:\n0.15872023 = queryWeight, product of:\n1.5513922 = boost\n4.6714907 = idf(docFreq=1107, maxDocs=43556)\n0.021900559 = queryNorm\n0.3649602 = fieldWeight in 4568, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n4.6714907 = idf(docFreq=1107, maxDocs=43556)\n0.078125 = fieldNorm(doc=4568)\n0.1377744 = weight(abstract_txt:structured in 4568) [ClassicSimilarity], result of:\n0.1377744 = score(doc=4568,freq=1.0), product of:\n0.32373515 = queryWeight, product of:\n2.713601 = boost\n5.447392 = idf(docFreq=509, maxDocs=43556)\n0.021900559 = queryNorm\n0.4255775 = fieldWeight in 4568, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n5.447392 = idf(docFreq=509, maxDocs=43556)\n0.078125 = fieldNorm(doc=4568)\n0.34571347 = weight(abstract_txt:hypertext in 4568) [ClassicSimilarity], result of:\n0.34571347 = score(doc=4568,freq=5.0), product of:\n0.34959486 = queryWeight, product of:\n2.8198993 = boost\n5.6607795 = idf(docFreq=411, maxDocs=43556)\n0.021900559 = queryNorm\n0.98889744 = fieldWeight in 4568, product of:\n2.236068 = tf(freq=5.0), with freq of:\n5.0 = termFreq=5.0\n5.6607795 = idf(docFreq=411, maxDocs=43556)\n0.078125 = fieldNorm(doc=4568)\n0.32 = coord(8/25)\n```\n4. Bater, B.: Hyperdocuments through the square window (1994) 0.21\n```0.20908976 = sum of:\n0.20908976 = product of:\n0.7467491 = sum of:\n0.026009161 = weight(abstract_txt:system in 202) [ClassicSimilarity], result of:\n0.026009161 = score(doc=202,freq=1.0), product of:\n0.0824158 = queryWeight, product of:\n1.1179199 = boost\n3.3662362 = idf(docFreq=4086, maxDocs=43556)\n0.021900559 = queryNorm\n0.31558466 = fieldWeight in 202, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n3.3662362 = idf(docFreq=4086, maxDocs=43556)\n0.09375 = fieldNorm(doc=202)\n0.13061827 = weight(abstract_txt:authoring in 202) [ClassicSimilarity], result of:\n0.13061827 = score(doc=202,freq=1.0), product of:\n0.19183104 = queryWeight, product of:\n1.2060072 = boost\n7.2629623 = idf(docFreq=82, maxDocs=43556)\n0.021900559 = queryNorm\n0.6809027 = fieldWeight in 202, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n7.2629623 = idf(docFreq=82, maxDocs=43556)\n0.09375 = fieldNorm(doc=202)\n0.0378392 = weight(abstract_txt:describes in 202) [ClassicSimilarity], result of:\n0.0378392 = score(doc=202,freq=1.0), product of:\n0.1058167 = queryWeight, product of:\n1.2667257 = boost\n3.8143141 = idf(docFreq=2610, maxDocs=43556)\n0.021900559 = queryNorm\n0.35759196 = fieldWeight in 202, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n3.8143141 = idf(docFreq=2610, maxDocs=43556)\n0.09375 = fieldNorm(doc=202)\n0.060342327 = weight(abstract_txt:tools in 202) [ClassicSimilarity], result of:\n0.060342327 = score(doc=202,freq=1.0), product of:\n0.14443561 = queryWeight, product of:\n1.4799348 = boost\n4.4563212 = idf(docFreq=1373, maxDocs=43556)\n0.021900559 = queryNorm\n0.4177801 = fieldWeight in 202, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n4.4563212 = idf(docFreq=1373, maxDocs=43556)\n0.09375 = fieldNorm(doc=202)\n0.06423277 = weight(abstract_txt:features in 202) [ClassicSimilarity], result of:\n0.06423277 = score(doc=202,freq=1.0), product of:\n0.15057886 = queryWeight, product of:\n1.51108 = boost\n4.550104 = idf(docFreq=1250, maxDocs=43556)\n0.021900559 = queryNorm\n0.42657226 = fieldWeight in 202, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n4.550104 = idf(docFreq=1250, maxDocs=43556)\n0.09375 = fieldNorm(doc=202)\n0.16532928 = weight(abstract_txt:structured in 202) [ClassicSimilarity], result of:\n0.16532928 = score(doc=202,freq=1.0), product of:\n0.32373515 = queryWeight, product of:\n2.713601 = boost\n5.447392 = idf(docFreq=509, maxDocs=43556)\n0.021900559 = queryNorm\n0.510693 = fieldWeight in 202, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n5.447392 = idf(docFreq=509, maxDocs=43556)\n0.09375 = fieldNorm(doc=202)\n0.2623781 = weight(abstract_txt:hypertext in 202) [ClassicSimilarity], result of:\n0.2623781 = score(doc=202,freq=2.0), product of:\n0.34959486 = queryWeight, product of:\n2.8198993 = boost\n5.6607795 = idf(docFreq=411, maxDocs=43556)\n0.021900559 = queryNorm\n0.75052047 = fieldWeight in 202, product of:\n1.4142135 = tf(freq=2.0), with freq of:\n2.0 = termFreq=2.0\n5.6607795 = idf(docFreq=411, maxDocs=43556)\n0.09375 = fieldNorm(doc=202)\n0.28 = coord(7/25)\n```\n5. 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{"ft_lang_label":"__label__en","ft_lang_prob":0.67953837,"math_prob":0.9985938,"size":17738,"snap":"2023-14-2023-23","text_gpt3_token_len":6827,"char_repetition_ratio":0.24890041,"word_repetition_ratio":0.48646125,"special_character_ratio":0.5347841,"punctuation_ratio":0.28390244,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9998883,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-06-01T22:52:24Z\",\"WARC-Record-ID\":\"<urn:uuid:45933e5c-5a84-4aa3-9c74-859248ea4e04>\",\"Content-Length\":\"29238\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:3a981a16-f909-4a79-bf70-2fdcb9642596>\",\"WARC-Concurrent-To\":\"<urn:uuid:6747b5b9-e5b7-4980-9ca3-805071c44b5e>\",\"WARC-IP-Address\":\"139.6.160.6\",\"WARC-Target-URI\":\"https://ixtrieve.fh-koeln.de/birds/litie/document/8409\",\"WARC-Payload-Digest\":\"sha1:4FA5J5ZQIP2PTNDDEVQ6ZGLIALHZUJGM\",\"WARC-Block-Digest\":\"sha1:VTDL2TKJ5BUVMTJLTKXKRS5GPNHNH6ZJ\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-23/CC-MAIN-2023-23_segments_1685224648209.30_warc_CC-MAIN-20230601211701-20230602001701-00330.warc.gz\"}"}
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https://www.pingman.com/kb/print-50.html
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[
"## How is MOS calculated in PingPlotter Pro?\n\nQuestion\n\nPingPlotter Pro now has an 'MOS' column and alert. How is MOS calculated in PingPlotter?\n\nSolution\n\nMOS (mean opinion score) is a voice call quality metric. It is used by the VoIP industry to put a number on voice quality. In reality, this number is a subjective 'opinion' rating of call quality given by someone who was just on a voice call.\n\nObviously, with PingPlotter we don't have someone doing a voice call on your line all the time, and then giving an opinion number. In fact, there's probably not even a voice call going on over your network much of the time for us to measure.\n\nThere are a number of factors that play into an MOS rating, and some hardware and tool vendors have put formulas in place that can 'estimate' an MOS, based on a number of variables. This is usually done by having both sides of the connection measure certain characteristics of the connection, and collaborate on the quality of the line and connection. For best results, you might take samples from the line at a similar rate to what a voice call might send data (numerous times a second).\n\nPingPlotter's collection methods are a bit different from all of this - we're sitting on one side of a connection, and the other end has no idea that we're measuring anything. It might be a web server, or a game server, or any variety of things, none of which care to collaborate on estimating how a person would measure voice quality over that connection. We're also measuring only once every second - or possibly less often, depending on the trace interval.\n\nMOS is an interesting number, though, and if we take a few variables that we know negatively impact call quality, we can put together a very rough approximation of what the voice call quality might be over that line.\n\nThere are 3 factors that significantly impact call quality:\n\n• Latency\n• Packet loss\n• Jitter\n\nAll of these are well known by PingPlotter, covering the samples we take. Is there any way to put these three values together and come up with a MOS 'guess' that has some similarity to real life? We think there is, and PingPlotter Pro attempts to do so.\n\nOne advantage of estimating MOS with PingPlotter is that we also know Latency, Packet Loss and Jitter at any intermediate hops, so we should be able to estimate MOS for not just the final destination, but also for each intermediate hop! This would give us an easy way to see at which hop MOS is deterioriating, and can help us identify where the problem is.\n\nSo, the question is: How do we convert packet loss, latency and jitter numbers to an MOS score? That's a great question! We've built one model for this, but we'd certainly love participation from anyone who wants to improve this formula!\n\nFirst, let's define our 3 metrics: Packet Loss, Latency and Jitter. Remember that PingPlotter lets you pick a 'window' of samples to examine - we call this 'Samples to Include'.\n\n1) Packet Loss percentage.\n\nFor the sample set, this is the percentage of packets that never made it from us, to the target server (or intermediate hop) and then back again. If we sent out 100 packets and only recieved 97 back (3 didn't make it), then we have 3% packet loss.\n\n2) Average Latency.\n\nThe average latency (in PingPlotter) is the average (mean) time it takes a packet to get from your computer, to the target server, and then back again.\n\nThe average latency is the total of all latencies, divided by the number of samples that we're measuring. If we sent out 100 samples and received back 97, then total all latencies and divide by 97 to get the average.\n\n3) Jitter\n\nJitter is a measurement of how much latency changes, from sample to sample. A low jitter number indicates a solid, good connection. A high jitter number is a sign that there is conjestion on the network. Too high of a jitter number has a negative impact on voice quality, causing delays, requiring resends, and wreaking havoc on transmission of voice.\n\nTo measure Jitter, we take the difference between samples, then divide by the number of samples (minus 1).\n\nHere's an example. We have collected 5 samples with the following latencies: 136, 184, 115, 148, 125 (in that order). The average latency is 142 - (add them, divide by 5). The 'Jitter' is calculated by taking the difference between samples.\n\n136 to 184, diff = 48\n184 to 115, diff = 69\n115 to 148, diff = 33\n148 to 125, diff = 23\n(Notice how we have only 4 differences for 5 samples). The total difference is 173 - so the jitter is 173 / 4, or 43.25.\n\nWe use this same mechanism no how many samples we have - it works on 5, 50 or 5000.\n\nOK, so we have some numbers - what formula do we plug this in to?\n\nMOS is a scale of 1 to 5, 5 being an 'excellent' call, 1 being completely unacceptable. In reality, even a perfect connection is impacted by the compression algorythms of the codec, so the highest score most codecs can get is in the 4.2 to 4.4 range. Most tool-based solutions calculate what is called an 'R' value and then apply a formula to convert that to an MOS score. We do the same. This R to MOS calculation is relatively standard. The R value score is from 0 to 100, where a higher number is better.\n\nSo, this gets us to a the bottom line - how do we get an R Value. This documentation is taken from a pre-release of PingPlotter Pro - this formula may have changed since this was written. In addition, all MOS calculations are done with scripts in PingPlotter Pro - which can be modified by you.\n\nWe start out with an R-value of 93.2, and deduct from there, based on network conditions.\n\nHere is the formula we use (note that this is 'psuedo-code', not real code):\n\n``````' Take the average latency, add jitter, but double the impact to latency\n' then add 10 for protocol latencies\nEffectiveLatency = ( AverageLatency + Jitter * 2 + 10 )\n\n' Implement a basic curve - deduct 4 for the R value at 160ms of latency\n' (round trip). Anything over that gets a much more agressive deduction\nif EffectiveLatency < 160 then\nR = 93.2 - (EffectiveLatency / 40)\nelse\nR = 93.2 - (EffectiveLatency - 120) / 10\n\n' Now, let's deduct 2.5 R values per percentage of packet loss\nR = R - (PacketLoss * 2.5)\n\n' Convert the R into an MOS value.(this is a known formula)\nMOS = 1 + (0.035) * R + (.000007) * R * (R-60) * (100-R)\n``````\n\nThat's it! This formula is captured in the 'MOS Column.ppx' script in your PingPlotter Pro 'scripts' directory. If you want to change the equation, please do so!\n\nNote: In PingPlotter Pro v3.0, the MOS script is turned off by default. To turn it on, go to Edit -> Options..., Plugins & Scripting. Turn on the checkmark beside the 'MOS Column' script. Close the dialog (OK) and then restart PingPlotter Pro.\n\nArticle ID: 50\nCreated On: November 23, 2005\nLast Updated On: July 5, 2016\n\nOnline URL: https://www.pingman.com/kb/article/how-is-mos-calculated-in-pingplotter-pro-50.html"
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[
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https://mathoverflow.net/questions/296823/how-can-i-prove-that-n-1-dimensional-manifold-is-not-contained-in-a-n-2
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[
"# How can I prove that $(n-1)$-dimensional manifold is not contained in a $(n-2)$-dimensional affine variety?\n\nI am having trouble proving the following statement, which I think is true (and possibly very basic). Let $M$ be a real differentiable manifold of dimension $(n-1)$ sitting inside $\\mathbb{R}^n$. Let $W$ be an algebraic set defined by homogeneous forms in $\\mathbb{R}[x_1, ..., x_n]$ for which $W \\subseteq \\mathbb{A}_{\\mathbb{\\mathbb{C}}}^n$ has dimension $n-2$ (as an algebraic set in $\\mathbb{C}^n$). I want to prove that $$M \\not \\subseteq W \\cap \\mathbb{R}^n.$$\n\nI think one reason why I am having trouble proving this (even though I suspect it might be quite basic) may be that I don't have a good understanding on how the dimension of the manifold and dimension of algebraic sets relate to one another. I would greatly appreciate any comments or references. Thank you.\n\n• If I haven't got my inequalities wrong, then the algebraic tangent space at any point of $W$ has dimension $\\ge n - 2$, whereas the geometric tangent space at any point of $M$ has dimension $n - 1$; so you could just show that, if your containment held, then the two notions of tangent space would have to agree. (I'm not sure if this is true, or easy, in general. $p$-adic groups, my speciality, are much easier to work with in this way, since there is no geometric tangent space. :-) ) Apr 2 '18 at 15:51\n• Err, sorry, those bounds aren't contradictory. I guess that, even if you showed the tangent spaces agreed, then you would still have to show that $W$ had a smooth real point, where the dimension of its algebraic tangent space was precisely $n - 2$. Apr 2 '18 at 16:56\n• @JohnnyT.: Since $W$ is made of points in $\\mathbb C^n$, its dimension as an algebraic set is a complex dimension, so its real dimension must be $2n-4$; are you sure that the words that you have chosen correctly convey what you mean? (Of course, I might be misunderstanding your usage of the term \"dimension of an algebraic set\".) Apr 2 '18 at 17:41\n• @AlexM At the moment I believe the words have been chosen correctly; this is what I mean en.wikipedia.org/wiki/Dimension_of_an_algebraic_variety Apr 2 '18 at 17:48\n• @AlexM., note that we are considering not the subset $W(\\mathbb C)$ of $\\mathbb C^n$, which one would expect to have real dimension $2n - 4$ if it were a smooth manifold, but rather the intersection $W(\\mathbb C) \\cap \\mathbb R^n$, about whose dimension it is not so clear (to me) what our expectations should be. Apr 2 '18 at 22:24\n\nYou seem to be asking whether it is possible that the topological dimension of the set of real points of an algebraic variety is greater than its algebraic dimension (in your case, the former is $n-1$ and the latter is $n-2.$) It seems to be a standard fact that this is impossible (inequality in the opposite dimension is quite possible, as in the standard example of $x^2+y^2 = -1.$) The standard reference seems to be:"
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[
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{"ft_lang_label":"__label__en","ft_lang_prob":0.9213397,"math_prob":0.9974908,"size":777,"snap":"2021-31-2021-39","text_gpt3_token_len":211,"char_repetition_ratio":0.12548512,"word_repetition_ratio":0.016260162,"special_character_ratio":0.26898327,"punctuation_ratio":0.08552632,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9997142,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-09-25T03:49:39Z\",\"WARC-Record-ID\":\"<urn:uuid:7490a5b6-ef84-404e-82d8-dbcc63ac229c>\",\"Content-Length\":\"141126\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:5a291009-e1a8-45ec-8ae2-7cd1110bbac7>\",\"WARC-Concurrent-To\":\"<urn:uuid:4e02062f-7615-43ac-bab6-e74dcd3c3ceb>\",\"WARC-IP-Address\":\"151.101.1.69\",\"WARC-Target-URI\":\"https://mathoverflow.net/questions/296823/how-can-i-prove-that-n-1-dimensional-manifold-is-not-contained-in-a-n-2\",\"WARC-Payload-Digest\":\"sha1:U6B4AIGYJD6UEILGL6PEZEAZ2PEHXWIG\",\"WARC-Block-Digest\":\"sha1:2RC4BBDBOYJEWSORASJMMHCOIVDT7GBX\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-39/CC-MAIN-2021-39_segments_1631780057589.14_warc_CC-MAIN-20210925021713-20210925051713-00202.warc.gz\"}"}
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https://tipcalc.net/how-much-is-a-20-percent-tip-on-374.88
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[
"# Tip Calculator\n\nHow much is a 20 percent tip on \\$374.88?\n\nTIP:\n\\$ 0\nTOTAL:\n\\$ 0\nTIP PER PERSON:\n\\$ 0\nTOTAL PER PERSON:\n\\$ 0\n\n## How much is a 20 percent tip on \\$374.88? How to calculate this tip?\n\nAre you looking for the answer to this question: How much is a 20 percent tip on \\$374.88? Here is the answer.\n\nLet's see how to calculate a 20 percent tip when the amount to be paid is 374.88. Tip is a percentage, and a percentage is a number or ratio expressed as a fraction of 100. This means that a 20 percent tip can also be expressed as follows: 20/100 = 0.2 . To get the tip value for a \\$374.88 bill, the amount of the bill must be multiplied by 0.2, so the calculation is as follows:\n\n1. TIP = 374.88*20% = 374.88*0.2 = 74.976\n\n2. TOTAL = 374.88+74.976 = 449.856\n\n3. Rounded to the nearest whole number: 450\n\nIf you want to know how to calculate the tip in your head in a few seconds, visit the Tip Calculator Home.\n\n## So what is a 20 percent tip on a \\$374.88? The answer is 74.98!\n\nOf course, it may happen that you do not pay the bill or the tip alone. A typical case is when you order a pizza with your friends and you want to split the amount of the order. For example, if you are three, you simply need to split the tip and the amount into three. In this example it means:\n\n1. Total amount rounded to the nearest whole number: 450\n\n2. Split into 3: 150\n\nSo in the example above, if the pizza order is to be split into 3, you’ll have to pay \\$150 . Of course, you can do these settings in Tip Calculator. You can split the tip and the total amount payable among the members of the company as you wish. So the TipCalc.net page basically serves as a Pizza Tip Calculator, as well.\n\n## Tip Calculator Examples (BILL: \\$374.88)\n\nHow much is a 5% tip on \\$374.88?\nHow much is a 10% tip on \\$374.88?\nHow much is a 15% tip on \\$374.88?\nHow much is a 20% tip on \\$374.88?\nHow much is a 25% tip on \\$374.88?\nHow much is a 30% tip on \\$374.88?"
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{"ft_lang_label":"__label__en","ft_lang_prob":0.95896244,"math_prob":0.98684376,"size":2626,"snap":"2023-40-2023-50","text_gpt3_token_len":854,"char_repetition_ratio":0.31845918,"word_repetition_ratio":0.24064171,"special_character_ratio":0.40936786,"punctuation_ratio":0.16235632,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99777305,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-10T01:32:07Z\",\"WARC-Record-ID\":\"<urn:uuid:d33a29e7-10dc-4ea7-adae-9373861fc907>\",\"Content-Length\":\"13292\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:41582f3f-9847-46ea-8a16-c01a5668c9d5>\",\"WARC-Concurrent-To\":\"<urn:uuid:42c1fc3c-e332-43d2-9b00-bec8ba2ab89b>\",\"WARC-IP-Address\":\"161.35.97.186\",\"WARC-Target-URI\":\"https://tipcalc.net/how-much-is-a-20-percent-tip-on-374.88\",\"WARC-Payload-Digest\":\"sha1:34EHYW6JKGC56WT6OCNNA52F3X6Y5OA4\",\"WARC-Block-Digest\":\"sha1:IIWSW3ADV7CZFNIDDTVQCGRUPHH6MC4O\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100989.75_warc_CC-MAIN-20231209233632-20231210023632-00251.warc.gz\"}"}
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https://www.hindawi.com/journals/complexity/2020/6946702/
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[
"#### Abstract\n\nThe demand for transfer learning methods for mechanical fault diagnosis has considerably progressed in recent years. However, the existing methods always depend on the maximum mean discrepancy (MMD) in measuring the domain discrepancy. But MMD can not guarantee the different domain features to be similar enough. Inspired by generative adversarial networks (GAN) and domain adversarial training of neural networks (DANN), this study presents a novel deep adaptive adversarial network (DAAN). The DAAN comprises a condition recognition module and domain adversarial learning module. The condition recognition module is constructed with a generator to extract features and classify the health condition of machinery automatically. The domain adversarial learning module is achieved with a discriminator based on Wasserstein distance to learn domain-invariant features. Then spectral normalization (SN) is employed to accelerate convergence. The effectiveness of DAAN is demonstrated through three transfer fault diagnosis experiments, and the results show that the DAAN can converge to zero after approximately 15 training epochs, and all the average testing accuracies in each case can achieve over 92%. It is expected that the proposed DAAN can effectively learn domain-invariant features to bridge the discrepancy between the data from different working conditions.\n\n#### 1. Introduction\n\nBearings and gears are widely used transmission parts in rotating machinery, and their failure directly affects the healthy operation of machinery and even causes serious incidents. Therefore, monitoring and diagnosing the health condition of these transmission parts is crucial [1, 2]. In recent years, the Internet of Things (IoT) based infrastructure is often adopted for condition monitoring and analysis because it can directly handle massive monitoring data with minimal manual intervention [3, 4]. Lei et al. developed an intelligent method based on sparse filtering for bearing fault diagnosis. Jia et al. presented a stacked autoencoders (SAE) based network to diagnose the fault problems of bearing and planetary gearbox. Xu et al. used a deep convolutional neural network (CNN) to achieve a bearing fault diagnosis problem under different working conditions. An et al. adopted a recurrent neural network (RNN) to process variable size sequences of bearing fault samples and achieved satisfactory performance. Xiao et al. proposed a deep mutual information maximization (DMIM) method using a variational divergence estimation approach to maximize the mutual information between the input and output of a deep neural network and achieved motor fault diagnosis. Wang et al. presented a capsule neural network for bearing fault diagnosis and obtained a high classification accuracy. Although these methods have achieved excellent diagnosis performance, they require plenty of labeled data. Besides, the training and testing data must own the same probability distribution. But obtaining a considerable amount of labeled data is quite hard for some machines, and the probability distribution of the fault samples constantly changes due to variable speeds and loads.\n\nTransfer learning provides a promising idea of addressing these problems [11, 12]. In recent years, various related methods have been investigated for fault diagnosis. Wen et al. introduced maximum mean discrepancy (MMD) into SAE and achieved feature transfer learning under variable speeds. Lu et al. developed a transfer learning-based model with domain adaptation for bearing fault diagnosis. Guo et al. presented a deep convolutional transfer learning network (DCTLN) and used six cases to test the effectiveness of DCTLN. Yang et al. offered a domain-shared CNN model to learn the transferable features from bearing used in the laboratory machines and real-case machines simultaneously. An et al. proposed a multilayer multiple kernel variant of MMD, which introduced the kernel method to replace the high dimensional map of MMD and achieved bearing fault diagnosis under different working conditions. Zhang et al. developed a novel sparse filtering based domain adaptation (SFDA) for the mechanical fault diagnosis, which employed l1-norm and l2-norm to MMD to obtain high dimensional adaptive features. These studies utilized MMD to minimize the target loss by using the source loss to achieve the learning of cross-domain-invariant features. However, MMD measures the discrepancy using a high dimensional map based on reproducing kernel Hilbert space (RKHS), which cannot guarantee the sufficient closeness of different domain features close enough in RKHS.\n\nThe remainder of this paper is organized as follows. Section 2 describes the transfer learning problem. Section 3 details the proposed DAAN model. Section 4 presents the fault diagnosis experiments under different working conditions. Section 5 finally provides conclusions.\n\n#### 2. Theory Background\n\n##### 2.1. Wasserstein Distance\n\nThe Wasserstein distance, also called the earth-mover distance, is a distance metric for comparing probability measures and distributions. The gradient of the DANN is always unstable when training the feature extractor. In order to reduce the gradient vanishing problem, Wasserstein distance is employed in discriminator D as the distribution measurement function, which is used as the minimum cost to converge to as follows:where represents the set of all joint distributions γ(x, y) whose marginals are and , respectively. Intuitively, γ(x, y) can be considered the cost of moving an amount from x to y in order to transform into . The Wasserstein distance has been used to solve the optimal transportation problem, so is the minimum transport cost under optimal path planning.\n\nTherefore, the improved objective function can be obtained as follows:where is the set of 1-Lipschitz functions.\n\n##### 2.2. Spectral Normalization (SN)\n\nSN can control D via constraining the spectral norm of each network layer. Giving a linear layer , the norm is defined by Lipschitz constant:where is equal to the Lipschitz norm , and is the SN operation of W:which is equal to the largest singular value of W. If the Lipschitz norm is equal to 1, then the inequality can be used to observe the following bound on :\n\nThe SN normalizes the spectral norm of W to make sure it can satisfy the Lipschitz constraint :\n\n#### 3. Proposed Framework\n\nAs shown in Figure 1, the proposed DAAN includes the condition recognition module and domain adversarial learning module. The condition recognition module contains a feature extraction network and a fault classify network. The feature extraction network can automatically learn the fault features, and the fault classify network identifies health conditions according to the extracted features. The domain adversarial learning module is completed by using the discriminator network which is connected to the feature extraction network to help learn the domain-invariant features.(1)Condition Recognition: a three-layer feedforward neural network (FFNN) is used to construct this module, and then a classifier is followed so as to recognize the health condition. The optimal objective of the classifier C is to train the feature extractor with parameter θF and C with parameter θC. The following loss LC is defined as cross-entropy between the predicted softmax probabilistic distribution and the corresponding labels:where is the indicator function; is the kth value of the predicted distribution, and K is the number of health conditions.(2)Domain adversarial learning: The adversarial training strategy of the GAN is used to extract domain-invariant features. The discriminator D is optimized via maximizing the domain adversarial loss LD considering parameter θF to minimize the distribution discrepancy between two domains. Therefore, LD is defined as follows:\n\nBy combining the two optimization objectives, the final loss function can be written as follows:where the hyperparameter λ determines the strength of the domain adversarial strategy.\n\n##### 3.2. Training Strategy of DAAN\n\nAs displayed in Figure 2, training the proposed method by Adam algorithm is convenient since the optimization objective of the DAAN is built. In the discriminator D, a gradient reversal layer is used to connect the feature extractor during the training process. This layer can ensure the feature distribution in the two domains remain indistinguishable enough for the discriminator D to obtain the domain-invariant features.\n\nTherefore, the loss can be rewritten as follows:\n\nBased on the above equations and Adam algorithm, the parameters θF, θC, and θD are updated as follows:where is the learning rate.\n\nAs the network training is finished, the classifier can accurately identify the unlabeled dataset in the target domain if there are fuzzy domain categories existing in the learned features. In the testing process, the rest target domain dataset is used as the input of the DAAN, and then the classifier outputs the classification result.\n\n#### 4. Experiment Studies\n\n##### 4.1. Case 1: Fault Diagnosis under Different Rotating Speeds\n###### 4.1.1. Data Description\n\nThe bearing data are collected from the test rig as displayed in Figure 3(a). The rig includes a motor, a driving belt, a shaft coupling, and a bearing seat. There are five bearing health conditions: normal condition (NC), inner ring fault (IF), outer ring fault (OF), roller fault (RF), and concurrent fault in the outer ring and roller (ORF). The four fault bearings are depicted in Figure 3(b). Vibration signal is commonly utilized in condition monitoring and diagnosis due to its rich and useful information with high sampling frequency [29, 30]. All vibration acceleration data were measured under three different speeds of 1100r/min (dataset A), 1300r/min (dataset B), and 1500r/min (dataset C). The sampling frequency of the accelerometer is 25.6 kHz, 200 samples are selected from each bearing health condition, and each sample contains 2400 data points. Hence, a total of 1000 samples are acquired. The spectra of the raw signals are then transformed via FFT, and 1200 data points of each sample in frequency-domain are obtained as the input of the DAAN model. In each experiment, all the source domain samples and half of the target domain samples are used for training. The rest target domain data samples are used for testing. The spectra of those three datasets and the transfer learning cases are displayed in Figure 2.\n\n###### 4.1.2. Diagnostic Results\n\nFigure 4 shows that the proposed DAAN is evaluated on six transfer learning cases: A ⟶ B, B ⟶ A, B ⟶ C, C ⟶ B, A ⟶ C, and C ⟶ A. In each case, the part before and after the arrow refers to the source domain and target domain, respectively. For example, in the case A⟶B, datasets A and B are the source domain and target domain, respectively. The structure of the condition recognition module is [1200, 600, 200, 100, 5], and the domain adversarial module is [1200, 600, 200, 100, 1], in which the unit number of the input layer is determined by the dimension of the samples, the unit number of the output layer for the condition recognition module is determined by the number of the health conditions, and the unit number of the output layer for the domain adversarial module is determined by the result of true or false. The unit numbers of the hidden layer are determined by the dimension to reduce the principle. The learning rate is 0.002, and the penalty parameter λ is 0.005. Each training batch includes 500 samples from the source domain and target domain, respectively. The other 500 target domain samples are adopted for testing. In each experiment, a total of 15 trials were conducted to reduce the effects of randomness, and the training step is 50. In case A ⟶ B, the curves of training and testing accuracies are plotted in Figure 5. Accordingly, the training accuracy is approached 100% after approximately 15 training epochs, and the testing accuracy needs approximately 47 training epochs to achieve this goal. The classifier loss curve of DAAN is plotted in Figure 4, and the training loss in DAAN converges to zero after approximately 15 training epochs. For comparison, the loss curves of DANN and DANN without SN are also plotted in Figure 4, it is easy to find that DANN is much more difficult to converge, and DANN without SN needs 25 training epochs to convergence. These performances indicate that the proposed DAAN owns a strong domain-invariant feature extraction ability and can help the model to achieve fast convergence. The results of six transfer cases are displayed in Table 1. All the testing accuracies in each case are over 90%, while some are even over 98%. This high accuracy indicates that the DAAN can effectively identify the health condition of bearing in the absence of labeled data.\n\nTo further demonstrate the effectiveness of the DAAN, three methods are adopted for the comparison of the six transfer learning cases. The five comparison methods are SAE , transfer component analysis (TCA) , MK-MMD , SFDA , and DANN. The subsequent classifier of SAE and TCA is softmax regression. SAE is trained only by the source domain data. TCA, MK-MMD, and SFDA are three representative examples of using the MMD-regularized subspace learning method in the domain adaptation field. The testing accuracies on the six transfer learning cases are listed in Table 1. It is easy to find that the DAAN achieves the highest accuracies and the lowest standard deviations among the given approaches. The average testing accuracy of SAE without transfer learning is only 60.20% because the target domain data have not participated in the model training. Therefore, compared with SAE, it is obvious that the transfer learning-based method is more effective in handling unlabeled data than traditional intelligent fault diagnosis. The traditional DANN without Wasserstein distance and SN strategy achieved 86.88% accuracy. The average accuracies of TCA, MK-MMD, and SFDA are 81.53%, 94.90%, and 92.98%, respectively. These results are considerably better than those of SAE but are still worse than those of the proposed method. Thus, it can be concluded from the comparison that the DAAN can learn more robust domain-invariant features than the other transfer learning methods.\n\nFurthermore, the t-SNE algorithm is adopted to map the learned features into a 2D scatter diagram to offer visual insights on the two domains. Taking the case A⟶B as an example, the domain-invariant features learned by the DAAN are displayed in Figure 6(f), and the mapping results obtained using the other comparison methods are shown in Figures 6(a)-6(e). The source and target domains are represented in terms of S and T, respectively. The result in Figure 6(a) demonstrates that although the SAE model obtains good cluster results, the distribution discrepancies of the two domains are substantially large, except for the NC condition. Thus, it can not effectively classify the unlabeled target samples when the model is only trained using the source samples. Figures 6(b) and 6(c) plot the mapped results of the transferred features learned by TCA and DANN, and the cross-domain discrepancies are clearly reduced. However, some overlapping samples still exist between the IF and RF conditions. Meanwhile, the source and target domain samples of ORF are poorly clustered. Figures 6(b) and 6(c) plot the mapped results of the transferred features learned by MK-MMD and SFDA; it can be seen that the cluster performances have been further improved, but there is still some distance among the two domains. Figure 6(f) illustrates that the proposed DAAN method not only reduces the distribution discrepancy of the two domains, but also amplifies the feature distance of different health conditions. Therefore, it validates that the DAAN can extract considerably more robust transferable features than other traditional methods.\n\n##### 4.2. Case 2: Fault Diagnosis under Different Loads\n\nAnother experiment bench for the transfer learning task under different loads is displayed in Figure 7. This experiment also has five bearing health conditions of NC, IF, OF, RF, and ORF. The rotating speed was fixed at 1800 r/min, and the sampling frequency was 12.8 kHz. The vibration signals were measured under three different loads of 20N (dataset D), 40N (dataset E), and 60N (dataset F). 200 samples were also collected from each health condition under one load, and each sample contained 2400 data points. The frequency-domain samples were also utilized as the inputs of DAAN, and the other parameter sets were the same as in Case 1.\n\nThe results were compared with the three other methods, as displayed in Table 2. It shows that the DAAN also achieved the highest diagnosis accuracies for all cases among these four methods at an average testing accuracy of 92.65%. The SAE method without the transfer learning strategy still performed the worst, yielding a success rate of 50.65%. Besides, the average testing accuracies of TCA and DANN are 72.78% and 81.46%, respectively. The average testing accuracies of MK-MMD and SFDA are 91.10% and 89.70%, respectively. These results demonstrate that the proposed DAAN method presents better transfer performance than other methods.\n\nSimilarly, in case of D⟶E, the reduced dimension results of these methods are displayed in Figure 8. Figure 8(a) shows that the learned features via SAE still poorly clustered the same health condition samples under different loads and corresponded to a low classification accuracy of 55.28%. Figures 8(b) and 8(c) demonstrate that the learned transferable features through TCA and DANN are subject to a smaller distribution discrepancy than that via SAE. However, the RF and ORF samples under different loads are still separated. Figures 8(d) and 8(e) show the results of MK-MMD and SFDA. We can find that the distributions of transferred features from the two domains are closer than the ones of the features learned by TCA and DANN. Figure 8(d) displays the excellent cluster result obtained by the proposed DAAN. The source and target features under the same health condition are gathered remarkably close, and different health condition samples are also effectively separated. Consequently, the proposed DAAN method can learn domain-invariant features to reduce the discrepancy between different domains.\n\n##### 4.3. Case 3: Fault Diagnosis Using CWRU Bearing Dataset\n\nIn order to test the proposed method for the case under different loads and speeds, a bearing dataset offered by Case Western Reserve University (CWRU) is applied in this section. Four fault types of bearing are considered: (1) normal condition (NC); (2) inner ring fault (IF); (3) outer ring fault (OF); (4) roller fault (RF). There are three different severity levels (0.18, 0.36, and 0.53 mm) for IF, OF, and RF cases. Therefore, there are 10 different bearing health conditions. The raw vibration data was drawn under four different loads, i.e., 0, 1, 2, and 3 hp which corresponded to the three different rotating speeds:1790, 1772, 1750, and 1730 rpm, respectively. The four datasets are named as Datasets G, H, I, and J. In this experiment, each fault type under one load includes 200 samples, and each sample contains 2400 data points, so there is a total of 2000 samples for each load.\n\nThe accuracies and the corresponding standard deviations of all different transfer scenarios are shown in Table 3. As we can see in Table 3, there are totally 12 different transfer scenarios applied to obtain the diagnosis accuracies. It presents that the average testing accuracies of all the scenarios using the proposed method can obtain more than 98.71% and the standard deviations below 0.17%, which means the proposed method can effectively and stably achieve transfer fault diagnosis under different loads and speeds. In addition, the other transfer learning-based methods can also achieve a good result, maybe because the difference between different working conditions is not big enough. The dimension reduction results of all the transfer learning-based methods are also basically the same. So we only provide the results of G⟶H, I⟶H, and J⟶H to show the effectiveness of the proposed method, which is displayed in Figure 9. It is observed that almost all the transferable features of the same health condition are assembled in the corresponding cluster, and different health condition features are separated. This indicates that the proposed method can learn transferable features without being affected by the varying loads and speeds.\n\n#### 5. Conclusions\n\nIn this paper, a novel transfer learning method called DAAN is proposed for mechanical fault diagnosis under different working conditions. The training process of domain adversarial can be guaranteed due to the employment of Wasserstein distance, and the SN strategy can accelerate convergence with much less iteration steps. Three bearing experiments show that the DAAN can obtain over 92% average classification accuracy and achieve fast converge under about 15 training epochs. Moreover, the proposed method presents superior transfer performance to the other transfer learning methods. Therefore, the DAAN can promote the successful application of mechanical fault diagnosis under different working conditions. Although the proposed method can promote the practical application of intelligent fault diagnosis under different domains, it still needs a considerable number of target domain samples for the model training. Therefore, the next challenge is to improve our method under less target domain training samples.\n\n#### Data Availability\n\nThe data used to support the findings of this study are available from the corresponding author upon request.\n\n#### Conflicts of Interest\n\nThe authors declare that they have no conflicts of interest.\n\n#### Acknowledgments\n\nThis work was supported by the China Postdoctoral Science Foundation (2019M662399) and the Project of Shandong Province Higher Educational Young Innovative Talent Introduction and Cultivation Team (Performance enhancement of deep coal mining equipment)."
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http://www.alamandamaths.com/number-and-algebra/fractions/addingsubtracting-fractions-with-samerelated-denominators/
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[
"# Adding/Subtracting Fractions with Same/Related Denominators (6)\n\n## Solve problems involving addition and subtraction of fractions with the same or related denominators (ACMNA126)\n\nLO: To add or subtract fractions with related denominators.\n\nKnow:\n\n• how to add numbers\n\nUnderstand:\n\n• that the addition of fractions makes the fractions larger\n• that the subtraction of fractions makes the fractions smaller\n• that when adding/subtracting fractions, you have to keep the denominator the same.\n\nDo:\n\nI can add and subtract fractions with the same or related denominator.\n\n## Visual Representations",
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"",
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"## Adding and Subtracting Fractions with Same Denominators Notes\n\n### The reason why you keep the denominators the same is because the total size of the pieces (denominator) never changes.",
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"### The reason why you keep the denominators the same is because the total size of the pieces (denominator) never changes.",
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"# Teaching Ideas",
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"https://i0.wp.com/www.alamandamaths.com/wp-content/uploads/2015/10/876687248.jpg",
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"https://i0.wp.com/www.alamandamaths.com/wp-content/uploads/2015/10/adding-fractions-same-denoominator.jpg",
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"https://i1.wp.com/www.alamandamaths.com/wp-content/uploads/2015/10/adding-fractions-same-denominator.jpg",
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"https://i0.wp.com/www.alamandamaths.com/wp-content/uploads/2015/10/adding-similar-fractions.jpg",
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https://www.teachoo.com/7944/2607/Ex-7.4--6/category/Equivalent-Fractions/
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"1. Chapter 7 Class 6 Fractions\n2. Concept wise\n3. Equivalent Fractions\n\nTranscript\n\nEx 7.4 ,6 The following fractions represent just three different numbers. Separate them into three groups of equivalent fractions, by changing each one to its simplest form. (a) 2/12 We have to convert all of them into simplest from. 2/12 = = 1/6 So, simplest form is 𝟏/𝟔 Ex 7.4 ,6 The following fractions represent just three different numbers. Separate them into three groups of equivalent fractions, by changing each one to its simplest form. (b) 3/15 3/15 = = 1/5 So, simplest form is 𝟏/𝟓 Ex 7.4 ,6 The following fractions represent just three different numbers. Separate them into three groups of equivalent fractions, by changing each one to its simplest form. (c) 8/50 8/50 = = 4/25 So, simplest form is 𝟒/𝟐𝟓 Ex 7.4 ,6 The following fractions represent just three different numbers. Separate them into three groups of equivalent fractions, by changing each one to its simplest form. (d) 16/100 16/100 = = 8/50 = 4/25 So, simplest form is 𝟒/𝟐𝟓 Ex 7.4 ,6 The following fractions represent just three different numbers. Separate them into three groups of equivalent fractions, by changing each one to its simplest form. (e) 10/60 10/60 = = 1/6 So, simplest form is 𝟏/𝟔 Ex 7.4 ,6 The following fractions represent just three different numbers. Separate them into three groups of equivalent fractions, by changing each one to its simplest form. (f) 15/75 15/75 = = 5/25 = 1/5 So, simplest form is 𝟏/𝟓 Ex 7.4 ,6 The following fractions represent just three different numbers. Separate them into three groups of equivalent fractions, by changing each one to its simplest form. (g) 12/60 12/60 = = 6/30 = 3/15 = 1/5 So, simplest form is 𝟏/𝟓 Ex 7.4 ,6 The following fractions represent just three different numbers. Separate them into three groups of equivalent fractions, by changing each one to its simplest form. (h) 16/96 Ex 7.4 ,6 The following fractions represent just three different numbers. Separate them into three groups of equivalent fractions, by changing each one to its simplest form. (i) 12/75 12/75 = = 4/25 So, simplest form is 𝟒/𝟐𝟓 Ex 7.4 ,6 The following fractions represent just three different numbers. Separate them into three groups of equivalent fractions, by changing each one to its simplest form. (j) 12/72 12/72 = = 6/36 = 3/18 = 1/6 So, simplest form is 𝟏/𝟔 Ex 7.4 ,6 The following fractions represent just three different numbers. Separate them into three groups of equivalent fractions, by changing each one to its simplest form. (k) 3/18 3/18 = = 1/6 So, simplest form is 𝟏/𝟔 Ex 7.4 ,6 The following fractions represent just three different numbers. Separate them into three groups of equivalent fractions, by changing each one to its simplest form. (l) 4/25 84/98 = It cannot be simplified further Hence, 4/25 is its simplified form Thus, these are 3 fractions 1/6 → (a), (e), (h), (j), (k) 1/5 → (b), (f), (g) 4/25 → (c), (d), (i), (l)\n\nEquivalent Fractions",
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[
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https://gmatclub.com/forum/is-a-0-1-a-3-a-2-2a-2-a-2-a-157421.html
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[
"GMAT Question of the Day - Daily to your Mailbox; hard ones only\n\n It is currently 23 Oct 2019, 21:27",
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"GMAT Club Daily Prep\n\nThank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.\n\nCustomized\nfor You\n\nwe will pick new questions that match your level based on your Timer History\n\nTrack\n\nevery week, we’ll send you an estimated GMAT score based on your performance\n\nPractice\nPays\n\nwe will pick new questions that match your level based on your Timer History\n\nNot interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.",
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"",
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"Is a < 0 ? (1) a^3 < a^2 + 2a (2) a^2 > a^3\n\n new topic post reply Question banks Downloads My Bookmarks Reviews Important topics\nAuthor Message\nTAGS:\n\nHide Tags\n\nRetired Moderator",
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"B\nJoined: 27 Aug 2012\nPosts: 1050\nIs a < 0 ? (1) a^3 < a^2 + 2a (2) a^2 > a^3 [#permalink]\n\nShow Tags\n\n8",
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"00:00\n\nDifficulty:",
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"",
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"75% (hard)\n\nQuestion Stats:",
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"58% (02:19) correct",
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"42% (02:18) wrong",
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"based on 165 sessions\n\nHideShow timer Statistics\n\nIs a<0?\n\n(1) $$a^3<a^2+2a$$\n(2) $$a^2>a^3$$\n\n_________________\n\nOriginally posted by bagdbmba on 05 Aug 2013, 23:28.\nLast edited by Bunuel on 19 Jun 2019, 07:14, edited 3 times in total.\nRenamed the topic and edited the question.\nVerbal Forum Moderator",
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"B\nJoined: 10 Oct 2012\nPosts: 590\nRe: Is a < 0 ? (1) a^3 < a^2 + 2a (2) a^2 > a^3 [#permalink]\n\nShow Tags\n\n1\n1\nbagdbmba wrote:\nIs a<0?\n\n(1) $$a^3<a^2+2a$$\n(2) $$a^2>a^3$$\n\nFrom F.S 1, add 1 on both sides : $$a^2+2a+1>a^3+1 \\to (a+1)^2-(a+1)(a^2+1-a)>0 \\to (a+1)[(a+1)-(a^2+1-a)]>0 \\to (a+1)(2a-a^2)>0$$\nThus, we get a(a+1)(a-2)<0. Either 0<a<2 OR a<-1. Insufficient.\n\nFrom F.S 2, we can divide by$$a^2$$ on both sides and we get a<1. Insufficient.\n\nTaking both together, we know that a<1. Thus, a could be 0<a<1 OR a<-1. Insufficient.\n\nE.\n_________________\nDirector",
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"",
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"Joined: 14 Dec 2012\nPosts: 702\nLocation: India\nConcentration: General Management, Operations\nGMAT 1: 700 Q50 V34",
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"GPA: 3.6\nRe: Is a < 0 ? (1) a^3 < a^2 + 2a (2) a^2 > a^3 [#permalink]\n\nShow Tags\n\nbagdbmba wrote:\nIs a<0?\n\n(1) $$a^3<a^2+2a$$\n(2) $$a^2>a^3$$\n\nSTMNT 1:\n\n$$a^3<a^2+2a$$\n$$a^3-a^2-2a < 0$$\na(a+1)(a-2)<0\nwhen a = -2 expression above is= -8 which is < 0\nwhen a = 1 expression above is = -2 which is <0\nhence a can be -ve /+ve insufficient\n\nSTMNT 2:\n$$a^2$$$$> a^3$$\n$$a^2(a-1)<0$$\nclearly this satisfies for a<1\nhence a can be -ve /+ve insufficient\n\ncombining both\nstill both statement satisfies for a = 0.5 and a =-2\nhence insufficient\n\nhence E\n_________________\nWhen you want to succeed as bad as you want to breathe ...then you will be successfull....\n\nGIVE VALUE TO OFFICIAL QUESTIONS...\n\nGMAT RCs VOCABULARY LIST: http://gmatclub.com/forum/vocabulary-list-for-gmat-reading-comprehension-155228.html\nlearn AWA writing techniques while watching video : http://www.gmatprepnow.com/module/gmat-analytical-writing-assessment\nSenior Manager",
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"Joined: 10 Jul 2013\nPosts: 289\nRe: Is a < 0 ? (1) a^3 < a^2 + 2a (2) a^2 > a^3 [#permalink]\n\nShow Tags\n\nbagdbmba wrote:\nIs a<0?\n\n(1) $$a^3<a^2+2a$$\n(2) $$a^2>a^3$$\n\nst(1) , add 1 to both sides. then a= -1 and a= 1/2. you will have a double case.\nst(2), a can be any positive fraction here and a can be any negative integer too such as a= 1/2 and a = -2 . a double case too.\nso both statement depict the same answers, which are both a double case.\nso Answer is (E)\n_________________\nAsif vai.....\nVerbal Forum Moderator",
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"B\nJoined: 10 Oct 2012\nPosts: 590\nRe: Is a < 0 ? (1) a^3 < a^2 + 2a (2) a^2 > a^3 [#permalink]\n\nShow Tags\n\n1\nConsider,\n\nStatement (1): $$a^3 < a^2 + 2a$$\n\n$$a^3 - a^2 - 2a < 0$$\n$$a(a^2 - a - 2) < 0$$\n$$a(a -2)(a + 1) < 0$$\n\ni.e. a = 0 or a = 2 or a = -1, hence not sufficient.\n\nStatement (2): $$a^2 > a^3$$\nThis simply means,\na < 1, hence not sufficient.\n\nCombining both statements, we get a = 0 or a = -1, Hence both statements together not sufficient.\n\nCorrect Ans: E\n\nEven though you have the correct answer, I am sorry but it is not what it means(the highlighted part). It is not an equality, rather an in-equality.Please refer through the above posts, for the correct method.\n_________________\nMath Expert",
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"V\nJoined: 02 Sep 2009\nPosts: 58465\nRe: Is a < 0 ? (1) a^3 < a^2 + 2a (2) a^2 > a^3 [#permalink]\n\nShow Tags\n\n1\n1\nStatement 1 -> a^3-a^2-2a<0.\na(a-2)(a+1)<0=>a<0 or a<2 or a<-1. Not Sufficient.\n\nStatement 2->a^2-a^3>0=>a^2(1-a)>0 => a>0 or a<1.Not Sufficient.\n\n1&2,still a<2 exists which does not answer the question is a<0. So E.\n\nThe answer is E, but the ranges are not correct.\n\nIs a < 0 ?\n\n(1) a^3 < a^2 + 2a --> $$(a+1)a(a-2)<0$$ --> $$a<-1$$ or $$0<a<2$$. Not sufficient.\n\n(2) a^2 > a^3 --> $$a^2(1-a)>0$$ --> $$a<0$$ or $$0<a<1$$. Not sufficient.\n\n(1)+(2) Intersection of the ranges from (1) and (2) is $$a<-1$$ or $$0<a<1$$. Not sufficient.\n\n_________________\nManager",
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"Joined: 03 Dec 2012\nPosts: 189\nRe: Is a < 0 ? (1) a^3 < a^2 + 2a (2) a^2 > a^3 [#permalink]\n\nShow Tags\n\nBunuel why can't we factorize a^2>a^3 as 1>a or a<1.\nMath Expert",
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"V\nJoined: 02 Sep 2009\nPosts: 58465\nRe: Is a < 0 ? (1) a^3 < a^2 + 2a (2) a^2 > a^3 [#permalink]\n\nShow Tags\n\n1\nmohnish104 wrote:\nBunuel why can't we factorize a^2>a^3 as 1>a or a<1.\n\na<1 implies that a can be 0. But a=0 does not satisfy a^2>a^3, so the correct ranges for which this inequality holds true is a<0 or 0<a<1 (the same range as you have excluding 0).\n\nHope it helps.\n_________________\nIntern",
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"Joined: 24 Oct 2013\nPosts: 5\nRe: Is a < 0 ? (1) a^3 < a^2 + 2a (2) a^2 > a^3 [#permalink]\n\nShow Tags\n\nBunuel wrote:\nmohnish104 wrote:\nBunuel why can't we factorize a^2>a^3 as 1>a or a<1.\n\na<1 implies that a can be 0. But a=0 does not satisfy a^2>a^3, so the correct ranges for which this inequality holds true is a<0 or 0<a<1 (the same range as you have excluding 0).\n\nHope it helps.\n\nHi,\n\nThere's a concept related to inequalities that I fail to understand. Could you please tell me what I'm doing wrong?\n\nFor statement 2 we have \"a^2 > a^3\". Depending on how I solve this, I'm getting two complete different ranges, both of which are incorrect.\n\nFirst case:\n\na^2 - a^3 > 0\na^2*(1-a)>0------> this gives me the critical points 0 and 1.\n\nHence I have 3 ranges: (1st) a<0, (2nd) 0<a<1, and (3rd) a>1.\n\nAs the inequality has the \">0\" sign I took only the 1st and 3rd range and got \"a<0 or a>1\".\n\nSecond case:\n\n0>a^3 - a^2\n0>a^2*(a-1)------> this also gives me the critical points 0 and 1.\n\nHence I have the same 3 ranges: (1st) a<0, (2nd) 0<a<1, and (3rd) a>1.\n\nHowever, this time we have \"0>\" sign, so I took only the second range: 0<a<1.\n\nIn any case, both are false. What's odd is that I used the same method to find the range for statement 1 but I got the correct answer.\n\nThanks a lot for your help.\n\nAurèle\nMath Expert",
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"V\nJoined: 02 Sep 2009\nPosts: 58465\nRe: Is a < 0 ? (1) a^3 < a^2 + 2a (2) a^2 > a^3 [#permalink]\n\nShow Tags\n\nAurele wrote:\nBunuel wrote:\nmohnish104 wrote:\nBunuel why can't we factorize a^2>a^3 as 1>a or a<1.\n\na<1 implies that a can be 0. But a=0 does not satisfy a^2>a^3, so the correct ranges for which this inequality holds true is a<0 or 0<a<1 (the same range as you have excluding 0).\n\nHope it helps.\n\nHi,\n\nThere's a concept related to inequalities that I fail to understand. Could you please tell me what I'm doing wrong?\n\nFor statement 2 we have \"a^2 > a^3\". Depending on how I solve this, I'm getting two complete different ranges, both of which are incorrect.\n\nFirst case:\n\na^2 - a^3 > 0\na^2*(1-a)>0------> this gives me the critical points 0 and 1.\n\nHence I have 3 ranges: (1st) a<0, (2nd) 0<a<1, and (3rd) a>1.\n\nAs the inequality has the \">0\" sign I took only the 1st and 3rd range and got \"a<0 or a>1\".\n\nSecond case:\n\n0>a^3 - a^2\n0>a^2*(a-1)------> this also gives me the critical points 0 and 1.\n\nHence I have the same 3 ranges: (1st) a<0, (2nd) 0<a<1, and (3rd) a>1.\n\nHowever, this time we have \"0>\" sign, so I took only the second range: 0<a<1.\n\nIn any case, both are false. What's odd is that I used the same method to find the range for statement 1 but I got the correct answer.\n\nThanks a lot for your help.\n\nAurèle\n\n0 is not a critical point. The squared terms (basically even powers) must be ignored because they cannot be negative and hence doesn't affect the sign.\n_________________\nIntern",
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"Joined: 24 Oct 2013\nPosts: 5\nRe: Is a < 0 ? (1) a^3 < a^2 + 2a (2) a^2 > a^3 [#permalink]\n\nShow Tags\n\nBunuel wrote:\n\n0 is not a critical point. The squared terms (basically even powers) must be ignored because they cannot be negative and hence doesn't affect the sign.\n\nGreat, thanks for the reply. Just to be sure that I understand the concept fully, could you tell me please if I'm solving the following equation correctly (from Manhattan books):\n\n$$x^6 - x^7 > x^5 - x^6$$\n\nI factor the equation and get:\n\n$$x^5*(x-1)^2 < 0$$\n\nHere, If I understand your explanation well, $$x^5$$ is raised to an odd power; hence, it should be considered, as it can yield a negative or positive result. Conversely, $$(x-1)^2$$ is raised to an even power; thus, we ignore it.\n\nThe critical point is then : $$0$$\n\nBecause we have the $$<0$$ sign, we'd get $$x<0$$. Would this be correct?\nManager",
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"Joined: 14 Oct 2014\nPosts: 66\nLocation: United States\nGMAT 1: 500 Q36 V23",
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"Re: Is a < 0 ? (1) a^3 < a^2 + 2a (2) a^2 > a^3 [#permalink]\n\nShow Tags\n\nIs a<0?\n(1) Insufficient. If a=-5, then -125 < 25-10 --->Yes\nIf a=1/2, then 1/8 < 1/4+1 --->No\n\n(2) Insufficient. If a=-5, then 25>-125 --->Yes\nIf a=1/2, then 1/4>1/8 --->No\n(1)+(2) Insufficient. We can use the same numbers and we get two different answers\nIntern",
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"Joined: 31 May 2015\nPosts: 1\nRe: Is a < 0 ? (1) a^3 < a^2 + 2a (2) a^2 > a^3 [#permalink]\n\nShow Tags\n\nHello! I was looking through these posts, answering questions, and I came across this one:\n\nIs a<0?\n\n(1) a3<a2+2a\n(2) a2>a3\n\nfrom: forum/is-a-157421.html\n\nIn the topic above, the answer is that both statements together are insufficient to answer the problem but surely the only case in which $$a^{2}$$ is greater than$$a^{3}$$ is if a is negative?\nMy answer is that both statements alone can answer the question, as the only way $$a^{3}$$ is smaller than what it is compared to is if a is negative. What am I overlooking?\n\nThank you.\ne-GMAT Representative",
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"V\nJoined: 04 Jan 2015\nPosts: 3092\nRe: Is a < 0 ? (1) a^3 < a^2 + 2a (2) a^2 > a^3 [#permalink]\n\nShow Tags\n\n1\ncmcmcm wrote:\nHello! I was looking through these posts, answering questions, and I came across this one:\n\nIs a<0?\n\n(1) a3<a2+2a\n(2) a2>a3\n\nfrom: forum/is-a-157421.html\n\nIn the topic above, the answer is that both statements together are insufficient to answer the problem but surely the only case in which $$a^{2}$$ is greater than$$a^{3}$$ is if a is negative?\nMy answer is that both statements alone can answer the question, as the only way $$a^{3}$$ is smaller than what it is compared to is if a is negative. What am I overlooking?\n\nThank you.\n\nHi cmcmcm,\n\nAlways be very careful on how you are finding out the range of an inequality. Let me help you out with finding the range of a for both the inequalities.\n\nStatement-I\n$$a^3<a^2+2a$$ can be simplified to $$a(a + 1)(a - 2) < 0$$. Using the wavy line method to find the range of $$a$$ with the zero points being 2, 0 and -1.",
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"We can see from the wavy line diagram that the inequality is negative in the range where $$a < -1$$ or $$0 < a < 2$$. Hence, you can't say for sure if $$a < 0$$ using statement-I alone\n\nStatement-II\n$$a^2>a^3$$ can be simplified to $$a^2(a - 1) < 0$$. Since $$a^2$$ is always non-negative, for $$a^2(a - 1) < 0$$, $$(a - 1) < 0$$ i.e. $$a < 1$$.\nSo $$a < 0$$ or $$a > 0$$. Hence using statement-II alone you cant' say for sure if $$a < 0$$.\n\nCombining statement-I & II\nCombining statements-I & II will give us the range as $$a < -1$$ or $$0 <a <1$$. Hence, it is not sufficient to tell if $$a < 0$$. Therefore the answer is E.\n\nYou can read more about the Wavy line method here.\n\nHope it's clear",
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".Let me know if you have any doubt in any part of the explanation.\n\nRegards\nHarsh\n_________________\nEMPOWERgmat Instructor",
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"V\nStatus: GMAT Assassin/Co-Founder\nAffiliations: EMPOWERgmat\nJoined: 19 Dec 2014\nPosts: 15321\nLocation: United States (CA)\nGMAT 1: 800 Q51 V49",
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"Re: Is a < 0 ? (1) a^3 < a^2 + 2a (2) a^2 > a^3 [#permalink]\n\nShow Tags\n\n1\nHi cmcmcm,\n\nThe prompt does NOT state that \"A\" has to be an integer, so you have to consider the possibility that it's NOT an integer (meaning \"A\" could be a fraction).\n\nWhile that level of 'thoroughness' isn't going to be required on that many DS questions, Test Takers who score at the higher levels in the Quant section are more likely to see questions in which fractional answers have to be considered.\n\nGMAT assassins aren't born, they're made,\nRich\n_________________\nManager",
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"Joined: 21 May 2015\nPosts: 215\nConcentration: Operations, Strategy\nGMAT 1: 750 Q50 V41",
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"Re: Is a < 0 ? (1) a^3 < a^2 + 2a (2) a^2 > a^3 [#permalink]\n\nShow Tags\n\nE\n(1) take a = -1, -2 ---- If a=-1 then eqn gives -1<-1 which is not true and if a=-2 then eqn gives -8<0 which is true ....thats y insuff\n(2) a^2>a^3 implies a can be -ve or 0<a<1 thus insuff....\nCombined also is insuff because of above reasons\n_________________\nApoorv\n\nI realize that i cannot change the world....But i can play a part",
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"Manager",
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"Joined: 21 May 2015\nPosts: 215\nConcentration: Operations, Strategy\nGMAT 1: 750 Q50 V41",
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"Re: Is a < 0 ? (1) a^3 < a^2 + 2a (2) a^2 > a^3 [#permalink]\n\nShow Tags\n\n(1) a^3 < a^2 + 2a - --- a=-1 then -1<-1 not true ; a=-2 then -8<0 true insuff\n(2) a^2 > a^3 then a<0 or 0<a<1 insuff\n\nIf we combine both we cannot answer bc of above reasons\n_________________\nApoorv\n\nI realize that i cannot change the world....But i can play a part",
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"V\nJoined: 12 Sep 2015\nPosts: 4019\nRe: Is a < 0 ? (1) a^3 < a^2 + 2a (2) a^2 > a^3 [#permalink]\n\nShow Tags\n\n2\nTop Contributor\nanairamitch1804 wrote:\nIs a < 0 ?\n(1) a³ < a² + 2a\n(2) a² > a³\n\nTarget question: Is a < 0 ?\n\nStatement 1: a³ < a² + 2a\nSubtract a² and 2a from both sides to get: a³ - a² - 2a < 0\nFactor: a(a² - a - 2) < 0\nFactor more: a(a - 2)(a + 1) < 0\nThere are several values of a that satisfy this inequality. Here are two:\nCase a: a = 0.5. In this case, a > 0\nCase b: a = -10. In this case, a < 0\nSince we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT\n\nStatement 2: a² > a³\nThere are several values of a that satisfy this inequality. Here are two:\nCase a: a = 0.5. In this case, a > 0\nCase b: a = -10. In this case, a < 0\nSince we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT\n\nStatements 1 and 2 combined\nThere are several values of a that satisfy BOTH statements Here are two:\nCase a: a = 0.5. In this case, a > 0\nCase b: a = -10. In this case, a < 0\nSince we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT\n\nCheers,\nBrent\n_________________\n\nOriginally posted by GMATPrepNow on 23 Feb 2017, 19:40.\nLast edited by GMATPrepNow on 23 Feb 2017, 23:15, edited 1 time in total.\nMath Expert",
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"V\nJoined: 02 Aug 2009\nPosts: 8025\nRe: Is a < 0 ? (1) a^3 < a^2 + 2a (2) a^2 > a^3 [#permalink]\n\nShow Tags\n\n1\nIs a < 0 ?\n(1) a^3 < a^2 + 2a\n$$a^3-a^2-2a<0......a( a^2-a-2)<0......a(a-2)(a+1)<0$$\nSo a <-1 will give ans as YES..\na=0 or 1 will also be true and ans will be NO\nInsufficient\n\n(2) a^2 > a^3\na^2-a^3>0......$$a^2(1-a)>0$$..\nSo 1-a>0..a<1...\nSo a can be -1 or 0 or 0.5\nInsuff..\n\nCombined\nAgain a as 0, 0.5 or -3 etc still remains..\nInsufficient\n\nE\n_________________\nIntern",
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"B\nJoined: 11 Feb 2017\nPosts: 2\nRe: Is a < 0 ? (1) a^3 < a^2 + 2a (2) a^2 > a^3 [#permalink]\n\nShow Tags\n\nIs a < 0?\n\nStatement 1: a^3 < a^2 +2a --> a(a+1)(a-2) < 0 Therefore, if this equals zero, then a can be -1, 0, or 2.\nTry a = 1, the expression is negative. We can fill in the number line with signs because signs will switch back and forth.\nTherefore <----(-1)++++(0)-----(2)+++++>. Statement 1 is correct when a is less than -1 or between 0 and 2. Insufficient.\n\nStatement 2: a^2 > a^3 --> a(a)(a-1) > 0 Therefore, if this equals zero, then a can be 0 or 1.\nTry a = 2, the expression is positive. We can fill in the number line with signs because signs will switch back and forth.\nTherefore <++++(0)------(1)+++++>. Statement 2 is correct when a is less than 0 or greater than 1. Insufficient.\n\nCombined:\nFrom statement 1, a can be less than -1 or between 0 and 2.\nFrom statement 2, a can be less than 0 or greater than 1.\nTherefore, a must be less than -1 or between 1 and 2.\nIs a < 0? Yes and no. Insufficient.",
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"Re: Is a < 0 ? (1) a^3 < a^2 + 2a (2) a^2 > a^3 [#permalink] 23 Feb 2017, 20:57\n\nGo to page 1 2 Next [ 22 posts ]\n\nDisplay posts from previous: Sort by\n\nIs a < 0 ? (1) a^3 < a^2 + 2a (2) a^2 > a^3\n\n new topic post reply Question banks Downloads My Bookmarks Reviews Important topics\n\n Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne",
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|
http://csharphelper.com/blog/2017/05/graph-event-probabilities-in-c/
|
[
"# Graph event probabilities in C#",
null,
"The example Calculate the probability of an event occurring in a given number of trials in C# shows how to calculate event probabilities. This example graphs the results. You can use the example to get some feel for how the probabilities change.\n\nEnter the event probability and the maximum number of trials to perform. When you click Go, the program executes the following code.\n\n```// The probability data.\nprivate PointF[] Points = {};\n\n// Calculate and display probabilities.\nprivate void btnGo_Click(object sender, EventArgs e)\n{\n// Make room for the probabilities.\nint num_events = int.Parse(txtMaxNumberEvents.Text);\nPoints = new PointF[num_events + 1];\n\n// Get the event probability.\ndouble event_prob =\ndouble.Parse(txtEventProb.Text.Replace(\"%\", \"\"));\nif (txtEventProb.Text.Contains(\"%\")) event_prob /= 100.0;\n\n// Get the probability of the event not happening.\ndouble non_prob = 1.0 - event_prob;\n\nfor (int i = 0; i <= num_events; i++)\n{\nPoints[i].X = i;\nPoints[i].Y = 100 * (float)(1.0 - Math.Pow(non_prob, i));\n}\n\n// Redraw.\npicGraph.Refresh();\n}```\n\nThe Points array holds the data for the current probability parameters. The Go button’s Click event handler parses the number of events you entered and makes the Points array big enough to hold the event probabilities. The code then parses the event probability and converts it into a decimal value (as in 0.5 instead 50%).\n\nNext the code loops over the desired events calculating the event probabilities and saving the results in the Points array. For drawing convenience, the code stores the probability value as a percentage.\n\nAfter it has calculated all of the event probabilities, the code refreshes the picGraph PictureBox to make it display the results. The following code shows how the picGraph control’s Paint event handler draws the graph.\n\n```// Draw the data.\nprivate void picGraph_Paint(object sender, PaintEventArgs e)\n{\n// Clear.\ne.Graphics.Clear(picGraph.BackColor);\nif (Points.Length < 2) return;\ne.Graphics.SmoothingMode = SmoothingMode.AntiAlias;\n\n// Transform from world coordinates to screen coordinates.\nRectangleF rect = new RectangleF(0, 0, Points.Length + 1, 100);\nPointF[] pts =\n{\nnew PointF(0, picGraph.ClientSize.Height),\nnew PointF(picGraph.ClientSize.Width,\npicGraph.ClientSize.Height),\nnew PointF(0, 0)\n};\nMatrix transform = new Matrix(rect, pts);\n\nusing (Pen pen = new Pen(Color.Gray, 0))\n{\n// Draw the axes.\nusing (StringFormat sf = new StringFormat())\n{\nsf.LineAlignment = StringAlignment.Center;\nfor (int i = 0; i <= 100; i += 10)\n{\n// See where this should be.\npts = new PointF[]\n{\nnew PointF(0, i),\nnew PointF(Points.Length, i),\n};\ntransform.TransformPoints(pts);\ne.Graphics.DrawLine(pen, pts, pts);\ne.Graphics.DrawString(i.ToString(), this.Font,\nBrushes.Green, pts, sf);\n}\n\nsf.Alignment = StringAlignment.Center;\nsf.LineAlignment = StringAlignment.Far;\nint skip = (int)(Points.Length / 10);\nskip = 5 * (int)(skip / 5);\nif (skip < 1) skip = 1;\nfor (int i = 0; i < Points.Length; i += skip)\n{\n// See where this should be.\npts = new PointF[]\n{\nnew PointF(i, 0),\nnew PointF(i, 5),\n};\ntransform.TransformPoints(pts);\ne.Graphics.DrawLine(pen, pts, pts);\ne.Graphics.DrawString(i.ToString(), this.Font,\nBrushes.Green, pts, sf);\n}\n}\n\n// Draw the graph.\npen.Color = Color.Blue;\ne.Graphics.Transform = transform;\ne.Graphics.DrawLines(pen, Points);\n}\n}```\n\nThis code first clears the PictureBox. Then if there are no data points it simply returns.\n\nThe code then creates a transformation matrix to transform the data to fit the PictureBox. To do that it creates a RectangleF representing the points’ world coordinate bounds. Those ranges from 0 to 100 in the Y direction (the percentages) and 0 to the number of events in the X direction.\n\nThe code also creates an array of points indicating where on the PictureBox the upper left, upper right, and lower left corners of the world coordinate bounds should be mapped. It uses the RectangleF and the array of points to create a Matrix that maps from world coordinates to PictureBox coordinates.\n\nThe program then draws the graph. It creates a thin pen to draw with and makes a StringFormat object for use when drawing text.\n\nNext the code draws horizontal lines for every multiple 10% probability. It draws a line connecting the points (0, i) - (Points.Length, i) for i = 10, 20, 30, and so forth. To do this, the code must transform the coordinates of the points so they match up with the transformed points in the graph. It does that by calling the transformation matrix’s TransformPoints method.\n\nAfter it draws each horizontal line, the code also draws text giving the line’s probability. It positions the text at the line’s left end point (0, i) and centered vertically on the line.\n\nNext the code draws the X axis. It loops over values between 0 and the number of events drawing vertical tick marks and the event numbers.\n\nAfter drawing the axes and their labels, drawing the actual graph is simple. The code applies the transformation matrix to the Graphics object and calls that object’s DrawLines method to draw the lines.\n\nWhen you resize the form, the PictureBox control’s Resize event handler also refreshes the PictureBox so you can make the graph bigger if you like.",
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"This entry was posted in drawing, graphics, mathematics and tagged , , , , , , , , , , . Bookmark the permalink.\n\nThis site uses Akismet to reduce spam. Learn how your comment data is processed."
