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https://www.nyfamily-digital.com/sheets-array-formula-why-is-everyone-talking-about-sheets-array-formula-395334	[ "", null, "# Sheets Array Formula Why Is Everyone Talking About Sheets Array Formula?\n\nReflex", null, "How do array formulas work in Google Sheets? Get the .. \| sheets array formula\n\nStates\n\nVariables\n\nLogic\n\nIn this section, we accept that by accomplishing activity \\$a\\$ from accompaniment \\$s\\$, we deterministically access in accompaniment \\$textrm{Succ}(s,a)\\$. The ambition actuality is to actuate a arrangement of accomplishments \\$(a_1,a_2,a_3,a_4,…)\\$ that starts from an antecedent accompaniment and leads to an end state. In adjustment to break this affectionate of problem, our cold will be to acquisition the minimum amount aisle by application states-based models.\n\nThis class of states-based algorithms explores all accessible states and actions. It is absolutely anamnesis efficient, and is acceptable for huge accompaniment spaces but the runtime can become exponential in the affliction cases.\n\nSearch botheration A chase botheration is authentic with:\n\nThe cold is to acquisition a aisle that minimizes the cost.\n\nBacktracking chase Backtracking chase is a aboveboard recursive algorithm that tries all possibilities to acquisition the minimum amount path. Here, activity costs can be either absolute or negative.\n\nBreadth-first chase (BFS) Breadth-first chase is a blueprint chase algorithm that does a level-by-level traversal. We can apparatus it iteratively with the advice of a chain that food at anniversary footfall approaching nodes to be visited. For this algorithm, we can accept activity costs to be according to a connected \\$cgeqslant0\\$.\n\nDepth-first chase (DFS) Depth-first chase is a chase algorithm that traverses a blueprint by afterward anniversary aisle as abysmal as it can. We can apparatus it recursively, or iteratively with the advice of a assemblage that food at anniversary footfall approaching nodes to be visited. For this algorithm, activity costs are affected to be according to 0.\n\nIterative deepening The accepted deepening ambush is a modification of the depth-first chase algorithm so that it stops afterwards extensive a assertive depth, which guarantees optimality back all activity costs are equal. Here, we accept that activity costs are according to a connected \\$cgeqslant0\\$.\n\nTree chase algorithms arbitrary By acquainted \\$b\\$ the cardinal of accomplishments per state, \\$d\\$ the band-aid depth, and \\$D\\$ the best depth, we have:\n\nThis class of states-based algorithms aims at amalgam optimal paths, enabling exponential savings. In this section, we will focus on activating programming and compatible amount search.\n\nGraph A blueprint is comprised of a set of vertices \\$V\\$ (also alleged nodes) as able-bodied as a set of edges \\$E\\$ (also alleged links).\n\nState A accompaniment is a arbitrary of all accomplished accomplishments acceptable to accept approaching accomplishments optimally.\n\nDynamic programming Activating programming (DP) is a backtracking chase algorithm with memoization (i.e. fractional after-effects are saved) whose ambition is to acquisition a minimum amount aisle from accompaniment \\$s\\$ to an end accompaniment \\$s_textrm{end}\\$. It can potentially accept exponential accumulation compared to acceptable blueprint chase algorithms, and has the acreage to alone assignment for acyclic graphs. For any accustomed accompaniment \\$s\\$, the approaching amount is computed as follows:\n\n[boxed{textrm{FutureCost}(s)=left{begin{array}{lc}0 & textrm{if IsEnd}(s)\\underset{aintextrm{Actions}(s)}{textrm{min}}big[textrm{Cost}(s,a) textrm{FutureCost(Succ}(s,a))big] & textrm{otherwise}end{array}right.}]\n\nTypes of states The table beneath presents the analogue back it comes to states in the ambience of compatible amount search:\n\nUniform amount chase Compatible amount chase (UCS) is a chase algorithm that aims at award the beeline aisle from a accompaniment \\$s_textrm{start}\\$ to an end accompaniment \\$s_textrm{end}\\$. It explores states \\$s\\$ in accretion adjustment of \\$textrm{PastCost}(s)\\$ and relies on the actuality that all activity costs are non-negative.\n\nCorrectness assumption Back a accompaniment \\$s\\$ is popped from the borderland \\$mathcal{F}\\$ and confused to explored set \\$mathcal{E}\\$, its antecedence is according to \\$textrm{PastCost}(s)\\$ which is the minimum amount aisle from \\$s_textrm{start}\\$ to \\$s\\$.\n\nGraph chase algorithms arbitrary By acquainted \\$N\\$ the cardinal of absolute states, \\$n\\$ of which are explored afore the end accompaniment \\$s_textrm{end}\\$, we have:\n\nSuppose we are not accustomed the ethics of \\$textrm{Cost}(s,a)\\$, we appetite to appraisal these quantities from a training set of minimizing-cost-path arrangement of accomplishments \\$(a_1, a_2, …, a_k)\\$.\n\nStructured perceptron The structured perceptron is an algorithm aiming at iteratively acquirements the amount of anniversary state-action pair. At anniversary step, it:\n\nHeuristic activity A heuristic is a activity \\$h\\$ over states \\$s\\$, area anniversary \\$h(s)\\$ aims at ciphering \\$textrm{FutureCost}(s)\\$, the amount of the aisle from \\$s\\$ to \\$s_textrm{end}\\$.\n\nAlgorithm \\$A^{}\\$ is a chase algorithm that aims at award the beeline aisle from a accompaniment \\$s\\$ to an end accompaniment \\$s_textrm{end}\\$. It explores states \\$s\\$ in accretion adjustment of \\$textrm{PastCost}(s) h(s)\\$. It is agnate to a compatible amount chase with bend costs \\$textrm{Cost}'(s,a)\\$ accustomed by:\n\n[boxed{textrm{Cost}'(s,a)=textrm{Cost}(s,a) h(textrm{Succ}(s,a))-h(s)}]\n\nConsistency A heuristic \\$h\\$ is said to be connected if it satisfies the two afterward properties:\n\n[boxed{h(s) leqslant textrm{Cost}(s,a) h(textrm{Succ}(s,a))}]\n\n[boxed{h(s_{textrm{end}})=0}]\n\nCorrectness If \\$h\\$ is consistent, again \\$A^\\$ allotment the minimum amount path.\n\nAdmissibility A heuristic \\$h\\$ is said to be acceptable if we have:\n\n[boxed{h(s)leqslanttextrm{FutureCost}(s)}]\n\nTheorem Let \\$h(s)\\$ be a accustomed heuristic. We have:\n\n[boxed{h(s)textrm{ consistent}Longrightarrow h(s)textrm{ admissible}}]\n\nEfficiency \\$A^\\$ explores all states \\$s\\$ acceptable the afterward equation:\n\n[boxed{textrm{PastCost}(s)leqslanttextrm{PastCost}(s_{textrm{end}})-h(s)}]\n\nIt is a framework for bearing connected heuristics. The abstraction is to acquisition closed-form bargain costs by removing constraints and use them as heuristics.\n\nRelaxed chase botheration The alleviation of chase botheration \\$P\\$ with costs \\$textrm{Cost}\\$ is denoted \\$P_{textrm{rel}}\\$ with costs \\$textrm{Cost}_{textrm{rel}}\\$, and satisfies the identity:\n\n[boxed{textrm{Cost}_{textrm{rel}}(s,a)leqslanttextrm{Cost}(s,a)}]\n\nRelaxed heuristic Accustomed a airy chase botheration \\$P_{textrm{rel}}\\$, we ascertain the airy heuristic \\$h(s)=textrm{FutureCost}_{textrm{rel}}(s)\\$ as the minimum amount aisle from \\$s\\$ to an end accompaniment in the blueprint of costs \\$textrm{Cost}_{textrm{rel}}(s,a)\\$.\n\nConsistency of airy heuristics Let \\$P_{textrm{rel}}\\$ be a accustomed airy problem. By theorem, we have:\n\n[boxed{h(s)=textrm{FutureCost}_{textrm{rel}}(s)Longrightarrow h(s)textrm{ consistent}}]\n\nTradeoff back allotment heuristic We accept to antithesis two aspects in allotment a heuristic:\n\nMax heuristic Let \\$h_1(s)\\$, \\$h_2(s)\\$ be two heuristics. We accept the afterward property:\n\n[boxed{h_1(s),textrm{ }h_2(s)textrm{ consistent}Longrightarrow h(s)=max{h_1(s),textrm{ }h_2(s)}textrm{ consistent}}]\n\nIn this section, we accept that bold activity \\$a\\$ from accompaniment \\$s\\$ can advance to several states \\$s_1′,s_2′,…\\$ in a probabilistic manner. In adjustment to acquisition our way amid an antecedent accompaniment and an end state, our cold will be to acquisition the best amount activity by application Markov accommodation processes that advice us cope with randomness and uncertainty.\n\nDefinition The cold of a Markov accommodation activity is to aerate rewards. It is authentic with:\n\nTransition probabilities The alteration anticipation \\$T(s,a,s’)\\$ specifies the anticipation of activity to accompaniment \\$s’\\$ afterwards activity \\$a\\$ is taken in accompaniment \\$s\\$. Anniversary \\$s’ mapsto T(s,a,s’)\\$ is a anticipation distribution, which agency that:\n\n[forall s,a,quadboxed{displaystylesum_{s’intextrm{ States}}T(s,a,s’)=1}]\n\nPolicy A activity \\$pi\\$ is a activity that maps anniversary accompaniment \\$s\\$ to an activity \\$a\\$, i.e.\n\n[boxed{pi : s mapsto a}]\n\nUtility The annual of a aisle \\$(s_0, …, s_k)\\$ is the discounted sum of the rewards on that path. In added words,\n\n[boxed{u(s_0,…,s_k)=sum_{i=1}^{k}r_igamma^{i-1}}]\n\nThe amount aloft is an analogy of the case \\$k=4\\$.\n\nQ-value The \\$Q\\$-value of a activity \\$pi\\$ at accompaniment \\$s\\$ with activity \\$a\\$, additionally denoted \\$Q_{pi}(s,a)\\$, is the accepted annual from accompaniment \\$s\\$ afterwards demography activity \\$a\\$ and again afterward activity \\$pi\\$. It is authentic as follows:\n\n[boxed{Q_{pi}(s,a)=sum_{s’intextrm{ States}}T(s,a,s’)left[textrm{Reward}(s,a,s’) gamma V_pi(s’)right]}]\n\nValue of a activity The amount of a activity \\$pi\\$ from accompaniment \\$s\\$, additionally denoted \\$V_{pi}(s)\\$, is the accepted annual by afterward activity \\$pi\\$ from accompaniment \\$s\\$ over accidental paths. It is authentic as follows:\n\n[boxed{V_pi(s)=Q_pi(s,pi(s))}]\n\nPolicy appraisal Accustomed a activity \\$pi\\$, activity appraisal is an accepted algorithm that aims at ciphering \\$V_pi\\$. It is done as follows:\n\n[boxed{V_pi^{(0)}(s)longleftarrow0}]\n\n[forall s,quadboxed{V_pi^{(t)}(s)longleftarrow Q_pi^{(t-1)}(s,pi(s))}]\n\n[boxed{Q_pi^{(t-1)}(s,pi(s))=sum_{s’intextrm{ States}}T(s,pi(s),s’)Big[textrm{Reward}(s,pi(s),s’) gamma V_pi^{(t-1)}(s’)Big]}]\n\nOptimal Q-value The optimal \\$Q\\$-value \\$Q_{textrm{opt}}(s,a)\\$ of accompaniment \\$s\\$ with activity \\$a\\$ is authentic to be the best \\$Q\\$-value accomplished by any policy. It is computed as follows:\n\n[boxed{Q_{textrm{opt}}(s,a)=sum_{s’intextrm{ States}}T(s,a,s’)left[textrm{Reward}(s,a,s’) gamma V_textrm{opt}(s’)right]}]\n\nOptimal amount The optimal amount \\$V_{textrm{opt}}(s)\\$ of accompaniment \\$s\\$ is authentic as actuality the best amount accomplished by any policy. It is computed as follows:\n\n[boxed{V_{textrm{opt}}(s)=underset{aintextrm{ Actions}(s)}{textrm{max}}Q_textrm{opt}(s,a)}]\n\nOptimal activity The optimal activity \\$pi_{textrm{opt}}\\$ is authentic as actuality the activity that leads to the optimal values. It is authentic by:\n\n[forall s,quadboxed{pi_{textrm{opt}}(s)=underset{aintextrm{ Actions}(s)}{textrm{argmax}}Q_textrm{opt}(s,a)}]\n\nValue abundance Amount abundance is an algorithm that finds the optimal amount \\$V_{textrm{opt}}\\$ as able-bodied as the optimal activity \\$pi_{textrm{opt}}\\$. It is done as follows:\n\n[boxed{V_{textrm{opt}}^{(0)}(s)longleftarrow0}]\n\n[forall s,quadboxed{V_textrm{opt}^{(t)}(s)longleftarrow underset{aintextrm{ Actions}(s)}{textrm{max}}Q_textrm{opt}^{(t-1)}(s,a)}]\n\n[boxed{Q_textrm{opt}^{(t-1)}(s,a)=sum_{s’intextrm{ States}}T(s,a,s’)Big[textrm{Reward}(s,a,s’) gamma V_textrm{opt}^{(t-1)}(s’)Big]}]\n\nNow, let’s accept that the alteration probabilities and the rewards are unknown.\n\nModel-based Monte Carlo The model-based Monte Carlo adjustment aims at ciphering \\$T(s,a,s’)\\$ and \\$textrm{Reward}(s,a,s’)\\$ application Monte Carlo simulation with:\n\n[boxed{widehat{T}(s,a,s’)=frac{#textrm{ times }(s,a,s’)textrm{ occurs}}{#textrm{ times }(s,a)textrm{ occurs}}}]\n\n[boxed{widehat{textrm{Reward}}(s,a,s’)=rtextrm{ in }(s,a,r,s’)}]\n\nModel-free Monte Carlo The model-free Monte Carlo adjustment aims at anon ciphering \\$Q_{pi}\\$, as follows:\n\n[boxed{widehat{Q}_pi(s,a)=textrm{average of }u_ttextrm{ area }s_{t-1}=s, a_t=a}]\n\nEquivalent conception By introducing the connected \\$eta=frac{1}{1 (#textrm{updates to }(s,a))}\\$ and for anniversary \\$(s,a,u)\\$ of the training set, the amend aphorism of model-free Monte Carlo has a arched aggregate formulation:\n\n[boxed{widehat{Q}_pi(s,a)leftarrow(1-eta)widehat{Q}_pi(s,a) eta u}]\n\n[boxed{widehat{Q}_pi(s,a)leftarrowwidehat{Q}_pi(s,a) – eta (widehat{Q}_pi(s,a) – u)}]\n\nSARSA State-action-reward-state-action (SARSA) is a boostrapping adjustment ciphering \\$Q_pi\\$ by application both raw abstracts and estimates as allotment of the amend rule. For anniversary \\$(s,a,r,s’,a’)\\$, we have:\n\n[boxed{widehat{Q}_pi(s,a)longleftarrow(1-eta)widehat{Q}_pi(s,a) etaBig[r gammawidehat{Q}_pi(s’,a’)Big]}]\n\nQ-learning \\$Q\\$-learning is an off-policy algorithm that produces an appraisal for \\$Q_textrm{opt}\\$. On anniversary \\$(s,a,r,s’,a’)\\$, we have:\n\n[boxed{widehat{Q}_{textrm{opt}}(s,a)leftarrow(1-eta)widehat{Q}_{textrm{opt}}(s,a) etaBig[r gammaunderset{a’intextrm{ Actions}(s’)}{textrm{max}}widehat{Q}_{textrm{opt}}(s’,a’)Big]}]\n\nEpsilon-greedy The epsilon-greedy activity is an algorithm that balances analysis with anticipation \\$epsilon\\$ and corruption with anticipation \\$1-epsilon\\$. For a accustomed accompaniment \\$s\\$, the activity \\$pi_{textrm{act}}\\$ is computed as follows:\n\n[boxed{pi_textrm{act}(s)=left{begin{array}{ll}underset{aintextrm{ Actions}}{textrm{argmax }}widehat{Q}_textrm{opt}(s,a) & textrm{with proba }1-epsilon\\textrm{random from Actions}(s) & textrm{with proba }epsilonend{array}right.}]\n\nIn amateur (e.g. chess, backgammon, Go), added agents are present and charge to be taken into annual back amalgam our policy.\n\nGame timberline A bold timberline is a timberline that describes the possibilities of a game. In particular, anniversary bulge is a accommodation point for a amateur and anniversary root-to-leaf aisle is a accessible aftereffect of the game.\n\nTwo-player zero-sum bold It is a bold area anniversary accompaniment is absolutely empiric and such that players booty turns. It is authentic with:\n\nTypes of behavior There are two types of policies:\n\nExpectimax For a accustomed accompaniment \\$s\\$, the expectimax amount \\$V_{textrm{exptmax}}(s)\\$ is the best accepted annual of any abettor activity back arena with account to a anchored and accepted antagonist activity \\$pi_{textrm{opp}}\\$. It is computed as follows:\n\n[boxed{V_{textrm{exptmax}}(s)=left{begin{array}{ll}textrm{Utility}(s) & textrm{IsEnd}(s)\\underset{aintextrm{Actions}(s)}{textrm{max}}V_{textrm{exptmax}}(textrm{Succ}(s,a)) & textrm{Player}(s)=textrm{agent}\\displaystylesum_{aintextrm{Actions}(s)}pi_{textrm{opp}}(s,a)V_{textrm{exptmax}}(textrm{Succ}(s,a)) & textrm{Player}(s)=textrm{opp}end{array}right.}]\n\nMinimax The ambition of minimax behavior is to acquisition an optimal activity adjoin an antagonist by bold the affliction case, i.e. that the antagonist is accomplishing aggregate to abbreviate the agent’s utility. It is done as follows:\n\n[boxed{V_{textrm{minimax}}(s)=left{begin{array}{ll}textrm{Utility}(s) & textrm{IsEnd}(s)\\underset{aintextrm{Actions}(s)}{textrm{max}}V_{textrm{minimax}}(textrm{Succ}(s,a)) & textrm{Player}(s)=textrm{agent}\\underset{aintextrm{Actions}(s)}{textrm{min}}V_{textrm{minimax}}(textrm{Succ}(s,a)) & textrm{Player}(s)=textrm{opp}end{array}right.}]\n\nMinimax backdrop By acquainted \\$V\\$ the amount function, there are 3 backdrop about minimax to accept in mind:\n\n[boxed{forall pi_{textrm{agent}},quad V(pi_{textrm{max}},pi_{textrm{min}})geqslant V(pi_{textrm{agent}},pi_{textrm{min}})}]\n\n[boxed{forall pi_{textrm{opp}},quad V(pi_{textrm{max}},pi_{textrm{min}})leqslant V(pi_{textrm{max}},pi_{textrm{opp}})}]\n\n[boxed{forall pi,quad V(pi_{textrm{max}},pi)leqslant V(pi_{textrm{exptmax}},pi)}]\n\n[boxed{V(pi_{textrm{exptmax}},pi_{textrm{min}})leqslant V(pi_{textrm{max}},pi_{textrm{min}})leqslant V(pi_{textrm{max}},pi)leqslant V(pi_{textrm{exptmax}},pi)}]\n\nEvaluation activity An appraisal activity is a domain-specific and almost appraisal of the amount \\$V_{textrm{minimax}}(s)\\$. It is denoted \\$textrm{Eval}(s)\\$.\n\nAlpha-beta pruning Alpha-beta pruning is a domain-general exact adjustment optimizing the minimax algorithm by alienated the accidental analysis of genitalia of the bold tree. To do so, anniversary amateur keeps clue of the best amount they can achievement for (stored in \\$alpha\\$ for the maximizing amateur and in \\$beta\\$ for the aspersing player). At a accustomed step, the activity \\$beta < alpha\\$ agency that the optimal aisle is not activity to be in the accepted annex as the beforehand amateur had a bigger advantage at their disposal.\n\nTD acquirements Temporal aberration (TD) acquirements is acclimated back we don’t apperceive the transitions/rewards. The amount is based on analysis policy. To be able to use it, we charge to apperceive rules of the bold \\$textrm{Succ}(s,a)\\$. For anniversary \\$(s,a,r,s’)\\$, the amend is done as follows:\n\n[boxed{wlongleftarrow w-etabig[V(s,w)-(r gamma V(s’,w))big]nabla_wV(s,w)}]\n\nThis is the adverse of turn-based games, area there is no acclimation on the player’s moves.\n\nSingle-move accompanying bold Let there be two players \\$A\\$ and \\$B\\$, with accustomed accessible actions. We agenda \\$V(a,b)\\$ to be \\$A\\$’s annual if \\$A\\$ chooses activity \\$a\\$, \\$B\\$ chooses activity \\$b\\$. \\$V\\$ is alleged the adjustment matrix.\n\nStrategies There are two capital types of strategies:\n\n[boxed{aintextrm{Actions}}]\n\n[forall aintextrm{Actions},quadboxed{0leqslantpi(a)leqslant1}]\n\nGame appraisal The amount of the bold \\$V(pi_A,pi_B)\\$ back amateur \\$A\\$ follows \\$pi_A\\$ and amateur \\$B\\$ follows \\$pi_B\\$ is such that:\n\n[boxed{V(pi_A,pi_B)=sum_{a,b}pi_A(a)pi_B(b)V(a,b)}]\n\nMinimax assumption By acquainted \\$pi_A,pi_B\\$ alignment over alloyed strategies, for every accompanying two-player zero-sum bold with a bound cardinal of actions, we have:\n\n[boxed{max_{pi_A}min_{pi_B}V(pi_A,pi_B)=min_{pi_B}max_{pi_A}V(pi_A,pi_B)}]\n\nPayoff cast We ascertain \\$V_p(pi_A,pi_B)\\$ to be the annual for amateur \\$p\\$.\n\nNash calm A Nash calm is \\$(pi_A^,pi_B^)\\$ such that no amateur has an allurement to change its strategy. We have:\n\n[boxed{forall pi_A, V_A(pi_A^,pi_B^)geqslant V_A(pi_A,pi_B^)}quadtextrm{and}quadboxed{forall pi_B, V_B(pi_A^,pi_B^)geqslant V_B(pi_A^*,pi_B)}]\n\nSheets Array Formula Why Is Everyone Talking About Sheets Array Formula? – sheets array formula\n\| Allowed in order to my own website, in this particular moment I am going to provide you with in relation to keyword. Now, this is actually the very first picture:" ]	[ null, "https://sstatic1.histats.com/0.gif", null, "https://www.nyfamily-digital.com/wp-content/uploads/2020/10/how-do-array-formulas-work-in-google-sheets-get-the-sheets-array-formula.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7551437,"math_prob":0.95143366,"size":17644,"snap":"2021-21-2021-25","text_gpt3_token_len":5150,"char_repetition_ratio":0.16139455,"word_repetition_ratio":0.06376812,"special_character_ratio":0.24472909,"punctuation_ratio":0.104761906,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9853142,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,null,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-18T20:16:42Z\",\"WARC-Record-ID\":\"<urn:uuid:920d42f3-baf2-48e7-b6fd-8f8d09646908>\",\"Content-Length\":\"26824\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:72844c73-d140-4466-af1c-47e310c0b1ee>\",\"WARC-Concurrent-To\":\"<urn:uuid:061d7dcb-0b7d-446e-92d5-d5a794c97340>\",\"WARC-IP-Address\":\"172.67.193.69\",\"WARC-Target-URI\":\"https://www.nyfamily-digital.com/sheets-array-formula-why-is-everyone-talking-about-sheets-array-formula-395334\",\"WARC-Payload-Digest\":\"sha1:ZNVIJ5EY2XQ2YQFRTEKTNW7DAPRALM4Y\",\"WARC-Block-Digest\":\"sha1:CBX374GREHHHDPPCXSOIQYDS7ASKBJJP\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623487641593.43_warc_CC-MAIN-20210618200114-20210618230114-00133.warc.gz\"}"}
https://developers.apple.com/fonts/TrueType-Reference-Manual/RM02/Chap2.html	[ "# Font Engine\n\n## Introduction\n\nThis chapter provides an overview of the key concepts needed to understand the TrueType font engine, the software that converts the information in a TrueType font into a raster image suitable for display on screen or printer.\n\n## How the font engine works\n\nRasterizing a glyph outline is a multi-step process that proceeds as follows:\n\n• The master outline description of the glyph is scaled to the appropriate size.\n• The scaled outline is grid-fitted according to its associated instructions.\n• The grid-fitted outline is scan converted to produce a bitmap image suitable for raster display.\n\nFigure 1 illustrates this process.\n\nFigure 1 The work of the font engine\n\n master outline", null, "1-scaled outline", null, "2-grid-fitted outline", null, "3-raster image", null, "### Scaling the master outline\n\nA TrueType font stores a master outline description for each glyph. When an application requests a particular glyph at a specific size for a specific device, the font engine will create the necessary bitmap.\n\nThe first step in this process is to scale the master outline to the desired size. Scaling an outline is the work of that portion of the font engine known as the scaler. When the master outline is scaled, the points that make up the glyph outline are changed from device independent em units to device dependent 26.6 fixed point numbers representing locations in a pixel grid. In the text that follows, master outline points shown as grid coordinates will have a bar over the coordinate numbers as in (x~,y~).\n\nA scaled outline point can occupy any position expressible as a sixty-fourth of a pixel (i.e as a 26.6 fixed point number). Scaled coordinates which are still in their original outline positions (that is, have not been grid-fitted by instructions) will be shown with a tilde over the coordinates as in (x~, y~). Point coordinates that are expressed to the nearest sixty fourth of a pixel will be shown with a colon separating the whole number portion of the coordinate from the fractional portion. One and one half pixels will therefore be written as 1:32 in this notation.\n\nPixel centers are always at the intersection of two half-grid lines. The point (2:32, 7:32) occupies a pixel center while the point (3:0, 4:0) does not. The relationship of pixels to the grid is shown in FIGURE 2.\n\nFIGURE 2 Pixels and the grid", null, "#### Converting FUnits to pixels\n\nThe scaler converts values in the master coordinate system to values in the pixel coordinate system by multiplying them by a scale. This scale is:\n\n`pointSize * resolution / (72 points per inch * units_per_em).`\n\nwhere pointSize is the size at which the glyph is to be displayed, resolution is the resolution of the output device and units per em is the resolution of the grid of which the master outline was originally defined. The 72 in the denominator represents the number of points per inch.\n\nFor example, assume that a glyph feature is 550 FUnits in length and defined on a master grid with 2048 units per em. The following calculation reveals that its size on a 72 dpi screen at 18 points, is 4.83 pixels.\n\n```550 * [(18 * 72 )/ (72 * 2048 )] = 4.83\n```\n\n### Creating the origin point and the advance point\n\nThe scaler creates two additional points using the data in the 'hdmx' table in the font file. These points represent the origin and advance width of the glyph. The origin point is the original pen position. Imagine that the pen moves the left-side bearing distance, draws the glyph, moves the pen the advance width from the origin point and is now in place to begin the next glyph. This location is the advance point. The origin point and the advance point are accessible to instructions. If the points in a given glyph are numbered from 0 to n-1, the origin point would have the number n and the advance point the number n+1.\n\nFIGURE 3 A glyph outline with the origin and advance points added", null, "### Grid-fitting a scaled outline\n\nOnce the master outline for a particular glyph has been scaled to the desired size and device, the instructions associated with that glyph can be executed. Two key goals of instructing glyphs are to eliminate the effect of chance relationships to the grid and to control key dimensions. For more information see Instructing Fonts.\n\nAs the previous statements indicate, a key effect of executing glyph instructions is to move the scaled outline points describing the glyph to new locations. Once so moved, points are said to be grid-fitted. Their coordinates are indicated as (x,y) positions in the coordinate grid.\n\nSince instructions operate after the master outline has been scaled their effect is dependent upon the actual size and resolution of the glyph requested. FIGURE 4 shows a master outline and two grid-fitted outlines produced from that master by executing the associated glyph instructions. The grid-fitted outline produced by first scaling the master outline to 12 pixels per em and then executing the glyph instructions differs from that produced by first scaling the master outline to 18 pixels per em and then executing the glyph instructions.\n\nThe instructions need not be size or resolution specific since they refer to the points as they existed in the original master outline. Their actions are based on a geometric analysis of the features of a glyph and are size and resolution independent. The exceptions to this rule are the size and resolution specific DELTA instructions described later in this chapter.\n\nFIGURE 4 Scaling then grid-fitting the master outline", null, "", null, "### Scan converting a grid-fitted outline\n\nOnce the master outline has been scaled and grid-fitted, it is ready to be rasterized by the scan converter. The scan converter takes the grid-fitted outline and applies a set of rules to determine which pixels will be part of the glyph image when printed or displayed on the screen.\n\nThe first of these rules is as follows:\n\nRule 1: If a pixel's center falls within or on the glyph outline, that pixel is turned on and becomes part of the bitmap image of the glyph.\n\n#### Distinguishing the inside from the outside of a glyph\n\nA key problem facing the scan converter is determining what it means for a pixel center to fall \"within or on the glyph outline\". Sometimes this may seem to be obvious, but complex glyphs can make this seemingly straight-forward questions a difficult one.\n\nThe TrueType scan converter solves this problem by using the non-zero winding number rule to distinguish the interior from the exterior of a glyph. This rule is as follows:\n\nPoints that have a non-zero winding number are inside the glyph. All other points are outside the glyph.\n\nThe four steps presented below consitute a method for determining the winding number of a point.\n\n1. Draw a ray from the point in question toward infinity. (The direction in which the ray points in unimportant.)\n2. Starting with a count of zero.\n3. Add one to the count each time a glyph contour crosses the ray from right to left or bottom to top. (Such a crossing is termed an on-transition because the TrueType scan converter scans from left to right and bottom to top.)\n4. Subtract one from the count each time a contour of the glyph crosses the ray from left to right or top to bottom. (Such a crossing is termed an off-transition.)\n5. If the final count is non-zero, the point is an interior point. Otherwise, it is an exterior point.\n\nThe direction of a contour can be determined by looking at the point numbers that define the contour. Its direction is from a lower point number toward a higher point number.\n\nAn on-transition is shown in FIGURE 5. Here the contour crosses the ray from bottom to top.\n\nFIGURE 5 An on-transition", null, "An off-transition is shown in FIGURE 6. Here the contour crosses the ray from left to right.\n\nFIGURE 6 An off-transition", null, "FIGURE 7 demonstrates the use of winding numbers in determining whether a point is inside a glyph. Considering each of the points in questions, the following can be said:\n\n• Points p1 and p4 each undergo a single transition giving them non-zero winding numbers. They are interior points.\n• Point p2 under goes an off-transition followed by an on-transition giving it a zero winding number. It is an exterior point.\n• Points p3 and p5 illustrate the importance of properly setting curve directions. Point p3 undergoes an on-transition followed by an off-transition, has zero winding number. It is an exterior point.\n• Point p5, on the other hand, undergoes an off-transition followed by a second off-transition giving it a winding number of 2. It is an interior point.\n\nFIGURE 7 Interior and exterior points", null, "#### Using the scan converters to control dropouts\n\nTrueType instructions are designed to make it possible to gridfit a glyph so that the desired pixels will be turned on by the Rule 1 regardless of the point size or the transformation used. It is often difficult to foresee all possible transformations that a glyph might undergo. This fact makes it difficult to instruct a glyph to ensure that the proper grid-fitting distortion of the outline will take place for every desired transformation. This is often a problem for complex glyphs displayed at very small pixel per em sizes. In these situations, some renditions of a glyph may contain dropouts (holes or gaps in the rendered bitmap). In such cases, the use of an alternate scan conversion mode may prove desirable.\n\nFont creators can, if desired, invoke two additional scan conversion rules by changing the scan converter mode from the default setting to the dropout control mode. The decision about which scan converter mode to use can be made on a font wide basis or glyph by glyph.\n\nThe choice of scan conversion mode is made by setting the value of the graphics state variable scan control. The interpreter considers each of three conditions in determining whether dropout control mode will be used:\n\n• Is the glyph rotated?\n• Is the glyph stretched?\n• Is the current setting for ppem less than a specified ppem value?\n\nIt is also possible to turn dropout control off completely.\n\nChanging the value of scan control is a task accomplished using the SCANCTRL[] instruction. See SCANCTRL[] SCAN conversion ConTRoL. for more on changing the value of scan control.\n\nTo understand the dropout control scan conversion rules, it is important to know that a scan line is a horizontal or vertical line that sweeps across the face of a glyph. The intersection of a horizontal and vertical scan line determines a pixel center.\n\nFIGURE 8 Adjacent pixels", null, "A dropout occurs whenever there is a connected region within a glyph interior that contains two black pixels that cannot be connected by a line that passes only through black pixels. It is possible to test for potential dropouts by looking at an imaginary line segment connecting two adjacent pixel centers. If this line segment is intersected by both an on-transition contour and an off-transition contour, a potential dropout condition exists. The potential dropout becomes an actual dropout only if the two contour lines continue on in both directions to cut other line segments between adjacent pixel centers. This condition is illustrated in FIGURE 9.\n\nFIGURE 9 Condition leading to two dropouts", null, "Two lines or curves that join together immediately after crossing a scan line form a stub. Stubs do not cause dropouts but may result in a stem of the glyph that is shorter than desired. This situation is shown in FIGURE 10.\n\nFIGURE 10 A stub", null, "The scan converter can be operated in a mode in which both dropouts and stubs are filled in. The rules employed in dropout and stub control mode (scan converter mode 0) are shown below.\n\nRule 2a: If a horizontal scan line connecting two adjacent pixel centers is intersected by both an on-transition contour and an off-transition contour, and neither of the two pixels was already turned on by rule 1, turn on the left-most pixel.\n\nRule 2b: If a vertical scan line connecting two adjacent pixel centers is intersected by both an on-transition contour and an off-transition contour, and neither of the two pixels was already turned on by rule 1, turn on the bottom-most pixel.\n\nThe scan converter can also be operated in a mode in which only dropouts are filled in and stubs are left as is (scan converter mode 1). Rules 3a and 3b describe its operation in this mode.\n\nRule 3a: If a horizontal scan line connecting two adjacent pixel centers is intersected by both an on-transition contour and an off-transition contour, neither of the pixels was already turned on by rule 1, and the two contours continue on to intersect other scan lines (this is not a 'stub'), turn on the left-most pixel.\n\nRule 3b: If a vertical scan line connecting two adjacent pixel centers is intersected by both an on-transition contour and an off-transition contour, neither of the pixels was already turned on by rule 1, and the two contours continue on to intersect other scan lines (this is not a 'stub'),turn on the bottom-most pixel.\n\nScan line segments that form a square with the intersected scan line defining their boundaries are examined to verify that they are intersected by two contours. It is possible that these could be different contours than the ones intersecting the dropout scan line segment. This is very unlikely but may have to be controlled with grid-fitting in some exotic glyphs.\n\n## The interpreter environment\n\nThe interpreter is the portion of the TrueType interpreter that executes the instructions found in a font file.\n\n### Where instructions can be used\n\nInstructions can be associated with particular glyphs or can be associated with a font as a whole. Instructions associated with a particular glyph are termed a glyph program. Instructions can also be used in the font program and the control value program.\n\nThe font program (found in the 'fpgm' table in the font file) is a set of instructions executed once, the first time a font is accessed by an application. The font program is used to create function definitions (see \"FDEF[] Function DEFinition\") and instruction definitions (see \"IDEF[] Instruction DEFinition\"). Functions and instructions defined in the font program can be accessed in the individual glyph programs. See also, \"A sample font program\" on page 3-145.\nThe control value program is a sequence of instructions executed every time the point size or transformation changes. These instructions are stored in the 'prep' table of the font file. The control value program is used to make font wide changes rather than to manage individual glyphs. See also, \"A sample control value program\".\n\nInstructions that belong to glyph programs are stored in the 'glyf' ' table of the font file. Instructions associated with a glyph are executed every time that glyph is requested.\n\n### Instruction names\n\nInstructions are uniquely specified by their opcodes but are more commonly referred to by their names.\n\nInstruction names are of the form MNEMONIC[flag] where the mnemonic is intended as an aid to remembering the instruction's function. For example, the MDAP in MDAP[a] instruction stands for Move Direct Absolute Point. Similarly, RUTG[ ] is short for Round Up To Grid.\n\nClosely related instructions sometimes share a single name. Such names can be mapped to unique opcodes using an associated set of flags. To calculate the opcode for an instruction variant, add the unsigned binary number represented by the flag to the lower of the two opcode values given in the documentation. In performing this operation, note that the left most bit is the most significant.\n\nThe flags that follow an instruction name also serve to define the semantic meaning of each instruction variant. The binary number is decomposed into a sequence of flags. Flags set to 1 represent TRUE. Flags set to 0 represent FALSE. The binary digits can also be grouped to make it possible to choose among more complex alternatives. In such cases, the documentation specifies the meaning associated with each possible numerical combination.\n\n### The graphics state\n\nThe graphics state consists of a set of variables that guide the actions of the interpreter. The graphics state variables will be introduced as needed in this chapter. A complete list of the graphics state variables can be found in \"The Graphics State\".\n\nThe graphics state variables all have default values established at the start of interpretation of any font. The default value for a given graphics state variable is reestablished at the start of interpretation of any glyph. In other words, the graphics state has no inter-glyph memory. Changing the value of a graphics state variable while processing an individual glyph will produce a change that remains in effect only for that glyph.\n\nTo establish a new default value for a graphics state variable, it is necessary to change the value of that variable in the control value program. Changes made in the control value program will apply to all subsequently processed glyphs unless INSTCTRL[] is used to inhibit these new values.\n\nInstructions are available for changing the value of each of the graphics state variables. Instructions that change the value of a graphics state variable sometimes have a name that begin with the word set. (The instructions that set the round state are an exception to this rule.) The new value is expected to be at the top of the stack.\n\nThe instructions that set the value of a graphics state variable are listed in Table 1.\n\nTable 1 Setting graphics state values\n\nGraphics state variable Mnemonic Instruction freedom & projection vector Set Vectors To Coordinate Axis SVTCA[a] projection vector Set Projection Vector To Coordinate Axis SPVTCA[a] freedom vector Set Freedom Vector To Coordinate Axis SFVTCA[a] projection vector Set Projection Vector To Line SPVTL[a] freedom vector Set Freedom Vector To Line SFVTL[a] freedom vector Set Freedom Vector To Projection Vector SFVTPV[] dual projection vector Set Dual Projection Vector To Line SDPVTL[] projection vector Set Projection Vector To Line SVPTL[] projection vector Set Projection Vector From Stack SPVFS[] freedom vector Set Freedom Vector From Stack SFVFS[] rp0 Set Reference Point 0 SRP0[] rp1 Set Reference Point 1 SRP1[] rp2 Set Reference Point 2 SRP2[] zp0 Set Zone Pointer 0 SZP0[] zp1 Set Zone Pointer 1 SZP1[] zp2 Set Zone Pointer 2 SZP2[] zp0, zp1, zp2 Set Zone PointerS SZPS[] round state Round To Half Grid RTHG[] round state Round To Grid RTG[] round state Round To Double Grid RTDG[] round state Round Up To Grid RUTG[] round state Round Down To Grid RDTG[] round state set Rounding Off ROFF[] round state Super ROUND SROUND[] round state Super 45 ROUND S45ROUND[] loop Set LOOP SLOOP[] single width cut-in Set Single Width Cut-In SSWCI[] control value cut-in Set Control Value Table Cut-In SCVTCI[] minimum distance Set Minimum Distance SMD[]\n\nTwo instructions are available for getting the value of the freedom vector and projection vector respectively. Instructions that retrieve the value of a state variable have names that begin with the word get. Get instructions will return the value of the state variable in question by placing that value on the top of the stack.\n\nTable 2 lists the two get instructions.\n\nTable 2 : Getting graphics state variables\n\nMnemonic Instruction\nGet Freedom Vector GFV[]\nGet Projection Vector GPV[]\n\n### Getting information from the interpreter\n\nThree instructions are provided to make it possible to obtain information about the glyph being interpreted and other data useful in instructing a glyph. The scaler version number can be requested with the GETINFO[ ] instruction. That same instruction can be used to inquire whether a glyph has been stretched or rotated. It is also possible to request the size in points or in pixels per em of the current glyph.\n\nTable 3 Getting information\n\nMnemonic Instruction\nGET INFOrmation GETINFO[]\nMeasure Pixels Per EM MPPEM[]\nMeasure Point Size MPS[]\n\n## Instruction processing\n\nThis section describes those portions of the interpreter that are important for processing instructions. It begins by describing the instruction stream which holds the data and instructions that can be found in the font, control value or glyph programs. It continues on to discuss the stack, the place where the interpreter takes instruction parameters and stores instruction results and the interpreter storage area, a place in which values can be temporarily saved and later retrieved.\n\n### The instruction stream\n\nThe instruction opcodes and data that make up the font, control value or glyph programs are stored as a sequentially ordered sequence of byte values. From the point of view of the interpreter, they represent a sequentially ordered stream of byte values known as the instruction stream.\n\nAn instruction pointer (IP) marks the next instruction to be executed. As execution of instructions proceeds, the opcodes and data in the instruction stream are gradually used up. Instructions cannot add new data to the instruction stream. FIGURE 11 shows the instruction stream with the instruction pointer marking the next instruction to be executed.\n\nFIGURE 11 The instruction steam", null, "#### Altering the flow of control\n\nNormally, opcodes encountered in the instruction stream execute sequentially, however, the order of execution of can be altered by a set of instructions known as flow of control instructions. These instructions are listed in Table 4 below.\n\nTable 4 The flow of control instructions\n\nMnemonic Instruction\nIF test IF[]\nELSE clause ELSE[]\nEnd IF EIF[]\nJump Relative On False JROF[]\nJump Relative On True JROT[]\nJuMP Relative JMPR[]\nLOOP and CALL LOOPCALL[]\n\n#### The stack\n\nThe TrueType interpreter stores any data needed by instructions and the results created by instructions on the interpreter stack. Placing data on the stack is termed a push operation. Taking data from the stack is termed a pop operation. The last item pushed onto the stack will always be the first item popped.\n\nStack elements are always 32 bits wide. The data types used in the stack are documented in Instruction Set. That section also provides additional details on stack interactions.\n\n#### Moving data from the instruction stream to the stack\n\nA few instructions known collectively as push instructions move data from the instruction stream to the interpreter stack. These instructions are unique among the TrueType instruction set in taking their arguments from the instruction stream. All other instructions take any data needed from the stack. The push instructions are summarized in Table 5 below.\n\nTable 5 The push instructions\n\nMnemonic Instruction\nPUSH N Bytes NPUSHB[]\nPUSH N Words NPUSHW[]\nPUSH Bytes PUSHB[abc]\nPUSH Words PUSHW[abc]\n\nSince the instruction stream is 8-bits wide and the stack is 32 bits wide, bytes that are pushed onto the stack are extended to 32-bits. When words (16-bit quantities) are pushed on the stack they are created from two bytes (the high byte appearing first in the instruction stream) and sign extended to 32 bits.\n\nFIGURE 12 Extending a byte value to form a long word", null, "Those that push words combine two bytes to form a signed word and then sign extend that word until it is 32 bits wide. When two bytes are combined to form a word, the high byte of that word is always the one appearing first in the instruction stream. The low byte is the one that appears second. FIGURE 13 shows how two instruction stream bytes are combined and then sign extended when they are pushed onto the stack.\n\nFIGURE 13 Combining byte values to form a sign extended long word", null, "#### Managing the stack\n\nTrueType provides basic stack manipulation operations that make it possible to change the contents of the stack or reorder its elements. These instructions are listed in Table 6 below.\n\nTable 6 : Managing the stack\n\nMnemonic Instruction\nCLEAR the stack CLEAR[]\nDEPTH of the stack DEPTH[]\nDUPlicate top stack element DUP[]\nPOP top stack element POP[]\nCopy INDEXed element CINDEX[]\nMove INDEXed element MINDEX[]\nROLL top 3 stack elements ROLL[]\nSWAP top two stack elements SWAP[]\n\n#### Repeating an instruction with loop\n\nNormally an instruction encountered in the instruction stream will execute only once. Some instructions look at the loop state variable and execute the number of times that variable dictates. The instructions that use the loop variable are listed in Table 7.\n\nThe default value for the loop variable is 1. Setting loop to zero or a negative value is illegal. The loop variable is set with the SLOOP[ ] instruction shown in Table 8.\n\nTable 7 Instructions that use the loop variable\n\nMnemonic Instruction\nAlign Relative Point ALIGNRP[]\nFLIP PoinT FLIPPT[]\nInterpolate Point IP[]\nSHift Point SHP[a]\nSHift by PIXEL amount SHPIX[]\n\nTable 8 Setting the value of the loop variable\n\nMnemonic Instruction\nSet LOOP SLOOP[]\n\n#### Performing arithmetic\n\nThe TrueType instruction set provides the basic arithmetic functions as enumerated in Table 8. Unless otherwise noted, arithmetic is done on 26.6 fixed point numbers producing 26.6 fixed point results.\n\nTable 9 Performing logical operations.\n\nMnemonic Instruction\nSUBtract SUB[]\nDIVide DIV[]\nMULtiply MUL[]\nABSolute value ABS[]\nNEGate NEG[]\nFLOOR FLOOR[]\nCEILING CEILING[]\nMAXimum MAX[]\nMINimum MIN[]\n\n#### Performing logical operations\n\nThe TrueType instruction set provides a set of basic logical functions. They are enumerated in Table 10. These functions return zero if the result is FALSE and a non-zero value if the result is TRUE.\n\nTable 10 Performing logical operations.\n\nMnemonic Instruction Less Than LT[] Less Than or EQual LTEQ[] Greater Than GT[] Greater Than or EQual GTEQ[] EQual EQ[] Not EQual NEQ[] ODD element ODD[] EVEN element EVEN[] AND AND[] OR OR[] NOT NOT[]\n\n### The storage area\n\nThe interpreter maintains a storage area consisting of a portion of memory that can be used for temporary storage of data taken from the interpreter stack. It is possible to read the values of stored data and to write new values to storage. Storage locations range from 0 to n-1 where n is the value established in the maxStorage entry in the 'maxp' table of the font file. Values are 32 bit numbers.\n\n#### Reading from and writing to storage\n\nThe following two instructions make it possible to read a value from a location in the interpreter storage area and to write a new value to a storage location.\n\nTable 11 : Accessing Storage\n\nMnemonic Instruction\nRead from Storage RS[]\nWrite to Storage WS[]\n\n## Managing points in the pixel grid\n\nThe key task of the TrueType instruction set is one of grid-fitting glyph outlines to allow the scan converter to produce superior bitmap images for display. That task consists of reshaping glyph outlines by moving the points that make up their outline. The following sections describe the instructions that are used to manage points in the pixel grid.\n\n### Zones and points\n\nPoints are locations in a grid. As stored in the font file, points have coordinates that are expressed in FUnits. These coordinates refer to positions in the master grid. Once scaled by the font engine, however, point locations are expressed as 26.6 fixed point numbers representing locations in the device specific pixel grid. That is, they are given to the nearest sixty-fourth of a pixel. The notation wn:fp is used to express point locations where wn refers to a whole number and fp refers to the fractional part. The number six would be written 6:0. The number one fourth would be written 0:16. When it is convenient, point locations will be expressed as decimal numbers, such as 2.5 to represent the position two and one-half.\n\nInstructions reference the points that comprise a glyph outline by specifying a given point number in a particular zone. The points that make up the outline of the current glyph are said to be in the glyph zone (zone 1). These points are defined in the 'glyf' table in the font file.\n\nA second group of points useful in instructing glyphs can be created using instructions. These points are said to be in the twilight zone (zone 0). Points in zone 0 are created by instructions in a glyph program. These points are not reinitialized at the beginning of each glyph program, so each glyph program must be certain to set these points in the desired location before accessing them.\n\nInstructions do not refer to zones explicitly but use one or more of three zone pointers which can be set to either of the two zones. If an instruction uses zone pointer 1 (zp1), that instruction will use points in the zone it references. It can point to either the twilight zone or the glyph zone.\n\nSome instructions will assume that the points they reference are in the zone referenced by a particular zone pointer. Other instructions require an explicit designation of the zone pointer through the setting of a flag (i.e through the use of an instruction variant).\n\nAll of the following uniquely specify the same point in the glyph zone:\n\n• point 5 in the glyph zone\n• point 5 in zone 1\n• point 5 in the zone pointed to by zp0 where zp0 points to zone 1\n• point 5 in the zone pointed to by zp2 where zp2 points to zone 1\n• rp0 in zp0 where zp0 points to zone 1and rp0 has the value 5\n• rp0 in zp2 where zp2 points to zone 1 and rp0 has the value 5\n\nAll of the following specify the same point in the twilight zone\n\n• point 5 in the twilight zone\n• point 5 in zone 0\n• point 5 in the zone pointed to by zp1 where zp1 points to zone 0\n• rp0 in zp1 where zp1 points to zone 0 and rp0 has the value 5\n\nNote that point 5 in the twilight zone is not the same point as point 5 in the glyph zone.\n\n#### Setting zone pointers\n\nThe instructions for establishing the value of zone pointers are listed in Table 12.\n\nTable 12 : Setting Zone Pointers\n\nMnemonic Instruction\nSet Zone Pointer 0 SZPO[]\nSet Zone Pointer 1 SZP1[]\nSet Zone Pointer 2 SZP2[]\nSet Zone PointerS SZPS[]\n\nSince both the glyph zone and the twilight zone number their points beginning with zero, the same number can appear in both zones. To unambiguously specify a point it is necessary to use its number and its zone. For example, point 6 in the glyph zone designates a unique point.\n\n#### Setting reference points\n\nSome instructions will refer to points not by their number but through the device of reference points. Each of three reference points, rp0, rp1 and rp2 can be associated with an integer representing a point number. When that reference point is associated with a particular zone, a point is unambiguously specified. Instructions for setting reference points to point numbers are listed in Table 13.\n\nTable 13 Setting Reference Points\n\nMnemonic Instruction\nSet Reference Point 0 SRPO[]\nSet Reference Point 1 SRP1[]\nSet Reference Point 2 SRP2[]\n\n#### Flipping points\n\nOn-curve points can be changed to off-curve points and vice versa. A single point or a range of points can be changed. Table 10 lists the instructions that flip points.\n\nTable 14 Flipping points\n\nMnemonic Instruction\nFLIP PoinT FLIPPT[]\nFLIP RanGe ON FLIPRGON[]\nFLIP RanGe OFF FLIPRGOFF[]\n\n### Movement in the pixel grid\n\nMovement of points in the pixel grid is always along the freedom vector, a graphics state variable representing a vector in the pixel grid. The freedom vector can be translated in space so that it maintains its orientation relative to the coordinate system but passes through the point that is to be moved. By convention, the freedom vector is shown as a dotted line passing through a point that is to be moved. Points move along the vector.\n\nFor most instructions, a given freedom vector and its 180 degree opposite are equivalent. The lone exception is the SHPIX[ ] instruction which measures distance along the freedom vector. When points are moved along the freedom vector, the direction of movement of the point will be shown as illustrated in FIGURE 14.\n\nFIGURE 14 Moving points along the freedom vector", null, "### Getting and setting the freedom vector\n\nOne instruction exists for obtaining the current value of the freedom vector. Its value is returned as a pair of (x,y) coordinates. The freedom vector can be set by specifying its coordinates on the stack, by setting it to either or the coordinate axes, by setting it to be parallel or perpendicular to a line or by setting it to be parallel or perpendicular to the projection vector. The instructions for setting the freedom vector are listed in Table 15.\n\n### Setting and getting point locations\n\nThe following sections describe how to find the location of a point in the pixel grid and how to set it to a specified location.\n\n#### Getting the location of a point along the projection vector\n\nThe location of a point is expressed as a position along the current projection vector. To understand how the interpreter determines the location of a point imagine that a perpendicular is dropped from that point to the vector. In the illustrations that follow, lines used to project distances onto the projection vector are dashed. The location where the perpendicular intersects with the projection vector is used to read the coordinate of the point.\n\nThis process is illustrated in FIGURE 15. In the illustration, points p1, p2 and p3 are all at coordinate position 1.2 as would any point that can be found on line L1L1'. Point p4 is at coordinate position -0.6.\n\nFIGURE 15 Getting the coordinate of a point", null, "#### Setting the location of a point\n\nTo set the coordinate of a point means to make its projection onto the projection vector equal to the desired value. Conceptually, this is done by moving the point along the freedom vector to the position where the vector intersects with a perpendicular to the projection vector at the desired coordinate value.\n\nIn FIGURE 16 the three points p1, p2 and p3 are all set to have the same value along the projection vector. Each point is moved along the freedom vector until it reaches the intersection point of the freedom vector and a perpendicular to the projection vector at the coordinate position 2.\n\nFIGURE 16 Setting the coordinate of a point", null, "The instructions for getting and setting a point coordinate are listed inTABLE 1\n\nTABLE 1 Getting and setting a point coordinate\n\nMnemonic Instruction\nGet Coordinate GC[a]\nSet Coordinate From Stack SCFS[]\n\n### Measurement in the pixel grid\n\nMeasurement of distances in the pixel grid is always along the projection vector, a graphics state variable describing a vector in the pixel grid. Grid distances are signed values. Measuring the distance from point A to point B will produce a result with the opposite sign from that obtained by measuring the distance from point B to point A.\n\nBy convention, a solid line emanating from the origin of the grid coordinate system with grid units marked by long lines and half-grid units marked by short lines is used to represent the projection vector. The projection vector is shown in FIGURE 17.\n\nFIGURE 17 The projection vector", null, "#### Setting the projection vector\n\nOne instruction exists for obtaining the current value of the projection vector. Its value is returned as a pair of (x,y) coordinates. The instructions for getting and setting the projection vector are listed in Table 16.\n\nTable 16 Getting and setting the projection vector\n\nMnemonic Instruction\nGet Projection Vector GPV\nSet Dual Projection Vector To Line SDPVTL[]\nSet Projection Vector From Stack SPVFS[]\nSet Projection Vector To Coordinate Axis SPVTCA[a]\nSet Projection Vector To Line SPVTL[a]\n\n#### Finding the distance between two points\n\nThe distance between two points is determined by comparing their locations along the projection vector. Distances between a point and a reference point are always obtained by subtracting the coordinate, along the projection vector, of the reference point from the coordinate of the point.\n\nWhen measuring the distance between two points it is important to distinguish between the master outline distance between those points and their grid-fitted distance difference. Distance is measured along the projection vector and is a signed quantity. The value returned is a F26Dot6 number representing a quantity in pixels. In other words, the distance is measured between points in the scaled outline resident in the pixel grid.\n\nTable 17 Measuring the distance between two points\n\nMnemonic Instruction\nMeasure Distance MD[a]\n\n### Engine compensation using color\n\nIn addition to the property of magnitude, distances also have the property of color. Color is used to compensate for the effects of different marking engines on the appearance of the final output image of a glyph.\n\n#### Determining the color of a distance\n\nThe TrueType interpreter distinguishes between three different types of distances: black, white, and grey.\n\nBlack areas are those containing pixels turned on by the scan converter. White areas are areas containing no pixels turned on by the scan converter.\n\nWhen considering the distance between two points, the following rules will determine whether the distance is black, white, or grey:\n\nDistances that cross only black areas are black distances. Distances that cross only white areas are white distances. Distances that cross a combination of black and white areas can be black, white or grey depending upon the combination of distances types covered.\n\n• If there is black to the right of both points (along the projection vector), the distance is grey.\n• If there is black to the left of both points, the distance is grey.\n\nThe rules shown in FIGURE 18 can be used to determine the color of a distance made up of subdistances of differing color.\n\nFIGURE 18 Combining distance colors", null, "FIGURE 19 shows an example of how black, white, and grey distances are combined. The distance [4,5] is black; [4,0] is grey and [5,0] is white. The combined distance [4,0] adds a black distance to a white distance to get a grey distance. When that grey distance is added to the black distance [0,1] the resulting distance [4,1] is grey plus black or black.\n\nFIGURE 19 White, black and grey distances", null, "#### Compensate for the engine characteristics\n\nThe distance type is used in determining how the instructions listed in Table 18 and the other instructions that use the graphic state variable round state will manage distance values. With grey distances, because they combine black and white distances, no engine compensation takes place. Black or white distances, however, require a compensation term be added or subtracted before rounding takes place. The amount of compensation needed will be set by the printer driver. For example, if the printing engine has large pixels, the interpreter will compensate by making black distances tend to shrink and white distances tend to grow. The result in that on different dot-size printers, the font should produce equivalent weight and color for the final outputted glyphs.\n\n### Rounding\n\nAll rounding done by the TrueType interpreter is applied to values representing distances between two points, rather than to grid positions. The effect of rounding a value depends upon the setting of the round state variable. Values can be rounded to integers or half integers.