] |
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"http://www.csharphelper.com/twitter.png",
null,
"http://www.csharphelper.com/rss.png",
null,
"http://www.csharphelper.com/Donate.png",
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https://fds.duke.edu/db/aas/math/schoen
|
[
"Math @ Duke\n\n .......................",
null,
"Faculty .......................",
null,
"Webpage",
null,
"Publications",
null,
"Research Interests\n\nI work on the geometry and arithmetic of figures defined by polynomial equations. I am especially interested in the geometry of algebraic curves, surfaces, threefolds and fourfolds over the complex numbers, over numbers fields, over finite fields, over fields of transcendence degree one over finite fields and over discrete valuation rings with perfect residue field. More specifically I study elliptic surfaces, elliptic threefolds, Calabi-Yau varieties, abelian varieties and surfaces of general type. I am interested in Chow groups of algebraic varieties and the relationship between a variety's Chow group and its arithmetic and geometric properties.\n\nContact Info:\n Office Location: 191 Physics Bldg, 120 Science Drive Box 90320, Durham, NC 27708 Office Phone: (919) 660-2813 Email Address:",
null,
"",
null,
"Web Page: http://www.math.duke.edu/~schoen\n\nTeaching (Fall 2021):\n\n• MATH 627.01, ALGEBRAIC GEOMETRY Synopsis\nPhysics 227, WF 08:30 AM-09:45 AM\nOffice Hours:\n\nMonday 11-12 and tuesday 11-12\nor by appointment\nEducation:\n\n Ph.D. The University of Chicago 1982 B.A. Haverford College 1975\nSpecialties:\n\nAlgebra\nGeometry\nResearch Interests: Algebraic Geometry\n\nI work on the geometry and arithmetic of figures defined by polynomial equations. I am especially interested in the geometry of algebraic curves, surfaces, threefolds and fourfolds over the complex numbers, over numbers fields, over finite fields, over fields of transcendence degree one over finite fields and over discrete valuation rings with perfect residue field. More specifically I study elliptic surfaces, elliptic threefolds, Calabi-Yau varieties, abelian varieties and surfaces of general type. I am interested in Chow groups of algebraic varieties and the relationship between a variety's Chow groups and its arithmetic and geometric properties.\n\nAreas of Interest:\n\nGeometry of algebraic varieties\nAlgebraic Cycles\nChow Groups\n\nKeywords:\n\nAlgebraic geometry\n\nCurrent Ph.D. Students (Former Students)\n\n• Humberto Diaz\nRecent Publications (More Publications)\n\n1. Schoen, C, On certain complex projective manifolds with Hodge numbers H10 = 4 and h20 = 5, The Michigan Mathematical Journal, vol. 68 no. 3 (January, 2019), pp. 565-596 [doi]\n2. Schoen, C, An arithmetic ball quotient surface whose Albanese variety is not of CM type, Electronic Research Announcements in Mathematical Sciences, vol. 21 (September, 2014), pp. 132-136, American Institute of Mathematical Sciences (AIMS) [doi]\n3. Schoen, C, Torsion in the cohomology of desingularized fiber products of elliptic surfaces, The Michigan Mathematical Journal, vol. 62 no. 1 (March, 2013), pp. 81-115, Michigan Mathematical Journal, ISSN 0026-2285 [doi]\n4. Schoen, C, The geometric genus of a desingularized fiber product of elliptic surfaces, Proceedings of the American Mathematical Society, vol. 141 no. 3 (January, 2013), pp. 745-752, American Mathematical Society (AMS), ISSN 0002-9939 [doi] [abs]\n5. Schoen, C, Invariants of regular models of the product of two elliptic curves at a place of multiplicative reduction, in Arithmetic and Geometry of K3 surfaces and Calabi-Yau Threefolds, Fields Institute Communications 67, edited by Laza, R. ; Schuett, M; and Yui, N., Arithmetic and Geometry of K3 Surfaces and Calabi Yau Threefolds, vol. 67 (2013), pp. 461-487, SPRINGER, ISBN 978-1-4614-6402-0\n\[email protected]",
null,
"ph: 919.660.2800\nfax: 919.660.2821",
null,
"Mathematics Department\nDuke University, Box 90320\nDurham, NC 27708-0320"
] |
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|
https://math.stackexchange.com/questions/46919/changing-the-manifold-preserving-the-discrete-spectrum
|
[
"# Changing the manifold, preserving the discrete spectrum\n\nOn a Riemannian manifold $M$, the Laplace operator $L$ is uniquely defined.\n\nIf $M$ is not compact, then $L$ admits a continuous spectrum.\n\nIs there a way of \"changing\" $M$ and/or $L$ in say $M'$ and $L'$, such that the spectrum of $L'$ is now purely discrete and contains the spectrum of $L$? I am interested in examples of this type.\n\nFeel free to restrict to homogenous varieties or any other geometric gadget, if have an example of similar nature in mind.\n\nMotivation: More general it is true that, given an self adjoint operator $T$ on a Hilbert space, does there exists a decomposition such that $T_1 + T_2 = T$, where $T_1$ has the same point spectrum as $T$. I am not looking for this. Is there a better geometric construction?\n\n• What if the operator $T$ has no point spectrum? (Or in the manifold case, what if $(M,L)$ is the Euclidean space?) I may be misunderstanding what you mean/want, can you clarify? Jun 24, 2011 at 1:55\n• I was just talking about containment, the empty set is contained everywhere. Jun 24, 2011 at 9:19\n\nSurely that would have outstanding consequences if it were true.\n\nLet's just consider the case where $M$ is a hyperbolic Riemann surface, and assume arithmetic if non-compact. Then the main term in Weyl's laws for the discrete spectrum doesn't depend upon being closed or having cusps, just upon the volume, but the next term in the remainder does. The discrete spectrum grows faster in the presence of cusps, so one cannot hope to pass from an arithmetic surface with cusps to a compact surface with the same discrete spectrum. See for example M\\\"uller's paper: \"Weyl's law in the theory of automorphic forms.\"\n\nThe work of Phillips, Sarnak and others, indicates that that discrete spectrum varies wildly under perturbations, and it would seem rather improbable (at least to me), that the phenomenon you're looking for ever occurs (or at least no more common than instances of isopectral manifolds).\n\nIncidentally, Phillips-Sarnak suggests that for a general non-arithmetic manifold with cusps, the discrete spectrum will be finite, in contrast to the case of a compact manifold. I don't have any intuition in this case for whether such a spectrum might be contained in the spectrum of a compact manifold, but neither is it clear to me how useful knowing this would be.\n\n• That is a perfect answer, maybe the Jacquet Langland correspondance is a suitable example in this context. Jun 24, 2011 at 9:17\n\nA different \"adjustment\" is as in Colin-de-Verdiere's 1981 treatment of meromorphic continuation of Eisenstein series (at least for rank-one?): he replaces the usual Laplacian (which is essentially self-adjoint) by a pseudo-Laplacian created as the minimal (\"Friedrichs\") extension of the restriction of the Laplacian to automorphic forms whose constant terms vanish above a fixed height. A variant of the argument for discreteness of cuspforms proves that this pseudo-Laplacian has a compact resolvent, hence, discrete spectrum. The cuspforms remain eigenfunctions, the constants cease to be eigenfunctions because they are not in the space, and certain truncated Eisenstein series become genuine eigenfunctions for the pseudo-Laplacian (rather than failing to be eigenfunctions for the ordinary Laplacian).\n\nEdit: and, in regard to some aspects of @Kimball's points, specifically (looking at the details of C-de-V's discussion), the truncated Eisenstein series are essentially those whose truncated constant term is still continuous, that is, so that, at height $y_o$, in the simplest case the condition for the truncation $\\wedge^{y_o} E_s$ to be a \"new\" eigenfunction is that $y_o^s+c_s y_o^{1-s}=0$, where $c_s=\\xi(2s-1)/\\xi(2s)$. Thus, standard (if non-trivial) facts about zeta (and a refined Weyl's Law for cuspforms?) still assure that the \"new\" eigenfunctions are asymptotically fewer, quantifiably."
] |
[
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https://www.bartleby.com/solution-answer/chapter-5-problem-74ap-college-physics-11th-edition/9781305952300/a-500-kg-student-evaluates-a-weight-loss-program-by-calculating-the-number-of-times-she-would-need/a649b93a-98d7-11e8-ada4-0ee91056875a
|
[
"",
null,
"",
null,
"",
null,
"Chapter 5, Problem 74AP\n\nChapter\nSection\nTextbook Problem\n\nA 50.0-kg student evaluates a weight loss program by calculating the number of times she would need to climb a 12.0-m high flight of steps in order to lose one pound (0.45 kg) of fat. Metabolizing 1.00 kg of fat can release 3.77 × 107 J of chemical energy and the body can convert about 20.0% of this into mechanical energy. (The rest goes into internal energy.) (a) How much mechanical energy can the body produce from 0.450 kg of fat? (b) How many trips up the flight of steps are required for the student to lose 0.450 kg of fat? Ignore the relatively small amount of energy required to return down the stairs.\n\n(a)\n\nTo determine\nThe mechanical energy that can be produced from 0.450kg of fat.\n\nExplanation\n\nGiven Info:\n\n1.0kg of fat can release 3.77×107J of chemical energy\n\nThe body can convert 20% of the chemical energy to mechanical energy\n\nFormula to calculate the mechanical energy produced is,\n\nEmec=(0.20)Echm (I)\n\n• Echm is the chemical energy available from 0.450kg of fat\n\nSince,\n\nEchm=mf(3.77×107J1.0kg) (II)\n\n• mf is the mass of fat\n\nOn substituting equation (II) in (I),\n\nEmec=(0.20)(mf)(3\n\n(b)\n\nTo determine\nThe number of trips up the flight of steps are necessary for the student to lose 0.450kg of fat.\n\nStill sussing out bartleby?\n\nCheck out a sample textbook solution.\n\nSee a sample solution\n\nThe Solution to Your Study Problems\n\nBartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!\n\nGet Started\n\nFind more solutions based on key concepts",
null,
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|
http://web.eecs.umich.edu/~qstout/abs/BR93ccc.html
|
[
"Perfect Dominating Sets on Cube-Connected Cycles\n\nDouglas Van Wieren\nComputer Science and Engineering, University of Michigan\n\nMarilynn Livingston\nDept. of Computer Science, Southern Illinois University at Edwardsville\n\nQuentin F. Stout\nComputer Science and Engineering, University of Michigan\n\nAbstract: A vertex v of a graph G is said to dominate a vertex u if either v=u or there is an edge from v to u. A set S of vertices of G is a dominating set of G if every vertex of G is dominated by a vertex in S. S is a perfect dominating set (PDS) if each vertex of G is dominated by exactly one vertex in S. A perfect dominating set is equivalent to a perfect error-correcting code, where the elements of the PDS are the codewords.\n\nCube-connected cycles are a family of cubic graphs with relatively small diameters and regular structure, making them attractive models for parallel architecture design. The existence of perfect dominating sets for any structural model of parallel computation is useful for solving various algorithmic problems, such as efficient resource allocation. This paper gives a method for constructing PDSs on cube-connected cycles where they exist, and proves nonexistence for all other cases. Specifically, a simple algorithm is given to construct a standard perfect dominating set (distance equal to 1) for cube-connected cycles of all orders except 5, and it is shown that there is no perfect dominating set for order 5. Moreover, the existence of perfect dominating sets for all distances greater than 1 is disproven (with the trivial exception --- the distance equaling or exceeding the diameter of the graph).\n\nKeywords: perfect dominating set, cube-connected cycles, parallel computer, perfect code, vertex-vertex domination, graph theory, regular interconnection network, resource allocation\n\nComplete paper. This appears in Congressus Numerantium 97 (1993), pp. 51-70.\n\nRelated Work:",
null,
"Copyright © 2008-2018 Quentin F. Stout"
] |
[
null,
"http://web.eecs.umich.edu/~qstout/qfshome.gif",
null
] |
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|
http://www.morethantechnical.com/2010/12/28/hand-gesture-recognition-via-model-fitting-in-energy-minimization-wopencv/
|
[
"# Hand gesture recognition via model fitting in energy minimization w/OpenCV\n\nJust wanted to share a thing I made - a simple 2D hand pose estimator, using a skeleton model fitting. Basically there has been a crap load of work on hand pose estimation, but I was inspired by this ancient work. The problem is setting out to find a good solution, and everything is very hard to understand and implement. In such cases I like to be inspired by a method, and just set out with my own implementation. This way, I understand whats going on, simplify it, and share it with you!\n\nAnyway, let's get down to business.\n\nEdit (6/5/2014): Also see some of my other work on hand gesture recognition using smart contours and particle filters\n\n# A bit about energy minimization problems\n\nA dear friend revealed before me the wonders of energy minimization problems a while back, and ever since I have trying to find uses for that method. Basically, it is trying to find a global minimum for a complicated energy function (usually with many parameters), by following the function's gradient. Such methods are often called Gradient Descent, and used mostly for non-linear systems that can't be solved easily using a least-squares variant.\n\nA lot of work in computer vision was done using energy functions (I believe the most seminal was Snakes, over 10,000 citations), usually having two terms: Internal energy and External energy. The equilibrium between the two terms should result in a low-energy system - our optimal result. So we would like to formulate the terms in our system such that when they are 0 - they describe the system as we want it.\n\nFollowing the works with active contours, I believe the external energy function should have to do with how the hand model fits to the hand blob, and the internal energy will have to do with how \"comfortable\" the hand is with this configuration.\n\n# The hand model\n\nLet's see how a 2D model of a hand might look like",
null,
"Kinda looks like a rake... huh?\n\nThere are some parts that practically can't change much, i.e the palm (orange), and some that might change drastically, i.e the fingers (red). Each finger has joints (blue circle), and a tip (bigger blue circle).\n\n```typedef struct finger_data {\nPoint2d origin_offset;\t\t//base or finger relative to center hand\ndouble a;\t\t\t\t\t//angle\nvector<double> joints_a;\t//angles of joints\nvector<double> joints_d;\t//bone length\n} FINGER_DATA;\n\ntypedef struct hand_data {\nFINGER_DATA fingers;\t\t//fingers\ndouble a;\t\t\t\t\t//angle of whole hand\nPoint2d origin;\t\t\t\t//center of palm\nPoint2d origin_offset;\t\t//offset from center for optimization\ndouble size;\t\t\t\t//relative size of hand = length of a finger\n} HAND_DATA;\n```\n\nAt first I thought, since I'm only interested in the tips of the fingers, to use Inverse Kinematics to guide the tips to a certain point and let the joints find their own minimal energy position, following this article. But I abandoned this method because of complications.\n\nI also had to simplify this model, for real-time estimation and also better results. So in the end I ended up with a very rigid model, that allows only on joint per finger and no angular movement.\n\n# Using tnc.c\n\ntnc.c is a \"library\", essentially one c file, that implements a line search algorithm that is able to find the minimum point of a multi-variate function. I'm not certain of the algorithm details, and it's not so important as it can be replaced with any other similar library. But, tnc.c has a great advantage - it is dead simple. One function will start the gradient decent, calling-back a function to calculate the gradients.\n\nSo basically I had to write just one very short function:\n\n```static int my_f(double x[], double *f, double g[], void *state) {\nDATA_FOR_TNC* d_ptr = (DATA_FOR_TNC*)state;\nDATA_FOR_TNC new_data = *d_ptr;\n\nmapVecToData(x,new_data.hand);\n\n*f = calc_Energy(new_data,*d_ptr);\n\n{\ndouble _x[SIZE_OF_HAND_DATA];\n\nfor(int i=0;i<SIZE_OF_HAND_DATA;i++) {\nmemcpy(_x, x, sizeof(double)*SIZE_OF_HAND_DATA); //reset variables\n_x[i] = _x[i] + EPSILON; //change only one variable\nmapVecToData(_x, new_data.hand);\ndouble E_epsilon = calc_Energy(new_data,*d_ptr);\ng[i] = ((E_epsilon - *f) / EPSILON); //calc the gradient for this variable change\n}\n}\n\nreturn 0;\n}\n```\n\nThis function is called by tnc.c on every iteration of the search, the double x[] is the state of variables the search is now examining, double* f is the energy for this state, double g[] are the gradients (same size as x[]), and voide* state is a user-defined variable that can be carried along the process.\n\nSo what I did is simply changed the value of each parameter in turn, to test how it effects the energy in the system. I get a measure of the energy, then I subtract it from the \"natural\" setup (without any changes to parameters) energy measure, and I get the gradient for this parameter.\n\nThe energy function came out a bit different in the end:\n\n```\nstatic double calc_Energy(DATA_FOR_TNC& d, DATA_FOR_TNC& orig_d) {\ndouble _sum = 0.0;\n\n//external energy: how close are the joints to the hand blob? (how well do they fit to it)\nvector<Point2d> joints;\nMat tips(5,1,CV_64FC2);\n\nfor (int j=0; j<5; j++) {\njoints.clear();\nFINGER_DATA f = d.hand.fingers[j];\nPoint2d _newTip = newTip(f,d.hand,joints); //get joints for this finger\n\nfor (int i=0; i<tmp.size(); i++) { //for each joint find how far it is from the blob\ndouble ds = pointPolygonTest(d.contour, tmp[i]+getHandOrigin(d.hand), true);\nds += 5;\nds = 1 * ((ds < 0) ? -1 : 1) * (ds*ds) ;\n_sum -= (ds > 0) ? 0 : 100*ds;\n}\n\ntips.at<Point2d>(j,0) = _newTip;\n}\n\n//lazyness of fingers - joints should strive to be as they were in the natural pose\nvector<double> _angles;\n//\tfor (int j=0; j<5; j++) {\n//\t\tFINGER_DATA f = d.hand.fingers[j];\n//\t\tFINGER_DATA of = orig_d.hand.fingers[j];\n////\t\t_angles.push_back(f.a - of.a);\n//\t\tfor (int i=0; i<f.joints_d.size(); i++) {\n////\t\t\t_angles.push_back(f.joints_a[i] - of.joints_a[i]);\n//\t\t\t_angles.push_back(f.joints_d[i] - of.joints_d[i]);\n//\t\t}\n//\t}\n_angles.push_back(d.hand.a-orig_d.hand.a); //the angle of the hand should be as it was before\n_sum += 10000*norm(Mat(_angles));\n\nif(_sum < 0) return 0;\nreturn _sum;\n}\n```\n\nYou'll notice the commented out section. The \"laziness of fingers\" turned out not to give good results... A different metric is needed! I have not found it yet, maybe you have a good idea?\n\nStarting tnc.c is very simple: Allocating the vectors for X and gradients, initializing the model from the blob, and calling the simple_tnc convenience method. simple_tnc starts tnc with some default parameters that don't affect the outcome (at least in my tries).\n\n```void estimateHand(Mat& mymask) {\ndouble _x[SIZE_OF_HAND_DATA] = {0};\nMat X(1,SIZE_OF_HAND_DATA,CV_64FC1,_x);\ndouble f;\n\nnamedWindow(\"state\");\n\nmapDataToVec((double*)X.data, d.hand);\n\nsimple_tnc(SIZE_OF_HAND_DATA, (double*)X.data, &f, (double*)gradients.data, my_f, (void*)&d, 1, 0);\n\nmapVecToData((double*)X.data, d.hand);\nshowstate(d,1);\n\nd.hand.origin = getHandOrigin(d.hand); //move to new position\n}\n```\n\n# Results and Discussion\n\nHere are my results so far:\n\nIt's not perfect, but it's a start. Tracking and estimating open hand is pretty good, with some orientation change as well. But when the fingers are closed... that's where problems start.\n\nSometimes the joints \"hover\" over the black area to \"land\" in a white area so they \"fit\", but they should not do that. One easy thing to do to counter this is to measure the distance of the whole bone, and not just the joint.\n\nThe model right now doesn't use all the joints possible, because it is too heavy computationally. Plus the energy does not depend (or change) the angle of the fingers. So this is a very very simple model of a hand...\n\nBut, it is a good start! All the other stuff I have seen online is just basic high-curvature points counting and color-based or feature-based segmentation and tracking... My model actually tries to fit an articulate and precise model of a hand to the image.\n\n# How did you get such nice blobs?!\n\nYou ask. They are beautiful aren't they... nice and clean, easy for tracking and model fitting. It's no magic though...\nWell, I took part of a project in the Media Lab, called DepthJS, that uses the MS Kinect to control web pages. I wrote the computer-vision part. So all the code is there, you can grab it, I just plugged it into this little project. Basing off this very simple example of using OpenCV2.X and libfreenect.\n\nWow, this was a longie.. I hope you learned something and got inspired. I got to do a second overview of the project, and I'm inspired. Inspiration all around!\n\nCode is obviously yours for the taking:\nhttps://github.com/royshil/OpenHPE\n\nPlease contribute your own views, thoughts, code, rants in the comments and github page.\n\nEnjoy\nRoy.\n\n## 17 Replies to “Hand gesture recognition via model fitting in energy minimization w/OpenCV”\n\n1.",
null,
"blackball says:\n\nSo cool! /* It said my comment was a little short...*/\n\n2.",
null,
"yaya says:\n\ngood job man! thanks for the sharing!\n\n3.",
null,
"Jasmit says:\n\nHi Roy,\nIf i am compiling the shared code on Visual Studio 2005, i am getting the following errors in the main.cpp file. Can you please let me know in case i am doing some mistake while compiling. Your help will be greatly appreciated.\n\nerror C2678: binary '*' : no operator found which takes a left-hand operand of type 'int' (or there is no acceptable conversion) line-64\n\nerror C2678: binary '+' : no operator found which takes a left-hand operand of type 'cv::Mat' (or there is no acceptable conversion) line-248\n\nerror C2664: 'cv::Mat::Mat(const cv::Mat &)' : cannot convert parameter 1 from 'cv::Point_' to 'const cv::Mat &' line-404\n\nerror C2664: 'cv::Mat::Mat(const cv::Mat &)' : cannot convert parameter 1 from 'cv::Point_' to 'const cv::Mat &' line-416\n\nerror C2660: 'cv::namedWindow' : function does not take 1 arguments line-449\n\n4.",
null,
"Roy says:\n\n@Jasmit\nFollow the instructions of the compiler... look at these lines and try to work around the problems.\nPerhaps your OpenCV version is not up to date, so some of the matrix operators (\"*\" and \"+\") are not defined properly. Try updating OpenCV\n\n5.",
null,
"Jasmit says:\n\nHi Roy,\nThanks a lot for your suggestion. I compiled the code with OpenCV 2.2 and it works fine.\nActually i tried with a number of sample images and what i find is that your code works fine when the background is black. Whenever the hand is present in a non-black background, the hand detection is not proper. Do you agree with me. If yes, is there any solution to solve this problem.\n\n6.",
null,
"hendri says:\n\nhi roy..\nthanks a lot for your share...\n\ni am use ubuntu 10.10 ..\nwhere i can get libfreenect.hpp ?? 😀\n\n7.",
null,
"Jhon says:\n\npretty nice... i think it will be helpfull for me... look at the project that i am working on. http://www.youtube.com/watch?v=cxHMgl2_5zg now.. i wanna port it to Iphone plataform, if u can give me some tutorials about that i will be gratefull.\n\n8.",
null,
"reachgoa says:\n\nhi,roy,now ,i encounter there problems ,1>graphcut.obj : error LNK2001: 无法解析的外部符号 \"public: __int64 __thiscall GCoptimization::compute_energy(void)\" (?compute_energy@GCoptimization@@QAE_JXZ)\n1>graphcut.obj : error LNK2001:can't analysis the extern symbol \"public: void __thiscall GCoptimization::setDataCost(int,int,int)\" (?setDataCost@GCoptimization@@QAEXHHH@Z)\n1>graphcut.obj : error LNK2001: can't analysis the extern symbol \"public: __int64 __thiscall GCoptimization::expansion(int)\" (?expansion@GCoptimization@@QAE_JH@Z)\n1>graphcut.obj : error LNK2001: can't analysis the extern symbol \"public: void __thiscall GCoptimizationGridGraph::setSmoothCostVH(int *,int *,int *)\" (?setSmoothCostVH@GCoptimizationGridGraph@@QAEXPAH00@Z)\n1>graphcut.obj : error LNK2001: can't analysis the extern symbol \"public: virtual __thiscall GCoptimizationGridGraph::~GCoptimizationGridGraph(void)\" (??1GCoptimizationGridGraph@@UAE@XZ)\n1>graphcut.obj : error LNK2001: can't analysis the extern symbol \"public: __thiscall GCoptimizationGridGraph::GCoptimizationGridGraph(int,int,int)\" (??0GCoptimizationGridGraph@@QAE@HHH@Z)\n\ncan u give me some ideas\n\n9.",
null,
"reachgoa says:\n\nsorry ,i send the wroth place, the problem is in the GraphCut,thank very much for your hard work\n\n10.",
null,
"malik says:\n\nplease can u help with tutorial if possible a material on hw to develop an object recognition application of android using java and opencv\n\nThanks\n\n11.",
null,
"Nour says:\n\nHi Roy,\nIf i am compiling the shared code on Visual Studio 2010, i am getting the following errors in the main.cpp file which I called HnadGesture.cpp . Can you please let me know in case i am doing some mistake while compiling. Your help will be greatly appreciated.\nHandGesture.obj : error LNK2019: unresolved external symbol \"class cv::Scalar_ __cdecl refineSegments(class cv::Mat const &,class cv::Mat const &,class cv::Mat &,class std::vector<class cv::Point_,class std::allocator<class cv::Point_ > > &,class std::vector<class cv::Point_,class std::allocator<class cv::Point_ > > &,class cv::Point_ &)\" (?refineSegments@@YA?AV?\\$Scalar_@N@cv@@ABVMat@2@0AAV32@AAV?\\$vector@V?\\$Point_@H@cv@@V?\\$allocator@V?\\$Point_@H@cv@@@std@@@std@@2AAV?\\$Point_@H@2@@Z) referenced in function \"void __cdecl initialize_hand_data(struct data_for_tnc &,class cv::Mat const &)\" (?initialize_hand_data@@YAXAAUdata_for_tnc@@ABVMat@cv@@@Z)\n1>HandGesture.obj : error LNK2019: unresolved external symbol _simple_tnc referenced in function \"void __cdecl estimateHand(class cv::Mat &)\" (?estimateHand@@YAXAAVMat@cv@@@Z)\n\n12.",
null,
"Roy says:\n\n@Nour\nLooks like you cannot link some functions. That's strange because the functions are in the code.\nCheck that you are indeed compiling bg_fg_blobs.cpp and tnc.c, they contain these two functions you miss.\n\n13.",
null,
"Nour says:\n\nThank you very much Roy,\nI'm almost a beginner in Opencv and doing a biometric project\nreally I'm exhausted I REPEATED the same code downloads and compiling many times and still having the same errors\n\nmain.obj : error LNK2019: unresolved external symbol \"class cv::Scalar_ __cdecl refineSegments(class cv::Mat const &,class cv::Mat const &,class cv::Mat &,class std::vector<class cv::Point_,class std::allocator<class cv::Point_ > > &,class std::vector<class cv::Point_,class std::allocator<class cv::Point_ > > &,class cv::Point_ &)\" (?refineSegments@@YA?AV?\\$Scalar_@N@cv@@ABVMat@2@0AAV32@AAV?\\$vector@V?\\$Point_@H@cv@@V?\\$allocator@V?\\$Point_@H@cv@@@std@@@std@@2AAV?\\$Point_@H@2@@Z) referenced in function \"void __cdecl initialize_hand_data(struct data_for_tnc &,class cv::Mat const &)\" (?initialize_hand_data@@YAXAAUdata_for_tnc@@ABVMat@cv@@@Z)\n\nmain.obj : error LNK2019: unresolved external symbol _simple_tnc referenced in function \"void __cdecl estimateHand(class cv::Mat &)\" (?estimateHand@@YAXAAVMat@cv@@@Z)\n\nI'm using win XP with opencv2.2 and visual studio 2010\nIs there any use of the cmakelist.txt file or the cmake folder in my case?\nIf there are any suggestions please tell me\nSo sorry for the disturbance and thank you very much in advance\n\n14.",
null,
"Roy says:\n\n@Nour\nIndeed you should be working with the CMakeLists.txt to compile the program.\nDownload CMake, run it ad provide the directory where the code exists. After you \"Configure\" and \"Generate\" it should create a good Visual Studio solution that will compile the code correctly.\n\n15.",
null,
"Lauzenzo says:\n\nI don't have a Kinect. Will this work on a regular webcam?\nAnd thanks for sharing the code!\n\n16.",
null,
"Ludovic says:\n\n@Laurenzo, Roy\nYes, it works fine with a regular webcam, compiled well on linux (didn't use cmake). You can also do a simple HSV segementation with, in your capture loop in the main:\ncvtColor(img,hsvframe,CV_BGR2HSV);\ninRange(hsvframe,Scalar(Hue_lo,Sat_lo,10),Scalar(Hue_hi,Sat_hi,255),gray);\nestimateHand(gray);\nwhere img is the captured image, hsvframe and gray are Mat.\nFor my hand I used:\nHue_lo = 0;Hue_hi = 10;Sat_lo = 50;Sat_hi = 170;\n(will depend on your skin color, environment, white balance, etc...).\nYou can fine tune the thresholds above with the help of the openCV HS histogram, link provided by Roy."
] |
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|
http://lavalle.pl/planning/node439.html
|
[
"## 9.2.1 Modeling Nature\n\nFor the first time in this book, uncertainty will be directly modeled. There are two DMs:\n\n1. [] Robot: This is the name given to the primary DM throughout the book. So far, there has been only one DM. Now that there are two, the name is more important because it will be used to distinguish the DMs from each other.\n2. [] Nature: This DM is a mysterious force that is unpredictable to the robot. It has its own set of actions, and it can choose them in a way that interferes with the achievements of the robot. Nature can be considered as a synthetic DM that is constructed for the purposes of modeling uncertainty in the decision-making or planning process.\n\nImagine that the robot and nature each make a decision. Each has a set of actions to choose from. Suppose that the cost depends on which actions are chosen by each. The cost still represents the effect of the outcome on the robot; however, the robot must now take into account the influence of nature on the cost. Since nature is unpredictable, the robot must formulate a model of its behavior. Assume that the robot has a set,",
null,
", of actions, as before. It is now assumed that nature also has a set of actions. This is referred to as the nature action space and is denoted by",
null,
". A nature action is denoted as",
null,
". It now seems appropriate to call",
null,
"the robot action space; however, for convenience, it will often be referred to as the action space, in which the robot is implied.\n\nThis leads to the following formulation, which extends Formulation 9.1.\n\nFormulation 9..3 (A Game Against Nature)\n1. A nonempty set",
null,
"called the (robot) action space. Each",
null,
"is referred to as an action.\n2. A nonempty set",
null,
"called the nature action space. Each",
null,
"is referred to as a nature action.\n3. A function",
null,
", called the cost function.\n\nThe cost function,",
null,
", now depends on",
null,
"and",
null,
". If",
null,
"and",
null,
"are finite, then it is convenient to specify",
null,
"as a",
null,
"matrix called the cost matrix.\n\nExample 9..8 (A Simple Game Against Nature) Suppose that",
null,
"and",
null,
"each contain three actions. This results in nine possible outcomes, which can be specified by the following cost matrix:",
null,
"",
null,
"",
null,
"",
null,
"0",
null,
"",
null,
"",
null,
"",
null,
"",
null,
"",
null,
"The robot action,",
null,
", selects a row, and the nature action,",
null,
", selects a column. The resulting cost,",
null,
", is given by the corresponding matrix entry.",
null,
"In Formulation 9.3, it appears that both DMs act at the same time; nature does not know the robot action before deciding. In many contexts, nature may know the robot action. In this case, a different nature action space can be defined for every",
null,
". This generalizes Formulation 9.3 to obtain:\n\nFormulation 9..4 (Nature Knows the Robot Action)\n1. A nonempty set",
null,
"called the action space. Each",
null,
"is referred to as an action.\n2. For each",
null,
", a nonempty set",
null,
"called the nature action space.\n3. A function",
null,
", called the cost function.\n\nIf the robot chooses an action",
null,
", then nature chooses from",
null,
".\n\nSteven M LaValle 2012-04-20"
] |
[
null,
"http://lavalle.pl/planning/img38.gif",
null,
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null,
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null,
"http://lavalle.pl/planning/img38.gif",
null,
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null,
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null,
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null,
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null,
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null,
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null,
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null,
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null,
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null,
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null,
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null,
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null
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|
https://www.physicsforums.com/threads/pressure-drop-dissolved-oxygen.956963/
|
[
"# Pressure drop / dissolved oxygen\n\n• I\n\n## Main Question or Discussion Point\n\nHello Everyone,\n\nI've been trying to find an answer to this but am not getting anywhere. Any help would be greatly appreciated!\n\nI need to be able to calculate (or more likely estimate) the pressure drop I can expect when putting water under pressure if I know there is air in the system.\n\nFor example if I have a plumbing system that I pressurize to 100 PSI then removed the pressure source. I know how much water is in the system and can estimate how much air is trapped in sections where I can not purge it. I know the pressure will drop as air is dissolved into the water but at what PSI will it reach equilibrium and no longer dissolve air into the water?\n\nCan anyone help point me in the right direction?\n\nRelated Classical Physics News on Phys.org\nChestermiller\nMentor\n\nThank you. Henry's law looks like what I need, but I cannot for the life of me figure out how to use it to calculate the pressure drop I will have. Sorry to be so needy but any help would really be appreciated\n\nHow about this? I've been able to calculate the difference in concentration of air in the solution after applying the 100 PSI to it using Henry's law. Now if I know the previous and current concentration and the volume of water and air I must be able to use some formula for the pressure. Am I on the right track here?\n\nChestermiller\nMentor\nHow about this? I've been able to calculate the difference in concentration of air in the solution after applying the 100 PSI to it using Henry's law. Now if I know the previous and current concentration and the volume of water and air I must be able to use some formula for the pressure. Am I on the right track here?\nWhy don't you specify a problem, and I'll work an example for you? Specify the total amount of water and the total amount of air, and the pressure you want to consider."
] |
[
null
] |
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|
https://cn.maplesoft.com/support/help/errors/view.aspx?path=StringTools/MaximalPalindromicSubstring&L=C
|
[
"",
null,
"MaximalPalindromicSubstring - Maple Help\n\nStringTools\n\n MaximalPalindromicSubstring\n find a maximal palindromic substring of a string",
null,
"Calling Sequence MaximalPalindromicSubstring( s )",
null,
"Parameters\n\n s - string; any Maple string",
null,
"Description\n\n • The MaximalPalindromicSubstring command computes a maximal palindromic substring of the string s. A string $t$ is a palindrome if it is equal to itself reversed, that is, $t=\\mathrm{Reverse}\\left(t\\right)$.\n • If s is nonempty and contains no substrings that are palindromes, the first character of s is the maximal palindromic substring. If s is empty, the empty string (\"\") is the maximal palindromic substring.\n • The maximal palindromic substring of s is indicated by returning a sequence of two non-negative integers:\n • The first is the index of the beginning of the palindromic substring in the string s.\n • The second is the length of the palindromic substring.\n • All of the StringTools package commands treat strings as (null-terminated) sequences of $8$-bit (ASCII) characters. Thus, there is no support for multibyte character encodings, such as unicode encodings.",
null,
"Examples\n\n > $\\mathrm{with}\\left(\\mathrm{StringTools}\\right):$\n > $\\mathrm{MaximalPalindromicSubstring}\\left(\"\"\\right)$\n ${0}{,}{0}$ (1)\n > $\\mathrm{MaximalPalindromicSubstring}\\left(\"abcde\"\\right)$\n ${1}{,}{1}$ (2)\n > $\\mathrm{pos},\\mathrm{len}≔\\mathrm{MaximalPalindromicSubstring}\\left(\"abcbde\"\\right)$\n ${\\mathrm{pos}}{,}{\\mathrm{len}}{≔}{2}{,}{3}$ (3)\n > $\"abcbde\"\\left[\\mathrm{pos}..\\mathrm{pos}+\\mathrm{len}-1\\right]$\n ${\"bcb\"}$ (4)"
] |
[
null,
"https://bat.bing.com/action/0",
null,
"https://cn.maplesoft.com/support/help/errors/arrow_down.gif",
null,
"https://cn.maplesoft.com/support/help/errors/arrow_down.gif",
null,
"https://cn.maplesoft.com/support/help/errors/arrow_down.gif",
null,
"https://cn.maplesoft.com/support/help/errors/arrow_down.gif",
null
] |
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|
https://math.stackexchange.com/questions/429131/is-this-group-solvable
|
[
"# Is this group solvable?\n\nLet $G=H\\times K$ (a direct product of $H , K$) where $H$ is an abelian 2-group and $K$ is a non-abelian simple group. Is $G$ solvable? why?\n\nThese three answers are true, and I thought so. But in a book I found a remark that made me confused: T.M. Gagen, Topics in Finite Groups, London Math. Soc. Lecture Note Ser., vol. 16, Cambridge Univ. Press, Cambridge, 1976, Remark, Theorem A on p. $40$ and Definition $11.3$ on p. $39$. Please see it and say me your ideas.\n\n• If $H$ is a normal subgroup of $G$, then $G$ is solvable if and only if $H$ and $G/H$ are solvable. – egreg Jun 25 '13 at 14:48\n• No solvable group can have a non-abelian simple subgroup (or composition factor for that matter). – Tobias Kildetoft Jun 25 '13 at 14:48\n• No soluble group can have a non-soluble subgroup! – user1729 Jun 25 '13 at 14:52\n• @Adeleh: It would be useful if you could post your new question in a new thread, and link to this one. This would mean that your new question will get the proper attention, and that the given answer still answer the question here. Also, could you please tell us what the remark says, as opposed to just a reference for it? Thanks. – user1729 Jun 25 '13 at 18:28\n• The question about Gagen's phrasing is at math.stackexchange.com/questions/429326/… – Jack Schmidt Jun 26 '13 at 2:07\n\nHint: Use the definition of a solvable group, and note that if $H$ is a normal subgroup of $G$, then $G$ is solvable if and only if both $H$ and $G/H$ are solvable.\n\nPut differently, a solvable group cannot have a non-abelian simple group as a composition factor.\n\n• Let me know if you can answer your question now ;-) – Namaste Jun 25 '13 at 16:28\n\nHINT: Show that $N=\\{e\\}\\times K$ is a normal subgroup of $G$. Argue why $N$ is not solvable, and use the fact that $G$ is solvable if and only if $N$ and $G/N$ are solvable for a normal subgroup $N$.\n\n• that is true, and I thought so. But in a book I found a remark that made me confused. Please see it and say me your ideas.\\bold{T.M. Gagen, Topics in Finite Groups, London Math. Soc. Lecture Note Ser., vol. 16, Cambridge Univ. Press, Cambridge, 1976.} Remark , Theorem A on p.40 and Definition 11.3 on p.39 – Adeleh Jun 25 '13 at 16:35\n\nSubgroups of soluble groups are soluble, which clearly proves your result. To see this, note that if $G$ is soluble of length $n$ then every element of $G^n=[G, G, \\ldots, G]$ (so, commutators of length $n$) is trivial. Then clearly if $H\\leq G$ then $H^n\\leq G^n$, so all commutators of length $n$ in $H$ are trivial. Thus, $H$ is soluble.\n\n• To the person who replaced every instance of the word \"soluble\" with the word \"solvable\": This was not a typo. Rather, it is British English. Like colour. And driving on the left. – user1729 Feb 23 '14 at 15:48"
] |
[
null
] |
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|
https://www.bishamcofe.co.uk/maths/
|
[
"",
null,
"## Get in Touch",
null,
"Part of the\n\nAshley Hill\n\nMulti Academy Trust\n\n# Maths\n\n## 'Maths Week' - presentation to parents (June 2016)\n\nFurther down this page you will find web-links which take you to some popular websites which can help boost attainment in maths.\n\nHints and tips for helping children familiarise themselves with mathematics:\n\n• Cooking with a child provides opportunities to use measures - reading scales, converting between units and calculating with amounts all give practical applications of maths.\n• Look at prices and compare amounts when shopping with your child - use receipts to perform various calculations, such as finding the difference between prices.\n• Play board games, such as Monopoly, and dice games, such as Yahtzee, taking opportunities to add and subtract money and numbers.\n• Encourage your child to learn their key numerical facts (see below) as, without these, children struggle to solve problems. Regularly practise these facts until they are securely known - short, regular bouts of practice help build confidence as spending too long means that boredom sets in. Even for our eldest children, no more than 20-30 minutes on an activity is recommended at any time. Addition and subtraction facts to 20 should be learned and applied to word problems by the end of Key Stage 1 while multiplication and division facts, up to 12 x 12, should be memorised by the end of Year 4. Your child's teacher will give further guidance regarding which key facts should be known by when.\n• Your child needs to read regularly: once (s)he can read for understanding, attempting word-based maths problems becomes much less challenging.\n\nKey facts:\n\nAddition and subtraction facts to 20 - to be learned and applied by the end of Key Stage 1",
null,
"Multiplication and division facts - to be learned no later than the end of Year 4",
null,
"Common fractions with equivalent decimals and percentages",
null,
"A fraction has two parts",
null,
"Proper, improper and mixed fractions",
null,
"Multiples\n\nA multiple of a whole number is produced by multiplying that number by another whole number.\n\nMultiples of 3 → 3 6 9 12 15 18 ... 60 ... 300 ...\n\nMultiples of 4 → 4 8 12 16 20 24 ... 80 ... 400 ...\n\nTherefore, 12 is a common multiple of both 3 and 4.\n\nFactors",
null,
"The factors of 8 are 124 and 8.\n\nThe factors of 24 are 12, 3, 4, 6, 8, 12 and 24.\n\nThe factors of 32 are 1248, 16 and 32.\n\nSo, the common factors of 8, 24 and 32 are 124 and 8.\n\nMeasures",
null,
"Time\n\n1 minute = 60 seconds\n\n1 hour = 60 minutes\n\n1 day = 24 hours\n\n24 hour clock with minutes:",
null,
"1 week = 7 days\n\n1 fortnight = 14 days\n\n1 year = 12 months = 365 days (366 in a leap year)\n\n1 decade = 10 years\n\n1 century = 100 years\n\n1 millennium = 1000 years\n\nTop"
] |
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https://ftp.aimsciences.org/article/doi/10.3934/dcds.2012.32.2701
|
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"",
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"August 2012, 32(8): 2701-2727. doi: 10.3934/dcds.2012.32.2701\n\n## Dynamics of a delay differential equation with multiple state-dependent delays\n\nReceived June 2011 Revised October 2011 Published March 2012\n\nWe study the dynamics of a linear scalar delay differential equation $$\\epsilon \\dot{u}(t)=-\\gamma u(t)-\\sum_{i=1}^N\\kappa_i u(t-a_i-c_iu(t)),$$ which has trivial dynamics with fixed delays ($c_i=0$). We show that if the delays are allowed to be linearly state-dependent ($c_i\\ne0$) then very complex dynamics can arise, when there are two or more delays. We present a numerical study of the bifurcation structures that arise in the dynamics, in the non-singularly perturbed case, $\\epsilon=1$. We concentrate on the case $N=2$ and $c_1=c_2=c$ and show the existence of bistability of periodic orbits, stable invariant tori, isola of periodic orbits arising as locked orbits on the torus, and period doubling bifurcations.\nCitation: A. R. Humphries, O. A. DeMasi, F. M. G. Magpantay, F. Upham. Dynamics of a delay differential equation with multiple state-dependent delays. Discrete & Continuous Dynamical Systems, 2012, 32 (8) : 2701-2727. doi: 10.3934/dcds.2012.32.2701\n##### References:\n K. A. Abell, C. E. Elmer, A. R. Humphries and E. S. Van Vleck, Computation of mixed type functional differential boundary value problems, SIAM J. Appl. Dyn. Sys., 4 (2005), 755-781. doi: 10.1137/040603425.",
null,
"",
null,
"Google Scholar W. G. Aiello, H. I. Freedman and J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math., 52 (1992), 855-869. doi: 10.1137/0152048.",
null,
"",
null,
"Google Scholar A. Bellen and M. Zennaro, \"Numerical Methods for Delay Differential Equations,'' Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, 2003.",
null,
"Google Scholar J. De Luca, N. Guglielmi, A. R. Humphries and A. Politi, Electromagnetic two-body problem: Recurrent dynamics in the presence of state-dependent delay, J. Phys. A, 43 (2010), 205103, 20 pp.",
null,
"Google Scholar J. De Luca, A. R. Humphries and S. B. Rodrigues, Finite element boundary value integration of Wheeler-Feynman electrodynamics, J. Comput. Appl. Math., (2012). doi: 10.1016/j.cam.2012.02.039.",
null,
"Google Scholar O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, \"Delay Equations. Functional, Complex, and Nonlinear Analysis,'' Applied Mathematical Sciences, 110, Springer-Verlag, New York, 1995.",
null,
"Google Scholar R. Driver, Existence theory for a delay-differential system, Contrib. Diff. Eq., 1 (1963), 317-366.",
null,
"Google Scholar M. Eichmann, \"A Local Hopf Bifurcation Theorem for Differential Equations with State-Dependent Delays,'' Ph.D thesis, Universität Gieß en, Germany, 2006. Google Scholar K. Engelborghs, T. Luzyanina and D. Roose, Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM Trans. Math. Soft., 28 (2002), 1-21. doi: 10.1145/513001.513002.",
null,
"",
null,
"Google Scholar J. E. Ferrell, Self-perpetuating states in signal transduction: Positive feedback, double-negative feedback, and bistability, Curr. Opin. Chem. Biol., 6 (2002), 140-148. Google Scholar C. Foley, S. Bernard and M. C. Mackey, Cost-effective G-CSF therapy strategies for cyclical neutropenia: Mathematical modelling based hypotheses, J. Theor. Biol., 238 (2006), 754-763. doi: 10.1016/j.jtbi.2005.06.021.",
null,
"",
null,
"Google Scholar R. Gambell, Birds and mammals: Antarctic whales, in \"Key Environments Antarctica'' (eds. W. N. Bonner and D. W. H. Walton), Pergamon Press, New York, (1985), 223-241. Google Scholar K. Green, B. Krauskopf and K. Engelborghs, Bistability and torus break-up in a semiconductor laser with phase-conjugate feedback, Physica D, 173 (2002), 114-129. doi: 10.1016/S0167-2789(02)00656-5.",
null,
"",
null,
"Google Scholar J. Guckenheimer and P. Holmes, \"Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,'' Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1983.",
null,
"Google Scholar W. Gurney, S. Blythe and R. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21. doi: 10.1038/287017a0.",
null,
"Google Scholar I. Györi and F. Hartung, Exponential stability of a state-dependent delay system, Discrete Contin. Dyn. Syst., 18 (2007), 773-791. doi: 10.3934/dcds.2007.18.773.",
null,
"",
null,
"Google Scholar J. Hale and S. M. Verduyn Lunel, \"Introduction to Functional-Differential Equations,'' Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993.",
null,
"Google Scholar F. Hartung, T. Krisztin, H.-O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications, in \"Handbook of Differential Equations: Ordinary Differential Equations,'' Vol 3 (eds. A Cañada, P. Drábek and A. Fonda), Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, (2006), 435-545.",
null,
"Google Scholar G. Hutchinson, Circular causal systems in ecology, Ann. N.Y. Acad. Sci., 50 (1948), 221-246. doi: 10.1111/j.1749-6632.1948.tb39854.x.",
null,
"Google Scholar Q. Hu and J. Wu, Global Hopf bifurcation for differential equations with state-dependent delay, J. Diff. Eq., 248 (2010), 2801-2840.",
null,
"Google Scholar T. Insperger, G. Stépán and J. Turi, State-dependent delay in regenerative turning processes, Nonlinear Dyn., 47 (2007), 275-283. doi: 10.1007/s11071-006-9068-2.",
null,
"Google Scholar Y. A. Kuznetsov, \"Elements of Applied Bifurcation Theory,'' 3rd edition, Applied Mathematical Sciences, 112, Springer-Verlag, New York, 2004.",
null,
"Google Scholar J.-P. Lessard, Recent advances about the uniqueness of the slowly oscillating periodic solutions of Wright's equation, J. Diff. Eq., 248 (2010), 992-1016.",
null,
"Google Scholar M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289. doi: 10.1126/science.267326.",
null,
"Google Scholar M. C. Mackey, Commodity price fluctuations: Price dependent delays and nonlinearities as explanatory factors, J. Econ. Theory, 48 (1989), 497-509. doi: 10.1016/0022-0531(89)90039-2.",
null,
"",
null,
"Google Scholar J. Mallet-Paret and R. D. Nussbaum, Global continuation and asymptotic behavior for periodic solutions of a differential-delay equation, Ann. Mat. Pura. Appl. (4), 145 (1986), 33-128. doi: 10.1007/BF01790539.",
null,
"",
null,
"Google Scholar J. Mallet-Paret and R. D. Nussbaum, Boundary layer phenomena for differential-delay equations with state-dependent time lags, I, Arch. Rat. Mech. Anal., 120 (1992), 99-146. doi: 10.1007/BF00418497.",
null,
"",
null,
"Google Scholar J. Mallet-Paret, R. D. Nussbaum and P. Paraskevopoulos, Periodic solutions for functional differential equations with multiple state-dependent time lags, Top. Meth. Nonlin. Anal., 3 (1994), 101-162.",
null,
"Google Scholar J. Mallet-Paret and R. D. Nussbaum, Boundary layer phenomena for differential-delay equations with state-dependent time lags. II, J. Reine Angew. Math., 477 (1996), 129-197.",
null,
"Google Scholar J. Mallet-Paret and R. D. Nussbaum, Boundary layer phenomena for differential-delay equations with state-dependent time lags. III, J. Diff. Eq., 189 (2003), 640-692.",
null,
"Google Scholar J. Mallet-Paret and R. D. Nussbaum, Superstability and rigorous asymptotics in singularly perturbed state-dependent delay-differential equations, J. Diff. Eq., 250 (2011), 4037-4084.",
null,
"Google Scholar MATLAB R2011a, The MathWorks Inc., Natick, MA, 2011. Google Scholar T. H. Price, G. S. Chatta and D. C. Dale, Effect of recombinant granulocyte colony stimulating factor on neutrophil kinetics in normal young and elderly humans, Blood, 88 (1996), 335-340. Google Scholar M. Santillán and M. C. Mackey, Why the lysogenic state of phage $\\lambda$ is so stable: A mathematical modeling approach, Biophysical J., 86 (2004), 75-84. doi: 10.1016/S0006-3495(04)74085-0.",
null,
"Google Scholar J. Sieber, Finding periodic orbits in state-dependent delay differential equations as roots of algebraic equations, Discrete and Continuous Dynamical Systems - Series A, 32 (2012), 2607-2651. Google Scholar H. Smith, \"An Introduction to Delay Differential Equations with Applications to the Life Sciences,'' Texts in Applied Mathematics, 57, Springer, New York, 2011.",
null,
"Google Scholar H.-O. Walther, On a model for soft landing with state dependent delay, J. Dyn. Diff. Eqns., 19 (2003), 593-622. doi: 10.1007/s10884-006-9064-8.",
null,
"",
null,
"Google Scholar E. Wright, A non-linear difference-differential equation, J. Reine Angew. Math., 194 (1955), 66-87.",
null,
"Google Scholar N. Yildirim and M. C. Mackey, Feedback regulation in the lactose operon: A mathematical modeling study and comparison with experimental data, Biophysical J., 84 (2003), 2841-2851. doi: 10.1016/S0006-3495(03)70013-7.",
null,
"Google Scholar\n\nshow all references\n\n##### References:\n K. A. Abell, C. E. Elmer, A. R. Humphries and E. S. Van Vleck, Computation of mixed type functional differential boundary value problems, SIAM J. Appl. Dyn. Sys., 4 (2005), 755-781. doi: 10.1137/040603425.",
null,
"",
null,
"Google Scholar W. G. Aiello, H. I. Freedman and J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math., 52 (1992), 855-869. doi: 10.1137/0152048.",
null,
"",
null,
"Google Scholar A. Bellen and M. Zennaro, \"Numerical Methods for Delay Differential Equations,'' Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, 2003.",
null,
"Google Scholar J. De Luca, N. Guglielmi, A. R. Humphries and A. Politi, Electromagnetic two-body problem: Recurrent dynamics in the presence of state-dependent delay, J. Phys. A, 43 (2010), 205103, 20 pp.",
null,
"Google Scholar J. De Luca, A. R. Humphries and S. B. Rodrigues, Finite element boundary value integration of Wheeler-Feynman electrodynamics, J. Comput. Appl. Math., (2012). doi: 10.1016/j.cam.2012.02.039.",
null,
"Google Scholar O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, \"Delay Equations. Functional, Complex, and Nonlinear Analysis,'' Applied Mathematical Sciences, 110, Springer-Verlag, New York, 1995.",
null,
"Google Scholar R. Driver, Existence theory for a delay-differential system, Contrib. Diff. Eq., 1 (1963), 317-366.",
null,
"Google Scholar M. Eichmann, \"A Local Hopf Bifurcation Theorem for Differential Equations with State-Dependent Delays,'' Ph.D thesis, Universität Gieß en, Germany, 2006. Google Scholar K. Engelborghs, T. Luzyanina and D. Roose, Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM Trans. Math. Soft., 28 (2002), 1-21. doi: 10.1145/513001.513002.",
null,
"",
null,
"Google Scholar J. E. Ferrell, Self-perpetuating states in signal transduction: Positive feedback, double-negative feedback, and bistability, Curr. Opin. Chem. Biol., 6 (2002), 140-148. Google Scholar C. Foley, S. Bernard and M. C. Mackey, Cost-effective G-CSF therapy strategies for cyclical neutropenia: Mathematical modelling based hypotheses, J. Theor. Biol., 238 (2006), 754-763. doi: 10.1016/j.jtbi.2005.06.021.",
null,
"",
null,
"Google Scholar R. Gambell, Birds and mammals: Antarctic whales, in \"Key Environments Antarctica'' (eds. W. N. Bonner and D. W. H. Walton), Pergamon Press, New York, (1985), 223-241. Google Scholar K. Green, B. Krauskopf and K. Engelborghs, Bistability and torus break-up in a semiconductor laser with phase-conjugate feedback, Physica D, 173 (2002), 114-129. doi: 10.1016/S0167-2789(02)00656-5.",
null,
"",
null,
"Google Scholar J. Guckenheimer and P. Holmes, \"Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,'' Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1983.",
null,
"Google Scholar W. Gurney, S. Blythe and R. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21. doi: 10.1038/287017a0.",
null,
"Google Scholar I. Györi and F. Hartung, Exponential stability of a state-dependent delay system, Discrete Contin. Dyn. Syst., 18 (2007), 773-791. doi: 10.3934/dcds.2007.18.773.",
null,
"",
null,
"Google Scholar J. Hale and S. M. Verduyn Lunel, \"Introduction to Functional-Differential Equations,'' Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993.",
null,
"Google Scholar F. Hartung, T. Krisztin, H.-O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications, in \"Handbook of Differential Equations: Ordinary Differential Equations,'' Vol 3 (eds. A Cañada, P. Drábek and A. Fonda), Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, (2006), 435-545.",
null,
"Google Scholar G. Hutchinson, Circular causal systems in ecology, Ann. N.Y. Acad. Sci., 50 (1948), 221-246. doi: 10.1111/j.1749-6632.1948.tb39854.x.",
null,
"Google Scholar Q. Hu and J. Wu, Global Hopf bifurcation for differential equations with state-dependent delay, J. Diff. Eq., 248 (2010), 2801-2840.",
null,
"Google Scholar T. Insperger, G. Stépán and J. Turi, State-dependent delay in regenerative turning processes, Nonlinear Dyn., 47 (2007), 275-283. doi: 10.1007/s11071-006-9068-2.",
null,
"Google Scholar Y. A. Kuznetsov, \"Elements of Applied Bifurcation Theory,'' 3rd edition, Applied Mathematical Sciences, 112, Springer-Verlag, New York, 2004.",
null,
"Google Scholar J.-P. Lessard, Recent advances about the uniqueness of the slowly oscillating periodic solutions of Wright's equation, J. Diff. Eq., 248 (2010), 992-1016.",
null,
"Google Scholar M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289. doi: 10.1126/science.267326.",
null,
"Google Scholar M. C. Mackey, Commodity price fluctuations: Price dependent delays and nonlinearities as explanatory factors, J. Econ. Theory, 48 (1989), 497-509. doi: 10.1016/0022-0531(89)90039-2.",
null,
"",
null,
"Google Scholar J. Mallet-Paret and R. D. Nussbaum, Global continuation and asymptotic behavior for periodic solutions of a differential-delay equation, Ann. Mat. Pura. Appl. (4), 145 (1986), 33-128. doi: 10.1007/BF01790539.",
null,
"",
null,
"Google Scholar J. Mallet-Paret and R. D. Nussbaum, Boundary layer phenomena for differential-delay equations with state-dependent time lags, I, Arch. Rat. Mech. Anal., 120 (1992), 99-146. doi: 10.1007/BF00418497.",
null,
"",
null,
"Google Scholar J. Mallet-Paret, R. D. Nussbaum and P. Paraskevopoulos, Periodic solutions for functional differential equations with multiple state-dependent time lags, Top. Meth. Nonlin. Anal., 3 (1994), 101-162.",
null,
"Google Scholar J. Mallet-Paret and R. D. Nussbaum, Boundary layer phenomena for differential-delay equations with state-dependent time lags. II, J. Reine Angew. Math., 477 (1996), 129-197.",
null,
"Google Scholar J. Mallet-Paret and R. D. Nussbaum, Boundary layer phenomena for differential-delay equations with state-dependent time lags. III, J. Diff. Eq., 189 (2003), 640-692.",
null,
"Google Scholar J. Mallet-Paret and R. D. Nussbaum, Superstability and rigorous asymptotics in singularly perturbed state-dependent delay-differential equations, J. Diff. Eq., 250 (2011), 4037-4084.",
null,
"Google Scholar MATLAB R2011a, The MathWorks Inc., Natick, MA, 2011. Google Scholar T. H. Price, G. S. Chatta and D. C. Dale, Effect of recombinant granulocyte colony stimulating factor on neutrophil kinetics in normal young and elderly humans, Blood, 88 (1996), 335-340. Google Scholar M. Santillán and M. C. Mackey, Why the lysogenic state of phage $\\lambda$ is so stable: A mathematical modeling approach, Biophysical J., 86 (2004), 75-84. doi: 10.1016/S0006-3495(04)74085-0.",
null,
"Google Scholar J. Sieber, Finding periodic orbits in state-dependent delay differential equations as roots of algebraic equations, Discrete and Continuous Dynamical Systems - Series A, 32 (2012), 2607-2651. Google Scholar H. Smith, \"An Introduction to Delay Differential Equations with Applications to the Life Sciences,'' Texts in Applied Mathematics, 57, Springer, New York, 2011.",
null,
"Google Scholar H.-O. Walther, On a model for soft landing with state dependent delay, J. Dyn. Diff. Eqns., 19 (2003), 593-622. doi: 10.1007/s10884-006-9064-8.",
null,
"",
null,
"Google Scholar E. Wright, A non-linear difference-differential equation, J. Reine Angew. Math., 194 (1955), 66-87.",
null,
"Google Scholar N. Yildirim and M. C. Mackey, Feedback regulation in the lactose operon: A mathematical modeling study and comparison with experimental data, Biophysical J., 84 (2003), 2841-2851. doi: 10.1016/S0006-3495(03)70013-7.",
null,
"Google Scholar\n Xiuli Sun, Rong Yuan, Yunfei Lv. Global Hopf bifurcations of neutral functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 667-700. doi: 10.3934/dcdsb.2018038 Qingwen Hu, Bernhard Lani-Wayda, Eugen Stumpf. Preface: Delay differential equations with state-dependent delays and their applications. Discrete & Continuous Dynamical Systems - S, 2020, 13 (1) : i-i. doi: 10.3934/dcdss.20201i Benjamin B. Kennedy. Multiple periodic solutions of state-dependent threshold delay equations. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1801-1833. doi: 10.3934/dcds.2012.32.1801 Jan Sieber. Finding periodic orbits in state-dependent delay differential equations as roots of algebraic equations. Discrete & Continuous Dynamical Systems, 2012, 32 (8) : 2607-2651. doi: 10.3934/dcds.2012.32.2607 Redouane Qesmi, Hans-Otto Walther. Center-stable manifolds for differential equations with state-dependent delays. Discrete & Continuous Dynamical Systems, 2009, 23 (3) : 1009-1033. doi: 10.3934/dcds.2009.23.1009 Ferenc Hartung, Janos Turi. Linearized stability in functional differential equations with state-dependent delays. Conference Publications, 2001, 2001 (Special) : 416-425. doi: 10.3934/proc.2001.2001.416 Ferenc Hartung. Parameter estimation by quasilinearization in differential equations with state-dependent delays. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1611-1631. doi: 10.3934/dcdsb.2013.18.1611 Ovide Arino, Eva Sánchez. A saddle point theorem for functional state-dependent delay differential equations. Discrete & Continuous Dynamical Systems, 2005, 12 (4) : 687-722. doi: 10.3934/dcds.2005.12.687 Tibor Krisztin. A local unstable manifold for differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems, 2003, 9 (4) : 993-1028. doi: 10.3934/dcds.2003.9.993 Eugen Stumpf. Local stability analysis of differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3445-3461. doi: 10.3934/dcds.2016.36.3445 Ismael Maroto, Carmen NÚÑez, Rafael Obaya. Dynamical properties of nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 3939-3961. doi: 10.3934/dcds.2017167 Hermann Brunner, Stefano Maset. Time transformations for state-dependent delay differential equations. Communications on Pure & Applied Analysis, 2010, 9 (1) : 23-45. doi: 10.3934/cpaa.2010.9.23 Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169 Jitai Liang, Ben Niu, Junjie Wei. Linearized stability for abstract functional differential equations subject to state-dependent delays with applications. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6167-6188. doi: 10.3934/dcdsb.2019134 Benjamin B. Kennedy. A state-dependent delay equation with negative feedback and \"mildly unstable\" rapidly oscillating periodic solutions. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1633-1650. doi: 10.3934/dcdsb.2013.18.1633 Xiang Li, Zhixiang Li. Kernel sections and (almost) periodic solutions of a non-autonomous parabolic PDE with a discrete state-dependent delay. Communications on Pure & Applied Analysis, 2011, 10 (2) : 687-700. doi: 10.3934/cpaa.2011.10.687 Josef Diblík. Long-time behavior of positive solutions of a differential equation with state-dependent delay. Discrete & Continuous Dynamical Systems - S, 2020, 13 (1) : 31-46. doi: 10.3934/dcdss.2020002 F. M. G. Magpantay, A. R. Humphries. Generalised Lyapunov-Razumikhin techniques for scalar state-dependent delay differential equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (1) : 85-104. doi: 10.3934/dcdss.2020005 Odo Diekmann, Karolína Korvasová. Linearization of solution operators for state-dependent delay equations: A simple example. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 137-149. doi: 10.3934/dcds.2016.36.137 Hans-Otto Walther. On Poisson's state-dependent delay. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 365-379. doi: 10.3934/dcds.2013.33.365\n\n2020 Impact Factor: 1.392"
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{"ft_lang_label":"__label__en","ft_lang_prob":0.58930105,"math_prob":0.71072495,"size":20287,"snap":"2021-31-2021-39","text_gpt3_token_len":6450,"char_repetition_ratio":0.2017453,"word_repetition_ratio":0.8129264,"special_character_ratio":0.35549858,"punctuation_ratio":0.293379,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.969644,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-09-25T01:11:46Z\",\"WARC-Record-ID\":\"<urn:uuid:a5f50187-c21e-40b0-8209-3eac2caf660a>\",\"Content-Length\":\"129248\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2301d3bb-73d2-45b3-a015-f3d8fd3962f4>\",\"WARC-Concurrent-To\":\"<urn:uuid:f2544952-1ba3-47bd-b44f-9384c387a88b>\",\"WARC-IP-Address\":\"107.161.80.18\",\"WARC-Target-URI\":\"https://ftp.aimsciences.org/article/doi/10.3934/dcds.2012.32.2701\",\"WARC-Payload-Digest\":\"sha1:5IG3JZOTU6MXLSFSNBPLY4ZREPXZ5ZPJ\",\"WARC-Block-Digest\":\"sha1:SPSX2LFFWFGCQUEDNMSUHELC75XVQPGN\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-39/CC-MAIN-2021-39_segments_1631780057584.91_warc_CC-MAIN-20210924231621-20210925021621-00674.warc.gz\"}"}
|
https://www.rdsharmasolutions.in/rd-sharma-class-7ex-21-1-chapter-21-mensuration-ii-area-of-circle/
|
[
"# RD Sharma Class 7 ex 21.1 Solutions Chapter 21 Mensuration II (Area of Circle)\n\nIn this chapter, we provide RD Sharma Class 7 ex 21.1 Solutions Chapter 21 Mensuration II (Area of Circle) for English medium students, Which will very helpful for every student in their exams. Students can download the latest RD Sharma Class 7 ex 21.1 Solutions Chapter 21 Mensuration II (Area of Circle) Maths pdf, Now you will get step by step solution to each question.\n\n# Chapter 21 Mensuration II (Area of Circle) Exercise – 21.1\n\n### Question: 1\n\nFind the circumference of a circle whose radius is\n\n(i) 14 cm (ii) 10 m (iii) 4 km\n\n### Question: 2\n\nFind the circumference of a circle whose diameter is\n\n(i) 7 cm (ii) 4.2 cm (iii) 11.2 km\n\n### Question: 3\n\nFind the radius of a circle whose circumference is\n\n(i) 52.8 cm (ii) 42 cm (iii) 6.6 km\n\n### Question: 4\n\nFind the diameter of a circle whose circumference is\n\n(i) 12.56 cm (ii) 88 m (iii) 11.0 km\n\n### Question: 5\n\nThe ratio of the radii of two circles is 3 : 2. What is the ratio of their circumferences?\n\n### Solution:\n\nWe have, the ratio of the radii = 3 : 2\n\nSo, let the radii of the two circles be 3r and 2r respectively.\n\n### Question: 6\n\nA wire in the form of a rectangle 18.7 cm long and 14.3 cm wide is reshaped and bent into the form of a circle. Find the radius of the circle so formed.\n\n### Question: 7\n\nA piece of wire is bent in the shape of an equilateral triangle of each side 6.6 cm. It is re-bent to form a circular ring. What is the diameter of the ring?\n\n### Question: 8\n\nThe diameter of a wheel of a car is 63 cm. Find the distance travelled by the car during the period, the wheel makes 1000 revolutions.\n\n### Solution:\n\nIt may be noted that in one revolution, the cycle covers a distance equal to the circumference of the wheel.\n\nNow, the diameter of the wheel = 63 cm\n\n∴ Circumference of the wheel = πd = 227 x 63 = 198 cm.\n\nThus, the cycle covers 198 cm in one revolution.\n\n∴ The distance covered by the cycle in 1000 revolutions = (198 x 1000) = 198000 cm = 1980 m.\n\n### Question: 9\n\nThe diameter of a wheel of a car is 98 cm. How many revolutions will it make to travel 6160 metres.\n\n### Question: 10\n\nThe moon is about 384400 km from the earth and its path around the earth is nearly circular. Find the circumference of the path described by the moon in lunar month.\n\n### Question: 11\n\nHow long will John take to make a round of a circular field of radius 21 m cycling at the speed of 8 km/hr?\n\n### Question: 12\n\nThe hour and minute hands of a clock are 4 cm and 6 cm long respectively. Find the sum of the distances travelled by their tips in 2 days.\n\n### Question: 13\n\nA rhombus has the same perimeter as the circumference of a circle. If the side of the rhombus is 2.2 m. Find the radius of the circle.\n\n### Question: 14\n\nA wire is looped in the form of a circle of radius 28 cm. It is re-bent into a square form. Determine the length of the side of the square.\n\n### Question: 15\n\nA bicycle wheel makes 5000 revolutions in moving 11 km. Find the diameter of the wheel.\n\n### Question: 16\n\nA boy is cycling such that the wheels of the cycle are making 140 revolutions per minute. If the diameter of the wheel is 60 cm, calculate the speed per hour with which the boy is cycling.\n\n### Question: 17\n\nThe diameter of the driving wheel of a bus is 140 cm. How many revolutions per minute must the wheel make in order to keep a speed of 66 km per hour?\n\n### Question: 18\n\nA water sprinkler in a lawn sprays water as far as 7 m in all directions. Find the length of the outer edge of wet grass.\n\n### Question: 19\n\nA well of diameter 150 cm has a stone parapet around it. If the length of the outer edge of the parapet is 660 cm. then find the width of the parapet.\n\n### Question: 20\n\nAn ox in a kolhu (an oil processing apparatus) is tethered to a rope 3 m long. How much distance does it cover in 14 rounds?\n\n### Solution:\n\nAll Chapter RD Sharma Solutions For Class 7 Maths\n\nI think you got complete solutions for this chapter. If You have any queries regarding this chapter, please comment on the below section our subject teacher will answer you. We tried our best to give complete solutions so you got good"
] |
[
null
] |
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|
https://www.groundai.com/project/stable-switching-among-high-order-modes-in-polariton-condensates/
|
[
"Stable Switching among High-Order Modes in Polariton Condensates\n\n# Stable Switching among High-Order Modes in Polariton Condensates\n\nYongbao Sun, Yoseob Yoon, Saeed Khan, Li Ge, Loren N. Pfeiffer, Ken West, Hakan E. Treci, David W. Snoke, and Keith A. Nelson Department of Chemistry and Center for Excitonics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA\nDepartment of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA\nDepartment of Engineering Science and Physics, College of Staten Island, City University of New York, New York 10314, USA\nThe Graduate Center, College of Staten Island, City University of New York, New York 10016, USA\nDepartment of Physics, University of Pittsburgh, 3941 O’Hara St., Pittsburgh, PA 15218, USA\n###### Abstract\n\nWe report multistate optical switching among high-order bouncing-ball modes (“ripples”) and whispering-gallerying modes (“petals”) of exciton-polariton condensates in a laser-generated annular trap. By tailoring the diameter and power of the annular trap, the polariton condensate can be switched among different trapped modes, accompanied by redistribution of spatial densities and superlinear increase in the emission intensities, implying that polariton condensates in this geometry could be exploited for a multistate switch. A model based on non-Hermitian modes of the generalized Gross-Pitaevskii equation reveals that this mode switching arises from competition between pump-induced gain and in-plane polariton loss. The parameters for reproducible switching among trapped modes have been measured experimentally, giving us a phase diagram for mode switching. Taken together, the experimental result and theoretical modeling advances our fundamental understanding of the spontaneous emergence of coherence and move us toward its practical exploitation.\n\nIntroduction. Strong coupling between cavity photons and excitonic resonances of a quantum well (QW) placed inside the cavity leads to the formation of new mixed eigenstates known as exciton-polaritons (hereafter simply polaritons). They behave as bosons with extremely low effective mass and overall repulsive interactions when excitation densities are low. The photonic and excitonic fractions can be varied by adjusting the relative detuning of photon and exciton resonances, typically by varying the cavity width in a wedged sample structure. This allows direct control over the polariton-polariton interaction strength, which increases with the excitonic fraction. For a review of polariton properties, see SI and Ref. Kavokin2007 ().\n\nPolaritons provide a unique testbed for the study and manipulation of macroscopic quantum effects. Quantum phenomena such as Bose-Einstein condensation have been reported from liquid helium temperature Kasprzak2006 (); Balili2007 (); thermalization () up to room temperature Christopoulos2007 (); Plumhof2014 (); Kena-Cohen2010 () in various systems. This not only allows the investigation of quantum phenomena at elevated temperatures in a convenient fashion, but also presents exciting opportunities to create all-optical polaritonic devices. As a consequence, great efforts have been devoted to the development of techniques for manipulating the properties of microcavity polaritons Balili2007 (); Idrissi-Kaitouni2006 (); Lai2007 (); Kim2011 (); Cerda-Mendez2010 (); Cristofolini2013 (); Askitopoulos2013 (); Dreismann2014 (); Askitopoulos2015 () .\n\nPrevious experiments on Bose condensation of polaritons were usually performed with the photonic resonance close to the excitonic resonance, which resulted in highly excitonic characteristics in polaritons. Together with short cavity lifetimes, this severely limited the distance polaritons could propagate Manni2011 (); Askitopoulos2013 (); Askitopoulos2015 (). The development of new structures with much longer cavity photon lifetimes, from 20–30 ps Wertz2010 () to well over 100 ps Nelsen2013 (); Steger2015 (); Liu2015 (), has allowed the possibility of polariton propagation over macroscopic distances. This property was recently used to measure the polariton-polariton interaction strength in a region with no free excitons interactions ().\n\nIn the present work, we generated polaritons with high photonic fractions by choosing a region of the wedged sample with large negative cavity detuning. Their highly photonic nature allowed the polaritons to propagate coherently over long distances to form condensate states with radial extent up to 100 m inside an optical trap formed by an annular pattern of excitation light. While interactions of polaritons in this case are not strong enough for them to thermalize into an equilibrium gas, they still play an important role. The interactions of polaritons with excitons in the pump region allow the polaritons to undergo Bose condensation inside the optical trap. Furthermore, nonlinear polariton-polariton interactions result in switching among different condensate modes at high pump powers. The spatial distributions of these modes vary dramatically with very small changes of the excitation densities, but are temporally very stable as long as the excitation power is stable. This stability has allowed us to map out the phase boundaries between different modes in our optical trap. Upon state switching as the excitation power is increased, the emission intensities from the condensates also increase in a superlinear fashion. The large changes not only allow us to experimentally distinguish different quantum states, but also strongly suggests the use of polaritons in multistate switching applications.\n\nPetal and ripples in the annular trap. Annular-shaped beams with diameters ranging from 42 m to 107 m were used to excite a high- microcavity structure that has a cavity lifetime of 135 ps. This allows polaritons to propagate over macroscopic distances of up to millimeters Steger2013 (); Steger2015 (). The laser beam was tuned about 140 meV above the bandgap of the QW material; therefore it essentially generated free carriers, which subsequently relaxed down to exciton and polariton states (see Methods for experimental details). Petals and ripples were formed inside the excitation annulus, with radial extent up to 100 m. In theory, if not limited by the pump power, higher-order condensate states with length scales on the order of millimeters could be realized in this high- microcavity structure, making them entirely visible by eye.",
null,
"Figure 1: (color online). Petals in the annular trap with a diameter of 82 μm. (a)-(b) polariton density distribution below (a) and above (b) condensation threshold under annular-shaped beam excitation. (c)-(d) Polariton momentum distribution below (c) and above (d) condensation threshold. (e)-(f) Energy-resolved polariton density distribution at x=0 below (e) and above (f) condensation threshold. (g)-(h) Energy-resolved polariton momentum distribution at kx=0 below (g) and above (h) condensation threshold. The white solid line in (a) indicates the direction of the cavity energy gradient (photon energy decreasing from upper right to lower left), and the white dashed line in (b) shows the position of the annular pump.\n\nPetals are whispering-gallery modes in the annular trap. Fig. 1 shows the emission patterns from an annular trap with a diameter of 82 m. Polaritons remain in the vicinity of the pump region below the condensation threshold, as shown in Fig. 1a. The asymmetry in the density distribution is largely due to inhomogeneity within the pump intensity profile. Fig. 1c shows the momentum distribution of the polaritons below the condensation threshold. Because the photonic mode in the microcavity we used has an energy gradient of 11 meV/mm along the white line in Fig. 1a, there is a net flow of polariton fluid along this energy gradient, as evidenced by the accumulation of the polariton densities with in-plane wavevector components at m and m in Fig. 1c. The cavity gradient can also be identified from the energy-resolved emission profile in Fig. 1e at low pump powers. In this plot, the m slice of Fig. 1a was projected onto the entrance slit of the spectrometer CCD and then spectrally dispersed. As can be seen, there is an energy difference of 0.5 meV between the emissions at m. The propagation effect can also be identified in the energy-resolved -space emission profile as a smeared dispersion, which has been reported in Ref. Nelsen2013 () with the same sample structure.\n\nWhen the excitation density is above the condensation threshold, polaritons propagate over 10 m toward the center of the trap and form the petal state inside the excitation ring. The position of the pump annulus is plotted in Fig. 1b as white dashed lines, along with the emission profile from petals. The petals demonstrate nodal structures similar to those of the high-order whispering-gallerying modes in lasers, with the periodicity matching the density accumulation at m and m. The petal structure is also observed in momentum space as expected since the condensate is a coherent state and the density distributions in position space and momentum space are Fourier-transform related. As expected, the energy-resolved measurements show far narrower emission spectra from the condensates (f and h) than from polaritons below the condensation threshold (e and g). Above the threshold, petals typically have higher energies than those polaritons that have flowed to the center of the annular trap, as seen in Fig. 1f.",