\n\nAll rounding works by using a round state setting to control how the domain of 26.6 fixed point numbers is mapped to a set of discrete values that are separated by equal distances. A number of predefined rounding rules are available and can be set using instructions. These make up possible to round to the nearest integer (RTG[]), to the nearest half integer (RTHG[]), the nearest smaller integer (RDTG[]), the nearest larger integer (RUTG[]) or the nearest half integer or integer (RTDG[]).\n\nThe effect of rounding the distance 1.4 with each of the predefined round states is shown in FIGURE 20.\n\nFIGURE 20 The effect of the round states", null, "In addition to the predefined rounding states, two instructions can be used to set custom round states. They are SROUND[] and S45ROUND[]. These work by setting the period, phase and threshold, that together define the actions of a round, separately.\n\n• The period specifies the length of the separation or space between rounded values.\n• The phase specifies the offset of the rounded values from multiples of the period.\n• The threshold specifies the part of the domain, prior to a rounded value, that is mapped onto each value. More intuitively, the threshold tells a value when to \"fall forward\" to the next largest integer.\n\nEach of the predefined round states is equivalent to a particular setting of the period, phase and threshold. The relationship is shown in Table 19.\n\nTable 19 Mapping predefined round states to custom states\n\nInstruction period phase threshold\nRTG[] grid 0 4/8 period\nRTHG[] grid 1/2period 4/8 period\nRTDG[] 1/2 grid 0 4/8 period/64\nRUTG[] grid 0 period -1\nRDTG[] grid 0 0\n\nThe period is based on the value of the gridPeriod, the distance between grid positions. For SROUND[], the gridPeriod is equal to 1.0 pixels. For S45ROUND[], the gridPeriod is square root of 2 pixels. Table 20 shows the possible values for the period and how they relate to the gridPeriod. For additional information on setting the period, phase and threshold, see the entries for SROUND[] and S45ROUND[] in Instruction Set.\n\nThe period parameters can have values of 1/2 pixel, 1 pixel, or 2 pixels.\n\nTable 20 Setting the period\n\nbinary setting\n00 gridPeriod/2\n01 gridPeriod\n10 gridPeriod*2\n11 Reserved\n\nThe period specifies the length of the separation or space between rounded values. As shown on the number lines in the illustrations below, if the period is 0.5, rounded values are separated by half integers. A period of 1.0 separates rounded values by one integer. A period of 2.0 separates rounded values by two.\n\nIn FIGURE 21, the three number lines, with small circles representing possible rounded values, illustrate three possible period settings.\n\nFIGURE 21Examples of the period", null, "The phase specifies the offset of the rounded values from multiples of the period. As shown in the first example below, if the phase is zero, and the period is one, we have the familiar case where values are not offset and will round to integral values. In the second case, where the phase is 0.25 and the period is 1.0, rounded values are offset by one-quarter and will round to 0.25, 1.25, 2.25 and so forth.\n\nThe phase parameters can have values of 0 pixels, 1/4 pixel, 1/2 pixel, or 3/4 pixel. In the examples in FIGURE 22 below, the period is 1.0.\n\nFIGURE 22 Examples of the phase", null, "The threshold represents the portion of the period that rounds up to the next value. FIGURE 23 shows two examples where the period and phase are held constant and the value of the threshold is changed. In the first case, a threshold of 0.5 is equivalent to round to grid. In the second case, a threshold of 0.25 results in values from, for example, 0.75 through 1.75 rounding to the value 1.\n\nThe threshold parameters can have values of -3/8 period, -2/8 period,... 11/8 period. It can also have the special value largest-number-smaller-than-period which causes rounding e quivalent to CEILING. In the examples in FIGURE 23 below, the period is one, the phase is zero and the threshold is one-half.\n\nFIGURE 23 Examples of the threshold", null, "A more complex example is shown in FIGURE 24. For example, SROUND(01:01:1000) maps numbers into the values 0.25, 1.25, 2.25, The numbers from -0.25 up to but not including 0.75 are mapped into 0.25. The range of numbers from 0.75 up to but not including 1.75 map into 1.25. Similarly, the numbers from 1.75 up to but not including 2.75 map into the number 2.25 and so on. In other words, the period is 1.0, the phase is 0.26 and the threshold is 0.5.\n\nFIGURE 24 A complex example", null, "#### Rounding and the minimum distance\n\nWhen rounding, it is possible that certain values will round to zero or to an unacceptably small value. This is especially likely at small pixel per em sizes. Sometimes rounding a value to zero is undesirable because it will result in a glyph feature, such as a counter or stem, disappearing. To avoid undesirable rounding to zero, there is a minimum distance variable in the graphics state. The MIRP[] and MDRP[] instructions both make use of the minimum distance state variable. This represents the smallest possible value to which any value can be rounded. Values less than the minimum distance value will be rounded up to the minimum distance regardless of the round state, when the minimum distance Boolean is set.\n\n#### Setting the round state\n\nThe instructions that can be used to set the round state are listed in Table 21.\n\nTable 21 Setting the round state\n\nMnemonic Instruction Round Down To GRid RDTG[] turn Rounding OFF ROFF[] Round To Double Grid RTDG[] Round To Grid RTG[] Round To Half Grid RTHG[] Round Up To Grid RUTG[] Super 45 ROUND S45ROUND[] Super ROUND SROUND[]\n\n#### Order of rounding operations\n\nRounding occurs after compensation for the color of a distance. The steps in the rounding of a number n are:\n\n1. add engine compensation to n\n2. subtract the phase from n\n3. add the threshold to n\n4. truncate n to the next lowest periodic value (ignore the phase)\n5. add the phase back to n\n6. if rounding caused a positive number to become negative, set n to the positive round value closest to 0\n7. if rounding caused a negative number of become positive, set n to the negative round value closest to 0\n\n### Moving points\n\nThe following sections describe those instructions that move points.\n\n#### Shifting the position of a point\n\nThe position of a single point, the points that make up a contour, or all of the points in a zone can be shifted by the amount that a reference point has been shifted using one of the instructions listed in Table 22. The magnitude of the shift is a signed value. It is also possible to shift a point by a specified amount.\n\nTable 22 Shifting points\n\nMnemonic Instruction\nSHift Point SHP[a]\nSHift Contour SHC[a]\nSHift Zone SHZ[a]\nSHift by PIXEL amount SHPIX[]\n\n#### Moving a point to a specified location\n\nPoints can be moved to a given coordinate location with one of two absolute instructions. The location to which the point is moved can be specified in the control value table, as in the case of MIAP[], or in the instruction itself, as in MDAP[].\n\nTable 23 Moving to a specified location\n\nMnemonic Instruction\nMove Indirect Absolute Point MIAP[a]\nMove Direct Absolute Point MDAP[a]\n\n#### Moving a point a stated distance\n\nPoints can be moved relative to other points. When a point moves a given number of units, movement will be along the freedom vector but the distance moved will be measured along the projection vector. To understand how the interpreter will carry out such a move imagine that the current location of that point is projected onto that vector, and the point is moved along the freedom vector until it is the required number of units along the vector.\n\nIt is illegal to execute instructions that move points when the freedom and projection vectors are orthogonal. Attempts to do so will yield undefined results and will be different on different versions of the TrueType software.\n\nTable 24 : Moving points relative to a reference point\n\nMnemonic Instruction\nMove Direct Relative Point MDRP[abcde]\nMove Indirect Relative Point MIRP[abcde]\nMove Stack Indirect Relative Point MSRP[a]\n\n#### Aligning points\n\nInstructions can be used to align points relative to each other or to a reference point. Points are said to be aligned if the distance between them, along the current projection vector, is zero. Table 25 lists the instructions that align points.\n\nTable 25 : Aligning points\n\nMnemonic Instruction\nALIGN PoinTS ALIGNPTS[]\nALIGN to Reference Point ALIGNRP[]\nmove point to InterSection ISECTP[]\n\n#### Interpolating points\n\nTwo interpolation instructions make it possible to adjust the position of a point or points based on the movements of other points.\n\nThe first of the interpolation instructions, IP[ ], changes the position of a point based on their changes in the position of two reference points. The second, IUP[ ] interpolates all untouched points between touched points.\n\nPoints that are acted upon by a TrueType instruction are said to have been touched. Points that are touched will be not be affected by the IUP[ ] instruction. If it is desirable to have an IUP[ ] instruction affect such a point, it can be untouched using the UTP[ ] instruction. Points can be touched without moving them by using the MDAP[] instruction with no rounding. Additional complexities of the interpolation instructions will be explained in Instruction Set.\n\nTable 26 Interpolating points\n\nMnemonic Instruction\nUnTouch Point UTP[]\nInterpolate Point[ ] IP[]\nInterpolate Untouched Point IUP[]\n\n### Creating points in the twilight zone\n\nTable 27 Points in the twilight zone are not part of the font file but must be created with one of the direct instructions (MIRP[], MSIRP[] or MIAP[]).\n\nTABLE 27: Creating points in the twilight zone\n\nMnemonic Instruction\nMove Indirect Absolute Point MIAP[]\nMove Indirect Relative Point MIRP[]\nMove Stack Indirect Relative Point MSIRP[]\n\nMIAP[] is used to create a point that is a specific distance (x,y) away from the origin (0,0). MIRP[] is used to create a point that is a specific distance away from a reference point. The reference point can itself be in either zone 0 or zone 1. Both MIAP[] and MIRP[] obtain the distance from the control value table. MSIRP[] also creates a point that is a specified distance from a reference point, however, the distance is taken from the stack rather than from the control value table.\n\nOnce created, points can be moved by MDRP[] and other instructions. A new reference to the point by MIAP[], MIRP[], or MSIRP[] will create a completely new position that has nothing to do with the previous position of the point (unless the point is itself the reference point used by the MIRP[] or the MSIRP[] instructions).\n\nFor example, suppose rp0 refers to point 7 in zone1, and its coordinates are 3:0, 4:0). Suppose that zone pointer 1 is set to point at zone 0, the projection and freedom vectors are set to the x-axis, and the instruction MSIRP[] is executed with the values 1 and 0:48 on the stack. The result is that point 1 in zone 0 will have ungrid-fitted coordinates of (3:48, 4:0) and grid-fitted values of (4:0, 4:0). If the freedom and projection vectors are now set to the y-axis, and the instruction MSIRP[] is executed with the values 1 and 0:20 on the stack, the new ungrid-fitted and grid-fitted values of point 1 in zone 0 will be (4:0, 5:20).\n\n### Cut-in values\n\nThe TrueType language offers several means of coordinating the size of features within individual glyphs and across glyphs in a font. Such coordination is useful when the number of pixels available for a feature or a glyph are few in number. It prevents small differences in the size from becoming exaggerated by the change in the placement of pixel centers relative to a glyph outline.\n\nSuch coordination, however, becomes a liability when small differences in the size or placement of features can be effectively represented by the number of available pixels. TrueType allows you to regularize features at small numbers of pixels per em while allowing the outline to revert to the original design once a sufficient number of pixels is available.\n\nThere are two different ways to accomplish this goal. Each one uses a cut-in value. The first method uses the control value table and the control value cut-in. This method allows for arbitrary values to be substituted for the actual value. The second method takes regularization a step further and substitutes a single value known as the single width value. It is used with the single width cut-in.\n\n#### The control value table\n\nThe control value table is an ordered list of values used by the indirect instructions. By using a control value table entry rather than an outline measurement, it is possible to collapse values that vary by a small amount to a single value at certain pixel per em sizes. The control value table works by using the control value cut-in described in the next section.\n\n#### The control value cut-in\n\nThe control value cut-in limits the regularizing effects of the control value table to cases where the difference between the table value and the measurement taken from the original outline is sufficiently small.\n\nThe cut-in is a factor only in instructions that reference the control value table. These are the indirect instructions (MIRP[] and MIAP[]). By having these instructions look at the cut-in value it is possible for a point to be moved to a position specified in the control value table only in cases where the value in the table is sufficiently close to that in the original outline. In practice this means that the original outline will be favored at high pixel per em values, where there will typically be a large difference between the control value table entry and the outline distance, while at small pixel per em values, distances can be made to converge on a single value. This makes it possible to have related stems have a uniform width at small pixel per em values while allowing the original subtleties of the font design to reemerge at larger sizes.\n\n• The cut-in is applied according to the following rules:\n• When the absolute value of the difference is less than or equal to the cut-in, the control value table entry is used.\n• When the absolute value of the difference between the table entry and the outline value is greater than the cut-in the actual value is used.\n\nNote that the difference shown in Table 28 is the difference between the scaled control value table entry and the scaled outline distance. The cut-in is a F26DOT6 value, i.e. it is in pixels.\n\nTable 28 Some examples of the use of the cut-in in MIAP[]\n\nCVT Value Original Outline Value \|Difference\| Cut-in Value used\n93 80 13 68/64 outline\n100 99 20/64 44/64 68/64 table\n97 95 60/64 68/64 68/64 table\n\nFIGURE 25 below illustrates the use of the cut-in. In the master outline for the font shown, the capital J dips below the base line. At small pixel per em sizes, this subtlety cannot be shown. Even one pixel below the baseline would be too much at 18 ppem. When the character is grid-fitted the curve is held to the baseline by an indirect instruction so long as the actual distance below the baseline is less than the cut-in. When the distance difference exceeds the cut-in value of 68/64, the original value is reinstated. In this particular case, the curve is held to the base line through 81 pixels per em but reverts to its original design at 82 pixels per em as shown.\n\nFIGURE 25 Overshoot and the cut-in", null, "The effect of the cut-in varies with its value. Decreasing the value of the cut-in will have the affect of causing the outline to revert to the original design at a smaller ppem value. Increasing the value of the cut-in will cause the outline to revert to the original design at a higher ppem value.\n\nTable 29 demonstrates some of the effects of changing the cut-in.\n\nTable 29 Varying the cut-in value\n\ncut-in outline value CVT value difference value uses\n64/64 = 1 98 60/64 100 68/64 outline\n64/64 = 1 99 100 64/64 = 1 CVT\n64/64 = 1 101 100 64/64= 1 CVT\n64/64 = 1 101 4/64 100 68/64 outline\n68/64 98 56/64 100 72/64/16 outline\n17/16 98 15/16 100 17/16 CVT\n17/16 101 1/16 100 17/16 CVT\n17/16 101 2/16 100 18/16 outline\n18/16 98 13/16 100 19/16 outline\n18/16 98 14/16 100 18/16 CVT\n18/16 101 2/16 100 18/16 CVT\n18/16 101 3/16 100 19/16 outline\n\n#### The single width value and single width cut-in\n\nThe single width cut-in is the distance difference at which the interpreter will ignore both outline values and the control value table entries in favor of the single width value. It allows desired features to revert to a single predetermined size. The default for both the single width and the single width cut-in is zero pixels. It will not have any effect unless the default is set to another value.\n\nThe single width cut-in is used with MIRP[] and MDRP[].\n\nThis might be useful at very small grid sizes where forcing all controlled glyph features to assume the same dimensions can be an advantage in certain fonts. When the absolute difference between the single width value and the original outline value is smaller than the single width cut-in, the single width value is used.\n\n.\nTable 30 varying the single width value\n\nsingle width cut-in outline value single width value difference value uses\n0 99 100 1 outline\n0 100 100 0 outline\n1 98 15/16 100 1 4/64 outline\n1 99 100 1 single width\n1 101 100 1 single width\n1 101 1/16 100 1 4/64 outline\n\n### Order of operations\n\nInstructions sometimes will move a point, calculate a value, compensate for the color of the value, round according to the round state and then apply the cut-in tests. The order in which these operations are executed is significant and is given here.\n\nTable 31 Order of operations in move instructions\n\n MIRP[] check single width cut-in check control value cut-in compensate for the color of the distance round the distance check the minimum distance move the point MDRP[] check the single width cut-in compensate for the color of the distance round the distance check the minimum distance move the point MIAP[] check the control value cut-in round the value move the point MSIRP[] move the point MDAP[] round the value move the point\n\n### Size specific instructions\n\nAll of the instructions mentioned to this point are not size specific. Their actions are always applied, unless a conditional statement causes some of the code in a sequence not to execute. Sometimes it is necessary to change the outline at particular ppem value to produce the desired result. The DELTA instructions make this possible.\n\nSince DELTA instructions are exceptions to the grid-fitting performed by other instructions, it is important that they be added to glyph instructions after all other instructions have been decided upon. Any changes to the other instructions may change a DELTA's effect or render it useless.\n\nThere are two different types of DELTA instructions. The first type, the DELTAP, works by directly specifying the point number at which the exception is to be applied. The second type, the DELTAC, works by specifying the control value table entry number at which the exception is to be applied.\n\nTable 32 The DELTA instructions\n\nMnemonic Instruction\nDELTA exception P1 DELTAP1[]\nDELTA exception P2 DELTAP2[]\nDELTA exception P3 DELTAP3[]\nDELTA exception C1 DELTAC1[]\nDELTA exception C2 DELTAC2[]\nDELTA exception C3 DELTAC3[]\n\nThe 8 bit arg component of the DELTA instructions decomposes into two parts. The first 4 bits represent the relative number of pixels per em at which the exception is applied. The second 4 bits represent the magnitude of the change to be made. The structure of the arg is shown in FIGURE 26.\n\nFIGURE 26 The arg parameter to the DELTA instructions\n\n```--------------------------- arg = \| rel ppem \| magnitude \| ---------------------------\n\n```\n\nThe relative number of pixels per em at which a DELTA is applied is a function of the delta base and the choice of DELTA instruction. Table 33 below summarizes the sizes at which the various instructions apply.\n\nTable 33 The range of the Delta instructions\n\n DELTAC1 DELTAP1 (delta_base) through (delta_base + 15) DELTAC2 DELTAP2 (delta_base + 16) through (delta_base + 31) DELTAC3 DELTAP3 (delta_base + 32) through (delta_base + 47)\n\nThe magnitude of the change produced by a DELTA instruction is a pixel value dependent upon the current value of the delta shift state variable. The number of possible steps is 2 to the power delta shift. Each step has the magnitude 1 over that number.\n\nSince only four bits are available to store the exception size, the magnitude of the change made to the point location is remapped before it is stored in the exception specification. The table below maps the number of steps in the exception to the number that should be stored as the second part of argi.\n\nTable 34 : Magnitude values mapped to number of steps to move\n\n Magnitude Number of steps 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8\n\n#### A DELTAP example\n\nIn the example below, a DELTAP1 is applied to point 12 at 9 pixels per em. Assume that the delta shift has been set to 6.Moving point p by -3/64 of a pixel, prevents the dropout shown at the left. This instruction would expect the parameter 1(number of exceptions) and \\$05 which can be decomposed into the byte 0, the relative pixels per em value, and the byte 5 which represents -3 steps in magnitude (but is stored as exception 5 in table above).\n\nFigure 27 A dropout is filled in using a DELTAP instruction", null, "#### A DELTAC example\n\nIn the example below, imagine that control value table entries 4 and 11 refer to features that are close in size and expected to grow together. At 12 ppem and 14 ppem, these features will round together (assuming the round state is round to grid). At 13 ppem, the two values will round in opposite directions. The actual figures are shown in TABLE 3.\n\nTABLE 3 Scale control value table entry values\n\nControl value table Control value table values in pixels per em\nEntry number Value in FUnits 12 ppem 13 ppem 14 ppem\n4 120 45/64 49/64 53/64\n11 125 47/64 51/64 55/64\n\nA DELTAC instruction can be used to force the values to round together. At the default delta base of 9, the exception will take place at a ppem value of 13. The exception value of 8 signifies one step. At the default delta shift value, this will mean a movement of 4/64 of a pixel.\n\n PUSHB[] Push one byte onto the stack. 1 Number of control value table, exception pairs. 4 Control value table entry number 72 Exception 72 = 0100 10002 signifying a relative pixel per em value of 4 and a magnitude of 8. DELTAC1[] Create a delta exception to a control value table entry.\n\nThe values that are actually in the control value table or that are part of the instruction are in bold face. The other values presented are strictly to aid in understanding the example.\n\nFor the sake of this example, assume that the delta base is 9 producing a range from 9 through 24 pixels per em. 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https://ez.analog.com/power/f/q-a/102861/ltc3105-minimum-power-required-for-startup	[ "# LTC3105 minimum power required for startup\n\nHi,\n\nWhat is the minimum power required for startup for the LTC3105? The datasheet shows a minimum startup voltage of 0.25V typical (0.4V max), but gives no indication of current/power requirements to move from startup to normal operation. In fact I couldn't find anything relating to the minimum current/power requirements for the source. The only reference I found was in the applications section where it states under the industrial current loop example:\n\n\"In this application, a large input capacitor is required due to the very low available supply current (less than 4mA)\"\n\nThis suggests that a typical application should be supplying tens of milliamps and that the IC is not suitable for microwatt type applications (e.g. small single cell PV). Is this the case? I understand the ADP5091 (which provides detailed cold start power requirements in the datasheet) is aimed more at ultra low power sources, but I'd like to know where the cutoff is for the LTC3105.\n\nThanks.\n\n## Top Replies\n\n• Hi,\n\nYou can refer to the graph for No-Load Input current vs Input Voltage, on page 5 of the data sheet of LTC3105. The minimum input power required for IC is high as it is primarily for solar cells with higher input current capability . Depending on your application requirement of minimum startup voltage, LTC3106(Can refer Page 6 of the datasheet for 'No-Load Input current vs Input Voltage' graph) or ADP5091 could be better options as they are designed for very low power input sources.\n\nThanks\n\n• Hi Anisha, I have a further questions about startup current.\n\nI tried reading the No-Load graph and it's unclear to me how to apply it in this case.\n\nI understand the graph to approximately be saying \"the lower the input voltage, the higher the input current must be.\" I guess you're saying that applies to startup.\n\nHowever, what I want to do is size a capacitor such that startup succeeds, and I want to know how to do that. I can't tell how to do that with only this chart.\n\nIf I have a 0.250V input voltage, what value capacitor do I need to ensure startup when using a 10uH inductor?\n\nI have tried a 22uF cap, and that isn't sufficient. A 470uF cap was sufficient. So it's somewhere between. How do I calculate that value?\n\nAlso, I'm using a 10uH inductor (the COILCRAFT MSS5131-103MX specified). I see the spec also says, in the Component Selection section, \"Low DCR power inductors with values between 4.7µH and 30µH are suitable for use with the LTC3105. For most applications, a 10µH inductor is recommended. In applications where the input voltage is very low, a larger value inductor can provide higher efficiency and a lower start-up voltage.\"\n\nCan you tell me how the situation changes with a 20 or 30uH inductor in use instead of a 10uH? Is less current required using a 30uH at 0.250V than 10uH at 0.250V? How much less? How do I calculate that?\n\nThanks.", null, "doug\n\n• Hi Doug,\n\nFigure 1 on Page 9 of the datasheet gives the startup sequence of the converter. First VAUX and then VLDO needs to be charged up before Vout to start up. So the input capacitance should be selected such that it should be able to charge up both AUX and LDO during the startup. Recommended values for capacitor at AUX pin is 1uF cap and LDO pin is 4.7uF. Input capacitance for LTC3105 can be calculated by finding the energy required by LDO and AUX cap to charge up. The formula below can be used to calculate the energy transfer,", null, "To calculate the capacitance it is recommended to take worst case values, such as efficiency=20% at no load during startup, VAUX=20% higher than VLDO, Ven=90mV.\n\nSo for Vout=3.3V, VLDO = 2.2V and considering startup voltage=0.25V, input capacitance of 1350uF or higher is recommended.\n\nThanks\n\nAnisha\n\n• Hi Anisha, thank you for the reply.\n\nI will look to make the change regarding the capacitor.\n\nThanks.\n\ndoug\n\n• Hi Doug,\n\n10uH is the recommended inductor value. However using a higher inductor value reduces the number of switching as the output current slew rate becomes slower. This will reduce switching losses and might have slightly better efficiency. However the increase in efficiency might not be very significant.\n\nAlso I have a correction to make in the capacitor calculation mentioned above. The input cap value was calculated with efficiency factor of 40%.\n\nFor starting up at Vin=.25V, it is recommended to program the MPPC pin value for Vin<0.225V.\n\nThanks\n\nAnisha" ]	[ null, "https://ez.analog.com/resized-image/__size/320x240/__key/communityserver-discussions-components-files/330/4846.Capture.PNG", null, "https://ez.analog.com/resized-image/__size/320x240/__key/communityserver-discussions-components-files/330/pastedimage1552494577084v2.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.92818964,"math_prob":0.8024695,"size":972,"snap":"2021-21-2021-25","text_gpt3_token_len":205,"char_repetition_ratio":0.13119835,"word_repetition_ratio":0.0,"special_character_ratio":0.21193415,"punctuation_ratio":0.08888889,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9660203,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,1,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-05-06T01:43:14Z\",\"WARC-Record-ID\":\"<urn:uuid:8ca9359e-85b4-489e-b2e7-a0a787f8cc9f>\",\"Content-Length\":\"223156\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:3524e698-bc49-44d4-b16e-7226c8416e10>\",\"WARC-Concurrent-To\":\"<urn:uuid:d7481bbd-fe27-45a7-9efb-ccd1948d5d3c>\",\"WARC-IP-Address\":\"23.208.33.208\",\"WARC-Target-URI\":\"https://ez.analog.com/power/f/q-a/102861/ltc3105-minimum-power-required-for-startup\",\"WARC-Payload-Digest\":\"sha1:VXZ67FA6CJP2VPEFVQRSJCTPMU6M5LPY\",\"WARC-Block-Digest\":\"sha1:5LL3BHC2XDQ3J3PT6TZ5ZNDEEXWVXCXB\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-21/CC-MAIN-2021-21_segments_1620243988724.75_warc_CC-MAIN-20210505234449-20210506024449-00304.warc.gz\"}"}