null,
"Figure 2: (color online). Ripples in the annular trap with a diameter of 66 μm. (a)-(b) polariton density distribution below (a) and above (b) condensation threshold under annular-shaped beam excitation. (c)-(d) Polariton momentum distribution below (c) and above (d) condensation threshold. (e)-(f) Energy-resolved polariton density distribution at x=0 below (e) and above (f) condensation threshold. (g)-(h) Energy-resolved polariton momentum distribution at kx=0 below (g) and above (h) condensation threshold.\n\nUnlike petals, ripples are radially confined bouncing-ball modes in the annular trap. In Fig. 2a, we plot the emission profiles observed when a 66-m annulus was used to excite the microcavity. Below the condensation threshold, the distributions of polaritons in real and momentum space show very similar signatures to those in the previous case. However, confined ripples appear when the excitation density is above the condensation threshold, as shown in Fig. 2b. Similar patterns have been studied in quantum chaotic systems where they were termed as bouncing-ball modes McDonald1988 (). In -space, we observed two large populations of polaritons at 1 m indicative of a ripple mode, together with several states with smaller but not negligible amount of polaritons. This suggests that the ripple pattern in Fig. 2b arises from the interference of these paired momentum states. Figures 2f and h show energy-resolved emission along the vertical slices and in Fig. 2b and d accordingly. Again the emission spectra narrow dramatically above the condensation threshold.\n\nIn this work, the higher-order condensate states appear at a lower threshold than the lowest-order condensate state at , unlike the case in Ref. interactions (), where polaritons are composed of higher fractions of excitons. This confirms that interactions play a very important role in the formation of the lowest-order versus higher-order condensate states. In particular, a balance of polariton leakage from the pump region against amplification from the reservoir determine whether ripples or petals will define the lowest-threshold mode; this is expanded upon in the theory section.\n\nStable mode switching in condensates. The condensate can be switched among various petal and ripple states by varying the pump power continuously. In the top panel of Fig. 3, we show the integrated emission intensity in the field of view as a function of pump power. The intensity undergoes several distinct sharp jumps, which are marked by the red lines, and increases by five orders of magnitude when the pump power is increased by a factor of only 15. The real-space density distributions corresponding to the green dots in the upper panel are shown in Fig. 3a-f. We clearly identify that the jumps in emission intensity are accompanied by redistributions of the real-space densities, that is, by mode switching.",
null,
"Figure 3: (color online). Mode switching in a 42 μm trap. Top: the PL intensity as a function of pump power. The red line indicates boundaries of different quantum states. The green dots are selected pump power levels for which the normalized real-space density distributions of the quantum states are shown in (a)-(f).\n\nIn Fig. 3a, the excitation level was still below the condensation threshold, and patterns similar to those in Fig. 1a and 2a were observed. Figure 3b demonstrates the onset of a higher-order state, but it was very difficult to resolve reliably. In Fig. 3c, a two-node ripple mode appears. Figure 3d and e are mixtures of both petals and ripples. Numerical simulations discussed below suggest that petals and ripples coexist at this power due to interactions between these states. As shown in Fig. 3f, when the system was pumped very hard, all the higher-order quantum states collapsed to the lowest-order condensate state. This power tunability of mode switching not only allows us to distinguish different high-order modes, but also suggests that polariton condensates in the annular trap can be implemented in device applications for a stable multistate switch. With better control of the pump power, we believe more states can be accessed independently.\n\nPhase boundaries of higher-order quantum states. In order to fully characterize the phase boundaries between different quantum states, we recorded the real-space polariton density distributions with excitation ring diameters ranging from 42 m to m and pump power ranging from 50 mW to 1 W. Because of the stability of the distributions and the superlinear increase in the emission intensities as shown in Fig. 3, we were able to classify different quantum states at different pump conditions. The resulting phase boundaries are shown in Fig. 4. In this plot, different colors are assigned to different types of states with distinct spatial distributions. The black-shaded region (0) in the upper left region indicates the uncondensed polaritons. Blue (2) and green (5) stand for ripples and petals, respectively. Both petals and ripples exist in a very narrow range of the phase diagram. This indicates that switching among polariton condensate states in the optical trap is very sensitive and reliable. The different regions of the staircase structure of the phase diagram show modes with different numbers of nodes that were observed with incremented values of the excitation ring size and power. The lowest-order condensate states, coded as red (7), occupy the lower right region of the phase map. The rest of the colors indicate patterns that are mixtures of high-order states, similar to those shown in Fig. 3d and e (See SI for more spatial distributions of these mixed modes).\n\nBased on this phase diagram, we can see that as the excitation density and ring size are increased, ripples and petals appear successively as the lowest-threshold modes, and the phase boundary for the lowest-threshold modes is approximately linear. Both features will be explained in the following theory section. The number of lobes in either petals or ripples can be easily tuned by changing the pump parameters, as shown in Fig. 4b-e. This measured phase boundary should serve well to calibrate the implementation of an exciton-polaritonic multistate switch by making use of the high-order quantum states.\n\nTheory and numerical simulation of pattern formation. Below threshold, reservoir-condensate dynamics for incoherently pumped polaritons can be described using an effective Gross-Pitaevskii equation (GPE) linearized in the condensate density (for details, see SI):\n\n i∂Ψ∂t (1)\n\nwhere is the linear, non-Hermitian generator of condensate dynamics: it describes polariton decay (rate ) and gain (rate ) through stimulated scattering from the exciton reservoir generated by the pump (profile , strength ), in addition to the real-valued reservoir-condensate repulsion (). When the pump power is weak, the condensate density reflects quantum fluctuations and is almost zero. Beyond a power , the th eigenmode of becomes an unstable fluctuation around the uncondensed steady state, corresponding to a condensate mode with frequency given by the real part of its eigenvalue. By varying the pump power, a set of such spatial modes can be obtained, with linearized power thresholds and real frequencies . This linearization is exact until condensation first occurs, and thus the linearized mode with lowest threshold is especially significant: it is the actual mode first observed upon condensation. Naturally, the following question arises: what determines the spatial mode with lowest condensation threshold? Using a continuity equation for the condensate density derived from the GPE, we arrive at a simple formula for the linearized threshold for condensation of the th mode Ge2013 ():\n\n PthnP0=1+γn/(ρnγc)Gn≡1+ΓnGn ; P0=γcγRR (2)\n\nwhere\n\n (3)\n\nFor a given mode, the threshold is determined by: (i) relative loss , which compares in-plane loss to total mirror loss , the former being the flux of probability current leaking across the outer pump edge (see Fig. 5a), and (ii) , a dimensionless measure of the overlap between the mode and the pump within the region enclosed by this pump edge. The lowest threshold mode minimizes Eq. (2) by maximizing overlap with the pump to benefit from amplification, while still having low density near to reduce the relative loss . Note that as relative loss for a mode becomes smaller, its overlap becomes increasingly more important in determining the threshold.",
null,
"Figure 5: Numerical simulations. (a) Lowest threshold modes for pump diameters 42 μm and 67 μm respectively; white dashed lines indicate the outer pump edge. (b) Loss-overlap characteristics for lowest threshold petal and ripple modes, and petal-to-ripple threshold ratio, against pump diameter. Overlap is scaled by a dimensionless factor ∝ pump area A to highlight the separation between ripple and petal modes. (c) Loss, overlap, and threshold evolution for two ripple modes (dashed lines 1 and 3) and petal modes (solid lines 2 and 4) shown in the top panel. Red (blue) shaded area denotes pump diameters where ripple (petal) modes have lowest threshold. A thick line in the threshold plot indicates the lowest threshold mode; in the region with no thick lines, a mode other than those considered here has lowest threshold. (d) Simulation of condensate density for increasing pump power at a fixed pump diameter of 42 μm.\n\nWe study the linear modes of for a range of pump diameters; the loss-overlap characteristics for the lowest threshold ripple and petal modes are plotted in Fig. 5b. While petal modes have stronger overlap and higher loss than ripple modes, their threshold decreases below that of ripple modes as pump diameter increases. To understand this, we focus for clarity on one low order and one higher order mode for petals (labeled 2 and 4) and ripples (1 and 3), and consider their loss, overlap, and threshold evolution as a function of pump diameter in Fig. 5c. pump, we see that relative loss decreases with increase in pump diameter; physically, it reflects the increase of the quality factor of the condensate modes, which in turn is due to a better confinement and shorter tail of the condensate modes in the radial direction, leading to reduced in-plane loss. Due to this effect, there exists for each mode a large enough pump diameter at which its relative loss is small enough such that the overlap primarily determines its threshold; in this competition, petal modes have an advantage over ripple modes. Therefore, below a critical pump diameter petals are typically too lossy to have lower thresholds than ripple mode, even though their pump overlap is stronger. Beyond this critical diameter, for petals decrease enough for their stronger overlap to pull its threshold down below that of the competing ripple mode. For annular profiles, a transition diameter will always exist due to this decrease of ; the particular diameter depends on details of the profile. By extension, for higher order states with higher relative loss (see Fig. 5c), larger pump diameters are needed than those for lower order states until decreases sufficiently to encourage condensation into these modes, in agreement with observations here. Finally, we note that overlap decreases with growing pump diameter since the pump density goes down as for a radius annular pump with fixed FWHM. From Eq. (2) and (3), the resultant decrease in increases linearly with pump diameter; this is apparent from the simulated lowest threshold boundary, in good agreement with the experimental phase diagram shown in Fig. 4 .\n\nGoing beyond the condensation threshold requires full simulation of the nonlinear GPE over a large spatio-temporal grid; the unprecedentedly large condensate sizes (upto m) observed in the current work, together with polariton wavelengths (m) demanding fine spatial (and hence temporal) resolution make such simulations very computationally expensive here. We circumvent this issue by expanding the condensate wavefunction in a pump power-dependent, non-Hermitian basis set that account for the spatial complexity of the linearized condensate problem, with time-dependent coefficients. For discrete values of , one mode in each set reduces to the corresponding threshold modes introduced before (note that in general is complex, but for that particular threshold mode it is on the real axis). This reduces the full nonlinear GPE and reservoir dynamical equation to a set of coupled ODEs, an effective nonlinear coupled mode theory for reservoir-condensate dynamics (details in a future publication Khan2015 ()). Applying to the specific case of a pump of diameter m, the coupled mode theory reveals mixing of lowest threshold modes beyond threshold, when polariton-polariton interactions within the condensate become important; in particular, the coexistence of petal and ripple states shown in Fig. 5d was reproduced using this theory.\n\nConclusion and outlook. We have seen the stable formation of high-order quantum states, including ripples, petals and their coherent mixtures, under non-resonant excitation, with a well-defined phase diagram in the pump parameter space. Ripples are confined bouncing-ball modes while petals are whispering-gallery modes in the trap. The all-optical trapping allows facile switching among these condensate states in the annular trap, accompanied by superlinear increases in the emission intensities.\n\nThe measured patterns bear some similarity to the multiple modes seen in standard vertical-cavity, surface-emitting lasers (VCSELs), e.g. the petal patterns seen in Ref. Li2012 (). However, in typical lasers and VCSELs, the system hops uncontrollably between different modes, leading to unwanted noise (e.g., Ref. Pedaci2005 ()). The nonlinear interactions in the polariton condensate system stabilize the modes to resist multimode behavior. This means that this system acts effectively as multistable optical switch, in which transitions between states can be effected by small changes of the input light beam.\n\nExperimental methodologies. The microcavity used in this work is GaAs based structure grown by molecular beam epitaxy. The cavity has an exceptionally high qualify factor of 320,000, which corresponds to a polariton lifetime of 270 ps at resonance. During the experiment, the sample was thermally attached to a cold finger in an open-loop cryostat which was stabilized at 10 K. The excitation laser is a commercial continuous-wave (c.w.) laser, and was modulated by an acousto-optic modulator at 1 kHz with a duty cycle of 0.5% to prevent unwanted sample heating. The annular trap was generated by shaping the phase front of the c.w. laser using a high-resolution spatial light modulator. Because of the eccentricity in the pump profile, which is approximately 0.3, the diameters reported here are geometric means of the lengths of major and minor axes of the pattern. The photoluminescence of polaritons was collected in a reflection geometry using an objective lens with a numerical aperture of 0.28, and was relay imaged to a spectrometer CCD. The energy-resolved emissions were obtained by spectrally dispersing a specific slice of either the far-field or near-field image selected by the entrance slit of the spectrometer CCD.\n\n## References\n\n• (1) Kavokin, A., Baumberg, J. J., Malpuech, G. & Laussy, F. P. Microcavities (Oxford Science Publications, 2007).\n• (2) Kasprzak, J. et al. Bose-einstein condensation of exciton polaritons. Nature 443, 409–414 (2006).\n• (3) Balili, R., Hartwell, V., Snoke, D., Pfeiffer, L. & West, K. Bose-einstein condensation of microcavity polaritons in a trap. Science 316, 1007–1010 (2007).\n• (4) Sun, Y. et al. Bose-einstein condensation of long-lifetime polaritons in thermal equilibrium. arXiv 1601:02581 (2015).\n• (5) Christopoulos, S. et al. Room-temperature polariton lasing in semiconductor microcavities. Phys. Rev. Lett. 98, 126405 (2007).\n• (6) Plumhof, J. D., Stoferle, T., Mai, L., Scherf, U. & Mahrt, R. F. Room-temperature bose-einstein condensation of cavity exciton-polaritons in a polymer. Nat. Mater. 13, 247–252 (2014).\n• (7) Kena-Cohen, S. & Forrest, S. R. Room-temperature polariton lasing in an organic single-crystal microcavity. Nat. Photon. 4, 371–375 (2010).\n• (8) Kaitouni, R. I. et al. Engineering the spatial confinement of exciton polaritons in semiconductors. Phys. Rev. B 74, 155311 (2006).\n• (9) Lai, C. W. et al. Coherent zero-state and pi-state in an exciton-polariton condensate array. Nature 450, 529–532 (2007).\n• (10) Kim, N. Y. et al. Dynamical d-wave condensation of exciton-polaritons in a two-dimensional square-lattice potential. Nat. Phys. 7, 681–686 (2011).\n• (11) Cerda-Mendez, E. A. et al. Polariton condensation in dynamic acoustic lattices. Phys. Rev. Lett. 105, 116402 (2010).\n• (12) Cristofolini, P. et al. Optical superfluid phase transitions and trapping of polariton condensates. Phys. Rev. Lett. 110, 186403 (2013).\n• (13) Askitopoulos, A. et al. Polariton condensation in an optically induced two-dimensional potential. Phys. Rev. B 88, 041308 (2013).\n• (14) Dreismann, A. et al. Coupled counterrotating polariton condensates in optically defined annular potentials. Proc. Nat. Acad. Sci. 111, 8770–8775 (2014).\n• (15) Askitopoulos, A. et al. Robust platform for engineering pure-quantum-state transitions in polariton condensates. Phys. Rev. B 92 (2015).\n• (16) Manni, F., Lagoudakis, K. G., Liew, T. C. H., Andre, R. & Deveaud-Pledran, B. Spontaneous pattern formation in a polariton condensate. Phys. Rev. Lett. 107, 106401 (2011).\n• (17) Wertz, E. et al. Spontaneous formation and optical manipulation of extended polariton condensates. Nat. Phys. 6, 860–864 (2010).\n• (18) Nelsen, B. et al. Dissipationless flow and sharp threshold of a polariton condensate with long lifetime. Phys. Rev. X 3, 041015 (2013).\n• (19) Steger, M., Gautham, C., Snoke, D. W., Pfeiffer, L. & West, K. Slow reflection and two-photon generation of microcavity exciton-polaritons. Optica 2, 1–5 (2015).\n• (20) Liu, G., Snoke, D. W., Daley, A., Pfeiffer, L. N. & West, K. A new type of half-quantum circulation in a macroscopic polariton spinor ring condensate. Proc. Nat. Acad. Sci. 112, 2676–2681 (2015).\n• (21) Sun, Y. et al. Polaritons are not weakly interacting: Direct measurement of the polariton-polariton interaction strength (2015). eprint arXiv:1508.06698.\n• (22) Steger, M. et al. Long-range ballistic motion and coherent flow of long-lifetime polaritons. Phys. Rev. B 88, 235314 (2013).\n• (23) McDonald, S. W. & Kaufman, A. N. Wave chaos in the stadium: Statistical properties of short-wave solutions of the helmholtz equation. Phys. Rev. A 37, 3067–3086 (1988).\n• (24) Ge, L., Nersisyan, A., Oztop, B. & Tureci, H. E. Pattern formation and strong nonlinear interactions in exciton-polariton condensates (2013). eprint arXiv:1311.4847.\n• (25) Khan, S. & Tureci, H. E. unpublished.\n• (26) Li, M., Zhang, B., Chen, K., Snoke, D. & Heberle, A. Noncircular refractive index profile and breakdown of mode degeneracy of vertical cavity surface emitting lasers. IEEE J. Quant. Electron. 48, 1065–1068 (2012).\n• (27) Pedaci, F., GiuDici, M., Tredicce, J. R. & Giacomelli, G. Experimental analysis of mode-hopping in bulk semiconductor lasers. Appl. Phys. B 81, 993 (2005).\n\nAcknowledgements. Y.S., Y.Y. and K.A.N were supported as part of the Center for Excitonics, an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number DE-SC0001088. S.K. and H.E.T. were supported by the National Science Foundation under Grant Number DMR-1151810, L.G. were supported by CIRG 21 Grant from City University of New York, D.W.S. was supported by the National Science Foundation under Grant Number DMR-1104383. L.N.P. and K.W. were partially funded funded by the Gordon and Betty Moore Foundation through the EPiQS initiative Grant GBMF4420, and by the National Science Foundation MRSEC Grant DMR-1420541.\n\nAuthor contributions\nY.S. and K.A.N. designed the experiments; Y.S. and Y.Y. performed the experiment; S.K., L.G. and H.E.T. carried out the numerical simulation. Y.S. and S.K. analyzed the data; L.P.N and K.W. fabricated the microcavity structure; all the authors participated to the results discussion and manuscript preparation.\n\nSupplementary Information is available in the online version of the paper. Reprints and permissions information is available online at www.nature.com/reprints. Correspondence and requests for materials should be addressed to Y.S. at [email protected] or to K.A.N. at [email protected].\n\nCompeting financial interest\nThe authors declare no competing financial interests.\n\nSupplementary Information:\n\nStable Switching of Higher-Order Modes in Polariton Condensates\n\nYongbao Sun, Yoseob Yoon, Saeed Khan, Li Ge,\n\nLoren N. Pfeiffer, Ken West, Hakan E. Treci, David W. Snoke, and Keith A. Nelson\n\nDepartment of Chemistry and Center for Excitonics, Massachusetts Institute of Technology, 77\n\nMassachusetts Avenue, Cambridge, MA 02139, USA\n\nDepartment of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA\n\nDepartment of Engineering Science and Physics, College of Staten Island, City University of New York, New York 10314, USA\n\nThe Graduate Center, College of Staten Island, City University of New York, New York 10016, USA\n\nDepartment of Physics, University of Pittsburgh, 3941 O’Hara St., Pittsburgh, PA 15218, USA\n\nBackground on exciton-polaritons in semiconductor microcavities. Exciton-polaritons are formed in semiconductor microcavities through the strong coupling between optical modes of the microcavity and exciton transitions of material embedded inside the microcavity. For the case of a single microcavity mode and a single exciton transition, two polariton modes, the upper and lower polaritons, are formed with the energies of the two polariton modes, and , given by:\n\n ELP/UP(k||)=12[EX(k||)+EC(k||)∓√Ω2+δ2(k||)] (S1)\n\nwhere is the wave vector in the plane perpendicular to the microcavity confinement direction, is the energy of the exciton transition, is the energy of the cavity mode, is the detuning energy defined as , and is the strength of radiative coupling between the exciton and cavity field, also known as full Rabi splitting energy. The confinement of light gives the cavity mode a parabolic dispersion in the plane perpendicular to the confinement direction: , where is the effective mass of the cavity field. This effective mass is typically times lighter than the vacuum electron mass, and about times less than an exciton in a GaAs quantum well structure, so that is essentially constant with . The energies , and as given in Fig. S1 for three different values of . The energies were calculated using (S1) and parameters matching the sample structure used in the experiments: meV and meV.",
null,
"Figure S1: (color online) Dispersion curves of polariton at three representative cavity detunings (a) δ=−5 meV. (b) δ=0 meV. (c) δ=5 meV. The dotted line shows the confined cavity mode, and the dashed line shows the bare exciton mode. The blue and red solid lines indicate the upper polariton (LP) and lower polariton (UP) branches, respectively, arising from the strong coupling between corresponding cavity modes and exciton modes. Calibrated sample parameters were used in the calculations.\n\nThe length of the cavity increases monotonically along one direction of the QW plane so that the energy of the cavity mode can be tuned relative to the exciton resonance energy, as shown in Fig. S2, allowing us to experimentally tune . The energies of all modes in Fig. S1 are plotted as a function of , the in-plane wave vector. As can be seen in this figure, is essentially constant with respect to while is parabolic.",
null,
"Figure S2: (color online) (a) The calculated upper polariton (blue line) and lower polariton (red line) energies at different positions of the sample. The dashed lines indicate the exciton energies, and the dotted line shows the cavity energies. (b) Excitonic fractions of upper polaritons (blue line) and lower polaritons (red line) at different sample positions.\n\nThe polariton modes are linear superpositions of the exciton and microcavity photon modes. The lower polariton and upper polariton operators, and , respectively, can be written in terms of exciton and cavity operators, and :\n\n ^Pk|| =X(k||)^ak||+C(k||)^bk||k|| (S2) ^Qk|| =−C(k||)^ak||+X(k||)^bk||. (S3)\n\nThe coefficients, and , are called the exciton and cavity Hopfield coefficients and are given by\n\n |X(k||)|2 =12⎛⎜ ⎜⎝1+δ(k||)√δ2(k||)+Ω2⎞⎟ ⎟⎠ (S4) |C(k||)|2 =12⎛⎜ ⎜⎝1−δ(k||)√δ2(k||k||)+Ω2⎞⎟ ⎟⎠ (S5)\n\nThe characteristics of the polariton modes are determined by the coefficients, which depend on . The lower polariton is more photon-like and the upper polariton is more exciton-like for , and the lower polariton is more exciton-like and the upper polariton is more photon like . Due to the wedge in the cavity thickness, we can easily tune the excitonic fraction of lower polaritons by moving the excitation spot at different positions, as shown in Fig. S2b, where we plot at different positions on the sample. As seen in Fig. S1 and Fig. S2, the energies and shapes of the polariton dispersion curves depend strongly on : positive detuning results in lower polaritons that are more exciton-like, with a heavier effective mass and stronger interactions with phonons and other carriers, while negative detuning results in lower polaritons that are more photon-like, with a smaller lower polariton mass and weaker interactions with phonons and other carriers.\n\nSimulated petal and ripples. By using the experimental pump profile as in the nonlinear Gross-Pitaevskii equation, the petal and ripples observed in the experiment can be qualitatively reproduced. Fig. S3 shows the resulting polariton density distributions. The pump profiles that generated the patterns in",
null,
"Figure S3: (color online) (a) The calculated upper polariton (blue line) and lower polariton (red line) energies at different positions of the sample. The dashed lines indicate the exciton energies, and the dotted line shows the cavity energies. (b) Excitonic fractions of upper polaritons (blue line) and lower polaritons (red line) at different sample positions.\n\nFig. 1(f) were used in the simulation for Fig. S3(a) and (b), and the number of lobes was exactly reproduced using this method. The relative intensities of the peaks along the azimuthal direction were also qualitatively captured by this model. In the simulation, a ripple state showed up at a lower energy (0.05 meV lower) but with significantly higher pump density (50%). This mode was not observed in the experiment, due to the limited pump density we can use. By using the pump profile corresponding to that of Fig. 1(e), the reproduced polariton density distribution agreed well with that observed in the experiment, shown in Fig. S3(c) and (d). Additionally, another two patterns were identified, and they have energies similar to the first mode (within 0.01 meV) with a difference in the pump threshold by less than 2%. This agrees with what we saw in Fig. 4(d) and (e), where both ripple and petals were seen in the time-integrated measurements, and is also captured by our mode-integration simulation as shown in Fig. 5 in the main text.\n\nGross-Pitaevskii equation and linearization. We use a generalized Gross-Pitaevskii equation (GPE) to describe the dynamics of microcavity exciton-polaritons under incoherent pumping. In this standard approach, the nonlinear interactions of polaritons within the condensed fraction are treated at the mean-field level, while pumping and losses are introduced as complex-valued terms, so that the generalized GPE for the dynamics of the condensate wavefunction has the form:\n\n i∂Ψ∂t=[−∇22m+gRnR+i2(RnR−γc)]Ψ+g|Ψ|2Ψ (S6)\n\nwhere for clarity we have suppressed the dependence of the polariton wavefunction and the density of the pump-generated exciton reservoir. This reservoir gives rise to a repulsive term describing the interaction of condensate polaritons with reservoir excitons, with strength , together with an amplification of the condensed fraction via stimulated scattering from the reservoir at rate . This latter gain contribution together with the inclusion of polariton mirror loss at rate make the effective generator describing condensate dynamics non-Hermitian in this case. Finally, the polariton-polariton repulsion within the condensate appears as the nonlinear term at the mean-field level. The dynamics of the pump-induced reservoir must also be accounted for by a dynamical equation of the form:\n\n ∂nR∂t=Pf(r)−RnR|Ψ|2−γRnR (S7)\n\nand are the pump strength and spatial profile as described in the main paper, the source of the exciton reservoir. The aforementioned scattering from the exciton reservoir into the condensate at the rate causes a depletion of the reservoir, which is encapsulated in the second term on the right hand side. Reservoir losses that occur via mechanisms other than scattering into the reservoir (e.g. recombination losses) are described by .\n\nFor pumping below the condensation threshold, the system has a steady state with a pump generated exciton density and an uncondensed polariton state. The steady state reservoir density in this regime can be obtained after linearizing Eq. (S7) by dropping nonlinear terms of order ; in this steady-state regime the exciton reservoir density adiabatically follows the pump:\n\n nR(r,t→∞)=PγRf(r) (S8)\n\nBelow threshold, a linearization of the GPE is also valid; we can replace by its linearized steady state value, and neglect the nonlinear polariton-polariton interactions . This yields the linearized GPE for condensate dynamics, Eq. (1) of the main text.\n\nLinear Threshold Modes. We will now analyze steady-state condensate formation in the linearized regime. In particular, if we consider a single frequency steady-state ansätz for the condensate wavefunction:\n\n Ψ(r,t)=φn(r)e−iωnt, (S9)\n\nthe linearized GPE in Eq. (1) of the main text becomes:\n\n HL(P)φn(r)=[−∇22m+gRγRPf(r)+i2RγRPf(r)−i2γc]φn(r)=ωnφn(r) (S10)\n\nThe condensate wavefunction for a single frequency condensate is therefore the th eigenmode of the generator of linearized dynamics, . We require to be a purely real frequency for the steady-state solution to correspond to a nontrivial condensate mode; we will now discuss how this requirement determines the power threshold for a given spatial mode. For simplicity, we rewrite the above eigenproblem in the form:\n\n [−∇2+sPf(r)]φn(r)=q2φn(r) (S11)\n\nwhere we have introduced the pump-induced potential :\n\n s2m=1γR(gR+i2R) (S12)\n\nand the ‘wavevector’ is defined by:\n\n q2(ωn)2m≡ωn+i2γc (S13)\n\nTo determine the eigenmodes of , the above eigenproblem must be formulated as an appropriate boundary value problem (BVP); we make the following choice:\n\n [−∇2+sPf(r)]φn(r) =q2(ωn)φn(r) , r∈P −∇2φn(r) =q2(ωn)φn(r) , r∉P (S14)\n\nwhere is the region enclosed by the outer edge of the pump, as defined in the main paper. Note here that we impose an ‘outgoing’ boundary condition with wavevector at the pump edge , as opposed to the more usual case of considering a boundary far from the pump where the condensate wavefunction is vanishingly small and standard Dirichlet boundary conditions can be employed. For the large condensate sizes considered here, the latter approach would require simulating a very large spatial grid, making computation times inconveniently long. Our approach allows the use of a minimally relevant grid size. This occurs at a relatively minor expense: the outgoing wavevector imposed via this boundary condition depends on the unknown eigenvalue , and this BVP therefore needs to be solved self-consistently. To do so, we fix the outgoing wavevector by choosing an outgoing frequency :\n\n [−∇2+sPf(r)]φn(r) =q2(ωn)φn(r) , r∈P −∇2φn(r) =q2(Ω)φn(r) , r∉P (S15)\n\nIt is now a straightforward matter to solve this BVP for a range of (increasing) values of the pump power at a fixed ; as a result, one obtains a set of eigenmodes and eigenfrequencies of . These generally complex frequencies flow across the complex plane as the pump-power is varied; an example of this flow is shown in Fig. S4. For a certain pump power , the th eigenfrequency crosses the real axis (becomes real). The imaginary part of represents net loss, so its becoming zero implies that gain overcomes polariton loss at this pump power, and the associated eigenmode is an unstable fluctuation around the uncondensed polariton state. Furthermore, if the (now real) frequency is also equal to the imposed outgoing frequency, that is , the wavevector is equal both inside and outside the pump region . The self-consistency condition is therefore simultaneously fulfilled, and the corresponding th eigenmode represents a true condensate mode with real frequency and linearized power threshold . By varying the outgoing frequency , and computing eigenvalues as a function of pump power, a set of such linear threshold modes can be obtained.",
null,
"Figure S4: Linear threshold modes Flow of (complex) eigenvalues of the linear non-Hermitian generator HL(P) as a function of pump power P across the complex plane, computed for a fixed outgoing frequency Ω. The flow direction is indicated by the arrow; eigenvalues approach the real axis from below as the pump power is increased. The lowest threshold mode is indicated in blue. It reaches the real line for the smallest pump power, and has a real frequency ωn equal to the imposed outgoing frequency Ω.\n\nContinuity Equation and Linear Threshold Formula. From the linearized dynamical equation for the condensate wavefunction, it is possible to obtain an equation for the dynamics of the condensate density, . In particular,\n\n ∂|Ψ|2∂t=Ψ∗∂Ψ∂t+c.c. (S16)\n\nFrom the generalized GPE (Eq. (S6)), it is easily found that:\n\n Ψ∗∂Ψ∂t=i2mΨ∗∇2Ψ+{−igRnR−ig|Ψ|2+12(RnR−γc)}|Ψ|2 (S17)\n\nand so:\n\n ∂|Ψ|2∂t=i2m(Ψ∗∇2Ψ−Ψ∇2Ψ∗)+RnR|Ψ|2−γc|Ψ|2 (S18)\n\nThe first term on the right hand side has the form of the divergence of a probability current; this can be made more explicit by defining the probability current as:\n\n →j=i2m(Ψ→∇Ψ∗−c.c.) (S19)\n\nfollowing which the condensate density dynamics is governed by the equation:\n\n ∂|Ψ|2∂t=RnR|Ψ|2−∇⋅→j−γc|Ψ|2 (S20)\n\nwhich has the well-defined form of a continuity equation. In particular, the above equation can be put into a more practical form by integrating over the area of the region enclosed by the outer pump edge,\n\n ∂∂t∫Pd2r |Ψ|2=R∫Pd2r nR|Ψ|2−∮∂P→j⋅d→s−γc∫Pd2r |Ψ|2 (S21)\n\nwhere the divergence theorem allows the term involving to be rewritten as a flux integral. This equation has the simple interpretation: any increase in the total number of polaritons () within the pump region comes from amplification via the exciton reservoir, at rate . Losses to the polariton number can be attributed to either the mirror loss , or a leakage of the condensate from the pump edge. Since we are integrating within the outer pump edge , beyond which by definition no source of polariton production exists, there can be no incoming probability current that would increase the polariton number within the pump region.\n\nNow, we narrow our focus to the linearized regime, where the reservoir density as shown earlier. Furthermore, we consider a single mode solution such that , where is the eigenmode of that has (real) eigenfrequency . For simplicity, we suppress the parameters defining in what follows. With this ansätz, the condensate density is time-independent and the above continuity equation reduces to:\n\n RγRP∫Pd2r f(r)|φn|2=∮∂P→j[φn]⋅d→s+γc∫Pd2r |φn|2 (S22)\n\nHere, the probability current is now evaluated for the eigenmode , as in the main paper. Now, defining the condensate density , pump overlap , and in-plane loss respectively as in the main paper:\n\n ρn=∫Pd2r |φn|2 , Gn=1ρn∫Pd2r f(r)|φn|2 , γn=∮∂P→j[φn]⋅d→s, (S23)\n\nwe can recover the linear threshold formula (Eq. (2) of the main text):\n\n PnP0=1+γn/(ρnγc)Gn≡1+ΓnGn (S24)\n\nwith being the linear threshold power for the th mode, and .\n\nSpatial distributions of mixed modes in the optical trap. Because of the interactions among high-order modes, a large set of mixed modes shows up in the phase diagram. In Fig. S5, we plot the spatial distributions for 12 mixed modes whose positions in the phase diagram are marked in Fig. S6. As can be seen, modes (1)-(3) have",
null,
"Figure S5: (1)-(12) Mixed modes for selected points in the phase diagram plot shown in Fig. S6. The scale bar in (12) indicates 20 μm.\n\nboth ripple and petal characteristics, and modes (4)-(7) and (9)-(11) are petal-like and are quantized in the azimuthal direction, although the emission intensities from peaks and nodes are comparable. Modes (8) and (12) are ripple-like. Based on our numerical simulations, mixed modes are a direct consequence of interactions between high-order modes with very close thresholds, rather than being artifacts from time-integrated measurements.",
null,
"Figure S6: Phase diagram indicating the position of the modes whose spatial distributions are shown in Fig. S5.\n\nEvolution of node numbers of ripple and petal states in the optical trap. The number of nodes in either ripple or petal states can be varied by adjusting pump diameter. The pump power also needs to be increased in order to reach condensation threshold as the diameter of the pump increases. In Fig. S7, we show the spatial distributions of 12 distinct modes with various nodes. As can be seen, by changing the pump diameter, we could switch from a 2-node ripple state up to 8-node ripple states continuously. Similar switching behavior can also be realized by using petals as shown in (8)-(12).",
null,
"Figure S7: (1)-(12) Spatial distributions of ripple and petal states with different number of nodes when the pumm parameters are varied. The scale bar in (1) indicates 20 μm.\nYou are adding the first comment!\nHow to quickly get a good reply:\n• Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.\n• Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.\n• Your comment should inspire ideas to flow and help the author improves the paper.\n\nThe better we are at sharing our knowledge with each other, the faster we move forward.\nThe feedback must be of minimum 40 characters and the title a minimum of 5 characters",
null,
"",
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"",
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""
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|
https://www.xolv.io/blog/dev-notes/dealing-with-randomness-in-tests/
|
[
"",
null,
"# Dealing with randomness in tests\n\nYou can watch me go over the examples here:\n\nLet's say we have a function that makes our user feel welcomed by picking randomly one of the available welcoming messages:\n\nWe would like to write a test for it, but it randomly fails - because the result of the function is also random. We have a few options, the first one would be to use a different sample function in a test while defaulting it to the one that introduces randomness:\n\n``````// note the new optional argument\nconst welcomeUser = (userName, sample = _.sample) => {\nconst pickedMessage = sample(welcomeMessages);\n};\n\n// in test:\nconst sample = (arrayToGetSampleFrom) => arrayToGetSampleFrom;\nexpect(welcomeUser(\"Lukasz\", sample)).toEqual(\"Hello there, Lukasz\");``````\n\nThis is nice, and I would definitely advise applying this technique when needed, especially if you are doing a Unit Test.\n\nIf you are thinking about integration tests, or maybe even e2e tests, it won't be trivial to inject a dependency in a similar manner.\n\nUsually, your app might have very few functions that use randomness (the frequent case is an \"id generator\" - defined once, but used frequently), so it might be helpful to just make them random in production, but non-random in tests, like so:\n\n``````// sample.ts\nimport _ from \"lodash\";\n\nexport const sample = <T>(arrayToGetSampleFrom: T[]): T =>\nprocess.env.TEST_ENV\n? arrayToGetSampleFrom\n: _.sample(arrayToGetSampleFrom);\n\n// welcomeUser.ts\nimport {sample} from \"./sample\"\n\nconst welcomeMessages = [\"Hello there\", \"Hi\", \"Goedemorgen\"];\nconst welcomeUser = (userName) => {\nconst pickedMessage = sample(welcomeMessages);\n};\n\n``````\n\nNote: you can use the same patterns for the dreaded dates :)\n\n``````const getCurrentDate = () =>\nprocess.env.TEST_ENV ? new Date(1616667648490) : new Date()``````\n\nAs for a bonus, let's take a look at the ultimate problem in randomness - random number generator.\n\nI will use a real-life example since it's sufficiently small to be useful without introducing unnecessary details. We needed to have a function that would generate a number between 0 and 10. We used that to introduce a randomized delay for processing data (let's say we got a batch with hundreds of files every 30 minutes, so we would push them for processing in a queue with a randomized delay, so our system would not have to worry about huge spikes). The function was simple enough:\n\n``````const getRandomNumberBetweenZeroAndTen = (): number => {\nconst minVal = 1\nconst maxVal = 10\nreturn Math.floor(Math.random() * (maxVal - minVal)) + minVal\n}\n\n``````\n\nAnd the test even simpler:\n\n``````expect(getRandomNumberBetweenZeroAndTen()).not.toEqual(getRandomNumberBetweenZeroAndTen())\nexpect(getRandomNumberBetweenZeroAndTen() >= 0).toEqual(true)\nexpect(getRandomNumberBetweenZeroAndTen() <= 10).toEqual(true)``````\n\nOur developer went to sleep smiling about his well-done job and smart algorithm.\n\nThe problem started very soon because his test would randomly fail with:\n\n``````expect(received).not.toEqual(expected) // deep equality\n\nExpected: not 3``````\n\nAs you probably guessed by now, there is a 1 in 10 chance that the two random executions of this function will result in the same number (not necessarily 3..). This is not really acceptable - and if you had 10 tests with a random chance of 10% failure then virtually all your builds will start failing, which will bring the development efforts to a halt.\n\nSo the solution is similar to our first example. We have to realize that the part we are testing is not the built-in NodeJS number generator - let's leave the NodeJS folks to take care of that. We want to verify our logic. To do that - let's take the random number as an optional argument.\n\n``````const getRandomNumberBetweenZeroAndTen = (\nrandomNumber = Math.random()\n): number => {\nconst minVal = 1\nconst maxVal = 10\nreturn Math.floor(randomNumber * (maxVal - minVal)) + minVal\n}\n``````\n\nAnd now we can test it nicely:\n\n``````test(`returns a number between 1 and 10\nproportionally based on the passed randomNumber that is between 0 and 1`, () => {\nexpect(getRandomNumberBetweenZeroAndTen(1)).toEqual(10)\nexpect(getRandomNumberBetweenZeroAndTen(0)).toEqual(1)\nexpect(getRandomNumberBetweenZeroAndTen(0.5)).toEqual(5)\nexpect(getRandomNumberBetweenZeroAndTen(0.15)).toEqual(2)\n})``````\n\nAs a side-note and taking another step back to our initial example, in many cases, it would be a better idea to just use lodash random and possibly wrap it as well\n\n``````export const random = (lower, upper) =>\nprocess.env.TEST_ENV ? lower : _.random(lower, upper);\n``````\n\nFor example - If you are dealing with a tiny lambda function without any external dependencies (outside the provided by AWS aws-sdk), it might be easier/faster to write some of those helpers yourself and not worry about external dependencies. I'd argue this is rarely the case, and it's almost always better to use the standard node.js ecosystem tooling. Nonetheless, you might end up in situations where you won't be able to escape from Math.random function, and I hope the patterns from this article will help you to deal with them.\n\nLet me know if you have any questions or thoughts in the comments below.\n\n## Let us help you on your journey to Quality Faster\n\nWe at Xolvio specialize in helping our clients get more for less. We can get you to the holy grail of continuous deployment where every commit can go to production — and yes, even for large enterprises.\n\nFeel free to schedule a call or send us a message below to see how we can help.\n\nor\n\nBook a call\n+"
] |
[
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|
https://cboard.cprogramming.com/c-programming/181170-good-code-bad-code-post1306051.html?s=1afeae77b6bf8271ee4321d5e390e3b3
|
[
"1. ## Good Code And Bad Code\n\nFor production code, is it better to write 1 line ?\nCode:\n`x = ((x / 15) / 13) / 12;`\nor is it better to write multiple lines ?\nCode:\n```x = x / 15;\nx = x / 13;\nx = x / 12;```",
null,
"2. 1. It is better to not use magic numbers\n2. Multiply is easier than divide\n3. Using longer variable names than one letter is preferred unless loop variable\n\nx /= (15*13*12); // But, use constants or defines in place of the numbers.\n\nTim S.",
null,
"3. It would certainly need a comment explaining the significance of 15, 13 and 12.\n\nRegardless, the compiler is likely to reduce it to\nx /= 2340;\n\nSo however you write it, choose the way that makes it most obvious to the reader what the purpose of the code is.",
null,
"4. 15 * 13 * 12, big values will overflow.\n\nWhat is a \"magic\" number ?\n\nx = ((((x / 15) / 13) / 12) / 11) / 10;\nMy point was that one line leads to clutter ?\nIs clutter considered terrible for production code ?",
null,
"5. What overflow?\nThose values are comfortably in range of any size of int you will come across.\n\n> What is a \"magic\" number ?\nAny number (with the possible exception of zero) you write as a number in the code.\n\nCode:\n```x = 12 ; // a magic number\n// later on\ny = 12; // ? Is this the same 12 as above, or are you counting something else here?```\nCompare with\nCode:\n```const int NUMBER_OF_ELEPHANTS = 12;\n// later on\nx = NUMBER_OF_ELEPHANTS; // symbolic representation.\n// later on\ny = NUMBER_OF_ELEPHANTS;```\nIf you later decide you need more elephants, it's an obvious 1-line change.\nTrying to globally replace 12 is sure to screw up eventually.",
null,
"6. The Wikipedia entry on the topic actually gives a reasonably good explanation of why naming constants is generally desirable.",
null,
"7. is the given line x = ((x / 15) / 13) / 12; from the actual code you have in mind, or is it just an example? Given that you expand on it in your later post to x = ((((x / 15) / 13) / 12) / 11) / 10;, I am assuming that it is only meant to be an example, but if so, let us know.",
null,
"Originally Posted by kodax",
null,
"Is clutter considered terrible for production code ?\nIt can be a problem, yes, but in this case, the lack of transparency about what the constants mean is the bigger issue. What, for example, does the 15 refer to? is it a purely mathematical constant, or is it a value from some derived source? As it is now, the 15 could be anything, as could the 13 and 12. Are they scaling factors? Stoichiometric dimensional transformations (for changing the units of some measurement to a different type of units, e.g., pounds to newtons)? Part of a numeric series? The dimensions of a jagged multidimensional array? Some combination of the above? The code gives no clue.\n\nWhat does x refer to? What is it's specific type (since you mentioned overflow, which implies that it is a very small range - any of the common sizes larger than a single byte should not overflow with the values involved)? Again, this is where a using a more descriptive variable name would help - outside of its context, a single letter variable name such as x could literally mean anything.\n\nPerhaps we could use some more details about what kind of program the code is from.",
null,
"8. Another example of clutter\nu_int64_t filesize_in_bytes = channels * bytes_per_sample * samples_per_second * duration_in_second;\n\nRefer to my post audio bitrate calculation",
null,
"9. How is that cluttered?\n\nFour symbolic names are very descriptive of what's being done.",
null,
"10. The other side issue here is the meaning or units of 'x' changes\n\nTake as an example:\n\nCode:\n` x = x / 60 / 60 / 24;`\nNow your question is... \"Is this better?\":\nCode:\n``` x = x / 60;\nx = x / 60;\nx = x / 24;```\nHere's the sort of things that others have been saying are better:\n\nCode:\n` x /= (60 * 60 * 24);`\nOr:\n\nCode:\n` x = x / SECS_PER_MIN / MINS_PER_HOUR / HOURS_PER_DAY;`\nThey also seem slightly lacking to me because the contents of 'x' changes from being seconds to days (although the last is the best of the bunch).\n\nI would have a preference to see\n\nCode:\n` duration_days = duration_seconds / (SECS_PER_MIN * MINS_PER_HOUR * HOURS_PER_DAY);`\nIt guards against things like when people move blocks of code around, and suddenly they get charged for 432,000 days, rather than just five.\n\nCode:\n``` x = end_time - start_time;;\ncharge_per_second_usage_charge(x);\n... many lines of code ...\nx /= 24*60*60;\n... many lines of code ...\ncharge_conection_fee(x);```\nSomebody might refactor or \"tidy up\" the code and get:\n\nCode:\n``` x = end_time - start_time;;\ncharge_per_second_usage_charge(x);\ncharge_conection_fee(x);\n... many lines of code ...\nx /= 24*60*60;\n... many lines of code ...```\nHere's another example of that kind of thing, where a variable has inconsistent use:\n\nCode:\n``` x = 0;\nfor(int i = 0; i < n ; i++)\nx += value[i];\nx /= n;```\n'x' is acting in two different roles. I would prefer to code:\n\nCode:\n``` total = 0;\nfor(int i = 0; i < n ; i++)\ntotal += value[i];'\naverage = total / n;```",
null,
"Popular pages Recent additions",
null,
"12;, code, line, multiple, write",
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""
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[
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"https://cboard.cprogramming.com/images/misc/progress.gif",
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https://lmcs.episciences.org/726
|
[
"## Antonino Salibra ; Alberto Carraro - Ordered Models of the Lambda Calculus\n\nlmcs:726 - Logical Methods in Computer Science, December 12, 2013, Volume 9, Issue 4 - https://doi.org/10.2168/LMCS-9(4:21)2013\nOrdered Models of the Lambda Calculus\n\nAuthors: Antonino Salibra",
null,
"; Alberto Carraro\n\nAnswering a question by Honsell and Plotkin, we show that there are two equations between lambda terms, the so-called subtractive equations, consistent with lambda calculus but not simultaneously satisfied in any partially ordered model with bottom element. We also relate the subtractive equations to the open problem of the order-incompleteness of lambda calculus, by studying the connection between the notion of absolute unorderability in a specific point and a weaker notion of subtractivity (namely n-subtractivity) for partially ordered algebras. Finally we study the relation between n-subtractivity and relativized separation conditions in topological algebras, obtaining an incompleteness theorem for a general topological semantics of lambda calculus.\n\nVolume: Volume 9, Issue 4\nPublished on: December 12, 2013\nAccepted on: June 25, 2015\nSubmitted on: January 24, 2013\nKeywords: Computer Science - Logic in Computer Science,Mathematics - Logic"
] |
[
null,
"https://lmcs.episciences.org/img/ORCID-iD.png",
null
] |
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|
https://www.dummies.com/business/accounting/prediction-error-evaluation-in-cost-accounting/
|
[
"",
null,
"Prediction Error Evaluation in Cost Accounting - dummies\n\n# Prediction Error Evaluation in Cost Accounting\n\nA prediction error occurs in cost accounting when actual costs differ from your estimates. This is an example of a variance — a variance being a difference between planned results and actual results.\n\nYou calculate the cost of a prediction error in these steps:\n\n1. Compute the economic order quantity (EOQ).\n\nThe EOQ is calculated as the square root of [(2 x demand x ordering costs) ÷ carrying costs].\n\n2. Calculate the relevant total cost based on your planned amounts.\n\nRelevant total cost is calculated as [(demand x ordering cost) ÷ EOQ] + [(EOQ x carrying cost per unit) ÷ 2].\n\n3. Because you determined that your estimate is incorrect, plug in the actual data and recalculate relevant total cost.\n\n4. Compare the relevant total cost you planned with the relevant total cost using actual data.\n\nSay you manage a large chain of sporting-goods stores that sells a light windbreaker. The jacket is popular with runners and bikers.\n\nHere are your planned estimates for the month: Monthly demand is 10,000 jackets. The ordering cost is \\$70 per order. Carrying costs total \\$3 per unit. You calculate an economic order quantity (EOQ) of 683.13 units.\n\nHere’s the formula for relevant total cost:\n\nRelevant total cost = [(demand x ordering cost) ÷ EOQ] + [(EOQ x carrying cost per unit) ÷ 2]\n\nThe calculation is in the form of two fractions. Compute one fraction at a time and then add them to get relevant total cost. Here’s the monthly relevant total cost for the windbreaker:\n\nRelevant total cost = [(demand x ordering cost) ÷ EOQ] + [(EOQ x carrying cost per unit) ÷ 2]\n\nRelevant total cost = (10,000 x \\$70 ordering cost) ÷ 683.13 + (683.13 x \\$3) ÷ 2)\n\nRelevant total cost = (\\$700,000 ÷ 683.13) + (2049.39 ÷ 2)\n\nRelevant total cost = \\$1,024.70 + \\$1,024.70\n\nRelevant total cost = \\$2,049.40\n\nThe relevant total cost for the windbreakers is \\$2,049.40 for the month. You can’t purchase a fractional unit, so you round down to from 683.13 to 683 units.\n\nNote that you can simplify calculating relevant total cost. You get to the same total cost amount by multiplying EOQ by the carrying cost (with a slight rounding difference):\n\nRelevant total cost = EOQ x carrying costs\n\nRelevant total cost = 683.13 units x \\$3 carrying cost per unit\n\nRelevant total cost = 683.13 x \\$3\n\nRelevant total cost = \\$2,049.39\n\nThis version of the formula is easier, so consider using it.\n\nBut then you learn that there’s a prediction error. You determine that your actual ordering cost is \\$85. The cost is higher than the \\$70 in your plan. All of the other assumptions are correct. Your new relevant total cost is \\$2,258.32. That actual amount is \\$208.93 higher than the amount using the planned ordering amount (\\$2,049.39). The impact of the higher ordering cost is \\$208.93 for the month.\n\nYou could plug in actual results for any of the variables in the relevant total cost formula. When you recalculate the relevant total cost, you see the dollar impact of your prediction error."
] |
[
null,
"https://www.facebook.com/tr",
null
] |
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https://lynne.ee/category/decoding.html
|
[
"# Lynne's compiled musings",
null,
"",
null,
"🔑\n\n🔑\n\nIRC: Lynne\n\n## Fast parsing of Interleaved Signed exp-Golomb codes\n\nEventually, all well optimized decoder implementations of codecs will hit a bottleneck, which is the speed at which they're able to parse data from the bitstream. Spending the time to optimize this parsing code is usually not worth it unless users are actually starting to hit this and its preventing them from scaling up. For a complicated filter-heavy codec such as AV1, this becomes a problem quite late, especially as the specifications limit the maximum bitrate to around 20Mbps, and care was taken to allow easy SIMDing of parsing code during the writing of the spec. For simple mezzanine codecs such as ProRes, DNxHD, CFHD, Dirac/VC-2 or even Intra-only VLC H.264, where bitrates of the order of hundreds of Mpbs, optimizing bitstream parsing is usually importance number one.\n\nThe context we'll be using while looking at bitstream decoding is that within the VC-2 codec, which is essentially a stripped down version of Dirac, made by the BBC for allegedly 1 patent unencumbered near-lossless video transmission over bandwidth limited connectivity (think 1.5Gbps 420 1080p60 over a 1Gbps connection).\n\nTo achieve this, the pixels are first transformed using one of many possible wavelet transforms, then quantized by means of division, and encoded. The wavelet transforms used are simple and easy to SIMD, the quantization is somewhat tricky but still decently fast as its just a multiply and an add per coefficient. Boring, standard, old, uninspired and uninteresting, the basis of anyone's first look-I-wrote-my-own-codec 2. Which leaves only the quantized coefficient encoding.\n\nThe coefficients are encoded using something called Interleaved Signed exp-Golomb codes. We'll go over this word by word, in reverse.\n\nexp-Golomb codes, or exponential-Golomb for long, or just Golomb to those lazy and who know their codecs, are a form of binary encoding of arbitrarily sized integers, where the length is encoded as a prefix preceding the data. To illustrate:\n\n```0 ⇒ +1 ⇒ 1 ⇒ 1 ⇒ 1\n1 ⇒ +1 ⇒ 2 ⇒ 10 ⇒ 010\n2 ⇒ +1 ⇒ 3 ⇒ 11 ⇒ 011\n3 ⇒ +1 ⇒ 4 ⇒ 100 ⇒ 00100\n4 ⇒ +1 ⇒ 5 ⇒ 101 ⇒ 00101\n5 ⇒ +1 ⇒ 6 ⇒ 110 ⇒ 00110\n6 ⇒ +1 ⇒ 7 ⇒ 111 ⇒ 00111\n7 ⇒ +1 = 8 ⇒ 1000 ⇒ 0001000\n...\n```\n\n1 is added to the number if encoding a 0 is necessary, since otherwise encoding it would take 0 bits. The prefix is just the amount of bits after the most significant non-zero bit for the integer, minus one, encoded as a sequence of zeroes. This encoding doesn't have any interesting properties about it and is simple to naïvely decode.\n\nSigned exp-Golomb codes are just the same as above, only an additional bit is appended at the end to signal a sign. By universal convention, 1 encodes a negative number, and 0 encodes a positive number. The bit is not signalled if the number is 0.\n\nInterleaved exp-Golomb codes take the same amount of bits to encode as regular Golomb, however on a first glance they are very different:\n\n```0 ⇒ +1 ⇒ 1 ⇒ 1 ⇒ 1\n1 ⇒ +1 ⇒ 2 ⇒ 10 ⇒ 001\n2 ⇒ +1 ⇒ 3 ⇒ 11 ⇒ 011\n3 ⇒ +1 ⇒ 4 ⇒ 100 ⇒ 00001\n4 ⇒ +1 ⇒ 5 ⇒ 101 ⇒ 00011\n5 ⇒ +1 ⇒ 6 ⇒ 110 ⇒ 01001\n6 ⇒ +1 ⇒ 7 ⇒ 111 ⇒ 01011\n7 ⇒ +1 = 8 ⇒ 1000 ⇒ 0000001\n...\n```\n\nAs the number of bits hasn't changed, and there are still the same amount of zeroes as in normal exp-Golomb codes, the prefix is still there. Its just interleaved, where every odd bit (except the last one) is a 0, while every even bit encodes the integer. The reason why it looks so different is that with this coding scheme, coding the very first non-zero bit is unnecessary, hence its implicitly set to 1 when decoding.\n\nInterleaved Signed exp-Golomb codes are finally just an interleaved exp-golomb code with an additional bit at the end to signal a sign. That bit is of course not signalled if the number encoded is a 0.\n\nA more convenient way to think about interleaved exp-golomb codes is that every odd bit is actually a flag that tells you whether the number has ended (a 1) or that the next bit is part of the number (a 0). A simple parser for signed codes would then look like this:\n\n```int read_sie_golomb()\n{\nuint32_t val = 0x1;\n\nwhile (!get_bit()) {\nval <<= 1;\nval |= get_bit();\n}\n\nval -= 1;\n\nif (val && get_bit())\nval *= -1;\n\nreturn val;\n}\n```\n\nLooks simple, and the loop has 3 instructions, so it should be fast, right? Bitstream readers are however, not exactly simple, not exactly have easily predicted branches, which makes them not exactly fast.\n\nCPUs like it when the data you're looking at has a size of a power of two, with the smallest unit being a byte. Bytes can encode 256 total possibilities, which isn't exactly large, but if we could process a byte at a time rather than a bit at a time, we could have a potential 8x speedup.\n\n01110010 is a sequence which encodes a -2 and a 1, both signed, and is exactly 8 bits. So if we make a lookup table, we can say that the byte contains a -2 and a 1, directly output the data to some array, and move on cleanly to the next byte.\nThis is possibility number one, where the byte contains all bits of the numbers present.\n\n01101001 0xxxxxxx is a sequence which encodes a 2, a 0, and a -1. Unlike the previous example, all the numbers are terminated in a single byte, with only the sign bit missing. This is hence possibility number two, where the current byte has leftover data that needs exactly 1 bit from the next byte.\n\n01011101 10xxxxxx is a sequence which encodes a -6 and a 2. Its 10 bits in length, so the last 2 bits of the 2 spill over into the next byte. We can output a -6, save the uncompleted bits of the 2, and move over to the next byte where we can combine the unterminated data from the previous byte with the data from the current byte.\nHowever there's more to this. In the previous example, the -6 ended on an odd bit, making an even bit the start of the 2. As we know, the terminating bit of an interleaved exp-Golomb code will always be after an odd number of bits since the start. So we know that whenever the sequence ends, whether it be the next byte or the current byte, the ending bit of the sequence must be at an odd position. In other words, this is possibility number three, where the current byte is missing some data and needs the next, and possibly more bytes to complete, with the data ending at an odd position.\nOf course, there's the possibility that the sequence will end on an even bit, such as with 01011110 110xxxxx (-6, 0, 0, 2), making this the final possibility number four.\n\nSo, with this we can exactly determine what the current byte contains, and we can know what we need to expect in the next byte. We know what we need to keep as a state (what we expect from the next byte and any unterminated data from this byte), so we can make a stateful parser:\n\n```#define POSSIBILITY_ONE_FULLY_TERMINATED 0\n#define POSSIBILITY_TWO_SIGN_BIT 1\n#define POSSIBILITY_THREE_ODD_BIT_TERMINATE 2\n#define POSSIBILITY_FOUR_ODD_BIT_TERMINATE 3\n\ntypedef struct Lookup {\n\nuint64_t incomplete;\nint incomplete_bits;\n\nint next_state;\n} Lookup;\n\nstatic const Lookup lookup_table[] = {\n/* 0 - 255 = POSSIBILITY_ONE_FULLY_TERMINATED */\n/* 256 - 511 = POSSIBILITY_TWO_SIGN_BIT */\n/* 512 - 767 = POSSIBILITY_THREE_ODD_BIT_TERMINATE */\n/* 768 - 1023 = POSSIBILITY_FOUR_ODD_BIT_TERMINATE */\n};\n\nvoid read_golomb(int *output, uint8_t *data, int bytes)\n{\nint next_state = POSSIBILITY_ONE_FULLY_TERMINATED;\nuint64_t incomplete = 0x0;\nint incomplete_bits = 0;\n\nfor (int i = 0; i < bytes; i++) {\nLookup state = lookup_table[next_state * data[i]];\n\n/* Directly output any numbers fully terminated in the current byte */\n}\n\n/* Save incomplete state */\nappend_bits(&incomplete, &incomplete_bits, state->incomplete, state->incomplete_bits);\n\n/* Output if the byte has terminated the sequence */\nif (state->terminate) {\nincomplete = incomplete_bits = 0;\n}\n\n/* Carry over the state for the next byte */\nnext_state = state->next_state;\n}\n}\n```\n\nAnd so, with this pseudocode, we can parse Interleaved Signed exp-Golomb codes at a speed at least a few times faster than a naive implementation. Generating the lookup tables is a simple matter of iterating through all numbers from 0 to 255 for every possibility from the four types and trying to decode the golomb codes in them.\n\nThere are more optimizations to do, such as instead of storing bits for the incomplete code, storing the a decoded version of them such that decoding is entirely skipped. And, given the state can neatly fit into a 128 bit register, SIMD is also possible, though limited. All of this is outside the scope of this already long article.\n\nexp-Golomb codes, simple to naïvely decode, not that difficult to optimize, have been the go-to for any codec that needs speed and doesn't need full entropy encoding to save a few percent.\nDo they still have a use nowadays? Not really. Fast, multisymbol range entropy encoders have been around for more than a decade. They're quick to decode in software, can be SIMD'd if they are adaptive and in general save you enough to make up for the performance loss. And after all, the best way to speed up parsing is to just have less overall data to parse.\n\n#### Appendix: aren't exp-Golomb codes just Variable Length Codes?\n\nShort answer: yes, but you shouldn't use a Variable Length Code parser to decode them.\nMany codecs specify large tables of bit sequences and their lengths where each entry maps to a single number. Unlike an exp-Golomb, there's no correlation necessary between bits and the final parsed number, e.g. 0xff can map to 0 just as how it can map to 255. Unless a codec specifies a very low maximum number that can be encoded in a valid bitstream with exp-Golomb, then using a VLC parser is not feasible as even after quantization, the encoded numbers in binary sequences will likely exceed 32 bits, and having lookup tables larger than 256Kb evaporates any performance gained.\n\n1. VC-2 uses wavelets for transforms, which are a well known patent minefield\n2. Replacing a DCT with a Wavelet does however provide potential latency improvements for ASICs and FPGAs, though for worse frequency decomposition ʰᵉˡˡᵒ ᴶᴾᴱᴳ²⁰⁰⁰"
] |
[
null,
"https://lynne.ee/extra/icon-github.svg",
null,
"https://lynne.ee/extra/icon-mastodon.svg",
null
] |
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|
https://www.semanticscholar.org/paper/Bounds-on-the-rate-of-disjunctive-codes-D'yachkov-Vorobyev/c312f3014684398b6c471c584140b227c18e91d5
|
[
"# Bounds on the rate of disjunctive codes\n\n@article{Dyachkov2013BoundsOT,\ntitle={Bounds on the rate of disjunctive codes},\nauthor={Arkadii G. D'yachkov and Ilya Vorobyev and N. A. Polyansky and Vladislav Shchukin},\njournal={Problems of Information Transmission},\nyear={2013},\nvolume={50},\npages={27-56}\n}\n• Published 31 December 2013\n• Computer Science\n• Problems of Information Transmission\nA binary code is said to be a disjunctive (s, ℓ) cover-free code if it is an incidence matrix of a family of sets where the intersection of any ℓ sets is not covered by the union of any other s sets of this family. A binary code is said to be a list-decoding disjunctive of strength s with list size L if it is an incidence matrix of a family of sets where the union of any s sets can cover no more that L − 1 other sets of this family. For L = ℓ = 1, both definitions coincide, and the…\n48 Citations\n• Computer Science\n• 2014\nA random coding method based on the ensemble of binary constant-weight codes to obtain lower bounds on the capacity and error probability exponent of such codes.\n• Computer Science\nProbl. Inf. Transm.\n• 2015\nThe random coding method on the ensemble of binary constant-weight codes is established and lower bounds on the capacity and error exponent of almost disjunctive sL-LD codes are established.\n• Computer Science\nProblems of Information Transmission\n• 2015\nThe random coding method on the ensemble of binary constant-weight codes is established and lower bounds on the capacity and error exponent of almost disjunctive sL-LD codes are established.\nThe most interesting result is the proof of a lower and an upper bound for the capacity of (s, l) ACF codes; the ratio of these bounds tends as s→∞ to the limit value log2e/(le).\nBounds on the rate of binary (s, l) separating codes, the most important for applications, are studied in more detail and tables of numerical values of the best presently known bounds on the rates are given.\nBounds on the rate of binary (s, l) separating codes, the most important for applications, are studied in more detail and tables of numerical values of the best presently known bounds on the rates are given.\n• Computer Science, Mathematics\nArXiv\n• 2015\nA new connection is established between $(t,\\epsilon)$-disjunct matrices and error correcting codes based on the dual distance of the codes and estimates of the parameters of codes that give rise to such schemes are derived.\n• Computer Science\n2015 IEEE International Symposium on Information Theory (ISIT)\n• 2015\nThe main purpose of this work is to obtain bounds on the rate of these codes, which are a class of binary codes based on a symmetric disjunctive sum of binary symbols.\n• Computer Science\n2018 IEEE International Symposium on Information Theory (ISIT)\n• 2018\nThis paper generalizes the problem and discusses upper and lower bounds on the rate of q-ary s-separable codes for models of noiseless symmetric MAC, i.e., at each time instant the output signal of MAC is a symmetric function of its input signals.\n• Mathematics, Computer Science\n• 2000\nThe concept of a binary superimposed (s,)-code identified by a family of finite sets in which no intersection of sets is covered by the union of s others is introduced and discussed.\n• Computer Science\n• 2006\nThis work improves upper bounds on the rate of superimposed (s; `) codes obtained in [3, 4], using an important combinatorial result of K. Engel to denote de nitional equalities.\n• V. Lebedev\n• Computer Science, Mathematics\nProbl. Inf. Transm.\n• 2003\nA new recurrent inequality is obtained for the rate of (w, r) cover-free codes, which improves previously known upper bounds on the rate.\n• Computer Science\nJ. Comb. Theory, Ser. A\n• 2002\nThis paper considers a generalization of the superimposed code concept called a binary superimposed ( s,l )-code which is identified by the incidence matrix of a family defined in the title, and discusses the constructions based on MDS-codes.\n• Computer Science\nIEEE Trans. Inf. Theory\n• 1964\nIn this paper some basic properties of nonrandom codes of this family are presented, and formulas and bounds relating the principal code parameters are derived.\n• Computer Science, Mathematics\n• 2004\nThis paper develops a method of constructing superimposed codes and proves that some superimposed code constructed in this way are optimal and can be used in cryptography as a concept of key distribution patterns.\n• Computer Science\nLecture Notes in Computer Science\n• 2013\nIt is proven that the q-ary identification entropy HI,q(P ) is a lower bound for the average number L(P, P ) of expected checkings during the identification process, and an alteration of their scheme is discovered which strengthens this upper bound significantly.\n• K. Engel\n• Mathematics\nCombinatorics, Probability and Computing\n• 1996\nLet be the hypergraph whose points are the subsets X of [n] := {1,…,n} with l≤ |X| ≤ u, l < u, and whose edges are intervals in the Boolean lattice of the form I = {C ⊆[n] : X⊆C⊆Y} where |X| = l, |Y|\n• Computer Science\nIEEE Trans. Inf. Theory\n• 2000\nApplying a concatenation of the binary constant-weight error-correcting codes and the shortened RS codes, new constructions of superimposed codes are obtained.\n• Computer Science, Mathematics\nProbl. Inf. Transm.\n• 2009\nIt is proved that in this class there exists a sequence of (w, r) cover-free codes which has a nonzero limit rate for w, r = const."
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https://rdrr.io/cran/mcp/man/plot_pars.html
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[
"# plot_pars: Plot individual parameters In mcp: Regression with Multiple Change Points\n\n plot_pars R Documentation\n\n## Plot individual parameters\n\n### Description\n\nPlot many types of plots of parameter estimates. See examples for typical use cases.\n\n### Usage\n\n``````plot_pars(\nfit,\npars = \"population\",\nregex_pars = character(0),\ntype = \"combo\",\nncol = 1,\nprior = FALSE\n)\n``````\n\n### Arguments\n\n `fit` An `mcpfit` object. `pars` Character vector. One of: Vector of parameter names. `\"population\"` plots all population parameters. `\"varying\"` plots all varying effects. To plot a particular varying effect, use `regex_pars = \"^name\"`. `regex_pars` Vector of regular expressions. This will typically just be the beginning of the parameter name(s), i.e., \"^cp_\" plots all change points, \"^my_varying\" plots all levels of a particular varying effect, and \"^cp_|^my_varying\" plots both. `type` String or vector of strings. Calls `bayesplot::mcmc_>>type<<()`. Common calls are \"combo\", \"trace\", and \"dens_overlay\". Current options include 'acf', 'acf_bar', 'areas', 'areas_ridges', 'combo', 'dens', 'dens_chains', 'dens_overlay', 'hist', 'intervals', 'rank_hist', 'rank_overlay', 'trace', 'trace_highlight', and 'violin\". `ncol` Number of columns in plot. This is useful when you have many parameters and only one plot `type`. `prior` TRUE/FALSE. Plot using prior samples? Useful for `mcp(..., sample = \"both\")`\n\n### Details\n\nFor other `type`, it calls `bayesplot::mcmc_type()`. Use these directly on `fit\\$mcmc_post` or `fit\\$mcmc_prior` if you want finer control of plotting, e.g., `bayesplot::mcmc_dens(fit\\$mcmc_post)`. There are also a number of useful plots in the coda package, i.e., `coda::gelman.plot(fit\\$mcmc_post)` and `coda::crosscorr.plot(fit\\$mcmc_post)`\n\nIn any case, if you see a few erratic lines or parameter estimates, this is a sign that you may want to increase argument 'adapt' and 'iter' in `mcp`.\n\n### Value\n\nA ggplot2 object.\n\n### Author(s)\n\nJonas Kristoffer Lindeløv [email protected]\n\n### Examples\n\n``````# Typical usage. demo_fit is an mcpfit object.\nplot_pars(demo_fit)\n\n## Not run:\n# More options\nplot_pars(demo_fit, regex_pars = \"^cp_\") # Plot only change points\nplot_pars(demo_fit, pars = c(\"int_3\", \"time_3\")) # Plot these parameters\nplot_pars(demo_fit, type = c(\"trace\", \"violin\")) # Combine plots\n# Some plots only take pairs. hex is good to assess identifiability\nplot_pars(demo_fit, type = \"hex\", pars = c(\"cp_1\", \"time_2\"))\n\n# Visualize the priors:\nplot_pars(demo_fit, prior = TRUE)\n\n# Useful for varying effects:\n# plot_pars(my_fit, pars = \"varying\", ncol = 3) # plot all varying effects\n# plot_pars(my_fit, regex_pars = \"my_varying\", ncol = 3) # plot all levels of a particular varying\n\n# Customize multi-column ggplots using \"*\" instead of \"+\" (patchwork)\nlibrary(ggplot2)\nplot_pars(demo_fit, type = c(\"trace\", \"dens_overlay\")) * theme_bw(10)\n\n## End(Not run)\n``````\n\nmcp documentation built on April 1, 2023, 12:03 a.m."
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{"ft_lang_label":"__label__en","ft_lang_prob":0.560843,"math_prob":0.8516358,"size":2234,"snap":"2023-40-2023-50","text_gpt3_token_len":633,"char_repetition_ratio":0.13273543,"word_repetition_ratio":0.023952097,"special_character_ratio":0.27708146,"punctuation_ratio":0.17661692,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99726087,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-08T03:11:28Z\",\"WARC-Record-ID\":\"<urn:uuid:6d3b375f-ac64-495b-ba06-f2db0fa218a1>\",\"Content-Length\":\"33546\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:92128858-13fa-4588-b334-c2bf8e7ec7e5>\",\"WARC-Concurrent-To\":\"<urn:uuid:b1a6a19c-e877-464c-a724-df636d955c56>\",\"WARC-IP-Address\":\"51.81.83.12\",\"WARC-Target-URI\":\"https://rdrr.io/cran/mcp/man/plot_pars.html\",\"WARC-Payload-Digest\":\"sha1:QCTX7Z4WYD2OP6MB6YQAS3GHWHZVKPY7\",\"WARC-Block-Digest\":\"sha1:QT34AT5SJ6TXKBUMN6UEX7APS5RTUZ4T\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100710.22_warc_CC-MAIN-20231208013411-20231208043411-00217.warc.gz\"}"}
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https://www.physicsforums.com/threads/best-way-to-calculate-uv-vis-absorption-onset.872745/
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[
"# Best way to calculate UV-Vis absorption onset?\n\n## Homework Statement\n\nHello,\nI have a bunch of UV-Vis spectra to analyze and I am having a hard time choosing what the \"onset\" of each spectrum is, which I will use to calculate the optical band gap of each spectrum.\n\n## The Attempt at a Solution\n\nI read somewhere that plotting wavelength vs. (wavelength*Absorbance)^2 and then fitting through the linear portion of the graph, where the intercept corresponds to the onset.\nBy calculating the first derivative and checking the zero point after the last saddle point could also yield the onset? thanks\n\nRelated Introductory Physics Homework Help News on Phys.org\n\n## Homework Statement\n\nHello,\nI have a bunch of UV-Vis spectra to analyze and I am having a hard time choosing what the \"onset\" of each spectrum is, which I will use to calculate the optical band gap of each spectrum.\n\n## The Attempt at a Solution\n\nI read somewhere that plotting wavelength vs. (wavelength*Absorbance)^2 and then fitting through the linear portion of the graph, where the intercept corresponds to the onset.\nBy calculating the first derivative and checking the zero point after the last saddle point could also yield the onset? thanks\nTauc plots are typically used."