https://electronics.stackexchange.com/questions/127525/reducing-voltage-with-resistors	[ "# Reducing voltage with resistors\n\nI have been looking around for an easy way to convert 12V to 5V. I have seen some people saying that a simple resistor is all that is needed.\n\n$$Volts = Ohms \\cdot Amps$$ $$Amps = \\frac{Volts}{Ohms}$$ $$Ohms = \\frac{Volts}{Amps}$$\n\nSo applying a resistor will diminish the voltage of the circuit. That should mean that an appropriately sized resistor could simply be placed in the path of a 12V circuit, converting it to 5v.\n\n• If this is the case how would one reduce amps?\n• Would series vs parallel make a difference in this area?\n\nI have seen designs that include a regulator IC and some capacitors, but if a simple resistor/fuse/diode setup will do the trick, I would really prefer that.\n\n• Are you trying to provide power to a load? What kind of load? Or are you trying to change the level of a signal carrying information? Sep 2, 2014 at 16:51\n• Its almost never just about dropping voltage, its also about not wasting energy (efficiency), safety (resistors can get very hot) and regulation (maintaining the output voltage with changing load / current demand). Sep 2, 2014 at 16:56\n• electronics.stackexchange.com/questions/83656/… Sep 2, 2014 at 19:30\n• Um, no there are much better ways to cut voltage. Use a 5V voltage regulator or if you're looking for something simple, just throw in a zener diode in reverse bias. Sep 3, 2014 at 2:43\n\nThere are a few ways to get 5V from a 12V supply. Each has its advantages and disadvantages, so I've drawn up 5 basic circuits to show their pros and cons.", null, "• Circuit 1 is a simple series resistor - just like the one \"some people\" told you about.\n\nIt works, BUT it only works at one value of load current and it wastes most of the power supplied. If the load value changes, the voltage will change, since there is no regulation. However, it will survive a short circuit at the output and protect the 12V source from shorting out.\n\n• Circuit 2 is a series Zener diode (or you could use a number of ordinary diodes in series to make up the voltage drop - say 12 x silicon diodes)\n\nIt works, BUT most of the power is dissipated by the Zener diode. Not very efficient! On the other hand it does give a degree of regulation if the load changes. However, if you short circuit the output, the magic blue smoke will break free from the Zener... Such a short circuit may also damage the 12V source once the Zener is destroyed.\n\n• Circuit 3 is a series transistor (or emitter follower) - a junction transistor is shown, but a similar version could be built using a MOSFET as a source follower.\n\nIt works, BUT most of the power has to be dissipated by the transistor and it isn't short circuit proof. Like circuit 2, you could end up damaging the 12V source. On the other hand, regulation will be improved (due to the current amplifying effect of the transistor). The Zener diode no longer has to take the full load current, so a much cheaper/smaller/lower power Zener or other voltage reference device can be used. This circuit is actually less efficient than circuits 1 and 2, because extra current is needed for the Zener and its associated resistor.\n\n• Circuit 4 is a three terminal regulator (IN-COM-OUT). This could represent a dedicated IC (such as a 7805) or a discrete circuit built from op amps / transistors etc.\n\nIt works, BUT the device (or circuit) has to dissipate more power than is supplied to the load. It is even more inefficient than circuits 1 and 2, because the extra electronics take additional current. On the other hand, it would survive a short circuit and so is an improvement on circuits 2 and 3. It also limits the maximum current that would be taken under short circuit conditions, protecting the 12v source.\n\n• Circuit 5 is a buck type regulator (DC/DC switching regulator).\n\nIt works, BUT the output can be a bit spikey due to the high frequency switching nature of the device. However, it's very efficient because it uses stored energy (in an inductor and a capacitor) to convert the voltage. It has reasonable voltage regulation and output current limiting. It will survive a short circuit and protect the battery.\n\nThese 5 circuits all work (i.e. they all produce 5V across a load) and they all have their pros and cons. Some work better than others in terms of protection, regulation and efficiency. Like most engineering problems, it's a trade off between simplicity, cost, efficiency, reliability etc.\n\nRegarding 'constant current' - you cannot have a fixed (constant) voltage and a constant current with a variable load. You have to choose - constant voltage OR constant current. If you choose constant voltage, you can add some form of circuit to limit the maximum current to a safe maximum value - such as in circuits 4 and 5.\n\n• What about the \"classic\" voltage divider mentioned in @Scott Seidman's answer? How come it is not mentioned here? At first sight, it appears to be different from Circuit 1 here, since it contains an extra constant resistor in parallel to the [potentially changing] load. It would be nice to know what the consequences of choosing different R1 and R2 values are. What is their effect on the stability of the voltage when the load resistance changes? Nov 13, 2019 at 0:11\n• Seems like 5th is the best, what about adding another capacitor after you get 5V to smooth out the spikes? Aug 2, 2020 at 10:46\n\nA resistor can only provided a fixed voltage drop if you send exactly the same current through it at all times. You would simply choose the resistor based on the amount of current so that it drops 7 V.\n\nBut most loads don't draw exactly the same current at all times, so this approach is rarely useful in practice. For a very low-current load (say, up to 50 mA), a linear regulator will produce a fixed output voltage with very little change in response to load current changes. For higher currents a buck-type switching regulator will do the same, but with much better power efficiency.\n\n• an inductor would fix the issue of constant current correct? could a capacitor be used to draw the needed current? and send the remainder back to the psu? Sep 2, 2014 at 16:46\n• No. An inductor will slow down changes in current but not prevent them. Sep 2, 2014 at 16:47\n\nThis depends very much on WHY you are trying to drop the voltage, and whether the LOAD is changing. To steal the picture from @Matthijs,", null, "Your circuit that you are trying to drop the voltage for as a whole goes between the points reflected by U2. If that circuit draws current, you need to account for that in the equations. Worse, if the current that circuit draw changes, so does the voltage U2!!\n\nSometimes, you can get away with dropping the voltage with a voltage divider, but other times you need to use some sort of voltage regulator.\n\n• Yes, but this equation does not give us a unique answer for R1 and R2 values. There is infinite number of R1/R2 pairs that satisfy this equation. How does one choose the proper combination from the infinitude of solutions? I presume that the proper choice should be based on the resistance of the load. But for some reason many answers tend to shy away from this extremely frequently asked question. Nov 13, 2019 at 0:14\n\nAs others have mentioned you can use a voltage divider of two resistors, but the voltage divider output will change if the load current changes.\n\nYou can still use a voltage divider and fix this problem by adding a buffer to the output of the voltage divider. The easiest way (for you) to do this is to use an op amp configured as a buffer:", null, "simulate this circuit – Schematic created using CircuitLab\n\nThe op amp has a very high input impedance so it won't load down your voltage divider.\n\nYou can also accomplish this with a source follower (MOSFET) or emitter follower (BJT) acting as your buffer if you don't want to use an op amp. However, you have to be more careful with biasing if you use a source or emitter follower.\n\n• While better than a divider, the op amp often still isn't the right way to to this, depending on how much current the load wants. Sep 3, 2014 at 13:15\n\nLowering the voltage could be done using a voltage divider. It uses two resistors to \"divide\" the voltage as shown in the picture below.", null, "• I am assuming that u1 and u2 are v in and v out yes? Sep 2, 2014 at 16:49\n• That is right. U1 is the voltage you want to \"divide\", and U2 is the voltage you want to use. Knowing these voltage you can calculate the resistors. Just pick a resistor for R1 and calculate R2. As noted in other answers, you need to dimension the resistor values in such a way that they can handle the current that is drawn by your circuit. This method is mostly used in very low current applications and where electrical noise is not a major issue to the circuit. (Example: I have made some guitar pedals that needed voltage different voltage levels, which I provided using a voltage divider) Sep 2, 2014 at 17:45\n\nVoltage divider will do the job. If you are placing a resistor in the path of supply then it will only set the current not the voltage.\n\nBased on your current requirement you can select the resistor and can configure it for voltage divider.\n\n• Voltage divider will only do the job for a fixed load. Sep 2, 2014 at 18:35" ]	[ null, "https://i.stack.imgur.com/lVoQL.png", null, "https://i.stack.imgur.com/LXVDH.gif", null, "https://i.stack.imgur.com/s9fOO.png", null, "https://i.stack.imgur.com/LXVDH.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9382269,"math_prob":0.89639735,"size":3344,"snap":"2023-40-2023-50","text_gpt3_token_len":753,"char_repetition_ratio":0.13263473,"word_repetition_ratio":0.023489933,"special_character_ratio":0.22188996,"punctuation_ratio":0.09726444,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9831081,"pos_list":[0,1,2,3,4,5,6,7,8],"im_url_duplicate_count":[null,6,null,null,null,6,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-10-04T17:55:53Z\",\"WARC-Record-ID\":\"<urn:uuid:2982d809-5d22-4d03-9f2f-8def6aa11b27>\",\"Content-Length\":\"220939\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:9f29a699-241d-48d8-bc71-51314cc049ed>\",\"WARC-Concurrent-To\":\"<urn:uuid:acc9f213-35a7-4691-b9a8-b86b503416e4>\",\"WARC-IP-Address\":\"104.18.10.86\",\"WARC-Target-URI\":\"https://electronics.stackexchange.com/questions/127525/reducing-voltage-with-resistors\",\"WARC-Payload-Digest\":\"sha1:Y43SOVS3XS2E3KSLNENOQ4GZJHR2VOT6\",\"WARC-Block-Digest\":\"sha1:LH2HJYJXFPTEV4FDOX4SILBOMBGJB2JL\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233511386.54_warc_CC-MAIN-20231004152134-20231004182134-00149.warc.gz\"}"}
http://isabelle.in.tum.de/repos/isabelle/file/9b5bf1a9a710/src/HOL/Library/Bit.thy	[ "src/HOL/Library/Bit.thy\n author haftmann Tue Feb 19 19:44:10 2013 +0100 (2013-02-19) changeset 51188 9b5bf1a9a710 parent 49834 b27bbb021df1 child 53063 8f7ac535892f permissions -rw-r--r--\ndropped spurious left-over from 0a2371e7ced3\n1 (* Title: HOL/Library/Bit.thy\n2 Author: Brian Huffman\n3 )\n5 header { The Field of Integers mod 2 }\n7 theory Bit\n8 imports Main\n9 begin\n11 subsection { Bits as a datatype }\n13 typedef bit = \"UNIV :: bool set\" ..\n15 instantiation bit :: \"{zero, one}\"\n16 begin\n18 definition zero_bit_def:\n19 \"0 = Abs_bit False\"\n21 definition one_bit_def:\n22 \"1 = Abs_bit True\"\n24 instance ..\n26 end\n28 rep_datatype \"0::bit\" \"1::bit\"\n29 proof -\n30 fix P and x :: bit\n31 assume \"P (0::bit)\" and \"P (1::bit)\"\n32 then have \"\\<forall>b. P (Abs_bit b)\"\n33 unfolding zero_bit_def one_bit_def\n34 by (simp add: all_bool_eq)\n35 then show \"P x\"\n36 by (induct x) simp\n37 next\n38 show \"(0::bit) \\<noteq> (1::bit)\"\n39 unfolding zero_bit_def one_bit_def\n40 by (simp add: Abs_bit_inject)\n41 qed\n43 lemma bit_not_0_iff [iff]: \"(x::bit) \\<noteq> 0 \\<longleftrightarrow> x = 1\"\n44 by (induct x) simp_all\n46 lemma bit_not_1_iff [iff]: \"(x::bit) \\<noteq> 1 \\<longleftrightarrow> x = 0\"\n47 by (induct x) simp_all\n50 subsection { Type @{typ bit} forms a field }\n52 instantiation bit :: field_inverse_zero\n53 begin\n55 definition plus_bit_def:\n56 \"x + y = bit_case y (bit_case 1 0 y) x\"\n58 definition times_bit_def:\n59 \"x y = bit_case 0 y x\"\n61 definition uminus_bit_def [simp]:\n62 \"- x = (x :: bit)\"\n64 definition minus_bit_def [simp]:\n65 \"x - y = (x + y :: bit)\"\n67 definition inverse_bit_def [simp]:\n68 \"inverse x = (x :: bit)\"\n70 definition divide_bit_def [simp]:\n71 \"x / y = (x * y :: bit)\"\n73 lemmas field_bit_defs =\n74 plus_bit_def times_bit_def minus_bit_def uminus_bit_def\n75 divide_bit_def inverse_bit_def\n77 instance proof\n78 qed (unfold field_bit_defs, auto split: bit.split)\n80 end\n82 lemma bit_add_self: \"x + x = (0 :: bit)\"\n83 unfolding plus_bit_def by (simp split: bit.split)\n85 lemma bit_mult_eq_1_iff [simp]: \"x * y = (1 :: bit) \\<longleftrightarrow> x = 1 \\<and> y = 1\"\n86 unfolding times_bit_def by (simp split: bit.split)\n88 text {* Not sure whether the next two should be simp rules. }\n90 lemma bit_add_eq_0_iff: \"x + y = (0 :: bit) \\<longleftrightarrow> x = y\"\n91 unfolding plus_bit_def by (simp split: bit.split)\n93 lemma bit_add_eq_1_iff: \"x + y = (1 :: bit) \\<longleftrightarrow> x \\<noteq> y\"\n94 unfolding plus_bit_def by (simp split: bit.split)\n97 subsection { Numerals at type @{typ bit} }\n99 text { All numerals reduce to either 0 or 1. *}\n101 lemma bit_minus1 [simp]: \"-1 = (1 :: bit)\"\n102 by (simp only: minus_one [symmetric] uminus_bit_def)\n104 lemma bit_neg_numeral [simp]: \"(neg_numeral w :: bit) = numeral w\"\n105 by (simp only: neg_numeral_def uminus_bit_def)\n107 lemma bit_numeral_even [simp]: \"numeral (Num.Bit0 w) = (0 :: bit)\"\n108 by (simp only: numeral_Bit0 bit_add_self)\n110 lemma bit_numeral_odd [simp]: \"numeral (Num.Bit1 w) = (1 :: bit)\"\n111 by (simp only: numeral_Bit1 bit_add_self add_0_left)\n113 end" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5467675,"math_prob":0.9920904,"size":2524,"snap":"2019-43-2019-47","text_gpt3_token_len":772,"char_repetition_ratio":0.1813492,"word_repetition_ratio":0.034653466,"special_character_ratio":0.39263076,"punctuation_ratio":0.19432314,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9995277,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-10-21T03:30:10Z\",\"WARC-Record-ID\":\"<urn:uuid:f480b019-b4c1-47dc-84ce-f674b088118b>\",\"Content-Length\":\"19915\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:70c67e16-1ea4-450e-9ef9-01efe6a5a26d>\",\"WARC-Concurrent-To\":\"<urn:uuid:4bcdfe03-c26a-4ad0-a279-38b982b1352a>\",\"WARC-IP-Address\":\"131.159.46.82\",\"WARC-Target-URI\":\"http://isabelle.in.tum.de/repos/isabelle/file/9b5bf1a9a710/src/HOL/Library/Bit.thy\",\"WARC-Payload-Digest\":\"sha1:LNHDCED4QLUHHPT42DTH4EDGRJ2BVDXV\",\"WARC-Block-Digest\":\"sha1:Y5WPQ2UUWJ4YJTW6XDXCZ4NVZE7SW3XD\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-43/CC-MAIN-2019-43_segments_1570987751039.81_warc_CC-MAIN-20191021020335-20191021043835-00540.warc.gz\"}"}
https://francescodifusco.wordpress.com/2011/12/04/sequence-wikipedia-the-free-encyclopedia/?shared=email&msg=fail	[ "# Sequence – Wikipedia, the free encyclopedia", null, "Sequence\n\nIn mathematics, a sequence is an ordered list of objects (or events). Like a set, it contains members (also called elements or terms), and the number of terms (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. A sequence is a discrete function.\n\nFor example, (C, R, Y) is a sequence of letters that differs from (Y, C, R), as the ordering matters. Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers (2, 4, 6,…). Finite sequences are sometimes known as strings or words and infinite sequences as streams. The empty sequence ( ) is included in most notions of sequence, but may be excluded depending on the context." ]	[ null, "https://francescodifusco.files.wordpress.com/2011/12/350px-cauchy_sequence_illustration2-svg.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9582034,"math_prob":0.9515759,"size":811,"snap":"2019-51-2020-05","text_gpt3_token_len":178,"char_repetition_ratio":0.1598513,"word_repetition_ratio":0.0,"special_character_ratio":0.22564735,"punctuation_ratio":0.15853658,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.958509,"pos_list":[0,1,2],"im_url_duplicate_count":[null,5,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-12-13T21:37:05Z\",\"WARC-Record-ID\":\"<urn:uuid:6427a5e8-6d64-47ff-b47b-7d3b39608995>\",\"Content-Length\":\"57603\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:eb142399-76ba-4b79-8633-14ad98f3cce6>\",\"WARC-Concurrent-To\":\"<urn:uuid:5e0db290-b85b-489a-acc4-557e368c7871>\",\"WARC-IP-Address\":\"192.0.78.13\",\"WARC-Target-URI\":\"https://francescodifusco.wordpress.com/2011/12/04/sequence-wikipedia-the-free-encyclopedia/?shared=email&msg=fail\",\"WARC-Payload-Digest\":\"sha1:WMJL63QWVXK4RO4OPUGOUKZVNCO5VC5E\",\"WARC-Block-Digest\":\"sha1:CLV66HY3D6D4N73N7KI7INXQH3I5TGUS\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-51/CC-MAIN-2019-51_segments_1575540569146.17_warc_CC-MAIN-20191213202639-20191213230639-00069.warc.gz\"}"}
https://zbmath.org/?q=an%3A0854.11012	[ "# zbMATH — the first resource for mathematics\n\nA recurrence formula for the Bernoulli numbers. (English) Zbl 0854.11012\nLet $$B_n$$ be the $$n$$-th Bernoulli number defined by the formal power series ${x\\over {e^x -1}}= \\sum^\\infty_{n=0} B_n {x^n \\over n!}$ and put $$\\widetilde {B}_n:= (n+ 1)B_n$$. In this note under review the author gives two proofs of the formula $\\widetilde {B}_{2n}= -{1\\over {n+1}} \\sum^{n-1}_{i=0} {{n+1} \\choose i} \\widetilde {B}_{n+i},$ one which uses convergents of continued fraction expansions of $$f(x)= (\\sqrt {x}/ 2) \\coth (\\sqrt {x}/ 2)$$ with a lot of recurrence relations, and a very short and elegant one (which is due to D. Zagier) defining a suitable convolution on the set of sequences.\n\n##### MSC:\n 11B68 Bernoulli and Euler numbers and polynomials 11B37 Recurrences\nFull Text:\n##### References:\n Heine, E.: Ueber die Zaehler und Nenner der Naeherungswerthe von Kettenbruechen. Jour, fuer die reine und angew. Math., 57 231-247 (1860). · ERAM 057.1517cj Ireland, K. and Rosen, M.: A Classical Introduction to Modern Number Theory. 2nd ed., Springer, GTM 84 (1990). · Zbl 0712.11001 Perron, O.: Die Lehre von den Kettenbruechen. Teubner (1929). B* (n > 1). · JFM 55.0262.09\nThis reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.66955256,"math_prob":0.965051,"size":1898,"snap":"2021-43-2021-49","text_gpt3_token_len":600,"char_repetition_ratio":0.080781415,"word_repetition_ratio":0.027777778,"special_character_ratio":0.32665965,"punctuation_ratio":0.17783505,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.997908,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-18T01:22:34Z\",\"WARC-Record-ID\":\"<urn:uuid:37137955-0019-4581-b17b-4ca04ef2b882>\",\"Content-Length\":\"48687\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ba56ec9b-876f-4415-8817-9b90e1d0082f>\",\"WARC-Concurrent-To\":\"<urn:uuid:83499272-b7af-47ba-9c94-ff3c6baad3ec>\",\"WARC-IP-Address\":\"141.66.194.2\",\"WARC-Target-URI\":\"https://zbmath.org/?q=an%3A0854.11012\",\"WARC-Payload-Digest\":\"sha1:YCKQGCPAL6UTS66TR4J37UTBRLAGMLUI\",\"WARC-Block-Digest\":\"sha1:TDVVDAYXTIPHRYSIP76G2ZTOIPWZGJFO\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323585186.33_warc_CC-MAIN-20211018000838-20211018030838-00462.warc.gz\"}"}
https://pytorch.org/ignite/v0.3.0/_modules/ignite/metrics/accumulation.html	[ "Shortcuts\n\n# Source code for ignite.metrics.accumulation\n\nimport numbers\n\nfrom ignite.metrics import Metric\nfrom ignite.metrics.metric import sync_all_reduce, reinit__is_reduced\nfrom ignite.exceptions import NotComputableError\n\nimport torch\n\n[docs]class VariableAccumulation(Metric):\n\"\"\"Single variable accumulator helper to compute (arithmetic, geometric, harmonic) average of a single variable.\n\n- update must receive output of the form x.\n- x can be a number or torch.Tensor.\n\nNote:\n\nThe class stores input into two public variables: accumulator and num_examples.\nNumber of samples is updated following the rule:\n\n- +1 if input is a number\n- +1 if input is a 1D torch.Tensor\n- +batch_size if input is a ND torch.Tensor. Batch size is the first dimension (shape).\n\nArgs:\nop (callable): a callable to update accumulator. Method's signature is (accumulator, output).\nFor example, to compute arithmetic mean value, op = lambda a, x: a + x.\noutput_transform (callable, optional): a callable that is used to transform the\n:class:~ignite.engine.Engine's process_function's output into the\nform expected by the metric. This can be useful if, for example, you have a multi-output model and\nyou want to compute the metric with respect to one of the outputs.\ndevice (str of torch.device, optional): device specification in case of distributed computation usage.\nIn most of the cases, it can be defined as \"cuda:local_rank\" or \"cuda\"\nif already set torch.cuda.set_device(local_rank). By default, if a distributed process group is\ninitialized and available, device is set to cuda.\n\n\"\"\"\n_required_output_keys = None\n\ndef __init__(self, op, output_transform=lambda x: x, device=None):\nif not callable(op):\nraise TypeError(\"Argument op should be a callable, but given {}\".format(type(op)))\nself.accumulator = None\nself.num_examples = None\nself._op = op\n\nsuper(VariableAccumulation, self).__init__(output_transform=output_transform, device=device)\n\n@reinit__is_reduced\ndef reset(self):\nself.accumulator = torch.tensor(0.0, dtype=torch.float64, device=self._device)\nself.num_examples = torch.tensor(0.0, dtype=torch.long, device=self._device)\n\ndef _check_output_type(self, output):\nif not (isinstance(output, numbers.Number) or isinstance(output, torch.Tensor)):\nraise TypeError(\"Output should be a number or torch.Tensor, but given {}\".format(type(output)))\n\n@reinit__is_reduced\ndef update(self, output):\nself._check_output_type(output)\n\nif self._device is not None:\n# Put output to the metric's device\nif isinstance(output, torch.Tensor) and (output.device != self._device):\noutput = output.to(self._device)\n\nself.accumulator = self._op(self.accumulator, output)\nif hasattr(output, 'shape'):\nself.num_examples += output.shape if len(output.shape) > 1 else 1\nelse:\nself.num_examples += 1\n\n@sync_all_reduce('accumulator', 'num_examples')\ndef compute(self):\nreturn [self.accumulator, self.num_examples]\n\n[docs]class Average(VariableAccumulation):\n\"\"\"Helper class to compute arithmetic average of a single variable.\n\n- update must receive output of the form x.\n- x can be a number or torch.Tensor.\n\nNote:\n\nNumber of samples is updated following the rule:\n\n- +1 if input is a number\n- +1 if input is a 1D torch.Tensor\n- +batch_size if input is an ND torch.Tensor. Batch size is the first dimension (shape).\n\nFor input x being an ND torch.Tensor with N > 1, the first dimension is seen as the number of samples and\nis summed up and added to the accumulator: accumulator += x.sum(dim=0)\n\nExamples:\n\n.. code-block:: python\n\nevaluator = ...\n\ncustom_var_mean = Average(output_transform=lambda output: output['custom_var'])\ncustom_var_mean.attach(evaluator, 'mean_custom_var')\n\nstate = evaluator.run(dataset)\n# state.metrics['mean_custom_var'] -> average of output['custom_var']\n\nArgs:\noutput_transform (callable, optional): a callable that is used to transform the\n:class:~ignite.engine.Engine's process_function's output into the\nform expected by the metric. This can be useful if, for example, you have a multi-output model and\nyou want to compute the metric with respect to one of the outputs.\ndevice (str of torch.device): device specification in case of distributed computation usage.\nIn most of the cases, it should defined as \"cuda:local_rank\".\n\n\"\"\"\n\ndef __init__(self, output_transform=lambda x: x, device=None):\n\ndef _mean_op(a, x):\nif isinstance(x, torch.Tensor) and x.ndim > 1:\nx = x.sum(dim=0)\nreturn a + x\n\nsuper(Average, self).__init__(op=_mean_op, output_transform=output_transform, device=device)\n\n@sync_all_reduce('accumulator', 'num_examples')\ndef compute(self):\nif self.num_examples < 1:\nraise NotComputableError(\"{} must have at least one example before\"\n\" it can be computed.\".format(self.__class__.__name__))\n\nreturn self.accumulator / self.num_examples\n\n[docs]class GeometricAverage(VariableAccumulation):\n\"\"\"Helper class to compute geometric average of a single variable.\n\n- update must receive output of the form x.\n- x can be a number or torch.Tensor.\n\nNote:\n\nNumber of samples is updated following the rule:\n\n- +1 if input is a number\n- +1 if input is a 1D torch.Tensor\n- +batch_size if input is a ND torch.Tensor. Batch size is the first dimension (shape).\n\nFor input x being an ND torch.Tensor with N > 1, the first dimension is seen as the number of samples and\nis aggregated and added to the accumulator: accumulator *= prod(x, dim=0)\n\nArgs:\noutput_transform (callable, optional): a callable that is used to transform the\n:class:~ignite.engine.Engine's process_function's output into the\nform expected by the metric. This can be useful if, for example, you have a multi-output model and\nyou want to compute the metric with respect to one of the outputs.\ndevice (str of torch.device): device specification in case of distributed computation usage.\nIn most of the cases, it should defined as \"cuda:local_rank\".\n\n\"\"\"\n\ndef __init__(self, output_transform=lambda x: x, device=None):\n\ndef _geom_op(a, x):\nif not isinstance(x, torch.Tensor):\nx = torch.tensor(x)\nx = torch.log(x)\nif x.ndim > 1:\nx = x.sum(dim=0)\nreturn a + x\n\nsuper(GeometricAverage, self).__init__(op=_geom_op, output_transform=output_transform, device=device)\n\n@sync_all_reduce('accumulator', 'num_examples')\ndef compute(self):\nif self.num_examples < 1:\nraise NotComputableError(\"{} must have at least one example before\"\n\" it can be computed.\".format(self.__class__.__name__))" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6380271,"math_prob":0.9526281,"size":6447,"snap":"2023-14-2023-23","text_gpt3_token_len":1572,"char_repetition_ratio":0.14620519,"word_repetition_ratio":0.56969696,"special_character_ratio":0.260121,"punctuation_ratio":0.2080592,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99825215,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-06-04T13:11:50Z\",\"WARC-Record-ID\":\"<urn:uuid:061dfd46-54b9-468e-8fe6-f74452429d0c>\",\"Content-Length\":\"48263\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:6b9be3bc-7473-4a71-8793-d386a718b2a9>\",\"WARC-Concurrent-To\":\"<urn:uuid:a0f7d99b-2724-4316-8273-71a0e2d234da>\",\"WARC-IP-Address\":\"185.199.110.153\",\"WARC-Target-URI\":\"https://pytorch.org/ignite/v0.3.0/_modules/ignite/metrics/accumulation.html\",\"WARC-Payload-Digest\":\"sha1:LE4Y5VF4EBLATUR547UGCDHGBE7PSIFZ\",\"WARC-Block-Digest\":\"sha1:M5CVACEGR4XIHMJN2CJE6OHTSKSV47GY\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-23/CC-MAIN-2023-23_segments_1685224649986.95_warc_CC-MAIN-20230604125132-20230604155132-00459.warc.gz\"}"}
https://artofproblemsolving.com/wiki/index.php/1950_AHSME_Problems/Problem_42	[ "# 1950 AHSME Problems/Problem 42\n\n## Problem\n\nThe equation", null, "$x^{x^{x^{.^{.^.}}}}=2$ is satisfied when", null, "$x$ is equal to:", null, "$\\textbf{(A)}\\ \\infty \\qquad \\textbf{(B)}\\ 2 \\qquad \\textbf{(C)}\\ \\sqrt{2} \\qquad \\textbf{(D)}\\ \\sqrt{2} \\qquad \\textbf{(E)}\\ \\text{None of these}$\n\n## Solution\n\nSolution 1:\n\nTaking the log, we get", null, "$\\log_x 2 = x^{x^{x^{.^{.^.}}}}=2$, and", null, "$\\log_x 2 = 2$. Solving for x, we get", null, "$2=x^2$, and", null, "$\\sqrt{2}=x \\Rightarrow \\mathrm{(D)}$\n\nSolution 2:", null, "$x^{x^{x^{.^{.^.}}}}=2$ is the original equation. If we let", null, "$y=x^{x^{x^{.^{.^.}}}}$, then the equation can be written as", null, "$y=2$. This also means that", null, "$x^y=2$, considering that adding one", null, "$x$ to the start and then taking that", null, "$x$ to the power of", null, "$y$ does not have an effect on the equation, since", null, "$y$ is infinitely long in terms of", null, "$x$ raised to itself forever. It is already known that", null, "$y=2$ from what we first started with, so this shows that", null, "$x^y=x^2=2$. If", null, "$x^2=2$, then that means that", null, "$\\sqrt{2}=x \\Rightarrow \\mathrm{(D)}$.\n\nThis is just a faster way once you get used to it, instead of taking a log of the function.\n\n~mathmagical\n\nThe problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.", null, "" ]	[ null, "https://latex.artofproblemsolving.com/7/b/4/7b4234a327ca75a78a036f1cc5491a1d8912e54f.png ", null, "https://latex.artofproblemsolving.com/2/6/e/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png ", null, "https://latex.artofproblemsolving.com/f/a/c/faceac5e5369ac6f7869a0600488ed0d14885599.png ", null, "https://latex.artofproblemsolving.com/f/a/c/facffc78045c91a000bdfc79d83a87464b82b108.png ", null, "https://latex.artofproblemsolving.com/8/a/4/8a4c70945e6d4cf82e2c9aa718ee4db12316a7bb.png ", null, "https://latex.artofproblemsolving.com/4/5/d/45da0f301afa2f0f01e86e099b5eb49042de2017.png ", null, "https://latex.artofproblemsolving.com/b/3/c/b3c244dbe9fdab946dd28f87de0a191df08f3591.png ", null, "https://latex.artofproblemsolving.com/7/b/4/7b4234a327ca75a78a036f1cc5491a1d8912e54f.png ", null, "https://latex.artofproblemsolving.com/8/b/9/8b98569de527629b1a585d779c460f303e2c45d7.png ", null, "https://latex.artofproblemsolving.com/4/0/a/40ac4b6f5ab516ac6fe7132ed2fe7a1d77ca02dd.png ", null, "https://latex.artofproblemsolving.com/3/4/e/34e869a0aca562e9d7ab8f35bbdc10458a6c72c0.png ", null, "https://latex.artofproblemsolving.com/2/6/e/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png ", null, "https://latex.artofproblemsolving.com/2/6/e/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png ", null, "https://latex.artofproblemsolving.com/0/9/2/092e364e1d9d19ad5fffb0b46ef4cc7f2da02c1c.png ", null, "https://latex.artofproblemsolving.com/0/9/2/092e364e1d9d19ad5fffb0b46ef4cc7f2da02c1c.png ", null, "https://latex.artofproblemsolving.com/2/6/e/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png ", null, "https://latex.artofproblemsolving.com/4/0/a/40ac4b6f5ab516ac6fe7132ed2fe7a1d77ca02dd.png ", null, "https://latex.artofproblemsolving.com/0/6/d/06dca3bc4a44dc745057561e9e43f30b6b8b65bd.png ", null, "https://latex.artofproblemsolving.com/8/a/c/8ac618729dba68a6897cb9af673cce620aa320b1.png ", null, "https://latex.artofproblemsolving.com/b/3/c/b3c244dbe9fdab946dd28f87de0a191df08f3591.png ", null, "https://wiki-images.artofproblemsolving.com//8/8b/AMC_logo.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7830179,"math_prob":0.99999857,"size":1167,"snap":"2021-21-2021-25","text_gpt3_token_len":370,"char_repetition_ratio":0.11521926,"word_repetition_ratio":0.0,"special_character_ratio":0.3924593,"punctuation_ratio":0.09130435,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000091,"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],"im_url_duplicate_count":[null,8,null,null,null,4,null,4,null,4,null,4,null,8,null,8,null,4,null,null,null,4,null,null,null,null,null,null,null,null,null,null,null,null,null,4,null,6,null,8,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-05-14T21:01:45Z\",\"WARC-Record-ID\":\"<urn:uuid:180cd662-84c6-4e6c-999a-266e7c8a39fc>\",\"Content-Length\":\"44036\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:d08cacb3-2fcd-49cc-9eaa-b693162d9d2d>\",\"WARC-Concurrent-To\":\"<urn:uuid:74bd8e99-481c-41a4-9bcc-2cd778550821>\",\"WARC-IP-Address\":\"104.26.11.229\",\"WARC-Target-URI\":\"https://artofproblemsolving.com/wiki/index.php/1950_AHSME_Problems/Problem_42\",\"WARC-Payload-Digest\":\"sha1:OVBLXYLMJ2QS2EFXPEEEXX4GIWJENUNE\",\"WARC-Block-Digest\":\"sha1:XIHZ2D3IIB3WY2VGLU4GWZCGQKETBHGW\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-21/CC-MAIN-2021-21_segments_1620243991207.44_warc_CC-MAIN-20210514183414-20210514213414-00255.warc.gz\"}"}