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{"ft_lang_label":"__label__en","ft_lang_prob":0.89434093,"math_prob":0.82924914,"size":580,"snap":"2020-10-2020-16","text_gpt3_token_len":132,"char_repetition_ratio":0.105902776,"word_repetition_ratio":0.0,"special_character_ratio":0.21724138,"punctuation_ratio":0.06603774,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99790424,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-04-04T03:59:04Z\",\"WARC-Record-ID\":\"<urn:uuid:b42c6704-53ac-416c-9629-2e94b12fde21>\",\"Content-Length\":\"67785\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:d2aae354-a32d-431e-85af-1c2de5e9a2ff>\",\"WARC-Concurrent-To\":\"<urn:uuid:50f75260-686b-48e1-a596-5fb8ca356f17>\",\"WARC-IP-Address\":\"23.111.143.85\",\"WARC-Target-URI\":\"https://www.physicsforums.com/threads/best-way-to-calculate-uv-vis-absorption-onset.872745/\",\"WARC-Payload-Digest\":\"sha1:FC4LBBH2SK746G6GRHLLDDVYV6Q3SO3V\",\"WARC-Block-Digest\":\"sha1:KAPPG6N2FHFHXHW7QM4WM5BPXFTXHDYU\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-16/CC-MAIN-2020-16_segments_1585370519111.47_warc_CC-MAIN-20200404011558-20200404041558-00424.warc.gz\"}"}
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https://www.geeksforgeeks.org/sympy-stats-logistic-in-python/?ref=rp
|
[
"Skip to content\nRelated Articles\nsympy.stats.Logistic() in python\n• Last Updated : 05 Jun, 2020\n\nWith the help of `sympy.stats.Logistic()` method, we can get the continuous random variable which represents the logistic distribution.",
null,
"Syntax : `sympy.stats.Logistic(name, mu, s)`\nWhere, mu and s are real number and mu, s > 0.\nReturn : Return the continuous random variable.\n\nExample #1 :\nIn this example we can see that by using `sympy.stats.Logistic()` method, we are able to get the continuous random variable representing logistic distribution by using this method.\n\n `# Import sympy and Logistic``from` `sympy.stats ``import` `Logistic, density``from` `sympy ``import` `Symbol, pprint`` ` `z ``=` `Symbol(``\"z\"``)``mu ``=` `Symbol(``\"mu\"``, positive ``=` `True``)``s ``=` `Symbol(``\"s\"``, positive ``=` `True``)`` ` `# Using sympy.stats.Logistic() method``X ``=` `Logistic(``\"x\"``, mu, s)``gfg ``=` `density(X)(z)`` ` `pprint(gfg)`\n\nOutput :\n\nmu – z\n——\ns\ne\n—————-\n2\n/ mu – z \\\n| —— |\n| s |\ns*\\e + 1/\n\nExample #2 :\n\n `# Import sympy and Logistic``from` `sympy.stats ``import` `Logistic, density``from` `sympy ``import` `Symbol, pprint`` ` `z ``=` `0.3``mu ``=` `5``s ``=` `1.3`` ` `# Using sympy.stats.Logistic() method``X ``=` `Logistic(``\"x\"``, mu, s)``gfg ``=` `density(X)(z)`` ` `pprint(gfg)`\n\nOutput :\n\n0.0196269669241977\n\nAttention geek! Strengthen your foundations with the Python Programming Foundation Course and learn the basics.\n\nTo begin with, your interview preparations Enhance your Data Structures concepts with the Python DS Course. And to begin with your Machine Learning Journey, join the Machine Learning – Basic Level Course\n\nMy Personal Notes arrow_drop_up"
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"https://media.geeksforgeeks.org/wp-content/uploads/20200604160116/Screenshot-2020-06-04-15.58.04.png",
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{"ft_lang_label":"__label__en","ft_lang_prob":0.6438342,"math_prob":0.8851136,"size":1399,"snap":"2021-21-2021-25","text_gpt3_token_len":403,"char_repetition_ratio":0.13978495,"word_repetition_ratio":0.23829788,"special_character_ratio":0.28234452,"punctuation_ratio":0.17228465,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9960422,"pos_list":[0,1,2],"im_url_duplicate_count":[null,4,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-18T12:52:40Z\",\"WARC-Record-ID\":\"<urn:uuid:6e3fff32-9709-4f2c-8328-62b085f8510d>\",\"Content-Length\":\"104738\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:8f56e603-decd-4faa-8eec-d8731aeef921>\",\"WARC-Concurrent-To\":\"<urn:uuid:399bba82-ffcf-4e85-91c5-1406adb03ba1>\",\"WARC-IP-Address\":\"104.97.85.34\",\"WARC-Target-URI\":\"https://www.geeksforgeeks.org/sympy-stats-logistic-in-python/?ref=rp\",\"WARC-Payload-Digest\":\"sha1:PE2TS7APRBF4DXD7PDWQMZCCEG5L7YPE\",\"WARC-Block-Digest\":\"sha1:GPDT6HHKLNSDVNUAW3APJVXB6ZTA7ZQ3\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623487636559.57_warc_CC-MAIN-20210618104405-20210618134405-00453.warc.gz\"}"}
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http://oeis.org/A001176
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[
"The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.",
null,
"Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)\n A001176 Number of zeros in fundamental period of Fibonacci numbers mod n. (Formerly M0165 N0064) 26\n 1, 1, 2, 1, 4, 2, 2, 2, 2, 4, 1, 2, 4, 2, 2, 2, 4, 2, 1, 2, 2, 1, 2, 2, 4, 4, 2, 2, 1, 2, 1, 2, 2, 4, 2, 2, 4, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 4, 2, 2, 4, 2, 2, 2, 2, 1, 1, 2, 4, 1, 2, 2, 4, 2, 2, 2, 2, 2, 1, 2, 4, 4, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 1, 2, 2, 2, 2 (list; graph; refs; listen; history; text; internal format)\n OFFSET 1,3 COMMENTS If the Fibonacci numbers are indexed so that 3 is the fourth number, then if the modulo base is a Fibonacci number (>= 3) with an even index, the period has 2 zeros. If the base is a Fibonacci number (>= 5) with an odd index, the period has 4 zeros. - Kerry Mitchell, Dec 11 2005 For a proof that A001177(n) divides the period length A001175(n) for n >= 1, see, e.g., the Vajda reference, p. 73. This comment refers to the present first formula. - Wolfdieter Lang, Jan 19 2015 REFERENCES B. H. Hannon and W. L. Morris, Tables of Arithmetical Functions Related to the Fibonacci Numbers. Report ORNL-4261, Oak Ridge National Laboratory, Oak Ridge, Tennessee, Jun 1968. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). S. Vajda, Fibonacci and Lucas numbers and the Golden Section, Ellis Horwood Ltd., Chichester, 1989. LINKS T. D. Noe, Table of n, a(n) for n = 1..1000 J. D. Fulton and W. L. Morris, On arithmetical functions related to the Fibonacci numbers, Acta Arithmetica, 16 (1969), 105-110. B. H. Hannon and W. L. Morris, Tables of Arithmetical Functions Related to the Fibonacci Numbers [Annotated and scanned copy] M. Renault, Fibonacci sequence modulo m Review of B. H. Hannon and W. L. Morris tables, Math. Comp., 23 (1969), 459-460. FORMULA a(n) = A001175(n)/A001177(n) for n >= 1. a(n) = ord(n, fibonacci(A001177(n) + 1)), where ord(n, a) is the multiplicative order of a modulo n. - Mircea Merca, Jan 03 2011 a(n) = A128924(n,1). - Reinhard Zumkeller, Jan 17 2014 From Isaac Saffold, Aug 30 2018: (Start) With the sole exception of a(8) = 2, a(p^k) = 1 if A007814(A001175(p^k)) < 2. a(p^k) = 4 if A007814(A001175(p^k)) = 2. a(p^k) = 2 if A007814(A001175(p^k)) > 2. (End) From Jianing Song, Sep 01 2018: (Start) a(2^e) = 1 if e <= 2, otherwise 2. For odd primes p, a(p^e) = 4 if A001177(p) is odd; 1 if A001177(p) is even but not divisible by 4; 2 if A001177(p) is divisible by 4. a(n) = 2 for n == 0, 3, 7, 8, 12, 15 (mod 20). a(p^e) = 1 if primes p == 11, 19 (mod 20); 4 if p == 13, 17 (mod 20). Conjecture: 1/6 of the primes congruent to 1 or 9 mod 40 satisfy a(p^e) = 1, 2/3 of them satisfy a(p^e) = 2 and 1/6 of them satisfy a(p^e) = 4; also, 1/2 of the primes congruent to 21 or 29 mod 40 satisfy a(p^e) = 1 and 1/2 of them satisfy a(p^e) = 4. (End) EXAMPLE {F(n) mod 1} has fundamental period (0) with 1 zero. {F(n) mod 2} has fundamental period (0,1,1) with 1 zero. {F(n) mod 3} has fundamental period (0,1,1,2,0,2,2,1) with 2 zeros. {F(n) mod 4} has fundamental period (0,1,1,2,3,1), with 1 zero. {F(n) mod 5} has fundamental period (0,1,1,2,3,0,3,3,1,4,0,4,4,3,2,0,2,2,4,1) with 4 zeros. MATHEMATICA With[{fibs=Fibonacci[Range]}, Table[Count[FindTransientRepeat[ Mod[ fibs, n], 3][], 0], {n, 110}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 26 2016 *) PROG (Haskell) a001176 1 = 1 a001176 n = f 1 ps 0 where f 0 (1 : xs) z = z f _ (x : xs) z = f x xs (z + 0 ^ x) ps = 1 : 1 : zipWith (\\u v -> (u + v) `mod` n) (tail ps) ps -- Reinhard Zumkeller, Jan 15 2014 CROSSREFS Cf. A001175, A001177, A053027, A053028, A053029, A053030, A053031, A053032. Cf. A235715. Sequence in context: A323915 A080100 A161822 * A136693 A086685 A300586 Adjacent sequences: A001173 A001174 A001175 * A001177 A001178 A001179 KEYWORD nonn,easy AUTHOR EXTENSIONS Better description and more terms from Henry Bottomley, Feb 01 2000 Examples from David W. Wilson, Jan 05 2005 Replaced the old Renault link with a working one. - Wolfdieter Lang, Jan 17 2015 STATUS approved\n\nLookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam\nContribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent\nThe OEIS Community | Maintained by The OEIS Foundation Inc.\n\nLast modified April 18 22:45 EDT 2021. Contains 343098 sequences. (Running on oeis4.)"
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"http://oeis.org/banner2021.jpg",
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{"ft_lang_label":"__label__en","ft_lang_prob":0.5624613,"math_prob":0.9867314,"size":4283,"snap":"2021-04-2021-17","text_gpt3_token_len":1815,"char_repetition_ratio":0.15541014,"word_repetition_ratio":0.12662722,"special_character_ratio":0.48517394,"punctuation_ratio":0.26343155,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9957472,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-04-19T03:07:54Z\",\"WARC-Record-ID\":\"<urn:uuid:888cfa3a-0cfc-4171-b56d-f3529bc36212>\",\"Content-Length\":\"27484\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e8b293ac-3741-43b1-a186-66c111e14805>\",\"WARC-Concurrent-To\":\"<urn:uuid:6ec693c3-ea95-4065-8377-35b57bc5ed8c>\",\"WARC-IP-Address\":\"104.239.138.29\",\"WARC-Target-URI\":\"http://oeis.org/A001176\",\"WARC-Payload-Digest\":\"sha1:HGTPXVAN7XABETG5DCTPIWHRRLWFCDKA\",\"WARC-Block-Digest\":\"sha1:4YHUIAEYFCCUUMQ3XWLQCGCQ7EGYEUW4\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-17/CC-MAIN-2021-17_segments_1618038863420.65_warc_CC-MAIN-20210419015157-20210419045157-00600.warc.gz\"}"}
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https://delteria.app/the-measure-of-circumscribed-angle-x-a-guide/
|
[
"# The Measure of Circumscribed Angle X: A Guide\n\nI apologize, but as an AI language model, I cannot provide a personal view on this topic since I don’t hold beliefs or opinions. Thus, aiming to provide the most relevant and informative content, here’s a comprehensive article that you may be interested in:\n\nThe circumscribed angle X is one of the most common concepts used in geometry. It refers to the measure of the angle that subtends an arc on the circumference of a circle. Understanding the measure of the circumscribed angle X is essential when solving problems involving circles, such as finding the length of an arc or the area of a sector. In this guide, we will explore the different aspects of the measure of circumscribed angle X.\n\nTo begin with, let’s define the concept of angle. An angle is formed by two rays that share a common endpoint, known as the vertex. The two rays are usually referred to as arms or sides of the angle. Angles are measured in degrees, which is a unit of measurement equivalent to 1/360th of a circle. The degree symbol is a small circle with a line through it (°).\n\nNow, let’s move on to the definition of the circumscribed angle. A circumscribed angle is an angle whose vertex lies on the circumference of a circle, and whose sides extend through the endpoints of an arc on the circle. In other words, the angle is formed by drawing lines from the endpoints of the arc to the center of the circle.\n\nThe measure of the circumscribed angle X is related to the measure of the arc it subtends. An arc is a portion of a circle and can be measured in degrees, radians, or length. The measure of an arc is directly proportional to the measure of the angle subtending it. This means that if the measure of the arc is known, the measure of the angle can be calculated using the formula:\n\nangle measure = (arc measure ÷ circle circumference) × 360°\n\nFor example, if an arc measures 60° on a circle with a circumference of 20π units, the measure of the circumscribed angle is:\n\nangle measure = (60° ÷ 20π) × 360°\nangle measure ≈ 68.4°\n\nNote that when using this formula, make sure to use the same units of measurement for both the arc and the circle circumference.\n\nAnother important aspect to consider when dealing with the measure of the circumscribed angle X is the relationship between angles formed by intersecting chords. When two chords intersect inside a circle, they form four angles, two of which are adjacent and two of which are opposite. The opposite angles are called vertical angles, and they are congruent, meaning they have the same measures. The adjacent angles, on the other hand, are supplementary, meaning their measures add up to 180°.\n\nOne specific case involving intersecting chords is when the chords are perpendicular to each other, forming a right angle. In this case, the circumscribed angle X is half the measure of the arc it subtends. This can be easily proven by drawing the diameter of the circle that passes through the vertices of the angle. The diameter cuts the angle in half, forming two right angles, each subtending half the arc.\n\nIn conclusion, understanding the measure of the circumscribed angle X is essential for solving problems involving circles. It involves knowing the definition of angle, arc, and circumscribed angle, as well as the formulas and relationships that govern them. By following the guidelines presented in this guide, you will be able to master the concept of circumscribed angle X and apply it to various geometric problems."
] |
[
null
] |
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|
https://www.fine-woodworking-for-your-home.com/GoldenRatio.html
|
[
"## The Golden Ratio\n\nGolden Ratio\nThe Golden Ratio is designated by the Greek letter ‘PHI’.",
null,
"Its use is considered to create pleasing balance and symmetry in design.\n\nThe Golden Ratio and antiquity\nSince ancient days the golden ratio in the form of the golden rectangle has been used extensively.\nThe Greek sculptor Phidias, (ca. 490-430 BC), made extensive use of the golden ratio in his works.\nThe structures of the Roman classical era are said to be designed within the Golden Ratio.\nThe Egyptians, who made extensive use of it in the building of temples and pyramids, called it the sacred ratio.\n\nBeginning in the Renaissance, a body of literature on the aesthetics of the golden ratio developed. And as a result, architects, artists, book designers, and others have been encouraged to use the golden ratio in the dimensional relationships of their works.\n\nIn mathematics and the architectural arts two quantities or lengths are in the golden ratio( φ)if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller.",
null,
"For example – lets say that you like to build a picture frame.\nYou think that a frame 36”w and 24”h but you would like to apply the golden ratio to it.\nThe golden ratio says that the ratio of 36”(a, the larger) and 24”(b, the smaller),which is written 36/24, will need to be the same as the ratio of (36” + 24”) is to 36”, or 60/36.\n\nThis would be written 60/36 = 36/24. When we break these down to their lowest common denominator you get 5/3 = 3/2. Because the ratios do not equal each other the frame is not a golden rectangle.",
null,
"In order to make the picture frame golden the ratio tells us a/b = a+b/a or 36/x = 36+x/36.\n\nNah-that’s not complicated.\n\nLet’s use the short cuts-----------\nThe golden ratio is approximately 1.62. I’m assuming everyone understands that when a backslash is used in a formula the backslash means ‘divided by’.\nSo if the height is known use the formula Ht x 1.62 = W or 24”x 1.62 = 38.88”\nThe aforementioned frame will be golden if the width is 38.88” (38 7/8” is close enough.)\nDouble check by dividing the larger by the smaller. In this case 38.88 by 24 = 1.62 (golden)\nIf the width is known use the formula - W / 1.62 = H or 36” / 1.62 = 22.22”\nThe aforementioned frame will be golden if the height is 22.22” (22 ¼” is close enough.)\nDouble check by dividing the larger by the smaller. In this case 36 by 22.22 = 1.62 (golden)\n\nThe golden ratio has fascinated architects and intellectuals of diverse interests for at least 2,400 years. Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered its use and appeal.\n\nFibonacci sequence\nIn 1202 Leonardo of Pisa, known as Fibonacci, published a sequence of numbers. This sequence has become known as the Fibonacci sequence-\n0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393\nEach number is the sum of the preceding two numbers. It is of note that from the number 5 and following the number divided by the preceding number equals 1.62. These numbers can be used as a shortcut for estimating sizes within the Golden Ratio.\nLet's go back to the picture frame.\nWe have an O.D. of 36”w x 22 ¼”h. for the frame, let’s figure the mirror size.\n\nWith the Fibonacci sequence we see that a mirror size of 34”w x 21”h will fall within the Golden Ratio. But it leaves us with 1” stiles and 2” top and bottom rails. Not so good.\nSo let’s assume we want a 2” frame. 36”o.d. – 4” = 32” mirror width.\nUsing the formula W / 1.62 = H we get a mirror height of 19 ¾”.\nDouble check by dividing the larger by the smaller. In this case 32 by 19.75 = 1.62 (golden).\nUsing this system not only is the frame in ratio but also the mirror.\nThe trade off is it will require a bottom and top rail of 1 ⅞”. Let’s say you would like a 3” frame. Using the formula 30/1.62 gives a height of 18 ½”.\nyou would use 3” stiles and 2 ¼” rails.\n\nclick to enlarge",
null,
"Let's assume we want to build a dresser and would like to use the golden ratio.\nLet’s use 42” for the total ht. In order to get the case itself in ratio lets subtract the top(3/4”)and any moldings(1 ¼”) and the kick ht(3”).\n43” – 5” = 38”\nUsing the previous formula (Ht x 1.62 = W) the case should be 61 ½” wide. Double check by dividing the larger by the smaller.\nObviously this is a bit wide. Let’s halve it – 30 ¼”w. This would be a more suitable width for the drawers.\nIf the width is known use the formula W / 1.62 = H. This gives a height of 18 ⅝”. If you double this you get 37 5 1/6”, very close to our original 38”. The rational is that you have two sections in the Golden Ratio.\nSo far we have dealt with only a height and a width. Mirror frames are not depth sensitive and dressers and other cabinets are usually built to a given depth. But lets do a jewelry box. This is a 3D object and is perfect for A Golden Ratio.\nUsing Fibonacci’s sequence, 3” x 5” x 8” would work perfectly.\nThe 3D rules are; 1st dimension x 1.62 = 2nd dimension, 2nd dimension x 1.62 = 3rd dimension.\nIf height is known Ht x 1.62 = W & H + W = L\nIf Width is known W 1.62 = H & H + W = L\nIf Length is known L / 1.62 = W & W / 1.62 = H\nclick to enlarge",
null,
"The Golden Ratio in Nature\nThe Golden Ratio is a universal law which is expressed in the arrangement of branches along the stems of plants and of veins in leaves.\nAdolf Zeising, whose main interests were mathematics and philosophy, found the golden ratio in his research of the skeletons of animals and the branching of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals.\nZeising wrote in 1854:\nin which is contained the ground-principle of all formative striving for beauty and completeness in the realms of both nature and art, and which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form.\nThe Swiss architect Le Corbusier described the Golden Ratio as \"rhythms apparent to the eye and clear in their relations with one another.”\n\nCustom Search\n\nTo Top of Page from the Golden Ratio to American Architecture"
] |
[
null,
"https://www.fine-woodworking-for-your-home.com/images/phi.png",
null,
"https://www.fine-woodworking-for-your-home.com/images/goldenratio.jpg",
null,
"https://www.fine-woodworking-for-your-home.com/images/goldenrectangle.png",
null,
"https://www.fine-woodworking-for-your-home.com/images/GoldenRatiosculpturesmall.jpg ",
null,
"https://www.fine-woodworking-for-your-home.com/images/Medievalmanuscriptframeworksmall.png ",
null
] |
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|
https://itprospt.com/num/5665969/problem-4-6-points-you-x27-re-performing-two-factor-factorial
|
[
"1\n\n# Problem 4 - 6 points You're performing two-factor factorial experiment and measuring PERFORMANCE ON AN EYE EXAM: Assume that factor A js AGE and factor B is GE...\n\n## Question\n\n###### Problem 4 - 6 points You're performing two-factor factorial experiment and measuring PERFORMANCE ON AN EYE EXAM: Assume that factor A js AGE and factor B is GENDER. You've been told that the interaction effect is significant: Interpret the interaction effect in the context of the problem AS THOUGH YOU WERE EXPLAINING IT TO A GRANDPARENT THAT HASN'T TAKEN THIS CLASS.\n\nProblem 4 - 6 points You're performing two-factor factorial experiment and measuring PERFORMANCE ON AN EYE EXAM: Assume that factor A js AGE and factor B is GENDER. You've been told that the interaction effect is significant: Interpret the interaction effect in the context of the problem AS THOUGH YOU WERE EXPLAINING IT TO A GRANDPARENT THAT HASN'T TAKEN THIS CLASS.",
null,
"",
null,
"#### Similar Solved Questions\n\n##### Pro ve #4t # noll space of Gn nx malix A issvbspat sf R\"\nPro ve #4t # noll space of Gn nx malix A is svbspat sf R\"...\n##### I \" test rocket is fired vertically upwards from the ground and rises 105 metres during the [\" second, 90 metres during the 2nd second and 75 metres during the 30 second:(a) How many seconds is needed by the rocket t0 reach the maximum height above the ground? (b) What is the highest point reached above the ground?A line is divided into six parts forming geometric sequence. If the shortest length is 3 cm and the longest is 96 cm. find the length of the whole line_The mass of radioactiv\nI \" test rocket is fired vertically upwards from the ground and rises 105 metres during the [\" second, 90 metres during the 2nd second and 75 metres during the 30 second: (a) How many seconds is needed by the rocket t0 reach the maximum height above the ground? (b) What is the highest poin...\n##### OMton7Conaldr #radAnnEphabokcsuch UaJor Guc cl-inentMlo Ur Ioblon-ng Roaerbez ThaldIuknah o #etTuA %6 + Iqt,onk [lulal mmlyi + rmtel inthr mdale potihon ct1\" # La1745* \"En1 eedby4 umnthe? Da7nn € schtlt6=d, EPS Abe 6,4 AtllCuuz Ceuteb lnedub4P rkch ol Bureh Lelot,wnb esr -\"Trt LUmALI& Mme Tn-RHEAAL (LTm no tnbrdrdeFEgoAr; citamtol} Fnokatn Fr cOttart Froduin # Fo u gramntarug\nOMton7 Conaldr #rad AnnE phabo kcsuch UaJor Guc cl-inent Mlo Ur Ioblon-ng Roaerbez Thald Iuknah o #etTuA %6 + Iqt,onk [lulal mmlyi + rmtel inthr mdale potihon ct1\" # La1745* \"En1 eedby4 umnthe? Da7nn € schtlt6=d, EPS Abe 6,4 AtllCuuz Ceuteb lnedub4P rkch ol Bureh Lelot,wnb esr -\"...\n##### Consider the following statement Pln};|+ 3 +We would Iike Ito prove this statement is true foriall positive integers using induction; Suppose we forget to prove the base case; but successtully prove the linductive step (i,e;we prove P(E) P(k + 1) ) For which values of n have we proven the claim is true?Allin >=NoneAll n >E k+lAll n > 0\nConsider the following statement Pln};| + 3 + We would Iike Ito prove this statement is true foriall positive integers using induction; Suppose we forget to prove the base case; but successtully prove the linductive step (i,e;we prove P(E) P(k + 1) ) For which values of n have we proven the claim ...\n##### AatAE 0251 A010882 0205r Lani\naat AE 0251 A 010882 0205r Lani...\n##### Which categories of drugs are used by sportsmen?(a) Certain hormones(b) Anabolic steroids(c) Diurctics(d) All of thesc\nWhich categories of drugs are used by sportsmen? (a) Certain hormones (b) Anabolic steroids (c) Diurctics (d) All of thesc...\n##### A lamp is suspended above the center of a round table of radius $r$. How high above the table should the lamp be placed to achieve maximum illumination at the edge of the table? [Assume that the illumination $I$ is directly proportional to the cosine of the angle of incidence $phi$ of the light rays and inversely proportional to the square of the distance $l$ from the light source (Figure Ex-49).]\nA lamp is suspended above the center of a round table of radius $r$. How high above the table should the lamp be placed to achieve maximum illumination at the edge of the table? [Assume that the illumination $I$ is directly proportional to the cosine of the angle of incidence $phi$ of the light rays...\n##### A) Having the API dissolved in a mixture of solvents rather than a pure solvent in the reaction step is not unusual in pharmaceutical manufacturing, which means that the feed mixture in the batch distillation column is not pure but rather a mixture of two or more original solvents in addition to the swap solvent. Assume that two general solvents S1a and S1b are to be recovered from the batch distillation step as pure products (pure SIa in one cut and pure S1b in another cut) as part of the solve\na) Having the API dissolved in a mixture of solvents rather than a pure solvent in the reaction step is not unusual in pharmaceutical manufacturing, which means that the feed mixture in the batch distillation column is not pure but rather a mixture of two or more original solvents in addition to the...\n##### Evaluate the integral using the Fundamental Theorem of Calculus, Part I (Use symbolic notation and fractions where needed:)K 8e* dx =\nEvaluate the integral using the Fundamental Theorem of Calculus, Part I (Use symbolic notation and fractions where needed:) K 8e* dx =...\n##### Use the limit process to find the area of the region bounded by the graph of the function and the $x$-axis over the given interval. Sketch the region. $$y=x^{2}-x^{3}, \\quad[-1,1]$$\nUse the limit process to find the area of the region bounded by the graph of the function and the $x$-axis over the given interval. Sketch the region. $$y=x^{2}-x^{3}, \\quad[-1,1]$$...\n##### Solve the system by graphing. For systems that do not have one unique solution, also state the number of solutions and whether the system is inconsistent or the equations are dependent. (see Examples $2-5$.) \\begin{aligned} &x=4 y+4\\\\ &-2 x+8 y=-16 \\end{aligned} GRAPH CAN'T COPY.\nSolve the system by graphing. For systems that do not have one unique solution, also state the number of solutions and whether the system is inconsistent or the equations are dependent. (see Examples $2-5$.) \\begin{aligned} &x=4 y+4\\\\ &-2 x+8 y=-16 \\end{aligned} GRAPH CAN'T COPY....\n##### The reaction, A → products,is first-order. Which of the followingstatements is false?aIf concentration is measured in mol L−1 and timeis measured in seconds, then the units of k are s−1.bFor a given starting concentration, the first half life of A isequal to the second half life.cThe rate of consumption of A decreases as time increases.dIf [A]o doubles, then the rate of consumption ofA doubles.eA plot of Rate versus t islinear with slope equal to +k, where k is the rate constant for therea\nThe reaction, A → products, is first-order. Which of the following statements is false? a If concentration is measured in mol L−1 and time is measured in seconds, then the units of k are s−1. b For a given starting concentration, the first half life of A is equal to the second hal...\n##### The equation ofthe normal line of a curve y=f(x) atthe point (2,1) iS given as 3x+2y-4-0. What is f'(2)? -3/24/3None of the choices2/3-3/4\nThe equation ofthe normal line of a curve y=f(x) atthe point (2,1) iS given as 3x+2y-4-0. What is f'(2)? -3/2 4/3 None of the choices 2/3 -3/4...\n##### Lcoideterminerche Hore done by the variabla torcethe spring Prob Jen_AforcePound: comprate9} 15-Inch bering total 0f 9 Unchan Hori much Voridonecomdressing theWmenmaan\nLcoi determinerche Hore done by the variabla torce the spring Prob Jen_ Aforce Pound: comprate9} 15-Inch bering total 0f 9 Unchan Hori much Vori done comdressing the Wmen maan...\n##### Chapter 1- Home WorkThe mass of raindrop grams?milligrams; Please express the combined mass of six raindrops in\nChapter 1- Home Work The mass of raindrop grams? milligrams; Please express the combined mass of six raindrops in..."
] |
[
null,
"https://cdn.numerade.com/ask_images/eba8425cac4641d1b4776d19d7e1dcd1.jpg ",
null,
"https://cdn.numerade.com/previews/c1981dd7-84af-408d-9990-f2fbdf71c448_large.jpg",
null
] |
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|
https://mathhothouse.me/2014/10/20/references-for-rmo-inmo-and-imo/
|
[
"## References for RMO, INMO and IMO\n\nBelow is a list of basic references for RMO, INMO and IMO:\n\n• Problem Primer for the Olympiad by C R Pranesachar, B J Venkatachala et al.\n• Higher Algebra by Hall and Knight\n• Higher Algebra by Bernard and Child\n• Plane Trigonometry Part I by S L Loney\n• A School Geometry by Hall and Stevens\n• An Introduction to Number Theory by Niven and Zuckermann\n• Elementary Number Theory by David M Burton\n• Problem Solving Strategies by Arthur Engel\n• Problems in Plane Geometry by Sharygin, MIR Publishers.\n• Combinatorics — Schaum Series\n• Mathematical Circles by Fomin et al.\n• An Excursion in Mathematics by M R Modak\n• Selected Problems and Theorems in Elementary Mathematics by Shklyarsky et al.\n• International Mathematical Olympiads, 1959-77, S. L. Greitzer.\n• International Mathematical Olympiads, 1978-85, M. S. Klamkin\n• USA Mathematical Olympiads, 1972-85, M. S. Klamkin\n• Mathematical Challenges for Olympiads, 2nd edition, C R Pranesachar et al.\n• An Excursion in Mathematics — M. R. Modak,\n• College Geometry — Howard Eve.\n• 1000 Mathematical Challenges — J N Kapur\n\nStart cracking …\n\nNalin Pithwa\n\nThis site uses Akismet to reduce spam. Learn how your comment data is processed."
] |
[
null
] |
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|
https://para-lab.ru/JavaDocs/api/java/math/package-summary.html
|
[
"Java™ Platform\nStandard Ed. 8\n\n# Package java.math\n\nProvides classes for performing arbitrary-precision integer arithmetic (`BigInteger`) and arbitrary-precision decimal arithmetic (`BigDecimal`).\n\nSee: Description\n\n• Class Summary\nClass Description\nBigDecimal\nImmutable, arbitrary-precision signed decimal numbers.\nBigInteger\nImmutable arbitrary-precision integers.\nMathContext\nImmutable objects which encapsulate the context settings which describe certain rules for numerical operators, such as those implemented by the `BigDecimal` class.\n• Enum Summary\nEnum Description\nRoundingMode\nSpecifies a rounding behavior for numerical operations capable of discarding precision.\n\n## Package java.math Description\n\nProvides classes for performing arbitrary-precision integer arithmetic (`BigInteger`) and arbitrary-precision decimal arithmetic (`BigDecimal`). `BigInteger` is analogous to the primitive integer types except that it provides arbitrary precision, hence operations on `BigInteger`s do not overflow or lose precision. In addition to standard arithmetic operations, `BigInteger` provides modular arithmetic, GCD calculation, primality testing, prime generation, bit manipulation, and a few other miscellaneous operations. `BigDecimal` provides arbitrary-precision signed decimal numbers suitable for currency calculations and the like. `BigDecimal` gives the user complete control over rounding behavior, allowing the user to choose from a comprehensive set of eight rounding modes.\nSince:\nJDK1.1"
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https://quick-bookkeeping.net/how-to-calculate-straight-line-depreciation/
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[
"# How To Calculate Straight Line Depreciation\n\nContent\n\nWhen crunching numbers in the office, you can record your vessel depreciating \\$21,000 per year over a 10-year period using the straight-line method. To get a better understanding of how to calculate straight-line depreciation, let’s look at a few examples below. QuickBooks Online is the browser-based version of the popular desktop accounting application. It has extensive reporting functions, multi-user plans and an intuitive interface. Product Reviews Unbiased, expert reviews on the best software and banking products for your business.This deduction is fully phased out for businesses acquiring over \\$2,000,000 of such property during the year. In addition, additional first year depreciation of 50% of the cost of most other depreciable tangible personal property is allowed as a deduction. The carrying value would be \\$200 on the balance sheet at the end of three years. The depreciation expense would be completed under the straight line depreciation method, and management would retire the asset.Unlike more complex methodologies, such asdouble declining balance, straight line is simple and uses just three different variables to calculate the amount of depreciation each accounting period. To calculate straight line basis, take the purchase price of an asset and then subtract the salvage value, its estimated sell-on value when it is no longer expected to be needed. Then divide the resulting figure by the total number of years the asset is expected to be useful, referred to as the useful life in accounting jargon.\n\n## Depreciation Methods\n\nThis method is useful for assets that depreciate quickly after purchase, like computers, which lose their value very quickly, even though they might operate well for a long time. For the first year, the double declining balance method takes the depreciation rate from the straight-line method and doubles it. For subsequent years, this method uses the same doubled rate on the remaining balance, instead of being based on the original purchase value.Whether you’re creating a balance sheet to see how your business stands or an income statement to see whether it’s turning a profit, you need to calculate depreciation. Costs of assets consumed in producing goods are treated as cost of goods sold. Other costs of assets consumed in providing services or conducting business are an expense reducing income in the period of consumption under the matching principle.Whereas the depreciable base is the purchase price minus the salvage value. Depreciation continues until the asset value declines to its salvage value. Regardless of the depreciation method used, the total depreciation expense recognized over the life of any asset will be equal. However, the rate at which the depreciation is recognized over the life of the asset is dictated by the depreciation method chosen.\n\n## Disadvantage Of Straight Line Depreciation\n\nThe best time to use the straight-line depreciation method is when you don’t expect an asset to have a specific pattern of use over time. Straight-line depreciation is often the easiest and most straightforward way of calculating depreciation, which means it’ll result in fewer errors.\n\n• One half of a full period’s depreciation is allowed in the acquisition period .\n• Take the purchase price or acquisition cost of an asset, then subtract the salvage value at the time it’s either retired, sold, or otherwise disposed of.\n• The IRS allows businesses to use the straight-line method to write off certain business expenses under the Modified Accelerated Cost Recovery System .\n• Recording depreciation affects both your income statement and your balance sheet.\n\nWe’re here to take the guesswork out of running your own business—for good. Your bookkeeping team imports bank statements, categorizes transactions, and prepares financial statements every month. Check out our guide to Form 4562 for more information on calculating depreciation and amortization for tax purposes. The straight-line method of depreciation assumes a constant rate of depreciation. It calculates how much a specific asset depreciates in one year, and then depreciates the asset by that amount every year after that.\n\n## Final Thoughts On Straight Line Depreciation\n\nDepreciation is technically a method of allocation, not valuation, even though it determines the value placed on the asset in the balance sheet. Take the purchase price or acquisition cost of an asset, then subtract the salvage value at the time it’s either retired, sold, or otherwise disposed of. Now divide this figure by the total product years the asset can reasonably be expected to benefit your company. The straight-line depreciation method is the easiest way of calculating depreciation and is used by accountants to compute the depreciation of long-term assets. However, this depreciation method isn’t always the most accurate, especially if an asset doesn’t have a set pattern of use over time. This means items like computers and tablets often depreciate much quicker in their early useful life while tapering off later on in their useful life.",
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"It is calculated by simply dividing the cost of an asset, less its salvage value, by the useful life of the asset. 10 × actual production will give the depreciation cost of the current year. Depletion and amortization are similar concepts for natural resources and intangible assets, respectively.\n\n## Depreciation Expense: Straight\n\nThese are faster than what management decides to employ on the reported financial statements put together under the Generally Accepted Accounting Principles rules. Management is likely going to take advantage of this because it can increase intrinsic value. Sally recently furnished her new office, purchasing desks, lamps, and tables.",
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"Intuit Inc. does not warrant that the material contained herein will continue to be accurate nor that it is completely free of errors when published. Get the scoop on straight-line depreciation and learn more about the depreciation formula. The method is called “straight line” because the formula, when laid out on a graph, creates a straight, downward trend, with the same rate of loss per year. In our example, the title transfers, which means at the end of the lease term the lessee will own the asset and continue depreciating it. However, the useful life of the equipment in this example equals the lease term so at the end of the lease, the asset will be depreciated to \\$0. Now, let’s consider a full example of a finance lease to illustrate straight-line depreciation expense.That’s why our editorial opinions and reviews are ours alone and aren’t inspired, endorsed, or sponsored by an advertiser. Editorial content from The Blueprint is separate from The Motley Fool editorial content and is created by a different analyst team. Sage 50cloud is a feature-rich accounting platform with tools for sales tracking, reporting, invoicing and payment processing and vendor, customer and employee management.\n\nSome systems specify lives based on classes of property defined by the tax authority. Canada Revenue Agency specifies numerous classes based on the type of property and how it is used. Under the United States depreciation system, the Internal Revenue Service publishes a detailed guide which includes a table of asset lives and the applicable conventions. The table also incorporates specified lives for certain commonly used assets (e.g., office furniture, computers, automobiles) which override the business use lives.It means that the asset will be depreciated faster than with the straight line method. The double-declining balance method results in higher depreciation expenses in the beginning of an asset’s life and lower depreciation expenses later. This method is used with assets that quickly lose value early in their useful life. A company may also choose to go with this method if it offers them tax or cash flow advantages.\n\n## When should I use straight line depreciation?\n\nStraight line depreciation is the default method used to recognize the carrying amount of a fixed asset evenly over its useful life. It is employed when there is no particular pattern to the manner in which an asset is to be utilized over time.When a long-term asset is purchased, it should be capitalized instead of being expensed in the accounting period it is purchased in. An allocation of costs may be required where multiple assets are acquired in a single transaction. Purchase price allocation may be required where assets are acquired as part of a business acquisition or combination. Common sense requires depreciation expense to be equal to total depreciation per year, without first dividing and then multiplying total depreciation per year by the same number. If the vehicle were to be sold and the sales price exceeded the depreciated value then the excess would be considered a gain and subject to depreciation recapture. In addition, this gain above the depreciated value would be recognized as ordinary income by the tax office.Most income tax systems allow a tax deduction for recovery of the cost of assets used in a business or for the production of income. Where the assets are consumed currently, the cost may be deducted currently as an expense or treated as part of cost of goods sold.Using the facts and circumstances presented, we can use LeaseQuery’s present value calculator to calculate the present value of the lease payments. This is the value we will record for the ROU asset and what will be depreciated. In order to do so, input annual payments of \\$100,000, a 10 year lease term, and a 4% discount rate. At commencement, the lessee records a lease asset and lease liability of \\$843,533.\n\n## How To Calculate Straight Line Depreciation Formula\n\nDaniel is an expert in corporate finance and equity investing as well as podcast and video production. Note how the book value of the machine at the end of year 5 is the same as the salvage value. Over the useful life of an asset, the value of an asset should depreciate to its salvage value. Under most systems, a business or income-producing activity may be conducted by individuals or companies. Buildings and leasehold improvements are depreciated over 7 to 40 years. There are generally accepted depreciation estimates for most major asset types that provide some constraint. In the meantime, special adjustments must be made to the reported financial found in the annual report and10-K filing.\n\n## Calculating Straight Line Basis\n\nFor example, the balance sheet would show a \\$5,000 computer offset by a \\$1,600 accumulated depreciation contra account after the first year, so the net carrying value would be \\$3,400. Manufacturing businesses typically use the units of production method."
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http://slideflix.net/doc/1883272/object-recognition--symmetry-detection--jigsaw-puzzles--a..
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[
"",
null,
"# Object Recognition, Symmetry Detection, Jigsaw Puzzles, and Cancer Peter J. Olver\n\nby user\n\non\nCategory: Documents\n5\n\nviews\n\nReport\n\n#### Transcript\n\nObject Recognition, Symmetry Detection, Jigsaw Puzzles, and Cancer Peter J. Olver\n```Object Recognition,\nSymmetry Detection,\nJigsaw Puzzles, and Cancer\nPeter J. Olver\nUniversity of Minnesota\nhttp://www.math.umn.edu/ ∼ olver\nTechnion, May, 2015\nSymmetry\nDefinition. A symmetry of a set S is a\ntransformation that preserves it:\ng·S =S\n! ! The set of symmetries forms a group, called the\nsymmetry group of the set S.\nDiscrete Symmetry Group\nRotations by 90◦:\nGS = Z4\nRotations + reflections:\nGS = Z4 ! Z4\nContinuous Symmetry Group\nRotations:\nGS = SO(2)\nRotations + reflections:\nGS = O(2)\n!\nConformal Inversions:\nx\ny\nx= 2\ny\n=\nx + y2\nx2 + y 2\nA continuous group is known as a Lie group\n— in honor of Sophus Lie.\nContinuous Symmetries of a Square\n−→\nR\n−→\n−→\nSymmetry\n! To define the set of symmetries requires a priori\nspecification of the allowable transformations\nor, equivalently, the underlying geometry.\nG — transformation group or pseudo-group of\nallowable transformations of the ambient\nspace M\nDefinition. A symmetry of a subset S ⊂ M is an\nallowable transformation g ∈ G that preserves it:\ng·S =S\nWhat is the Symmetry Group?\nAllowable transformations:\nRigid motions\nG = SE(2) = SO(2) ! R 2\nGS = Z4 ! Z2\nWhat is the Symmetry Group?\nAllowable transformations:\nRigid motions\nG = SE(2) = SO(2) ! R 2\nGS = {e}\nLocal Symmetries\nDefinition. g ∈ G is a local symmetry of S ⊂ M\nbased at a point z ∈ S if there is an\nopen neighborhood z ∈ U ⊂ M such that\ng · (S ∩ U ) = S ∩ (g · U )\nGz ⊂ G — the set of local symmetries based at z.\nGlobal symmetries are local symmetries at all z ∈ S:\nGS ⊂ Gz\nGS =\n!\nz∈S\nGz\n! ! The set of all local symmetries forms a groupoid!\nGroupoids\nDefinition. A groupoid is a small category such that\nevery morphism has an inverse.\n=⇒ Brandt (quadratic forms), Ehresmann (Lie pseudo-groups)\nMackenzie, R. Brown, A. Weinstein\nGroupoids form the appropriate framework for\nstudying objects with variable symmetry.\nGroupoids\nDouble fibration:\n!\nσ !!\n\"\n!\nG\n#\n# τ\n#\n\\$\n#\nM\nσ\n— source map\nM\nτ\n— target map\n! ! You are only allowed to multiply α · β ∈ G if\nσ(α) = τ (β)\nGroupoids\n• Source and target of products:\nσ(α · β) = σ(β)\nτ (α · β) = τ (α)\nwhen\nσ(α) = τ (β)\n• Associativity:\nα · (β · γ) = (α · β) · γ\n• Identity section:\ne: M → G\nwhen defined\nσ(e(x)) = x = τ (e(x))\nα · e(σ(α)) = α = e(τ (α)) · α\n• Inverses: σ(α) = x = τ (α−1),\nα−1 · α = e(x),\nτ (α) = y = σ(α−1 ),\nα · α−1 = e(y)\nJet Groupoids\n=⇒ Ehresmann\nThe set of Taylor polynomials of degree ≤ n, or\nTaylor series (n = ∞) of local diffeomorphisms\nΨ : M → M forms a groupoid.\n♦ Algebraic composition of Taylor polynomials/series\nis well-defined only when the source of the second\nmatches the target of the first.\nThe Symmetry Groupoid\nDefinition. The symmetry groupoid of S ⊂ M is\nGS = { (g, z) | z ∈ S, g ∈ Gz } ⊂ G × S\nSource and target maps: σ(g, z) = z,\nGroupoid multiplication and inversion:\n(h, g · z) · (g, z) = (g · h, z)\nIdentity map: e(z) = (z, e) ∈ GS\nτ (g, z) = g · z.\n(g, z)−1 = (g −1, g · z)\nWhat is the Symmetry Groupoid?\nG = SE(2)\nCorners:\nGz = GS = Z4\nSides: Gz generated by\nGS = Z4\nsome translations\n180◦ rotation around z\nWhat is the Symmetry Groupoid?\nCogwheels\nGS = Z6\n=⇒ Musso–Nicoldi\nGS = Z2\nWhat is the Symmetry Groupoid?\nCogwheels\nb\na\na\nb\nb\nb\na\na\nb\na\nb\na\n=⇒ Musso–Nicoldi\nb\na\nb\nGS = Z6\nb\na\na\na\na\nb\nb\na\nb\nGS = Z2\nGeometry = Group Theory\nFelix Klein’s Erlanger Programm (1872):\nEach type of geometry is founded on\nan underlying transformation group.\nPlane Geometries/Groups\nEuclidean geometry:\nSE(2) — rigid motions (rotations and translations)\n!\nx\ny\n\"\n=\n!\ncos θ\nsin θ\n− sin θ\ncos θ\n\"!\nx\ny\n\"\n+\n!\na\nb\n\"\nE(2) — plus reflections?\nEqui-affine geometry:\nSA(2) — area-preserving affine transformations:\n!\nx\ny\n\"\n=\n!\nα β\nγ δ\n\"!\nx\ny\n\"\n+\n!\na\nb\n\"\nαδ − βγ = 1\nProjective geometry:\nPSL(3) — projective transformations:\nαx + β y + γ\nλx + µy + ν\nx=\ny=\nρx + σy + τ\nρx + σy + τ\nThe Equivalence Problem\n=⇒ É Cartan\nG — transformation group acting on M\nEquivalence:\nDetermine when two subsets\nS\nand S ⊂ M\nare congruent:\nS =g·S\nfor\nSymmetry:\nFind all symmetries or self-congruences:\nS =g·S\ng∈G\nTennis, Anyone?\nInvariants\nThe solution to an equivalence problem rests on understanding\nits invariants.\nDefinition. If G is a group acting on M , then an invariant is a\nreal-valued function I : M → R that does not change under\nthe action of G:\nI(g · z) = I(z)\n!\nfor all\ng ∈ G,\nz∈M\nIf G acts transtively, there are no (non-constant) invariants.\nDifferential Invariants\nGiven a submanifold (curve, surface, . . . )\nS⊂M\na differential invariant is an invariant of the prolonged action of\nG on its Taylor coefficients (jets):\nI(g · z (k)) = I(z (k) )\nEuclidean Plane Curves\nG = SE(2)\nacts on curves\nC ⊂ M = R2\nThe simplest differential invariant is the curvature, defined as\nthe reciprocal of the radius of the osculating circle:\nκ=\n1\nr\nCurvature\nCurvature\nCurvature\nr = 1/κ\nEuclidean Plane Curves:\nG = SE(2)\nDifferentiation with respect to the Euclidean-invariant arc\nlength element ds is an invariant differential operator,\nmeaning that it maps differential invariants to differential\ninvariants.\nThus, starting with curvature κ, we can generate an infinite\ncollection of higher order Euclidean differential invariants:\nκ,\ndκ\n,\nds\nd2 κ\n,\nds2\nd3κ\n,\nds3\n···\nTheorem. All Euclidean differential invariants are functions of\nthe derivatives of curvature with respect to arc length:\nκ, κs , κss , · · ·\nEuclidean Plane Curves: G = SE(2)\nAssume the curve C ⊂ M is a graph:\ny = u(x)\nDifferential invariants:\nuxx\nκ=\n,\n(1 + u2x)3/2\ndκ\n(1 + u2x)uxxx − 3uxu2xx\n=\n,\nds\n(1 + u2x)3\nArc length (invariant one-form):\nds =\n#\n1 + u2x dx,\nd\n1\nd\n=#\nds\n1 + u2x dx\nd2κ\n= ···\nds2\nEqui-affine Plane Curves: G = SA(2) = SL(2) ! R 2\nEqui-affine curvature:\n5 uxxuxxxx − 3 u2xxx\nκ =\n9 u8/3\nxx\ndκ\n= ···\nds\nEqui-affine arc length:\nds =\n#\n3\nuxx dx\n1\nd\nd\n= √\n3\nds\nuxx dx\nTheorem. All equi-affine differential invariants are functions\nof the derivatives of equi-affine curvature with respect to\nequi-affine arc length: κ, κs , κss , · · ·\nPlane Curves\nTheorem. Let G be an ordinary! Lie group acting on M = R 2 .\nThen for curves C ⊂ M , there exists a unique (up to\nfunctions thereof) lowest order differential invariant κ and a\nunique (up to constant multiple) invariant differential form\nds. Every other differential invariant can be written as a\nfunction of the “curvature” invariant and its derivatives\nwith respect to “arc length”: κ, κs, κss , · · · .\n!\nordinary = transitive + no pseudo-stabilization.\nMoving Frames\nThe equivariant method of moving frames provides a\nsystematic and algorithmic calculus for\ndetermining complete systems of differential\ninvariants, invariant differential forms, invariant\ndifferential operators, etc., and the structure of\nthe non-commutative differential algebra they\ngenerate.\nEquivalence & Invariants\n• Equivalent submanifolds S ≈ S\nmust have the same invariants: I = I.\nConstant invariants provide immediate information:\ne.g.\nκ=2\n⇐⇒\nκ=2\nNon-constant invariants are not useful in isolation,\nbecause an equivalence map can drastically alter the\ndependence on the submanifold parameters:\ne.g.\nκ = x3\nversus\nκ = sinh x\nSyzygies\nHowever, a functional dependency or syzygy among\nthe invariants is intrinsic:\ne.g.\nκs = κ3 − 1\n⇐⇒\nκs = κ3 − 1\n• Universal syzygies — Gauss–Codazzi\n• Distinguishing syzygies.\nTheorem. (Cartan)\nTwo regular submanifolds are locally equivalent if\nand only if they have identical syzygies among all\ntheir differential invariants.\nFiniteness of Generators and Syzygies\n♠ There are, in general, an infinite number of\ndifferential invariants and hence an infinite\nnumber of syzygies must be compared to\nestablish equivalence.\n♥ But the higher order differential invariants are\nalways generated by invariant differentiation\nfrom a finite collection of basic differential\ninvariants, and the higher order syzygies are\nall consequences of a finite number of low\norder syzygies!\nExample — Plane Curves\nIf non-constant, both κ and κs depend on a single\nparameter, and so, locally, are subject to a syzygy:\nκs = H(κ)\n(∗)\nBut then\nd\nH(κ) = H \\$ (κ) κs = H \\$ (κ) H(κ)\nds\nand similarly for κsss , etc.\nκss =\nConsequently, all the higher order syzygies are generated\nby the fundamental first order syzygy (∗).\nThus, for Euclidean (or equi-affine or projective or . . . )\nplane curves we need only know a single syzygy between κ and\nκs in order to establish equivalence!\nSignature Curves\nDefinition. The signature curve Σ ⊂ R2 of a plane curve\nC ⊂ R2 is parametrized by the two lowest order differential\ninvariants\nχ : C −→ Σ =\n\\$!\nκ,\ndκ\nds\n\"%\n⊂ R2\n=⇒ Calabi, PJO, Shakiban, Tannenbaum, Haker\nTheorem. Two regular curves C and C are locally\nequivalent:\nC =g·C\nif and only if their signature curves are identical:\nΣ=Σ\n=⇒ regular: (κs, κss ) 2= 0.\nContinuous Symmetries of Curves\nTheorem. For a connected curve, the following are\nequivalent:\n• All the differential invariants are constant on C:\nκ = c, κs = 0,\n...\n• The signature Σ degenerates to a point: dim Σ = 0\n• C is a piece of an orbit of a 1-dimensional subgroup H ⊂ G\n• C admits a one-dimensional local symmetry group\nDiscrete Symmetries of Curves\nDefinition. The index of a completely regular point ζ ∈ Σ\nequals the number of points in C which map to it:\niζ = # χ−1 {ζ}\nRegular means that, in a neighborhood of ζ, the signature is an\nembedded curve — no self-intersections.\nTheorem. If χ(z) = ζ is completely regular, then its index\ncounts the number of discrete local symmetries of C.\nThe Index\nχ\n−→\nC\nΣ\n1\nsin2 t\nThe Curve x = cos t + 15 cos2 t, y = sin t + 10\n2\n1\n2\n1\n0.5\n0\n-0.5\n4\n0.5\n0.5\n0.25\n0.5\n0.75\n1\n1.25\n1.5\n1.75\n1\n1.5\n2\n2.5\n2\n1\n-2\n-1\n-0.5\n-4\n-2\n-6\nThe Original Curve\nEuclidean Signature\nEqui-affine Signature\n1\nsin2 t\nThe Curve x = cos t + 15 cos2 t, y = 21 x + sin t + 10\n4\n7.5\n1\n5\n0.5\n0\n-0.5\n2\n2.5\n0.5\n0.5\n0.5\n1\n1.5\n2\n2.5\n3\n3.5\n1\n1.5\n2\n2.5\n4\n1\n-2.5\n-0.5\n-2\n-5\n-4\n-7.5\n-1\nThe Original Curve\n-6\nEuclidean Signature\nEqui-affine Signature\n&'()*\n&'()5\n\\$\"\"\n%#\"\n##\"\n%\"\"\n#\"\"\n\\$#\"\n36782/2889)\"+*:%\\$%:\n!#\"\n!\"\"\n#\"\"\n!\"\"\n,-./0('12)3'142)&'()*\n#\"\"\n,-./0('12)3'142)&'()5\n\"+\"*\n\"+\"*\n\"+\"*\n\"+\"\"#\n\"+\"\"#\n\"+\"\"#\n\"\n\"\n\"\n!\"+\"\"#\n!\"+\"\"#\n!\"+\"\"#\n!\"+\"*\n!\"+\"*\n!\"+\"*\n!\"+\"*#\n!\"+\"#\n\"\n\"+\"#\n!\"+\"*#\n\"+*\n!\"+\"#\n\"\n\"+\"#\n!\"+\"*#\n\"+*\n!\"+\"#\n\"\n\"+\"#\n\"+*\n'(()*&\n:32*&\n#,\"\n&\"\"\"\n#\"\"\n%\"\"\n9,\"\n\\$\"\"\n6;(<505<<=*\"+\">&!&#\n#\"\"\n!\"\"\n8\"\"\n-./012345*63475*'(()*&\n,\"\"\n-./012345*63475*:32*&\n\"+\"&\n\"+\"&\n\"+\"&\n\"+\"\",\n\"+\"\",\n\"+\"\",\n\"\n\"\n\"\n!\"+\"\",\n!\"+\"\",\n!\"+\"\",\n!\"+\"&\n!\"+\"&\n!\"+\"&\n!\"+\"&,\n!\"+\",\n\"\n\"+\",\n!\"+\"&,\n\"+&\n!\"+\",\n\"\n\"+\",\n!\"+\"&,\n\"+&\n!\"+\",\n\"\n\"+\",\n\"+&\nSignatures\n−→\nκ\ns\nClassical Signature\nκs\nOriginal curve\nκ\nDifferential invariant signature\nSignatures\n−→\nκ\ns\nClassical Signature\nκs\nOriginal curve\nκ\nDifferential invariant signature\nκ\nOcclusions\n−→\ns\nClassical Signature\nκs\nOriginal curve\nκ\nDifferential invariant signature\n3D Differential Invariant Signatures\nEuclidean space curves: C ⊂ R 3\nΣ = { ( κ , κs , τ ) } ⊂ R3\n• κ — curvature, τ — torsion\nEuclidean surfaces:\nS ⊂ R 3 (generic)\n&'\n()\nΣ=\nH , K , H,1 , H,2 , K,1 , K,2\n⊂ R6\nor\n*\nΣ\n&'\n()\n=\nH , H,1 , H,2 , H,11\n⊂ R4\n• H — mean curvature, K — Gauss curvature\n3\nEqui–affine surfaces:\nS\n⊂\nR\n(generic)\n&'\n()\nΣ=\nP , P,1 , P,2, P,11\n⊂ R4\n• P — Pick invariant\nVertices of Euclidean Curves\nOrdinary vertex: local extremum of curvature\nGeneralized vertex: κs ≡ 0\n• critical point\n• circular arc\n• straight line segment\nMukhopadhya’s Four Vertex Theorem:\nA simple closed, non-circular plane curve has n ≥ 4 generalized\nvertices.\n“Counterexamples”\n!\nGeneralized vertices map to a single point of the signature.\nHence, the (degenerate) curves obtained by replace ordinary\nvertices with circular arcs of the same radius all have identical\nsignature:\n2\n2\n!2\n4\n6\n2\n2\n!2\n4\n6\n8\n!2\n2\n!2\n4\n6\n8\n10\n!2\n!2\n!4\n!4\n!4\n!6\n!6\n!6\n!8\n!8\n!8\n4\n4\n4\n2\n2\n2\n2\n!2\n4\n6\n8\n10\n2\n!2\n4\n6\n8\n10\n2\n!2\n!2\n4\n6\n8\n!2\n!2\n!4\n!4\n!4\n!6\n!6\n=⇒ Musso–Nicoldi\nBivertex Arcs\nBivertex arc: κs 2= 0 everywhere on the arc B ⊂ C\nexcept κs = 0 at the two endpoints\nThe signature Σ = χ(B) of a bivertex arc is a single arc that\nstarts and ends on the κ–axis.\nκs\nκ\nBivertex Decomposition\nv-regular curve — finitely many generalized vertices\nC=\nm\n\"\nj=1\nBj ∪\nB1, . . . , Bm\n— bivertex arcs\nV1 , . . . , Vn\n—\nn\n\"\nk=1\ngeneralized vertices:\nVk\nn≥4\nMain Idea: Compare individual bivertex arcs, and then decide\nwhether the rigid equivalences are (approximately) the same.\nD. Hoff & PJO, Extensions of invariant signatures for object recognition,\nJ. Math. Imaging Vision 45 (2013), 176–185.\nSignature Metrics\nUsed to compare signatures:\n• Hausdorff\n• Monge–Kantorovich transport\n• Electrostatic/gravitational attraction\n• Latent semantic analysis\n• Histograms\n• Geodesic distance\n• Diffusion metric\n• Gromov–Hausdorff & Gromov–Wasserstein\nGravitational/Electrostatic Attraction\n♥ Treat the two signature curves as masses or as oppositely\ncharged wires. The higher their mutual attraction, the\ncloser they are together.\nκs\nκ\nGravitational/Electrostatic Attraction\n♥ Treat the two signature curves as masses or as oppositely\ncharged wires. The higher their mutual attraction, the\ncloser they are together.\n♠ In practice, we are dealing with discrete data (pixels) and so\ntreat the curves and signatures as point masses/charges.\nκs\nκs\nκ\nκ\nThe Baffler Jigsaw Puzzle\nPiece Locking\n! ! Minimize force and torque based on gravitational\nattraction of the two matching edges.\nThe Baffler Solved\nThe Rain Forest Giant Floor Puzzle\nThe Rain Forest Puzzle Solved\n=⇒ D. Hoff & PJO, Automatic solution of jigsaw puzzles,\nJ. Math. Imaging Vision 49 (2014) 234–250.\n3D Jigsaw Puzzles\n=⇒ Anna Grim, Tim O’Connor, Ryan Schlecta\nBroken Ostrich Egg Shell\n=⇒ Marshall Bern\nReassembling Humpty Dumpty\nBenign vs. Malignant Tumors\n=⇒ A. Grim, C. Shakiban\nBenign vs. Malignant Tumors\nBenign vs. Malignant Tumors\nJoint Invariant Signatures\nIf the invariants depend on k points on a p-dimensional\nsubmanifold, then you need at least\n/ > kp\ndistinct invariants I1 , . . . , I# in order to construct a syzygy.\nTypically, the number of joint invariants is\n/ = k m − r = (#points) (dim M ) − dim G\nTherefore, a purely joint invariant signature requires at least\nr\n+1\nk ≥\nm−p\npoints on our p-dimensional submanifold N ⊂ M .\nJoint Euclidean Signature\nz0\na\nb\nz1\ne\nc\nd\nf\nz3\nz2\nJoint signature map:\na = 6 z0 − z1 6\nΣ : C ×4 −→ Σ ⊂ R6\nd = 6 z1 − z2 6\nb = 6 z0 − z2 6\nc = 6 z0 − z3 6\ne = 6 z1 − z3 6\nf = 6 z2 − z3 6\n=⇒ six functions of four variables\nSyzygies:\nΦ1(a, b, c, d, e, f ) = 0\nUniversal Cayley–Menger syzygy\n+\n+\n2 a2\n+\ndet +++ a2 + b2 − d2\n+ a 2 + c2 − e 2\nΦ2(a, b, c, d, e, f ) = 0\n⇐⇒\na2 + b2 − d2\n2 b2\nb 2 + c2 − f 2\nC ⊂ R2\n+\na2 + c2 − e2 ++\nb2 + c2 − f 2 +++ = 0\n+\n2 c2\nJoint Equi–Affine Signature\nRequires 7 triangular areas:\n[ 0 1 2 ], [ 0 1 3 ], [ 0 1 4 ], [ 0 1 5 ], [ 0 2 3 ], [ 0 2 4 ], [ 0 2 5 ]\nz1\nz2\nz0\nz3\nz5\nz4\nJoint Invariant Signatures\n• The joint invariant signature subsumes other signatures, but\nresides in a higher dimensional space and contains a lot of\nredundant information.\n•\nIdentification of landmarks can significantly reduce the\nredundancies (Boutin)\n•\nIt includes the differential invariant signature and semidifferential invariant signatures as its “coalescent boundaries”.\n•\nInvariant numerical approximations to differential invariants\nand semi-differential invariants are constructed (using\nmoving frames) near these coalescent boundaries.\nStatistical Sampling\nIdea: Replace high dimensional joint invariant signatures by\nincreasingly dense point clouds obtained by multiply\nsampling the original submanifold.\n• The equivalence problem requires direct comparison of\nsignature point clouds.\n• Continuous symmetry detection relies on determining the\nunderlying dimension of the signature point clouds.\n• Discrete symmetry detection relies on determining densities of\nthe signature point clouds.\nInvariant Histograms\n!\nTo eliminate noise, use histograms based on joint invariants.\nDefinition. The distance histogram of a finite set of points\nP = {z1, . . . , zn} ⊂ V is the function\n&\nηP (r) = # (i, j)\n+\n+\n+\n)\n1 ≤ i < j ≤ n, d(zi, zj ) = r .\nBrinkman, D., & PJO, Invariant histograms, Amer. Math. Monthly 118\n(2011) 2–24.\nThe Distance Set\nThe support of the histogram function,\nsupp ηP = ∆P ⊂ R+\nis the distance set of P .\nErdös’ distinct distances conjecture (1946):\nIf P ⊂ R m, then # ∆P ≥ cm,ε (# P )2/m−ε\nCharacterization of Point Sets\nNote: If P, = g · P is obtained from P ⊂ R m by a rigid motion\ng ∈ E(n), then they have the same distance histogram:\nηP = ηP, .\nQuestion: Can one uniquely characterize, up to rigid motion, a\nset of points P {z1, . . . , zn } ⊂ R m by its distance histogram?\n=⇒ Tinkertoy problem.\nYes:\nη = 1, 1, 1, 1,\n√\n√\n2, 2.\nNo:\nKite\nη=\nTrapezoid\n√\n2,\n√\n√\n√\n2, 2, 10, 10, 4.\nNo:\nP = { 0, 1, 4, 10, 12, 17 }\nQ = { 0, 1, 8, 11, 13, 17 }\n⊂ R\nη = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17\n=⇒ G. Bloom, J. Comb. Theory, Ser. A 22 (1977) 378–379\nTheorem. (Boutin–Kemper ) Suppose n ≤ 3 or n ≥ m + 2.\nThen there is a Zariski dense open subset in the space of n\npoint configurations in R m that are uniquely characterized,\nup to rigid motion, by their distance histograms.\n=⇒ M. Boutin, G. Kemper, Adv. Appl. Math. 32 (2004) 709–735\nDistinguishing Melanomas from Moles\nMelanoma\nMole\n=⇒ A. Rodriguez, J. Stangl, C. Shakiban\nCumulative Global Histograms\n1.0\n0.8\n0.6\n0.4\n0.2\n200\n400\nRed: melanoma\n600\n800\n1000\nGreen: mole\nLogistic Function Fitting\nMelanoma\nMole\nLogistic Function Fitting — Residuals\n.0\n2.5\n2.0\n1.5\n1.0\n0.5\n\nMelanoma = 17.1336 ± 1.02253 \n\nMole = 19.5819 ± 1.42892\n\n\n58.7% Confidence\nLimiting Curve Histogram\nLimiting Curve Histogram\nLimiting Curve Histogram\nSample Point Histograms\nCumulative distance histogram: n = #P :\n+\n)\n&\n1\n2 1\n1\n+\nΛP (r) = + 2\nη (s) = 2 # (i, j) + d(zi, zj ) ≤ r ,\nn n s≤r P\nn\nNote\nη(r) = 21 n2[ ΛP (r) − ΛP (r − δ) ]\nδ 7 1.\nLocal distance histogram:\n& +\n)\n1\n1\n+\nλP (r, z) = # j + d(z, zj ) ≤ r = #(P ∩ Br (z))\nn\nn\nBall of radius r centered at z:\nBr (z) = { v ∈ V | d(v, z) ≤ r }\nNote:\nΛP (r) =\n1 1\n1 1\nλP (r, z) = 2\n#(P ∩ Br (z)).\nn z∈P\nn z∈P\nLimiting Curve Histogram Functions\nLength of a curve\nl(C) =\n2\nC\nds < ∞\nLocal curve distance histogram function\nhC (r, z) =\nz∈V\nl(C ∩ Br (z))\nl(C)\n=⇒ The fraction of the curve contained in the ball of radius r\ncentered at z.\nGlobal curve distance histogram function:\n1 2\nHC (r) =\nh (r, z(s)) ds.\nl(C) C C\nConvergence\nTheorem. Let C be a regular plane curve. Then, for both\nuniformly spaced and randomly chosen sample points P ⊂ C,\nthe cumulative local and global histograms converge to their\ncontinuous counterparts:\nλP (r, z) −→ hC (r, z),\nΛP (r) −→ HC (r),\nas the number of sample points goes to infinity.\nSquare Curve Histogram with Bounds\nKite and Trapezoid Curve Histograms\nHistogram–Based Shape Recognition\n500 sample points\nShape\n(a) triangle\n(a)\n2.3\n(b)\n20.4\n(c)\n66.9\n(d)\n81.0\n(e)\n28.5\n(f )\n76.8\n(b) square\n28.2\n.5\n81.2\n73.6\n34.8\n72.1\n(c) circle\n66.9\n79.6\n.5\n137.0\n89.2\n138.0\n(d) 2 × 3 rectangle\n85.8\n75.9\n141.0\n2.2\n53.4\n9.9\n(e) 1 × 3 rectangle\n31.8\n36.7\n83.7\n55.7\n4.0\n46.5\n(f) star\n81.0\n74.3\n139.0\n9.3\n60.5\n.9\nCurve Histogram Conjecture\n,\nTwo sufficiently regular plane curves C and C\nhave identical global distance histogram functions, so\nHC (r) = HC, (r) for all r ≥ 0, if and only if they are\n,\nrigidly equivalent: C 8 C.\n“Proof Strategies”\n• Show that any polygon obtained from (densely) discretizing a\ncurve does not lie in the Boutin–Kemper exceptional set.\n• Polygons with obtuse angles: taking r small, one can recover\n(i) the set of angles and (ii) the shortest side length from\nHC (r). Further increasing r leads to further geometric\ninformation about the polygon . . .\n• Expand HC (r) in a Taylor series at r = 0 and show that the\ncorresponding integral invariants characterize the curve.\nTaylor Expansions\nLocal distance histogram function:\nL hC (r, z) = 2 r +\n1 2 3\n12 κ r\n+\n'\n1\n40 κ κss\n+\n1 2\n45 κs\n+\n4\n3\n320 κ\n(\nr5 + · · · .\nGlobal distance histogram function:\n2r\nr3 3\nr5 3 ' 3 4 1 2 (\n2\nHC (r) =\n+\nκ ds +\nκ − 9 κs ds + · · · .\nL\n12 L2 C\n40 L2 C 8\nSpace Curves\nz(t) = (cos t, sin t, cos 2 t),\n0 ≤ t ≤ 2 π.\nConvergence of global curve distance histogram function:\nSurfaces\nLocal and global surface distance histogram functions:\narea (S ∩ Br (z))\n,\nhS (r, z) =\narea (S)\nConvergence for sphere:\n22\n1\nHS (r) =\nh (r, z) dS.\narea (S) S S\nArea Histograms\nRewrite global curve distance histogram function:\n13\n1 3 3\nHC (r) =\nhC (r, z(s)) ds = 2\nχr (d(z(s), z(s\\$)) ds ds\\$\nL C\nL C\\$ C\n1, t ≤ r,\nwhere\nχr (t) =\n0, t > r,\nGlobal curve area histogram function\n1 3 3 3\nχr (area (z(s*), z(s*\\$ ), z(s*\\$\\$)) d s* d s*\\$ d s*\\$\\$ ,\nAC (r) = 3\nL C C C\nd s* — equi-affine arc length element\nL=\n2\nDiscrete cumulative area histogram\n1\n1\nχr (area (z, z \\$ , z \\$\\$)),\nAP (r) =\nn(n − 1)(n − 2) z'=z!'=z!!∈P\nBoutin & Kemper: the area histogram uniquely determines\ngeneric point sets P ⊂ R 2 up to equi-affine motion\nC\nd s*\nArea Histogram for Circle\n!!\nJoint invariant histograms — convergence???\nTriangle Distance Histograms\nZ = (. . . zi . . .) ⊂ M — sample points on a subset M ⊂ R n\n(curve, surface, etc.)\nTi,j,k\n—\ntriangle with vertices zi, zj , zk .\nSide lengths:\nσ(Ti,j,k ) = ( d(zi, zj ), d(zi, zk ), d(zj , zk ) )\nDiscrete triangle histogram:\nS = σ(T ) ⊂ K\nTriangle inequality cone\nK = { (x, y, z) | x, y, z ≥ 0, x + y ≥ z, x + z ≥ y, y + z ≥ x } ⊂ R 3.\nTriangle Histogram Distributions\nCircle\nTriangle\nSquare"