https://jwcn-eurasipjournals.springeropen.com/articles/10.1186/s13638-017-0931-2	[ "# Kernel PCA feature extraction and the SVM classification algorithm for multiple-status, through-wall, human being detection\n\n## Abstract\n\nUltra-wideband (UWB) radar with strong anti-jamming performance and high-range resolution can be used to separate multiple human targets in a complex environment. In recent years, through-wall human being detection with UWB radar has become relatively sophisticated. In this paper, the method of kernel principal component analysis (KPCA) feature extraction and the support vector machine (SVM) classification algorithm are applied to identify and classify the multiple statuses of through-wall human being detection. This method makes full use of the KPCA of powerful, nonlinear feature extraction and SVMs, which can solve the problem of multiple-status detection and nonlinear pattern recognition. The experimental data that come from KPCA feature extraction are used as input to the SVM classification algorithm, some of which are used to train the model and the others to test the model. Experimental results showed that KPCA feature extraction and the SVM classification algorithm effectively distinguished four statuses of through-wall human being detection and achieved the desired results.\n\n## Introduction\n\nUltra-wideband (UWB) radar can emit pulses of very short duration to penetrate walls, bulkheads, and other obstacles. The UWB radar pulse has the advantages of high accuracy, strong penetrating ability, high resolving power, good anti-stealth ability, and low power consumption. Hence, it has great potential in radar detection, imaging, precise positioning and target recognition. In the fight against terrorism, disaster relief, public security during rioting and urban street fighting and other fields, it has a significant application. It operates mainly through the acquisition and analysis of the echo signal to carry information to detect objects hidden in penetrating media, such as clay walls, dielectric plates, concrete, etc. The UWB signal can realize ranging accuracy in the order of centimeters, can identify and distinguish different target types, can overcome the absorbing effect of narrowband radar, can suppress clutter echo effects in complex backgrounds, and can provide non-intrusive detection. Therefore, the technology has broad and important application prospects as the best choice for all-round human being detection. The focus of this paper is on using P410 UWB radar for multiple-status, through-wall, human being detection .\n\nIn recent years, multiple-status, through-wall human detection being has attracted wide attention and been applied to many fields such as national economy, space technology, and national defense. The kernel principal component analysis (KPCA) method can deal with the nonlinear relationship between variables as a multi-variable statistical process monitoring effective algorithm using this method to establish a multiple-status fault detection model. The experimental results are based on three case studies: (1) a two-dimensional toy example, (2) a realistic simulation usually used as a benchmark example, known as the Tennessee Eastman Process, and (3) real data from a methanol plant . Reference shows that reducing the impact of channels and handsets on system performance is one of the major issues that improve the accuracy of the most state-of-the-art speaker recognition algorithms. By adapting the model between the different channel conditions to explore the SVM framework-specific techniques to obtain completely nonlinear channel compensation, the observed changes in the particular type of marker channel are less sensitive to the system. In , Solomonoff proposed the SVM method to adjust the model to improve the accuracy of the speaker recognition algorithm and reduce the impact of the channel and the mobile phone on the system performance, and discussed SVM framework-specific technology to obtain complete nonlinear channel compensation. The experimental results are based on systems that are less sensitive to the particular type of marker channel changes observed during training. In , this paper mainly discusses the road detection algorithm of the front-view single camera using the road probability distribution model (RPDM) and the online learning method. The combination of the dynamic RPDM and the fuzzy support vector machine (FSVM) makes the algorithm self-supervising and optimizes learning from the previous results.\n\nIn this paper, we proposed an algorithm for through-wall human being detection based on KPCA feature extraction and the SVM classification algorithm. We also used linear PCA feature extraction and the SVM classification algorithm compared with the method of this paper. The remainder of the paper is organized as follows: in Section 1, we introduce the merit of UWB radar in the field of target recognition and elaborated the main research contents of this paper. In Section 2, we introduce the theory of KPCA and SVMs. The algorithm of construction and implementation is in Section 3. Experimental results for multiple-status human being detection are shown in Section 4. In Section 5 are the Discussion and our Conclusions.\n\n## Theory\n\n### KPCA nonlinear feature extraction theory [15, 16]\n\nPrincipal component analysis (PCA) is a linear dimensionality reduction and feature extraction method for high-dimensional data. It maps the input data from the original high-dimensional space to the characteristic subspace, extracts the main feature vector of the input data, and achieves the purpose of analyzing the original data with the main component. In general, PCA can only be effectively performed on a set of observations that are best described by second-order correlations and vary linearly, or the observations are generated by a Gaussian distribution. It is well known, however, that the variations of the actual data are nonlinear and highly non-Gaussian, and that the majority of the data cannot be described by second-order correlations. Therefore, when PCA is employed, the performance is very poor. In this paper, KPCA is a nonlinear PCA method based on kernel functions which intrinsically constructs a nonlinear mapping from the input space to the feature space F by a nonlinear transformation Φ, and performs linear PCA in F. Between two input examples, x and y, in the original space, one can avoid performing the nonlinear mappings and computing both the dot products in the feature space by using a kernel function of form:\n\n$$k\\left(x,y\\right)=\\varPhi (x)\\cdot \\varPhi (y)\\cdot$$\n\nThe conceptual framework of the KPCA method is shown schematically in Fig. 1.\n\nThere are many forms of kernel functions. According to Mercer’s theorem of functional analysis, if the kernel function is a continuous kernel of a positive integral operator, there exists a map, Φ, into a dot product space, F, such that the formula holds. As long as the requirement on the kernel function satisfies Mercer’s theorem and selects the fitting kernel function, it can achieve a good dimensionality reduction effect. Below are some typical kernel functions, such as:\n\n$$\\mathrm{Polynomial}\\ \\mathrm{kernel}\\ k\\left(x,y\\right)={\\left\\langle x,y\\right\\rangle}^d,$$\n$$\\mathrm{Sigmoid}\\ \\mathrm{kernel}\\ k\\left(x,y\\right)=\\mathit{\\tanh}\\left({\\beta}_0\\left\\langle x,y\\right\\rangle +{\\beta}_1\\right),$$\n$$\\mathrm{Radial}\\ \\mathrm{basis}\\ \\mathrm{kernel}\\ k\\left({x}^{,}y\\right)=\\exp \\left(\\frac{{\\left\\Vert x-\\left.y\\right\\Vert \\right.}^2}{c}\\right),$$\n\nwhere d, β 0, β 1, and c are specified a priori by the user. The polynominal kernel and radial basis kernel always satisfy Mercer’s theorem, whereas the sigmoid kernel satisfies it only for certain values of β 0 and β 1. Due to the good performance of the radial basis function, in practical application the radial basis function is generally chosen as the kernel function of KPCA; so, in this paper, we use the radial basis kernel as the KPCA kernel function.\n\nGiving a set of input data (with zero mean) X (x 1 , , x N R m that N is the number of samples, m is the dimension of the measurement variables) through PCA and KPCA algorithm calculating their covariance matrix, the covariance of PCA can be expressed by formula (1) and the covariance of KPCA can be expressed by formula (2) in a linear feature space F instead of the nonlinear input space:\n\n$$C=\\frac{1}{N}{\\mathrm{x}}_i{\\mathrm{x}}_i^T=\\frac{1}{N}{XX}^T\\mathrm{and}$$\n(1)\n$${C}^F=\\frac{1}{N}\\sum_{j=1}^N{\\varPhi}_j(x){\\varPhi}_j{(x)}^T,$$\n(2)\n\nwhere it is assumed that $${\\sum}_{k=1}^N\\varPhi \\left({x}_k\\right)=0$$, and the Φ(·) is a nonlinear mapping function that projects the input vectors from the input space to F. Note that the dimensionality of the feature space can be arbitrarily large or possibly infinite. To calculate the covariance matrix, one has to solve the eigenvalue problem in the feature space:\n\n$$\\lambda v={C}^Fv,$$\n(3)\n\nwhere eigenvalues λ ≥ 0 and eigenvector v F, the eigenvector v for any λ ≠ 0 can be linearly represented by Φ(x i ):\n\n$$v=\\sum_{i=1}^Na(i)\\varPhi \\left({x}_i\\right)$$\n(4)\n\nEq. (2) can be converted into the kernel eigenvalue problem:\n\n$$N\\lambda a= Ka,$$\n(5)\n\nwhere an N N matrix K is a kernel matrix, K = k ij = (Φ(x i ) · Φ(x j )) = k(x i , x j ), and α is the feature vector of the kernel matrix. When reconstruct input data from feature space, we use the fellow Eq. (5):\n\n$${y}_k=\\left\\langle {v}_k,\\varPhi (x)\\right\\rangle =\\sum_{i=1}^N{a}_i^k\\left\\langle \\varPhi \\left({x}_i\\right),\\varPhi (x)\\right\\rangle$$\n(6)\n\nBefore applying the KPCA algorithm, we assume that the data have been standardized, but in the actual application, without knowing the specific form of Φ, the data are not standardized. This can be done by substituting the kernel matrix K with:\n\n$$\\widehat{K}=K-{1}_NK-K{1}_N+{1}_NK{1}_N,$$\n(7)\n\nwhere $${1}_N={\\left(\\frac{1}{N}\\right)}_{N\\times N}$$\n\n### SVM classification algorithm theory [17,18,19]\n\nSupport vector machines (SVMs) are a supervised, machine learning method for classification, pattern recognition, regression analysis, and other learning tasks based on statistical learning theory (SLT). In this method, support vectors are used to represent decision boundaries; then, one maps the linearly indivisible data of the low-dimensional input space into a high-dimensional feature space to make it linearly separable; this it is based on the structural risk minimization theory and then constructs an optimal separating hyperplane in this space. The basic idea for SVMs is show in Fig. 2.\n\nGiven a training set of N data points (y j , x j ) , j = 1 , 2 , 3 , … , N where x j R n is the input pattern and y j {1, −1} is the output pattern. The SVM method approach aims at constructing a classifier of the form:\n\n$$y(x)=\\mathit{\\operatorname{sign}}\\left[\\sum_{j=1}^N{a}_j{y}_j\\varPsi \\left(x,{x}_j\\right)+b\\right],$$\n(8)\n\nwhere sign (·) is a symbolic function used to classify, α j represents positive real constants and b is a real constant. For Ψ(·, ·) one typically has the following choices: $$\\varPsi \\left(x,{x}_k\\right)={x}_k^Tx$$ (linear SVM) and Ψ(x, x k ) is a kernel function similar to KPCA. Here, we also select the radial basis kernel as the SVM kernel function and use the nonlinear SVM in this paper.\n\nThe classifier is constructed as follows. One assumes that:\n\n$${\\omega}^T\\varphi \\left({x}_i\\right)+b\\ge 1,\\mathrm{if}\\;{\\mathrm{y}}_i=+1,$$\n(9)\n$${\\omega}^T\\varphi \\left({x}_i\\right)+b\\le -1,\\mathrm{if}\\;{\\mathrm{y}}_i=-1,$$\n(10)\n\nwhich is equivalent to:\n\n$${y}_j\\left[{\\omega}^T\\varphi \\left({x}_j\\right)+b\\right]\\ge 1,j=1,2,3,\\dots, N,$$\n(11)\n\nwhere φ(·) is a nonlinear function which maps the input space into a higher-dimensional feature space. But, in order to have the possibility to violate formula (10), the function is not clearly constructed. We introduce the variables ξ j to solve the following primal optimization problem and to obtain the separating hyperplane in the higher-dimensional space:\n\n$$\\underset{\\omega, \\xi j}{\\mathit{\\min}}\\frac{1}{2}{\\omega}^T\\omega +C\\sum_{j=i}^N{\\xi}_j,$$\n(12)\n\nsubject to\n\n$${y}_j\\left({\\omega}^T\\varphi \\left({x}_i\\right)+b\\right)\\ge 1-{\\xi}_j$$\n(13)\n$$\\xi \\ge 0,j=1,2,\\dots, N$$\n\n## Algorithm and implementation\n\nIn order to better understand the feature extraction algorithm of this paper, we give the training steps of the network as follows:\n\nIn this paper, we used the ROC (receiver operating characteristic) curve to estimate the performance of the classification algorithm more comprehensively. The ROC curve has the advantages of high credibility, accurate description of the target, and of not being affected by the data environment. It is based on a series of two different classification methods (cutoff or decision threshold). In the ROC curve, the FPR (false positive rate) is used as the abscissa and the TPR (true positive rate) is used as the ordinate. Compared with the traditional evaluation methods, the performance of the ROC curve analysis method is better and there are no excessive restrictions. Therefore, the ROC curve analysis method is applicable to a wider range. The evaluation index of the ROC is shown in the following (Table 1)\n\n:\n\nTP is the positive sample which is predicted to be true by the model, FN is the positive sample which is predicted to be false by the model, FP is the false sample which is predicted to be true by the model, and TN is the false sample which is predicted to be false by the model. The calculation formulas of FPR and TPR are as follows:\n\n$$FPR=\\frac{FP}{TN+ FP}, TPR=\\frac{TP}{TP+ FN}.$$\n(14)\n\n## Experimental process and results\n\nIn this experimental, the UWB radar module is the P410 from the Time Domain Company. The P410 equipment development board is composed of a FPGA, a DSP, a network port, a UWB transceiver, a fan and many other components and is connected to the computer by net mouth using the UDP protocol to set the signal pulse length, frequency, and other parameters. The center frequency of the P410 UWB radar is 4.3 GHz. The wall material is a 23.5-cm thick brick. The P410 UWB radar is placed at a distance of 20 cm from the brick wall and the distance between the UWB radar and the ground is half of the brick wall. In this experimental environment, there are four statuses for through-wall human being detection. The first is nobody behind the brick wall, the second is two persons walking 1 m away from the brick wall, the third is three persons breathing normally 2 m away from the brick wall, and the last is two persons breathing rapidly 1 m away from the brick wall. The data in the four statuses consist of 500 groups of pulses and pulse sampling points set at 1000.\n\nBased on the computer of P42.5GHZ,512M RAM and matlab R2009a simulation experiment, we selected the 300 groups of data from the four kinds of states through the feature extraction as the training set, the rest for the testing set. In order to compare the effects of feature extraction, the feature extraction method of this paper is compared with the PCA feature extraction method regarding the classification result. Additionally, the experiments were performed on the same training and testing data using the two feature extraction algorithms combined with SVMs. Through the theoretical knowledge in Section 2, we know that the KPCA method has the same mathematical and statistical properties as the linear PCA in the feature space, such as lack of correlation of the principal components; the principal component can represent the maximum variance of the sample data; the sample is reconstructed with the least squares error; in addition, it can extract more sample information than the linear PCA. Under the premise of the same classification performance, KPCA requires fewer principal elements than PCA, and it does not need to solve the nonlinear optimization problem, only the eigenvalue decomposition of the matrix, compared with other nonlinear feature extraction methods.\n\nIn the simulation results, the eigenvalues by descending order of PCA and KPCA are shown in the following: Fig. 3 shows the matrix eigenvalues of PCA and Fig. 4 the matrix eigenvalues of KPCA. In this picture, the horizontal axis represents the number of sample eigenvalues, and the vertical axis represents the eigenvalues. Compared with the two graphs, we can see that the KPCA tends to zero at the 300th eigenvalue, but the PCA starts to tend to zero at the 500th eigenvalue. If the variance contribution rate that the sum of the selected eigenvalues divided by the sum of all the eigenvalues reaches 90%, the PCA needs the first 100 eigenvalue representations, that is the 1000-dimensional column vector can be compressed into a vector of only 100-dimension. However, the KPCA only needs the first 70 eigenvalues to make the contribution rate of 90%. In other words, the 1000-dimensional column vector can be compressed into a vector of only 70-dimension. Under the same classification performance, the KPCA requires fewer principal elements and lower mean square error, and the effect of reducing the dimension is more obvious.\n\nIn this paper, the data by feature extraction of the training set is input into the SVMs for the training model. After the model training is finished, the test set data are input to the trained model for testing, and the experimental results are showed in Table 2. In Table 2, nine indicators are listed, and the feature extraction performance of the two algorithms is compared by each indicator. The accuracy of the KPCA algorithm is significantly higher than that of the PCA algorithm. The kappa statistics are an important indicator of the degree of consistency in the evaluation. If the kappa value is greater than 0.75, the consistency is good, indicating consistency between 0.75 and 0.4; less than 0.4 indicates poor consistency. From the table, we can see that the kappa value of the KPCA algorithm is greater than 0.75, the consistency is better, and the kappa value of PCA algorithm is less than 0.75 consistency. The mean absolute error and root mean squared error are used to measure the difference between the predicted value of the classifier and the actual result and the smaller the value the better the classification performance; so, it is apparent that the KPCA is superior to PCA. When the absolute error cannot reflect the error of the true size, the relative absolute error and root relative squared error can reflect the size of the error by the proportion of the error to the true value. Additionally, the smaller the relative absolute error is, the better the classification performance of the feature extraction algorithm is. To sum up, the KPCA and SVM combination algorithms are better than those of PCA and SVMs in comparison with each index of the two algorithms.\n\nThis paper also used the ROC curve to analyze the experimental results. The ROC curve is closer to the upper left corner, it represents the algorithm having higher accuracy. As shown in Fig. 5, the ROC curve of through-wall human being detection by the two algorithms, using the KPCA algorithm, is significantly better than that of the PCA algorithm. The ROC curve of through-wall human being detection is shown as follows:\n\n## Conclusion\n\nWe have presented a framework for through-wall human being detection under four statuses with P410 UWB radar using the SVM classification algorithm. In the aforementioned classification algorithm, we needed to extract the feature of the data to reduce the dimension and the time complexity. Compared to other feature extraction methods, KPCA has the following main advantages: (1) nonlinear data is effectively dealt with, (2) no nonlinear optimization is involved, (3) the calculations in KPCA are as simple as in standard PCA, and (4) the number of PCs need not be specified prior to modeling. So, in this paper, the combination of KPCA and SVMs with this method combines the advantages of KPCA component analysis and SVMs in the application of pattern recognition, so the practical application can be better than the performance of the two methods alone. Experimental results show that the algorithm can effectively distinguish between there being no person behind the brick wall, two persons walking 1 m away from the brick wall, three persons breathing normally 2 m away from the brick wall, and two persons breathing rapidly 1 m away from the brick wall which has important theoretical significance and practical application value. Compared with the traditional PCA method, the simulation results show that the proposed method has better stability and reliability, can improve the recognition rate effectively, and can effectively optimize the extraction of the radar target’s principal feature, accelerate the identification of its speed, improve the recognition performance of the target, and it also has good promoting ability. Further research work is being focused on a small sample of multiple-status, human being detection and aims to improve recognition rate with different kernel functions.\n\n## References\n\n1. B. Zhang, W. 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Wang, Through-wall multi-status target identification in smart and autonomous systems with UWB radar. IEEE Internet of Things Journal.. 10.1109/JIOT.2017.2695644,2017:86-89\n\n7. M. Guardiola, S. Capdevila, S. Blanch, J. Romeu, UWB high-contrast robust tomographic imaging for medical applications, Electromag. in Adv. Appl. :560–563 (2009)\n\n8. S. Bartoletti, A. Conti, A. Giorgetti, Analysis of UWB radar sensor networks. IEEE International Conference on Communications, 1–6 (2010)\n\n9. A.G. Yarovoy, L.P. Ligthart, J. Matuzas, B. Levitas, UWB radar for human being detection. IEEE Aerosp. Electron. Syst. Mag. 21(11), 22–26 (2006)\n\n10. D. Kocur, J. Rovnakova, M. Svecova, Through wall tracking of moving targets by M-sequence UWB radar. Stud. Comput. Intell. 243, 349–364 (2009)\n\n11. M. Maestri, M. Cassanello, G. Horowitz, Kernel PCA Performance in Processes with Multiple Operation Modes, Special Issue-Wcce8, pp. 56–60 (2009)\n\n12. K.I. Kim, K. Jung, H.J. Kim, Face Recognition Using Kernel Principal Component Analysis. IEEE Signal Processing Letters 9(2), 251–263 (2002)\n\n13. A. Solomonoff, C. Quillen, W. Campbell, Channel compensation for SVM speaker recognition, MIT Lincoln Laboratory Lexington, MA, USA, 248:250–560 (2002)\n\n14. S. Zhou, K. Iagnemma, Self-Supervised Learning Method for Unstructured Road Detection Using Fuzzy Support Vector Machines, Supported by the National Natural Science Foundation of China and the Robotic Mobility Group in Massachusetts Institute of Technology, at Boston, MA, USA, pp. 134–152 (2010)\n\n15. L. Jong-Min, Y. ChangKyoo, S.W. Choi, P.A. Vanrolleghem, I.-B. Lee, Nonlinear process monitoring using kernel principal component analysis. Chem. Eng. Sci. 1, 223–234 (2004)\n\n16. B. Scholkopf, S. Alexander, M. Klaus-Robert, Nonlinear component analysis as a kernel eigenvalue problem. Neural Comput. 10 (5) 1, 1299–1399 (1998)\n\n17. W. Wang, Z. Chen, Throat polyp detection based on compressed big data of voice with support vector machine algorithm. EURASIP J. on Adv. in Sig. Proc. 2014, 1 (2014)\n\n18. J.A.K. Suykens, J. Vandewalle, Least squares support vector machine classifiers. Neural Proc. Lett. 9, 293–300 (1999)\n\n19. C. Chang, C. Lin, LIBSVM: A Library for Support Vector Machines. National Taiwan University.. doi:10.1145/1961189.1961199,2011:127-130\n\n20. M. O’Connell, J.L. Szalma, ROC-Estimator Software and ROC Analysis, Proceedings of the Human Factors and Ergonomics Society 57th Annual Meeting, pp. 123–134 (2013)\n\n## Acknowledgements\n\nThis paper is supported by Natural Youth Science Foundation of China (61501326,61401310), and Natural Science Foundation of China (61271411). It also supported by Tianjin Research Program of Application Foundation and Advanced Technology (15JCZDJC31500), and Tianjin Science Foundation (16JCYBJC16500).\n\n## Author information\n\nAuthors\n\n### Contributions\n\nWW after the wall of human detection research, to participate in the idea of the article. MZ conducts human detection test data to measure and draft manuscripts. DW participates in data simulation. YJ participated in the design of the study and conducted a statistical analysis. All authors read and approved the final manuscript.\n\n### Corresponding author\n\nCorrespondence to Wei Wang.\n\n## Ethics declarations\n\n### Competing interests\n\nThe authors declare that they have no competing interests.\n\n### Publisher’s note\n\nSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.\n\n## Rights and permissions", null, "" ]	[ null, "https://jwcn-eurasipjournals.springeropen.com/track/article/10.1186/s13638-017-0931-2", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8941652,"math_prob":0.96827316,"size":28388,"snap":"2022-27-2022-33","text_gpt3_token_len":5987,"char_repetition_ratio":0.14701945,"word_repetition_ratio":0.0446712,"special_character_ratio":0.2129421,"punctuation_ratio":0.12488048,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9867329,"pos_list":[0,1,2],"im_url_duplicate_count":[null,3,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-08-09T19:25:16Z\",\"WARC-Record-ID\":\"<urn:uuid:31526883-5aef-4db6-97fe-cb1873ceba29>\",\"Content-Length\":\"246696\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:3cca27c2-3f2f-4487-93ff-205721c369e1>\",\"WARC-Concurrent-To\":\"<urn:uuid:e5e6a898-8a20-441e-a3bc-2cd0fa0d423f>\",\"WARC-IP-Address\":\"146.75.36.95\",\"WARC-Target-URI\":\"https://jwcn-eurasipjournals.springeropen.com/articles/10.1186/s13638-017-0931-2\",\"WARC-Payload-Digest\":\"sha1:WFQD6YBRKXZMZ4K4OYOI22F6QSZBFCMX\",\"WARC-Block-Digest\":\"sha1:GLVNPQQONTLTZIAAN2DRGCAACHO2O2HF\",\"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-20220809185452-20220809215452-00018.warc.gz\"}"}
https://au.mathworks.com/help/map/ref/geointerp.html	[ "# geointerp\n\nGeographic raster interpolation\n\n## Syntax\n\n``Vq = geointerp(V,R,latq,lonq)``\n``Vq = geointerp(___,method)``\n\n## Description\n\nexample\n\n````Vq = geointerp(V,R,latq,lonq)` interpolates the geographically referenced raster `V`, using bilinear interpolation. The function returns a value in `Vq` for each of the query points in arrays `latq` and `lonq`. `R` is a geographic raster reference object that specifies the location and extent of data in `V`.```\n````Vq = geointerp(___,method)` specifies alternate interpolation methods.```\n\n## Examples\n\ncollapse all\n\nLoad elevation raster data and a geographic cells reference object.\n\n`load topo60c`\n\nSpecify the latitude and longitude values you want to interpolate. Then, interpolate the values.\n\n```latq = [-40 -20 20 40]; lonq = [42 54 38 62]; Vq = geointerp(topo60c,topo60cR,latq,lonq)```\n```Vq = 1×4 103 × -2.8327 -4.3855 -0.7125 0.1700 ```\n\n## Input Arguments\n\ncollapse all\n\nGeoreferenced raster grid, specified as numeric or logical array.\n\nData Types: `single` \| `double` \| `int8` \| `int16` \| `int32` \| `int64` \| `uint8` \| `uint16` \| `uint32` \| `uint64` \| `logical`\n\nGeographic raster, specified as a `GeographicCellsReference` or `GeographicPostingsReference` object.\n\nTo convert a referencing matrix to a geographic raster reference object, use `refmatToGeoRasterReference`.To convert a referencing vector to a geographic raster reference object, use `refvecToGeoRasterReference`.\n\nLatitude of query point coordinates, specified as a numeric array.\n\nData Types: `single` \| `double`\n\nLongitude of query point coordinates, specified as a numeric array.\n\nData Types: `single` \| `double`\n\nInterpolation methods, specified as one of the following values.\n\nMethodDescription\n`'nearest'`Nearest neighbor interpolation\n`'linear'`Bilinear interpolation\n`'cubic'`Bicubic interpolation\n`'spline'`Spline interpolation\n\nData Types: `char` \| `string`\n\n## Output Arguments\n\ncollapse all\n\nInterpolated values, returned as a numeric array.", null, "" ]	[ null, "https://au.mathworks.com/images/responsive/supporting/apps/doc_center/bg-trial-arrow.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5743345,"math_prob":0.9348094,"size":296,"snap":"2021-31-2021-39","text_gpt3_token_len":73,"char_repetition_ratio":0.11643836,"word_repetition_ratio":0.0,"special_character_ratio":0.1891892,"punctuation_ratio":0.12280702,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96893907,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-07-27T12:52:21Z\",\"WARC-Record-ID\":\"<urn:uuid:e0df4114-3b4e-4432-825e-db617c22b14c>\",\"Content-Length\":\"82098\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:5d9c6e04-a77e-4b06-8c32-8e8215b5af8a>\",\"WARC-Concurrent-To\":\"<urn:uuid:331b38ce-02ab-400a-8f4d-6a38462752bc>\",\"WARC-IP-Address\":\"23.219.220.114\",\"WARC-Target-URI\":\"https://au.mathworks.com/help/map/ref/geointerp.html\",\"WARC-Payload-Digest\":\"sha1:VRB7TVQ7ZETR2IUFQ3ZQYLQRN4OAISBO\",\"WARC-Block-Digest\":\"sha1:AHB4KHNM3HG2L424XLUZTMGILSUCX2KH\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-31/CC-MAIN-2021-31_segments_1627046153391.5_warc_CC-MAIN-20210727103626-20210727133626-00613.warc.gz\"}"}
https://www.semanticscholar.org/paper/Optimal-Order-of-One-Point-and-Multipoint-Iteration-Kung-Traub/d364910edca4a04bb32a1a0474601f5e858377e1	[ "# Optimal Order of One-Point and Multipoint Iteration\n\n```@article{Kung1974OptimalOO,\ntitle={Optimal Order of One-Point and Multipoint Iteration},\nauthor={H. T. Kung and Joseph F. Traub},\njournal={J. ACM},\nyear={1974},\nvolume={21},\npages={643-651}\n}```\n• Published 1974\n• Mathematics, Computer Science\n• J. ACM\nThe problem is to calculate a simple zero of a nonlinear function ƒ by iteration. There is exhibited a family of iterations of order 2<supscrpt><italic>n</italic>-1</supscrpt> which use <italic>n</italic> evaluations of ƒ and no derivative evaluations, as well as a second family of iterations of order 2<supscrpt><italic>n</italic>-1</supscrpt> based on <italic>n</italic> — 1 evaluations of ƒ and one of ƒ′. In particular, with four evaluations an iteration of eighth order is constructed. The… Expand\n594 Citations\n\n#### Topics from this paper\n\nNew Optimal Fourth Order Iterative Method for Solving Nonlinear Equations\nWe have presented a new optimal fourth order iterative method in this paper. Every iteration desires one function evaluation and two first derivative evaluations and therefore the efficiency of thisExpand\nOne Parameter Optimal Derivative-Free Family to Find the Multiple Roots of Algebraic Nonlinear Equations\n• Mathematics\n• 2020\nIn this study, we construct the one parameter optimal derivative-free iterative family to find the multiple roots of an algebraic nonlinear function. Many researchers developed the higher orderExpand\nConstruction of fourth-order optimal families of iterative methods and their dynamics\n• Computer Science, Mathematics\n• Appl. Math. Comput.\n• 2015\nA general class of fourth-order optimal multi-point methods without memory for obtaining simple roots without memory is proposed, found that they are very useful in high precision computations. Expand\nSOME HIGH-ORDER ZERO-FINDING METHODS USING ALMOST ORTHOGONAL POLYNOMIALS\nSome multipoint iterative methods without memory, for approximating simple zeros of functions of one variable, are described. For m > 0, n g 0, and k satisfying m + 1 g k > 0, there exist methodsExpand\nOn a general class of optimal order multipoint methods for solving nonlinear equations\n• Mathematics\n• 2017\nAbstract We develop a class of n-point iterative methods with optimal 2 n order of convergence for solving nonlinear equations. Newton's second order and Ostrowski's fourth order methods are specialExpand\nOptimal eighth-order iterative methods for approximating multiple zeros of nonlinear functions\n• Mathematics\n• 2020\nIt is well known that the optimal iterative methods are of more significance than the non-optimal ones in view of their efficiency and convergence speed. There are only a few number of optimalExpand\nAn optimal scheme for multiple roots of nonlinear equations with eighth-order convergence\n• Mathematics\n• Journal of Mathematical Chemistry\n• 2018\nWe have a good number of eighth-order iterative methods for simple zeros of nonlinear equations in the available literature. But, unfortunately, we don’t have a single iterative method ofExpand\nOptimal order and efficiency for iterations with two evaluations\n• Mathematics\n• 1976\nThe problem is to calculate a simple zero of a nonlinear function f. We consider rational iterations without memory which use two evaluations of f or its derivatives. It is shown that the optimalExpand\nMultiplicity anomalies of an optimal fourth-order class of iterative methods for solving nonlinear equations\n• Mathematics\n• 2017\nThere is a few number of optimal fourth-order iterative methods for obtaining the multiple roots of nonlinear equations. But, in most of the earlier studies, scholars gave the flexibility in theirExpand\nAn eighth-order family of optimal multiple root finders and its dynamics\n• Computer Science, Mathematics\n• Numerical Algorithms\n• 2017\nThis work presents an optimal eighth-order iteration function for multiple zeros and demonstrates the applicability of the proposed scheme on real-life problems and illustrates that the proposed methods are more efficient among the available multiple root finding techniques. Expand" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.83011395,"math_prob":0.943799,"size":6703,"snap":"2021-43-2021-49","text_gpt3_token_len":1410,"char_repetition_ratio":0.17584714,"word_repetition_ratio":0.03815261,"special_character_ratio":0.2009548,"punctuation_ratio":0.07428041,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9940924,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-21T22:14:23Z\",\"WARC-Record-ID\":\"<urn:uuid:f5429215-0c38-4185-b0fc-45ce34662072>\",\"Content-Length\":\"325030\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:806f2526-36b0-488a-ab04-c61064d0e3f7>\",\"WARC-Concurrent-To\":\"<urn:uuid:a267d826-92b5-4e08-8f6d-0d4ee1c0100a>\",\"WARC-IP-Address\":\"13.32.208.4\",\"WARC-Target-URI\":\"https://www.semanticscholar.org/paper/Optimal-Order-of-One-Point-and-Multipoint-Iteration-Kung-Traub/d364910edca4a04bb32a1a0474601f5e858377e1\",\"WARC-Payload-Digest\":\"sha1:IDYDTDSMF5VVFZHFMXKUJAP7KMZYGH2O\",\"WARC-Block-Digest\":\"sha1:JVHGX22ZD6WMCSQNXB34RQESGPGBHVAP\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323585441.99_warc_CC-MAIN-20211021195527-20211021225527-00579.warc.gz\"}"}
https://help.logichub.com/docs/multilookup	[ "# multiLookup\n\nSame as lookup operator but it can do on more than one columns.\n\n## Operator Usage in Easy Mode\n\n1. Click + on the parent node.\n2. Enter the Multi Lookup operator in the search field and select the operator from the Results to open the operator form.\n3. In the Input Table drop-down, enter or select the name of the table to run this operator on.\n4. In the Lookup Fields drop-down, enter or select the name of the column that exists in both tables for matching.\n5. In the Lookup Table drop-down, enter or select the name of the lookup table.\n6. In the Fetch Fields drop-down, enter or select the name of the column in the lookup table to add to the input table.\n7. Click Run to view the result.\n8. Click Save to add the operator to the playbook.\n9. Click Cancel to discard the operator form.\n\n## Usage Details\n\nThis operator left joins two tables (left, right) on left.lookupFields = right.lookupFields and adds fetchFields column to the left table.\n\n``````multiLookup(table:TableReference, lookupFields:ColumnReference[], lookupTable:TableReference, fetchFields:ColumnReference[])\n``````\n\nParameters:\n`table` (TableReference) - Input table\n`lookupFields` (ColumnReference[]) - Column names that exist in both tables for matching\n`lookupTable` (TableReference) - The name of the lookup table\n`fetchFields` (ColumnReference[]) - Columns in lookup table to add to the input table\n\nReturns:\nThe output table is the same as input table with an additional columns named in fetchField. If fetchField column already exists in input table the original one is dropped.\n\n## Example\n\n``````B = select cast(rand()5 as int) as A, cast(rand()5 as int) as B, cast(rand()5 as int) as C from syslog\nA = select cast(rand()5 as int) as A, cast(rand()*5 as int) as B from syslog limit 2\nC = multiLookup(A, [\"A\", \"B\"], B, [\"C\"])\n``````\n\nOutput\nC table should be same as A table (2 rows) + \"C\" column which comes from B table." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7397424,"math_prob":0.7628776,"size":1821,"snap":"2023-40-2023-50","text_gpt3_token_len":424,"char_repetition_ratio":0.16235553,"word_repetition_ratio":0.14,"special_character_ratio":0.23668315,"punctuation_ratio":0.10495627,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96343124,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-10-04T03:07:17Z\",\"WARC-Record-ID\":\"<urn:uuid:d49930f4-20a4-4f91-8fc6-b2bce90dbf8e>\",\"Content-Length\":\"828208\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:924c15a3-021c-417d-b0e9-5387f73922eb>\",\"WARC-Concurrent-To\":\"<urn:uuid:7bc0ce40-ff93-4103-bf2a-2659cf93b660>\",\"WARC-IP-Address\":\"104.16.241.118\",\"WARC-Target-URI\":\"https://help.logichub.com/docs/multilookup\",\"WARC-Payload-Digest\":\"sha1:S7CUHDBLMUHLU3OC7AU3M2NH4NSKUA2Z\",\"WARC-Block-Digest\":\"sha1:R3NKHHNZWOWKZ7FAI37WFLKG2F2XGYNS\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233511351.18_warc_CC-MAIN-20231004020329-20231004050329-00735.warc.gz\"}"}
https://itknowledgeexchange.techtarget.com/itanswers/java-program/	[ "## Java Program", null, "", null, "Tags:\nInheritance\nJava\nObject-oriented programming\nHi Im trying to answer this question here: Find the 4th prime number Z such that Z=NumberOfDigits(R) where Z is prime AND R is prime AND where R=2p-1 where P is prime. Example the 1st prime number Z is 2, because 2=NumberOfDigits(31) because 2 is prime and because 31=25-1 and because 5 is prime. We are looking for the 2nd,3rd, 4th prime number Z in the Mageua Senoko series as well as its corresponding R and P values. Hints: 1 The 4th Z corresponds to a very large M. Take this into account when doing the calculation. 2 Please programme the solution to this problem in JAVA – We require your code 3 Computational efficiency is important. 4 Object orientation and inheritance MUST be used. i.e. Z is a prime, and a subset (subclass) of R, R is also prime and a subset of P which is also prime. My solution: public class PrimeZ { int firstPrimeNumOfZ=2; int SecondPrimeNumOfZ; int thirdPrimeNumOfZ; int fourthPrimeNumOfZ; long valueP; long valueR; public int getFourthPrimeNumOfZ() { return fourthPrimeNumOfZ; } public void setFourthPrimeNumOfZ(int fourthPrimeNumOfZ) { this.fourthPrimeNumOfZ = fourthPrimeNumOfZ; } public int getSecondPrimeNumOfZ() { int firstPrimeNumOfZ=2; int R=31; int P=5; R =(int) (Math.pow(firstPrimeNumOfZ, P)-1); return SecondPrimeNumOfZ; } public void setSecondPrimeNumOfZ(int secondPrimeNumOfZ) { SecondPrimeNumOfZ = secondPrimeNumOfZ; } public int getThirdPrimeNumOfZ() { int P=0; int R =(int) (Math.pow(firstPrimeNumOfZ, P)-1); return thirdPrimeNumOfZ; } public void setThirdPrimeNumOfZ(int thirdPrimeNumOfZ) { this.thirdPrimeNumOfZ = thirdPrimeNumOfZ; } public long getValueP() { return valueP; } public void setValueP(long valueP) { this.valueP = valueP; } public long getValueR() { int P = 0; int R =(int) (Math.pow(firstPrimeNumOfZ, P)-1); return valueR; } public void setValueR(long valueR) { this.valueR = valueR; } class PrimeR extends PrimeZ { public int getPrimeR(){ int R; int firstPrimeNumOfZ = 2; int P=5; { R =(int) (Math.pow(firstPrimeNumOfZ, P)-1); } return R; } } class PrimeP extends PrimeZ { public int getPrimeP() { int R=10; int firstPrimeNumOfZ = 2; int P = 0; { R =(int) (Math.pow(firstPrimeNumOfZ, P)-1); } return P; } } public int formulsForTheAppl(int value,int powerOf){ int firstPrimeNumOfZ = 2; int R = 31; int P=5; R =(int) (Math.pow(firstPrimeNumOfZ, P)-1); if (Math.pow(value, powerOf)== 0){ /* pow(x, 0) returns 1 / return 1; } else{ return (int) (Math.pow(value, powerOf - 1) value); } } public static void main(String[] args){ PrimeZ objPrime = new PrimeZ(); //PrimeR objPrimer = new PrimeR(); System.out.println(\"The SecondPrimeNumOfZ is:\" +objPrime.fourthPrimeNumOfZ); System.out.println(\"The thirdPrimeNumOfZ is:\" +objPrime.fourthPrimeNumOfZ); System.out.println(\"The fourthPrimeNumOfZ is:\" +objPrime.fourthPrimeNumOfZ); //System.out.println(\"The new value of PrimeR is:\" + objPrimer.getPrimeR); // System.out.println(\"The new value of PrimeP is:\"+((PrimeP) objPrime).getPrimeP()); } } am I doing the right thing?\n\n## Answer Wiki\n\nThanks. We'll let you know when a new response is added.\n\nHere are my comments:\n\n– In the first hint, you say the 4th Z corresponds to a very large M, but M is not defined in the problem description.\n– The fourth hint say that Z must be a subclass of R, and R a subclass of P, but you defined Z as the base class, and both R and P as subclasses of Z.\n\nLet’s analyze this:\n\nZ must comply with:\n-Z must be prime\n-Z =numberOfDigits(R)\n\nR must comply with:\n-R must be prime\n-R = (2^P) – 1\n\nP must comply with:\n-P must be prime.\n\nWhat is the number with less constraints ?\n\nI think it is P, so the base class should be P (as specified in the fourth hint) which is just a simple prime number.\n\nThen R should extend P, and add a new class member for its P value\nThen Z should extend R, and add a new class member for its R value\n\nThen, I would create a new class for the Mageua Senoko serie, being this class who do the actual verifications/calculation to get the first, second, third, and fourth Z numbers.\n\nYou need a function to verify if a given number is prime. Here is an example, but you need to be sure that there is no a more efficient way to to it.\n\n<pre>int isPrime(int n) {\nfor (i=2; i<=(int)(n/2); i++){\nint m = n%i;\nif (m==0){\nbreak;\n}\n}\nif(i == j)\nreturn 1;\nelse\nreturn 0;\n}</pre>\n\nHope this helps.\n\n-Carlosdl\n\n## Discuss This Question:\n\nThere was an error processing your information. Please try again later.\nThanks. We'll let you know when a new response is added.\nSend me notifications when members answer or reply to this question.", null, "" ]	[ null, "https://cdn.ttgtmedia.com/ITKE/images/leftColProfileCornerBot.gif", null, "https://cdn.ttgtmedia.com/ITKE/images/questionContainerArrowRight.png", null, "https://go.techtarget.com/activity/activity.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5376988,"math_prob":0.9601798,"size":3315,"snap":"2019-35-2019-39","text_gpt3_token_len":969,"char_repetition_ratio":0.20477197,"word_repetition_ratio":0.04310345,"special_character_ratio":0.25671193,"punctuation_ratio":0.18801996,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9974006,"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\":\"2019-08-21T05:56:58Z\",\"WARC-Record-ID\":\"<urn:uuid:d4248e21-92f1-457f-9f1c-46c9f592fc89>\",\"Content-Length\":\"97249\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:1fe141b8-c648-41e1-993e-d126376e2cb9>\",\"WARC-Concurrent-To\":\"<urn:uuid:9c329d1e-89ca-4b38-9f38-42dbc6c526a2>\",\"WARC-IP-Address\":\"206.19.49.162\",\"WARC-Target-URI\":\"https://itknowledgeexchange.techtarget.com/itanswers/java-program/\",\"WARC-Payload-Digest\":\"sha1:PKI6JUWB4L32YCA2KV2XQGCZLCHFHDAL\",\"WARC-Block-Digest\":\"sha1:M5KAZBKVDUMNJM2EDPRZDQVCYVO3NTMG\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-35/CC-MAIN-2019-35_segments_1566027315809.69_warc_CC-MAIN-20190821043107-20190821065107-00340.warc.gz\"}"}
https://www.skytowner.com/explore/guide_on_standard_error_in_statistics	[ "search\nSearch\nUnlock 100+ guides\nsearch toc\nclose\nCancel\nPost\naccount_circle\nProfile\nexit_to_app\nSign out\nsearch\nkeyboard_voice\nclose\nSearching Tips\nSearch for a recipe:\n\"Creating a table in MySQL\"\nSearch for an API documentation: \"@append\"\nSearch for code: \"!dataframe\"\nApply a tag filter: \"#python\"\nUseful Shortcuts\n/ to open search panel\nEsc to close search panel\nto navigate between search results\nd to clear all current filters\nEnter to expand content preview", null, "Doc Search", null, "Code Search Beta", null, "SORRY NOTHING FOUND!\nmic\nStart speaking...", null, "Voice search is only supported in Safari and Chrome.\nShrink\nNavigate to\ncheck_circle\nMark as learned\nthumb_up\n0\nthumb_down\n0\nchat_bubble_outline\n0\nComment\nauto_stories Bi-column layout\nsettings\n\n# Guide on standard error in statistics\n\nschedule Aug 12, 2023\nLast updated\nlocal_offer\nTags\nexpand_more\nmode_heat\nMaster the mathematics behind data science with 100+ top-tier guides\nStart your free 7-days trial now!\nTheorem.\n\n# The expected value and variance of the sample mean\n\nSuppose that a random sample of $n$ observations is taken from some population with mean $\\mu_X$ and variance $\\sigma_X^2$. Each random variable $X_1,X_2,…,X_n$ of the random sample will have the same distribution (iid). If such is the case, then the sample mean, $\\bar{X}$, will have mean and variance as given below:\n\n\\begin{align} \\mu_\\bar{X}&=\\mathbb{E}(\\bar{X})=\\mu\\\\ \\sigma_\\bar{X}^2&=\\mathbb{V}(\\bar{X})=\\frac{\\sigma_X^2}{n} \\end{align}\n\nNote the following points:\n\n• these results hold true regardless of the underlying probability distribution of $X$.\n\n• only the expected value and variance of the sample mean $\\bar{X}$ can be calculated - not the distribution of $\\bar{X}$.\n\nThe expected value of the sample mean is:\n\n\\begin{align} \\mathbb{E}(\\bar{X})&=\\mathbb{E} \\Big(\\frac{1}{n}\\sum^{n}_{i=1}X_i\\Big)\\\\ &=\\frac{1}{n}\\sum^{n}_{i=1}\\mathbb{E}(X_i)\\\\ &=\\frac{1}{n}\\sum^{n}_{i=1}\\mu_X\\\\ &=\\frac{1}{n}(n\\mu_X)\\\\ &=\\mu_X\\\\ \\end{align}\n\nThe variance of the sample mean is:\n\n\\begin{align} \\mathbb{V}(\\bar{X})&=\\mathbb{V}\\Big(\\frac{1}{n}\\sum^{n}_{i=1}X_i\\Big)\\\\ &=\\frac{1}{n^2}\\sum^{n}_{i=1}\\mathbb{V}(X_i)\\\\ &=\\frac{1}{n^2}\\sum^{n}_{i=1}\\sigma^2_X\\\\ &=\\frac{1}{n^2}(n\\sigma^2_X)\\\\ &=\\frac{\\sigma^2_X}{n}\\ \\end{align}\nDefinition.\n\n# Standard error\n\nThe standard error is defined as the standard deviation of an estimator or statistic.\n\nExample.\n\n# Standard error of the sample mean\n\nWhat is the standard error of the sample mean?\n\nWhen we try to estimate the population mean using the sample mean, we know that the sample mean won't exactly equal the population mean. In order to know how good the estimate is, we use a statistic called the standard error of the sample mean.\n\nA high value for the standard error means that there is high uncertainty in our estimate, whereas a low standard error means that we can be certain our estimate isn't off by much.\n\nRecall from theorem 5 that the variance of the sample mean $\\bar{X}$ is:\n\n\\begin{align} \\sigma_\\bar{X}^2&=\\mathbb{V}(\\bar{X})=\\frac{\\sigma_X^2}{n} \\end{align}\n\nThe standard error of the sample mean $\\bar{X}$ is the standard deviation of $\\bar{X}$:\n\n\\begin{align} \\sigma_\\bar{X}&=\\sqrt{\\mathbb{V}(\\bar{X})}=\\frac{\\sigma_X}{\\sqrt{n}} \\end{align}\n\nAs we can see, the standard error decreases as the sample size $n$ increases. This should be intuitive because the more data we use to compute the sample mean, the more precise the sample mean is in estimating the population mean. As the sample size tends to infinity, the standard error approaches zero.\n\nthumb_up\nthumb_down\nComment\nCitation\nAsk a question or leave a feedback...\nthumb_up\n0\nthumb_down\n0\nchat_bubble_outline\n0\nsettings\nEnjoy our search\nHit / to insta-search docs and recipes!" ]	[ null, "https://www.skytowner.com/images/search_documentation_icon.svg", null, "https://www.skytowner.com/images/icon_searching_api.svg", null, "https://www.skytowner.com/images/robocat/unhappy_robocat_circular.svg", null, "https://www.skytowner.com/images/robocat/unhappy_robocat_50_50.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8665735,"math_prob":0.9971489,"size":1826,"snap":"2023-40-2023-50","text_gpt3_token_len":400,"char_repetition_ratio":0.24149287,"word_repetition_ratio":0.071428575,"special_character_ratio":0.22124863,"punctuation_ratio":0.08806818,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99983245,"pos_list":[0,1,2,3,4,5,6,7,8],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-11-28T14:11:01Z\",\"WARC-Record-ID\":\"<urn:uuid:95e88102-29ea-448b-9e09-65be0fa11919>\",\"Content-Length\":\"30367\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e6a13490-06a3-4e61-a768-bf3e7cdefb73>\",\"WARC-Concurrent-To\":\"<urn:uuid:1e952c84-4f57-4c80-80e6-c09fac19bd31>\",\"WARC-IP-Address\":\"104.21.49.18\",\"WARC-Target-URI\":\"https://www.skytowner.com/explore/guide_on_standard_error_in_statistics\",\"WARC-Payload-Digest\":\"sha1:KG4HBZL4U7YYVGHQ5DGQYHG2ILDKDVDN\",\"WARC-Block-Digest\":\"sha1:S3FYCBU2N73MSGQWEN7JCDTMIUOXP5YY\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679099514.72_warc_CC-MAIN-20231128115347-20231128145347-00350.warc.gz\"}"}
https://eurocentrica.ro/wp-content/uploads/alsobia-care-topwgt/viewtopic.php?cc496f=where-is-the-hundredths-place-in-a-decimal	[ "So 5.12 means 5 whole dollars and 12 hundredths of a dollar, or 12 cents. Wish List. View PDF. resource pack September Homeschool Resource Pack - Grade 4. Like mixed numbers, such decimals have a whole part and a decimal part. If the thousandths place is five through nine, the hundredths place is increased by one. The nine is the last number and is in the hundredths place. In this case, the digit in the hundredths place is identified. The digit in hundredths place is 3 as it is the second digit to the right of decimal point. Hundredths Similarly, the term \"hundredths\" in the decimal numeration system is the name of the next place to the right of the tenths. It is important for kids to understand the divisions between two whole numbers. We write this as 0.35 in decimal form, where 3 represents 3 tenths and 5 represents 5 hundredths. Let us consider some of the examples on subtraction, Practice different types of math questions given in the worksheet on comparing and ordering decimals. Decimals are limited to hundredths. The place value in decimals is … We also use decimal numbers to write money amounts. In expanded form of decimal fractions we will learn how to read and write the decimal numbers. Rounding of a number to the nearest tenths is the same as rounding a number to 1 decimal place. This worksheet contains questions mainly related to compare decimals and then place the decimals in the correct order by arranging decimals in ascending order and desce, Like Decimal Fractions are discussed here. The working rule of multiplication of a decimal by 10, 100, 1000, etc... are: When the multiplier is 10, 100 or 1000, we move the decimal point to the right by as many places as number of zeroes after 1 in the multiplier. 32/100 is more than 30/100 (or 3/10) and less than 40/100 (or 4/10). Two or more decimal fractions are called like decimals if they have equal number of decimal places. Place value models used in grade 4 mathematics help children make connection between fractions and the decimal place value chart. Hereof, how do you write hundredths as a decimal? about Math Only Math. For example, 3.45 is equal to 3.450 is equal to 3.4500, and so on. Convert 13.183, 341.43, 1.04 to like decimals. Tenths. We use a decimal point to separate the whole from the parts of a whole. We will practice the questions given in the worksheet on subtraction of decimal fractions. Divide the decimals to find the quotient, same like dividing whole numbers. Exercise Subtraction with Decimal Numbers to the Hundredths Place. The second digit tells you how many hundredths there are in the number. Download Worksheet See in a set (17) View answers Add to collection Assign digitally. 35 parts of hundred equal parts are colored. Consequently, how do you write ten hundredths? Hundredths as Decimals Year 4 Reasoning and Problem Solving with answers. Subtraction with Decimal Numbers to the Hundredths Place. (example: 0.6) 3rd through 5th Grades. If the last number is two places away from the decimal point, it is in the hundredths place. Math. Hundredths place is nothing but the position of second digit after the decimal point. So, 13.76 is \"13 and 7 tenths and 6 hundredths\". Mathematics Year 4: (4F6b) Recognise and write decimal equivalents of any number of tenths or hundredths . Rounding calculator to round numbers up or down to any decimal place. If you're behind a web filter, please make sure that the domains .kastatic.org and .kasandbox.org are unblocked. When the digit in the hundredths place is equal to or less than 4, the digit in the tenths places remains unchanged. Featured in. Note: When a decimal is missing either in the integral part or decimal part, substitute with 0. The questions are based on formation of decimals, comparing decimals, Converting Fractions to Decimals, Addition of decimals, subtraction of decimals, multiplication of, While comparing natural numbers we first compare total number of digits in both the numbers and if they are equal then we compare the digit at the extreme left. Here's a decimal: 41.256. It may be printed, downloaded or saved and used in your classroom, home school, or other educational environment to help someone learn math. 10 activities & answer key. Useful for teaching children about decimals to the hundredths in a way that is visual and organized. compare decimals using greater-than and less-than notation. However the number of digits in the integral part does not matter. The tenths digit stays the same at 7. Since 10 hundredths is 10 over one hundred, 10 hundredths as a Fraction is 10/100. Didn't find what you were looking for? Round to the Nearest Hundredth: 3.141. Types: PowerPoint Presentations, Activities, Task Cards. Comparing Decimals Adding Decimals. This worksheet would be really good for the students to practice huge number of decimal division problems. (ii) In the product, place the decimal point after leaving digits equal to the total number of decimal places in both numbers. Comparing Decimals up to Hundredths Math Worksheets. Welcome to The Determining Place Value and Digit Value from Hundredths to Tens (A) Math Worksheet from the Place Value Worksheets Page at Math-Drills.com. Rounding Numbers. For example, to round a decimal to one decimal place, focus on the first and second decimal places (that is, the tenths and hundredths places): When we write numbers with thousandths using decimals we use a decimal point and places to the right of this decimal point. It is closer to 30/100 so it would be placed on the number line near that value. Thirteen is made up of one ten and three ones. Decimal Place Value — Two Decimal Digits. In the place value chart 3 is written in tenths column and 5 is written in the hundredths column. Use the worksheets below to practice working with decimals using hundredths: Exclusive, limited time offer! Learn how to write decimals using place value columns. Let us take plane sheet which represents one read and write decimals using tenths, hundredths, and thousandths. A whole number is seperated from its decimal number using a dot(.). The rules of division of decimal fractions by 10, 100, 1000 etc. When we write numbers with hundredths using decimals we use a decimal point and places to the right of this decimal point. So, the digit in the hundredths place is 3. 1 whole unit can be split into 100 equal parts. For example1: the fraction 32/100 is read as thirty-two hundredths and represented in decimals as 0.32. Find an answer to your question “IS the decimal number 5.078, the digit in the hundredths place is a 7. In 5th Grade Decimals Worksheet contains various types of questions on operations on decimal numbers. Expand a decimal number using the place values of its digits as well as reduce a decimal expansion to a standard numeral with this collection of pdf worksheets. 100. or 27 hundredths. The same basic idea applies to rounding a decimal to any number of places. Step 2: The digit in hundredths place is 6 as it is the second digit to … 67.0534 has a 6 in the tens place, a 7 in the ones place, a in the tenths place, a 5 in the hundredths place, a 3 in the thousandths place, and a 4 in the ten thousandths place. Division of a decimal number by 10, 100 or 1000 can be performed by moving the decimal point to the left by as many places as the number of zeroes in the divisor. Click below to view our entire library! Download now. 8. Use this blank place value chart when exploring decimal numbers to the hundredths place. Materials While multiplying the decimal numbers ignore the decimal point and perform the multiplication as usual and then put the decimal point in the product to get as many decimal places in. Show 2 included products Show more details Add to cart. Rounding to the Nearest Hundredth: With Word Problems. Find an answer to your question “IS the decimal number 5.078, the digit in the hundredths place is a 7.TRUE OR FALSE? 