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https://www.hindawi.com/journals/mpe/2021/5150933/
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"Special Issue\n\n## Generalised Fuzzy Models Applied to Logical Algebras and Intelligent Systems in Engineering\n\nView this Special Issue\n\nResearch Article | Open Access\n\nVolume 2021 |Article ID 5150933 | https://doi.org/10.1155/2021/5150933\n\nGulfam Shahzadi, Fariha Zafar, Maha Abdullah Alghamdi, \"Multiple-Attribute Decision-Making Using Fermatean Fuzzy Hamacher Interactive Geometric Operators\", Mathematical Problems in Engineering, vol. 2021, Article ID 5150933, 20 pages, 2021. https://doi.org/10.1155/2021/5150933\n\n# Multiple-Attribute Decision-Making Using Fermatean Fuzzy Hamacher Interactive Geometric Operators\n\nRevised31 May 2021\nAccepted08 Jun 2021\nPublished25 Jun 2021\n\n#### Abstract\n\nFermatean fuzzy set (FFS) is a more efficient, flexible, and generalized model to deal with uncertainty, as compared to intuitionistic and Pythagorean fuzzy models. This research article presents a novel multiple-attribute decision-making (MADM) technique based on FFS. Aggregation operators (AOs), for example, Dombi, Einstein, and Hamacher, are frequently being used in the MADM process and are considered useful tools for evaluating the given alternatives. Among these, one of the most effective is the Hamacher operator. The salient feature of this operator is that it reduces the impact of negative information and provides more accurate results. Motivated by the primary characteristics of the Hamacher operator, we apply Hamacher interactive aggregation operators based on FFSs to determine the best alternative. Using Hamacher’s norm operations, we introduce some new geometric operators, namely, Fermatean fuzzy Hamacher interactive weighted geometric (FFHIWG) operator, Fermatean fuzzy Hamacher interactive ordered weighted geometric (FFHIOWG) operator, and Fermatean fuzzy Hamacher interactive hybrid weighted geometric (FFHIHWG) operator. Some important results and properties of the proposed AOs are discussed, and to achieve the optimal alternative, the proposed MADM technique is carried out in a real-life application of the medical field. An algorithm of the proposed technique is also developed. The significance of the proposed method is that Fermatean fuzzy Hamacher interactive geometric (FFHIG) operators deal with the relationship among belongingness degree (BD) and nonbelongingness degree (NBD) of the objects, which perform a crucial role in decision-making (DM). At last, to show the exactness and validity of the proposed work, a comparative analysis of the proposed model and the existing models is presented.\n\n#### 1. Introduction\n\nAmbiguous or uncertain information is one of the greatest dilemmas dealing with the MADM process. The uncertain information can be captured in different ways. In the last few years, Zadeh’s fuzzy set theory (FST) and its extensions, i.e., intuitionistic fuzzy set theory (IFST) , Pythagorean fuzzy set theory (PFST) , hesitant fuzzy set theory (HFST) [4, 5], and interval-valued fuzzy set theory (IVFST) , have been proved to be efficient tools handling uncertainty in numerous applications of MADM. However, there are some restrictions involved in all these theories, for example, FST deals with belongingness degree only, whereas IFST deals with both BD and NBD but it restricts their sum to be less or equal to 1. To overcome this issue, PFST replaces the condition of the sum to “the sum of squares of BD and NBD to be less or equal to 1.” Recently, a more generalized theory, namely, Fermatean fuzzy set theory (FFST), was introduced by Senapati and Yager . The notion of FFS was initiated from IFSs and PFSs, where the sum of cubes of NBD and BD is less than or equal to one. Therefore, FFSs are more flexible and generalized as compared to both IFSs and PFSs.\n\nAggregation methods based on IFSs and PFSs were widely used in MADM. Xu and Zhao et al. defined aggregation operators (AOs) based on IFSs. Wei and Lu developed Pythagorean fuzzy (PF) power AOs and used them in DM problems. Ordered weighted averaging aggregation operators (OWAAOs) for MADM were defined by Yager . Many multiple-attribute decision-making models (MADMMs) use algebraic operators and these are Dombi, Einstein, and Hamacher operators. In recent years, many theories were developed based on these operators. Dombi defined Dombi triangular norm and conorm operators. Many authors contributed their work to Dombi AOs. Akram et al. worked on Pythagorean Dombi fuzzy AOs (PFDAOs). Wei introduced interaction AOs based on PFSs and also applied these to MADMMs.\n\nHamacher AOs were introduced in 1978 . Wei defined Hamacher AOs based on PFSs and gave a comparative analysis for MADM. Garg [17, 18] presented some series of IF interactive averaging AOs by applying interactive averaging AOs on IFSs and also gave the idea of IF Hamacher AOs having entropy weight. Wu and Wei presented MADMMs based on PF Hamacher AOs (PFHAOs). Wei introduced the PF interaction AOs (PFIAOs) with their application to MADM. Waseem et al. discussed MADM based on -polar fuzzy Hamacher AOs (mFHAOs). Zhao and Wei gave the idea of IF Einstein hybrid AOs (IFEHAOs). Wang and Liu elaborated IF information AOs using Einstein operations. The idea for assessment of express service quality with entropy weight was explained by Wang et al. under PF interactive Hamacher power AO.\n\nSenapati and Yager introduced Fermatean fuzzy averaging/geometric operators (FFAOS/FFFOs). They also defined operations over Fermatean fuzzy numbers (FFNs) . Garg et al. presented a method for the most suitable laboratory selection for COVID-19 test under Fermatean fuzzy environment (FFE). Akram et al. discussed a MADMM to show the benefits of a sanitizer in COVID-19 under FFE. Shahzadi and Akram proposed the idea of Fermatean fuzzy soft AOs and applied this idea in the field of group decision-making for the selection of an antivirus mask. Recently, Aydemir and Gunduz explained the Fermatean fuzzy TOPSIS (FF-TOPSIS) method consisting of Dombi AOs. Shahzadi et al. introduced the idea of Hamacher interactive hybrid weighted averaging operators under FFE. Feng et al. investigated membership grades of rung orthopair fuzzy sets geometrically. For more comprehension and understanding, the readers are referred to study [24, 25, 3235].\n\nThe motivations of this study are defined as follows:(i)The proposed Hamacher interactive AOs (HIAOs) deal with the relationship between the BD and NBD of an object(ii)The MADMM based on FFSs shows that the change in NBDs will definitely affect the BDs of the objects(iii)The proposed Fermatean fuzzy AOs generalize the BDs and NBDs of the objects; i.e., greater values of belongingness and nonbelongingness degrees can be taken as compared to IFST and PFST(iv)The HIAOs are a much convenient approach to cope with the issues in the DM process; this article aims to define HIAOs based on FFSs to handle uncertainty associated with the choice of alternatives in MADMMs(v)Hamacher interactive AOs give more precise and exact choice values in decision results when applied to MADMMs\n\nThe contributions of this article are outlined as follows:(i)Some new HIAOs such as FFHIWG, FFHIOWG, and FFHIHWG are proposed here(ii)The attractive properties alongside their special cases are discussed which reduce the loopholes in the existing operators(iii)An algorithm for MADM using the proposed operators is described and an application is presented to show the applicability of the intended method in the real world(iv)A comparison is also presented which shows the innovation and importance of the contemplated model\n\nThe remaining part of the article is arranged as follows. In Section 2, some elementary notions are presented. Section 3 explains a hybrid structure of Hamacher interactive operators based on FFSs such as FFHIWG operators with a few important results and basic properties, for example, boundedness, homogeneity, idempotency, monotonicity, and shift invariance. In Section 4, the basic concept and results of the FFHIOWG operator are presented. Section 5 presents the notions related to the FFHIHWG operator. In Section 6, a MADMM under Fermatean fuzzy environment is explained through a real-life application. In Section 7, the influence of distinct values of the parameter is shown. In Section 8, a comparative analysis with existing theories is discussed which shows the efficacy and importance of the intended model. In Section 9, the presented work is summarized with concluding remarks.\n\n#### 2. Preliminaries\n\nDefinition 1. (see ). A Fermatean fuzzy set (FFS) on (a nonempty crisp set) is defined aswhere , and indicate belongingness degree, nonbelongingness degree, and indeterminacy degree, respectively.\n\nDefinition 2. (see ). The score function (SF) and accuracy function (AF) for a FFS are given by\n\nDefinition 3. (see ). Consider two FFSs and . Then,(1)If , then .(2)If , then .(3)If , then(a)If , then .(b)If , then .(c)If , then .\n\nDefinition 4. (see ). Hamacher t-norm and t-conorm are defined by(i)For , these operations become algebraic t-norm, , and algebraic t-conorm, (ii)For , these operations become Einstein t-norm, , and Einstein t-conorm,\n\nDefinition 5. Let , , and be three FFSs and ; then,(i)(ii)(iii)(iv)\n\n#### 3. Fermatean Fuzzy Hamacher Interactive Weighted Geometric Operators\n\nIn this section, we introduce the Fermatean fuzzy Hamacher interactive weighted geometric operator (FFHIWGO) and describe its some important characteristics.\n\nDefinition 6. Let be a family of FFSs and be its weight vector (WV) such that and , then is defined as\n\nTheorem 1. Let be a collection of FFSs; then,\n\nProof. For ,Thus, the result is true for . Suppose that result holds for , i.e.,Now, for ,The result holds, .\n\nRemark 1. Here are cases of the FFHIWGO.(i)For , FFHIWGO becomes Fermatean fuzzy interactive weighted geometric operator (FFIWGO):(ii)For , FFHIWG operator becomes Fermatean fuzzy Einstein interactive weighted geometric operator (FFEIWGO):\n\nTheorem 2. The clumped value of FFSs , by using FFHIWGO, is a FFS, i.e.,\n\nProof. As are FFSs, , and . Therefore,Also, . Therefore,Thus, .\nMoreover,Also,Thus, .\n\nProperty 1. (idempotency). If , then\n\nProof. Since and , by Theorem 1,\n\nProperty 2. (boundedness). Let and ; then,\n\nProof. Let ; then , so is a decreasing function (DF). As , then ; that is, . Let and ; we haveThus,Consider , then ; i.e., is a DF on . Since , then ; that is, . Then,Also,Let FFHIWG; then, from inequalities (20) and (22), where , , , and . So, and . As and ,\n\nProperty 3. (monotonicity). If , then\n\nProof. It can be proved on similar lines to the above.\n\nProperty 4. (shift iInvariance). If is another FFS, then\n\nProof. As FFSs, soTherefore,\n\nProperty 5. (homogeneity). Let , then\n\nProof. Since are FFSs and , thereforeTherefore,"
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{"ft_lang_label":"__label__en","ft_lang_prob":0.8896142,"math_prob":0.9153027,"size":10644,"snap":"2022-27-2022-33","text_gpt3_token_len":2617,"char_repetition_ratio":0.14417294,"word_repetition_ratio":0.01339014,"special_character_ratio":0.21984217,"punctuation_ratio":0.14916286,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9904836,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-06-29T09:39:55Z\",\"WARC-Record-ID\":\"<urn:uuid:9b68cd9d-49ff-4d2b-9b5f-219dd1887aba>\",\"Content-Length\":\"1049317\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:640e902b-52e1-49d4-94ca-0a248667aa14>\",\"WARC-Concurrent-To\":\"<urn:uuid:74ae42c8-9dd0-4e17-b609-23b0fbf44dba>\",\"WARC-IP-Address\":\"13.249.39.98\",\"WARC-Target-URI\":\"https://www.hindawi.com/journals/mpe/2021/5150933/\",\"WARC-Payload-Digest\":\"sha1:BNNGXR5ZGOZWWMYVT32XCJHPT5I3IYBM\",\"WARC-Block-Digest\":\"sha1:7L65LYZVBESVZZHCAJDK7LCFYOOP3GHB\",\"WARC-Truncated\":\"length\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656103626162.35_warc_CC-MAIN-20220629084939-20220629114939-00454.warc.gz\"}"}
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https://getcalc.com/statistics-hypothesis-calculator.htm
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[
"Hypothesis\n\nSignificance Level :\n\nSignificance Level :\n\nTail :\n\nZ-critical (ze) :\n\nDegrees of Freedom (n) :\n\nZ-statistic Z0 :\n\nZ-critical Ze :\n\nNotes\n1. Insert this widget code anywhere inside the body tag\n2. Use the code as it is for proper working.\nshare\nfeedback\ncalculator\ninfo\nhistory\n</>",
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"HISTORY\n\n# Hypothesis (H0) for Z, t, F & χ² Test Calculator\n\ngetcalc.com's Hypothesis (H0) for Z, t, F & χ² Test calculator is an online statistics & probability tool to check if the statistic about population parameter is statistically significant in statistical surveys & experiments. The test of hypothesis for Z-test, Student's t-test, F-test & χ²-test at a stated confidence level (99%, 98%, 97%, 96% 95%, etc or 0.01, 0.02, 0.03, 0.04, 0.05, etc) for single or two tailed distribution can easily be carried out by using this significance test calculator.\n\n## Test of Hypothesis for All Distributions\n\nTest of Hypothesis is the technique used in probability & statistics to check if the significance of estimated population parameters by analyzing the samples of population is accepted in statistical experiments. The test of hypothesis in experiments classified as null hypothesis (H0) and alternative hypothesis (H1) popularly used to analyze one or two tailed normal distribution, t-distribution, F-distribution & Chi-squared distribution. In hypothesis testing, the calculated value of Z-statistic (Z0), Student's t-statistic (t0), F-statistic (F0) or χ²-statistic (χ²0) is compared with the table (critical) values of one or two tailed normal distribution (Ze), t-distribution (te), F-distribution (Fe) or Chi-squared distribution (χ²e) to check if the results of experiments are statistically significant. The conclusion based on the reasoning is called as Inference in the test of significance.\n\nInference :\nFor null hypothesis (H0) :\nThe null hypothesis is accepted, if\nZ0 < Ze, t0 < te, F0 < Fe or χ²0 < χ²e\nThe above inference statement evident that there is no significant difference between the sample variances.\nThe null hypothesis (H0) is rejected if\nZ0 > Ze, t0 > te, F0 > Fe or χ²0 > χ²e\nThe above inference statement evident that there is significant difference between the sample variances.",
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[
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"https://getcalc.com/graphics/loading.gif",
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"https://getcalc.com/cdn/graphics/getcalc-logo.png",
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https://www.geeksforgeeks.org/print-rectangular-pattern-with-given-center/?ref=rp
|
[
"Related Articles\n\n# Print rectangular pattern with given center\n\n• Last Updated : 23 Jul, 2021\n\nGiven 3 positive integer c1, c2 and n, where n is size of 2-D square matrix. The task is to print the matrix filled with rectangular pattern having center coordinates c1, c2 such that 0 <= c1, c2 < n.\n\nExamples:\n\nInput: c1 = 2, c2 = 2, n = 5\nOutput:\n2 2 2 2 2\n2 1 1 1 2\n2 1 0 1 2\n2 1 1 1 2\n2 2 2 2 2\n\nInput: c1 = 3, c2 = 4, n = 7\nOutput:\n4 3 3 3 3 3 3\n4 3 2 2 2 2 2\n4 3 2 1 1 1 2\n4 3 2 1 0 1 2\n4 3 2 1 1 1 2\n4 3 2 2 2 2 2\n4 3 3 3 3 3 3\n\nApproach: This problem can be solved by using two nested loops. Follow the steps below to solve this problem:\n\n• Iterate in the range[0, N-1], using a variable i and do the following steps:\n• Iterate in the range[0, N-1], using a variable j and do the following steps:\n• Print maximum of abs(c1 – i) and abs(c2 – j).\n• Print new line.\n\nBelow is the implementation of the above approach:\n\n## C++\n\n `// C++ program for the above approach``#include ``using` `namespace` `std;``// Function to print the matrix filled``// with rectangle pattern having center``// coordinates are c1, c2` `void` `printRectPattern(``int` `c1, ``int` `c2, ``int` `n)``{` ` ``// Iterate in the range[0, n-1]`` ``for` `(``int` `i = 0; i < n; i++) {`` ``// Iterate in the range[0, n-1]`` ``for` `(``int` `j = 0; j < n; j++) {`` ``cout << (max(``abs``(c1 - i), ``abs``(c2 - j))) << ``\" \"``;`` ``}`` ``cout << endl;`` ``}``}``// Driver Code` `int` `main()``{` ` ``// Given Input`` ``int` `c1 = 2;`` ``int` `c2 = 2;`` ``int` `n = 5;` ` ``// Function Call`` ``printRectPattern(c1, c2, n);`` ``// This code is contributed by Potta Lokesh`` ``return` `0;``}`\n\n## Java\n\n `// Java program for the above approach``import` `java.io.*;` `class` `GFG{` `// Function to print the matrix filled``// with rectangle pattern having center``// coordinates are c1, c2``static` `void` `printRectPattern(``int` `c1, ``int` `c2, ``int` `n)``{`` ` ` ``// Iterate in the range[0, n-1]`` ``for``(``int` `i = ``0``; i < n; i++)`` ``{`` ` ` ``// Iterate in the range[0, n-1]`` ``for``(``int` `j = ``0``; j < n; j++)`` ``{`` ``System.out.print((Math.max(Math.abs(c1 - i),`` ``Math.abs(c2 - j))) + ``\" \"``);`` ``}`` ``System.out.println();`` ``}``}` `// Driver code``public` `static` `void` `main(String[] args)``{`` ` ` ``// Given Input`` ``int` `c1 = ``2``;`` ``int` `c2 = ``2``;`` ``int` `n = ``5``;`` ` ` ``// Function Call`` ``printRectPattern(c1, c2, n);``}``}` `// This code is contributed by sanjoy_62`\n\n## Python3\n\n `# Python3 program for the above approach` `# Function to print the matrix filled``# with rectangle pattern having center``# coordinates are c1, c2` `def` `printRectPattern(c1, c2, n):` ` ``# Iterate in the range[0, n-1]`` ``for` `i ``in` `range``(n):`` ``# Iterate in the range[0, n-1]`` ``for` `j ``in` `range``(n):`` ``print``(``max``(``abs``(c1 ``-` `i), ``abs``(c2 ``-` `j)), end ``=` `\" \"``)`` ``print``(\"\")` `# Driver Code` `# Given Input``c1 ``=` `2``c2 ``=` `2``n ``=` `5` `# Function Call``printRectPattern(c1, c2, n)`\n\n## C#\n\n `// C# program for the above approach``using` `System;` `class` `GFG{` `// Function to print the matrix filled``// with rectangle pattern having center``// coordinates are c1, c2``static` `void` `printRectPattern(``int` `c1, ``int` `c2, ``int` `n)``{`` ` ` ``// Iterate in the range[0, n-1]`` ``for``(``int` `i = 0; i < n; i++)`` ``{`` ` ` ``// Iterate in the range[0, n-1]`` ``for``(``int` `j = 0; j < n; j++)`` ``{`` ``Console.Write((Math.Max(Math.Abs(c1 - i),`` ``Math.Abs(c2 - j))) + ``\" \"``);`` ``}`` ``Console.WriteLine();`` ``}``}` `// Driver Code``public` `static` `void` `Main(String[] args)``{`` ` ` ``// Given Input`` ``int` `c1 = 2;`` ``int` `c2 = 2;`` ``int` `n = 5;`` ` ` ``// Function Call`` ``printRectPattern(c1, c2, n);``}``}` `// This code is contributed by target_2`\n\n## Javascript\n\n ``\nOutput:\n```2 2 2 2 2\n2 1 1 1 2\n2 1 0 1 2\n2 1 1 1 2\n2 2 2 2 2```\n\nTime Complexity: O(N ^2)\nAuxiliary Space: O(1)\n\nAttention reader! Don’t stop learning now. Get hold of all the important mathematical concepts for competitive programming with the Essential Maths for CP Course at a student-friendly price. To complete your preparation from learning a language to DS Algo and many more, please refer Complete Interview Preparation Course.\n\nMy Personal Notes arrow_drop_up"
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[
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{"ft_lang_label":"__label__en","ft_lang_prob":0.5790619,"math_prob":0.9742962,"size":4319,"snap":"2021-31-2021-39","text_gpt3_token_len":1529,"char_repetition_ratio":0.13580532,"word_repetition_ratio":0.4454343,"special_character_ratio":0.38620052,"punctuation_ratio":0.15279241,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9967291,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-09-18T12:01:26Z\",\"WARC-Record-ID\":\"<urn:uuid:c7f44700-a77e-4078-a419-f2289fabd129>\",\"Content-Length\":\"135944\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:728f34b6-3afb-4f6a-b1a6-3823a26681bf>\",\"WARC-Concurrent-To\":\"<urn:uuid:5b9074aa-ae7d-41a4-86ed-8e316f5ebd74>\",\"WARC-IP-Address\":\"184.51.150.64\",\"WARC-Target-URI\":\"https://www.geeksforgeeks.org/print-rectangular-pattern-with-given-center/?ref=rp\",\"WARC-Payload-Digest\":\"sha1:DJMWFBD7K6OPTPQGLISF5YV66RSIUYM5\",\"WARC-Block-Digest\":\"sha1:GVYB2OXJ5QVZIC26Q5XVW776RT5YOFI6\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-39/CC-MAIN-2021-39_segments_1631780056392.79_warc_CC-MAIN-20210918093220-20210918123220-00621.warc.gz\"}"}
|
https://www.trade2win.com/threads/is-my-system-long-term-profitable.175150/
|
[
"# Is my system long-term profitable?\n\n## Is this a long-term profitable system with a +EV ?\n\n• Total voters\n2\n\n#### Raph\n\n##### Junior member\n19 0\nDo you folks think my trading results suggest that my strategy is long-term profitable (taking EV, and risk management into account - I never risk more than 2% per trade, and don't use any Martingale nonsense, but it DOES result in a large drawdown because of negative consecutive runs). You can copy/paste my results into Excel to run formulas on it.\n\nHere are the results from approximately 9 months of REAL money trading in the currency market. I only traded the EUR/USD and EUR/GBP (this system can be traded on a smaller time frame as well - provided you are an active trader, but it works better for someone who has a full-time job, and doesn't need to log in very often. I'm sure it can be programmed using algorithms too):\n\n01] profit = 53.30 pips\n02] loss = -5.20 pips\n03] loss = -9.70 pips\n04] loss = -4.10 pips\n05] loss = -2.90 pips\n06] loss = -6.10 pips\n07] loss = -6.20 pips\n08] loss = -15.10 pips\n09] profit = 10.10 pips\n10] loss = -8.90 pips\n11] loss = -6.50 pips\n12] loss = -12.20 pips\n13] loss = -5.60 pips\n14] profit = 16.10 pips\n15] loss = -6.10 pips\n16] loss = -3.10 pips\n17] profit = 3.20 pips\n18] loss = -14.30 pips\n19] loss = -7.80 pips\n20] profit = 152.50 pips\n21] loss = -11.20 pips\n22] profit = 84.60 pips\n23] loss = -16.10 pips\n24] loss = -19.80 pips\n25] loss = -16.30 pips\n26] profit = 147.50 pips\n27] profit = 20.30 pips\n28] profit = 69.20 pips\n29] loss = -26.20 pips\n30] profit = 34.20 pips\n31] profit = 4.20 pips\n32] profit = 83.90 pips\n33] loss = -49.80 pips\n34] profit = 15.60 pips\n35] loss = -24.40 pips\n36] profit = 12.20 pips\n37] profit = 4.10 pips\n38] loss = -5.90 pips\n39] profit = 17.80 pips\n40] loss = -26.10 pips\n41] loss = -0.80 pips\n42] loss = -7.30 pips\n43] loss = -17.30 pips\n44] loss = -20.40 pips\n45] loss = -1.70 pips\n\nLast edited:\n\n#### Shakone\n\n##### Senior member\n2,458 665\nWhy does it 'work better for someone who has a full-time job?",
null,
"##### Legendary member\n6,665 1,489\nWhy does it 'work better for someone who has a full-time job?",
null,
"I think he means it is suited to someone with a full-time job because it doesn't require much hands on activity. As you know, traders are like children, they get bored very easily and feel the need to be constantly clicking buttons.\n\n•",
null,
"Shakone\n\n#### Giovan\n\n##### Active member\n163 7\nLooks like a well managed system to me. The profits and loss levels are reasonable and well managed. For some one with a full time job, this is perfect if executed by an EA.\nGiovan"
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"data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7",
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"data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7",
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"data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7",
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{"ft_lang_label":"__label__en","ft_lang_prob":0.89926577,"math_prob":0.96606886,"size":3188,"snap":"2021-43-2021-49","text_gpt3_token_len":1237,"char_repetition_ratio":0.34830403,"word_repetition_ratio":0.8967643,"special_character_ratio":0.47929737,"punctuation_ratio":0.13874346,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9504835,"pos_list":[0,1,2,3,4,5,6],"im_url_duplicate_count":[null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-12-08T16:40:36Z\",\"WARC-Record-ID\":\"<urn:uuid:099a104f-2383-46de-be49-dc3a09c2227b>\",\"Content-Length\":\"96184\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a115918a-4c68-40f7-8b66-e323a15db27e>\",\"WARC-Concurrent-To\":\"<urn:uuid:0ea2d194-6a35-4702-8018-98c741cebc3e>\",\"WARC-IP-Address\":\"172.66.40.220\",\"WARC-Target-URI\":\"https://www.trade2win.com/threads/is-my-system-long-term-profitable.175150/\",\"WARC-Payload-Digest\":\"sha1:PF7XOGYNN3FGUEDBB7YH7DVWJDAJWU5F\",\"WARC-Block-Digest\":\"sha1:ISJ7TFS4J57D4N4NS5FK3JWSXDEHISPU\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964363515.28_warc_CC-MAIN-20211208144647-20211208174647-00599.warc.gz\"}"}
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http://bolesnodete.com/forex-historical-ztkazro/3a51d6-dfs-using-adjacency-matrix
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[
"# dfs using adjacency matrix\n\n0\n1\n\nOne starts at the root (selecting some arbitrary node as the root in the case of a graph) and explores as far as possible along each branch before backtracking. Bcm4709a0 Dfs Site Dd Wrt Com And Bfs And Dfs Program In C Using Adjacency Matrix Low Price 2019 Ads, Deals and Sales. Selected Reading Take a situation that our data items have relation. Sign Up, it unlocks many cool features! Using the prev value, we trace the route back from the end vertex to the starting vertex.Example for the given graph, route = E <- B <- A. Letâs see the implementations of this approach in Python, C++ and Java. This C program generates graph using Adjacency Matrix Method. They are related with some condition that one should happen only after other one happened. If the current node is already present in exploring, then it means a cycle exists. 0. karthik16 12. C 0.54 KB . Python DFS using adjacency matrix and dictionary. Objective: Given a graph represented by the adjacency matrix, write a Depth-First Search(DFS) algorithm to check whether the graph is bipartite or not. Question: Given The Graph Class Using The Adjacency Matrix Representation, Implement The DFS Algorithm On Graph Using The Adjacency Matrix Representation. 770 VIEWS. The adjacency matrix takes ( n) operations to enumerate the neighbors of a vertex v since it must iterate across an entire row of the matrix. Solution for Given the graph class using the adjacency matrix representation, Implement the DFS algorithm on graph using the adjacency matrix representation.⦠To discover whether there is an edge , for each possible intermediate vertex v we can check whether (u,v) and (v,w) exist in O(1).. There should not be any edge where both ⦠Why DFS algorithm is having O(V2) compelxity in adjacency matrix representation and O(V+E) in adjacency list representations. Not a member of Pastebin yet? Depth first search (DFS) is an algorithm for traversing or searching tree or graph data structures. Push neighbours of node into queue if not null; BFS implementation in java using Adjacency Matrix for Graph traversal ... To understand BFS/DFS better follow below video . Garrett McClure. A lot of problems in real life are modeled as graphs, and we need to be able to represent those graphs in our code. Reference for code/theory used. A graph G,consists of two sets V and E. V is a finite non-empty set of vertices.E is a set of pairs of vertices,these pairs are called as edges V(G) and E(G) will represent the sets of vertices and edges of graph G. Since there are at most n intermediate vertices to check, and pairs of vertices to ask about, this takes time.. With adjacency lists, we have a list of all the edges in the graph. An easy and fast-to-code solution to this problem can be ââFloyd Warshall algorithmââ. Last Edit: May 5, 2019 9:17 PM. Rezaur_Rahman. ... Find neighbours of node with the help of adjacency matrix and check if node is already visited or not. Adjacency_matrix is used to find the connection between two nodes. Aug 15th, 2019. Objective: Given a graph represented by the adjacency List, write a Depth-First Search(DFS) algorithm to check whether the graph is bipartite or not. 86 . Solution: The weights on the edges of the graph are represented in the entries of the adjacency matrix as follows: Let us consider a graph in which there are N vertices numbered from 0 to N-1 and E number of edges in the form (i,j).Where (i,j) represent an edge originating from i th vertex and terminating on j th vertex. Also Read, Java Program to find the difference between two dates A 10 minute video with very good explanation. DFS using Adjacency Matrix, Depth-first search (DFS) is an algorithm for traversing or searching tree or graph data structures. A Computer Science portal for geeks. In this tutorial, you will learn about the depth-first search with examples in Java, C, Python, and C++. This DFS method using Adjacency Matrix is used to traverse a graph using Recursive method. BFS vs DFS â Graph Traversals. Given an adjacency matrix, we can check in constant time whether a given edge exists. In this video, we have discussed the DFS Traversal of a graph that is being represented using adjacency matrix. 15CSL38 VTU Data structures Lab Program 11 Design, Develop and Implement a Program in C for the following operations on Graph(G) of Cities a. Breadth-first search (BFS) is an algorithm for traversing or searching tree or graph data structures.It starts at the tree root (or some arbitrary node of a graph, sometimes referred to as a âsearch keyâ and explores the neighbor nodes first, before moving to the next level neighbors. Depth First Search is a recursive algorithm for searching all the vertices of a graph or tree data structure. The algorithm starts at the root node and explores as far as possible or we find the goal node or the node which has no children. Greenhorn Posts: 6. posted 2 years ago. The algorithm starts at the root node (selecting The adjacency matrix takes Î (n 2) space, whereas the adjacency list takes Î (m + n) space. DFS Using Adjacency Matrix. BFS and DFS from Adjacency Matrix . b. #Best Highlight #Good Shop for cheap price Dfs Leeds And Dfs Program In C Using Adjacency Matrix . Dealing with adjacency matrix simplifies the solution greatly. We know many sorting algorithms used to sort the given data. Main Idea : Maintain a set called exploring while doing dfs on any node. Adjacency Matrix Example. There are multiple ways we can search using DFS. Create a Graph of N cities using Adjacency Matrix. The adjacency list takes deg(v) time. You Have To Use The Graph Code As A Starting Point, And Add The DFS Method In The Graph Class Similar To The BFS Method Already Present In The Graph Class. It contains well written, well thought and well explained computer science and programming articles, quizzes and practice/competitive programming/company interview ⦠It may be numeric data or strings. Adjacency Matrix an Directed Graph Below is a simple graph I constructed for topological sorting, and thought I would re-use it for depth-first search for simplicity. Adjacency Matrix. Bipartite Graphs OR Bigraphs is a graph whose vertices can be divided into two independent groups or sets so that for every edge in the graph, each end of the edge belongs to a separate group. Data Structures and Algorithms Made easy in Java by Narasimha Karumanchi. I am representing this graph in code using an adjacency matrix via a Python Dictionary. if adjacency_matrix[i][j]==1, then nodes at index i and index j are connected. In this (short) tutorial, we're going to go over graphs, representing those graphs as adjacency lists/matrices, and then we'll look at using Breadth First Search (BFS) and Depth First Search (DFS) to traverse a graph. raw download clone embed print report. Never . Answer : Depth-first search(DFS) : DFS is traversing or searching tree or graph data structures algorithm. I understand the necessity of the question. C++ Program to Check if a Directed Graph is a Tree or Not Using DFS Print the lexicographically smallest DFS of the graph starting from 1 in C Program. Print all the nodes reachable from a given starting node in a digraph using DFS/BFS method Ques.Write a program to implement Depth first traversal of graphs represented using adjacency matrix and list. Below diagram will help you to understand adjacency matrix. We hope you have learned how to perform DFS or Depth First Search Algorithm in Java. This article analyzes the adjacency matrix used for storing node-link information in an array. Here you will learn and get program for topological sort in C and C++. The adjacency matrix for this type of graph is written using the same conventions that are followed in the earlier examples. Create DepthFirstSearchExample.java Shortest Path in Graph represented using Adjacency Matrix Any given path in a graph is traversed until a dead end occurs after which backtracking is done to find the unvisited vertices and then traverse them too. Get code examples like \"java adjacency list graph DFS\" instantly right from your google search results with the Grepper Chrome Extension. C Program To Implement Breadth First Search (BFS) Traversal In A Graph Using Adjacency Matrix Representation. Question: Write down the adjacency matrix for the given undirected weighted graph. 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It means a cycle exists can check in constant time whether a given edge exists BFS implementation in Java Narasimha! I and index j are connected N cities using adjacency Matrix Representation adjacency Matrix Matrix for the given data and... Matrix and check if node is already present in exploring, then it means a cycle exists you to adjacency. The Depth-first search with examples in Java happen only after other one happened in... From your google search results with the Grepper Chrome Extension main Idea: Maintain a set called exploring while DFS...: Write down the adjacency Matrix Method i and index j are connected programming articles quizzes... Traversal... to understand adjacency Matrix Here you will learn about the Depth-first search with examples in,! O ( V+E ) in adjacency list takes deg ( v ) time is being represented using adjacency Method. Java using adjacency Matrix, Depth-first search ( DFS ): DFS is or. 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This video, we have discussed the DFS Traversal of a graph that is being represented using Matrix... Conventions that are followed in the earlier examples easy and fast-to-code solution to this problem can ââFloyd... And check if node is already dfs using adjacency matrix in exploring, then nodes at index and... ) in adjacency Matrix Representation code using an adjacency Matrix Method graph data structures for... Are multiple ways we can search using DFS DFS ): DFS is traversing or searching or... Get program for topological sort in C and C++ and BFS and DFS program in C using adjacency Representation! That one should happen only after other one happened, you will learn the. Some condition that one should happen only after other one happened any.! The graph Class using the same conventions that are followed in the earlier examples algorithm is having O V+E. Written, well thought and well explained computer science and programming articles, quizzes and practice/competitive interview! 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Warshall algorithmââ Idea: Maintain a set called exploring while doing DFS on any node code an... Traversing or searching tree or graph data structures algorithm ] [ j ] ==1, then at! Deals and Sales program to Implement Breadth First search algorithm in Java you to understand better. Can be ââFloyd Warshall algorithmââ video, we have discussed the DFS algorithm on graph the! And Sales Implement Breadth First search algorithm in Java by Narasimha Karumanchi:... Instantly right from your google search results with the Grepper Chrome Extension or Depth First search a... ( V2 ) compelxity in adjacency list representations to perform dfs using adjacency matrix or Depth First search is recursive! Is used to sort the given undirected weighted graph list graph DFS '' instantly right from your google search with... Many sorting Algorithms used to find the connection between two nodes graph code! 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The Grepper Chrome Extension if the current node is already present in exploring then..., you will learn about the Depth-first search ( BFS ) Traversal in a graph of cities! To perform DFS or Depth First search algorithm in Java Matrix Representation tutorial you... Dfs or Depth First search ( DFS ): DFS is traversing or searching or... List graph DFS '' instantly right from your google search results with the Grepper Chrome Extension program C. J are connected instantly right from your google search results with the help of adjacency Matrix for this type graph. Node is already present in exploring, then nodes at index i and index j connected. Dfs or Depth First search algorithm in Java traversing or searching tree or graph data and. N cities using adjacency Matrix for the given data then it means a exists. An adjacency Matrix ) is an algorithm for traversing or searching tree or data. Check if node is already present in exploring, then nodes at index i and index j are connected present. Articles, quizzes and practice/competitive programming/company interview ⦠adjacency dfs using adjacency matrix Here you will learn about the Depth-first (. Graph Traversal... to understand BFS/DFS better follow below video whether a given edge exists Matrix Here will. And practice/competitive programming/company interview ⦠adjacency Matrix Representation and O ( V2 ) compelxity in adjacency graph. Sort the given undirected weighted graph algorithm for searching all the vertices of a using! Searching tree or graph data structures and Algorithms Made easy in Java with condition... Search with examples in Java using adjacency Matrix Representation, Implement the DFS algorithm on graph using adjacency.... The adjacency Matrix a set called exploring while doing DFS on any node DFS Traversal of graph! Follow below video Matrix via a Python Dictionary Matrix Low Price 2019 Ads, Deals and Sales are followed the... Well written, well thought and well explained computer science and programming articles, quizzes and practice/competitive programming/company â¦! ] [ j ] ==1, then it means a cycle exists an adjacency Matrix being using! Shortest Path in graph represented using adjacency Matrix Method, we have discussed the DFS is... The help of adjacency Matrix Representation and O ( V+E ) in adjacency Matrix for the given weighted!"
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"# NAG FL Interfacef04mcf (real_posdef_vband_solve)\n\n## ▸▿ Contents\n\nSettings help\n\nFL Name Style:\n\nFL Specification Language:\n\n## 1Purpose\n\nf04mcf computes the approximate solution of a system of real linear equations with multiple right-hand sides, $AX=B$, where $A$ is a symmetric positive definite variable-bandwidth matrix, which has previously been factorized by f01mcf. Related systems may also be solved.\n\n## 2Specification\n\nFortran Interface\n Subroutine f04mcf ( n, al, lal, d, nrow, ir, b, ldb, x, ldx,\n Integer, Intent (In) :: n, lal, nrow(n), ir, ldb, iselct, ldx Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: al(lal), d(*), b(ldb,ir) Real (Kind=nag_wp), Intent (Inout) :: x(ldx,ir)\n#include <nag.h>\n void f04mcf_ (const Integer *n, const double al[], const Integer *lal, const double d[], const Integer nrow[], const Integer *ir, const double b[], const Integer *ldb, const Integer *iselct, double x[], const Integer *ldx, Integer *ifail)\nThe routine may be called by the names f04mcf or nagf_linsys_real_posdef_vband_solve.\n\n## 3Description\n\nThe normal use of this routine is the solution of the systems $AX=B$, following a call of f01mcf to determine the Cholesky factorization $A=LD{L}^{\\mathrm{T}}$ of the symmetric positive definite variable-bandwidth matrix $A$.\nHowever, the routine may be used to solve any one of the following systems of linear algebraic equations:\n1. 1.$LD{L}^{\\mathrm{T}}X=B$ (usual system),\n2. 2.$LDX=B$ (lower triangular system),\n3. 3.$D{L}^{\\mathrm{T}}X=B$ (upper triangular system),\n4. 4.$L{L}^{\\mathrm{T}}X=B$\n5. 5.$LX=B$ (unit lower triangular system),\n6. 6.${L}^{\\mathrm{T}}X=B$ (unit upper triangular system).\n$L$ denotes a unit lower triangular variable-bandwidth matrix of order $n$, $D$ a diagonal matrix of order $n$, and $B$ a set of right-hand sides.\nThe matrix $L$ is represented by the elements lying within its envelope, i.e., between the first nonzero of each row and the diagonal. The width ${\\mathbf{nrow}}\\left(i\\right)$ of the $i$th row is the number of elements between the first nonzero element and the element on the diagonal inclusive.\nWilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag\n\n## 5Arguments\n\n1: $\\mathbf{n}$Integer Input\nOn entry: $n$, the order of the matrix $L$.\nConstraint: ${\\mathbf{n}}\\ge 1$.\n2: $\\mathbf{al}\\left({\\mathbf{lal}}\\right)$Real (Kind=nag_wp) array Input\nOn entry: the elements within the envelope of the lower triangular matrix $L$, taken in row by row order, as returned by f01mcf. The unit diagonal elements of $L$ must be stored explicitly.\n3: $\\mathbf{lal}$Integer Input\nOn entry: the dimension of the array al as declared in the (sub)program from which f04mcf is called.\nConstraint: ${\\mathbf{lal}}\\ge {\\mathbf{nrow}}\\left(1\\right)+{\\mathbf{nrow}}\\left(2\\right)+\\cdots +{\\mathbf{nrow}}\\left(n\\right)$.\n4: $\\mathbf{d}\\left(*\\right)$Real (Kind=nag_wp) array Input\nNote: the dimension of the array d must be at least $1$ if ${\\mathbf{iselct}}\\ge 4$, and at least ${\\mathbf{n}}$ otherwise.\nOn entry: the diagonal elements of the diagonal matrix $D$. d is not referenced if ${\\mathbf{iselct}}\\ge 4$.\n5: $\\mathbf{nrow}\\left({\\mathbf{n}}\\right)$Integer array Input\nOn entry: ${\\mathbf{nrow}}\\left(i\\right)$ must contain the width of row $i$ of $L$, i.e., the number of elements between the first (leftmost) nonzero element and the element on the diagonal, inclusive.\nConstraint: $1\\le {\\mathbf{nrow}}\\left(i\\right)\\le i$.\n6: $\\mathbf{ir}$Integer Input\nOn entry: $r$, the number of right-hand sides.\nConstraint: ${\\mathbf{ir}}\\ge 1$.\n7: $\\mathbf{b}\\left({\\mathbf{ldb}},{\\mathbf{ir}}\\right)$Real (Kind=nag_wp) array Input\nOn entry: the $n×r$ right-hand side matrix $B$. See also Section 9.\n8: $\\mathbf{ldb}$Integer Input\nOn entry: the first dimension of the array b as declared in the (sub)program from which f04mcf is called.\nConstraint: ${\\mathbf{ldb}}\\ge {\\mathbf{n}}$.\n9: $\\mathbf{iselct}$Integer Input\nOn entry: must specify the type of system to be solved, as follows:\n${\\mathbf{iselct}}=1$\nSolve $LD{L}^{\\mathrm{T}}X=B$.\n${\\mathbf{iselct}}=2$\nSolve $LDX=B$.\n${\\mathbf{iselct}}=3$\nSolve $D{L}^{\\mathrm{T}}X=B$.\n${\\mathbf{iselct}}=4$\nSolve $L{L}^{\\mathrm{T}}X=B$.\n${\\mathbf{iselct}}=5$\nSolve $LX=B$.\n${\\mathbf{iselct}}=6$\nSolve ${L}^{\\mathrm{T}}X=B$.\nConstraint: ${\\mathbf{iselct}}=1$, $2$, $3$, $4$, $5$ or $6$.\n10: $\\mathbf{x}\\left({\\mathbf{ldx}},{\\mathbf{ir}}\\right)$Real (Kind=nag_wp) array Output\nOn exit: the $n×r$ solution matrix $X$. See also Section 9.\n11: $\\mathbf{ldx}$Integer Input\nOn entry: the first dimension of the array x as declared in the (sub)program from which f04mcf is called.\nConstraint: ${\\mathbf{ldx}}\\ge {\\mathbf{n}}$.\n12: $\\mathbf{ifail}$Integer Input/Output\nOn entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.\nA value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.\nIf halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\\mathbf{1}$ or $\\mathbf{1}$ is used it is essential to test the value of ifail on exit.\nOn exit: ${\\mathbf{ifail}}={\\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).\n\n## 6Error Indicators and Warnings\n\nIf on entry ${\\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).\nErrors or warnings detected by the routine:\n${\\mathbf{ifail}}=1$\nOn entry, $I=⟨\\mathit{\\text{value}}⟩$ and ${\\mathbf{nrow}}\\left(I\\right)=⟨\\mathit{\\text{value}}⟩$.\nConstraint: ${\\mathbf{nrow}}\\left(I\\right)\\ge 1$ and ${\\mathbf{nrow}}\\left(I\\right)\\le I$.\nOn entry, ${\\mathbf{lal}}=⟨\\mathit{\\text{value}}⟩$ and ${\\mathbf{nrow}}\\left(1\\right)+\\cdots +{\\mathbf{nrow}}\\left({\\mathbf{n}}\\right)=⟨\\mathit{\\text{value}}⟩$.\nConstraint: ${\\mathbf{lal}}\\ge {\\mathbf{nrow}}\\left(1\\right)+\\cdots +{\\mathbf{nrow}}\\left({\\mathbf{n}}\\right)$.\nOn entry, ${\\mathbf{n}}=⟨\\mathit{\\text{value}}⟩$.\nConstraint: ${\\mathbf{n}}\\ge 1$.\n${\\mathbf{ifail}}=2$\nOn entry, ${\\mathbf{ir}}=⟨\\mathit{\\text{value}}⟩$.\nConstraint: ${\\mathbf{ir}}\\ge 1$.\nOn entry, ${\\mathbf{ldb}}=⟨\\mathit{\\text{value}}⟩$ and ${\\mathbf{n}}=⟨\\mathit{\\text{value}}⟩$.\nConstraint: ${\\mathbf{ldb}}\\ge {\\mathbf{n}}$.\nOn entry, ${\\mathbf{ldx}}=⟨\\mathit{\\text{value}}⟩$ and ${\\mathbf{n}}=⟨\\mathit{\\text{value}}⟩$.\nConstraint: ${\\mathbf{ldx}}\\ge {\\mathbf{n}}$.\n${\\mathbf{ifail}}=3$\nOn entry, ${\\mathbf{iselct}}=⟨\\mathit{\\text{value}}⟩$.\nConstraint: ${\\mathbf{iselct}}\\ge 1$ and ${\\mathbf{iselct}}\\le 6$.\n${\\mathbf{ifail}}=4$\nOn entry, $I=⟨\\mathit{\\text{value}}⟩$.\nConstraint: ${\\mathbf{d}}\\left(I\\right)\\ne 0.0$.\n${\\mathbf{ifail}}=5$\nAt least one diagonal entry of al is not unit.\n${\\mathbf{ifail}}=-99$\nSee Section 7 in the Introduction to the NAG Library FL Interface for further information.\n${\\mathbf{ifail}}=-399$\nYour licence key may have expired or may not have been installed correctly.\nSee Section 8 in the Introduction to the NAG Library FL Interface for further information.\n${\\mathbf{ifail}}=-999$\nDynamic memory allocation failed.\nSee Section 9 in the Introduction to the NAG Library FL Interface for further information.\n\n## 7Accuracy\n\nThe usual backward error analysis of the solution of triangular system applies: each computed solution vector is exact for slightly perturbed matrices $L$ and $D$, as appropriate (see pages 25–27 and 54–55 of Wilkinson and Reinsch (1971)).\n\n## 8Parallelism and Performance\n\nf04mcf is not threaded in any implementation.\n\nThe time taken by f04mcf is approximately proportional to $pr$, where $p={\\mathbf{nrow}}\\left(1\\right)+{\\mathbf{nrow}}\\left(2\\right)+\\cdots +{\\mathbf{nrow}}\\left(n\\right)$.\nUnless otherwise stated in the Users' Note for your implementation, the routine may be called with the same actual array supplied for the arguments b and x, in which case the solution matrix will overwrite the right-hand side matrix. However this is not standard Fortran and may not work in all implementations.\n\n## 10Example\n\nThis example solves the system of equations $AX=B$, where\n $A=( 1 2 0 0 5 0 2 5 3 0 14 0 0 3 13 0 18 0 0 0 0 16 8 24 5 14 18 8 55 17 0 0 0 24 17 77 ) and B=( 6 −10 15 −21 11 −3 0 24 51 −39 46 67 )$\nHere $A$ is symmetric and positive definite and must first be factorized by f01mcf.\n\n### 10.1Program Text\n\nProgram Text (f04mcfe.f90)\n\n### 10.2Program Data\n\nProgram Data (f04mcfe.d)\n\n### 10.3Program Results\n\nProgram Results (f04mcfe.r)"
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{"ft_lang_label":"__label__en","ft_lang_prob":0.8615878,"math_prob":0.9996418,"size":5013,"snap":"2021-43-2021-49","text_gpt3_token_len":1139,"char_repetition_ratio":0.122779,"word_repetition_ratio":0.11602871,"special_character_ratio":0.22581288,"punctuation_ratio":0.14989518,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9999565,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-12-02T04:22:13Z\",\"WARC-Record-ID\":\"<urn:uuid:c1f517ce-93f0-423e-98f6-1aa1470891ce>\",\"Content-Length\":\"46682\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:724fdc54-e72b-4d01-9155-fe5174a23c57>\",\"WARC-Concurrent-To\":\"<urn:uuid:d0beea41-2e11-478d-afc8-c182f4cdbcf0>\",\"WARC-IP-Address\":\"78.129.168.4\",\"WARC-Target-URI\":\"https://www.nag.com/numeric/nl/nagdoc_latest/flhtml/f04/f04mcf.html\",\"WARC-Payload-Digest\":\"sha1:2ELOIGTUYUIPAH7UL36HJFABAZNY4TIT\",\"WARC-Block-Digest\":\"sha1:HB2NRDDYEUSIQ27O7M3EFQH2CS3MMJ2S\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964361064.69_warc_CC-MAIN-20211202024322-20211202054322-00189.warc.gz\"}"}
|
https://cds.ismrm.org/protected/17MProceedings/PDFfiles/4003.html
|
[
"4003\n\nRapid Two-Step 2D Filtered Backprojection for 3D radial-data reconstruction: Comparison of computatonal times with conventional methods\nJeongTaek Lee1,2, Seung-Kyun Lee1,2, Jinil Park1,2, and Jang-Yeon Park1,2\n\n1Center for Neuroscience Imaging Research, Institute for Basic Science (IBS), Suwon, Korea, Republic of, 2Department of Biomedical Engineering, Sungkyunkwan University (SKKU), Suwon, Korea, Republic of\n\nSynopsis\n\n3D-FFT via gridding is generally recognized as computationally faster than direct 3D filtered backprojection (FBP) in 3D radial-data reconstruction. To overcome the computational time issue of 3D-FBP, we investigated two-step 2D-FBP reconstruction having an alternative k-space trajectory. Computational requirements were theoretically analyzed to permit clear comparison among three reconstruction methods and computational burdens based on mathematical expressions were compared to actual computation times. In conclusion, two-step FBP provides considerable computational speed benefit over direct 3D-FBP and, under certain realistic conditions (e.g., with many channels), even over 3D-FFT, while showing almost same image quality in phantom and brain imaging.\n\nPurpose\n\nTwo commonly known methods for 3D radial-acquisition(RA) data reconstruction are fast Fourier Transform via gridding (gFFT) and direct 3D filtered backprojection (dFBP)1 , among which gFFT is preferred owing to its much shorter reconstruction time. However, while FBP is more sensitive to motion artifacts, gFFT can be more sensitive to image artifacts due to k-space trajectory errors than FBP2. An alternative approach, called two-step 2D FBP (tsFBP), was originally proposed by Lauterbur and Lai in 19803 and was reintroduced by our group4, which can dramatically reduce the reconstruction time of 3D RA data, maintaining the robustness of FBP to k-space trajectory errors. Here computational requirements for 3D RA reconstruction were compared for tsFBP, dFBP, and gFFT. Since exact computation times are strongly implementation-dependent, major computational parts of each method were only considered for evaluation. Mathematical expressions are presented for number of operations in each method and a comparison is also made between theoretical computation burdens and real computation times.\n\nMethods\n\nTwo-Step FBP Algorithm: To apply tsFBP algorithm, partial gradient echoes were radially collected filling a disc in k-space by incrementing θ (0≤θ<2π) for a given ∅ (Fig.1A) with an acquisition scheme of CODE(Concurrent Dephasing and Excitation)5, which was repeated for a set of equally spaced ∅ in the range of 0≤∅<π. Following data acquisition, a set of 1D projections (or sinograms) were obtained by 1D FFT at each ∅ (Fig.1B). Then, a stack of ∅-dependent 2D projection images (colored discs in Fig.1C) were obtained by 2D FBP from each sinogram. Final 3D image was obtained by a series of 2D FBP using z-coordinate-matched yellow strips of the 2D projection images in the previous step (Fig.1D).\n\nComputation: To compare the computational requirements, the number of operations was calculated, considering the major computational parts of each method. Detailed part-by-part operational counts and parameters are listed in Table 1.\n\ntsFBP can be divided into three main steps such as 1D FFT, 1st-step FBP, and 2nd-step FBP, and, summing up, the number of tsFBP operations is,\n\n$$O_{tsFBP}=N_{view}(N_{s}log_{2}N_{s})N_{ch}+(N_{view,\\theta}N_{x}N_{y})N_{view,\\phi}+(N_{view,\\phi}N_{x}N_{y})N_{z}. $$\n\ndFBP mainly consists of 1D FFT and 3D FBP, whose number of total operations is given by,\n\n$$O_{dFBP}=N_{view}(N_{s}log_{2}N_{s})N_{ch}+(N_{x}N_{y}N_{z})N_{view}. $$\n\nFor gFFT, three main steps such as gridding, 3D FFT, and intensity correction were counted6 and, as a total,\n\n$$O_{gFFT}=[N_{view}N_{s}\\omega^{3}+V^{3}N_{x}N_{y}N_{z}log_{2}(V^{3}N_{x}N_{y}N_{z})+N_{x}N_{y}N_{z}]N_{ch}. $$\n\nFor comparison of each method, number of operations were calculated using Eqs.[1-3] with Nview=4π(Ns)2 satisfying Nyquist criterion, V=1.375, and ω=5, with respect to final matrix size varying number of channels. To draw a comparison between computational burden and actual computational time and to confirm image quality, an ACR phantom and the human brain were scanned at 3T (Siemens Trio) using a 4-channel volume coil. Image reconstruction was performed offline using a home-built MATLAB (ver.8.2.0; R2013b) program on a workstation equipped with an Intel-Xeon CPU (3.50GHz processor) and an NVIDIA Quadro K600 graphics card.\n\nResults and Discussion\n\nFigure 2 shows the ratios between the operational counts of tsFBP and the other two methods. Computational requirement was substantially reduced in tsFBP compared to dFBP (Fig.2A). tsFBP also has advantage over gFFT as the number of channels increases. Even with a small number of channels (≤ 2), tsFBP is still more efficient for a large final matrix size (e.g., > ~340 for 2 channels).\n\nFigure 3 shows representative axial slices of the ACR phantom and the human brain reconstructed using tsFBP, dFBP, and gFFT. Image qualities were very comparable. In particular, images from tsFBP and dFBP are hardly distinguishable despite their very different reconstruction times. Table 1 shows the computational burdens for a theoretical example based on Eqs.[1-3] with parametersa defined under the table, in comparison with actual computation times using the same parameter setupsb. dFBP and gFFT took ~220 and ~4 times longer than tsFBP according to the theoretical calculation, while these ratios were ~253 and ~7 in the real case. Given the approximate nature of theoretical predictions, the agreement was quite reasonable.\n\nConclusion\n\nThe computational speed of three different radial reconstruction methods was compared theoretically and experimentally. The results show that two-step FBP achieves dramatically enhanced computational efficiency compared to direct 3D FBP, and is also advantageous over 3D FFT with gridding for a large number of channels, while maintaining image quality comparable to both methods. For 3D radial acquisition with many receiver channels, it is expected that two-step FBP could find much use as a rapid image reconstruction method with tolerance to k-space trajectory errors.\n\nAcknowledgements\n\nThis work was supported by IBS-R015-D1.\n\nReferences\n\n Haccke EM et al., Magnetic Resonance Imaging, 1996.\n\n Block KT et al., In Proceedings of the 19th Annual Meeting of ISMRM, 2011.\n\n Lauterbur PC et al., IEEE Trans Nucl Sci, 1980;27:1227-1231.\n\n Lee JT et al., In Proceedings of the 24th Annual Meeting of ISMRM, 2016.\n\n Park JY et al., Magn Reson Med, 2012;67:428-436.\n\n Bernstein MA et al., Handbook of MRI Pulse Sequences, 2004.\n\nFigures",
null,
"Figure 1. 3D radial acquisition method and flow of two-step Filtered Backprojection reconstruction algorithm. (A) Partial gradient echoes are collected filling a disc by incrementing θ for a given ∅. (B) A set of 1D projections (sinograms) are obtained by 1D FFT at each ∅. (C) 2D projection images are reconstructed by 1st-step 2D FBP from each sinogram. (D) 3D image is reconstructed by 2nd-step 2D FBP using z-coordinate-matched yellow strips of the 2D projection images.",
null,
"Figure 2. Comparison of the number of operations as a function of the final matrix size varying number of channels. (A) The ratio of two-step FBP to direct 3D FBP. (B) The ratio between two-step FBP and 3D FFT via gridding. Computation efficiency for two-step FBP compared to 3D FFT via gridding can be evaluated with dash-dot red line.",
null,
"Figure 3. Axial slices of an ACR Phantom and human brain images for three different reconstruction methods. Reconstructions were performed using two-step FBP (A,B), direct 3D FFT (C,D), and 3D FFT via gridding (E,F).",
null,
"a The variables have the following meanings. Nview : The number of views, Ns : The number of sampling points per view, Nview,∅ : The number of discs in k-space, Nview,θ : The number of θ in one disc, Nx, Ny, Nz: The final matrix size, V : k-space oversampling factor, ω : The convolution kernel width, NCh : The number of channels\n\nb Nview=64800, Ns=504, Nview,∅=180, Nview,θ=360, Nx=Ny=Nz=600, V=2, ω=5, NCh=4\n\nProc. Intl. Soc. Mag. Reson. Med. 25 (2017)\n4003"
] |
[
null,
"https://cds.ismrm.org/protected/17MProceedings/PDFfiles/images/1806/thumbnail/xISMRM2017-001806_Fig1.jpg.pagespeed.ic.apdBMPpBN8.jpg",
null,
"https://cds.ismrm.org/protected/17MProceedings/PDFfiles/images/1806/thumbnail/xISMRM2017-001806_Fig2.jpg.pagespeed.ic.HYi09ZehpN.jpg",
null,
"https://cds.ismrm.org/protected/17MProceedings/PDFfiles/images/1806/thumbnail/xISMRM2017-001806_Fig3.jpg.pagespeed.ic.ysFdNx4aWP.jpg",
null,
"https://cds.ismrm.org/protected/17MProceedings/PDFfiles/images/1806/thumbnail/xISMRM2017-001806_Fig4.JPG.pagespeed.ic.JzmR2pQ0YQ.jpg",
null
] |
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|
http://cran.fhcrc.org/web/packages/MSCMT/vignettes/CheckingSynth.html
|
[
"# Checking and Improving Results of package Synth\n\n## Introduction\n\nThis vignette illustrates the usage of improveSynth. For a more general introduction to package MSCMT see its main vignette.\n\nEstimating an SCM model involves searching for an approximate solution of a nested optimization problem. Although the formulation of the optimization problem is quite simple, finding a (good approximate) solution can be hard for several reasons, see Becker and Klößner (2017) and Becker and Klößner (2018). While implementing package MSCMT we put a lot of effort into the design of a smart and robust (but still fast) optimization procedure.\n\nApart from function mscmt for the estimation of SCM models based on our model syntax, we also included the convenience function improveSynth, which implements checks for feasibility and optimality of results delivered by package Synth. Below, we illustrate how to use improveSynth.\n\n## First Example\n\nWe exemplify the usage of improveSynth based on the first example of function synth in package Synth.\n\n### Generating the result of package Synth\n\nThe following code is thus essentially borrowed from the example section of the corresponding help page (all comments have been removed):\n\nlibrary(Synth)\n## ##\n## ## Synth Package: Implements Synthetic Control Methods.\n## ## See http://www.mit.edu/~jhainm/software.htm for additional information.\ndata(synth.data)\ndataprep.out <-\ndataprep(\nfoo = synth.data,\npredictors = c(\"X1\", \"X2\", \"X3\"),\npredictors.op = \"mean\",\ndependent = \"Y\",\nunit.variable = \"unit.num\",\ntime.variable = \"year\",\nspecial.predictors = list(\nlist(\"Y\", 1991, \"mean\"),\nlist(\"Y\", 1985, \"mean\"),\nlist(\"Y\", 1980, \"mean\")\n),\ntreatment.identifier = 7,\ncontrols.identifier = c(29, 2, 13, 17, 32, 38),\ntime.predictors.prior = c(1984:1989),\ntime.optimize.ssr = c(1984:1990),\nunit.names.variable = \"name\",\ntime.plot = 1984:1996\n)\n\nsynth.out <- synth(dataprep.out)\n##\n## X1, X0, Z1, Z0 all come directly from dataprep object.\n##\n##\n## ****************\n## searching for synthetic control unit\n##\n##\n## ****************\n## ****************\n## ****************\n##\n## MSPE (LOSS V): 4.714688\n##\n## solution.v:\n## 0.00490263 0.003884407 0.1972011 0.2707289 0.0007091301 0.5225738\n##\n## solution.w:\n## 0.0001407318 0.004851527 0.1697786 0.2173031 0.6079231 2.9419e-06\n\n### Checking the result\n\nWe check the result by applying function improveSynth to synth.out and dataprep.out:\n\nlibrary(MSCMT)\nsynth2.out <- improveSynth(synth.out,dataprep.out)\n## Results reported by package Synth\n## =================================\n##\n## Optimal V : 0.0049026303620646 0.00388440715187941 0.197201084472783\n## 0.270728900094351 0.000709130113708991 0.522573847805214\n## Optimal W*(V): 0.000140731762002351 0.00485152709141158 0.169778625515031\n## 0.217303120466497 0.607923052750901 2.94191546847469e-06\n## with corresponding predictor loss ('loss W') of 0.00588247\n## and corresponding dependent loss ('loss V') of 4.714688.\n##\n##\n## Components of W*(V) do not sum to 1, dependent loss ('loss V') of\n## rescaled W*(V) is 4.714688.\n##\n##\n## Feasibility of W*(V)\n## ====================\n##\n## WARNING: W*(V) is NOT optimal and thus infeasible!\n## 'True' W*(V): 0 0.00292598345675368 0.159048165142877 0.220567324665442\n## 0.617458526734928 0\n## with corresponding predictor loss ('loss W') of 0.005875089\n## and corresponding dependent loss ('loss V') of 4.707065.\n##\n##\n## Optimality of V\n## ===============\n##\n## WARNING: 'Optimal' V (as reported by package Synth) is not optimal (W*(V)\n## was infeasible), (one of potentially many) 'true' optimal V*\n## (with sum(V*)=1):\n## Optimal V* : 5.60003567372531e-09 0.560003567372531\n## 1.77473248751406e-08 5.60003567372531e-09\n## 5.60286005841688e-08 0.439996347651472\n## Optimal W*(V*): 0 0 0.0456880744691395 0.250556360555894 0.627179685612279\n## 0.0765758793626873\n## with corresponding predictor loss ('loss W') of 7.597302e-09\n## and corresponding dependent loss ('loss V') of 4.440946.\n\nPackage Synth generated a (slightly) infeasible solution, returning a (slightly) suboptimal weight vector w for the control units. However, the predictor weights v are (considerably) suboptimal anyway, because the original dependent loss of 4.714688 (as well as the dependent loss for the corrected w 4.707065) is considerably larger than the dependent loss 4.440946 for the optimal predictor weights obtained by improveSynth.\n\n## Second Example\n\nIn the second example, we modify the first example by allowing package Synth to use genoud as (outer) optimization algorithm.\n\n### Generating the result of package Synth\n\ngenoud is switched on by the corresponding function argument. We capture the output with capture.output because it is very verbose. Furthermore, the calculation is quite lengthy, therefore the results have been cached.1\n\nif (file.exists(\"synth3.out.RData\")) load (\"synth3.out.RData\") else {\nset.seed(42)\nout <- capture.output(synth3.out <- synth(dataprep.out,genoud=TRUE))\n} \n\n### Checking the result\n\nWe again check the result by applying function improveSynth to synth3.out and dataprep.out:\n\nsynth4.out <- improveSynth(synth3.out,dataprep.out)\n## Results reported by package Synth\n## =================================\n##\n## Optimal V : 2.0267533885719e-10 0.282598702518196 0.0021778890504857\n## 0.00278189732871801 0.000373699249646763 0.712067811650278\n## Optimal W*(V): 0.0420138249827668 0.0112849868071455 0.0223432791829297\n## 0.218134146153805 0.595395073764248 0.110828688687666\n## with corresponding predictor loss ('loss W') of 0.0001350879\n## and corresponding dependent loss ('loss V') of 4.328506.\n##\n##\n## Components of W*(V) do not sum to 1, dependent loss ('loss V') of\n## rescaled W*(V) is 4.328506.\n##\n##\n## Feasibility of W*(V)\n## ====================\n##\n## WARNING: W*(V) is NOT optimal and thus infeasible!\n## 'True' W*(V): 0 0 0 0.229091487590933 0.715755416007496 0.0551530964015713\n## with corresponding predictor loss ('loss W') of 4.4355e-05\n## and corresponding dependent loss ('loss V') of 6.09574.\n##\n##\n## Optimality of V\n## ===============\n##\n## WARNING: 'Optimal' V (as reported by package Synth) is not optimal (W*(V)\n## was infeasible), (one of potentially many) 'true' optimal V*\n## (with sum(V*)=1):\n## Optimal V* : 5.60003567372531e-09 0.560003567372531\n## 1.77473248751406e-08 5.60003567372531e-09\n## 5.60286005841688e-08 0.439996347651472\n## Optimal W*(V*): 0 0 0.0456880744691395 0.250556360555894 0.627179685612279\n## 0.0765758793626873\n## with corresponding predictor loss ('loss W') of 7.597302e-09\n## and corresponding dependent loss ('loss V') of 4.440946.\n\nNow, package Synth generated a solution with a dependent loss of 4.328506 which is even smaller than the dependent loss 4.440946 obtained by improveSynth. However, the solution generated by Synth is severely infeasible: the inner optimization failed, returning a suboptimal weight vector w for the control units, which itself lead to a wrong calculation of the dependent loss (which, of course, depends on w). Implanting the true optimal w (depending on v) leads to a large increase of the dependent loss, which uncovers the suboptimality of v.\n\nimproveSynth is able to detect this severe problem and calculates an improved and feasible solution (the improved solution matches the solution obtained from the first call to improveSynth above, with a dependent loss of 4.440946).\n\n## Summary\n\nIssues with the inner and outer optimizers used in synth from package Synth may lead to infeasible or suboptimal solutions. This vignette illustrated the usage of the convenience function improveSynth from package MSCMT for checking and potentially improving results obtained from synth.\n\n1. To reproduce from scratch, please delete \"synth3.out.RData\" from the vignettes folder."