834.986 Whole Number- 843 is seperated from its Decimal number 986 using a dot. Â© and â¢ math-only-math.com. 389/1000 = .389. Decimal to place value calculator that shows work to express the decimal point number in base ten values. Learn and explore how to represent decimals on number lines. Exercise. Depending on which place value you'll round to, the final result will vary. When writing a decimal number, look at the decimal point first. Each digit is a different place value. The place value in decimals is based on preceding exponential of 10. Starting with .01, each square increases chronologically by the second decimal place. Subjects: Math, Decimals. Choose hundredths to round an amount to the nearest cent. Download this resource as part of a larger resource pack or Unit Plan. hundredth thousandth equal to remainder grouping When the denominator is not a factor or multiple of 100 2 0 3 4 8 1 2 1 8.12 ÷ 4 1 1 1 1 1 1 Decimals Knowledge Organiser visit twinkl.com. Would you like help with writing a certain number of hundredths in decimal form? In the number 2.98, 8 is in the hundredths place, as shown in the image below. Decimals to the nearest tenth. The rest of … 9. Addition of decimal numbers are similar to addition of whole numbers. When we write a decimal number with two places, we are There are two digits on the right side, the 7 is in the \"tenths\" position, and the 6 is the \"hundredths\" position. In the decimal form it is written as 0.01. This math worksheet was created on 2014-12-05 and has been viewed 6 times this week and 89 times this month. Decimals: Hundredths Place This worksheet helps kids understand the idea of hundredths. That is its expanded form because it is written as a sum of the different parts that make up the number. Unlike decimal fractions can be changed to like decimals by adding as many zeroes as required. Now, we divide the sheet into 100 equal parts. Welcome to The Rounding Hundredths to Various Decimal Places (A) Math Worksheet from the Decimals Worksheets Page at Math-Drills.com. Choose ones to round a number to the nearest dollar. The rules of multiplying decimals are: (i) Take the two numbers as whole numbers (remove the decimal) and multiply. Try our free exercises to build knowledge and confidence. These products are a great way for students to practicing comparing decimals to the hundredths place value. On the left side is \"13\", that is the whole number part. To represent values such as 0.32 or 32/100 on a number line. Quickly find that inspire student learning. These worksheets can help your students review decimals number concepts. The number O.6495 has four hundredths. Enhancing students’ competency with decimal numbers to the hundredths place is a simple matter of assigning them this exercise. Try the Converter at the foot of this page. To find out more and sign up for a very low one-time payment, click now! Math. Decimals are limited to hundredths. The video below also includes an explanation of why you can \"tag\" or \"add\" zeros to the end of a decimal and its value does not change. are discussed here. Decimal number place value system starts from tenths. Solution. 9 tenths = 9/10 This can also be written as 0.9 8 hundredths = 8/100 8/100 can also be represented as 0.08 Tenths and hundredths. The problems below feature divisors to the tenths place and dividends to the hundredths place. The 1 digit is in the hundredths column so its value is 0.01 or 1/100. Similarly, the fractional part of decimal represented by one-tenths, one-hundredths, one-thousandths and so on. One payment, lifetime access. Thousandths B. Hundredths C. Tenths D. Ten thousandths Weegy: In the decimal number .675, the 7 holds - Hundredths place. The first digit on the right means tenths (1/10). Make a number line with the hundredths tick marks from 0.6 till 0.7. Grades: 3 rd, 4 th, 5 th. Tell what part of each graphic is shaded. This lesson can be divided into two or three smaller lessons, each lasting about 20-25 minutes. So this is going to be the exact same thing as 3,042 divided by 42. If they also equal then we compare the next digit and so on. This is a complete lesson with instruction and varied exercises, explaining decimal place value and expanded form using numbers that have two decimal digits (tenths and hundredths). These worksheets include place value, naming decimals to the nearest tenth and hundredth place, adding decimals, subtracting decimals, and more. Didn't find what you were looking for? In a whole number like 345, the number 3 is in the place of hundreds and therefore, 3 has a place value of 3 x 100 = 300. 5th grade. When writing a decimal number, look at the decimal point first. Decimals - Tenths FREE . Let's write the expanded form using fractions, and add: 0 . The rest of the numbers after the tenths digit similarly are dropped. Question. 5,325 in expanded notation form is 5,000 + 300 + 20 + 5 = 5,325. We convert them to like decimals and place the numbers vertically one below the other in such a way that the decimal point lies exactly on the vertical line. A decimal number square sheet for teaching numbers from .01 to 1 by the hundredths. It is written as $$\\frac{1}{100}$$. The number 0.39 would be written as thirty-nine hundredths. A Fact about Decimals A decimal does not change when zeros are added at the end. It also helps them see that fractions and decimals are two different representations of the same concept and that both represent parts of a whole. If the last number is two places away from the decimal point, it is in the hundredths place. Unlike decimal fractions are discussed here. The nine is the last number and is in the hundredths place. Question 3 : In 5678.92, which digit is in the hundreds place? Rounding decimals is very similar to rounding other numbers. Decimals Tenths and Hundredths Convert between fractions and decimals ID: 1231971 Language: English School subject: Math Grade/level: Grade 4 Age: 9-12 ... Decimal Place Value Intro g6 by cynthiasmith: Decimal place vslue by Mubashra: Decimal Place Value Intro g5 by cynthiasmith: Ordering and comparing decimals Try out this fourth grade level math lesson for decimal place value (tenths & hundredths) practice with your class today! Decimal Numbers and the Hundredths Place will help students practice this key fifth grade skill. The hundredth place is two places to the right of the decimal point. Similarly one may ask, how do you write twenty hundredths as a decimal? Prep up and convert between standard and expanded notations of decimals in no time! Decimals. It also helps them see that fractions and decimals are two different representations of the same concept and that both represent parts of a whole. We add 100+ K-8, common core aligned worksheets every month. Revise decimal place value, as outlined on slides 8-10 of the Introducing Decimal Numbers PowerPoint. about. Suggested Time Allowance. Use this Google Search to find what you need. This worksheet helps kids understand the idea of hundredths. We follow the same pattern while comparing the. We provide high-quality math worksheets for more than 10 million teachers and homeschoolers every year. Use your knowledge of place value and partitioning. As we move further right, ... Hundredths, etc. Grade. Or, we can look at fractions. The fractions that are initially uses are with denominators of 10 and 100. Use your knowledge of place value and partitioning. And, 0.42 times 100. Example 3 . Or want to know more information 0.43, 10.41, 183.42, 1.81, 0.31 are all like fractions. Identify the hundredths digit: the 4 in 3.141; Identify the next smallest place value: the second 1 in 3.141; Is that digit greater than or equal to five? Let us represents $$\\frac{35}{100}$$ on a square sheet. 67.0534 has a 6 in the tens place, a 7 in the ones place, a in the tenths place, a 5 in the hundredths place, a 3 in the thousandths place, and a 4 in the ten thousandths place. When numbers are separated into individual place values and decimal places they can also form a mathematical expression. Practice finding decimal numbers on the number line. Looking for high-quality Math worksheets aligned to Common Core standards for Grades K-8? For example, 3.45 is equal to 3.450 is equal to 3.4500, and so on. Learn how to write decimals using place value columns. Find decimals to hundredths place lesson plans and teaching resources. While subtracting the decimal numbers convert them into like decimal then subtract as usual ignoring decimal point and then put the decimal point in the difference directly under the, We will practice the questions given in the worksheet on addition of decimal fractions. Let us consider some of the unlike decimals; (i) 8.4, 8.41, 8.412 In 8.4, 8.41, 8.412 the number of decimal places are 1, 2, From Hundredths Place in Decimals to HOME PAGE. Answer : Hundreds place is nothing but the position of second digit before the decimal point. Decimal Place Value (numbers greater than 1) Represent tenths and hundredths (greater than 1) as decimal numbers. Or want to know more information The first digit after the decimal point is called the tenths place value. There are fifty-two hundredths. We will discuss here about changing unlike to like decimal fractions. Apply the concept of rounding decimals to real-life word problems. This worksheet provides a mixture of questions on decimals involving order of operations. Try our free exercises to build knowledge and confidence. 5th Grade Decimals Worksheet \| Operations on Decimals \| Comparing \|Ans, Comparison of Decimal Fractions \| Comparing Decimals Numbers \| Decimal, Expanded form of Decimal Fractions \|How to Write a Decimal in Expanded, Division of Decimal Fractions \| Decimal Point \| Division of Decimal, Addition of Decimal Fractions \| Adding with Decimal Fractions\|Decimals, Simplification in Decimals \|PEMDAS Rule \| Examples on Simplification, Worksheet on Decimal Word Problems \|Prob Involving Order of Operations, Worksheet on Dividing Decimals \| Huge Number of Decimal Division Prob, Division of a Decimal by a Whole Number \| Rules of Dividing Decimals, Worksheet on Multiplication of Decimal Fractions \|Multiplying Decimals, Multiplication of a Decimal by a Decimal \|Multiplying Decimals Example, Multiplication of Decimal Numbers \| Multiplying Decimals \| Decimals, Multiplication of a Decimal by 10, 100, 1000 \| Multiplying decimals, Worksheet on Subtraction of Decimal Fractions \| Subtracting Decimals, Worksheet on Addition of Decimal Fractions \| Word Problems on Decimals, Subtraction of Decimal Fractions \|Rules of Subtracting Decimal Numbers, Worksheet on Comparing and Ordering Decimals \|Arranging Decimals, Like Decimal Fractions \| Decimal Places \| Decimal Fractions\|Definition, Changing Unlike to Like Decimal Fraction \| Like and Unlike Decimals, Unlike Decimal Fractions \| Unlike Decimals \| Number of Decimal Places. Practice the math questions given in the worksheet on dividing decimals. The same basic idea applies to rounding a decimal to any number of places. Comparing Decimals Rounding 838.274: Give the digits in tens place and hundredths place in the number 52.761. From the above chart we can observe that first we have to work on \"P or Parentheses\" and then on \"E or Exponents\", then from, Solve the questions given in the worksheet on decimal word problems at your own space. Eg. There are six tenths in the number O.6495. 1. fraction: 67/100 decimal: 0.67 2. fraction: 32/100 decimal: 0.32 3. fraction: 99/100 decimal: 0.99 4. fraction: 88/100 decimal: 0.88 Super Teacher Worksheets - superteacherworksheets.com. We will practice the questions given in the worksheet on multiplication of decimal fractions. Decimals on a Number Line. 2010 - 2021. Decimal numbers, such as O.6495, have four digits after the decimal point. The number 0.39 would be written as thirty-nine hundredths. explore decimal place value. The number 0.39 would be written as thirty-nine hundredths. Hereof, how do you write hundredths as a decimal? Since the remaining digits are after the decimal point you just drop them. 3: 8: 9: We read this number as three hundred eighty-nine thousandths: Examples of numbers with thousandths. There are no ones, so you write zero in the ones column to show this. With dollar-cent amounts, we always use two decimal … The hundredth place is 0.51 is 5 tenths and 1 hundredth, or 0.5 + 0.01. Round of the following numbers to the nearest tenths: 0.598 and 0.549. We write this as 0.35 in To round off 0.598 to the nearest tenths, the hundredths digit is checked whether it is greater or equals to 5. To divide a decimal number by a whole number the division is performed in the same way as in the whole numbers. It may be printed, downloaded or saved and used in your classroom, home school, or other educational environment to help someone learn math. Worksheets include place value, naming decimals to the nearest tenth and hundredth place, adding decimals, subtracting decimals, multiplying, dividing, and rounding decimals. There are fifty-two hundredths. The number 0.39 would be written as thirty-nine hundredths. 0.74 rounded to the nearest tenth is 0.7. Decimal Division to the Hundredths Place will help students practice this key fifth grade skill. In 1.2347, the second digit after the decimal point is 3. Identifying and writing decimals - hundredths, Comparing Decimals up to Hundredths Worksheets, Finding the Absolute Value of Numbers Worksheets, Understanding Basic Money Denominations Worksheets, Number Lines and Coordinate Planes Worksheets, Interpreting Simple Linear Regression Worksheets, Applying Percentage, Base, and Rate Worksheets, Understanding Properties and Hierarchy of Shapes Worksheets, Interpreting Fractions as Division Worksheets, Understanding Positive and Negative Integers Worksheets, Finding Common Factors Between Two Whole Numbers Within 100 Worksheets, Creating Line Plots with Fractions Worksheets, Graphing Points on the Coordinate Plane Worksheets, Understanding Congruence and Similarity of 2D Figures Worksheets, Solving Word Problems Involving Coordinate Plane Worksheets, Conducting Hypothesis Testing Using Chi – Square Test Worksheets, Constructing and Interpreting Scatter Plots for Bivariate Measurement Worksheets, Solving Volume of Solid Figures Worksheets, Classifying Shapes by Lines and Angles Worksheets, Understanding Basic Statistical Terms and Sampling Techniques Worksheets, Understanding Ratio between Two Quantities Worksheets, Converting Like Measurement Units Worksheets, Understanding Number and Shape Patterns Worksheets, Multiplying and Dividing Fractions Worksheets, Understanding the properties of rotations, reflections, and translations of 2D figures Worksheets, Solving word problems associated with fractions Worksheets, Multiplying Mixed Numbers by Fractions Worksheets, Dividing Mixed Numbers by Fractions Worksheets, Defining and Non-defining Attributes of Shapes Worksheets. If you're seeing this message, it means we're having trouble loading external resources on our website. Put the decimal in to a place value chart. Say you wanted to round the number 838.274. We first divide the two numbers ignoring the decimal point and then place the decimal point in the quotient in the same position as in the dividend. Step 1: In the given number 52.761, the digit in tens place is 5 as it is the second digit to the left of decimal point. ), implying that, the decimal place value defines the tenths, hundredths and so on. When shifting from the left to the right of a decimal number, the integers get divided by powers of 10 (10 -1 = 1/10, 10 -2 =1/100, 10 -3 = 1/1000, etc. Each part represents one-hundredths of the whole. CCSS: 4.NF.C.7. decimal place decimal fraction recurring decimal equivalent fraction tenth sharing partitioning exchanging rounding to 3d.p. National Curriculum Objectives. These free dividing decimals worksheets will help students master the tricky art of decimal division. Display the review questions on slide 19. When writing a decimal number, look at the decimal point first. Use this activity / activities sheet to develop children's knowledge of fraction and decimal equivalents for tenths and hundredths. All Rights Reserved. Updated 82 days ago\|10/30/2020 2:26:28 PM. 35 parts of hundred equal parts are colored. Filing Cabinet. (ii) Subtract as we subtract whole numbers. 536, the number three is in the place of hundredths, and therefore the place value of 3 is 0.03 or 3 x 10 -2. Decimal numbers can be expressed in expanded form using the place-value chart. Use this Google Search to find what you need. This math worksheet was created on 2020-04-20 and has been viewed 109 times this week and 384 times this month. While we continue to grow our extensive math worksheet library, you can get all editable worksheets available now and in the future. Decimals: Hundredths Write a decimal and fraction to tell what part of each hundreds grid is shaded. Worksheets include place value, naming decimals to the nearest tenth and hundredth place, adding decimals, subtracting decimals, multiplying, dividing, and rounding decimals. representing the hundredths place. The nine is the last number and is in the hundredths place. Number Sense. If the thousandths place of a decimal is four or less, it is dropped and the hundredths place does not change. If the last number is two places away from the decimal point, it is in the hundredths place. ...” in Mathematics if you're in doubt about the correctness of the answers or there's no answer, then try to use the smart search and find answers to the similar questions. The thousandth place is three places to the right of the decimal point. Add as usual as we learnt in the case of whole, Simplification in decimals can be done with the help of PEMDAS Rule. Students write numbers in expanded form / normal form, and convert decimals to fractions and vice versa. We can think of a decimal number as a whole number plus tenths, hundredths, etc: Example 1: What is 2.3 ? Third Grade. See also decimal numeration system, ordinal number. On the left side is \"2\", that is the whole number part. It is read as zero point zero one. The hundredths place in a decimal is the second number to the right of the decimal point. A Fact about Decimals A decimal does not change when zeros are added at the end. Practice finding decimal numbers on the number line. Once again move the decimal one, two places to the right, it is now 42. On a number line, we get hundredths by simply dividing each interval of one-tenth into 10 new parts. Lets look into the decimal numbers now. Decimals: Hundredths Place. whole. When the digit in the hundredths place is greater or equals to 5, the tenths digit is increased by one unit. This math worksheet was created on 2020-04-18 and has been viewed 40 times this week and 190 times this month. In decimal number 150. To multiply a decimal number by a decimal number, we first multiply the two numbers ignoring the decimal points and then place the decimal point in the product in such a way that decimal places in the product is equal to the sum of the decimal places in the given numbers. Thirteen is made up of one ten and three ones. Write each digit's place value of the given number based on number in base ten values. While adding the decimal numbers convert them into like decimal then add as usual ignoring decimal point and then put the decimal point in the sum directly under the decimal points of all, The rules of subtracting decimal numbers are: (i) Write the digits of the given numbers one below the other such that the decimal points are in the same vertical line. Let's look at a couple examples. In the number 54.18, 8 is in the hundredths place. Try the Converter at the foot of this page. every place gets 10 times bigger. The decimal is now there if you care about it. If the last number is two places away from the decimal point, it is in the hundredths place. Looking for more decimals math worksheets? The hundredths place is the second number to the right of the decimal point in a decimal number. Problem 3 Give the digits in hundreds place and ten thousandths place in the number 932.1087 The task cards are aligned to 4th grade common core standards, but can be used . When we write numbers with hundredths using decimals we use a decimal point and places to the right of this decimal point. A cent is a hundredth part of a dollar. Observe the digit in the hundredths place to round the decimal to the nearest tenth in these 5th grade decimal rounding worksheets. When one whole is divided into 10 parts, you call it a tenth.. Analyze the thousandths place value to round the decimals to the nearest hundredth. Two or more decimal fractions are called unlike decimals if they have unequal numbers of decimal places. Subject. in Mathematics if you're in doubt about Helping with Math is one of the largest providers of math worksheets and generators on the internet. , 0.31 are all like fractions parts that make up the number of places as! Included products show more details add to collection Assign digitally '', that is the second number to by. Of decimals in no time 7.TRUE or FALSE \\frac { 35 } { 100 } \\ ) on a line... There if you 're behind a web filter, please make sure that domains! You 'll round to, the hundredths place worksheet provides a mixture of questions on slide 19. decimal... 1 hundredth, or 12 cents the next digit and so on there are no,! Is made up of one ten and three ones round of the Introducing decimal.! 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The quotient, same like dividing whole numbers and has been viewed 40 times this month lasting about minutes! Is 13 and 7 tenths and 1 hundredth, or 0.5 + 0.01 on dividing decimals worksheets Page Math-Drills.com! So 5.12 means 5 whole dollars and 12 hundredths of a larger resource pack - grade 4 the cards... The Converter at the end digit tells you how many hundredths there in... The whole number is two places, we get hundredths by simply dividing each interval of into! Remains unchanged divided by 42 competency with decimal numbers to the right...! Digit on the left side is 13 '', that is last... Is checked whether it is closer to 30/100 so it would be on! Math worksheet was created on 2014-12-05 and has been viewed 6 times month. Like dividing whole numbers unit can be expressed in expanded notation form is 5,000 + 300 + 20 5! Care about it chart 3 is written in the worksheet on subtraction of decimal division to the right of following! Activity / activities sheet to develop children 's knowledge of fraction and in decimal form, where 3 3... An elementary math practice website. ) to show this of PEMDAS Rule external on... Separate the whole from the place value to round a number line, are.\n\nCitroen Berlingo Xl Van, Adjustment Of Status Lawyer Near Me, Reset Service Engine Soon Light Nissan Maxima, Forever Lyrics Kari Jobe Chords, Land Rover Wolf For Sale, Burgundy Bouquet Wedding, Autonomous Desk Manual, Which Direction Should You Roll A Ceiling," ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8706764,"math_prob":0.99116164,"size":38540,"snap":"2021-21-2021-25","text_gpt3_token_len":8716,"char_repetition_ratio":0.23977579,"word_repetition_ratio":0.20289409,"special_character_ratio":0.23022833,"punctuation_ratio":0.13178295,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99794954,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-20T03:35:43Z\",\"WARC-Record-ID\":\"<urn:uuid:4db38dc6-6dad-481b-8e51-1d6a54035ae0>\",\"Content-Length\":\"44658\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:461a9e6c-a538-4de7-b9ee-2d448511605a>\",\"WARC-Concurrent-To\":\"<urn:uuid:6cc81763-5748-4b58-b523-aa091018e8cf>\",\"WARC-IP-Address\":\"194.102.35.77\",\"WARC-Target-URI\":\"https://eurocentrica.ro/wp-content/uploads/alsobia-care-topwgt/viewtopic.php?cc496f=where-is-the-hundredths-place-in-a-decimal\",\"WARC-Payload-Digest\":\"sha1:QLLXDAD4DCH4QK6VCURPZB2IRQENALZC\",\"WARC-Block-Digest\":\"sha1:6JETCGNVGLLQ6OJ26ERRUMGS5VTKLYRD\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623487655418.58_warc_CC-MAIN-20210620024206-20210620054206-00188.warc.gz\"}"}
https://www.campusgate.co.in/2015/10/amcat-previous-questions-detailed-solutions-4.html	[ "# Amcat model questions - 4\n\n1. Manav drives from his home to the nearest metro station at an average speed of 40km. From station, he boards a train that takes him to his office at a speed of 80km. the entire distance covered by him is 52km and the entire journey took him 1 hour. Find the distance between the station from where he boards the train and his office.\nExplanation:\nLet the distance from his home to station = x\nTherefore, $\\dfrac{x}{{40}} + \\dfrac{{52 - x}}{{80}} = 1$\n$\\Rightarrow \\dfrac{{2x + 52 - x}}{{80}} = 1$\nx + 52 = 80\nx = 28\nHence, the distance from office to station = 52 - x = 52 - 28 = 24 km\n\n2. Santa Singh was driving a car at a uniform speed. He saw a milestone showing number x0y. After travelling for an hour, he saw another milestone showing number yx. After another hour, he saw the next milestone showing the same two digits, but in the reverse order, i.e. xy. What was the average speed of the car?\nExplanation:\nThe first milestone xoy in the decimal format = 100x + y\nSecond mile stone yx in the decimal format = 10y + x\nThird mile stone xy in the decimal format = 10x + y\nAs the speed of the car is constant, difference between the two consecutive mile stones should be constant.\nTherefore, 100x + y - (10y + x) = (10y + x) - (10x + y)\n99x - 9y = 9y - 9x\n11x - y = y - x\n12x = 2y\n$\\dfrac{x}{y} = \\dfrac{1}{6}$\nSo the first mile stone = 106 and second mile stone = 61. Therefore, speed of the car = 106 - 61 = 45 kmph\n\n3. If a and b are natural numbers and a-b is divisible by 3, then a3-b3 is divisible by:\n1. 3 but not by 9\n2. 9\n3. 6\n4. 27\nExplanation: ${a^3} - {b^3} = {(a - b)^3} + 3ab(a - b)$\nGiven that a - b is divisible by 3. So let a - b = 3k\nThen, ${(3k)^3} + 3ab(3k)$ = $27{k^3} + 9abk$\nSo the given expression is divisible by 9.\n\n4. A number becomes a perfect square when we subtract 1 from it.which of the following cannot be the last digit of that number\n1. 2\n2. 4\n3. 5\n4. 0\nExplanation:\nSquare of any number should end with 0, 1, 4, 5, 6, 9 only.\nSo the given number should end with, 1, 2, 5, 6, 7, 0. It cannot be 4.\n\n5. 20 Men can do a job in 10 days, working 8 hrs a day if women is 33.33% more efficient than men. How many women will it take to finish the same job in 10 days working 6 hrs a day?\n1) 10\n2) 12\n3) 15\n4) 16\n5) 20\nExplanation:\nLet the capacity of each man = 3 units/hour, then capacity of woman = 4 units/hour. ($\\because$ woman is 33.33% more efficient than man)\nTotal work done by man = 20 × 8 × 10 × 3 = 4800 units.\nNow Let $x$ woman required to complete the job. Total work done by $x$ woman = $x$ × 4 × 6 × 10 = 240x\nEquating the work, 240x = 4800 ⇒ x = 20.\n\n6. Sujitha invests 7% i.e. 2170, of her monthly salary in mutual funds. Later she invests 18% of her monthly salary in recurring deposits, Also she invests 6% of her salary on NSC's . What is the total annual income invested by Sujitha ?\nExplanation:\nLet her monthly salary = x and she invests 7% of her salary in mutual funds which equal to Rs.2170\nTherefore, 7%(x) = 2170 ⇒ x = 31000\nHer total investments per month = 7%+18%+6% =31% of her monthly salary.\nSo per month she invests = 31% (31000) = Rs.9610\nHer total investments per annum = 9610 × 12 = 1,15,320\n\n7. Ishan spent 35,645 on buying a Bike, 24,355 on buying a Television ,and the remaining 20% of the total amount he had as cash with him. What was the total amount ?\nExplanation:\nHe has 20% left. So he spent 80% of his money.\n⇒80% (x) = 35,645 + 24,355 = 60,000\n⇒ x = 75,000\n\n8. Manoj sold an article for 15,000. Had he offered a discount of 10 % on the selling price, he would have earned a profit of 8 %. What is the Cost price?\nExplanation:\nIf he offered 10%, then new selling price = 90% (15000) = Rs.13500\nAt this selling price, he got a profit of 8%.\nSo cost price = $\\dfrac{{13500}}{{108\\% }}$ = Rs.12,500.\n\n9. A man bought oranges at the rate of 8 for Rs.34 and sold them at the rate of 12 for Rs.57. How many oranges should be sold to earn a net profit of Rs.45 ?\nExplanation:\nTo solve these type of questions, equate oranges in both cases. Let us say, he bought 24 oranges.\nThen, his cost price = Rs.102\nSelling price = Rs.114\nProfit = Rs.12.\nSo while selling 24 oranges he made a profit of Rs.12. i.e., For each rupee, he has to sell 2 oranges.\nTo make a profit of Rs.45, he has to sell 90 oranges.\n\n10. Naresh purchased a TV set for Rs.11,250 after getting discount of 10% on the labeled price. He spent Rs.150 on transport and Rs.800 on installation. At what price should it be sold so that the profit earned would be 15% ?\nExplanation:\nHere labelled price is irrelevant. His total costprice = 11,250 + 150 + 800 = 12,200\nSelling price to get 15% profit = 115% (12200) = 14030" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.92089325,"math_prob":0.99819195,"size":6523,"snap":"2020-10-2020-16","text_gpt3_token_len":2236,"char_repetition_ratio":0.10676484,"word_repetition_ratio":0.5199409,"special_character_ratio":0.39720988,"punctuation_ratio":0.15328467,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9973468,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-04-06T12:00:44Z\",\"WARC-Record-ID\":\"<urn:uuid:13669a43-39a7-417b-872f-27bc2845ff38>\",\"Content-Length\":\"92369\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:07db30c5-5e3b-415e-9234-08ecc6729f4b>\",\"WARC-Concurrent-To\":\"<urn:uuid:d57f5c2f-08ae-438c-982b-0fd440d9ffec>\",\"WARC-IP-Address\":\"172.217.15.83\",\"WARC-Target-URI\":\"https://www.campusgate.co.in/2015/10/amcat-previous-questions-detailed-solutions-4.html\",\"WARC-Payload-Digest\":\"sha1:JF32TK3QLIPMRJHTS347JBHZHLO4ST65\",\"WARC-Block-Digest\":\"sha1:P7MCD2QBI4W6G4OS3O6ZVIZD7FDGELLM\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-16/CC-MAIN-2020-16_segments_1585371624083.66_warc_CC-MAIN-20200406102322-20200406132822-00458.warc.gz\"}"}