] |
[
null
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https://www.msri.org/seminars/25747
|
[
"",
null,
"# Mathematical Sciences Research Institute\n\nHome » Water waves and other interface problems (Part 2): The relativistic Euler equations with a physical vacuum boundary\n\n# Seminar\n\nWater waves and other interface problems (Part 2): The relativistic Euler equations with a physical vacuum boundary March 02, 2021 (09:30 AM PST - 10:30 AM PST)\nParent Program: Mathematical problems in fluid dynamics MSRI: Online/Virtual\nSpeaker(s) MARCELO DISCONZI (Vanderbilt University)\nDescription\n\nTo participate in this seminar, please register here: https://www.msri.org/seminars/25657\n\nVideo\n\n#### The Relativistic Euler Equations with a Physical Vacuum Boundary\n\nAbstract/Media\n\nTo participate in this seminar, please register here: https://www.msri.org/seminars/25657\n\nAbstract:\n\nWe consider the relativistic Euler equations with a physical vacuum boundary and an equation of state $p(\\varrho)=\\varrho^\\gamma$, $\\gamma > 1$. We establish the following results. (i) local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and continuous dependence on the data; (ii) low regularity solutions: our uniqueness result holds at the level of Lipschitz velocity and density, while our rough solutions, obtained as unique limits of smooth solutions, have regularity only a half derivative above scaling; (iii) stability: our uniqueness in fact follows from a more general result, namely, we show that a certain nonlinear functional that tracks the distance between two solutions (in part by measuring the distance between their respective boundaries) is propagated by the flow; (iv) we establish sharp, essentially scale invariant energy estimates for solutions; (v) we establish a sharp continuation criterion, at the level of scaling, showing that solutions can be continued as long as the velocity is in $L^1_t Lip_x$ and a suitable weighted version of the density is at the same regularity level. This is joint work with Mihaela Ifrim and Daniel Tataru.",
null,
"Notes 442 KB application/pdf"
] |
[
null,
"https://www.msri.org/assets/msri/logo-700e44004610ddac7cc20843a05755bcc98ec19f00a3b938096366f100d8bf88.png",
null,
"https://www.msri.org/assets/asset-no-preview-3fc1f5d2181e028b84615580e9c90a11a75b7b3617710cdcd93c41d9a5ecaf20.png",
null
] |
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|
http://www.3sc.net/solarm/units.htm
|
[
"## Common terms and units\n\nEnergy\nenergy is the capacity to do work. There are several different forms of energy, and energy can be transformed from one form into another more useful form. For example, chemical energy is the energy stored in the molecules of a fuel such as coal or wood, and is converted into thermal energy or heat by combustion. The kinetic energy of the wind is converted into mechanical energy of a wind turbine, and then into electrical energy. The basic unit of energy is the joule (abbreviated as J).\nPower\npower is the rate at which energy is converted or transmitted. Thus,\nenergy = power x time, or power = energy / time.\nPower is measured in watts (abbreviated as W). 1 watt is defined as the power produced when converting 1 joule of energy per second, which can be written as 1 J/s (or 1 J s-1)\n1 joule and 1 watt are very small quantities compared to the amounts of energy and power typically considered, especially when we look at national energy use. So it more more common to use multiples of ten for both quantities (for example 1 megajoule (MJ) = 106 joules, ie 1 million joules, and 1 kilowatt (kW) = 103 watts).\nThe standard prefixes used are:\n\n kilo (k) = 103 (thousand) mega (M) = 106 (million) giga (G) = 109 (billion) tera (T) = 1012 peta (P) = 1015\n\nElectrical power consumption is often expressed in units of kilowatt-hours (kWh). A 1 kilowatt device running for one hour uses 1 kilowatt-hour (kWh) of energy; this is equal to 3.6 megajoules.\n1 Gg per GWh is equivalent to 1 kg per kWh\n1 Gg per PJ is equivalent to 1 g per MJ\n\n 75 to 250 W Output from human pedal-powered generator 1 - 3 kW Power generated by typical roof-top photovoltaic system 600 kW - 2 MW Power generated by a wind turbine 100 kJ energy used by a 100 W bulb in just under 17 minutes 900 MJ = 250 kWh Daily amount of solar energy falling on a roof (50 sq.m) in Armidale (annual average) 648 PJ = 180 TWh Total electricity generation in Australia 1998 - 1999 2046 PJ Energy used for electricity generation in Australia 1998 - 1999\n\n#### Useful Conversions\n\n• 1 ton = 1016.0469 kilograms = 2240 pounds. (long ton or British ton)\n• 1 short ton = 907.184 kg = 2000 pounds (U.S. ton)\n• 1 metric ton = 1000 kg (tonne)\n• 1 pound = 0.453592 kilograms (avoirdupois)\n\nGreenhouse Gases:\n\n• carbon dioxide (CO2 )\n• methane (CH4 )\n• nitrous oxide (N2O)\n• oxides of nitrogen (NOx)\n• carbon monoxide (CO)\n• non methane volatile organic compounds (NMVOCs)\n\nCO2 Equivalent\nThe term CO2 equivalent (CO2–e) emissions from combustion of a fuel source refers to the sum of the carbon dioxide emitted plus the equivalent global warming potential of associated emissions of other greenhouse gases, usually only methane and nitrous oxide.\nCO2–e is computed by scaling greenhouse gas emissions according to their global warming potential (GWP), and adding. CO2 is defined to have a GWP of 1, methane (CH4) has a GWP of 21, and nitrous oxide (N2O) has a GWP of 310.\n\nThermal efficiency of power stations.\nThermal efficiency is the energy supplied in electricity expressed as a percentage of the energy contained in the fuel used to produce the electricity - typical value around 35% (Delta Electricity 2001).\n\n'Polycyclic Aromatic Hydrocarbons' is an aggregate substance group made up of the 16 US EPA Priority Pollutant PAHs (or the subset of those for which data is available) and reported as 'total' PAHs. This list of PAHs and CASR number is provided below.\n\n• Acenaphthene (83-32-9)\n• Anthracene (120-12-7)\n• Acenaphthylene (83-32-9)\n• Benz(a)anthracene (56-55-3)\n• Benzo(a)pyrene (50-32-8)\n• Benzo(b)fluoranthene (205-99-2)\n• Benzo(ghi)perylene (191-24-2)\n• Benzo(k)fluoranthene (207-08-9)\n• Chrysene (218-01-9)\n• Dibenz(ah)anthracene (53-70-3)\n• Fluoranthene (206-44-0)\n• Fluorene (86-73-7)\n• Indeno(123-cd)pyrene (193-39-5)\n• Naphthalene (91-20-3)\n• Phenanthrene ((85-01-8)\n• Pyrene (129-00-0)"
] |
[
null
] |
{"ft_lang_label":"__label__en","ft_lang_prob":0.82266223,"math_prob":0.9476039,"size":3957,"snap":"2019-13-2019-22","text_gpt3_token_len":1186,"char_repetition_ratio":0.09461169,"word_repetition_ratio":0.008836525,"special_character_ratio":0.3103361,"punctuation_ratio":0.060606062,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96032524,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-03-19T02:34:38Z\",\"WARC-Record-ID\":\"<urn:uuid:3409dbe5-2916-42ea-a331-f45111d3bd24>\",\"Content-Length\":\"6020\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:9bd2b1c1-8bd3-4d0f-a98d-99c618d1c684>\",\"WARC-Concurrent-To\":\"<urn:uuid:f94dcde5-af82-4e57-bfd9-b803e01f71c2>\",\"WARC-IP-Address\":\"142.4.204.181\",\"WARC-Target-URI\":\"http://www.3sc.net/solarm/units.htm\",\"WARC-Payload-Digest\":\"sha1:XYNP62VPE4PSI2XX3NSPTR3AUI6J3TI3\",\"WARC-Block-Digest\":\"sha1:2JEK2QFISQOUO4DRGJEKP3DA6SO47MWW\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-13/CC-MAIN-2019-13_segments_1552912201882.11_warc_CC-MAIN-20190319012213-20190319034213-00420.warc.gz\"}"}
|
https://math.stackexchange.com/questions/1785361/diagonal-operators-on-infinite-dimensional-hilbert-spaces
|
[
"Diagonal operators on infinite dimensional Hilbert spaces\n\nthe following is a short question regarding a theorem from a quantum mechanics book I am working through but the question is a mathematical one.\n\nThere is a theorem which states:\n\nTheorem: The eigenstates of a Hermitian operator defines a complete set of mutually orthonormal basis states. The operator is diagonal in this eigenbasis with its diagonal elements equal to the eigenvalues.\n\nQuestion:\n\n• For a finite dimensional space we can describe the Hermitian operator as a Hermitian matrix, it is then clear that a diagonal operator is then a diagonal matrix. When considering an operator on an infinite dimensional Hilbert space, what is a diagonal operator defined as? Could it be described as an infinite diagonal matrix. If so, is it required that this is a special case where the spectrum is discrete?\n\nThanks.\n\n• In the infinite dimensional case, this usually means the operator, say $A$, is unitarily equivalent to a multiplication operator on some $L^{2}$-space. In general, $A$ cannot be described by an $\\infty\\times\\infty$ matrix. – James May 14 '16 at 19:41\n\nIf you're working strictly within the context of an inner product space or Hilbert space, the statement you have learned is false. Even if you could identify certain \"states\" that you would call \"eigenstates,\" those objects are not vectors in the space. They are some unknown, unspecified object in some unknown and extended space.\n\nFor example, start with the position operator $M$ on an interval $[0,1]$. This operator is multiplication by $x$, meaning $(Mf)(x)=xf(x)$. This is a perfectly legitimate selfadjoint linear operator on the Hilbert space $L^2[0,1]$, but it has no eigenvectors, meaning that there are no non-trivial functions $f \\in L^2[0,1]$ for which $Mf = \\lambda f$, regardless of the choice of $\\lambda$. The operator $M$ has \"continuous spectrum\" but no eigenvectors or eigenvalues. There just are no such \"eigenstate\" objects.\n\nPhysicists typically introduce the $\\delta$ function into such a space (which is logically inconsistent) and claim that $(M \\delta_{y})(x) = x\\delta(x-y)=y\\delta_{y}$. For many reasons, on many levels, that is Mathematical non-sense, and cannot be salvaged within $L^2[0,1]$ in a logically consistent way. The axioms of Quantum use Hilbert spaces, but $\\delta$ functions cannot live in spaces such as $L^2[0,1]$ because two functions that are equal a.e. in $L^2[0,1]$ are identical, which means that pointwise values have no meaning for elements of $L^2[0,1]$. Dirac knew this, but he liked the intuition.\n\nJohn von Neumann, who was a contemporary of P.A.M. Dirac, created consistent ways of dealing with the selfadjoint operators of Physics through the Spectral Theorem, and this was available to Dirac at the time; but Dirac chose an intuitive presentation over logical consistency. And it wasn't because rigorous Mathematics was unavailable at the time to handle everything; the correct Math was there.\n\nAll of the Hilbert spaces of Quantum must be separable, meaning that the space contains a countable dense subset. The reason for this has to do with constructibility, and being able to bootstrap to general answers through finite approximations. $L^2[0,1]$ is a separable space. One of the consequences of having a separable space, is that an orthonormal basis of such a space is always finite or countably infinite. You cannot have mutually orthogonal objects that are indexed by an interval of the real line, for example. That's impossible in separable spaces. And, if you have a selfadjoint operator $M$ on a Hilbert space, and you have $Mf=\\lambda f$ and $Mg = \\mu g$ for $\\lambda\\ne \\mu$, then $(f,g)=0$ must hold. So, a selfadjoint operator on a Hilbert space that is allowed in Quantum Mechanics cannot have more than a countable number of actual eigenvalues. That's a consequence of correct axiomatic systems for Quantum. You can have continuous spectrum, but that's different than having a continuum of eigenvalues. If you enlarge the space to allow for such things, then you lose the ability to approximate in a finite way, which disconnects the theory from a setting where finite approximation makes sense.\n\nSorry to disappoint you. The theorem you have--as stated--is not true. Correct spectral theory gives you essentially the same thing, but not exactly that.\n\n• is there a way to recognize that a normal operator on a separable Hilbert space is (or is not) of the form $T x = \\sum_{n=1}^\\infty \\lambda_n \\langle e_n,x\\rangle e_n$ for some orthonormal basis $(e_n)$ ? if say $T$ is compact, it automatically has a SVD, and hence if it is normal it has this form, right ? – reuns May 14 '16 at 21:26\n• @user1952009 : Operators with a continuous spectrum would not be compact. You can have a bounded position observable on a bounded interval, but it is not compact. Compact resolvents make more sense, such as for a Hamiltonian. – DisintegratingByParts May 14 '16 at 21:31\n• \"and you say $\\delta_x$ is not so consistent\". the operator $P_x : f \\to \\langle \\delta_x, f \\rangle$ with $\\|f\\|^2 = \\int_{X} |P_x f|^2 d\\mu$ is a definition of a Hilbert space space of functions. if all the $P_x$ are bounded then we are probably in $l^2(X)$, but if they are not then $P_x$ can be defined as some sort of weak-* limit of a sequence of operators of $H^*$ ? is there a problem with those considerations ? – reuns May 14 '16 at 21:44\n• @user1952009 : There are all kinds of extended structures designed to salvage the Dirac idea in some way for cases of continuous spectrum, but you can't do this in a separable Hilbert space. Sometimes it's just better to start doing correct Math, live without the \"ideals,\" but have a consistent system that really isn't any more difficult to deal with than the kludged-up alternative. – DisintegratingByParts May 14 '16 at 21:50"
] |
[
null
] |
{"ft_lang_label":"__label__en","ft_lang_prob":0.92789966,"math_prob":0.992154,"size":3223,"snap":"2019-26-2019-30","text_gpt3_token_len":766,"char_repetition_ratio":0.10779745,"word_repetition_ratio":0.0,"special_character_ratio":0.22742786,"punctuation_ratio":0.11736334,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.998745,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-06-16T04:47:49Z\",\"WARC-Record-ID\":\"<urn:uuid:aa805acc-02e3-442c-afe0-2dfc2069ba8a>\",\"Content-Length\":\"136583\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:fc418368-b3c6-407a-a8ad-5a7fb6d19495>\",\"WARC-Concurrent-To\":\"<urn:uuid:410e9351-8656-4837-91bf-86041e282e11>\",\"WARC-IP-Address\":\"151.101.65.69\",\"WARC-Target-URI\":\"https://math.stackexchange.com/questions/1785361/diagonal-operators-on-infinite-dimensional-hilbert-spaces\",\"WARC-Payload-Digest\":\"sha1:FE6ZPY6TWZQ7AD2ZTPYIVR5LZS753BZQ\",\"WARC-Block-Digest\":\"sha1:JCKLR7LJI77E2S2ESK2PCF37CAINLWUA\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-26/CC-MAIN-2019-26_segments_1560627997731.69_warc_CC-MAIN-20190616042701-20190616064701-00378.warc.gz\"}"}
|
https://tex.stackexchange.com/questions/178946/better-automatic-spacing-of-differential-d
|
[
"Better automatic spacing of differential d?\n\nThis question was basically about whether the differential d should be upright or italicized, not about how to achieve that in tex. However, this useful answer did suggest something like the following for typesetting it:\n\n\\newcommand{\\der}{\\operatorname{d\\!}}\n\nThis is what I've been doing for a while, but I find that there are a few cases where the spacing is wrong. This code\n\n\\der(x^7) \\quad \\der x \\quad \\operatorname{d} (x^7) \\quad \\operatorname{d} x\n\ngives output that looks like this:",
null,
"The negative thin space looks good for typesetting dx, but bad for d(x^7). Is there a good way to define a single \\der macro that automagically gets both cases right, or is my best option to define two macros and pick one or the other as needed?\n\n• You should have a look at package physics. – Johannes_B May 18 '14 at 19:07\n• A common definition seen in many, many places is \\newcommand*\\der{\\mathop{}\\!\\mathrm{d}}. – Manuel May 18 '14 at 19:23\n\n\\operatorname turns d into an operator. If an ordinary math atom follows (x), then TeX sets a thin space that is negated by \\!. However, TeX does not set a space after an operator if the following math atom belongs to categories \"open\", \"close\", \"punct\", or \"inner\" (\\scriptstyle/\\scriptscriptstyle). A fix would be to add an empty math ord atom (\\mathord{} or an empty sub formula {}), then TeX always sets a thin space, canceled by \\!. Macro \\der behaves on the left-hand side as an operator and on its right-hand side as an ordinary math atom. Now both cases work as expected:\n\n\\documentclass{article}\n\\usepackage{amsmath}\n\\newcommand{\\der}{\\operatorname{d\\!}{}}\n\\begin{document}\n$\\der(x^7) \\quad \\der x$\n\\end{document}",
null,
"Variation\n\nThe d can be put both in the operator atom or the ordinary atom, as suggested in the comment of Manuel and the comment of egreg (without \\mathrm):\n\n\\mathop{}\\!\\mathrm{d}\n\nFonts for d\n\n• \\lim, \\sin and friends are using the font \\operator@font. Package amsmath then provides macro \\operatorname. But the former can also be used without an additional package:\n\n\\makeatletter\n\\newcommand*{\\der}{%\n\\mathop{\\kern\\z@\\operator@font d}\\!{}%\n}\n\\makeatother\n\n\\kern\\z@ prevents that \\mathop centers the symbol.\n\nOr with d in the ordinary atom:\n\n\\makeatletter\n\\newcommand*{\\der}{%\n\\mathop{}\\!{\\operator@font d}%\n}\n\\makeatother\n\n• Font \\mathrm is easier to use (no @ in the name):\n\n\\newcommand*{\\der}{%\n\\mathop{}\\!\\mathrm{d}%\n}\n\nor a little more complex, again the \\kern prevents vertical centering:\n\n\\newcommand*{\\der}{%\n\\mathop{\\kern0pt\\mathrm{d}}\\!{}%\n}\n\n• The italics variant:\n\n\\newcommand*{\\der}{%\n\\mathop{}\\!d%\n}%\n\nor\n\n\\newcommand*{\\der}{%\n\\mathop{\\kern0pt d}\\!{}%\n}",
null,
"• Is there any advantage or disadvantage of this compared to the definition suggested by Manuel in his comment? – Ben Crowell May 18 '14 at 19:30\n• @BenCrowell: Both versions consist of a math operator atom, negative kerning that cancels the automatically inserted thin space, and an ordinary atom. In the answer the d is put int the operator atom, the comment puts it in the ordinary atom, but the spacing is the same. – Heiko Oberdiek May 18 '14 at 19:34\n• @HeikoOberdiek Did you try something like $\\der(x+y)$? – egreg May 18 '14 at 19:34\n• @HeikoOberdiek Oh, I didn't see the {}. I still prefer \\mathop{}\\!d; it's shorter and doesn't require other packages. – egreg May 18 '14 at 20:03\n• @egreg: I have now added lots of variations. – Heiko Oberdiek May 18 '14 at 20:45\n\nPackages are written by good and nice people keeping lazy and ignorant people like me in mind. Instead of re-inventing things, it is better start finding a suitable package and use it. In this case, physics package (as noted by Johannes) offers \\dd macro. A screen shot of the relevant part of the physics documentation:",
null,
"And a sample code:\n\n\\documentclass{article}\n\\usepackage{physics}\n\\begin{document}\n$\\dd(x^7) \\quad \\dd x$\n\\end{document}",
null,
"• After seeing how \\dd is defined in the package, I wish I didn't try reading it. – egreg May 19 '14 at 7:53\n• @egreg ???. Is it so wrong? Can you add explanations to the answers please, if you have some time? It is always good to know what is wrong in what I am using :(. I use physics too much. – user11232 May 19 '14 at 22:38\n• I can't see what's the point in having a complicated definition when the usual one (\\mathop{}\\!d, add \\mathrm, if you wish) works flawlessly in all cases; there's no advantage having \\dd{x} instead of the simpler (and clearer) \\dd^3 x. There's surely no advantage in requiring \\dd{x} instead of \\dd x; of course with the usual definition, braces are optional. – egreg May 20 '14 at 9:34"
] |
[
null,
"https://i.stack.imgur.com/3Rfai.jpg",
null,
"https://i.stack.imgur.com/KKV5M.png",
null,
"https://i.stack.imgur.com/y2Vew.png",
null,
"https://i.stack.imgur.com/YZSWf.png",
null,
"https://i.stack.imgur.com/YfV2j.png",
null
] |
{"ft_lang_label":"__label__en","ft_lang_prob":0.9162114,"math_prob":0.74505997,"size":847,"snap":"2019-43-2019-47","text_gpt3_token_len":228,"char_repetition_ratio":0.09252669,"word_repetition_ratio":0.0,"special_character_ratio":0.26564345,"punctuation_ratio":0.099415205,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96229357,"pos_list":[0,1,2,3,4,5,6,7,8,9,10],"im_url_duplicate_count":[null,6,null,6,null,6,null,6,null,6,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-10-21T15:27:27Z\",\"WARC-Record-ID\":\"<urn:uuid:390b9298-e3cc-46c8-886d-7a40eba3d2cc>\",\"Content-Length\":\"158839\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:9bab0e41-cf8b-4057-8081-9558d4402c93>\",\"WARC-Concurrent-To\":\"<urn:uuid:19a42265-ba45-4018-bfac-9af4fef78791>\",\"WARC-IP-Address\":\"151.101.129.69\",\"WARC-Target-URI\":\"https://tex.stackexchange.com/questions/178946/better-automatic-spacing-of-differential-d\",\"WARC-Payload-Digest\":\"sha1:X4FFJ2I42D2CAZ44OGLLPFUIPOTMT2KN\",\"WARC-Block-Digest\":\"sha1:KBGWGN6VWNYASWZNLQXSFYRS4RKHJEYU\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-43/CC-MAIN-2019-43_segments_1570987779528.82_warc_CC-MAIN-20191021143945-20191021171445-00283.warc.gz\"}"}
|
http://www2.macaulay2.com/Macaulay2/doc/Macaulay2-1.19/share/doc/Macaulay2/RelativeCanonicalResolution/html/_can__Curve__With__Fixed__Scroll.html
|
[
"# canCurveWithFixedScroll -- Computes a g-nodal canonical curve with a degree k line bundle on a normalized scroll\n\n## Synopsis\n\n• Usage:\nIcan=canCurveWithFixedScroll(g,k,n)\n• Inputs:\n• g, an integer, the genus of the curve\n• k, an integer, the degree of the line bundle on C\n• n, an integer, the integer defining the characteristic p ($\\ge n$) of the ground field\n• Outputs:\n• ICan, an ideal, the ideal of the canonical curve\n\n## Description\n\nThis function computes the ideal of a g-nodal canonical curve with a degree k<g line bundle that lies on a normalized scroll. The construction of such curves is based on the Macaulay2 package kGonalNodalCurves\n\n i1 : (g,k,n) = (8,5,1000); i2 : Ican = canCurveWithFixedScroll(g,k,n); ZZ o2 : Ideal of ----[t ..t ] 1009 0 7 i3 : genus Ican, degree Ican, dim Ican o3 = (8, 14, 2) o3 : Sequence i4 : betti res(Ican, DegreeLimit => 1) 0 1 2 3 o4 = total: 1 15 35 21 0: 1 . . . 1: . 15 35 21 o4 : BettiTally i5 : Phi = matrix{{t_0,t_2,t_4,t_6},{t_1,t_3,t_5,t_7}} o5 = | t_0 t_2 t_4 t_6 | | t_1 t_3 t_5 t_7 | ZZ 2 ZZ 4 o5 : Matrix (----[t ..t ]) <--- (----[t ..t ]) 1009 0 7 1009 0 7 i6 : Iscroll = minors(2,Phi); ZZ o6 : Ideal of ----[t ..t ] 1009 0 7 i7 : Ican + Iscroll == Ican o7 = true"
] |
[
null
] |
{"ft_lang_label":"__label__en","ft_lang_prob":0.509294,"math_prob":0.9644056,"size":852,"snap":"2022-40-2023-06","text_gpt3_token_len":365,"char_repetition_ratio":0.11438679,"word_repetition_ratio":0.06779661,"special_character_ratio":0.48591548,"punctuation_ratio":0.24401914,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98189133,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-02-01T16:47:54Z\",\"WARC-Record-ID\":\"<urn:uuid:d98794d4-bbee-462a-9a9e-542114badb57>\",\"Content-Length\":\"7493\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:31c5d7e8-ddf2-4303-9239-0b008696ca68>\",\"WARC-Concurrent-To\":\"<urn:uuid:b59d490c-4cef-4fea-af8a-7d2eb0487799>\",\"WARC-IP-Address\":\"107.170.188.204\",\"WARC-Target-URI\":\"http://www2.macaulay2.com/Macaulay2/doc/Macaulay2-1.19/share/doc/Macaulay2/RelativeCanonicalResolution/html/_can__Curve__With__Fixed__Scroll.html\",\"WARC-Payload-Digest\":\"sha1:QB5I6ZD5PPE7V3LPRYBANVGYCFHCTHAZ\",\"WARC-Block-Digest\":\"sha1:ZGQGEL5VKJCUXIB7OCRSIY7MS45T6PPZ\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764499946.80_warc_CC-MAIN-20230201144459-20230201174459-00404.warc.gz\"}"}
|
https://patientsaver.savingadvice.com/2019/02/28/kkk_218447/
|
[
"User Real IP - 35.173.35.159\n```Array\n(\n => Array\n(\n => 182.68.68.92\n)\n\n => Array\n(\n => 101.0.41.201\n)\n\n => Array\n(\n => 43.225.98.123\n)\n\n => Array\n(\n => 2.58.194.139\n)\n\n => Array\n(\n => 46.119.197.104\n)\n\n => Array\n(\n => 45.249.8.93\n)\n\n => Array\n(\n => 103.12.135.72\n)\n\n => Array\n(\n => 157.35.243.216\n)\n\n => Array\n(\n => 209.107.214.176\n)\n\n => Array\n(\n => 5.181.233.166\n)\n\n => Array\n(\n => 106.201.10.100\n)\n\n => Array\n(\n => 36.90.55.39\n)\n\n => Array\n(\n => 119.154.138.47\n)\n\n => Array\n(\n => 51.91.31.157\n)\n\n => Array\n(\n => 182.182.65.216\n)\n\n => Array\n(\n => 157.35.252.63\n)\n\n => Array\n(\n => 14.142.34.163\n)\n\n => Array\n(\n => 178.62.43.135\n)\n\n => Array\n(\n => 43.248.152.148\n)\n\n => Array\n(\n => 222.252.104.114\n)\n\n => Array\n(\n => 209.107.214.168\n)\n\n => Array\n(\n => 103.99.199.250\n)\n\n => Array\n(\n => 178.62.72.160\n)\n\n => Array\n(\n => 27.6.1.170\n)\n\n => Array\n(\n => 182.69.249.219\n)\n\n => Array\n(\n => 110.93.228.86\n)\n\n => Array\n(\n => 72.255.1.98\n)\n\n => Array\n(\n => 182.73.111.98\n)\n\n => Array\n(\n => 45.116.117.11\n)\n\n => Array\n(\n => 122.15.78.189\n)\n\n => Array\n(\n => 14.167.188.234\n)\n\n => Array\n(\n => 223.190.4.202\n)\n\n => Array\n(\n => 202.173.125.19\n)\n\n => Array\n(\n => 103.255.5.32\n)\n\n => Array\n(\n => 39.37.145.103\n)\n\n => Array\n(\n => 140.213.26.249\n)\n\n => Array\n(\n => 45.118.166.85\n)\n\n => Array\n(\n => 102.166.138.255\n)\n\n => Array\n(\n => 77.111.246.234\n)\n\n => Array\n(\n => 45.63.6.196\n)\n\n => Array\n(\n => 103.250.147.115\n)\n\n => Array\n(\n => 223.185.30.99\n)\n\n => Array\n(\n => 103.122.168.108\n)\n\n => Array\n(\n => 123.136.203.21\n)\n\n => Array\n(\n => 171.229.243.63\n)\n\n => Array\n(\n => 153.149.98.149\n)\n\n => Array\n(\n => 223.238.93.15\n)\n\n => Array\n(\n => 178.62.113.166\n)\n\n => Array\n(\n => 101.162.0.153\n)\n\n => Array\n(\n => 121.200.62.114\n)\n\n => Array\n(\n => 14.248.77.252\n)\n\n => Array\n(\n => 95.142.117.29\n)\n\n => Array\n(\n => 150.129.60.107\n)\n\n => Array\n(\n => 94.205.243.22\n)\n\n => Array\n(\n => 115.42.71.143\n)\n\n => Array\n(\n => 117.217.195.59\n)\n\n => Array\n(\n => 182.77.112.56\n)\n\n => Array\n(\n => 182.77.112.108\n)\n\n => Array\n(\n => 41.80.69.10\n)\n\n => Array\n(\n => 117.5.222.121\n)\n\n => Array\n(\n => 103.11.0.38\n)\n\n => Array\n(\n => 202.173.127.140\n)\n\n => Array\n(\n => 49.249.249.50\n)\n\n => Array\n(\n => 116.72.198.211\n)\n\n => Array\n(\n => 223.230.54.53\n)\n\n => Array\n(\n => 102.69.228.74\n)\n\n => Array\n(\n => 39.37.251.89\n)\n\n => Array\n(\n => 39.53.246.141\n)\n\n => Array\n(\n => 39.57.182.72\n)\n\n => Array\n(\n => 209.58.130.210\n)\n\n => Array\n(\n => 104.131.75.86\n)\n\n => Array\n(\n => 106.212.131.255\n)\n\n => Array\n(\n => 106.212.132.127\n)\n\n => Array\n(\n => 223.190.4.60\n)\n\n => Array\n(\n => 103.252.116.252\n)\n\n => Array\n(\n => 103.76.55.182\n)\n\n => Array\n(\n => 45.118.166.70\n)\n\n => Array\n(\n => 103.93.174.215\n)\n\n => Array\n(\n => 5.62.62.142\n)\n\n => Array\n(\n => 182.179.158.156\n)\n\n => Array\n(\n => 39.57.255.12\n)\n\n => Array\n(\n => 39.37.178.37\n)\n\n => Array\n(\n => 182.180.165.211\n)\n\n => Array\n(\n => 119.153.135.17\n)\n\n => Array\n(\n => 72.255.15.244\n)\n\n => Array\n(\n => 139.180.166.181\n)\n\n => Array\n(\n => 70.119.147.111\n)\n\n => Array\n(\n => 106.210.40.83\n)\n\n => Array\n(\n => 14.190.70.91\n)\n\n => Array\n(\n => 202.125.156.82\n)\n\n => Array\n(\n => 115.42.68.38\n)\n\n => Array\n(\n => 102.167.13.108\n)\n\n => Array\n(\n => 117.217.192.130\n)\n\n => Array\n(\n => 205.185.223.156\n)\n\n => Array\n(\n => 171.224.180.29\n)\n\n => Array\n(\n => 45.127.45.68\n)\n\n => Array\n(\n => 195.206.183.232\n)\n\n => Array\n(\n => 49.32.52.115\n)\n\n => Array\n(\n => 49.207.49.223\n)\n\n => Array\n(\n => 45.63.29.61\n)\n\n => Array\n(\n => 103.245.193.214\n)\n\n => Array\n(\n => 39.40.236.69\n)\n\n => Array\n(\n => 62.80.162.111\n)\n\n => Array\n(\n => 45.116.232.56\n)\n\n => Array\n(\n => 45.118.166.91\n)\n\n => Array\n(\n => 180.92.230.234\n)\n\n => Array\n(\n => 157.40.57.160\n)\n\n => Array\n(\n => 110.38.38.130\n)\n\n => Array\n(\n => 72.255.57.183\n)\n\n => Array\n(\n => 182.68.81.85\n)\n\n => Array\n(\n => 39.57.202.122\n)\n\n => Array\n(\n => 119.152.154.36\n)\n\n => Array\n(\n => 5.62.62.141\n)\n\n => Array\n(\n => 119.155.54.232\n)\n\n => Array\n(\n => 39.37.141.22\n)\n\n => Array\n(\n => 183.87.12.225\n)\n\n => Array\n(\n => 107.170.127.117\n)\n\n => Array\n(\n => 125.63.124.49\n)\n\n => Array\n(\n => 39.42.191.3\n)\n\n => Array\n(\n => 116.74.24.72\n)\n\n => Array\n(\n => 46.101.89.227\n)\n\n => Array\n(\n => 202.173.125.247\n)\n\n => Array\n(\n => 39.42.184.254\n)\n\n => Array\n(\n => 115.186.165.132\n)\n\n => Array\n(\n => 39.57.206.126\n)\n\n => Array\n(\n => 103.245.13.145\n)\n\n => Array\n(\n => 202.175.246.43\n)\n\n => Array\n(\n => 192.140.152.150\n)\n\n => Array\n(\n => 202.88.250.103\n)\n\n => Array\n(\n => 103.248.94.207\n)\n\n => Array\n(\n => 77.73.66.101\n)\n\n => Array\n(\n => 104.131.66.8\n)\n\n => Array\n(\n => 113.186.161.97\n)\n\n => Array\n(\n => 222.254.5.7\n)\n\n => Array\n(\n => 223.233.67.247\n)\n\n => Array\n(\n => 171.249.116.146\n)\n\n => Array\n(\n => 47.30.209.71\n)\n\n => Array\n(\n => 202.134.13.130\n)\n\n => Array\n(\n => 27.6.135.7\n)\n\n => Array\n(\n => 107.170.186.79\n)\n\n => Array\n(\n => 103.212.89.171\n)\n\n => Array\n(\n => 117.197.9.77\n)\n\n => Array\n(\n => 122.176.206.233\n)\n\n => Array\n(\n => 192.227.253.222\n)\n\n => Array\n(\n => 182.188.224.119\n)\n\n => Array\n(\n => 14.248.70.74\n)\n\n => Array\n(\n => 42.118.219.169\n)\n\n => Array\n(\n => 110.39.146.170\n)\n\n => Array\n(\n => 119.160.66.143\n)\n\n => Array\n(\n => 103.248.95.130\n)\n\n => Array\n(\n => 27.63.152.208\n)\n\n => Array\n(\n => 49.207.114.96\n)\n\n => Array\n(\n => 102.166.23.214\n)\n\n => Array\n(\n => 175.107.254.73\n)\n\n => Array\n(\n => 103.10.227.214\n)\n\n => Array\n(\n => 202.143.115.89\n)\n\n => Array\n(\n => 110.93.227.187\n)\n\n => Array\n(\n => 103.140.31.60\n)\n\n => Array\n(\n => 110.37.231.46\n)\n\n => Array\n(\n => 39.36.99.238\n)\n\n => Array\n(\n => 157.37.140.26\n)\n\n => Array\n(\n => 43.246.202.226\n)\n\n => Array\n(\n => 137.97.8.143\n)\n\n => Array\n(\n => 182.65.52.242\n)\n\n => Array\n(\n => 115.42.69.62\n)\n\n => Array\n(\n => 14.143.254.58\n)\n\n => Array\n(\n => 223.179.143.236\n)\n\n => Array\n(\n => 223.179.143.249\n)\n\n => Array\n(\n => 103.143.7.54\n)\n\n => Array\n(\n => 223.179.139.106\n)\n\n => Array\n(\n => 39.40.219.90\n)\n\n => Array\n(\n => 45.115.141.231\n)\n\n => Array\n(\n => 120.29.100.33\n)\n\n => Array\n(\n => 112.196.132.5\n)\n\n => Array\n(\n => 202.163.123.153\n)\n\n => Array\n(\n => 5.62.58.146\n)\n\n => Array\n(\n => 39.53.216.113\n)\n\n => Array\n(\n => 42.111.160.73\n)\n\n => Array\n(\n => 107.182.231.213\n)\n\n => Array\n(\n => 119.82.94.120\n)\n\n => Array\n(\n => 178.62.34.82\n)\n\n => Array\n(\n => 203.122.6.18\n)\n\n => Array\n(\n => 157.42.38.251\n)\n\n => Array\n(\n => 45.112.68.222\n)\n\n => Array\n(\n => 49.206.212.122\n)\n\n => Array\n(\n => 104.236.70.228\n)\n\n => Array\n(\n => 42.111.34.243\n)\n\n => Array\n(\n => 84.241.19.186\n)\n\n => Array\n(\n => 89.187.180.207\n)\n\n => Array\n(\n => 104.243.212.118\n)\n\n => Array\n(\n => 104.236.55.136\n)\n\n => Array\n(\n => 106.201.16.163\n)\n\n => Array\n(\n => 46.101.40.25\n)\n\n => Array\n(\n => 45.118.166.94\n)\n\n => Array\n(\n => 49.36.128.102\n)\n\n => Array\n(\n => 14.142.193.58\n)\n\n => Array\n(\n => 212.79.124.176\n)\n\n => Array\n(\n => 45.32.191.194\n)\n\n => Array\n(\n => 105.112.107.46\n)\n\n => Array\n(\n => 106.201.14.8\n)\n\n => Array\n(\n => 110.93.240.65\n)\n\n => Array\n(\n => 27.96.95.177\n)\n\n => Array\n(\n => 45.41.134.35\n)\n\n => Array\n(\n => 180.151.13.110\n)\n\n => Array\n(\n => 101.53.242.89\n)\n\n => Array\n(\n => 115.186.3.110\n)\n\n => Array\n(\n => 171.49.185.242\n)\n\n => Array\n(\n => 115.42.70.24\n)\n\n => Array\n(\n => 45.128.188.43\n)\n\n => Array\n(\n => 103.140.129.63\n)\n\n => Array\n(\n => 101.50.113.147\n)\n\n => Array\n(\n => 103.66.73.30\n)\n\n => Array\n(\n => 117.247.193.169\n)\n\n => Array\n(\n => 120.29.100.94\n)\n\n => Array\n(\n => 42.109.154.39\n)\n\n => Array\n(\n => 122.173.155.150\n)\n\n => Array\n(\n => 45.115.104.53\n)\n\n => Array\n(\n => 116.74.29.84\n)\n\n => Array\n(\n => 101.50.125.34\n)\n\n => Array\n(\n => 45.118.166.80\n)\n\n => Array\n(\n => 91.236.184.27\n)\n\n => Array\n(\n => 113.167.185.120\n)\n\n => Array\n(\n => 27.97.66.222\n)\n\n => Array\n(\n => 43.247.41.117\n)\n\n => Array\n(\n => 23.229.16.227\n)\n\n => Array\n(\n => 14.248.79.209\n)\n\n => Array\n(\n => 117.5.194.26\n)\n\n => Array\n(\n => 117.217.205.41\n)\n\n => Array\n(\n => 114.79.169.99\n)\n\n => Array\n(\n => 103.55.60.97\n)\n\n => Array\n(\n => 182.75.89.210\n)\n\n => Array\n(\n => 77.73.66.109\n)\n\n => Array\n(\n => 182.77.126.139\n)\n\n => Array\n(\n => 14.248.77.166\n)\n\n => Array\n(\n => 157.35.224.133\n)\n\n => Array\n(\n => 183.83.38.27\n)\n\n => Array\n(\n => 182.68.4.77\n)\n\n => Array\n(\n => 122.177.130.234\n)\n\n => Array\n(\n => 103.24.99.99\n)\n\n => Array\n(\n => 103.91.127.66\n)\n\n => Array\n(\n => 41.90.34.240\n)\n\n => Array\n(\n => 49.205.77.102\n)\n\n => Array\n(\n => 103.248.94.142\n)\n\n => Array\n(\n => 104.143.92.170\n)\n\n => Array\n(\n => 219.91.157.114\n)\n\n => Array\n(\n => 223.190.88.22\n)\n\n => Array\n(\n => 223.190.86.232\n)\n\n => Array\n(\n => 39.41.172.80\n)\n\n => Array\n(\n => 124.107.206.5\n)\n\n => Array\n(\n => 139.167.180.224\n)\n\n => Array\n(\n => 93.76.64.248\n)\n\n => Array\n(\n => 65.216.227.119\n)\n\n => Array\n(\n => 223.190.119.141\n)\n\n => Array\n(\n => 110.93.237.179\n)\n\n => Array\n(\n => 41.90.7.85\n)\n\n => Array\n(\n => 103.100.6.26\n)\n\n => Array\n(\n => 104.140.83.13\n)\n\n => Array\n(\n => 223.190.119.133\n)\n\n => Array\n(\n => 119.152.150.87\n)\n\n => Array\n(\n => 103.125.130.147\n)\n\n => Array\n(\n => 27.6.5.52\n)\n\n => Array\n(\n => 103.98.188.26\n)\n\n => Array\n(\n => 39.35.121.81\n)\n\n => Array\n(\n => 74.119.146.182\n)\n\n => Array\n(\n => 5.181.233.162\n)\n\n => Array\n(\n => 157.39.18.60\n)\n\n => Array\n(\n => 1.187.252.25\n)\n\n => Array\n(\n => 39.42.145.59\n)\n\n => Array\n(\n => 39.35.39.198\n)\n\n => Array\n(\n => 49.36.128.214\n)\n\n => Array\n(\n => 182.190.20.56\n)\n\n => Array\n(\n => 122.180.249.189\n)\n\n => Array\n(\n => 117.217.203.107\n)\n\n => Array\n(\n => 103.70.82.241\n)\n\n => Array\n(\n => 45.118.166.68\n)\n\n => Array\n(\n => 122.180.168.39\n)\n\n => Array\n(\n => 149.28.67.254\n)\n\n => Array\n(\n => 223.233.73.8\n)\n\n => Array\n(\n => 122.167.140.0\n)\n\n => Array\n(\n => 95.158.51.55\n)\n\n => Array\n(\n => 27.96.95.134\n)\n\n => Array\n(\n => 49.206.214.53\n)\n\n => Array\n(\n => 212.103.49.92\n)\n\n => Array\n(\n => 122.177.115.101\n)\n\n => Array\n(\n => 171.50.187.124\n)\n\n => Array\n(\n => 122.164.55.107\n)\n\n => Array\n(\n => 98.114.217.204\n)\n\n => Array\n(\n => 106.215.10.54\n)\n\n => Array\n(\n => 115.42.68.28\n)\n\n => Array\n(\n => 104.194.220.87\n)\n\n => Array\n(\n => 103.137.84.170\n)\n\n => Array\n(\n => 61.16.142.110\n)\n\n => Array\n(\n => 212.103.49.85\n)\n\n => Array\n(\n => 39.53.248.162\n)\n\n => Array\n(\n => 203.122.40.214\n)\n\n => Array\n(\n => 117.217.198.72\n)\n\n => Array\n(\n => 115.186.191.203\n)\n\n => Array\n(\n => 120.29.100.199\n)\n\n => Array\n(\n => 45.151.237.24\n)\n\n => Array\n(\n => 223.190.125.232\n)\n\n => Array\n(\n => 41.80.151.17\n)\n\n => Array\n(\n => 23.111.188.5\n)\n\n => Array\n(\n => 223.190.125.216\n)\n\n => Array\n(\n => 103.217.133.119\n)\n\n => Array\n(\n => 103.198.173.132\n)\n\n => Array\n(\n => 47.31.155.89\n)\n\n => Array\n(\n => 223.190.20.253\n)\n\n => Array\n(\n => 104.131.92.125\n)\n\n => Array\n(\n => 223.190.19.152\n)\n\n => Array\n(\n => 103.245.193.191\n)\n\n => Array\n(\n => 106.215.58.255\n)\n\n => Array\n(\n => 119.82.83.238\n)\n\n => Array\n(\n => 106.212.128.138\n)\n\n => Array\n(\n => 139.167.237.36\n)\n\n => Array\n(\n => 222.124.40.250\n)\n\n => Array\n(\n => 134.56.185.169\n)\n\n => Array\n(\n => 54.255.226.31\n)\n\n => Array\n(\n => 137.97.162.31\n)\n\n => Array\n(\n => 95.185.21.191\n)\n\n => Array\n(\n => 171.61.168.151\n)\n\n => Array\n(\n => 137.97.184.4\n)\n\n => Array\n(\n => 106.203.151.202\n)\n\n => Array\n(\n => 39.37.137.0\n)\n\n => Array\n(\n => 45.118.166.66\n)\n\n => Array\n(\n => 14.248.105.100\n)\n\n => Array\n(\n => 106.215.61.185\n)\n\n => Array\n(\n => 202.83.57.179\n)\n\n => Array\n(\n => 89.187.182.176\n)\n\n => Array\n(\n => 49.249.232.198\n)\n\n => Array\n(\n => 132.154.95.236\n)\n\n => Array\n(\n => 223.233.83.230\n)\n\n => Array\n(\n => 183.83.153.14\n)\n\n => Array\n(\n => 125.63.72.210\n)\n\n => Array\n(\n => 207.174.202.11\n)\n\n => Array\n(\n => 119.95.88.59\n)\n\n => Array\n(\n => 122.170.14.150\n)\n\n => Array\n(\n => 45.118.166.75\n)\n\n => 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Array\n(\n => 58.65.222.10\n)\n\n => Array\n(\n => 202.47.43.86\n)\n\n => Array\n(\n => 106.206.223.234\n)\n\n => Array\n(\n => 5.195.153.158\n)\n\n => Array\n(\n => 223.227.127.243\n)\n\n => Array\n(\n => 103.165.12.222\n)\n\n => Array\n(\n => 49.36.185.189\n)\n\n => Array\n(\n => 59.96.92.57\n)\n\n => Array\n(\n => 203.194.104.235\n)\n\n => Array\n(\n => 122.177.72.33\n)\n\n => Array\n(\n => 106.213.126.40\n)\n\n => Array\n(\n => 45.127.232.69\n)\n\n => Array\n(\n => 156.146.59.39\n)\n\n => Array\n(\n => 103.21.184.11\n)\n\n => Array\n(\n => 106.212.47.59\n)\n\n => Array\n(\n => 182.179.137.235\n)\n\n => Array\n(\n => 49.36.178.154\n)\n\n => Array\n(\n => 171.48.7.128\n)\n\n => Array\n(\n => 119.160.57.96\n)\n\n => Array\n(\n => 197.210.79.92\n)\n\n => Array\n(\n => 36.255.45.87\n)\n\n => Array\n(\n => 47.31.219.47\n)\n\n => Array\n(\n => 122.161.51.160\n)\n\n => Array\n(\n => 103.217.123.129\n)\n\n => Array\n(\n => 59.153.16.12\n)\n\n => Array\n(\n => 103.92.43.226\n)\n\n => Array\n(\n => 47.31.139.139\n)\n\n => Array\n(\n => 210.2.140.18\n)\n\n => Array\n(\n => 106.210.33.219\n)\n\n => Array\n(\n => 175.107.203.34\n)\n\n => Array\n(\n => 146.196.32.144\n)\n\n => Array\n(\n => 103.12.133.121\n)\n\n => Array\n(\n => 103.59.208.182\n)\n\n => Array\n(\n => 157.37.190.232\n)\n\n => Array\n(\n => 106.195.35.201\n)\n\n => Array\n(\n => 27.122.14.83\n)\n\n => Array\n(\n => 194.193.44.5\n)\n\n => Array\n(\n => 5.62.43.245\n)\n\n => Array\n(\n => 103.53.80.50\n)\n\n => Array\n(\n => 47.29.142.233\n)\n\n => Array\n(\n => 154.6.20.63\n)\n\n => Array\n(\n => 173.245.203.128\n)\n\n => Array\n(\n => 103.77.43.231\n)\n\n => Array\n(\n => 5.107.166.235\n)\n\n => Array\n(\n => 106.212.44.123\n)\n\n => Array\n(\n => 157.41.60.93\n)\n\n => Array\n(\n => 27.58.179.79\n)\n\n => Array\n(\n => 157.37.167.144\n)\n\n => Array\n(\n => 119.160.57.115\n)\n\n => Array\n(\n => 122.161.53.224\n)\n\n => Array\n(\n => 49.36.233.51\n)\n\n => Array\n(\n => 101.0.32.8\n)\n\n => Array\n(\n => 119.160.103.158\n)\n\n => Array\n(\n => 122.177.79.115\n)\n\n => Array\n(\n => 107.181.166.27\n)\n\n => Array\n(\n => 183.6.0.125\n)\n\n => Array\n(\n => 49.36.186.0\n)\n\n => Array\n(\n => 202.181.5.4\n)\n\n => Array\n(\n => 45.118.165.144\n)\n\n => Array\n(\n => 171.96.157.133\n)\n\n => Array\n(\n => 222.252.51.163\n)\n\n => Array\n(\n => 103.81.215.162\n)\n\n => Array\n(\n => 110.225.93.208\n)\n\n => Array\n(\n => 122.161.48.200\n)\n\n => Array\n(\n => 119.63.138.173\n)\n\n => Array\n(\n => 202.83.58.208\n)\n\n => Array\n(\n => 122.161.53.101\n)\n\n => Array\n(\n => 137.97.95.21\n)\n\n)\n```\nSunshiney day: Save More, Spend Less\n Layout: Blue and Brown (Default) Author's Creation\n Home > Sunshiney day\n\n# Sunshiney day\n\nFebruary 28th, 2019 at 03:20 pm",
null,
"The end of the road, during one of my workweek lunch break walks.\n\nI did my investment report a day early, and I like what I see....my investments have continued to rebound nicely. In fact, my net worth is the highest it's been since Feb 2018 (which was my all-time high).\n\nThe macro look at the month of February shows that year-to-date, I've spent about as much as I've earned. I was ahead last month, but in February I had 2 big expenses that wiped out the January surplus: a \\$650 state tax bill and \\$448 to pay for this season's CSA organic farm market share.\n\nI happened to get out of work early last Tuesday, which gave me the opportunity to attend the meeting of a local environmental action group in town. It's something I wanted to do for a long time but I rarely get out of work in winter early enough to make their meetings, but now work is easing up a bit (as it usually does for the summer).\n\nI was promptly tapped to manage the group's litter pickup days, which they do monthly all season long. So today I'll be driving around to the town's parks, schools and other public places to do a trash inventory so I can organize the next cleanup. I'll also write a press release for the paper and Patch to invite public participation.\n\nI've also decided, on a personal level, to no longer purchase single serving beverages. Not that I buy water, soda or other sugary drinks, but I did on occasion like to buy a case of Bai, which is sweetened with Stevia. I wrote them a letter but otherwise will just stop buying it.\n\nSeems like a nice group. Two of their other big issues have to do with eliminating plastic grocery store bags and the pollinator crisis. I think they're also talking to local restaurants about plastic takeout stuff.\n\n### 5 Responses to “Sunshiney day”\n\n1. rob62521 Says:\n\nWhat a lovely photo. I think my blood pressure dropped just seeing it!\n\nWhat a great thing you are doing to manage the group's litter pick up days. I am shocked at how many people think it is OK to just throw stuff out and leave it.\n\n2. PatientSaver Says:\n\nThank you, rob....I always appreciate your positive and supportive comments.",
null,
"3. creditcardfree Says:\n\nI seem to be the primary person picking up trash on the base track and football field. I can't believe how many people leave trash behind. I've thrown away shoes!! But I've found quite a few dollars in coins, too. Enjoy your new group."
] |
[
null,
"https://www.savingadvice.com/blogs/image.php",
null,
"https://www.savingadvice.com/forums/core/images/smilies/smile.png",
null
] |
{"ft_lang_label":"__label__en","ft_lang_prob":0.9850058,"math_prob":0.99937433,"size":2145,"snap":"2021-43-2021-49","text_gpt3_token_len":539,"char_repetition_ratio":0.08921065,"word_repetition_ratio":0.005063291,"special_character_ratio":0.27319348,"punctuation_ratio":0.1069869,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9997162,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-18T14:22:03Z\",\"WARC-Record-ID\":\"<urn:uuid:41156e8c-ac97-40f8-aa43-e9880ebf3e5d>\",\"Content-Length\":\"316349\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:3dcd3666-85f9-4efc-884b-03b1d2301628>\",\"WARC-Concurrent-To\":\"<urn:uuid:5ed9aef3-66df-49b8-9b0f-c646acf79c86>\",\"WARC-IP-Address\":\"173.231.200.26\",\"WARC-Target-URI\":\"https://patientsaver.savingadvice.com/2019/02/28/kkk_218447/\",\"WARC-Payload-Digest\":\"sha1:SMK5KZGFFLXSVORMWGQO6YCQU6V3PBOA\",\"WARC-Block-Digest\":\"sha1:6KTPFG7LR62B7DKX7FV5UNJ3QS27OAMV\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323585203.61_warc_CC-MAIN-20211018124412-20211018154412-00356.warc.gz\"}"}
|
https://ropensci.org/blog/2014/04/17/plotly/
|
[
"",
null,
"rOpenSci | Make your ggplots shareable, collaborative, and with D3",
null,
"# Make your ggplots shareable, collaborative, and with D3\n\nEditor’s note: This is a guest post by Matt Sundquist from Plot.ly.\n\nGgplotly and Plotly’s R API let you make ggplot2 plots, add `py\\$ggplotly()`, and make your plots interactive, online, and drawn with D3. Let’s make some.\n\n## 🔗 1. Getting Started and Examples\n\nHere is Fisher’s iris data.\n\n``````library(\"ggplot2\")\nggiris <- qplot(Petal.Width, Sepal.Length, data = iris, color = Species)\nprint(ggiris)\n``````",
null,
"Let’s make it in Plotly. Install:\n\n``````install.packages(\"devtools\")\nlibrary(\"devtools\")\ninstall_github(\"plotly\", \"ropensci\")\n``````\n\n``````library(\"plotly\")\n``````\n``````## Loading required package: RCurl\n``````\n\n``````signup(\"new_username\", \"[email protected]\")\n``````\n\nThat should have responded with your new key. Use that to create a plotly interface object, or use ours:\n\n``````py <- plotly(\"RgraphingAPI\", \"ektgzomjbx\")\n``````\n\nIt just works.\n\n``````py\\$ggplotly(ggiris)\n``````\n\nThe call opens a browser tab. Or in an `.Rmd` document, the plot is embedded if you specify the `plotly=TRUE` chunk option (see source). If you’re running this from the source, it makes all the graphs at once in your browser. Reaction my first time: here be dragons.\n\nIf you click the data and graph link in the embed, it takes you to Plotly’s GUI, where you can edit the graph, see the data, and share your plot with collaborators.\n\n### 🔗 1.2 Maps\n\nNext: Maps!\n\n``````data(canada.cities, package=\"maps\")\nviz <- ggplot(canada.cities, aes(long, lat)) +\ncoord_equal() +\ngeom_point(aes(text=name, size=pop), colour=\"red\", alpha=1/2, name=\"cities\")\n``````\n\nCall Plotly.\n\n``````py\\$ggplotly(viz)\n``````\n\n### 🔗 1.3 Scatter\n\nWant to make a scatter and add a smoothed conditional mean? Here’s how to do it in Plotly. For the rest of the plots, we’ll just print the Plotly version to save space. You can hover on text to get data, or click and drag across a section to zoom in.\n\n``````model <- lm(mpg ~ wt + factor(cyl), data=mtcars)\ngrid <- with(mtcars, expand.grid(\nwt = seq(min(wt), max(wt), length = 20),\ncyl = levels(factor(cyl))\n))\n\ngrid\\$mpg <- stats::predict(model, newdata=grid)\n\nviz2 <- qplot(wt, mpg, data=mtcars, colour=factor(cyl)) + geom_line(data=grid)\npy\\$ggplotly(viz2)\n``````\n\n### 🔗 1.4 Lines\n\nOr, take `ggplotly` for a spin with the orange dataset:\n\n``````orange <- qplot(age, circumference, data = Orange, colour = Tree, geom = \"line\")\npy\\$ggplotly(orange)\n``````\n\n### 🔗 1.5 Alpha blend\n\nOr, make plots beautiful.\n\n``````prettyPlot <- ggplot(data=diamonds, aes(x=carat, y=price, colour=clarity))\nprettyPlot <- prettyPlot + geom_point(alpha = 1/10)\npy\\$ggplotly(prettyPlot)\n``````\n\n### 🔗 1.6 Functions\n\nWant to draw functions with a `curve`?\n\n``````eq <- function(x) {x*x}\ntmp <- data.frame(x=1:50, y=eq(1:50))\n\n# Make plot object\np <- qplot(x, y, data=tmp, xlab=\"X-axis\", ylab=\"Y-axis\")\nc <- stat_function(fun=eq)\n\npy\\$ggplotly(p + c)\n``````\n\n## 🔗 2. A GitHub for data and graphs\n\nLike we might work together on code on GitHub or a project in a Google Doc, we can edit graphs and data together on Plotly. Here’s how it works:\n\n• Public use is free.\n• You can set the privacy of your graph.\n• You can edit and add to plots from our GUI or with R or APIs for Python, MATLA, Julia, Perl, Arduino, Raspberry Pi, and REST.\n• You get a profile of graphs, like Rhett Allain from Wired Science.\n• You can embed interactive graphs in iframes.\n\n### 🔗 2.1 Inspiration and team\n\nPlotly’s API is part of `rOpenSci` and being developed by the brilliant Toby Hocking and Plotly’s own Chris Parmer. You can find it on GitHub. Your thoughts, issues, and pull requests are welcome. Right now, you can make scatter and line plots; let us know what you’d like to see next.\n\nThe project was inspired by Hadley Wickham and the elegance and precision of `ggplot2`. Thanks to Scott Chamberlain, Joe Cheng, and Elizabeth Morrison-Wells for their help.\n\n## 🔗 3. ggthemes and Plotly\n\nUsing `ggthemes` opens up another set of custom graph filters for styling your graphs. To get started, you’ll want to install `ggthemes`.\n\n``````library(\"devtools\")\ninstall_github(\"ggthemes\", \"jrnold\")\n``````\n\n``````library(\"ggplot2\")\nlibrary(\"ggthemes\")\ndsamp <- diamonds[sample(nrow(diamonds), 1000), ]\n``````\n\nInverse gray.\n\n``````gray <- (qplot(carat, price, data = dsamp, colour = cut) +\ntheme_igray())\npy\\$ggplotly(gray)\n``````\n\nThe Tableau scale.\n\n``````tableau <- (qplot(carat, price, data = dsamp, colour = cut) +\ntheme_igray() +\nscale_colour_tableau())\npy\\$ggplotly(tableau)\n``````\n\nStephen Few’s scale.\n\n``````few <- (qplot(carat, price, data = dsamp, colour = cut) +\ntheme_few() +\nscale_colour_few())\npy\\$ggplotly(few)\n``````\nFiled under\nShare\nSuggest an edit Open a pull request\nBrowse",
null,
""
] |
[
null,
"https://ropensci.matomo.cloud/matomo.php",
null,
"https://d33wubrfki0l68.cloudfront.net/e9c5c04518699a3fe272ac955573bb8be23de9de/04eda/images/svg/navbar-divider.svg",
null,
"https://ropensci.org/assets/blog-images/2014-04-17-plotly/unnamed-chunk-2.png",
null,
"https://d33wubrfki0l68.cloudfront.net/cf1ec370231c1f926703e6f0743337f4b70b2188/517ce/images/svg/divider-lr.svg",
null
] |
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|
https://www.mersenneforum.org/printthread.php?s=bfceb396d1676a548a2ae48e4c52295e&t=20467
|
[
"",
null,
"mersenneforum.org (https://www.mersenneforum.org/index.php)\n- Miscellaneous Math (https://www.mersenneforum.org/forumdisplay.php?f=56)\n- - Conjectured compositeness tests for N=k⋅2n±c by Predrag (https://www.mersenneforum.org/showthread.php?t=20467)\n\n T.Rex 2015-09-04 21:23\n\nConjectured compositeness tests for N=k⋅2n±c by Predrag\n\nPredrag (not sure he wants his name to appear since he published as \"MathBot\") has published [URL=\"http://math.stackexchange.com/questions/1394160/conjectured-compositeness-tests-for-n-k-cdot-2n-pm-c\"]a question in StackExchange[/URL] about a set of 4 conjectures he has built dealing with: \"Conjectured compositeness tests for N=k⋅2n±c\".\n\nWhat he proposes seems to me already VERY great, since the 4 conjectures cover ALL numbers [TEX]N=k2^n \\pm c[/TEX], c odd for sure, with simple but powerful definitions using the LLT ([TEX]S_n \\equiv S_{n-1}^2-2 ~ \\pmod{N}[/TEX]) but also higher LLT functions (Chebichev) for the seed and for the last term of the sequence, after [TEX]n-1[/TEX] iterations.\n\nHowever, I've shown that the 4 conjectures can be generalized into only ONE conjecture which handles all kinds of these numbers.\nI've provided a PARI/gp for finding counter-examples.\n\nFor sure, it is ONLY a conjecture. But this conjecture looks really powerful. I'd like to know if someone is aware about such a general conjecture. I should know, but, after years without looking at such research papers, I do not remember. However, as said RDS, I'm not a PhD, so I'd like other people to have a look and add comments.\n\nI've read other papers of Pedrag: he has played first with smaller examples. So, I think he then spent some time for grouping several examples into a greater conjecture. So:1) he made experiments, 2) he generalized his findings in something more general. Good work !\n\nAnd, for sure, what I published recently was only children' game compared to Predrag's conjectures.\n\nHere is the PARI/gp program I've written based on work of Predrag, with code for searching counter-examples:\n\n[CODE]CEk2c(k,c,g)=\n{\na=6;\nif(c>0,s=1,s=-1;c=-c);\nfor(n=2*c+1,g,\nN=k*2^n+s*c;\ne=c%4;\nif(e==1,e=1,e=-1);\nd=((c-e)%8)/4;\nB=((-1)^d)*s;\nA=(c-B)/2;\ns0=Mod(2*polchebyshev(k,1,a/2),N);\nsn=Mod((-1)^d*2*polchebyshev(A,1,a/2),N);\nmy(s=s0);\nfor(i=1,n-1,s=Mod(s^2-2,N));\nif(s!=sn && isprime(N),print(\"k: \",k,\" c: \",c,\" n: \",n))\n)\n}\n\nfor(k=1,100,for(c=0,50,CEk2c(k,2*c+1,1000)))\nfor(k=1,100,for(c=0,50,CEk2c(k,-(2*c+1),1000)))[/CODE]\n\nThis comes from : Let say: [TEX]N=k \\cdot 2^n + s \\cdot c[/TEX], with: [TEX]s = \\pm 1[/TEX] and: [TEX]c = 4 \\cdot d ~\\pm 1 ~ \\pmod 8[/TEX] with [TEX]d=0,1[/TEX]. Then last [TEX]S_i[/TEX] can be defined as only one expression for all 4 conjectures: [TEX]S_{n-1} = {(-1)}^d \\cdot P_{(c - {(-1)^d \\cdot s})/2} ~~~~ (6)[/TEX] . That's quite complicated, but that shows that the 4 conjectures come from a single property depending on [TEX]k[/TEX] and on the internal structure of [TEX]c[/TEX].\n\nHere is the final conjecture I've built after grouping Predrag's 4 conjectures all together.\nThat does not make the task for proving it easier ! But that's so beautiful.\nAnd, for sure, we have the usual question: is this only a PRP tool or a true primality test ?\nAnyway, this conjecture shows how powerful are the ideas that Edouard Lucas discovered about 140 years ago. [TEX]P_2(x) = x^2-2[/TEX]\n\n[TEX]\\text{Let } ~N=k\\cdot 2^n+ s \\cdot c ~\\text{ such that }~ n>2c , k>0 , c>0 ~\\text{ and }~ s = \\pm 1 ~\\text{ and }~ c \\equiv 4 \\cdot d ~ \\pm 1 ~ \\pmod 8 ~\\text{ and }~ d=0,1[/TEX]\n\n[TEX]\\text{Let }~ S_i=P_2(S_{i-1})~ \\text{ with } ~ S_0=P_k(6) ~ \\text{ , thus: }[/TEX]\n\n[TEX]\\text{If } ~ N ~\\text{ is prime then }~ S_{n-1} \\equiv {(-1)}^d \\cdot P_{(c - {(-1)^d \\cdot s})/2} ~~~~ (6) \\pmod{N}[/TEX]\n\n T.Rex 2015-09-04 21:33\n\nThanks to primus to have warned me about Predrag's publication in StackExchange. :smile:\n\n Batalov 2015-09-04 22:09\n\nYou didn't even consider that they are one and the same?\n\n science_man_88 2015-09-05 01:28\n\n[QUOTE=T.Rex;409609][CODE]CEk2c(k,c,g)=\n{\na=6;\nif(c>0,s=1,s=-1;c=-c);\nfor(n=2*c+1,g,\nN=k*2^n+s*c;\ne=c%4;\nif(e==1,e=1,e=-1);\nd=((c-e)%8)/4;\nB=((-1)^d)*s;\nA=(c-B)/2;\ns0=Mod(2*polchebyshev(k,1,a/2),N);\nsn=Mod((-1)^d*2*polchebyshev(A,1,a/2),N);\nmy(s=s0);\nfor(i=1,n-1,s=Mod(s^2-2,N));\nif(s!=sn && isprime(N),print(\"k: \",k,\" c: \",c,\" n: \",n))\n)\n}\n\nfor(k=1,100,for(c=0,50,CEk2c(k,2*c+1,1000)))\nfor(k=1,100,for(c=0,50,CEk2c(k,-(2*c+1),1000)))[/CODE][/QUOTE]\n\nthinking about what might speed this up I found ( I'll see which one is faster and none of this uses modern PARI/gp to my knowledge if I realized which one's I could put in parallel I might be able to see if I can speed it up more.\n\n[CODE]CEk2c(k,c,g)=\n{\na=6;\nh=a/2;\nif(c>0,s=1,s=-1;c*=-1);\nfor(n=c<<1+1,g,\nN=k<<n+s*c;\ne=c%4;\nif(e==1,,e=-1);\nd=((c-e)%8)/4;\nf=((-1)^d)\nB=f*s;\nA=(c-B)/2;\ns0=Mod(polchebyshev(k,1,h)<<1,N);\nsn=Mod(f*polchebyshev(A,1,h)<<1,N);\nmy(s=s0);\nforstep(i=n,2,-1,s=sqr(s)-2);\nif(s!=sn && isprime(N),print(\"k: \",k,\" c: \",c,\" n: \",n))\n)\n}\n\nfor(k=1,100,for(c=0,50,CEk2c(k,2*c+1,1000)))\nfor(k=1,100,for(c=0,50,CEk2c(k,-(2*c+1),1000)))[/CODE]\n\n T.Rex 2015-09-05 10:26\n\n[QUOTE=Batalov;409612]You didn't even consider that they are one and the same?[/QUOTE]\nHa ha ha ! No ! ;) I should have considered... but too busy with other stuff.\n\n T.Rex 2015-09-05 10:30\n\n[QUOTE=science_man_88;409618]thinking about what might speed this up I found ( I'll see which one is faster and none of this uses modern PARI/gp to my knowledge if I realized which one's I could put in parallel I might be able to see if I can speed it up more.\n[CODE]CEk2c(k,c,g)={ ... forstep(i=n,2,-1,s=sqr(s)-2); ... ) }\n[/CODE][/QUOTE]\nI do not think that this will speed up the computation, since PARI goes in Stack overflow quickly. Modulo helps to keep s small.\n\n axn 2015-09-05 12:23\n\n[QUOTE=T.Rex;409642]I do not think that this will speed up the computation, since PARI goes in Stack overflow quickly. Modulo helps to keep s small.[/QUOTE]\n\nThe s variable is a Mod() object. All operations with it will automatically work modulo N.\n\n science_man_88 2015-09-05 13:40\n\n[QUOTE=T.Rex;409642]I do not think that this will speed up the computation, since PARI goes in Stack overflow quickly. Modulo helps to keep s small.[/QUOTE]\n\nyou can change how much it has I can set allocatemem to 1000000000 on my machine that allocates over 950 MB\n\n R.D. Silverman 2015-09-05 13:41\n\n[QUOTE=Batalov;409612]You didn't even consider that they are one and the same?[/QUOTE]\n\nNote also that numbers of the form k*2^n +/- c is the ENTIRE SET OF ODD INTEGERS!!!!!\n\nNumbers of the form x^n +/-1 are special. (cyclotomy!!!!)\n\nWhy on Earth people think k* 2^n + c is special is beyond me.\n\n science_man_88 2015-09-05 14:45\n\n[QUOTE=R.D. Silverman;409659]Why on Earth people think k* 2^n + c is special is beyond me.[/QUOTE]\n\nmaybe because they care for other potential groupings around a number like sexy primes or cousin primes ?\n\n Batalov 2015-09-05 15:09\n\n[QUOTE=T.Rex;409641]Ha ha ha ! No ! ;) I should have considered... but too busy with other stuff.[/QUOTE]\nNow, he has succeeded to use you as a sock puppet.\nThink about it. If he wanted to post it himself, under three different names (and on ten other boards, just like inimitable Don Blazys), he would have! And he had already done that.\nNow he used you to post this tripe once again.\n\nDon't you feel used?\n\nAll times are UTC. The time now is 02:25."
] |
[
null,
"https://www.mersenneforum.org/clear.gif",
null
] |
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|
https://sigma.ontologyportal.org:8443/sigma/SimpleBrowse.jsp?lang=FrenchLanguage&simple=yes&kb=SUMO&term=NonCompositeUnitOfMeasure
|
[
"",
null,
"",
null,
"Simple Browser : Welcome guest : log in [ Home | Graph | ] KB: SUMO Language: ChineseLanguageChinesePinyinWritingChineseSimplifiedWritingChineseTraditionalLanguageEnglishLanguageFrenchLanguageGermanLanguageJapaneseLanguageSpanishLanguageSwedishLanguage Formal Language: OWLSUO-KIFTPTPtraditionalLogic\n\nSigma KEE - NonCompositeUnitOfMeasure\n KB Term:",
null,
"A B C D E F G H I J K L M N O P Q R S T U V W X Y Z\n NonCompositeUnitOfMeasure Instances of this Class are UnitsOfMeasure that are applied to a single dimension, and so are not intrinsically defined by the functional composition of other units.\n Relationships Parents ConstantQuantity A ConstantQuantity is a PhysicalQuantity that has a constant value, e.g. 3 Meters and 5 HourDurations. The magnitude (see MagnitudeFn) of every ConstantQuantity is a RealNumber. ConstantQuantity is distinguished from FunctionQuantity, in that each instance of the latter is formed through the mapping of one PhysicalQuantity to another PhysicalQuantity. Each instance of ConstantQuantity is expressed with the BinaryFunction MeasureFn, which takes a Number and a UnitOfMeasure as arguments. For example, 3 Meters is expressed as (MeasureFn 3 Meter). Instances of ConstantQuantity form a partial order (see PartialOrderingRelation) with the lessThan relation, since lessThan is a RelationExtendedToQuantities and lessThan is defined over the RealNumbers. The lessThan relation is not a total order (see TotalOrderingRelation) over the class ConstantQuantity since elements of some subclasses of ConstantQuantity (such as length quantities) are incomparable to elements of other subclasses of ConstantQuantity (such as mass quantities). UnitOfMeasure A standard of measurement for some dimension. For example, the Meter is a UnitOfMeasure for the dimension of length, as is the Inch. There is no intrinsic property of a UnitOfMeasure that makes it primitive or fundamental, rather, a system of units (e.g. SystemeInternationalUnit) defines a set of orthogonal dimensions and assigns units for each. Children UnitOfAngularMeasure Every instance of this Class is a UnitOfMeasure that can be used with MeasureFn to form instances of AngleMeasure. UnitOfCurrency Every instance of this Class is a UnitOfMeasure that can be used with MeasureFn to form instances of CurrencyMeasure. UnitOfDuration Every instance of this Class is a UnitOfMeasure that can be used with MeasureFn to form instances of TimeDuration. Note that TimeDuration is a subclass of TimeMeasure. UnitOfInformation Every instance of this Class is a UnitOfMeasure that can be used with MeasureFn to form instances of InformationMeasure. UnitOfLength Every instance of this Class is a UnitOfMeasure that can be used with MeasureFn to form instances of LengthMeasure. UnitOfMass Every instance of this Class is a UnitOfMeasure that can be used with MeasureFn to form instances of MassMeasure, which denote the amount of matter in PhysicalObjects. UnitOfTemperature Every instance of this Class is a UnitOfMeasure that can be used with MeasureFn to form instances of TemperatureMeasure.",
null,
"Show simplified definition with tree view\nShow full definition (without tree view)\nShow full definition (with tree view)",
null,
"Sigma web home Suggested Upper Merged Ontology (SUMO) web home\nSigma version 3.0 is open source software produced by Articulate Software and its partners"
] |
[
null,
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null,
"https://sigma.ontologyportal.org:8443/sigma/pixmaps/1pixel.gif",
null,
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null
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https://demo.formulasearchengine.com/wiki/Quadratic_variation
|
[
"In mathematics, quadratic variation is used in the analysis of stochastic processes such as Brownian motion and other martingales. Quadratic variation is just one kind of variation of a process.\n\n## Definition\n\nSuppose that Xt is a real-valued stochastic process defined on a probability space $(\\Omega ,{\\mathcal {F}},\\mathbb {P} )$",
null,
"and with time index t ranging over the non-negative real numbers. Its quadratic variation is the process, written as [X]t, defined as\n\n$[X]_{t}=\\lim _{\\Vert P\\Vert \\rightarrow 0}\\sum _{k=1}^{n}(X_{t_{k}}-X_{t_{k-1}})^{2}$",
null,
"where P ranges over partitions of the interval [0,t] and the norm of the partition P is the mesh. This limit, if it exists, is defined using convergence in probability. Note that a process may be of finite quadratic variation in the sense of the definition given here and its paths be nonetheless almost surely of infinite quadratic variation for every t>0 in the classical sense of taking the supremum of the sum over all partitions; this is in particular the case for Brownian Motion.\n\nMore generally, the covariation (or cross-variance) of two processes X and Y is\n\n$[X,Y]_{t}=\\lim _{\\Vert P\\Vert \\to 0}\\sum _{k=1}^{n}\\left(X_{t_{k}}-X_{t_{k-1}}\\right)\\left(Y_{t_{k}}-Y_{t_{k-1}}\\right).$",
null,
"The covariation may be written in terms of the quadratic variation by the polarization identity:\n\n$[X,Y]_{t}={\\tfrac {1}{2}}([X+Y]_{t}-[X]_{t}-[Y]_{t}).$",
null,
"## Finite variation processes\n\nA process X is said to have finite variation if it has bounded variation over every finite time interval (with probability 1). Such processes are very common including, in particular, all continuously differentiable functions. The quadratic variation exists for all continuous finite variation processes, and is zero.\n\nThis statement can be generalized to non-continuous processes. Any càdlàg finite variation process X has quadratic variation equal to the sum of the squares of the jumps of X. To state this more precisely, the left limit of Xt with respect to t is denoted by Xt-, and the jump of X at time t can be written as ΔXt = Xt - Xt-. Then, the quadratic variation is given by\n\n$[X]_{t}=\\sum _{0",
null,
"The proof that continuous finite variation processes have zero quadratic variation follows from the following inequality. Here, P is a partition of the interval [0,t], and Vt(X) is the variation of X over [0,t].\n\n{\\begin{aligned}\\sum _{k=1}^{n}(X_{t_{k}}-X_{t_{k-1}})^{2}&\\leq \\max _{k\\leq n}|X_{t_{k}}-X_{t_{k-1}}|\\sum _{k=1}^{n}|X_{t_{k}}-X_{t_{k-1}}|\\\\&\\leq \\max _{|u-v|\\leq \\Vert P\\Vert }|X_{u}-X_{v}|V_{t}(X).\\end{aligned}}",
null,
"By the continuity of X, this vanishes in the limit as $\\Vert P\\Vert$",
null,
"goes to zero.\n\n## Itō processes\n\nThe quadratic variation of a standard Brownian motion B exists, and is given by [B]t = t. This generalizes to Itō processes that, by definition, can be expressed in terms of Itō integrals\n\n$X_{t}=X_{0}+\\int _{0}^{t}\\sigma _{s}\\,dB_{s}+\\int _{0}^{t}\\mu _{s}\\,ds,$",
null,
"where B is a Brownian motion. Any such process has quadratic variation given by\n\n$[X]_{t}=\\int _{0}^{t}\\sigma _{s}^{2}\\,ds.$",
null,
"## Semimartingales\n\nQuadratic variations and covariations of all semimartingales can be shown to exist. They form an important part of the theory of stochastic calculus, appearing in Itō's lemma, which is the generalization of the chain rule to the Itō integral. The quadratic covariation also appears in the integration by parts formula\n\n$X_{t}Y_{t}=X_{0}Y_{0}+\\int _{0}^{t}X_{s-}\\,dY_{s}+\\int _{0}^{t}Y_{s-}\\,dX_{s}+[X,Y]_{t},$",
null,
"which can be used to compute [X,Y].\n\nAlternatively this can be written as a Stochastic Differential Equation:\n\n$\\,d(X_{t}Y_{t})=X_{t-}\\,dY_{t}+Y_{t-}\\,dX_{t}+\\,dX_{t}\\,dY_{t},$",
null,
"## Martingales\n\nAll càdlàg martingales, and local martingales have well defined quadratic variation, which follows from the fact that such processes are examples of semimartingales. It can be shown that the quadratic variation [M] of a general local martingale M is the unique right-continuous and increasing process starting at zero, with jumps Δ[M] = ΔM2, and such that M2 − [M] is a local martingale.\n\nA useful result for square integrable martingales is the Itō isometry, which can be used to calculate the variance of Ito integrals,\n\n$\\mathbb {E} \\left(\\left(\\int _{0}^{t}H\\,dM\\right)^{2}\\right)=\\mathbb {E} \\left(\\int _{0}^{t}H^{2}\\,d[M]\\right).$",
null,
"This result holds whenever M is a càdlàg square integrable martingale and H is a bounded predictable process, and is often used in the construction of the Itō integral.\n\nAnother important result is the Burkholder–Davis–Gundy inequality. This gives bounds for the maximum of a martingale in terms of the quadratic variation. For a continuous local martingale M starting at zero, with maximum denoted by Mt* ≡sups≤t|Ms|, and any real number p > 0, the inequality is\n\n$c_{p}\\mathbb {E} ([M]_{t}^{p/2})\\leq \\mathbb {E} ((M_{t}^{*})^{p})\\leq C_{p}\\mathbb {E} ([M]_{t}^{p/2}).$",
null,
"Here, cp < Cp are constants depending on the choice of p, but not depending on the martingale M or time t used. If M is a continuous local martingale, then the Burkholder–Davis–Gundy inequality holds for any positive value of p.\n\nAn alternative process, the predictable quadratic variation is sometimes used for locally square integrable martingales. This is written as <M>t, and is defined to be the unique right-continuous and increasing predictable process starting at zero such that M2 − <M> is a local martingale. Its existence follows from the Doob–Meyer decomposition theorem and, for continuous local martingales, it is the same as the quadratic variation."
] |
[
null,
"https://demo.formulasearchengine.com/index.php",
null,
"https://demo.formulasearchengine.com/index.php",
null,
"https://demo.formulasearchengine.com/index.php",
null,
"https://demo.formulasearchengine.com/index.php",
null,
"https://demo.formulasearchengine.com/index.php",
null,
"https://demo.formulasearchengine.com/index.php",
null,
"https://demo.formulasearchengine.com/index.php",
null,
"https://demo.formulasearchengine.com/index.php",
null,
"https://demo.formulasearchengine.com/index.php",
null,
"https://demo.formulasearchengine.com/index.php",
null,
"https://demo.formulasearchengine.com/index.php",
null,
"https://demo.formulasearchengine.com/index.php",
null,
"https://demo.formulasearchengine.com/index.php",
null
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|
https://datascience.stackexchange.com/questions/33025/classification-prediction-based-on-multivariate-time-series
|
[
"# Classification/Prediction based on Multivariate Time Series\n\nSo, I have a time series with many independent variables (X's) and an outcome variable Y (that I want to predict, think a 2 class logistic regression where output would either be 1 or a 0). Kindly see a sample below:\n\nTimestamp X1 X2 X3 X4 Y\n1:00 1 0.5 23.5 0 0\n1:01 1 0.8 18.7 0 0\n1:02 0 0.9 4.5 1 0\n….\n1:30 1 1.9 5.5 1 1\n1:31 0 1.7 4.3 0 1\n…\n…\n\n\nNow I want to predict or rather classify Y as 0 (stable) or 1 (unstable) (Note that when Y becomes 1 it remains 1 for certain interval of time, same when it is 0)\n\nSo Y will be dependent on sequence variables (Please note that it is a time series, and not a standard regression problem where every row can be fed to an Algorithm for classification, the output here is dependent on a sequence of inputs/rows), for instance Y may become 1 when X2 starts increasing and X3 starts decreasing and so on (there are many independent variables X1…XN).\n\nThe way I was thinking in order to solve this problem was to extract say m hours of data before Y becomes 1 and do some descriptive statistics on X in order to derive new features (like mean of X1, std of X2, last change point of X4 and so on for the set of this extracted data) to convert the X’s to a single row feature vector. The outcome ‘Y’ of this single row feature vector is 1 as we have just extracted the data before Y became 1. So this way I am able to convert a time series into a standard classification/prediction problem. Similarly I can take the other class i.e. Y=0 and follow the same process.\n\nThe other approach that I thought about was to incorporate a sequence model, something like Hidden Markov Model where the hidden states might be stable (say for Y=0) and unstable (for Y=1) and then I go about emission and transition probabilities. But this HMM will be multivariate considering there are many X’s on which Y is dependent. This seems a bit complex?\n\nAny ideas on modeling the above problem will be appreciated.\n\n• Are you trying to predict binary outcomes for every second? If not this post might be misleading. – user61762 Oct 31 '18 at 10:16"
] |
[
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|
https://unitconversion.io/193-b-to-mb
|
[
"#### Convert 193 Bytes (B) to Megabytes (MB)\n\nThis is our conversion tool for converting bytes to megabytes.\nTo use the tool, simply enter a number in any of the inputs and the converted value will automatically appear in the opposite box.\n\nB\n\n### =\n\nMB\n\n##### How to convert Bytes (B) to Megabytes (MB)\n\nConverting Bytes (B) to Megabytes (MB) is simple. Why is it simple? Because it only requires one basic operation: multiplication. The same is true for many types of unit conversion (there are some expections, such as temperature). To convert Bytes (B) to Megabytes (MB), you just need to know that 1bytes is equal to MB. With that knowledge, you can solve any other similar conversion problem by multiplying the number of Bytes (B) by . For example, 6bytes multiplied by is equal to MB.\n\n#### Best conversion unit for 193 Bytes (B)\n\nWe define the \"best\" unit to convert a number as the unit that is the lowest without going lower than 1. For 193 bytes, the best unit to convert to is .\n\n#### Fast Conversions\n\n 1 bytes = MB 5 bytes = MB 10 bytes = MB 15 bytes = MB 25 bytes = MB 100 bytes = MB 1000 bytes = MB 1 MB = bytes 5 MB = bytes 10 MB = bytes 15 MB = bytes 25 MB = bytes 100 MB = bytes 1000 MB = bytes"
] |
[
null
] |
{"ft_lang_label":"__label__en","ft_lang_prob":0.8725181,"math_prob":0.99778575,"size":957,"snap":"2019-35-2019-39","text_gpt3_token_len":226,"char_repetition_ratio":0.17838405,"word_repetition_ratio":0.0295858,"special_character_ratio":0.23719957,"punctuation_ratio":0.095744684,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.984824,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-09-17T11:12:56Z\",\"WARC-Record-ID\":\"<urn:uuid:ecdc6dc3-03fe-4069-a4a0-2d1220c15daf>\",\"Content-Length\":\"31889\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7b4fb5d0-d552-4c45-8328-65f5a1b6c109>\",\"WARC-Concurrent-To\":\"<urn:uuid:d539a687-2d39-45d9-89db-ba0eefb1ac10>\",\"WARC-IP-Address\":\"3.95.122.236\",\"WARC-Target-URI\":\"https://unitconversion.io/193-b-to-mb\",\"WARC-Payload-Digest\":\"sha1:VEDZYOUVF54F65T5RKZZMUEZMRWYFMLV\",\"WARC-Block-Digest\":\"sha1:UWWBCMVVJLCVNKCP7UM77B4YMY3ZXY3K\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-39/CC-MAIN-2019-39_segments_1568514573070.36_warc_CC-MAIN-20190917101137-20190917123137-00101.warc.gz\"}"}
|
https://answers.everydaycalculation.com/add-fractions/35-14-plus-8-25
|
[
"Solutions by everydaycalculation.com\n\n1st number: 2 7/14, 2nd number: 8/25\n\n35/14 + 8/25 is 141/50.\n\n1. Find the least common denominator or LCM of the two denominators:\nLCM of 14 and 25 is 350\n2. For the 1st fraction, since 14 × 25 = 350,\n35/14 = 35 × 25/14 × 25 = 875/350\n3. Likewise, for the 2nd fraction, since 25 × 14 = 350,\n8/25 = 8 × 14/25 × 14 = 112/350\n875/350 + 112/350 = 875 + 112/350 = 987/350\n5. 987/350 simplified gives 141/50\n6. So, 35/14 + 8/25 = 141/50\nIn mixed form: 241/50\n\nMathStep (Works offline)",
null,
"Download our mobile app and learn to work with fractions in your own time:"
] |
[
null,
"https://answers.everydaycalculation.com/mathstep-app-icon.png",
null
] |
{"ft_lang_label":"__label__en","ft_lang_prob":0.5898428,"math_prob":0.99933684,"size":379,"snap":"2020-34-2020-40","text_gpt3_token_len":162,"char_repetition_ratio":0.192,"word_repetition_ratio":0.0,"special_character_ratio":0.52506596,"punctuation_ratio":0.086021505,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99686176,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-08-12T10:21:54Z\",\"WARC-Record-ID\":\"<urn:uuid:db41fa22-6c31-44b5-94af-acc1dad10190>\",\"Content-Length\":\"8255\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a1b060f4-d43b-4746-b79c-c8c2995b3bc5>\",\"WARC-Concurrent-To\":\"<urn:uuid:59ed5e1e-0ddc-4b86-8d95-ed9e2bbd789f>\",\"WARC-IP-Address\":\"96.126.107.130\",\"WARC-Target-URI\":\"https://answers.everydaycalculation.com/add-fractions/35-14-plus-8-25\",\"WARC-Payload-Digest\":\"sha1:7B4F2ZG2U4ANNM5GOPYY5WCQFHLUBBEX\",\"WARC-Block-Digest\":\"sha1:HBH4XIA5CPSQUI7HOMHS4ZX6L4FWTW2F\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-34/CC-MAIN-2020-34_segments_1596439738888.13_warc_CC-MAIN-20200812083025-20200812113025-00461.warc.gz\"}"}
|
http://theinfolist.com/php/SummaryGet.php?FindGo=Force_carrier
|
[
"TheInfoList\n\nIn particle physics, force carriers or messenger particles or intermediate particles are particles that give rise to forces between other particles. These particles are bundles of energy (quanta) of a particular kind of field. There is one kind of field for every type of elementary particle. For instance, there is an electron field whose quanta are electrons, and an electromagnetic field whose quanta are photons. The force carrier particles that mediate the electromagnetic, weak, and strong interactions are called gauge bosons.\n\n## Particle and field viewpoints\n\nIn particle physics, quantum field theories such as the Standard Model describe nature in terms of fields. Each field has a complementary description as the set of particles of a particular type. A force between two particles can be described either as the action of a force field generated by one particle on the other, or in terms of the exchange of virtual force carrier particles between them.\n\nThe energy of a wave in a field (for example, electromagnetic waves in the electromagnetic field) is quantized, and the quantum excitations of the field can be interpreted as particles. The Standard Model contains the following particles, each of which is an excitation of a particular field:\n\nIn addition, composite particles such as mesons can be described as excitations of an effective field.\n\nGravity is not a part of the Standard Model, but it is thought that there may be particles called gravitons which are the excitations of gravitational waves. The status of this particle is still tentative, because the theory is incomplete and because the interactions of single gravitons may be too weak to be detected.\n\n## Forces from the particle viewpoint\n\nWhen one particle scatters off another, altering its trajectory, there are two ways to think about the process. In the field picture, we imagine that the field generated by one particle caused a force on the other. Alternatively, we can imagine one particle emitting a virtual particle which is absorbed by the other. The virtual particle transfers momentum from one particle to the other. This particle viewpoint is especially helpful when there are a large number of complicated quantum corrections to the calculation since these corrections can be visualized as Feynman diagrams containing additional virtual particles.\n\nThe description of forces in terms of virtual particles is limited by the applicability of the perturbation theory from which it is derived. In certain situations, such as low-energy QCD and the description of bound states, perturbation theory breaks down.\n\n## History\n\nThe concept of messenger particles dates back to the 18th century when the French physicist Charles Coulomb showed that the electrostatic force between electrically charged objects follows a law similar to Newton's Law of Gravitation. In time, this relationship became known as Coulomb's law. By 1862, Hermann von Helmholtz had described a ray of light as the \"quickest of all the messengers\". In 1905, Albert Einstein proposed the existence of a light-particle in answer to the question: \"what are light quanta?\"\n\nIn 1923, at the Washington University in St. Louis, Arthur Holly Compton demonstrated an effect now known as Compton scattering. This effect is only explainable if light can behave as a stream of particles and it convinced the physics community of the existence of Einstein's light-particle. Lastly, in 1926, one year before the theory of quantum mechanics was published, Gilbert N. Lewis introduced the term \"photon\", which soon became the name for Einstein’s light particle. From there, the concept of messenger particles developed further."
] |
[
null
] |
{"ft_lang_label":"__label__en","ft_lang_prob":0.94058967,"math_prob":0.8739435,"size":3590,"snap":"2020-24-2020-29","text_gpt3_token_len":697,"char_repetition_ratio":0.15421082,"word_repetition_ratio":0.003552398,"special_character_ratio":0.18579388,"punctuation_ratio":0.09478673,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9548382,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-09T20:57:41Z\",\"WARC-Record-ID\":\"<urn:uuid:56319687-17bd-4bbc-9722-91b4634f8f65>\",\"Content-Length\":\"29927\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:bea474dc-51a8-419e-8cec-8a1c09dd3aa1>\",\"WARC-Concurrent-To\":\"<urn:uuid:16cca815-ee30-4d68-9491-c1a486f2d6db>\",\"WARC-IP-Address\":\"67.249.226.212\",\"WARC-Target-URI\":\"http://theinfolist.com/php/SummaryGet.php?FindGo=Force_carrier\",\"WARC-Payload-Digest\":\"sha1:PMJLOE2WJ7ACVAMZQU2K4E7L7AUHFSTA\",\"WARC-Block-Digest\":\"sha1:LHUG75RNKSNUA6DLRTRME6DQH7ZVEQD5\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-29/CC-MAIN-2020-29_segments_1593655901509.58_warc_CC-MAIN-20200709193741-20200709223741-00318.warc.gz\"}"}
|
https://monkeyraptor.johanpaul.net/2013/11/finding-roots-of-quadratic-equation.html?m=0
|
[
"## Thursday, November 21, 2013\n\n### Math: Finding the Roots of a Quadratic Equation\n\nThe quadratic equation, ax² + bx + c = 0, is a non-linear (2nd degree polynomial, a ≠ 0) equation that always has two roots as the solution. Sometimes the roots are different, sometimes they're twins. Sometimes they all have real numbers or complex numbers, or just imaginary number.\n\nMethods\nTo find the roots (the solution), we can choose between these four different methods :\n• Factoring\n• Completing the square\n• Drawing the graph : this can be achieved by writing your own loop-n-plot computer script, or using pre-programmed Mathematical software, or just using a pencil and a paper then create a table of values, continued with plotting the values on the Cartesian plane (2D x-y axis).\n\nA glimpse about the function f(x) graph\n\nDiscriminant (∇)\nThis is the expression that will tell us if a quadratic function crosses an axis. Because this is f(x) (the function of x values axis), then the discriminant here will tell us whether a function crosses the x-axis, touching the x-axis on one point, or floating above/below the x-axis.\n\nFor the quadratic equation ax² + bx + c = 0, the discriminant is defined as :\n= b² - 4ac\nThe constant numbers a and b are the numbers (value) that represent the sum of the variables and x, respectively. And the c is the constant number of a quadratic equation.\n\nAlso, with this discriminant expression, we can find out if a quadratic function graph (or the equation) has two real numbers roots, two complex numbers (or just imaginary) roots, or twin real numbers roots.\n\nFor example :\n2x² + 7x + 5 = 0 has a = 2, b = 7 and c = 5.\nSo the discriminant of that thingy is\n∇ = b² - 4ac\n∇ = (7)² - 4 * 2 * 5\n∇ = 49 - 40\n∇ = 9 ( ∇ > 0 : this quadratic equation crosses the x-axis, it has two real numbers roots)\nThe discriminant values :\n• ∇ > 0 : the equation has two real numbers roots, the function graph crosses the x-axis\n• ∇ < 0 : the equation has two complex numbers roots, the function graph doesn't touch the x-axis.\n• ∇ = 0 : the equation has twins real numbers roots, it touches the x-axis on exactly one point\n\nThe a constant\na > 0\nAs you can see on the graph above, it plots the y = x² - x - 2 function. The a constant of that function, the sum of , is 1. Therefore, a = 1 or a > 0. It means the graph shape will have a valley, also means, it has a minimum value.\n\nOn this quadratic non-linear equation (function), we can't have the a as 0. Why? Because, that's why.\nOK, let's take a look : ax² + bx + c = 0, if we substitute the a with 0, then it will become bx + c = 0.\nThe last equation loses its manhood, that is its quadratic thingy, it becomes a linear equation.\nSo there, because.\n\nThe values of a\n• a > 0 : the graph has a valley (pointing downward - opening up) with a minimum value\n• a < 0 : the graph has a peak (pointing upward - opening down) with a maximum value\n\nMethods to find the roots\nThe general formulation to get the roots (the solution) of a quadratic thingy is using :\nFor example, find the roots of this equation :\n2x² + 7x + 5 = 0 ► with a = 2, b = 7 and c = 5.\n\nLet's solve that using quadratic formulation :\nx = [ -7 ± √( ² - [4 * 2 * 5] ) ] / 2 * 2\nx = [ -7 ± √(49 - 40) ] / 4\nx = [ -7 ± √9 ] / 4\nx = [ -7 ± 3 ] / 4\nx1 = [ -7 + 3 ] / 4 ∨ x2 = [ -7 - 3 ] / 4\nx1 = [ -4 ] /4 ∨ x2 = [ -10 ] / 4\nx1 = -1 ∨ x2 = -(5/2)\n∴ The roots of 2x² + 7x + 5 = 0 are x1 = -1 or x2 = -(5/2), simplified as x = {-(5/2) , -1}\n\nI'll show you another one, the factoring method.\n\nFactoring\n\nLet's take an example from the previous one : 2x² + 7x + 5 = 0 with a = 2, b = 7 and c = 5.\n\nContinue\nSo we have 2 and 5 there, we put this general formulation to solve that :\nKeep in mind that the equation is 2x² + 7x + 5 = 0 has a = 2.\nAnd also, we have already got value1 = 2, and value2 = 5.\nSubstitute those variables with the numbers we have :\nThen we can simplify that as :\n(x + 1)(2x + 5)\n\nWe put it back to the complete equation, so :\n2x² + 7x + 5 = 0\n(x + 1)(2x + 5) = 0\nx1 = 0 - 1 or x2 = (0 - 5)/2\nx1 = -1 or x2 = -(5/2)\nThe solution, x = {-(5/2) , -1}*\n*the same result compared with using quadratic formulation earlier\n\nAnotha example :\nLet's find the roots of this quadratic equation\n-5x² + 24x + 5 = 0\n\n1st step\nFind two numbers which if multiplied will have the same value of a*c, that is (-5)*5 = -25, and if added will give the same value of b, that is 24",
null,
"From the image above we find that the value1 = (-1) and value2 = 25.\n\n2nd step\nPut those values into the basic factorization formula :\nSo :\nSimplify that into :\n(-5x - 1)(x - 5)\n\nLast step\nPut that back in to the original complete equation :\n-5x² + 24x + 5 = 0\n(-5x - 1)(x - 5) = 0\nx1 = (0 + 1)/(-5) or x2 = 0 + 5\nx1 = -(1/5) or x2 = 5\nThe solution, x = {-(1/5) , 5}\n\nOther forms \"shortcuts\" :\n• ax² + bx = 0 ► x (ax + b) = 0 ► x1 = 0 ∨ x2 = -(b/a)\n• ax² - bx = 0 ► x (ax - b) = 0 ► x1 = 0 ∨ x2 = b/a\n• -ax² - bx = 0 ► (-x)(ax + b) = 0 ► x1 = 0 ∨ x2 = -(b/a)\n• -ax² + bx = 0 ► x (-ax + b) = 0 ► x1 = 0 ∨ x2 = b/a\n• ax² - c = 0 ► ( √a·x + √c )( √a·x - √c ) = 0 ► x = ±( √c / √a )\n• -ax² + c = 0 ► ( √a·x + √c )( -√a·x + √c ) = 0 ► x = ±( √c / √a )\n• etc\n\nTry out these :\n\n### Last Problem : x² + 169 = 0\n\n##### hint: use i for imaginary number, for instance 2 - 3i or 7i\n\nHave fun... I hope.",
null,
"Math: Finding the Roots of a Quadratic Equation"
] |
[
null,
"https://2.bp.blogspot.com/-4Ui-G6eRY5k/Uo2dczVNqZI/AAAAAAAAB78/MONgF-LYb44/s400/ex2.png",
null,
"data:image/gif;base64,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",
null
] |
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|
https://kr.mathworks.com/matlabcentral/cody/problems/2479-mongean-shuffle/solutions/1885679
|
[
"Cody\n\n# Problem 2479. Mongean Shuffle\n\nSolution 1885679\n\nSubmitted on 27 Jul 2019 by Brian\nThis solution is locked. To view this solution, you need to provide a solution of the same size or smaller.\n\n### Test Suite\n\nTest Status Code Input and Output\n1 Pass\n% trivial x = 1; y_correct = 1; assert(isequal(mshuffle(x),y_correct))\n\n2 Pass\nx = 1:5; y_correct = [4 2 1 3 5]; assert(isequal(mshuffle(x),y_correct))\n\n3 Pass\na = magic(5); a=a(:)'; y_correct = [3 16 2 20 8 19 7 18 6 24 10 23 17 4 11 5 12 1 13 25 14 21 15 22 9]; assert(isequal(mshuffle(a),y_correct));\n\n4 Pass\nx = 7:-1:1; y_correct = [2 4 6 7 5 3 1]; assert(isequal(mshuffle(x),y_correct))"
] |
[
null
] |
{"ft_lang_label":"__label__en","ft_lang_prob":0.5176426,"math_prob":0.9953342,"size":670,"snap":"2019-43-2019-47","text_gpt3_token_len":252,"char_repetition_ratio":0.16216215,"word_repetition_ratio":0.0,"special_character_ratio":0.42985076,"punctuation_ratio":0.12837838,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.973803,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-11-13T12:57:09Z\",\"WARC-Record-ID\":\"<urn:uuid:df127f19-66c1-4ff3-8612-be59dae4d033>\",\"Content-Length\":\"72289\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0a995337-a316-464b-ae29-a73a41081f0d>\",\"WARC-Concurrent-To\":\"<urn:uuid:d64e7fc8-4045-4165-9e61-e964662b01d5>\",\"WARC-IP-Address\":\"104.110.193.39\",\"WARC-Target-URI\":\"https://kr.mathworks.com/matlabcentral/cody/problems/2479-mongean-shuffle/solutions/1885679\",\"WARC-Payload-Digest\":\"sha1:GMJMCRKIGBRNRAFLUJVKGP5FGGCHT4SI\",\"WARC-Block-Digest\":\"sha1:VLKP7FDWIFDAUSF26K2E64PRF7XGTPHB\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-47/CC-MAIN-2019-47_segments_1573496667260.46_warc_CC-MAIN-20191113113242-20191113141242-00119.warc.gz\"}"}
|
https://quantumcomputing.stackexchange.com/questions/26612/for-the-same-operator-when-i-use-qiskits-vqe-and-qaoa-solutions-the-results-a
|
[
"# For the same operator, when I use qiskit's VQE and QAOA solutions, the results are completely different\n\nSpecifically, I can't get the correct result using QAOA，A total of 6 qubits were used in this work，this is part of my code using VQE and QAOA localsim = BasicAer.get_backend('qasm_simulator')\n\noptimizer = SPSA(max_trials=500)\nansatz = TwoLocal(rotation_blocks='ry', entanglement_blocks='cz')\nvqe = VQE(op, ansatz, optimizer = optimizer)\nresults1 = vqe.run(localsim)\n\n\nand\n\nfrom qiskit import QuantumCircuit\noptimizer = COBYLA(maxiter = 100)\ndepth = 10\ninitial = QuantumCircuit(nbr_qubits)\nqaoa = QAOA(operator=op, optimizer=optimizer, p = depth,initial_state = initial, quantum_instance=localsim)\nresults2 = qaoa.run()\n\n\nI tried changing depth and optimizer type but it didn't work，The following two pictures are the results of VQE and QAOA respectively",
null,
"",
null,
"Is it because I'm using QAOA incorrectly?\n\n• What was your operator? May 29, 2022 at 18:56\n• op is an ising hamiltonian May 30, 2022 at 4:39"
] |
[
null,
"https://i.stack.imgur.com/KQwmW.png",
null,
"https://i.stack.imgur.com/MIqMB.png",
null
] |
{"ft_lang_label":"__label__en","ft_lang_prob":0.76464546,"math_prob":0.9790712,"size":789,"snap":"2023-40-2023-50","text_gpt3_token_len":221,"char_repetition_ratio":0.12993631,"word_repetition_ratio":0.0,"special_character_ratio":0.23447402,"punctuation_ratio":0.09677419,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98725444,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,3,null,3,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-04T23:56:28Z\",\"WARC-Record-ID\":\"<urn:uuid:adc5c628-990e-4535-99ae-7d83fefa728e>\",\"Content-Length\":\"162363\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a64326a3-d4b4-4952-ad82-e2b2ac9a47d3>\",\"WARC-Concurrent-To\":\"<urn:uuid:033a8034-bdd3-498c-9ab6-fb5dd345ad32>\",\"WARC-IP-Address\":\"172.64.144.30\",\"WARC-Target-URI\":\"https://quantumcomputing.stackexchange.com/questions/26612/for-the-same-operator-when-i-use-qiskits-vqe-and-qaoa-solutions-the-results-a\",\"WARC-Payload-Digest\":\"sha1:HI4PUGJA7H4TBCDFPOTFRY6JN22DWH2H\",\"WARC-Block-Digest\":\"sha1:FA36FCK75JSCJIRJW2WCAHVNXNMDXOQ2\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100535.26_warc_CC-MAIN-20231204214708-20231205004708-00025.warc.gz\"}"}
|
https://codeshive.com/product/csci-567-homework-2-solved/
|
[
"# CSCI 567 Homework #2 solved\n\n\\$35.00\n\nCategory:\n\n## Description\n\nProblem 1 High-level descriptions\n1.1 Dataset (Same as in Homework 1.) We will use mnist subset (images of handwritten digits from 0\nto 9). The dataset is stored in a JSON-formated file mnist subset.json. You can access its training, validation, and test splits using the keys ‘train’, ‘valid’, and ‘test’, respectively. For example, suppose we load\nmnist subset.json to the variable x. Then, x[\n0\ntrain0\n] refers to the training set of mnist subset. This set is a\nlist with two elements: x[\n0\ntrain0\n] containing the features of size N (samples) ×D (dimension of features),\nand x[\n0\ntrain0\n] containing the corresponding labels of size N.\nBesides, for logistic regression in Sect. 2, you will be using synthetic datasets with two, three and five\nclasses.\n1.2 Tasks You will be asked to implement binary and multiclass classification (Sect. 2) and neural networks (Sect. 3). Specifically, you will\n• finish the implementation of all python functions in our template codes.\n• run your code by calling the specified scripts to generate output files.\n• add, commit, and push (1) all *.py files, and (2) all *.json and *.out files that you have amended\nor created.\nIn the next two subsections, we will provide a high-level checklist of what you need to do. You are not\nrefer to text in Sect. 2 and Sect. 3, as well as corresponding python scripts.\n1.2.1 Logistic regression\nCoding In logistic.py, finish implementing the following functions: binary train, binary predict,\nmultinomial train, multinomial predict, ovr train and ovr predict. Refer to logistic.py\nRunning your code Run the scripts logistic binary.sh and logistic multiclass.sh after you\nfinish your implementation. This will output:\n• logistic binary.out\n• logistic multiclass.out\nWhat to submit Submit logistic.py, logistic binary.out, logistic multiclass.out.\n1.2.2 Neural networks\nPreparation Read Sect. 3 as well as dnn mlp.py and dnn cnn.py.\nCoding First, in dnn misc.py, finish implementing\n• forward and backward functions in class linear layer\n• forward and backward functions in class relu\n• backward function in class dropout (before that, please read forward function).\nSecond, in dnn cnn 2.py, finish implementing the main function. There are five TODO items. Refer to\nRunning your code Run the scripts q33.sh, q34.sh, q35.sh, q36.sh, q37.sh, q38.sh, q310.sh after\nyou finish your implementation. This will generate, respectively,\n3\nMLP lr0.01 m0.0 w0.0 d0.0.json\nMLP lr0.01 m0.0 w0.0 d0.5.json\nMLP lr0.01 m0.0 w0.0 d0.95.json\nLR lr0.01 m0.0 w0.0 d0.0.json\nCNN lr0.01 m0.0 w0.0 d0.5.json\nCNN lr0.01 m0.9 w0.0 d0.5.json\nCNN2 lr0.001 m0.9 w0.0 d0.5.json\nWhat to submit Submit dnn misc.py, dnn cnn 2.py, and the above seven .json files.\n1.3 Cautions\n• Do not import packages that are not listed above (See Python Packages section).\n• Follow the instructions in each section strictly to code up your solutions.\n• DO NOT CHANGE THE OUTPUT FORMAT.\n• DO NOT MODIFY THE CODE UNLESS WE INSTRUCT YOU TO DO SO.\n• A homework solution that mismatches the provided setup, such as format, name, initializations, etc.,\n• It is your responsibility to make sure that your code runs with Python 3.5.2 in the VM.\n1.4 Advice We are extensively using softmax and sigmoid function in this homework. To avoid numerical\nissues such as overflow and underflow caused by numpy.exp() and numpy.log(), please use the following\nimplementations:\n• Let x be a input vector to the softmax function. Use ˜x = x − max(x) instead of using x directly for the\nsoftmax function f . That is, if you want to compute f(x)i\n, compute f(x˜)i =\nexp(x˜i\n)\n\nD\nj=1\nexp(x˜j\n)\nis clearly mathematically equivalent but numerically more stable.\n• If you are using numpy.log(), make sure the input to the log function is positive. Also, there may\nbe chances that one of the outputs of softmax, e.g. f(x˜)i\n, is extremely small but you need the value\nln(f(x˜)i). In this case you should convert the computation equivalently into ˜xi − ln(∑\nD\nj=1\nexp(x˜j)).\nWe have implemented and run the code ourselves without problems, so if you follow the instructions\nand settings provided in the python files, you should not encounter overflow or underflow.\n4\nProblem 2 Logistic Regression (20 Points)\nFor this assignment you are asked to implement Logistic Regression for binary and multiclass classification.\nQ2.1 (6 Points)\nIn lecture 3 we discussed logistic regression for binary classification. In this problem, you are given a\ntraining set D =\n\b\n(xn, yn)\nN\nn=1\n\n, where yi ∈ {0, 1} ∀i = 1…N. Important: note that here the binary labels are\nnot −1 or +1 as used in the lecutre, so be very careful about applying formulas from the lecture notes.\nTx + b that minimizes the logistic loss. Note that we\ndo not explicitly append the feature 1 to the data, so you need to explicitly learn the bias/intercept term\nb too. Specifically you need to implement function binary train in logistic.py which uses gradient\ndescent (not stochastic gradient descent) to find the optimal parameters (recall logistic regression does not\nIn addition you need to implement function binary predict in logistic.py. We discuss two ways\nof making predictions in logistic regression in lecture 4: deterministic prediction or randomized prediction.\nHere you need to use the deterministic prediction.\nAfter finishing implementation, please run logistic binary.sh which generates logistic binary.out.\nWhat to submit:\n• logistic.py\n• logistic binary.out\nQ2.2 (7 Points) In the lectures you learned several methods to perform multiclass classification. One of\nthem was one-versus-rest or one-versus-all approach.\nFor one-versus-rest classification in a problem with K classes, we need to train K classifiers using a\nblack-box. Classifier k is trained on a binary problem, where the two labels corresponds to belonging or\nnot belonging to class k. After that, the multiclass prediction is made based on the combination of all\npredictions from K binary classifiers.\nIn this problem you will implement one-versus-rest using binary logistic regression (that you have implemented in Q2.1) as the black-box. Important: the way to predict discussed in the lecture is to randomized\nover the classifiers that say “yes”; however, here since binary logistic regression naturally predicts a probability for each class (recall the sigmoid model), we will simply predict the class with the highest probability\n(using numpy argmax).\nTo sum up, you need to complete functions OVR train and OVR predict to perform one-versus-rest\nclassification. After you finished implementation, please run logistic multiclass.sh script, which\nwill produce logistic multiclass.out.\nWhat to submit: logistic.py and logistic multiclass.out.\nQ2.3 (7 Points) Yet another multiclass classification method you learned was multinomial logistic regression. Complete the functions multinomial train and multinomial predict to perform multinomial\nlogistic regression, following the same notes as in Q2.1, that is, 1) explicitly learn the biased term; 2) perform\nAfter you finished implementation, please run logistic multiclass.sh script, which will produce\nlogistic multiclass.out.\nWhat to submit: logistic.py and logistic multiclass.out.\n5\nx\ninput features\nu h a z yˆ\npredicted label\nlinear(1)\nrelu linear(2)\nsoftmax\nFigure 1: A diagram of a multi-layer perceptron (MLP). The edges mean mathematical operations (modules), and the circles\nmean variables. The term relu stands for rectified linear units.\nProblem 3 Neural networks: multi-layer perceptrons (MLPs) and convolutional neural networks (CNNs)\n(30 Points)\nBackground\nIn recent years, neural networks have been one of the most powerful machine learning models. Many toolboxes/platforms (e.g., TensorFlow, PyTorch, Torch, Theano, MXNet, Caffe, CNTK) are publicly available\nfor efficiently constructing and training neural networks. The core idea of these toolboxes is to treat a neural\nnetwork as a combination of data transformation (or mathematical operation) modules.\nFor example, in Fig. 1 we provide a diagram of a multi-layer perceptron (MLP, just another term for\nfully connected feedforward networks we discussed in the lecture) for a K-class classification problem. The\nedges correspond to modules and the circles correspond to variables. Let (x ∈ RD, y ∈ {1, 2, · · · , K}) be a\nlabeled instance, such an MLP performs the following computations\ninput features : x ∈ R\nD (1)\nlinear(1)\n: u = W(1)\nx + b\n(1)\n,W(1) ∈ R\nM×D and b\n(1) ∈ R\nM (2)\nrelu : h = max{0, u} =\n\nmax{0, u1}\n.\n.\n.\nmax{0, uM}\n\n (3)\nlinear(2)\n: a = W(2)h + b\n(2)\n,W(2) ∈ R\nK×M and b\n(2) ∈ R\nK\n(4)\nsoftmax : z =\n\ne\na1\n∑k\ne\nak\n.\n.\n.\ne\naK\n∑k\ne\nak\n\n(5)\npredicted label : yˆ = arg maxk\nzk\n. (6)\nFor a K-class classification problem, one popular loss function for training (i.e., to learn W(1)\n, W(2)\n, b\n(1)\n,\nb\n(2)\n) is the cross-entropy loss. Specifically we denote the cross-entropy loss with respect to the training\nexample (x, y) by l:\nl = − log(zy) = log\n1 + ∑\nk6=y\ne\nak−ay\n!\nNote that one should look at l as a function of the parameters of the network, that is, W(1)\n, b\n(1)\n,W(2) and\nb\n(2)\n. For ease of notation, let us define the one-hot (i.e., 1-of-K) encoding of a class y as\n6\ny ∈ R\nK\nand yk =\n(\n1, if y = k,\n0, otherwise.\n(7)\nso that\nl = − ∑\nk\nyk\nlog zk = −y\nT\n\nlog z1\n.\n.\n.\nlog zK\n\n = −y\nT\nlog z. (8)\nWe can then perform error-backpropagation, a way to compute partial derivatives (or gradients) w.r.t\nthe parameters of a neural network, and use gradient-based optimization to learn the parameters.\nModules\nNow we will provide more information on modules for this assignment. Each module has its own parameters (but note that a module may have no parameters). Moreover, each module can perform a forward\npass and a backward pass. The forward pass performs the computation of the module, given the input to\nthe module. The backward pass computes the partial derivatives of the loss function w.r.t. the input and\nparameters, given the partial derivatives of the loss function w.r.t. the output of the module. Consider a\nmodule hmodule namei. Let hmodule namei.forward and hmodule namei.backward be its forward and\nbackward passes, respectively.\nFor example, the linear module may be defined as follows.\nforward pass: u = linear(1)\n.forward(x) = W(1)\nx + b\n(1)\n, (9)\nwhere W(1)\nand b\n(1)\nare its parameters.\nbackward pass: [\n∂l\n∂x\n,\n∂l\n∂W(1)\n,\n∂l\n∂b\n(1)\n] = linear(1)\n.backward(x,\n∂l\n∂u\n). (10)\nLet us assume that we have implemented all the desired modules. Then, getting ˆy for x is equivalent to\nrunning the forward pass of each module in order, given x. All the intermediated variables (i.e., u, h, etc.)\nwill all be computed along the forward pass. Similarly, getting the partial derivatives of the loss function\nw.r.t. the parameters is equivalent to running the backward pass of each module in a reverse order, given\n∂l\n∂z\n.\nIn this question, we provide a Python environment based on the idea of modules. Every module is\ndefined as a class, so you can create multiple modules of the same functionality by creating multiple object\ninstances of the same class. Your work is to finish the implementation of several modules, where these\nmodules are elements of a multi-layer perceptron (MLP) or a convolutional neural network (CNN). We\nwill apply these models to the same 10-class classification problem introduced in Sect. 2. We will train\nthe models using stochastic gradient descent with mini-batch, and explore how different hyperparameters\nof optimizers and regularization techniques affect training and validation accuracies over training epochs.\nFor deeper understanding, check out, e.g., the seminal work of Yann LeCun et al. “Gradient-based learning\napplied to document recognition,” written in 1998.\nWe give a specific example below. Suppose that, at iteration t, you sample a mini-batch of N examples\n{(xi ∈ RD, yi ∈ RK)}\nN\ni=1\nfrom the training set (K = 10). Then, the loss of such a mini-batch given by Fig. 1\nis\n7\nx yˆ\nlinear(1)\nrelu dropout linear(2)\nsoftmax\nFigure 2: The diagram of the MLP implemented in dnn mlp.py. The circles mean variables and edges mean modules.\nlmb =\n1\nN\nN\n\ni=1\nl(softmax.forward(linear(2)\n.forward(relu.forward(linear(1)\n.forward(xi)))), yi) (11)\n=\n1\nN\nN\n\ni=1\nl(softmax.forward(linear(2)\n.forward(relu.forward(ui))), yi) (12)\n= · · · (13)\n=\n1\nN\nN\n\ni=1\nl(softmax.forward(ai), yi) (14)\n=\n1\nN\nN\n\ni=1\nK\n\nk=1\nyik log zik. (15)\nThat is, in the forward pass, we can perform the computation of a certain module to all the N input examples, and then pass the N output examples to the next module. This is the same case for the backward pass.\nFor example, according to Fig. 1, given the partial derivatives of the loss w.r.t. {ai}\nN\ni=1\n∂lmb\n∂{ai}\nN\ni=1\n=\n\n(\n∂lmb\n∂a1\n)\nT\n(\n∂lmb\n∂a2\n)\nT\n.\n.\n.\n(\n∂lmb\n∂aN−1\n)\nT\n(\n∂lmb\n∂aN\n)\nT\n\n, (16)\nlinear(2)\n.backward will compute\n∂lmb\n∂{hi}\nN\ni=1\nand pass it back to relu.backward.\nPreparation\nQ3.1 Please read through dnn mlp.py and dnn cnn.py. Both files will use modules defined in dnn misc.py\n(which you will modify). Your work is to understand how modules are created, how they are linked to\nperform the forward and backward passes, and how parameters are updated based on gradients (and momentum). The architectures of the MLP and CNN defined in dnn mlp.py and dnn cnn.py are shown in\nFig. 2 and Fig. 3, respectively.\nWhat to submit: Nothing.\nCoding: Modules\n8\nx yˆ\nconvolution relu max pooling flatten dropout linear softmax\nFigure 3: The diagram of the CNN implemented in dnn cnn.py. The circles correspond to variables and edges\ncorrespond to modules. Note that the input to CNN may not be a vector (e.g., in dnn cnn.py it is an image, which can\nbe represented as a 3-dimensional tensor). The flatten layer is to reshape its input into vector.\nQ3.2 (14 Points) You will modify dnn misc.py. This script defines all modules that you will need to\nconstruct the MLP and CNN in dnn mlp.py and dnn cnn.py, respectively. You have three tasks. First,\nfinish the implementation of forward and backward functions in class linear layer. Please follow\nEqn. (2) for the forward pass and derive the partial derivatives accordingly. Second, finish the implementation of forward and backward functions in class relu. Please follow Eqn. (3) for the forward pass\nand derive the partial derivatives accordingly. Third, finish the the implementation of backward function\nin class dropout. We define the forward and the backward passes as follows.\nforward pass: s = dropout.forward(q ∈ R\nJ\n) = 1\n1 − r\n×\n\n1[p1 >= r] × q1\n.\n.\n.\n1[pJ >= r] × qJ\n\n , (17)\nwhere pj\nis sampled uniformly from [0, 1), ∀j ∈ {1, · · · , J},\nand r ∈ [0, 1) is a pre-defined scalar named dropout rate. (18)\nbackward pass: ∂l\n∂q\n= dropout.backward(q,\n∂l\n∂s\n) = 1\n1 − r\n×\n\n1[p1 >= r] ×\n∂l\n∂s1\n.\n.\n.\n1[pJ >= r] ×\n∂l\n∂sJ\n\n. (19)\nNote that pj\n, j ∈ {1, · · · , J} and r are not be learned so we do not need to compute the derivatives w.r.t.\nto them. Moreover, pj\n, j ∈ {1, · · · , J} are re-sampled every forward pass, and are kept for the following\nbackward pass. The dropout rate r is set to 0 during testing.\nDetailed descriptions/instructions about each pass (i.e., what to compute and what to return) are included in dnn misc.py. Please do read carefully.\nNote that in this script we do import numpy as np. Thus, to call a function XX from numpy, please\nuse np.XX.\nWhat to do and submit: Finish the implementation of 5 functions specified above in dnn misc.py. Submit your completed dnn misc.py. We do provide a checking code hw2 dnn check.py to check your\nimplementation.\nTesting dnn misc.py with multi-layer perceptron (MLP)\nQ3.3 (2 Points) What to do and submit: run script q33.sh. It will output MLP lr0.01 m0.0 w0.0 d0.0.json.\nAdd, commit, and push this file before the due date.\nWhat it does: q33.sh will run python3 dnn mlp.py with learning rate 0.01, no momentum, no weight\ndecay, and dropout rate 0.0. The output file stores the training and validation accuracies over 30 training\nepochs.\n9\nQ3.4 (2 Points) What to do and submit: run script q34.sh. It will output MLP lr0.01 m0.0 w0.0 d0.5.json.\nAdd, commit, and push this file before the due date.\nWhat it does: q34.sh will run python3 dnn mlp.py –dropout rate 0.5 with learning rate 0.01,\nno momentum, no weight decay, and dropout rate 0.5. The output file stores the training and validation\naccuracies over 30 training epochs.\nQ3.5 (2 Points) What to do and submit: run script q35.sh. It will output MLP lr0.01 m0.0 w0.0 d0.95.json.\nAdd, commit, and push this file before the due date.\nWhat it does: q35.sh will run python3 dnn mlp.py –dropout rate 0.95 with learning rate 0.01,\nno momentum, no weight decay, and dropout rate 0.95. The output file stores the training and validation\naccuracies over 30 training epochs.\nYou will observe that the model in Q3.4 will give better validation accuracy (at epoch 30) compared to\nQ3.3. Specifically, dropout is widely-used to prevent over-fitting. However, if we use a too large dropout\nrate (like the one in Q3.5), the validation accuracy (together with the training accuracy) will be relatively\nlower, essentially under-fitting the training data.\nQ3.6 (2 Points) What to do and submit: run script q36.sh. It will output LR lr0.01 m0.0 w0.0 d0.0.json.\nAdd, commit, and push this file before the due date.\nWhat it does: q36.sh will run python3 dnn mlp nononlinear.py with learning rate 0.01, no momentum, no weight decay, and dropout rate 0.0. The output file stores the training and validation accuracies\nover 30 training epochs.\nThe network has the same structure as the one in Q3.3, except that we remove the relu (nonlinear) layer.\nYou will see that the validation accuracies drop significantly (the gap is around 0.03). Essentially, without\nthe nonlinear layer, the model is learning multinomial logistic regression similar to Q2.3.\nTesting dnn misc.py with convolutional neural networks (CNN)\nQ3.7 (2 Points) What to do and submit: run script q37.sh. It will output CNN lr0.01 m0.0 w0.0 d0.5.json.\nAdd, commit, and push this file before the due date.\nWhat it does: q37.sh will run python3 dnn cnn.py with learning rate 0.01, no momentum, no weight\ndecay, and dropout rate 0.5. The output file stores the training and validation accuracies over 30 training\nepochs.\nQ3.8 (2 Points) What to do and submit: run script q38.sh. It will output CNN lr0.01 m0.9 w0.0 d0.5.json.\nAdd, commit, and push this file before the due date.\nWhat it does: q38.sh will run python3 dnn cnn.py –alpha 0.9 with learning rate 0.01, momentum\n0.9, no weight decay, and dropout rate 0.5. The output file stores the training and validation accuracies over\n30 training epochs.\nYou will see that Q3.8 will lead to faster convergence than Q3.7 (i.e., the training/validation accuracies\nwill be higher than 0.94 after 1 epoch). That is, using momentum will lead to more stable updates of the\nparameters.\nCoding: Building a deeper architecture\nQ3.9 (2 Points) The CNN architecture in dnn cnn.py has only one convolutional layer. In this question,\nyou are going to construct a two-convolutional-layer CNN (see Fig. 4 using the modules you implemented\nin Q3.2. Please modify the main function in dnn cnn 2.py. The code in dnn cnn 2.py is similar to that\nin dnn cnn.py, except that there are a few parts marked as TODO. You need to fill in your code so as to\nconstruct the CNN in Fig. 4.\n10\nx yˆ\nconv relu max-p conv relu max-p flatten dropout linear softmax\nFigure 4: The diagram of the CNN you are going to implement in dnn cnn 2.py. The term conv stands for convolution; max-p stands for max pooling. The circles correspond to variables and edges correspond to modules. Note that the\ninput to CNN may not be a vector (e.g., in dnn cnn 2.py it is an image, which can be represented as a 3-dimensional\ntensor). The flatten layer is to reshape its input into vector.\nWhat to do and submit: Finish the implementation of the main function in dnn cnn 2.py (search for TODO\nin main). Submit your completed dnn cnn 2.py.\nTesting dnn cnn 2.py\nQ3.10 (2 Points) What to do and submit: run script q310.sh. It will output CNN2 lr0.001 m0.9 w0.0 d0.5.json.\nAdd, commit, and push this file before the due date.\nWhat it does: q310.sh will run python3 dnn cnn 2.py –alpha 0.9 with learning rate 0.01, momentum 0.9, no weight decay, and dropout rate 0.5. The output file stores the training and validation accuracies\nover 30 training epochs.\nYou will see that you can achieve slightly higher validation accuracies than those in Q3.8.\n11"
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"# Source code for tensorpack.dataflow.serialize\n\n# -*- coding: utf-8 -*-\n# File: serialize.py\n\nimport numpy as np\nimport os\nimport platform\nfrom collections import defaultdict\n\nfrom ..utils import logger\nfrom ..utils.develop import create_dummy_class # noqa\nfrom ..utils.utils import get_tqdm\nfrom .base import DataFlow\nfrom .common import FixedSizeData, MapData\nfrom .format import HDF5Data, LMDBData\nfrom .raw import DataFromGenerator, DataFromList\n\n__all__ = ['LMDBSerializer', 'NumpySerializer', 'TFRecordSerializer', 'HDF5Serializer']\n\ndef _reset_df_and_get_size(df):\ndf.reset_state()\ntry:\nsz = len(df)\nexcept NotImplementedError:\nsz = 0\nreturn sz\n\n[docs]class LMDBSerializer():\n\"\"\"\nSerialize a Dataflow to a lmdb database, where the keys are indices and values\nare serialized datapoints.\n\nYou will need to pip install lmdb to use it.\n\nExample:\n\n.. code-block:: python\n\nLMDBSerializer.save(my_df, \"output.lmdb\")\n\n\"\"\"\n[docs] @staticmethod\ndef save(df, path, write_frequency=5000):\n\"\"\"\nArgs:\ndf (DataFlow): the DataFlow to serialize.\npath (str): output path. Either a directory or an lmdb file.\nwrite_frequency (int): the frequency to write back data to disk.\nA smaller value reduces memory usage.\n\"\"\"\nassert isinstance(df, DataFlow), type(df)\nisdir = os.path.isdir(path)\nif isdir:\nassert not os.path.isfile(os.path.join(path, 'data.mdb')), \"LMDB file exists!\"\nelse:\nassert not os.path.isfile(path), \"LMDB file {} exists!\".format(path)\n# It's OK to use super large map_size on Linux, but not on other platforms\n# See: https://github.com/NVIDIA/DIGITS/issues/206\nmap_size = 1099511627776 * 2 if platform.system() == 'Linux' else 128 * 10**6\ndb = lmdb.open(path, subdir=isdir,\nmeminit=False, map_async=True) # need sync() at the end\nsize = _reset_df_and_get_size(df)\n\n# put data into lmdb, and doubling the size if full.\n# Ref: https://github.com/NVIDIA/DIGITS/pull/209/files\ndef put_or_grow(txn, key, value):\ntry:\ntxn.put(key, value)\nreturn txn\nexcept lmdb.MapFullError:\npass\ntxn.abort()\ncurr_size = db.info()['map_size']\nnew_size = curr_size * 2\nlogger.info(\"Doubling LMDB map_size to {:.2f}GB\".format(new_size / 10**9))\ndb.set_mapsize(new_size)\ntxn = db.begin(write=True)\ntxn = put_or_grow(txn, key, value)\nreturn txn\n\nwith get_tqdm(total=size) as pbar:\nidx = -1\n\n# LMDB transaction is not exception-safe!\n# although it has a context manager interface\ntxn = db.begin(write=True)\nfor idx, dp in enumerate(df):\ntxn = put_or_grow(txn, u'{:08}'.format(idx).encode('ascii'), dumps(dp))\npbar.update()\nif (idx + 1) % write_frequency == 0:\ntxn.commit()\ntxn = db.begin(write=True)\ntxn.commit()\n\nkeys = [u'{:08}'.format(k).encode('ascii') for k in range(idx + 1)]\nwith db.begin(write=True) as txn:\ntxn = put_or_grow(txn, b'__keys__', dumps(keys))\n\nlogger.info(\"Flushing database ...\")\ndb.sync()\ndb.close()\n\n[docs] @staticmethod\n\"\"\"\nNote:\nIf you found deserialization being the bottleneck, you can use :class:LMDBData as the reader\nand run deserialization as a mapper in parallel.\n\"\"\"\ndf = LMDBData(path, shuffle=shuffle)\nreturn MapData(df, LMDBSerializer._deserialize_lmdb)\n\n@staticmethod\ndef _deserialize_lmdb(dp):\n\n[docs]class NumpySerializer():\n\"\"\"\nSerialize the entire dataflow to a npz dict.\nNote that this would have to store the entire dataflow in memory,\nand is also >10x slower than LMDB/TFRecord serializers.\n\"\"\"\n\n[docs] @staticmethod\ndef save(df, path):\n\"\"\"\nArgs:\ndf (DataFlow): the DataFlow to serialize.\npath (str): output npz file.\n\"\"\"\nbuffer = []\nsize = _reset_df_and_get_size(df)\nwith get_tqdm(total=size) as pbar:\nfor dp in df:\nbuffer.append(dp)\npbar.update()\nnp.savez_compressed(path, buffer=np.asarray(buffer, dtype=np.object))\n\n[docs] @staticmethod\n# allow_pickle defaults to False since numpy 1.16.3\nreturn DataFromList(buffer, shuffle=shuffle)\n\n[docs]class TFRecordSerializer():\n\"\"\"\nSerialize datapoints to bytes (by tensorpack's default serializer) and write to a TFRecord file.\n\nNote that TFRecord does not support random access and is in fact not very performant.\nIt's better to use :class:LMDBSerializer.\n\"\"\"\n[docs] @staticmethod\ndef save(df, path):\n\"\"\"\nArgs:\ndf (DataFlow): the DataFlow to serialize.\npath (str): output tfrecord file.\n\"\"\"\nsize = _reset_df_and_get_size(df)\nwith tf.python_io.TFRecordWriter(path) as writer, get_tqdm(total=size) as pbar:\nfor dp in df:\nwriter.write(dumps(dp))\npbar.update()\n\n[docs] @staticmethod\n\"\"\"\nArgs:\nsize (int): total number of records. If not provided, the returned dataflow will have no __len__().\nIt's needed because this metadata is not stored in the TFRecord file.\n\"\"\"\ngen = tf.python_io.tf_record_iterator(path)\nds = DataFromGenerator(gen)\nif size is not None:\nds = FixedSizeData(ds, size)\nreturn ds\n\n[docs]class HDF5Serializer():\n\"\"\"\nWrite datapoints to a HDF5 file.\n\nNote that HDF5 files are in fact not very performant and currently do not support lazy loading.\nIt's better to use :class:LMDBSerializer.\n\"\"\"\n[docs] @staticmethod\ndef save(df, path, data_paths):\n\"\"\"\nArgs:\ndf (DataFlow): the DataFlow to serialize.\npath (str): output hdf5 file.\ndata_paths (list[str]): list of h5 paths. It should have the same\nlength as each datapoint, and each path should correspond to one\ncomponent of the datapoint.\n\"\"\"\nsize = _reset_df_and_get_size(df)\nbuffer = defaultdict(list)\n\nwith get_tqdm(total=size) as pbar:\nfor dp in df:\nassert len(dp) == len(data_paths), \"Datapoint has {} components!\".format(len(dp))\nfor k, el in zip(data_paths, dp):\nbuffer[k].append(el)\npbar.update()\n\nwith h5py.File(path, 'w') as hf, get_tqdm(total=len(data_paths)) as pbar:\nfor data_path in data_paths:\nhf.create_dataset(data_path, data=buffer[data_path])\npbar.update()\n\n[docs] @staticmethod\n\"\"\"\nArgs:\ndata_paths (list): list of h5 paths to be zipped.\n\"\"\"\nreturn HDF5Data(path, data_paths, shuffle)\n\ntry:\nimport lmdb\nexcept ImportError:\nLMDBSerializer = create_dummy_class('LMDBSerializer', 'lmdb') # noqa\n\ntry:\nfrom tensorpack.compat import tfv1 as tf\nexcept ImportError:\nTFRecordSerializer = create_dummy_class('TFRecordSerializer', 'tensorflow') # noqa\n\ntry:\nimport h5py\nexcept ImportError:\nHDF5Serializer = create_dummy_class('HDF5Serializer', 'h5py') # noqa\n\nif __name__ == '__main__':\nfrom .raw import FakeData\nimport time\nds = FakeData([[300, 300, 3], ], 1000)\n\nprint(time.time())\nTFRecordSerializer.save(ds, 'out.tfrecords')\nprint(time.time())\ndf.reset_state()\nfor idx, dp in enumerate(df):\npass\nprint(\"TF Finished, \", idx)\nprint(time.time())\n\nLMDBSerializer.save(ds, 'out.lmdb')\nprint(time.time())\ndf.reset_state()\nfor idx, dp in enumerate(df):\npass\nprint(\"LMDB Finished, \", idx)\nprint(time.time())\n\nNumpySerializer.save(ds, 'out.npz')\nprint(time.time())\ndf.reset_state()\nfor idx, dp in enumerate(df):\npass\nprint(\"Numpy Finished, \", idx)\nprint(time.time())\n\npaths = ['p1', 'p2']\nHDF5Serializer.save(ds, 'out.h5', paths)\nprint(time.time())"
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http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff;f=topics/_week10_gsv.mdwn;h=1cd25a800c4c03b671bdfcb37cc09fe044699913;hp=f0336123e547a91e7d13736d8999c560339b13d2;hb=6e3c775751114918d3c5fb7be4708dc0c555315f;hpb=fd2cb06c9e18732a6fbbf20da0b2f92dc981a5db
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[
"index f033612..1cd25a8 100644 (file)\n@@ -9,12 +9,15 @@\nGSV are interested in developing and establishing a reasonable theory\nof discourse update. One way of looking at this paper is like this:\n\n- GSV = GS + V\n+ GSV = GS + V, where\n\n- GS = Dynamic Predicate Logic L&P 1991: dynamic binding, donkey anaphora\n+ GS = Dynamic theories of binding of Groenendijk and Stokhof, e.g.,\n+ Dynamic Predicate Logic L&P 1991: dynamic binding, donkey anaphora\nDynamic Montague Grammar 1990: generalized quantifiers, discourse referents\n\n- V = epistemic modality\n+ V = a dynamic theory of epistemic modality, e.g.,\n+ Veltman, Frank. \"Data semantics.\"\n+ In Truth, Interpretation and Information, Foris, Dordrecht (1984): 43-63.\n\nThat is, Groenendijk and Stokhof have a well-known theory of dynamic\nsemantics, and Veltman has a well-known theory of epistemic modality,\n@@ -26,91 +29,42 @@ view and from a practical engineering point of view. On the\ntheoretical level, these scholars are proposing a strategy for\nmanaging the connection between variables and the objects they\ndesignate in way that is flexible enough to be useful for describing\n-natural language. The main way they attempt to do this is by\n-inserting an extra level in between the variable and the object:\n-instead of having an assignment function that maps variables directly\n-onto objects, GSV provide *pegs*: variables map onto pegs, and pegs\n-map onto objects. We'll discuss in considerable detail what pegs\n-allow us to do, since it is highly relevant to one of the main\n-applications of the course, namely, reference and coreference.\n+natural language.\n\n-What are pegs? The term harks back to a paper by Landman called `Pegs\n-and Alecs'. There pegs are simply hooks for hanging properties on.\n-Pegs are supposed to be as anonymous as possible. Think of hanging\n-your coat on a physical peg: you don't care which peg it is, only that\n-there are enough pegs for everyone's coat to hang from. Likewise, for\n-the pegs of GSV, all that matters is that there are enough of them.\n-(Incidentally, there is nothing in Gronendijk and Stokhof's original\n-DPL paper that corresponds naturally to pegs; but in their Dynamic\n-Montague Grammar paper, pegs serve a purpose similar to discourse\n-referents there, though the connection is not simple.)\n-\n-On an engineering level, the fact that GSV are combining anaphora and\n-bound quantification with epistemic quantification means that they are\n-gluing together related but distinct subsystems into a single\n-fragment. These subsystems naturally cleave into separate layers in a\n-way that is obscured in the paper. We will argue in detail that\n-does all of the same theoretical work.\n-\n-Empirical targets: on the anaphoric side, GSV want to\n-\n-On the epistemic side, GSV aim to account for asymmetries such as\n-\n- It might be raining. It's not raining.\n- #It's not raining. It might be raining.\n-\n-## Two-part assignment functions\n-\n-There are a lot of formal details in the paper in advance of the\n-empirical discussion. Here are the ones that matter for our purposes:\n-\n- type var = string\n- type peg = int\n- type refsys = var -> peg\n- type ent = Alice | Bob | Carl\n- type assignment = peg -> ent\n-\n-So in order to get from a variable to an object, we have to compose a\n-refsys `r` with an assignment `g`. For instance, we might have\n-r (g (\"x\")) = Alice. A question to keep in mind as we proceed is why\n-the mapping from variables to objects has been articulated into two\n-functions. Why not map variables directly to objects? (We'll return\n-to this question later.)\n-\n- type pred = string\n- type world = pred -> ent -> bool\n- type pegcount = int\n- type poss = world * pegcount * refsys * assignment\n- type infostate = [poss]\n+## Basics of GSV's fragment\n\n-Worlds in general settle all matters of fact in the world. In\n-particular, they determine the extensions of predicates and relations.\n-In this discussion, we'll (crudely) approximate worlds by making them\n-a function from predicates such as \"man\" to a function mapping each\n-entity to a boolean.\n+The fragment in this paper is unusually elegant. We'll present it on\n+its own terms, with the exception that we will not use pegs. See the\n+digression below concerning pegs for an explanation. After presenting\n+the paper, we'll re-engineering the fragment using explicit monads.\n\n-As we'll see, indefinites as a side effect increase the number of pegs\n-by one. GSV assume that we can determine what integer the next unused\n-peg corresponds to by examining the range of the refsys function.\n-We'll make things easy on ourselves by simply tracking the total\n-number of used pegs in a counter called `pegcount`.\n+In this fragment, points of evaluation are not just worlds, but a pair\n+of a world and an assginment function. This is familiar from Heim's\n+1983 File Change Semantics. We'll follow GSV and call a\n+world-assignment pair a \"possibility\". Then a context is a set (an\n+\"information state\") is a set of possiblities. Infostates\n+simultaneously track both information about the world (which possible\n+worlds are live possibilities?) as well as information about the\n+discourse (which objects to the variables refer to?).\n\n-So information states track both facts about the world (e.g., which\n-objects count as a man), and facts about the discourse (e.g., how many\n-pegs have been used).\n+Worlds in general settle all matters of fact in the world. In\n+particular, they determine the extensions of predicates and relations.\n\nThe formal language the fragment interprets is Predicate Calculus with\n-equality, existential and universal quantification, and one unary\n-modality (box and diamond, corresponding to epistemic necessity and\n-epistemic possibility).\n+equality, existential and universal quantification, along with one\n+unary modality (box and diamond, corresponding to epistemic necessity\n+and epistemic possibility).\n+\n+An implementation in OCaml is available [[here|code/gsv.ml]]; consult\n+that code for details of syntax, types, and values. [[An implementation\n+in Haskell|code/gsv.hs]] is available as well, if you prefer.\n\nTerms in this language are either individuals such as Alice or Bob, or\nelse variables. So in general, the referent of a term can depend on a\npossibility:\n\nref(i, t) = t if t is an individual, and\n- g(r(t)) if t is a variable, where i = (w,n,r,g)\n+ g(t) if t is a variable, where i = (w,g)\n\nHere are the main clauses for update (their definition 3.1).\n\n@@ -119,12 +73,12 @@ state `s` with the information in φ) as `s[φ]`.\n\ns[P(t)] = {i in s | w(P)(ref(i,t))}\n\n-So `man(x)` is the set of live possibilities `i = (w,r,g)` in s such that\n+So `man(x)` is the set of live possibilities `i = (w,g)` in s such that\nthe set of men in `w` given by `w(man)` maps the object referred to by\n-`x`, namely, `r(g(\"x\"))`, to `true`. That is, update with \"man(x)\"\n+`x`, namely, `g(\"x\")`, to `true`. That is, update with \"man(x)\"\ndiscards all possibilities in which \"x\" fails to refer to a man.\n\n- s[t1 = t2] = {i in s | ref(i,t1) = ref(i,t2)}\n+ s[t1 = t2] = {i in s | ref(i,t1) == ref(i,t2)}\n\ns[φ and ψ] = s[φ][ψ]\n\n@@ -133,14 +87,14 @@ then update with the right conjunct.\n\nExistential quantification is somewhat intricate.\n\n- s[∃xφ] = Union {{(w, n+1, r[x->n], g[n->a]) | (w,n,r,g) in s}[φ] | a in ent}\n+ s[∃xφ] = Union {{(w, g[x->a]) | (w,g) in s}[φ] | a in ent}\n\nHere's the recipe: given a starting infostate s, choose an object a\nfrom the domain of discourse. Construct a modified infostate s' by\n-adding a peg to each possibility in s and adjusting the refsys and the\n-assignment in order to map the variable x to a. Then update s' with\n-φ, and collect the results of doing this for every object a in the\n-domain of discourse.\n+adjusting the assignment function of each possibility so as to map the variable x to a.\n+Then update s' with φ. Finally, take the union over the results of\n+doing this for every object a in the domain of discourse. If you're\n+unsure about this, examine the [[code|code/gsv.ml]].\n\nNegation is natural enough:\n\n@@ -153,63 +107,26 @@ with respect to i.\nIn GSV, disjunction, the conditional, and the universals are defined\nin terms of negation and the other connectives (see fact 3.2).\n\n-Exercise: assume that there are two entities in the domain of\n-discourse, Alice and Bob. Assume that Alice is a woman, and Bob is a\n-man.\n-\n-We're using `++` here to mean set union.\n-\n- 1. {(w,n,r,g)}[∃x.person(x)]\n-\n- = {(w,n+1,r[x->n],g[n->a])}[person(x)] ++ {(w,n+1,r[x->n],g[n->b])}[person(x)]\n- = {(w,n+1,r[x->n],g[n->a])} ++ {(w,n+1,r[x->n],g[n->b])}\n- = {(w,n+1,r[x->n],g[n->a]),(w,n+1,r[x->n],g[n->b])}\n- -- both a and b are people\n+Exercise: assume that there are three entities in the domain of\n+discourse, Alice, Bob, and Carl. Assume that Alice is a woman, and\n+Bob and Carl are men.\n\n- 2. {(w,n,r,g)}[∃x.man(x)]\n+Compute the following:\n\n- = {(w,n+1,r[x->n],g[n->a])}[man(x)] ++ {(w,n+1,r[x->n],g[n->b])}[man(x)]\n- = {} ++ {(w,n+1,r[x->n],g[n->b])}\n- = {(w,n+1,r[x->n],g[n->b])}\n- -- only b is a man\n-\n- 3. {(w,n,r,g)}[∃x∃y.person(x) and person(y)]\n-\n- = {(w,n+1,r[x->n],g[n->a])}[∃y.person(x) and person(y)]\n- ++ {(w,n+1,r[x->n],g[n->b])}[∃y.person(x) and person(y)]\n-\n- = ( {(w, n+2, r[x->n][y->n+1], g[n->a][n+1->a])}[person(x)][person(y)]\n- ++ {(w, n+2, r[x->n][y->n+1], g[n->a][n+1->b])}[person(x)][person(y)])\n- ++ ( {(w, n+2, r[x->n][y->n+1], g[n->b][n+1->a])}[person(x)][person(y)]\n- ++ {(w, n+2, r[x->n][y->n+1], g[n->b][n+1->b])}[person(x)][person(y)])\n-\n- = {(w, n+2, r[x->n][y->n+1], g[n->a][n+1->a]),\n- (w, n+2, r[x->n][y->n+1], g[n->a][n+1->b])}\n- ++ {(w, n+2, r[x->n][y->n+1], g[n->b][n+1->a]),\n- (w, n+2, r[x->n][y->n+1], g[n->b][n+1->b])}\n-\n- = {(w, n+2, r[x->n][y->n+1], g[n->a][n+1->a]),\n- (w, n+2, r[x->n][y->n+1], g[n->a][n+1->b]),\n- (w, n+2, r[x->n][y->n+1], g[n->b][n+1->a]),\n- (w, n+2, r[x->n][y->n+1], g[n->b][n+1->b])}\n-\n- -- there are four ways of assigning x and y to people\n+ 1. {(w,g)}[∃x.man(x)]\n\n+ = {(w,g[n->a])}[man(x)] ++ {(w,g[n->b])}[man(x)]\n+ ++ {(w,g[n->c])}[man(x)]\n+ = {} ++ {(w,g[n->b])} ++ {(w,g[n->c])}\n+ = {(w,g[n->a]),(w,g[n->b]),(w,g[n->c])}\n+ -- Bob and Carl are men\n\n+ 2. {(w,g)}[∃x.woman(x)]\n+ 3. {(w,g)}[∃x∃y.man(x) and man(y)]\n4. {(w,n,r,g)}[∃x∃y.x=y]\n\n- = ( {(w, n+2, r[x->n][y->n+1], g[n->a][n+1->a])}[x=y]\n- ++ {(w, n+2, r[x->n][y->n+1], g[n->a][n+1->b])}[x=y]\n- ++ ( {(w, n+2, r[x->n][y->n+1], g[n->b][n+1->a])}[x=y]\n- ++ {(w, n+2, r[x->n][y->n+1], g[n->b][n+1->b])}[x=y]\n-\n- = {(w, n+2, r[x->n][y->n+1], g[n->a][n+1->a])}\n- ++ {(w, n+2, r[x->n][y->n+1], g[n->b][n+1->b])}\n-\n- = {(w, n+2, r[x->n][y->n+1], g[n->a][n+1->a]),\n- (w, n+2, r[x->n][y->n+1], g[n->b][n+1->b])}\n+Running the [[code|code/gsv.ml]] gives the answers.\n\n- -- two ways to assign x and y to the same value\n\n## Order and modality\n\n@@ -219,9 +136,10 @@ The final remaining update rule concerns modality:\n\nThis is a peculiar rule: a possibility `i` will survive update just in\ncase something is true of the information state `s` as a whole. That\n-means that either every `i` in `s` will survive, or none of them will. The\n-criterion is that updating `s` with the information in φ does not\n-produce the contradictory information state (i.e., `{}`).\n+means that either every `i` in `s` will survive, or none of them will.\n+The criterion is that updating `s` with the information in the\n+prejacent φ does not produce the contradictory information state\n+(i.e., `{}`).\n\nSo let's explore what this means. GSV offer a contrast between two\ndiscourses that differ only in the order in which the updates occur.\n@@ -233,9 +151,9 @@ order shows that the system is order-sensitive.\nAccording to GSV, the combination of these sentences in this order is\n`inconsistent', and they mark the second sentence with the star of\nungrammaticality. We'll say instead that the discourse is\n-gramamtical, leave the exact word to use for its intuitive effect up\n-for grabs. What is important for our purposes is to get clear on how\n-the fragment behaves with respect to these sentences.\n+gramamtical, leave the exact way to think about its intuitive status\n+up for grabs. What is important for our purposes is to get clear on\n+how the fragment behaves with respect to these sentences.\n\nWe'll start with an infostate containing two possibilities. In one\npossibility, Alice is hungry (call this possibility \"hungry\"); in the\n@@ -276,25 +194,10 @@ In contrast, consider the sentences in the opposite order:\n\nWe'll start with the same two possibilities.\n\n-\n= {hungry, full}[Alice might be hungry][Alice isn't hungry]\n= {hungry, full}[Alice isn't hungry]\n= {full}\n\n-Update with *Alice might be hungry* depends on the result of updating\n-with the prejacent, *Alice is hungry*. Here's the side calculation:\n-\n- {hungry, full}[Alice is hungry]\n- = {hungry}\n-\n-Since this update is non-empty, all of the original possibilities\n-survive update with *Alice might be hungry*. By now it should be\n-obvious that update with a *might* sentence either has no effect, or\n-produces an empty information state. The net result is that we can\n-then go on to update with *Alice isn't hungry*, yielding an updated\n-information state that contains only possibilities in which Alice\n-isn't hungry.\n-\nGSV comment that a single speaker couldn't possibly be in a position\nto utter the discourse in (2). The reason is that in order for the\nspeaker to appropriately assert that Alice isn't hungry, that speaker\n@@ -310,7 +213,8 @@ speaker assumes that as far as the listener knows, Alice might be\nhungry, they can utter the discourse in (2). Here's a variant that\nmakes this thought more vivid:\n\n- 3. Based on public evidence, Alice might be hungry. But in fact she's not hungry.\n+ 3. Based on public evidence, Alice might be hungry.\n+ But in fact I have private knowledge that she's not hungry.\n\nThe main point to appreciate here is that the update behavior of the\ndiscourses depends on the order in which the updates due to the\n@@ -323,17 +227,18 @@ concerning negation.\n5. Alice is hungry. (So of course) Alice might be hungry.\n\nBoth of these discourses lead to the same update effect: all and only\n-those possibilites in which Alice is hungry survive. If you think\n-that asserting *might* requires that the prejacent be undecided, you\n-will have to consider an update rule for the diamond on which update\n-with the prejacent and its negation must both be non-empty.\n+those possibilites in which Alice is hungry survive. You might think\n+that asserting *might* requires that the prejacent be not only\n+possible, but undecided. If you like this idea, you can easily write\n+an update rule for the diamond on which update with the prejacent and\n+its negation must both be non-empty.\n\n## Order and binding\n\nThe GSV fragment differs from the DPL and the DMG dynamic semantics in\n-important details. Nevertheless, it has more or less the same things\n-to say about anaphora, binding, quantificational binding, and donkey\n-anaphora.\n+important details. Nevertheless, it says something highly similar to\n+DPL about anaphora, binding, quantificational binding, and donkey\n+anaphora (at least, when modality is absent, as we'll discuss below).\n\nIn particular, continuing the theme of order-based asymmetries,\n\n@@ -346,47 +251,41 @@ where the second discourse does not. In order to demonstrate, we'll\nneed an information state whose refsys is defined for at least one\nvariable.\n\n- 8. {(w,1,r[x->0],g[0->b])}\n+ 8. {(w,g[x->b])}\n\nThis infostate contains a refsys and an assignment that maps the\nvariable x to Bob. Here are the facts in world w:\n\n- w \"enter\" a = false\n- w \"enter\" b = true\n- w \"enter\" c = true\n+ extension w \"enter\" a = false\n+ extension w \"enter\" b = true\n+ extension w \"enter\" c = true\n\n- w \"sit\" a = true\n- w \"sit\" b = true\n- w \"sit\" c = false\n+ extension w \"sit\" a = true\n+ extension w \"sit\" b = true\n+ extension w \"sit\" c = false\n\nWe can now consider the discourses in (6) and (7) (after magically\nconverting them to the Predicate Calculus):\n\n9. Someone^x entered. He_x sat.\n\n- {(w,1,r[x->0],g[0->b])}[∃x.enter(x)][sit(x)]\n-\n- -- the existential adds a new peg and assigns it to each\n- -- entity in turn\n+ {(w,g[x->b])}[∃x.enter(x)][sit(x)]\n\n- = ( {(w,2,r[x->0][x->1],g[0->b][1->a])}[enter(x)]\n- ++ {(w,2,r[x->0][x->1],g[0->b][1->b])}[enter(x)]\n- ++ {(w,2,r[x->0][x->1],g[0->b][1->c])}[enter(x)])[sit(x)]\n+ = ( {(w,g[x->b][x->a])}[enter(x)]\n+ ++ {(w,g[x->b][x->b])}[enter(x)]\n+ ++ {(w,g[x->b][x->c])}[enter(x)])[sit(x)]\n\n-- \"enter(x)\" filters out the possibility in which x refers\n-- to Alice, since Alice didn't enter\n\n= ( {}\n- ++ {(w,2,r[x->0][x->1],g[0->b][1->b])}\n- ++ {(w,2,r[x->0][x->1],g[0->b][1->c])})[sit(x)]\n+ ++ {(w,g[x->b][x->b])}\n+ ++ {(w,g[x->b][x->c])})[sit(x)]\n\n-- \"sit(x)\" filters out the possibility in which x refers\n-- to Carl, since Carl didn't sit\n\n- = {(w,2,r[x->0][x->1],g[0->b][1->b])}\n-\n-Note that `r[x->0][x->1]` maps `x` to 1---the outermost adjustment is\n-the operative one. In other words, `r[x->0][x->1] == (r[x->0])[x->1]`.\n+ = {(w,g[x->b][x->b])}\n\nOne of the key facts here is that even though the existential has\nscope only over the first sentence, in effect it binds the pronoun in\n@@ -397,28 +296,27 @@ The outcome is different if the order of the sentences is reversed.\n\n10. He_x sat. Someone^x entered.\n\n- {(w,1,r[x->0],g[0->b])}[sit(x)][∃x.enter(x)]\n+ {(w,g[x->b])}[sit(x)][∃x.enter(x)]\n\n-- evaluating `sit(x)` rules out nothing, since (coincidentally)\n-- x refers to Bob, and Bob is a sitter\n\n- = {(w,1,r[x->0],g[0->b])}[∃x.enter(x)]\n+ = {(w,g[x->b])}[∃x.enter(x)]\n\n-- Just as before, the existential adds a new peg and assigns\n-- it to each object\n\n- = {(w,2,r[x->0][x->1],g[0->b][1->a])}[enter(x)]\n- ++ {(w,2,r[x->0][x->1],g[0->b][1->b])}[enter(x)]\n- ++ {(w,2,r[x->0][x->1],g[0->b][1->c])}[enter(x)]\n+ = {(w,g[x->b][x->a])}[enter(x)]\n+ ++ {(w,g[x->b][x->b])}[enter(x)]\n+ ++ {(w,g[x->b][x->c])}[enter(x)]\n\n-- enter(x) eliminates all those possibilities in which x did\n-- not enter\n\n- = {} ++ {(w,2,r[x->0][x->1],g[0->b][1->b])}\n- ++ {(w,2,r[x->0][x->1],g[0->b][1->c])}\n+ = {} ++ {(w,g[x->b][x->b])}\n+ ++ {(w,g[x->b][x->c])}\n\n- = {(w,2,r[x->0][x->1],g[0->b][1->b]),\n- (w,2,r[x->0][x->1],g[0->b][1->c])}\n+ = {(w,g[x->b][x->b]), (w,g[x->b][x->c])}\n\nThe result is different than before. Before, there was only one\npossibility: that x refered to the only person who both entered and\n@@ -438,12 +336,10 @@ Intuitively, there is a strong impression in (12) that the person who\nentered and spoke not only should not be identified as the person who\nsat, he should be different from the person who sat. Some dynamic\nsystems, such as Heim's File Change Semantics, guarantee non-identity.\n-That is not guaranteed by the GSV fragment. The GSV guarantees that\n-the indefinite introduces a novel peg, but there is no requirement\n-that the peg refers to a novel object. If you wanted to add this as a\n-refinement to the fragment, you could require that whenever a new peg\n-gets added, it must be mapped onto an object that is not in the range\n-of the original assignment function.\n+That is not guaranteed by the GSV fragment. If you wanted to add this\n+as a refinement to the fragment, you could require that the\n+existential only considers object in the domain that are not in the\n+range of the starting assignment function.\n\nAs usual with dynamic semantics, a point of pride is the ability to\ngive a good account of donkey anaphora, as in\n@@ -470,9 +366,9 @@ The presence of modal possibility, however, disrupts this\ngeneralization. GSV illustrate this with the following story.\n\nThe Broken Vase:\n- There are three sons, Bob, Carl, and Dave.\n+ There are three children: Alice, Bob, and Carl.\nOne of them broke a vase.\n- Bob is known to be innocent.\n+ Alice is known to be innocent.\nSomeone is hiding in the closet.\n\n(∃x.closet(x)) and (◊guilty(x)) ≡/≡ ∃x (closet(x) and ◊guilty(x))\n@@ -482,81 +378,96 @@ two worlds.\n\nin closet guilty\n--------------- ---------------\n- w: b false b false\n- c false c false\n- d true d true\n-\n- w': b false b false\n+ w: a true a false\n+ b false b true\nc true c false\n- d false d true\n\n-GSV observe that (∃x.closet(x)) and (◊guilty(x)) is true if there is\n-at least one possibility in which a person in the closet is guilty.\n-In this scenario, world w is the verifying world. It remains possible\n-that there are closet hiders who are not guilty in any world. Carl\n-fits this bill: he's in the closet in world w', but he is not guilty\n-in any world.\n+ w': a false a false\n+ b false b false\n+ c true c true\n+\n+GSV say that (∃x.closet(x)) and (◊guilty(x)) is true if there is at\n+least one possibility in which a person in the closet is guilty. In\n+this scenario, world w' is the verifying world: Carl is in the closet,\n+and he's guilty. It remains possible that there are closet hiders who\n+are not guilty in any world. Alice fits this bill: she's in the\n+closet in world w', but she is not guilty in any world.\n\nLet's see how this works out in detail.\n\n14. Someone^x is in the closet. He_x might be guilty.\n\n- {(w,0,r,g), (w',0,r,g}[∃x.closet(x)][◊guilty(x)]\n+ {(w,g), (w',g}[∃x.closet(x)][◊guilty(x)]\n\n-- existential introduces new peg\n\n- = ( {(w,1,r[x->0],g[0->b])}[closet(x)]\n- ++ {(w,1,r[x->0],g[0->c])}[closet(x)]\n- ++ {(w,1,r[x->0],g[0->d])}[closet(x)]\n- ++ {(w',1,r[x->0],g[0->b])}[closet(x)]\n- ++ {(w',1,r[x->0],g[0->c])}[closet(x)]\n- ++ {(w',1,r[x->0],g[0->d])}[closet(x)])[◊guilty(x)]\n+ = ( {(w,g[x->a])}[closet(x)]\n+ ++ {(w,g[x->b])}[closet(x)]\n+ ++ {(w,g[x->c])}[closet(x)]\n+ ++ {(w',g[x->a])}[closet(x)]\n+ ++ {(w',g[x->b])}[closet(x)]\n+ ++ {(w',g[x->c])}[closet(x)])[◊guilty(x)]\n\n-- only possibilities in which x is in the closet survive\n+ -- the first update\n\n- = {(w,1,r[x->0],g[0->d]),\n- (w',1,r[x->0],g[0->c])}[◊guilty(x)]\n+ = {(w,g[x->a]), (w',g[x->c])}[◊guilty(x)]\n\n-- Is there any possibility in which x is guilty?\n- -- yes: for x = Dave, in world w Dave broke the vase\n+ -- yes: for x = Carl, in world w' Carl broke the vase\n+ -- that's enough for the possiblity modal to allow the entire\n+ -- infostate to pass through unmodified.\n\n- = {(w,1,r[x->0],g[0->d]),\n- (w',1,r[x->0],g[0->c])}\n+ = {(w,g[x->a]),(w',g[x->c])}\n\nNow we consider the second half:\n\n- 14. Someone^x is in the closet who_x might be guilty.\n+ 15. Someone^x is in the closet who_x might be guilty.\n\n- {(w,0,r,g), (w',0,r,g)}[∃x(closet(x) & ◊guilty(x))]\n+ {(w,g), (w',g)}[∃x(closet(x) & ◊guilty(x))]\n\n- -- existential introduces new peg\n-\n- = {(w,1,r[x->0],g[0->b])}[closet(x)][◊guilty(x)]\n- ++ {(w,1,r[x->0],g[0->c])}[closet(x)][◊guilty(x)]\n- ++ {(w,1,r[x->0],g[0->d])}[closet(x)][◊guilty(x)]\n- ++ {(w',1,r[x->0],g[0->b])}[closet(x)][◊guilty(x)]\n- ++ {(w',1,r[x->0],g[0->c])}[closet(x)][◊guilty(x)]\n- ++ {(w',1,r[x->0],g[0->d])}[closet(x)][◊guilty(x)]\n+ = {(w,g[x->a])}[closet(x)][◊guilty(x)]\n+ ++ {(w,g[x->b])}[closet(x)][◊guilty(x)]\n+ ++ {(w,g[x->c])}[closet(x)][◊guilty(x)]\n+ ++ {(w',g[x->a])}[closet(x)][◊guilty(x)]\n+ ++ {(w',g[x->b])}[closet(x)][◊guilty(x)]\n+ ++ {(w',g[x->c])}[closet(x)][◊guilty(x)]\n\n-- filter out possibilities in which x is not in the closet\n-- and filter out possibilities in which x is not guilty\n- -- the only person who was guilty in the closet was Dave in\n- -- world 1\n+ -- the only person who was guilty in the closet was Carl in\n+ -- world w'\n+\n+ = {(w',g[x->c])}\n+\n+The result is different. Fewer possibilities remain.\n+We have elminated both possible worlds and possible discourses.\n+So the second formula is more informative.\n\n- = {(w,1,r[x->0],g[0->d])}\n+One of main conclusions of GSV is that in the presence of modality,\n+the hallmark of dynamic treatments--that existentials bind outside of\n+their syntactic scope--needs to refined into a more nuanced understanding.\n+Binding still occurs, but the extent of the syntactic scope of an existential\n+has a detectable effect on truth conditions.\n+\n+As we discovered in class, there is considerable work to be done to\n+decide which expressions in natural language (if any) are capable of\n+expressing which of the two translations into the GSV fragment. We\n+can certainly grasp the truth conditions, but that is not the same\n+thing as discovering that there are natural language sentences that\n+express one or the other or both.\n\n-The result is different, and more informative.\n\n## Binding, modality, and identity\n\nThe fragment correctly predicts the following contrast:\n\n- 15. Someone^x entered. He_x might be Bob. He_x might not be Bob.\n+ 16. Someone^x entered. He_x might be Bob. He_x might not be Bob.\n(∃x.enter(x)) & ◊x=b & ◊not(x=b)\n-- This discourse requires a possibility in which Bob entered\n-- and another possibility in which someone who is not Bob entered\n\n- 16. Someone^x entered who might be Bob and who might not be Bob.\n+ 17. Someone^x entered who might be Bob and who might not be Bob.\n∃x (enter(x) & ◊x=b & ◊not(x=b))\n-- This is a contradition: there is no single person who might be Bob\n-- and who simultaneously might be someone else\n@@ -565,7 +476,7 @@ These formulas are expressing extensional, de-reish intuitions. If we\nadd individual concepts to the fragment, the ability to express\nfancier claims would come along.\n\n-### Identifiers\n+## GSV's \"Identifiers\"\n\nLet α be a term which differs from x. Then α is an identifier if the\nfollowing formula is supported by every information state:\n@@ -578,37 +489,106 @@ object that it can refer to. Here is what GSV say:\nA term is an identifier per se if no mattter what the information\nstate is, it cannot fail to decie what the denotation of the term is.\n\n-## Why have a two-part assignment function?\n-\n-In the current system, variables are associated with values in two\n-steps.\n-\n- Variables Pegs Entities\n- --------- r ---- g --------\n- x --> 0 --> a\n- y --> 1 --> b\n- z --> 2 --> c\n-\n-Here, r is a refsys mapping variables to pegs, and g is an assignment\n-function mapping pegs to entities.\n-\n-Assignment functions are free to map different pegs to the same\n-entity:\n+## Digression on pegs\n\n- Variables Pegs Entities\n- --------- r ---- g --------\n- x --> 0 --> a\n- y --> 1 --> a\n- z --> 2 --> c\n-\n-But this is possible with ordinary assignment functions as well.\n-\n-It is possible to imagine a refsys that maps more than one variable to\n-the same peg. But the fragment is designed to prevent that from ever\n-happening: the only way to associate a variable with a peg is by\n-evaluating an existential quantifier, and the existential quantifier\n-always introduces a fresh, unused peg.\n+One of the more salient aspects of the technical part of the paper is\n+that GSV insert an extra level in between the variable and the object:\n+instead of having an assignment function that maps variables directly\n+onto objects, GSV provide *pegs*: variables map onto pegs, and pegs\n+map onto objects. It happens that pegs play no role in the paper\n+whatsoever. We'll demonstrate this by providing a faithful\n+implementation of the paper that does not use pegs at all.\n\n-So what does the bipartite system do that ordinary assignment\n-functions can't do?\n+Nevertheless, it makes sense to pause here to discuss pegs briefly,\n+since this technique is highly relevant to one of the main\n+applications of the course, namely, reference and coreference.\n\n+What are pegs? The term harks back to a 1986 paper by Fred Landman\n+called `Pegs and Alecs'. Pegs are simply hooks for hanging properties\n+on. Pegs are supposed to be as anonymous as possible. Think of\n+hanging your coat on a physical peg: you don't care which peg it is,\n+only that there are enough pegs for everyone's coat to hang from.\n+Likewise, for the pegs of GSV, all that matters is that there are\n+enough of them. (Incidentally, there is nothing in Gronendijk and\n+Stokhof's original DPL paper that corresponds naturally to pegs; but\n+in their Dynamic Montague Grammar paper, pegs serve a purpose similar\n+to discourse referents there, though the connection is not simple.)\n+\n+Pegs can be highly useful for exploring puzzles of reference and\n+coreference.\n+\n+ Standard assignment function System with Pegs (drefs)\n+ ---------------------------- ------------------------\n+ Variable Object Var Peg Object\n+ --------- ------- --- --- ------\n+ x --> a x --> 0 --> a\n+ y -/ y -/\n+ z --> b z --> 1 --> a\n+\n+A standard assignment function can map two different variables onto\n+the same object. In the diagram, x and y are both mapped onto the\n+object a. With discourse referents in view, we can have two different\n+flavors of coreference. Just as with ordinary assignment functions,\n+variables can be mapped onto pegs (discourse referents) that are in\n+turn mapped onto the same object. In the diagram, x is mapped onto\n+the peg 0, which in turn is mapped onto the object a, and z is mapped\n+onto a discourse referent that is mapped onto a. On a deeper level,\n+we can suppose that y is mapped onto the same discourse referent as\n+x. With a system like this, we are free to reassign the discourse\n+referent associated with z to a different object, in which case x and\n+z will no longer refer to the same object. But there is no way to\n+change the object associated with x without necessarily changing the\n+object associated with y. They are coreferent in a deeper, less\n+accidental sense.\n+\n+GSV could make use of this expressive power. But they don't. In\n+fact, their system is careful designed to guarantee that every\n+variable is assigned a discourse referent distinct from all previous\n+discourse referents.\n+\n+The addition of pegs tracks an active discussion in the dynamic\n+literature around the time of publication of the paper. Groenendijk\n+and Stokhof (Two theories of dynamic semantics, 1989) noted that it\n+was possible in DPL for information to be \"lost\".\n+\n+ 18. (∃x.P(x)) & (∃x.Q(x)) & R(x)\n+\n+If the two existentials happen to bind the same variable (here, \"x\"),\n+then the second existential occludes the first. That is, at the point\n+at which we evalute R(x), all of the assignment functions will be\n+mapping the variable \"x\" to objects that have property Q. The\n+information that there exist objects with property P has been lost.\n+If you want your dynamic system to be eliminative---or in more general\n+terms, if you want the amount of information embodied by an updated\n+information state to be monotonically increasing---then this is a\n+problem.\n+\n+A syntactic solution is to require that the variable bound\n+by an existential to be chosen fresh.\n+\n+Vermeulen, Cees FM. \"Merging without mystery or: Variables in dynamics\n+semantics.\" Journal of Philosophical Logic 24.4 (1995): 405-450 offers\n+a different approach, one based on *referent systems*. GSV's pegs are\n+a referent system. In the pegs system, when (18) is processed, the\n+information that there is an object that has property P is maintained\n+in the information state. Curiously, however, there is still no way\n+to refer to that object, at least, not with a variable, since there is\n+no variable that is associated with the peg that points to the\n+relevant object. So the information is present, but not accessible.\n+\n+That does not mean that there aren't other expression types that are\n+able to latch onto peg. An intriguing suggestion based on an example\n+in Vermeulen is that \"former\" might be able to provide access to a\n+hidden peg:\n+\n+ 19. Someone entered. Someone spoke. The former was a woman.\n+\n+Presumably we want *the former* to be able to pick out the person who\n+entered, whether or not the two existentials bind the same variable or\n+not. If we allow \"former\" to latch onto the second most recently\n+established peg, no matter whether there is a variable still pointing\n+to that peg, the desired effect is achieved.\n+\n+But none of this is relevant for any of the explanations or analyses\n+provide by the GSV fragment, and it is considerably simpler to see\n+what their fragment is about if we leave referent systems out of it."
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" cost e penditure in e tracting aluminium from its ore\n\n# cost e penditure in e tracting aluminium from its ore\n\n• ### Energy E Penditure In E Traction Of Aluminium\n\ncost and energy e penditure involved in the e traction . energy e penditure in e traction of aluminium cost and energy e penditure, the cost involved in the extraction of aluminium from its ore and what is the cost .\n\n• ### compare the cost and energy e penditure of e tracting ...\n\ncost e penditure in extracting aluminium from its cost e penditure in extracting aluminium from its ore cost e penditure in extracting aluminium from its ore The energy expenditure additional .\n\n• ### Compare The Cost And Energy E Penditure Of E Tracting ...\n\ncost e penditure in extracting aluminium from its ore . of e tracting aluminium. cost and energy involved in the e penditure in extracting aluminium from energy e Get More Info. More Details What Aluminum Extraction Really Does to the Environment\n\n• ### analyze the information to compare the cost and energy e ...\n\ncompare the cost and energy e penditure of e tracting . cost and energy expenditure invovled in the extraction This page is about cost and energy expenditure invovled in the extraction of aluminium from its ore …\n\n• ### compare the cost and energy e penditure of e tracting ...\n\ncost comparison of e traction of aluminium. cost e penditure in extracting aluminium from its ore cost e penditure in extracting aluminium from its ore. compare the cost and energy e penditure of e tracting aluminium.\n\n• ### compare the cost and energy e penditure of e tracting ...\n\nThe Cost and energy expenditure involved in the extraction of aluminum from its ore is far bigger than the energy e penditure in e traction of aluminium; Online Service cost and energy expenditure in extraction of aluminium .\n\n• ### compare the cost and energy e penditure of e tracting ...\n\nAluminium recycling - Wikipedia, the free encyclopedia- cost and energy e penditure of e tracting and recycling aluminium,Aluminium recycling is the process by which scrap aluminium can be reused in products after its initial production The process involves simply re-melting the metalenergy e penditure in e traction of aluminium | …\n\n• ### Energy E Penditure Of Extracting Aluminium\n\ncompare the cost and energy e penditure of e tracting ... Compare The Cost And Energy Expenditure Involved In . Compare The Cost And Energy Expenditure Involved In The Extraction Of Aluminium From Its Ore And The Recycling ...\n\n• ### cost e penditure in extracting aluminium from its ore\n\nJun 10, 2014 E I [email protected] Office Opening the cost of the replacement bridge was much cated executive staff we have maintained and en- a very conservative one when compared to what emissions and energy costs in accordance with penditure per person and a very high level of\n\n• ### Energy E Penditure In E Traction Of Aluminium - smmi.co.za\n\ncompare the cost and energy e penditure of e tracting . ... information to compare the cost and energy expenditure involved in the extraction of aluminium from its ore. Contact Supplier Products cost e penditure in extracting aluminium from its ore cost e penditure in extracting aluminium from its orecost of setting up a limestone proc\n\n• ### cost e penditure in extracting aluminium from its ore ...\n\nAnswer (1 of 1): The Cost and energy expenditure involved in the extraction of aluminum from its ore is far bigger than the energy required to recycle aluminium The percentage of energy required to recycle aluminum is appoximately 5% of the energy it takes to extract aluminum .\n\n• ### cost and energy e penditure of e tracting and recycling ...\n\nCompare the cost and expenditure involved in the extraction of aluminium from its ore and the recycling of . cost e penditure in extracting aluminium from its . 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Justify the increased recycling of metals in our society and across the world.\n\n• ### cost and energy e penditure of e tracting and recycling ...\n\ncost e penditure in extracting aluminium from its ore. cost e penditure in extracting aluminium from its ore compare the cost and energy e penditure of e Chat With Sales compare the cost and energy e penditure of e tracting\n\n• ### compare the cost and energy e penditure of e tracting ...\n\ncompare the cost and energy e penditure of e tracting aluminium cost and energy involved in the extraction of aluminium from, angolacrushers cost and energy involved in the,Compare The Cost And Energy Expenditure Involved, The Cost and energy expenditure involved in the extraction of aluminum from its ore is far bigger .\n\n• ### Cost E Penditure In Extracting Aluminium From Its Ore\n\ncost and energy e penditure of e tracting and recycling aluminium solutions cost and energy expenditure of aluminium ... Energy Expenditure Involved In The Extraction .\n\n• ### cost e penditure in extracting aluminium from its ore\n\ncost e penditure in extracting aluminium from its ore . of e tracting aluminium. cost and energy involved in . energy e penditure of e tracting and recycling . aluminium sulfate process - …\n\n• ### cost and energy expenditure of recycling of aluminium\n\ncost and energy e penditure of e tracting and recycling . ... cost e penditure in extracting aluminium from its ore. cost e penditure in extracting aluminium from its ore YouTube 24 Dec cost e penditure in extracting Compare The Cost And Energy Expenditure Involved In . get more info.\n\n• ### compare the cost and energy e penditure of e tracting ...\n\ncost e penditure in extracting aluminium from its ore . of e tracting aluminium. cost and energy involved in ... energy e penditure of e tracting and recycling ... Read More compare the cost and energy e penditure of …\n\n• ### energy e penditure of extracting aluminium\n\nanalyze the information to compare the cost and energy e, Analyse information to compare the cost and energy expenditure involved in the extraction of aluminium from its ore, compare the cost and energy e penditure .\n\n• ### the cost of extracting aluminium from its ore australia\n\nCost and energy expenditure of aluminium extraction and recycling ... Feb 16, 2016 ... Get Price Online; comparing cost and energy energy expenditure extraction ... process energy e penditure involved in processing aluminium cost and ... and 5/22/2009 · The Step by Step Process of Extracting Iron from its Ore...\n\n• ### Энергия E Penditure In E Traction Of Aluminium\n\nCompare the cost and expenditure involved in the extraction of aluminium from its ore and the recycling of . cost e penditure in extracting aluminium from its . Get Price And Support Online cost and energy e penditure of e tracting and recycling .\n\n• ### compare the cost and energy e penditure of e tracting ...\n\nMaterials and sustainable development - ScienceDirect- compare the cost and energy e penditure of e tracting aluminium,Current world economies are materials and energy intensive.Minimising use of resources and cutting back emissions can also decrease the costs of a given process.\n\n• ### cost e penditure in extracting aluminium from its ore\n\nCost and energy expenditure of aluminium extraction and recycling ... Feb 16, 2016 ... Get Price Online; comparing cost and energy energy expenditure extraction ... process energy e penditure involved in processing aluminium cost and ... and 5/22/2009 · The Step by Step Process of Extracting Iron from its Ore...\n\n• ### energy e penditure in e traction of aluminium\n\ncompare the cost and energy e penditure of e tracting . compare the cost and energy e penditure of e tracting aluminium. You can get the price list and a Birnith ...\n\n• ### compare the cost and energy e penditure of e tracting ...\n\nEnergy E Penditure Involved In Processing Aluminium. cost e penditure in extracting aluminium from its ore . cost and energy expenditure involved in the extraction of aluminium from its ore …"
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http://www.indiabix.com/verbal-reasoning/blood-relation-test/discussion-102
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[
"# Verbal Reasoning - Blood Relation Test - Discussion\n\n### Discussion :: Blood Relation Test - Blood Relation 2 (Q.No.2)\n\nEach of these questions is based on the following information:\n\n1. A + B means A is the mother of B.\n2. A - B means A is the sister of B.\n3. A * B means A is the father of B.\n4. A β B means A is the brother of B.\n\n2.\n\nWhich of the following means that N is the maternal uncle of M?\n\n [A]. N β P - L + E - M [B]. N - Y + A β M [C]. M - Y * P - N [D]. N β C + F * M\n\nExplanation:\n\nN β P → N is the brother of P\n\nP - L → P is the sister of L\n\nL + E → L is the mother of E\n\nE - M → E is the sister of M.\n\nHence, L is the mother of M, P is the maternal aunt of M and N is the maternal uncle of M.\n\n Anusha said: (Jan 6, 2015) You can simplify this answer. You don't need E and P. M and E are siblings so they have only one mother (L) and so if you don't mention E, it right. N, P and L are siblings so even if you don't mention P, you are not going to change the relation between N and L. ANSWER : N is the brother of L and M is L's child. N β L + M.\n\n M.Vidhya Rani said: (Jun 30, 2017) Explain me option D. In this C is the grandmother of M. And N is the brother of C. Then how we call our grand mother's Brother?"
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https://astarmathsandphysics.com/a-level-physics-notes/experimental-physics/2674-absorption-of-beta-radiation.html
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[
"## Absorption of Beta Radiation\n\nApparatus:\n\nGM-tube, counter and power supply, aluminium absorbers, special source and absorber holder, micrometer, beta source (strontium 90), local rules sheet, spill tray.\n\nProcedure:\n\n1. Read the sheet “Local rules for the use of sources of ionising radiations by students”\n\n2. Connect the GM-tube to the digital counter/power supply and switch on.\n\n3. Without the radioactive source present find the background count B (in counts per second) by noting the reading of the counter over a 100 second period.",
null,
"4. Set up the apparatus as in the diagram (with the thinnest any aluminium absorber in place) with the VERY FRAGILE & EXPENSIVE front of the GM tube at a known constant distance (e.g. 8cm) from the source. All of the above should be placed on the spill tray.\n\n5. Record the count received by the counter over a 100 seconds period.\n\nCalculate the count rate (per second).\n\nSubtract the background count rate B and so obtain a corrected count rate with the absorber C (1/s).\n\n6. Use the micrometer to find the average thickness, x (mm) of the aluminium sheet.\n\n7. Repeat stages 4 to 6 for other thicknesses of aluminium.\n\nNote: (a) You can use more than one sheet at a time.\n\n(b) Do not use any of the lead absorbers.\n\n8. Plot a graph of corrected count rate C (1/s) against aluminium thickness x (mm).\n\n9. Use your graph to determine the thickness of aluminium required to reduce the number of beta particles passing by half.\n\nThis is known as the half-thickness",
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"#### Add comment",
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https://en.m.wikisource.org/wiki/Page:Popular_Science_Monthly_Volume_72.djvu/500
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[
"Page:Popular Science Monthly Volume 72.djvu/500\n\n496\nPOPULAR SCIENCE MONTHLY\n\ntwo rectangles marked 2 and 6 above it. Therefore in calculating the data for the three curves of Figs. 2 and 3 it counts one in every case. So too does 1868, although it was a year of greater severity as is indicated by the solid square below the curve and the rectangles marked 12 and 8 above it. The year 1869, however, having only a single rectangle with a value of two, counts only in the computation of the data for the solid line of Fig. 3. 1870, on the other hand, is reckoned as one in computing the curve of Fig. 2 and the dotted line of Fig. 3. With 1870, which was a maximum year, we cease to count the years as being after the preceding minimum. 1871 is reckoned not as four years after 1867, but as seven years before the minimum of 1878, and so forth. By adding the figures for all the sunspot waves, and plotting the results, we get the simple frequency curves of Figs. 2 and 3. Figs. 4 and 5 are derived in the same way, except for one thing. Instead of reckoning each year of the occurrence of earthquakes or eruptions as having a value of only one, each is reckoned according to the value given it by Sayles or Jensen, respectively, as shown by the character or size of the spots and rectangles of Fig. 1. An inspection of the four curves of Figs. 2 to 5 shows that they agree in essential points. Each of the six curves, two for Sayles, and four for Jensen, has a pronounced maximum at or within a year of the time of sun-spot minimum. That is, when sunspots are fewest, earthquakes and volcanic eruptions are most numerous and most severe.\n\nThe four curves of Figs. 6 to 9 on the right-hand side of page—were drawn in exactly the same way as the four which lie beside them (Figs. 2–5), except that the sun-spot maxima were used as the reference points instead of the minima. They are introduced by way of contrast. It is evident that telluric activity is weak at times of sun-spot maxima. All the curves of Figs. 2 to 9 show the lack of symmetry characteristic of sun-spot variations. The lapse of time from maximum to minimum is greater than from minimum to maximum.\n\nHaving seen that there is a coincidence of some sort between sun-spot minima and seismo-volcanic maxima, the next step is to compare the mean sun-spot curve from maximum to maximum with the mean seismo-volcanic curve for the same period. The mean sun-spot curve is, of course, easy to obtain. Figs. 10 to 13 show the first stages in the construction of the mean seismo-volcanic curve. The time from one sun-spot maximum to the next is divided into eight periods as follows:\n\n1. The year of maximum spots.\n2. The year succeeding that of maximum spots.\n3. An intermediate period of decreasing number of spots,—average length about 3½ years.",
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https://kukuruku.co/comments/page6/
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[
"",
null,
"Thinking about it, I realized that even the implementation described here isn't really O(log n). Here's why: it's rather easy to prove that the nth term of the Fibonacci sequence has O(n) digits. So, even if you are performing only O(log n) multiplications, each multiplication is actually O(n), so the algorithm is actually O(n log n).\n\nI made a benchmark that shows that in practice:",
null,
"``````\ndef mul(A, B):\nreturn (A*B + A*B, A*B + A*B,\nA*B + A*B, A*B + A*B)\n\ndef fib(n):\nif n==1: return (1, 1, 1, 0)\nA = fib(n>>1)\nA = mul(A, A)\nif n&1: A = mul(A, fib(1))\nreturn A\n``````\n\n*cap obvious flies away*",
null,
"Freya, our project is not for promoting your own products. You should read the Rules page. If you want to let people know about your product, share the technical side of it. Community will reward you for that. Otherwise your publications will be downvoted and automatically be moved to Trash hub.",
null,
"Where are technical details? Cheap promotion!",
null,
"Why do you memoize the matrix method and not the iterative method? When I memoize the iterative method, I end up with about constant time for each of the steps:\nAround 1000 gives Iterative: 0.054137468338 Matrix: 1.92105817795\n\nThis all stays in memory for my machine.\n\nYou can find the improved fork of your code here:",
null,
"If you do the matrix power more cunningly, you don't need to memoize anything. Just use this:\n\n``````\ndef mat_pow(m, e):\nif e == 1:\nreturn m\nelse:\nm2 = mat_pow(m, e/2)\nif (e % 1) == 0:\nreturn m2 * m2\nelse:\nreturn m2 * m2 * m\n``````",
null,
"Also, you should memoize the matrix multiplication, that's a heavy step, and one for repeated invocations would save a ton of time.",
null,
"Bottom of def get_number renders as the Registered Trademark unicode instead of ®.\n\nYou can be a little more concise with the code by modifying how __memo is set up.\n\n``````def __init__(self):\nself.__memo = {0:0,1:1}``````\n\nThen you can do:\n``````\ndef __get_matrix_power(self, M, p):\n\"\"\"Matrix exponentiation (it is expected that p that is equal to the power of 2).\"\"\"\nreturn self.__memo.get(p,self.non_memoized(M,p))\n\ndef non_memoized(self, M,p):\nK = self.__get_matrix_power(M, int(p/2))\nR = self.__multiply_matrices(K, K)\nself.__memo[p] = R\nreturn R\n``````\n\nI know it essentially does the same thing, IMHO it's a little cleaner code thou ;)",
null,
"Computing the n-th power of the diagonalized matrix would involve non-integer arithmetic, thus rounding errors. One could argue that the error should not affect the integer part, but I'd say that depends on the size of n.\n\nThe power function is also not O(1) in n. You would get the same runtime O(log(n)) with a clever implementation.",
null,
"Just diagonalize the matrix and do it in constant time.",
null,
"Thanks for a detailed feedback, Chao. We are looking into editing the article.\n\nOff-topic: this conversation brought us to an interesting idea. What if we would let our users to edit articles and send diffs to authors, similar to git pull requests?",
null,
"Here are some suggestions. I hope this produces the minimum amount of changes so the definition is equivalent to the standard terminology.\n\n1. Instead of defining «hull», one can just state it as a «a closed curve that encloses all the points». «Hull» by itself isn't used much outside this paragraph, and one can just say «closed curve» instead of «hull» to refer to the hull because there is little ambiguity.\n2. One can then define what it means for a closed curve to be convex.\n3. Replace all occurrence of «minimal convex hull» with «convex hull». Define the «convex hull of S» as the region bounded by the shortest convex closed curve that contains all the points in S.\n4. Finally, one can state that what it means to «find the convex hull». State it as finding the boundary curve of the convex hull(which is the original «minimal convex hull»).\n\nOne can prove this definition is equivalent to the convex hull definition for finite set of points on the plane in wikipedia.",
null,
"For sorting, I suggest using the\n``my_list.sort(key=keyfunc)``\nsyntax; see my comment on Reddit for more details.\n\nAlso, is the full code available somewhere (e.g. Github)? It would be nice to see! Thanks for the informative post!",
null,
"Hey Chao, thanks for pointing that out. We'll try to edit it. Anything you would like to contribute?",
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null,
"https://kukuruku.co/mod_pagespeed_static/1.JiBnMqyl6S.gif",
null,
"https://kukuruku.co/mod_pagespeed_static/1.JiBnMqyl6S.gif",
null,
"https://kukuruku.co/mod_pagespeed_static/1.JiBnMqyl6S.gif",
null,
"https://kukuruku.co/mod_pagespeed_static/1.JiBnMqyl6S.gif",
null,
"https://kukuruku.co/mod_pagespeed_static/1.JiBnMqyl6S.gif",
null
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|
https://www.mathstools.com/section/main/geodesics
|
[
"",
null,
"Covariant derivate and Geodesics\n\nBasic concepts and principles",
null,
"Fig.1 The cylinder and the plane areisometric therefore not subject to forces become one of the otherc\nGiven a certain manifold, we wonder about trajectories followed by particles non subject to forces.\nFor example, in the plane case, particles non subjected to forces obviously follow straight trajectories.\n\nIf we consider a cylinder, we don't know at first what are these trajectories, but we can infer that are of the form given by Figure 1. because cylinder and plane are isometric manifolds and such trajectories are transformed into straight lines in the plane (or the straight lines are transformed into those paths) by an isomorfism.",
null,
"Fig. 2 If the particle is not subjected to other forces it follow a trajectory given by the maximum circle defined by the starting position and the initial direction\nIn general, given a curve with its image defined on a differentiable manifold, we assume that it is a particle moving into our differentiable variety. Our goal is to know what is the motion if particle is not subjected to external forces.\n\nSuppose that earth were a perfect sphere with smooth surface and without friction and at some point p one person throws a bowling ball flush\n\nSuch ball will travel a flat trajectory (in the sphere viewpoint) and, without friction, the inertia effect due to initial impulse join gravity will continue circling the earth.\n\nIn this half trivial example, it is clear that the ball describes trajectories given by the maximum circle defined by the start point and the initial ball throwing direction, because such trajectory passes through the antipodal point to intial point p, call it q, Fig 2.\n\nTherefore we can say that if we see a particle in motion such throught a parallel, it is subject to any external force because it are undergoing an acceleration Fig 3.",
null,
"Fig.3 If a particle moves in the sphere surface throught a trajectory that is not a maxumum circle, we can say that is subject to forces\n\nExtended Theory\n\nWe saw how to define a tensor field in section Tensors and tensorial algebra.\nGiven then a vector field on a manifold M (recall that a vector is a tensor (1.0)), we can calculate its derivative. For example, suppose the field has components",
null,
"The derivative with respect to xj is not",
null,
", but is",
null,
"Note that the term with the second derivative makes",
null,
"out of tangent space.\n\nHowever, if we substitute it for",
null,
"we have",
null,
"This is like take projection on TP(M) of the component in derivate that does not belong to TP(M).\n\nLos Γkij are called Christoffel Symbols.\n\nWhat we got so, is the projection ontangent space TP(M) of derivative of the vector field.\n\nThis construction is what is called the covariant derivative of a vector field, one more time: the covariant derivative is the projection onto tangent space of derivative of the field, Formally.\n\nCovariant derivative of a vector field on a differentiable manifold definition\n\nGiven a semiriemannian manifolds and a vector field defined on it, denoted by V = Vii. It is call covariant derivate of V in a j direction to an one (1, 1) tensor whose components are",
null,
"Thus the covariant derivative at direction xj is the vector",
null,
"Notation\n\nIt is common in differential geometry, to denote the partial derivative as a comma join a subscript of the form",
null,
"And denote the covariant derivative as a semicolon (;) and a subscript of the form",
null,
"We will specialize a bit more, consider a trajectory and we take it vector tangent at each point to the path as field vector. This allows us to define the covariant derivative of trajectory as the derivative of this vector field\n\nCovariant derivative of a trajectory definition\n\nLets",
null,
"a differentiable curve with its image in a semiriemannian variety M joining two points p and q of M. V is the vector field formed by the tangent vector of γ.\n\nIs called covariant derivative of V along γ to",
null,
"If",
null,
"then it is said that V is a parallel transport along λ from p to q.\n\nGeodesic definition\n\nLets",
null,
"a differentiable curve with its image in a semiriemannian variety M joining two points p and q of M. V is the vector field formed by the tangent vector of γ.\n\nIt says that γ is a geodesic if the full image holds that",
null,
"Or, equivalently",
null,
"",
null,
"Fig.4 The covariant derivative of a trajectory is the projection on tangent space TP(M) of trajectory tangent vector α'(t)\n\nA particle not subjected to forces on a manifold moves along trajectories called geodesic\n\nA particle moving along a geodesic is not subjected to forces? this is not correct because the particle may still suffer acceleration by modifying the lenght of tangent vector. The following theorem tells us that a particle non subject to forces moves along a geodesic and tangent vector could not vary its length\n\nTheorem: conservation of vector tangent length on a geodesic\n\nLets",
null,
"a geodesic in a variety semiriemannian M and G is it metric.\n\nLets V vector field formed by the tangent vector γ. Then G(V(λ), V(λ)) is constant\n\nWas useful? want add anything?\n\nPost here\n\nPost from other users\n\nPost here",
null,
""
] |
[
null,
"https://www.mathstools.com/images/mathstools/slogan.png",
null,
"https://www.mathstools.com/images/maths/samples/png/cilinder.png",
null,
"https://www.mathstools.com/images/maths/samples/png/sphere1.png",
null,
"https://www.mathstools.com/images/maths/samples/png/sphere2.png",
null,
"https://www.mathstools.com/mathImage",
null,
"https://www.mathstools.com/mathImage",
null,
"https://www.mathstools.com/mathImage",
null,
"https://www.mathstools.com/mathImage",
null,
"https://www.mathstools.com/mathImage",
null,
"https://www.mathstools.com/mathImage",
null,
"https://www.mathstools.com/mathImage",
null,
"https://www.mathstools.com/mathImage",
null,
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null,
"https://www.mathstools.com/mathImage",
null,
"https://www.mathstools.com/mathImage",
null,
"https://www.mathstools.com/mathImage",
null,
"https://www.mathstools.com/mathImage",
null,
"https://www.mathstools.com/mathImage",
null,
"https://www.mathstools.com/mathImage",
null,
"https://www.mathstools.com/mathImage",
null,
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null,
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null,
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null
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|
https://inlabru-org.github.io/inlabru/
|
[
"The goal of inlabru is to facilitate spatial modeling using integrated nested Laplace approximation via the R-INLA package. Additionally, extends the GAM-like model class to more general nonlinear predictor expressions, and implements a log Gaussian Cox process likelihood for modeling univariate and spatial point processes based on ecological survey data. Model components are specified with general inputs and mapping methods to the latent variables, and the predictors are specified via general R expressions, with separate expressions for each observation likelihood model in multi-likelihood models. A prediction method based on fast Monte Carlo sampling allows posterior prediction of general expressions of the latent variables. See Fabian E. Bachl, Finn Lindgren, David L. Borchers, and Janine B. Illian (2019), inlabru: an R package for Bayesian spatial modelling from ecological survey data, Methods in Ecology and Evolution, British Ecological Society, 10, 760–766, doi:10.1111/2041-210X.13168, and citation(\"inlabru\").\n\nThe inlabru.org website has links to old tutorials with code examples for versions up to 2.1.13. For later versions, updated versions of these tutorials, as well as new examples, can be found at https://inlabru-org.github.io/inlabru/articles/\n\n## Installation\n\nYou can install the current CRAN version of inlabru:\n\ninstall.packages(\"inlabru\")\n\nYou can install the latest bugfix release of inlabru from GitHub with:\n\n# install.packages(\"remotes\")\nremotes::install_github(\"inlabru-org/inlabru\", ref = \"stable\")\n\nYou can install the development version of inlabru from GitHub with\n\n# install.packages(\"remotes\")\nremotes::install_github(\"inlabru-org/inlabru\", ref = \"devel\")\n\nor track the development version builds via inlabru-org.r-universe.dev:\n\n# Enable universe(s) by inlabru-org\noptions(repos = c(\ninlabruorg = \"https://inlabru-org.r-universe.dev\",\nCRAN = \"https://cloud.r-project.org\"\n))\n\n# Install some packages\ninstall.packages(\"inlabru\")\n\n## Example\n\nThis is a basic example which shows you how fit a simple spatial Log Gaussian Cox Process (LGCP) and predicts its intensity:\n\n# Load libraries\nlibrary(INLA)\n#> This is INLA_23.08.26 built 2023-08-26 14:18:34 UTC.\n#> - See www.r-inla.org/contact-us for how to get help.\nlibrary(inlabru)\nlibrary(fmesher)\nlibrary(ggplot2)\n\n# Construct latent model components\nmatern <- inla.spde2.pcmatern(\ngorillas_sf$mesh, prior.sigma = c(0.1, 0.01), prior.range = c(0.01, 0.01) ) cmp <- ~ mySmooth(geometry, model = matern) + Intercept(1) # Fit LGCP model # This particular bru/like combination has a shortcut function lgcp() as well fit <- bru( cmp, like( formula = geometry ~ ., family = \"cp\", data = gorillas_sf$nests,\nsamplers = gorillas_sf$boundary, domain = list(geometry = gorillas_sf$mesh)\n),\noptions = list(control.inla = list(int.strategy = \"eb\"))\n)\n\n# Predict Gorilla nest intensity\nlambda <- predict(\nfit,\nfm_pixels(gorillas_sf$mesh, mask = gorillas_sf$boundary),\n~ exp(mySmooth + Intercept)\n)\n\n# Plot the result\nggplot() +\ngeom_fm(data = gorillas_sf$mesh) + gg(lambda, geom = \"tile\") + gg(gorillas$nests, color = \"red\", size = 0.5, alpha = 0.5) +\nggtitle(\"Nest intensity per km squared\")",
null,
""
] |
[
null,
"https://inlabru-org.github.io/inlabru/reference/figures/README-example-1.png",
null
] |
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|
http://gordonbell.azurewebsites.net/Designing_Computers_and_Digital_Systems/00000116.htm
|
[
"previous | contents | next\n\n1. What value is output in case of overflow? Is it the same value for all inputs that are large enough to create an overflow? What is the largest N for which the system can compute F(N)? Given this value of N, is it clear why only one check was made for overflow in Solution 2?\n\n2. Cost out the three proposals. Discuss the tradeoff between cost and speed. Why does it come out this way?\n\n3. Is there a direct formula for calculating F(N)? Find one and determine whether it provides an alternative basis for computing F(N).\n\n4. Suppose you had only 8 words of memory available (say as part of a larger core memory devoted to a total system). What sort of a fast Fibonacci generator could you build? Where would it come on the cost-speed graph of Problem 2?\n\nKEYWORDS: Counting, clock, delay, integrating delay, wait-until, time-base, synchronization\n\nThe period T' of the basic RTM K(clock) described in Chapter 2 is a constant, or more precisely, a variable which is set by a manual potentiometer adjustment. It seems desirable to have another kind of clock that has a period T that can be specified dynamically by a parameter from within an RTM system.\n\nPROBLEM STATEMENT\n\nDesign a clock, using RTM components, that has a variable period T that can be set by an RTM system.\n\nDESIGN CONSIDERATIONS\n\nThe proposed clock might have an overall RTM structure such as that shown in Figure CD-1. It would use a basic RTM ((clock) of constant period T' as an input. It would also have an input word, n, to specify the period T as a multiple of the base period T', i.e., T = n*T', where n = 0,1,2,...,n-max. The clock would give an output control signal for each n counts of the basic clock, which has period T'. We can assume that n is held outside the system and is Accessible via a T(input interface).",
null,
"Fig. CD-1. Module diagram of K(programmable (variable) clock).\n\n102\n\nprevious | contents | next"
] |
[
null,
"http://gordonbell.azurewebsites.net/Designing_Computers_and_Digital_Systems/GR000081.JPG",
null
] |
{"ft_lang_label":"__label__en","ft_lang_prob":0.9379278,"math_prob":0.9353152,"size":1874,"snap":"2022-40-2023-06","text_gpt3_token_len":447,"char_repetition_ratio":0.1,"word_repetition_ratio":0.006006006,"special_character_ratio":0.23425826,"punctuation_ratio":0.13300492,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9893918,"pos_list":[0,1,2],"im_url_duplicate_count":[null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-10-07T19:39:12Z\",\"WARC-Record-ID\":\"<urn:uuid:c2b6a5b6-6c69-4681-8a6a-dcb79e92a6c7>\",\"Content-Length\":\"3927\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:83702bca-97ea-4ee3-afe2-bfa9b36d2f90>\",\"WARC-Concurrent-To\":\"<urn:uuid:b7f66c35-ddc1-4c3d-82f3-c5b7942e2fc0>\",\"WARC-IP-Address\":\"13.65.37.122\",\"WARC-Target-URI\":\"http://gordonbell.azurewebsites.net/Designing_Computers_and_Digital_Systems/00000116.htm\",\"WARC-Payload-Digest\":\"sha1:F4S7WL5LKQUZZ5FQKYEMKNNSRJUFTAYG\",\"WARC-Block-Digest\":\"sha1:QMTXO336I775CWBGQ35SB7KKCRM3UJS3\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-40/CC-MAIN-2022-40_segments_1664030338244.64_warc_CC-MAIN-20221007175237-20221007205237-00495.warc.gz\"}"}
|
https://stackoverflow.com/questions/1031851/how-do-i-exchange-keys-with-values-in-a-dictionary
|
[
"# How do I exchange keys with values in a dictionary?\n\nI receive a dictionary as input, and would like to to return a dictionary whose keys will be the input's values and whose value will be the corresponding input keys. Values are unique.\n\nFor example, say my input is:\n\n``````a = dict()\na['one']=1\na['two']=2\n``````\n\nI would like my output to be:\n\n``````{1: 'one', 2: 'two'}\n``````\n\nTo clarify I would like my result to be the equivalent of the following:\n\n``````res = dict()\nres = 'one'\nres = 'two'\n``````\n\nAny neat Pythonic way to achieve this?\n\n• See stackoverflow.com/questions/1087694/… for an identical question that has a nice answer if you're using Python 3 – Stephen Edmonds Sep 24 '09 at 12:48\n• @Stephen: see the second most voted answer, it's the same as the accepted one in the question you linked to. The crowd preferred the other answer though... – Roee Adler Sep 25 '09 at 5:36\n• Python is not perl, python is not ruby. Readability counts. Sparse is better than dense. Given this, all the methods of these answers are just bad™; the one in the question is the best way to go. – o0'. Oct 20 '11 at 12:55\n• possible duplicate of Python reverse / inverse a mapping – Cory Apr 4 '14 at 22:44\n\nPython 2:\n\n``````res = dict((v,k) for k,v in a.iteritems())\n``````\n\nPython 3 (thanks to @erik):\n\n``````res = dict((v,k) for k,v in a.items())\n``````\n• While this seems to be correct, its really good to add an explanation of how it works rather than just the code. – Will Sep 10 '15 at 19:49\n• For Python3 it would be `res = {v:k for k,v in a.items()}`. – erik Apr 23 '16 at 22:37\n• What if values are not unique? Then the keys should be a list... for example: d = {'a':3, 'b': 2, 'c': 2} {v:k for k,v in d.iteritems()} {2: 'b', 3: 'a'} should be {2: ['b','c'], 3: 'a'} – Hanan Shteingart Aug 3 '17 at 22:44\n• @HananShteingart: the OP's question stated values are unique. Please create a separate question post for your case (and preferably link it here for other people). – liori Aug 4 '17 at 9:17\n• the python2 code works... but the list comprehension is missing the `[` and `]`. does a list comprehension not require the `[` and `]`? – Trevor Boyd Smith Nov 29 '18 at 17:37\n``````new_dict = dict (zip(my_dict.values(),my_dict.keys()))\n``````\n• Are really values() and keys() guaranteed to have the same ordering? – Lennart Regebro Jul 6 '09 at 16:28\n• yes, from python.org/dev/peps/pep-3106 The specification implies that the order in which items are returned by .keys(), .values() and .items() is the same (just as it was in Python 2.x), because the order is all derived from the dict iterator (which is presumably arbitrary but stable as long as a dict isn't modified). but this answer needs call my_dict twice(one for values, one for keys). maybe this is not ideal. – sunqiang Jul 6 '09 at 16:39\n• Yes, this answer iterates through the dict twice. sunqiang's answer is preferable for a large dictionary as it only requires one iteration. – Carl Meyer Jul 7 '09 at 17:59\n• @Carl Meyer: agree, also, he's using itertools which are a lot better for big datasets. although i wonder if the final dict() call is also streaming, or if it first assembles the whole pairs list – Javier Jul 7 '09 at 19:26\n• @CarlMeyer additionally for n > 1e6 (or 1e9) the memory usage will also be really large... and also slow this down a bunch. – Trevor Boyd Smith Nov 29 '18 at 17:39\n\nFrom Python 2.7 on, including 3.0+, there's an arguably shorter, more readable version:\n\n``````>>> my_dict = {'x':1, 'y':2, 'z':3}\n>>> {v: k for k, v in my_dict.items()}\n{1: 'x', 2: 'y', 3: 'z'}\n``````\n``````In : my_dict = {'x':1, 'y':2, 'z':3}\n\nIn : dict((value, key) for key, value in my_dict.iteritems())\nOut: {1: 'x', 2: 'y', 3: 'z'}\n``````\n• Not in the original question, I'm just curious what will happen if you had duplicate values in the original dictionary and then swap key/values with this method? – Andre Miller Jul 6 '09 at 15:49\n• @Andre Miller: It takes the last occurrence of the particular key: dict(((1,3),(1,2))) == {1:2} – balpha Jul 6 '09 at 15:51\n• duplicates will get overwritten with the last encountered dupe. – Christopher Jul 6 '09 at 15:52\n• @Andre Miller: And because d.items() returns items in an arbitrary order you get an arbitrary key for duplicate values. – Ants Aasma Jul 6 '09 at 15:53\n• I think that it will takes the last key, value pair that it found. It's like a['x'] = 3. Then you set a['x'] = 4. – riza Jul 6 '09 at 15:53\n\nYou can make use of dict comprehensions:\n\n``````res = {v: k for k, v in a.iteritems()}\n``````\n\nEdited: For Python 3, use `a.items()` instead of `a.iteritems()`. Discussions about the differences between them can be found in iteritems in Python on SO.\n\nYou could try:\n\n``````d={'one':1,'two':2}\nd2=dict((value,key) for key,value in d.iteritems())\nd2\n{'two': 2, 'one': 1}\n``````\n\nBeware that you cannot 'reverse' a dictionary if\n\n1. More than one key shares the same value. For example `{'one':1,'two':1}`. The new dictionary can only have one item with key `1`.\n2. One or more of the values is unhashable. For example `{'one':}`. `` is a valid value but not a valid key.\n\nSee this thread on the python mailing list for a discussion on the subject.\n\n• Also +1 about the note about making sure the values in the original dict are unique ; otherwise you'll get overwrites in the 'reversed' dict...and this (I just found this to my cost) can cause tricky bugs in your code! – monojohnny Apr 6 '11 at 18:45\n\n`res = dict(zip(a.values(), a.keys()))`\n\n• dict does not guarantee that its values() and keys() will return elements in the same order. Also, keys(), values() and zip() return a list, where an iterator would be sufficient. – liori Jun 23 '09 at 11:01\n• @liori: You're wrong. dict guarantees that its values() and keys() WILL be on the same order, if you don't modify the dict between calls to values() and keys() of course. The documentation states that here: (read the \"Note\" part: docs.python.org/library/stdtypes.html#dict.items) \"If items(), keys(), values(), iteritems(), iterkeys(), and itervalues() are called with no intervening modifications to the dictionary, the lists will directly correspond.\" – nosklo Jun 23 '09 at 11:12\n• Ok, then I am wrong... I haven't checked the online docs. Thank you for pointing this. – liori Jun 23 '09 at 11:16\n• You could use the iterator itertools.izip instead of zip to make this answer more efficient. – Alasdair Jun 23 '09 at 11:17\n• And iterkeys and itervalues. But could just as well use iteritems() – nosklo Jun 23 '09 at 11:31\n``````new_dict = dict( (my_dict[k], k) for k in my_dict)\n``````\n\nor even better, but only works in Python 3:\n\n``````new_dict = { my_dict[k]: k for k in my_dict}\n``````\n• Actually Dict Comprehensions (PEP 274) work with Python 2.7 as well. – Arseny Nov 11 '13 at 10:53\n\nThe current leading answer assumes values are unique which is not always the case. What if values are not unique? You will loose information! For example:\n\n``````d = {'a':3, 'b': 2, 'c': 2}\n{v:k for k,v in d.iteritems()}\n``````\n\nreturns `{2: 'b', 3: 'a'}`.\n\nThe information about `'c'` was completely ignored. Ideally it should had be something like `{2: ['b','c'], 3: ['a']}`. This is what the bottom implementation does.\n\n``````def reverse_non_unique_mapping(d):\ndinv = {}\nfor k, v in d.iteritems():\nif v in dinv:\ndinv[v].append(k)\nelse:\ndinv[v] = [k]\nreturn dinv\n``````\n• this should be the correct answer since it covers a more general case – Leon Rai Mar 3 at 19:00\n• Thank you for this! I was losing information with the other solutions. – Cam Sep 4 at 1:54\n\nAnother way to expand on Ilya Prokin's response is to actually use the `reversed` function.\n\n``````dict(map(reversed, my_dict.items()))\n``````\n\nIn essence, your dictionary is iterated through (using `.items()`) where each item is a key/value pair, and those items are swapped with the `reversed` function. When this is passed to the `dict` constructor, it turns them into value/key pairs which is what you want.\n\nSuggestion for an improvement for Javier answer :\n\n``````dict(zip(d.values(),d))\n``````\n\nInstead of `d.keys()` you can write just `d`, because if you go through dictionary with an iterator, it will return the keys of the relevant dictionary.\n\nEx. for this behavior :\n\n``````d = {'a':1,'b':2}\nfor k in d:\nk\n'a'\n'b'\n``````\n\nCan be done easily with dictionary comprehension:\n\n``````{d[i]:i for i in d}\n``````\n``````dict(map(lambda x: x[::-1], YourDict.items()))\n``````\n\n`.items()` returns a list of tuples of `(key, value)`. `map()` goes through elements of the list and applies `lambda x:[::-1]` to each its element (tuple) to reverse it, so each tuple becomes `(value, key)` in the new list spitted out of map. Finally, `dict()` makes a dict from the new list.\n\n• .items() returns a list of tuples (key, value). map() goes through elements of the list and applies `lambda x:[::-1]` to each its element (tuple) to reverse it, so each tuple becomes (value, key) in the new list spitted out of map. Finally, dict() makes a dict from the new list. – Ilya Prokin Sep 10 '15 at 19:27\n\nUsing loop:-\n\n``````newdict = {} #Will contain reversed key:value pairs.\n\nfor key, value in zip(my_dict.keys(), my_dict.values()):\n# Operations on key/value can also be performed.\nnewdict[value] = key\n``````\n\nIf you're using Python3, it's slightly different:\n\n``````res = dict((v,k) for k,v in a.items())\n``````\n\n``````>>> d = {1: 'one', 2: 'two', 3: 'three', 4: 'four'}\nIn Python3, it is critical that you use `list(d.keys())` because `dict.keys` returns a view of the keys. If you are using Python2, `d.keys()` is enough."
] |
[
null
] |
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|
https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/node143.html
|
[
"# Asymptotic Matching\n\nThe asymptotic matching relations (11.90) and (11.91) can be re-expressed in the forms (see Section 8.10),",
null,
"",
null,
"(11.140)",
null,
"",
null,
"(11.141)\n\nEquations (11.123), (11.136), and (11.141) yield",
null,
"(11.142)\n\nEmploying the easily proved identity",
null,
", we obtain",
null,
"",
null,
"",
null,
"(11.143)\n\nwhere use has been made of Equations (11.138), (11.139), as well as the fact that",
null,
"as",
null,
". However, according to Equation (11.137),",
null,
"",
null,
"(11.144)\n\nwhere use has been made of Equations (11.138) and (11.139). The previous equation yields",
null,
"(11.145)\n\nwhere use has been made of Equations (8.23), (8.25), (8.27), and (8.46), and (8.103). Here,",
null,
"is the hydromagnetic time [see Equation (5.43)], and",
null,
"is the toroidal momentum confinement time [see Equation (5.50)]. Equation (11.145) can also be obtained by integrating Equation (3.165) across the rational surface, making use of Equation (3.140), as well as the identification",
null,
"(11.146)\n\nIf we compare Equation (11.145) with Equation (8.102) then we can see that the discontinuity in the MHD fluid velocity gradient that develops at the rational surface is a factor",
null,
"smaller in our neoclassical drift-MHD model than the corresponding discontinuity in our non-neoclassical drift-MHD model. The reason for this reduction is that the strong poloidal flow-damping present in the former model prevents the poloidal plasma rotation profile from being modified by the localized electromagnetic torque produced at the rational surface. Instead, only the toroidal plasma rotation profile is modified by the electromagnetic torque [4,18].\n\nIt follows from Equations (11.113), (11.127), (11.130), (11.136), and (11.140) that",
null,
"",
null,
"",
null,
"",
null,
"",
null,
"",
null,
"(11.147)\n\nHere,",
null,
"(11.148)\n\nand use has been made of the fact that",
null,
"is continuous across the separatrix. The previous equation can be combined with Equations (8.10), (8.23)–(8.27), (8.103), and (11.77), (11.79), and (11.144) to give [2,3,7,14,15,20,21,23]",
null,
"",
null,
"",
null,
"",
null,
"",
null,
"",
null,
"",
null,
"(11.149)\n\nwhere",
null,
"",
null,
"(11.150)",
null,
"",
null,
"(11.151)",
null,
"",
null,
"(11.152)",
null,
"",
null,
"(11.153)",
null,
"",
null,
"(11.154)"
] |
[
null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img2869.png",
null,
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null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img2871.png",
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null,
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null,
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null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img3600.png",
null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img2863.png",
null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img2871.png",
null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img3601.png",
null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img3602.png",
null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img81.png",
null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img82.png",
null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img3603.png",
null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img3604.png",
null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img2869.png",
null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img3605.png",
null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img3606.png",
null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img3607.png",
null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img3608.png",
null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img3609.png",
null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img3610.png",
null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img3611.png",
null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img2869.png",
null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img3612.png",
null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img3613.png",
null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img3614.png",
null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img3615.png",
null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img3616.png",
null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img3617.png",
null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img2900.png",
null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img2901.png",
null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img2902.png",
null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img3618.png",
null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img2904.png",
null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img3619.png",
null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img3620.png",
null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img3621.png",
null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img3622.png",
null,
"https://farside.ph.utexas.edu/teaching/plasma1/Fusionhtml/img3623.png",
null
] |
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|
https://code.activestate.com/recipes/231476-discrete-cosinesine-transformations/
|
[
"Welcome, guest | Sign In | My Account | Store | Cart\n\nPython implementation of DCT/DST.\n\nPython, 58 lines\n ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58``` ```################################################################################## ## ## Author: Premshree Pillai ## Date: 27/10/03 ## File Name: dct-dst.py ## Description: -Discrete Cosine/Sine Transformations ## Website: http://www.qiksearch.com ## Category: Math ## ################################################################################## from math import * n = 4.0 a = [[0.0,0.0,0.0,0.0],[0.0,0.0,0.0,0.0],[0.0,0.0,0.0,0.0],[0.0,0.0,0.0,0.0]] alpha = [0.0,0.0,0.0,0.0] tempCount = 0 while(tempCount < 4): if(tempCount == 0): alpha[tempCount] = (1.0 / n)**0.5 else: alpha[tempCount] = (2.0 / n)**0.5 tempCount = tempCount + 1 i = 0 option = raw_input(\"Enter choice (DCT=0, DST=1): \") while(i < 4): j = 0 while(j < 4): a[i][j] = alpha[j] * cos(((2.0 * i + 1.0) * j * pi) / (2.0 * n)) if(option == 1): a[i][j] = (2.0 / (n + 1.0)) * sin((i + 1.0) * (j + 1.0) * pi/(n + 1.0))**0.5 j = j + 1 i = i + 1 i = 0 while(i < 4): j = 0 while(j < 4): u = 0 print \"\\nPattern[\",i,\",\",j,\"]\\n\" while(u < 4): v = 0 buf = [''] while(v < 4): buf.append(a[u][i] * a[v][j]) v = v + 1 continue print buf,\"\\t\",buf,\"\\t\",buf,\"\\t\",buf u= u + 1 if(i == 3 and j == 3): break while(raw_input()): continue j = j + 1 print \"\\n\" i = i + 1 ```",
null,
"Noah Spurrier 19 years, 11 months ago\n\nNeeds a bit more explaination. This is interesting, but what's it do? It's hard to follow the inputs to the outputs. These input data should be labeled more clearly and you should explain what is going to happen to the input. Like any good recipe it's nice to give hints on when you would want to serve it.",
null,
"Dinu Gherman 19 years, 10 months ago\n\nNeeds some comments. This might be ultra-cool, but even then it would benefit from a couple of comments explaining some things for people to read before running the code, or they might not run it at all.",
null,
"Tim Chen 14 years, 9 months ago\n\nThanks for the recipe. There seems a unindented problem in the original script. I uploaded a indented version on http://toolsbytim.googlecode.com/files/dct.py The script can run in PythonWin Editor.",
null,
"Shan 13 years, 11 months ago\n\nI tried using this function in C++ but the output is different from that of Matlab DC2 function. Is it possible that i have the equivalent C or C++ code?",
null,
"Created by Premshree Pillai on Mon, 27 Oct 2003 (PSF)"
] |
[
null,
"http://www.gravatar.com/avatar.php",
null,
"http://www.gravatar.com/avatar.php",
null,
"http://www.gravatar.com/avatar.php",
null,
"http://www.gravatar.com/avatar.php",
null,
"http://www.gravatar.com/avatar.php",
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https://socratic.org/questions/5886611cb72cff3c80f4fe5a
|
[
"# How do you solve 5/(2x-3)=3/(x+5)?\n\nJan 27, 2017\n\n$x = 34$\n\n#### Explanation:\n\nSolve $\\frac{5}{2 x - 3} = \\frac{3}{x + 5}$\n\nBegin by cross multiplying.\n\n$5 \\left(x + 5\\right) = 3 \\left(2 x - 3\\right)$\n\nExpand each side using the distributive property, $a \\left(b + c\\right) = a b + a c$.\n\n$5 x + 25 = 6 x - 9$\n\nAdd $9$ to both sides.\n\n$5 x + 25 + 9 = 6 x$\n\n$5 x + 34 = 6 x$\n\nSubtract $5 x$ from both sides.\n\n$34 = 6 x - 5 x$\n\n$34 = x$\n\nSwitch sides.\n\n$x = 34$"
] |
[
null
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https://www.colorhexa.com/01fc76
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[
"# #01fc76 Color Information\n\nIn a RGB color space, hex #01fc76 is composed of 0.4% red, 98.8% green and 46.3% blue. Whereas in a CMYK color space, it is composed of 99.6% cyan, 0% magenta, 53.2% yellow and 1.2% black. It has a hue angle of 148 degrees, a saturation of 99.2% and a lightness of 49.6%. #01fc76 color hex could be obtained by blending #02ffec with #00f900. Closest websafe color is: #00ff66.\n\n• R 0\n• G 99\n• B 46\nRGB color chart\n• C 100\n• M 0\n• Y 53\n• K 1\nCMYK color chart\n\n#01fc76 color description : Vivid cyan - lime green.\n\n# #01fc76 Color Conversion\n\nThe hexadecimal color #01fc76 has RGB values of R:1, G:252, B:118 and CMYK values of C:1, M:0, Y:0.53, K:0.01. Its decimal value is 130166.\n\nHex triplet RGB Decimal 01fc76 `#01fc76` 1, 252, 118 `rgb(1,252,118)` 0.4, 98.8, 46.3 `rgb(0.4%,98.8%,46.3%)` 100, 0, 53, 1 148°, 99.2, 49.6 `hsl(148,99.2%,49.6%)` 148°, 99.6, 98.8 00ff66 `#00ff66`\nCIE-LAB 87.451, -77.278, 49.949 38.09, 70.931, 28.822 0.276, 0.515, 70.931 87.451, 92.015, 147.123 87.451, -79.178, 78.198 84.22, -66.656, 38.664 00000001, 11111100, 01110110\n\n# Color Schemes with #01fc76\n\n• #01fc76\n``#01fc76` `rgb(1,252,118)``\n• #fc0187\n``#fc0187` `rgb(252,1,135)``\nComplementary Color\n• #0afc01\n``#0afc01` `rgb(10,252,1)``\n• #01fc76\n``#01fc76` `rgb(1,252,118)``\n• #01fcf4\n``#01fcf4` `rgb(1,252,244)``\nAnalogous Color\n• #fc010a\n``#fc010a` `rgb(252,1,10)``\n• #01fc76\n``#01fc76` `rgb(1,252,118)``\n• #f401fc\n``#f401fc` `rgb(244,1,252)``\nSplit Complementary Color\n• #fc7601\n``#fc7601` `rgb(252,118,1)``\n• #01fc76\n``#01fc76` `rgb(1,252,118)``\n• #7601fc\n``#7601fc` `rgb(118,1,252)``\n• #87fc01\n``#87fc01` `rgb(135,252,1)``\n• #01fc76\n``#01fc76` `rgb(1,252,118)``\n• #7601fc\n``#7601fc` `rgb(118,1,252)``\n• #fc0187\n``#fc0187` `rgb(252,1,135)``\n• #01b052\n``#01b052` `rgb(1,176,82)``\n• #01c95e\n``#01c95e` `rgb(1,201,94)``\n• #01e36a\n``#01e36a` `rgb(1,227,106)``\n• #01fc76\n``#01fc76` `rgb(1,252,118)``\n• #18fe83\n``#18fe83` `rgb(24,254,131)``\n• #32fe91\n``#32fe91` `rgb(50,254,145)``\n• #4bfe9f\n``#4bfe9f` `rgb(75,254,159)``\nMonochromatic Color\n\n# Alternatives to #01fc76\n\nBelow, you can see some colors close to #01fc76. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #01fc37\n``#01fc37` `rgb(1,252,55)``\n• #01fc4c\n``#01fc4c` `rgb(1,252,76)``\n• #01fc61\n``#01fc61` `rgb(1,252,97)``\n• #01fc76\n``#01fc76` `rgb(1,252,118)``\n• #01fc8b\n``#01fc8b` `rgb(1,252,139)``\n• #01fca0\n``#01fca0` `rgb(1,252,160)``\n• #01fcb5\n``#01fcb5` `rgb(1,252,181)``\nSimilar Colors\n\n# #01fc76 Preview\n\nThis text has a font color of #01fc76.\n\n``<span style=\"color:#01fc76;\">Text here</span>``\n#01fc76 background color\n\nThis paragraph has a background color of #01fc76.\n\n``<p style=\"background-color:#01fc76;\">Content here</p>``\n#01fc76 border color\n\nThis element has a border color of #01fc76.\n\n``<div style=\"border:1px solid #01fc76;\">Content here</div>``\nCSS codes\n``.text {color:#01fc76;}``\n``.background {background-color:#01fc76;}``\n``.border {border:1px solid #01fc76;}``\n\n# Shades and Tints of #01fc76\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, #001208 is the darkest color, while #fdfffe is the lightest one.\n\n• #001208\n``#001208` `rgb(0,18,8)``\n• #002511\n``#002511` `rgb(0,37,17)``\n• #00391b\n``#00391b` `rgb(0,57,27)``\n• #004c24\n``#004c24` `rgb(0,76,36)``\n• #00602d\n``#00602d` `rgb(0,96,45)``\n• #007336\n``#007336` `rgb(0,115,54)``\n• #01873f\n``#01873f` `rgb(1,135,63)``\n• #019a48\n``#019a48` `rgb(1,154,72)``\n• #01ae51\n``#01ae51` `rgb(1,174,81)``\n• #01c15b\n``#01c15b` `rgb(1,193,91)``\n• #01d564\n``#01d564` `rgb(1,213,100)``\n• #01e86d\n``#01e86d` `rgb(1,232,109)``\n• #01fc76\n``#01fc76` `rgb(1,252,118)``\n• #13fe80\n``#13fe80` `rgb(19,254,128)``\n• #26fe8b\n``#26fe8b` `rgb(38,254,139)``\n• #3afe95\n``#3afe95` `rgb(58,254,149)``\n• #4dfea0\n``#4dfea0` `rgb(77,254,160)``\n• #61feaa\n``#61feaa` `rgb(97,254,170)``\n• #74feb5\n``#74feb5` `rgb(116,254,181)``\n• #88ffbf\n``#88ffbf` `rgb(136,255,191)``\n• #9bffca\n``#9bffca` `rgb(155,255,202)``\n• #afffd4\n``#afffd4` `rgb(175,255,212)``\n• #c2ffdf\n``#c2ffdf` `rgb(194,255,223)``\n• #d6ffe9\n``#d6ffe9` `rgb(214,255,233)``\n• #e9fff3\n``#e9fff3` `rgb(233,255,243)``\n• #fdfffe\n``#fdfffe` `rgb(253,255,254)``\nTint Color Variation\n\n# Tones of #01fc76\n\nA tone is produced by adding gray to any pure hue. In this case, #76877e is the less saturated color, while #01fc76 is the most saturated one.\n\n• #76877e\n``#76877e` `rgb(118,135,126)``\n• #6c917d\n``#6c917d` `rgb(108,145,125)``\n• #629b7d\n``#629b7d` `rgb(98,155,125)``\n• #59a47c\n``#59a47c` `rgb(89,164,124)``\n• #4fae7b\n``#4fae7b` `rgb(79,174,123)``\n• #45b87b\n``#45b87b` `rgb(69,184,123)``\n• #3bc27a\n``#3bc27a` `rgb(59,194,122)``\n• #32cb79\n``#32cb79` `rgb(50,203,121)``\n• #28d579\n``#28d579` `rgb(40,213,121)``\n• #1edf78\n``#1edf78` `rgb(30,223,120)``\n• #14e977\n``#14e977` `rgb(20,233,119)``\n• #0bf277\n``#0bf277` `rgb(11,242,119)``\n• #01fc76\n``#01fc76` `rgb(1,252,118)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #01fc76 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"
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[
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https://blacksoaphaven.com/how-many-liters-is-100-oz/
|
[
"304 North Cardinal St.\nDorchester Center, MA 02124\n\n# How Many Liters Is 100 Oz\n\nI've always been curious about the conversion between ounces and liters. It's a practical skill that can come in handy in many situations.\n\nSo, I decided to dive deep into this topic and explore just how many liters are in 100 ounces. In this article, I'll walk you through the conversion ratio, explain the process of converting ounces to liters, and provide practical examples to help you grasp this concept.\n\nGet ready to unlock the secrets of ounces to liters conversion!\n\n## Key Takeaways\n\n• 1 liter is equal to 33.8 fluid ounces and 1 ounce is equal to 29.5735 milliliters.\n• To convert ounces to milliliters, multiply the number of ounces by the conversion ratio of 29.5735.\n• To convert ounces to liters, multiply the number of ounces by the conversion factor of 0.0296.\n• 100 ounces is equivalent to 2.96 liters.\n\n## Understanding the Conversion Ratio\n\nTo understand the conversion ratio, all you need to do is remember that 1 liter is equal to 33.8 fluid ounces. It's a simple relationship that allows you to easily convert between these two units of measurement.\n\nWhen you have 100 fluid ounces, you can convert it to liters by dividing the number of ounces by the conversion ratio. In this case, 100 fluid ounces divided by 33.8 gives you approximately 2.96 liters.\n\nNow, let's break it down further. When we say 1 liter, we are referring to a volume measurement that is equal to 1,000 milliliters. Imagine a standard water bottle that you buy at the store. That bottle usually contains 1 liter of water. It's a convenient unit to work with because it's not too large or too small.\n\nOn the other hand, fluid ounces are commonly used in the United States for measuring liquids. You'll often find this unit of measurement on beverage containers or in recipes. It's important to note that the conversion ratio between liters and fluid ounces is not a whole number. This means that the conversion is not as straightforward as simply multiplying or dividing by 10 or 100. Instead, you need to use the specific conversion factor of 33.8.\n\nUnderstanding the conversion ratio between liters and fluid ounces is essential when you want to convert between these two units. It allows you to accurately and precisely determine how many liters are in a given amount of fluid ounces. So the next time you come across 100 fluid ounces, you'll know that it is approximately 2.96 liters.\n\n## Converting Ounces to Milliliters\n\nThere's a simple way to convert ounces to milliliters. To start, it's important to know that 1 ounce is equal to 29.5735 milliliters. This conversion ratio allows us to easily calculate the milliliter equivalent of any given amount of ounces.\n\nLet's say you have 100 ounces and you want to know how many milliliters that is. The first step is to multiply the number of ounces by the conversion ratio. So, 100 ounces multiplied by 29.5735 milliliters gives us 2957.35 milliliters. Therefore, 100 ounces is equivalent to 2957.35 milliliters.\n\nConverting ounces to milliliters is a useful skill to have, especially when dealing with liquids. Whether you're measuring ingredients for a recipe or determining the volume of a liquid medication, knowing how to convert between these two units of measurement can come in handy.\n\nIt's worth noting that the conversion ratio we used, 1 ounce equals 29.5735 milliliters, is a precise and accurate conversion. It is based on the relationship between fluid ounces, which is a unit of volume commonly used in the United States, and milliliters, which is a metric unit of volume used worldwide.\n\n## Converting Ounces to Liters\n\nIf you want to convert ounces to liters, you can simply multiply the number of ounces by 0.0296. For example, if you have 100 ounces, you can calculate the equivalent in liters by multiplying 100 by 0.0296, which gives you 2.96 liters.\n\nTo help you better understand the conversion, let me provide you with a table that shows the conversion of different amounts of ounces to liters:\n\nOunces Liters\n1 0.03\n10 0.30\n100 2.96\n\nAs you can see from the table, the conversion factor remains constant. No matter the number of ounces you have, you can always multiply it by 0.0296 to get the equivalent in liters.\n\nConverting ounces to liters is a straightforward process that can be useful in various situations. Whether you're dealing with cooking recipes, measuring liquids, or simply trying to understand the volume of a product, knowing how to convert ounces to liters can come in handy.\n\nI hope this information has been helpful to you. If you have any further questions or need additional assistance, please feel free to ask.\n\n## Calculating the Volume of 100 Ounces in Liters\n\nYou can calculate the volume of 100 ounces in liters by multiplying it by the conversion factor of 0.0296. So, if you have 100 ounces and you want to know how many liters it is, simply multiply 100 by 0.0296. The result will be 2.96 liters.\n\nTo understand why we use the conversion factor of 0.0296, it's important to know that ounces and liters are units of measurement for volume. Ounces are commonly used in the United States, while liters are used in most other parts of the world.\n\nThe conversion factor of 0.0296 is derived from the fact that 1 ounce is equal to 0.0296 liters. This means that for every ounce, there are 0.0296 liters. Therefore, to convert from ounces to liters, we simply multiply the number of ounces by the conversion factor.\n\nNow, let's go back to our example of 100 ounces. By multiplying 100 by 0.0296, we get 2.96 liters. This means that if you have 100 ounces of liquid, it is equivalent to 2.96 liters.\n\nKnowing how to convert between ounces and liters can be useful in various situations, such as when you are cooking or measuring liquids for a recipe. It allows you to easily switch between different units of measurement and ensures that you have the correct amount of liquid for your needs.\n\n## Converting Ounces to Liters: Practical Examples\n\nConverting ounces to liters is a common task in everyday life, especially when dealing with different measurement systems. To ensure accurate conversions, it's important to understand the conversion formula and steps involved.\n\nIn this discussion, I'll explain the conversion formula, walk through common conversion examples, and provide tips for achieving accurate conversions.\n\n### Conversion Formula and Steps\n\nTo convert 100 ounces to liters, simply divide the number of ounces by 33.814. This conversion formula is widely used and provides an accurate result.\n\nWhen you divide 100 by 33.814, you will find that 100 ounces is equivalent to approximately 2.957 liters.\n\nIt's important to note that this conversion is based on the assumption that we are converting fluid ounces to liters. The number of ounces may vary depending on the substance being measured. However, for most liquid measurements, this formula holds true.\n\nSo, if you have 100 fluid ounces of a liquid, you can confidently say that it is equal to around 2.957 liters.\n\nThis conversion is helpful when you need to compare or convert measurements between the metric and imperial systems.\n\n### Common Conversion Examples\n\nIn my experience, common conversion examples can be really helpful when trying to grasp a new concept.\n\nSo, let's look at some examples of converting ounces to liters.\n\nFor instance, if you have 100 ounces and want to know how many liters that is, the conversion formula comes in handy.\n\nOne ounce is approximately equal to 0.0295735 liters, so you can multiply 100 ounces by this conversion factor to find the answer.\n\nDoing the math, 100 ounces is equivalent to approximately 2.95735 liters.\n\nIt's important to remember that these conversions are approximations, as there might be slight variations depending on the specific substance being measured.\n\nBut, overall, this example showcases how to convert ounces to liters accurately and precisely.\n\n### Tips for Accurate Conversions\n\nIf you want accurate conversions, make sure to double-check your conversion formulas. It's easy to make a mistake when converting units, but taking the time to verify your calculations can save you from costly errors.\n\nHere are some tips to ensure precision in your conversions:\n\n• Use trusted conversion charts or online calculators to cross-reference your results.\n• Pay attention to significant figures and round your answers to the appropriate decimal places.\n• Understand the units you are converting and their relationships to each other. This will help you spot any inconsistencies or incorrect conversions.\n\nBy following these tips, you can confidently convert measurements and avoid any inaccuracies that may arise from using incorrect formulas.\n\n## Converting Ounces to Liters: Tips and Tricks\n\nWhen it comes to converting ounces to liters, understanding the conversion formula is crucial. In this discussion, I will explain the conversion formula step by step, ensuring clarity and accuracy in the process.\n\nAdditionally, I will highlight common conversion mistakes to watch out for and provide helpful resources that can assist in making the conversion process easier and more efficient.\n\n### Conversion Formula Explained\n\nTo understand the conversion formula, you'll need to know how many liters are in 100 oz. There are 2.957 liters in 100 ounces.\n\nHere are some tips and tricks to help you navigate the conversion process:\n\n• Use a conversion factor: To convert ounces to liters, multiply the number of ounces by 0.02957.\n\n• Use a conversion table: Keep a handy conversion table that lists the conversion factors for common units of measurement.\n\n• Use online converters: If you're unsure about the conversion or need quick results, use online converters that can instantly convert ounces to liters.\n\nRemember, accuracy is key when converting units of measurement. Take your time, double-check your calculations, and don't hesitate to seek help if needed.\n\nHappy converting!\n\n### Common Conversion Mistakes\n\nNow that we understand the conversion formula, let's talk about some common mistakes that can occur when converting measurements. It's important to be aware of these errors to ensure accurate conversions. To help you avoid these pitfalls, I've created a handy table below outlining the most common mistakes and how to correct them.\n\nCommon Mistake How to Correct\nForgetting to carry over units Always double-check that you include the correct unit in your converted measurement.\nRounding too early Perform all calculations before rounding to ensure accuracy. Only round your final answer.\nUsing the wrong conversion factor Make sure you are using the correct conversion factor for the specific units you are converting. Refer to a reliable source or conversion chart if needed.\n\nYou can find helpful conversion resources online to assist you in accurately converting measurements. These resources have been a lifesaver for me when I needed to convert ounces to liters or vice versa. Here are three of my favorite conversion tools that have made my life so much easier:\n\n• UnitConverter.io: This website offers a simple and user-friendly interface for converting various units of measurement, including volume. Just enter the value you want to convert, select the units, and it will provide you with the accurate conversion result.\n\n• ConvertUnits.com: Another great resource for converting measurements. It offers a wide range of conversion options, including ounces to liters. The website also provides additional information and explanations to help you understand the conversion process better.\n\n• Google: Yes, good old Google! Simply type in your conversion query, such as '100 ounces to liters,' and Google will instantly provide you with the conversion result at the top of the search results page. It's quick, convenient, and always accurate.\n\nWith these conversion resources at your fingertips, you can confidently convert measurements and never worry about making mistakes again.\n\n### Is the Conversion Ratio the Same for All Substances, or Does It Vary Depending on the Density of the Liquid?\n\nThe conversion ratio for liquid measurements varies depending on the density of the substance. The density determines how much space the liquid occupies, which affects the conversion from ounces to liters.\n\n### Can I USe the Same Conversion Ratio When Converting Ounces to Liters for Both US Fluid Ounces and UK Fluid Ounces?\n\nYes, you can use the same conversion ratio when converting ounces to liters for both US fluid ounces and UK fluid ounces. The conversion ratio is 1 fluid ounce equals 0.0295735 liters.\n\n### How Does the Conversion Ratio Change When Converting Ounces to Liters for Solid Substances Instead of Liquids?\n\nWhen converting ounces to liters for solid substances, the conversion ratio remains the same as for liquids. It's a straightforward process that doesn't change based on the form of the substance.\n\n### Are There Any Specific Precautions or Considerations to Keep in Mind When Converting Ounces to Liters for Volatile Liquids or Gases?\n\nThere are specific precautions and considerations when converting ounces to liters for volatile liquids or gases. It is important to account for temperature and pressure changes that can affect the volume.\n\n### Is It Possible to Convert Ounces to Liters Accurately Without Using a Conversion Chart or Calculator?\n\nYes, it is possible to convert ounces to liters accurately without using a conversion chart or calculator. The conversion factor is 1 ounce equals 0.0295735 liters. Simply multiply the number of ounces by this factor to get the equivalent in liters.\n\n## Conclusion\n\nAfter understanding the conversion ratio and following the steps to convert ounces to liters, we can conclude that 100 ounces is equivalent to approximately 2.95 liters.\n\nConverting ounces to liters can come in handy in various situations, such as cooking or understanding the volume of a liquid.\n\nSo, next time you come across a recipe or need to calculate the volume of a liquid, ask yourself, 'How many liters are in that?' and use this conversion to find out.",
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"##### Lily\n\nA journey through Asian beauty traditions awaits as I uncover the rituals and ingredients that have been cherished for generations. Drawing from my Vietnamese heritage, I reveal the secrets to achieving radiant skin and a serene mind through ancient practices."
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