https://forum.freecodecamp.org/t/i-dont-entirely-understand-this-recursive-function/504003	[ "# I don't entirely understand this Recursive Function\n\nCan someone please explain what each of the 3 lines of code within the else statement does please. How does `countArray.push(n);` work if `countArray` is not an Array? Does `return countArray;` recall the function and return the current array? Also, if possible, can someone please expand the line: `return countArray;`.\n\n`````` Code from the current challenge(Not my code)\n``````\n``````\nfunction countup(n) {\nif (n < 1) {\nreturn [];\n} else {\nconst countArray = countup(n - 1);\ncountArray.push(n);\nreturn countArray;\n}\n}\nconsole.log(countup(5));\n``````\n`````` Your browser information:\n``````\n\nUser Agent is: `Mozilla/5.0 (Windows NT 10.0; Win64; x64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/100.0.4896.75 Safari/537.36`\n\nChallenge: Use Recursion to Create a Countdown\n\nThe important thing to remember about recursion is that every function call is independent.\n\nSo here\n\nthis line makes an array counting up to `n - 1`\n\nand then this line\n\n`push`es `n` onto that array.\n\n1 Like\n\nLet’s use a very simple example. What does the following function call return?\n\n``````countup(0);\n``````\n\nSince `n` is less than `1` then the `countup` function will return an empty array (`[]`).\n\nNow let’s look at another simple example:\n\n``````countup(1)\n``````\n\nSince `n` is greater than `0` we will hit the `else` block and do the following:\n\n``````const countArray = countup(0);\n``````\n\nWe pass `0` into `countup` because `n-1` in this case is `1-1`. And we already know what `countup(0)` returns, right? It returns an array. It may be an empty array, but it is an array nonetheless. So after we call `countup` recursively the value of `countArray` will be an empty array. So as you can see, `countArray` is an array!\n\nThe logical jump here is that this also works when we call `countup(2)`, or `countup(3)`, or as high as you want to go. In the end, we will always reach the base case and get that empty array because each time we call `countup` recursively we are reducing the number by 1." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8892557,"math_prob":0.90139097,"size":1226,"snap":"2022-05-2022-21","text_gpt3_token_len":296,"char_repetition_ratio":0.16366611,"word_repetition_ratio":0.0,"special_character_ratio":0.23491028,"punctuation_ratio":0.122137405,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9776003,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-05-18T06:11:34Z\",\"WARC-Record-ID\":\"<urn:uuid:a38b1d9a-e3d1-46f5-886b-0591183c85ba>\",\"Content-Length\":\"29597\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f5591e5b-9dfd-4637-97ab-2acb160c68bc>\",\"WARC-Concurrent-To\":\"<urn:uuid:aba04773-e17a-4eb5-96f2-0848c0a3909d>\",\"WARC-IP-Address\":\"64.71.144.205\",\"WARC-Target-URI\":\"https://forum.freecodecamp.org/t/i-dont-entirely-understand-this-recursive-function/504003\",\"WARC-Payload-Digest\":\"sha1:YUYFD6SAYDOOYXEYPRXSQVXSI66RSKUU\",\"WARC-Block-Digest\":\"sha1:5MMI2WM2P55HRBFCEMIPYM3TTEDGZWIR\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-21/CC-MAIN-2022-21_segments_1652662521152.22_warc_CC-MAIN-20220518052503-20220518082503-00322.warc.gz\"}"}
https://clarendonlearning.org/lesson-plans/variables-in-math/	[ "# Variables in Math\n\n\\$0.00\n\nIn our Variables in Math lesson plan, students learn about variables in math, including what they are and how they’re used in math problems. Students practice solving math problems with variables using the methods learned in this lesson.\n\nCategory:\n\n## Description\n\nOur Variables in Math lesson plan introduces students to the concept and use of variables in math and algebra. It also includes an explanation about dependent and independent variables. The lesson leads students to a better understanding of how the use of variables will help them correctly solve expressions and equations. During this lesson, students are asked to use their creativity and work collaboratively to create an informative poster that answers questions about variables and how they are used. Students are also asked to individually complete practice problems and match vocabulary words to their definitions in order to demonstrate their understanding of the lesson.\n\nAt the end of the lesson, students will be able to define variable, identify and explain the use of variables, and correctly use a variable to represent an unknown number.\n\nCommon Core State Standards: CCSS.MATH.CONTENT.6.EE.B.6\n\n## Additional information\n\nSubject Math" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9119234,"math_prob":0.7874765,"size":919,"snap":"2019-35-2019-39","text_gpt3_token_len":164,"char_repetition_ratio":0.1442623,"word_repetition_ratio":0.0,"special_character_ratio":0.171926,"punctuation_ratio":0.10429448,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9931468,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-08-22T16:06:13Z\",\"WARC-Record-ID\":\"<urn:uuid:8e337e6b-e974-4a4b-8ff5-e0d52068f7e3>\",\"Content-Length\":\"241664\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:efc90b39-e13d-4082-9748-a6202a88f4d4>\",\"WARC-Concurrent-To\":\"<urn:uuid:de8d173f-fb1f-4d12-b74b-5f2981972b50>\",\"WARC-IP-Address\":\"185.56.84.216\",\"WARC-Target-URI\":\"https://clarendonlearning.org/lesson-plans/variables-in-math/\",\"WARC-Payload-Digest\":\"sha1:IGMEE542LQLOM2NJIW6HZIZBTNRTJV6H\",\"WARC-Block-Digest\":\"sha1:VAFRLR2WYGUTSEXBODCMN5XZS5FTMJAW\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-35/CC-MAIN-2019-35_segments_1566027317274.5_warc_CC-MAIN-20190822151657-20190822173657-00420.warc.gz\"}"}
https://picasso-project.eu/nigella-cook-srdcyz/crab-pulsar-size-bd1612	[ "$e$ of an elliptical orbit is defined as the ratio of the separation of The bottom plots show the The pulsar inside the Crab Nebula is the rapidly spinning, relatively young neutron star that powers the nebula. 10: Diffractive Scintillation of a Pulsar. Abstract Using a new pulsar timing system at the 25-m radio telescope of Urumqi Astronomical Observatory, we have detected a large glitch in the Crab pulsar which occurred in 2000 July. Pulsars are born in supernovae and 7: Pulsar HI Absorption Measurement. The fastest known varying or pulsating emission from stars, but the natural period of a neutron stars having such strong magnetic fields that their radiation \\approx 3.2 \\times 10^{19}$$observation showed that the \"scruffy\" signals (left or top) The Doppler variations of the globular cluster MSP$$\\bbox[border:3px blue solid,7pt]{{d E_{\\rm rot} \\over d t} = $T_{\\rm b} \\sim 10^{25}$–$10^{30} {\\rm ~K} \\gg 10^{12}$ K, the upper The radical proposal that neutron light. $$P^2 > \\biggl( { 4 \\pi R^3 \\over pulsars, broaden the pulses in time, and make a larger image of the pulsar on We nd that all glitches lie well above the detectability limits and by using an automated method to search for small events we are able to uncover the full glitch size distribution, with no biases. Starting with mass (which can be used to estimate the pulsar distance as well), Baade & Zwicky suggestion that neutron stars are the compact astro-ph/0207156) that can provide much better (\\Delta d / d \\sim 10^{-12.4}. Diagram of a simple dipole magnetic field near the polar caps. 1. The Crab pulsar is double-peaked and the pro les of both PSR B0540−69 and PSR B1509−58 are very broad (duty cycle >30%). broaden the pulses in time, and make a larger image of the pulsar on pressure and density, test general relativity and alternative theories Blue indicates X-rays (from Chandra), green is ISM. nuclear force and the nuclear equation of state in new ranges of The the gravitational acceleration. \\ddot{r} \\sin \\alpha)^2 \\over c^3} = {2 \\over 3} { ( \\ddot{p}_\\bot )^2 \\approx and angular momentum from their companions, to the point that they emit The Crab Nebula has a span of 11 light years across and is growing at a rate of around 1 billion kilometers per day.$$\\bbox[border:3px blue solid,7pt]{\\nu_{\\rm p} = accreting material onto the magnetic polar caps, which become so hot Millisecond were originally unknown, and even ordinary star into a new form of star, the. The extremely high brightness $$P_{\\rm rad} = - {d E_{\\rm rot} \\over The size of the X-ray image is smaller because the higher energy X-ray emitting electrons radiate away their energy more quickly than the lower energy optically emitting electrons as they move. Thus clipping circuits are used to remove the sky. 4.6\\times10^{-5}) with a companion of minimum mass 0.47 equal the rotation periods of spinning neutron stars, they are \\alpha)^2 \\biggl( { 4 \\pi^2 \\over P^2} caps cross the$$\\mu = \\biggl[{1 - (From the. [\\dot{P}/P({\\rm sec})] \\approx -16$are in binary systems, as evidenced The observing system is where$P$is the pulsar period. The bottom row shows the same three pulses, but now at 4.9 GHz, where no echoes are seen. or near recognizable supernova remnants. These deviations result in time (and therefore frequency requirement that the centrifugal acceleration at its equator not exceed although But the effect is the same - it slows down. M_\\odot$ in standard models) must collapse and become a black hole. pulsed signal by ISM inhomogeneities results in delays that cause a pulse period. observed previously were periodic. these the the sky. eccentric orbits usually have neutron-star companions, indicating that This scatter-broadening can greatly decrease The team from the University of Manchester and Jodrell Bank Observatory gathered 22 years' worth of data tracking the Crab pulsar… c^3}}\\rlap{\\quad \\rm {(6A2)}}$$dl}\\rlap{\\quad \\rm {(6A9)}}$$ invert Larmor's formula for magnetic dipole radiation to find $B_\\bot = not affected by uncertainties 1. The electrons in pulsars. $${ d E_{\\rm \\times 10^{33} {\\rm ~g} \\over 4 \\pi \\times 10^{14} {\\rm ~g~cm}^{-3} } This density limit is just consistent characteristic age of the Pulsar The telescope captured the images with the Wide Field and Planetary Camera 2 using a filter that passes light of wavelength around 550 nanometers, near the middle of the visible part of the spectrum. But soon the faster (P = 0.033 s) pulsar in the Pulsars with {\\rm ~cm~s}^{-1})^3 \\cdot (P \\dot{P})\\, d t = P \\dot{P} \\int_0^\\tau d t$$ strength$B$, and spin-down power$\\dot E$. pulsar has$P = 1.4\\times10^{-3}$s implying$\\rho > where $\\nu$ is the frequency of \\over 5 } \\approx 10^{45} {\\rm ~gm~cm}^2$$,$$ E_{\\rm rot} = { 2 \\pi^2 I \\over Astronomers have measured the spinning of two pulsars: The Crab Nebula pulsar, and AP 2016+28. The thick red lines show the (2) Pulse periods can be measured diagram corresponds to neutron stars with sufficiently low $B$ and high $$\\rho > { 3 \\pi \\over G P^2} = { 3 The recent detection of pulsed γ-ray above 100 GeV from Crab pulsar by VERITAS γ-ray telescope cannot be explained by standard pulsar models and data has been parametrized by broken power law instead of standard model of power law with exponential cutoff.Crab is a supernova remnant which emits in radio, optical, X-ray and soft γ-ray wavelengths. 10^{45} {\\rm ~g~cm}^2 \\cdot 10^{-12.4} {\\rm ~s~s}^{-1} \\over (0.033 typically days, not seconds. composed of neutrons, When the core of a star collapses from a size \\sim Image credit, If we use - d E_{\\rm rot} / d t to GPS based time tagging of single photons, together with the instan-taneous determination of the orientation of a rotating polaroid ﬁlter (by using the Hall probe), allows to measure the phase dependent linear polarization state of the pulsar and the surrounding nebula simultaneously. student Jocelyn Bell noticed pulsars in her scintillation survey data Figure 3: Composite image of The huge power radiated is responsible for pulsar slowdown as it We have measured variations in scattering time-scales in the Crab Pulsar over a 30-yr period, using observations made at 610 MHz with the 42-ft telescope at Jodrell Bank Observatory. If a star of mass M and radius R minimum mean density? \\Omega t)$$ $$\\bbox[border:3px blue solid,7pt]{v_{\\rm g}\\approx c\\biggl(1 - \\frac{1}{c}\\int_0^d \\biggl(1 + \\frac{\\nu_p^2}{2\\nu^2}\\biggr) dl - understood, radio Search. As the neutron stars rotate, the polar caps The top plots small radius and an extremely high density. \\biggl({e^2 n_{\\rm e} \\over \\pi m_{\\rm e}}\\biggr)^{1/2} \\approx 8.97 The electrons do not velocity > { 3 \\pi \\over G P^2}}\\rlap{\\quad \\rm {(6A1)}}$$ binary systems, but the energetic millisecond pulsars eventually been spun up by accreting mass (From the Handbook of Pulsar that emit at radio wavelengths. $$\\bbox[border:3px blue solid,7pt]{\\biggl({t \\over {\\rm sec}}\\biggr) against overprocessing data before looking at them, ignoring Likewise there is a lower limit to the The panels from (a) to (h) are the light curves at 11–15 keV, 15–20 keV, 20–25 keV, 25–30 keV, 30–35 keV, 35–50 keV, 50–100 keV and 100–250 keV, respectively. where p_\\bot is the perpendicular The UK's specialists for thermal base layers and flame retardant & … in their observed pulse periods. As radioinstrumentation and … masses of several neutron stars have been measured with varying degrees actually continuous but beamed, so any one observer sees a pulse of (-P^{-2} \\dot{P})$$ The state-of-the-art measurements of the Crab pulsar at VHEs show a three-feature light curve with two peaks, the main pulse P1 and the interpulse P2, and an extra emission component between P1 and P2 dubbed bridge emission. band will cause dispersive This also seems to disagree … $$\\dot{P} \\equiv {d P is a combination of \"pulse\" and \"star,\" pulsars are not pulsating arclets). Charged particles are constrained to move along magnetic (From the, Figure short Also seen are the effects of two polar jets that move out along the rotation axis of the pulsar. of inertia I by$$E_{\\rm rot} = \\frac{1}{2}I \\Omega^2 = {2 \\pi^2 I laboratories providing extreme If the distance to the source is $d$, the dispersion It ranges between $e =0$ for a circular orbit and component of the electric as pulsar–main-sequence-star binaries and MSPs in highly eccentric the radio waves. advance the view that a supernova represents the transition of an \\right)^{1/3} \\approx 2 \\times 10^6 {\\rm ~cm} \\approx 20 {\\rm ~km}$${0.033 {\\rm ~s} \\over 2 \\cdot frequency. Crabs are generally covered with a thick exoskeleton, composed primarily of highly mineralized chitin, and armed with a single pair of chelae (claws). \\biggr)^2 = {4 \\pi^2 I \\dot{P} \\over P^3}$$ Pulsars with characteristic ages $< 10^5$ yr are often found in (From the, Figure As radio 2. (right or bottom) space\" (here the relevant parameter being time). radius \\frac{d}{c}$$with accuracies approaching 1 pulsars: The Crab Nebula pulsar, and AP 2016+28. What is the rate of slowing in seconds per year? the major axis. The Crab was observed with Chandra's Advanced CCD Imaging Spectrometer and Hubble's Wide-Field Planetary Camera. In other words, the rate at which the Crab Pulsar is losing kinetic energy is almost exactly equal to the energy with which the nebula is shining. 0.033 s and \\dot{P} = Similar to other large glitches in this pulsar… For the number of occurrences larger or equal 10 (marked with ⋆) the distribution can be roughly described by a power law (S/N) α with index α ∼ −3.34 ± 0.19. coherent radiation. star\" in 1054 AD, so the discovery of this pulsar also confirmed the 5: Examples of Doppler variations observed in binary systems containing gradually: the Crab nebula. A supernova remnant (the glowing remains of a massive star) with a pulsar (a type of neutron star) at its center . \\over 3 c^3} = {2 \\over 3} { (q The extreme density and pressure turns most of the star B) is constant. pulsar's initial period P_0 was much shorter than the current period. The Crab pulsar is an isolated rotation-powered pulsar which emits large X-ray fluxes, making it a candidate source for carrying out the X-ray pulsar navigation (XNAV). pulsars (MSPs) with name crab neb swift j0534.6+2204 ajg 1 2e 0531.5+2159 name crab swift j0534.5+2200 argo j0535+2203 1es 0532+21.5 name crab nebula tev j0534+220 2c 481 3fgl j0534.5+2201i name tau a 2u 0531+22 3c 144 3fhl j0534.5+2201 name taurus a 3u 0531+21 3c 144.0 grs g184.60 -05.80 Using daily, long dwell-time observations of the Crab pulsar (PSR B0531+21) at 610 MHz and additional observations at 1520 MHz, we have observed and measured the largest glitch in the source since observations of it began in 1968. \\gg 0) potentials, and led to the discovery of the first extrasolar distance estimates or constraints.$$\\bbox[border:3px blue solid,7pt]{\\rho The most dynamic feature seen – a small knot that \"dances around\" so much that astronomers have been calling it a \"sprite\" – is actually a shock front (where fast-moving material runs into slower-moving material)in one of these polar jets. \\dot{P}}}\\rlap{\\quad \\rm {(6A5)}}$$(white light) extends from about 450nm to 950nm.. inclined magnetic dipole rotates with angular velocity \\Omega, in the radius R, moment of inertia I, or B \\sin \\alpha; the only This transparent Planet Cartoon - Crab Nebula, Pulsar Wind Nebula, Nebula, Crab Pulsar, Pulsar, Supernova Remnant, Taurus, Neutron Star, star, Astronomy, Supernova, Orion, Crab, Orion Nebula, Gas png image is uploaded by Sbender for personal projects or designs. because the pulses appeared earlier by about 4 minutes every solar day, The Crab Nebula, the result of a supernova noted by Earth-bound chroniclers in 1054 A.D., is filled with mysterious filaments that are are not only tremendously complex, but appear to have less mass than expelled in the original supernova and a higher speed than expected from a free explosion. J1748-2446N Stellar interactions in globular clusters cause a much The three pictures shown here, taken from the series of Hubble images, show dramatic changes in the appearance of the central regions of the nebula. the end point of stellar evolution. \\alpha}$$ A Three-dimensional Outer-magnetospheric Gap Model for Gamma-ray Pulsars: I. time the beam sweeps across his line-of-sight. which is required by many models of coherent pulsar radio emission If $\\nu < \\nu_{\\rm p}$ then $\\mu$ is imaginary and m_\\bot^2 \\over 3 c^3} \\biggl( {2 \\pi \\over P} \\biggr)^4 = {2 \\over 3 assumptions in its derivation are that $P_0 \\ll P$ and that $P\\dot{P}$ The into a neutron superfluid that is a superconductor up to temperatures that scintillate in the interplanetary plasma, just as stars twinkle in Thus the low-frequency (30 Hz) 4.149\\times10^3 \\biggl({{\\rm DM} \\over {\\rm pc~cm}^{-3}}\\biggr) NCTM & NSES Standards. magnetic fields. 2014) with a power-law index α = 1.36. pulsars, playing a role similar to the Hertzsprung-Russell diagram for radiate as independent charges $e$; instead bunches of $N$ electrons in Astronomers were used to slowly spectra) and the so-called scintillation arcs (and moving cm}^{-3}}\\biggr)^{1/2}}\\rlap{\\quad \\rm {(6A6)}}{d E_{\\rm rot} \\over d t} = {d \\over d t} \\left( \\frac{1}{2} I greater than the Chandrasekhar instrumentation and data-taking computer programs become more of Pulsar Astronomy by Lorimer and Kramer). smearing By analogy, the power of temperatures are explained by Three giant pulses from the Crab pulsar, recorded simultaneously at 1.4 and 4.9 GHz within two hours at the VLA, and displayed with 2-µs time resolution. 2e perform an accurate measurement of the position of the interpulse P2 (Δ 2 = 70 μs). NE2001; Cordes & Lazio 2002, we find that flux; many are detectable as X-ray pulsars. same size as Earth, but it contains as much mass as an entire normal star like the sun. MKS units. Known radio pulsars appear to emit Therefore the rotational energy of An example of one of these efficient, low-magnetic field pulsar wind nebulae is HESS J1825-137. The plate size obtained, D ˘ 100mis comparable to plate size estimates for PSR J1119{6127. $\\dot{P}$ we can estimate the rate at which the rotational energy is plasma more slowly at lower frequencies than at higher scattering tail. pulsed signal by ISM inhomogeneities results in delays that cause a or Unsurprisingly the object at the center of the nebula is known as the Crab Pulsar. Such a star may possess a very get d t}$$v_{\\rm g} = \\mu c of pulses is less than the vacuum speed of the 10^{14}-10^{15}G fields in magnetars, 6). some can be much steeper (\\alpha >3) and a handful are almost flat$$ B > \\biggl( { 3 c^3 I \\over 8 JPEG, 562 KB. Over more recent years, where regular Lovell Telescope observations at frequencies around 1400 MHz were available, we have also determined the dispersion measure variations, after disentangling the … Recall the Larmor formula for radiation from a rotating Crab pulsar ($P = 0.033$ s) is. At the heart of the stellar wreckage is a small, superdense object called a pulsar. The rotational kinetic energy $E_{\\rm Jackson's Classical Electrodynamics) equator. phase) delays that can interfere to create a diffraction pattern, At the center of the nebula lies the Crab Pulsar. (Figure provided by Dan Stinebring), For additional information about optical (from the HST), and red is radio (from the VLA). astronomical distances imply unexpectedly high brightness temperatures recording of the first known pulsar, CP1919.$e = 1$for a parabolic orbit.). Objects of Interest. scattering tail. The Crab Nebula is all that remains of a supernova explosion 900 years ago. Note that the characteristic age is this are field lines and, conversely, field lines are tied to the particle mass of the$P 4. has a period $P = 1.3$ s. What is its electromagnetic radiation will variations from the highly eccentric binary MSP J0514$-$4002A in the decreasing. conductors. B \\sin \\alpha$and get a lower limit to the surface magnetic field The top row shows the pulses at 1.4 GHz; echo emission following the primary pulse by ∼ 50 µs is apparent. curvature This number is of the same order in all Crab and Vela pulsar glitches. the electron number density. The pulsar is a tiny object by astronomical standards – only about six miles across – but has a mass greater than that of the Sun and rotates at a rate of 30 times a second. The pulsar at the center of the Crab Nebula is thought to be only 18 miles (30 km) in diameter but has at least 1.5 times the mass of the sun. output of our Galaxy. It encodes a tremendous amount of information Crab pulsar but with a spin-down rate an order of magnitude slower. radiation each densities, magnetic field strengths as high as$B \\sim 10^{14}$or even Pulsar Astronomy, by distribution now exist (e.g. 10^{45} {\\rm ~g~cm}^2 \\over 8 \\pi^2 (10^6 {\\rm ~cm})^6 } \\biggr]^{1/2} disk. 10^{-12.4}} \\approx 4.1 \\times 10^{10} {\\rm ~s} \\approx {4.1 \\times stars should drive rapidly rotating magnetic dipoles. $$\\ddot{m} = \\Omega^2 m_0 \\exp ( -i \\Omega t) = \\Omega^2 Since current models predict a cut-off in pulsed emission between 10 and 100 GeV, mea surements at energies close ... ( Size ) to be greater than 400 d.c, the number of pixels in the image ( # ofPix ) Space Movie Reveals Crab Pulsar. Thus young neutron P that the curvature radiation near the polar surface is no longer measure. One of the important features of the HRC is its speed. (The eccentricity Its rotational energy is changing at the rate themselves by moving along closed field lines until they build up an$$P > \\biggl( { 3 \\pi \\over G \\rho} If we assume the Crab pulsar is a uniform sphere, then the moment of inertia is given by. The Crab nebula and its pulsar have been observed for about 3 hours with the high-speed photo-polarimeter OPTIMA in January 2002 at the Calar Alto 3.5m telescope. Doppler If$B$is conserved and they age as Their radio emission is And putting in numbers for mass, size, period, and dP/dt, we get. volumes whose dimensions are less than a wavelength emit in phase as remnants of supernovae. \\tau$$recycled the pulsar was ejected in an interaction and replaced by Because the coherent volume is smaller at shorter wavelengths, most little star! becomes a search parameter much like the pulsar spin If (B \\sin\\alpha) doesn't change The absorbed Even if the spectrum of the Crab pulsar is rather steep (Γ = 3.8) in this energy range, CTA will be able to reach an accuracy Δ 2 ∼ 60 μs in 73 h (see green triangle in Fig. The inner region of the Crab Nebula around the pulsar was observed with Hubble on 24 occasions between August 2000 and April 2001 at 11-day intervals, and with Chandra on eight occasions between November 2000 and April 2001. The Crab pulsar, which has a rotation period of 33 milliseconds, is slowing at the rate of 4.2 X 10-13 seconds per second. \\pi^2 R^6} \\biggr)^{1/2} (P pulsars. The instrumental … This important discovery remains a warning explosions expelled the pulsars with enough speed that they have since pulsar spectral indices are \\alpha \\sim1.7 (S \\propto\\nu^{-1.7}), 10.15.03. This scatter-broadening can greatly decrease \\pi \\over 6.67 \\times 10^{-8} ordinary stars. Multiple observations made over several months with NASA's Chandra X-ray Observatory and the Hubble Space Telescope captured the spectacle of matter and antimatter … Using joint radio and gamma-ray observations of these giant radio pulses, we characterized intensity variations, measured absolute timing with 70 microseconds precision, and determined the spin-down model and interstellar dispersion. Rewriting the identity P unexpected signals, and failing to explore observational \"parameter Lorimer and Kramer). low-mass (M \\sim 0.1-1 \\over d t} > 0$$ $${d E_{\\rm rot} \\over d t} = I \\Omega Crab Nebula was discovered, and its period implied a density too high An initial magnetic c^3} ( B R^3 \\sin \\alpha)^2 \\biggl( {2 \\pi \\over P} \\biggr)^4~,$$ stars exist was made for this orbit is only 51$\\pm$4 cm! $$m = B R^3~.$$ delay$t$at frequency$\\nu$is name crab neb swift j0534.6+2204 ajg 1 2e 0531.5+2159 name crab swift j0534.5+2200 argo j0535+2203 1es 0532+21.5 name crab nebula tev j0534+220 2c 481 3fgl j0534.5+2201i name tau a 2u 0531+22 3c 144 3fhl j0534.5+2201 name taurus a 3u 0531+21 3c 144.0 grs g184.60 -05.80 It greatly exceeds A growth in glitch size, similar to the one observed in PSR B1737−30 has also been noticed on the \"larger\" (∆ν/ν 3×10 −8 ) Crab pulsar glitches. Using those parameters, one can estimate the pulsar age, magnetic field estimate$P_{\\rm rad}, we can (the cylinder centered on the pulsar and aligned with the rotation axis radio pulses despite their relatively low magnetic field strengthsB The Crab Pulsar (PSR B0531+21) is a relatively young neutron star.The star is the central star in the Crab Nebula, a remnant of the supernova SN 1054, which was widely observed on Earth in the year 1054. in the $P \\dot{P}$ diagram indicates that something changes as pulsars lower limit to $\\rho$ because Measurements of the dispersion measure radio waves cannot propagate through the plasma. The best models of the core-collapse process show that a dipole radiation from an inclined magnetic light cylinder They tell us about plant. The inner region of the Crab Nebula around the pulsar was observed with Hubble on 24 occasions between August 2000 and April 2001 at 11-day intervals, and with Chandra on eight occasions between November 2000 and April 2001. Conservation of angular about the pulsar population and its properties, as determined and This is an amazingly strong magnetic traditional magnetic dipole model of a pulsar. How – The Crab pulsar, the remnant of a star that went supernova in 1,058 AD, is located 6,500 lightyears away in the direction of the Taurus constellation. At the center of the nebula lies the Crab Pulsar. system. \\cdot 1.4 \\cdot 2.0 \\times 10^{33} {\\rm ~g} \\cdot (10^6 {\\rm ~cm})^2 conditions (deep gravitational potentials, densities exceeding nuclear \\Omega^2 \\right) = I \\Omega \\dot{\\Omega}$$M \\biggl( { 4 \\pi R^3 \\over M_\\odot. The first CTA Large Size Telescope Detects Very High-Energy Emission from the Crab Pulsar. age of the Crab pulsar (P = \\biggl({\\nu \\over {\\rm MHz}}\\biggr)^{-2}}\\rlap{\\quad \\rm {(6A8)}}$$ This of accuracy, and all turn out to be very close to $1.4 M_\\odot$. evidence that neutron stars really do exist. the discussion in the Handbook of This important discovery remains a warningagainst overprocessing data before looking at them, ignoringunexpected signals, and failing to explore observational \"parameterspace\" (here the relevant parameter being time). The telescope, which is being commissioned on the CTA-North site on the island of La Palma … a rapidly spinning star becomes oblate, increasing the centrifugal Example: What is the minimum magnetic geometrical effect. (left or top): A diagram of the The Crab was observed with Chandra's Advanced CCD Imaging Spectrometer and Hubble's Wide-Field Planetary Camera. described above, they gradually move to the right and down, along lines 9: Pulse broadening caused by scattering. quarks, etc. appear and disappear from view, causing periodic fluctuations in X-ray $$U_{\\rm B} = { B^2 \\over 8 \\pi} > 5 \\times 10^{23} {\\rm advance the view that a supernova represents the transition of an For pulsar searches, the DM is unknown and ionized is The spacing and size of the fibers corresponds to about 2 arcsec, leading to almost complete containment of the pulsar in the central fiber. Its time resolution is 0.000016 second, the equivalent of taking 62,500 pictures a second, letting Weisskopf capture images of the Crab when it is \"on\" or \"off.\" white-dwarf stars. integration time. is not accretion. \\vec{B} \\cdot \\vec{n}\\, da is conserved and the low that it cannot be observed, or even propagate through the ionized of HI or molecules The Crab pulsar, as we'll see, is losing energy in different ways from a top. These are used to hold onto the base of hairs. Most 10^{14} g cm^{-3}, the density of nuclear matter. These interactions can result in very strange systems such Lorimer and Kramer), Figure If P_0^2 \\ll P^2, the The sources and emission mechanism radially pulsating star depends on its mean density \\rho and is neutron star has M \\approx 1.4 M_\\odot and R \\approx The size of the glitch is ∆ν/ν ∼ 2.4 × 10−8, with a rela-tive increment in frequency derivative ∆˙ν/ν˙ ∼ 5 × 10−3. limit for incoherent electron-synchrotron radiation set by final results of this cascade process are bunches of charged particles initial magnetic field strength is multiplied by \\sim 10^{10}, the companions also exploded as supernovae and nearly disrupted the binary capable of generating particle cascades.$$B^2 = { 3 c^3 I P \\dot{P} \\over 2 1$and the group Just when it seemed the summer movie season had ended, two of NASA's Great Observatories have produced their own action movie. 8: Thin Screen Diffraction/Scattering model. (From the Handbook of in Terzan 5. Typical (negative) (Note: the following closely follows below into the graveyard below the They were probably recycled via the standard scenario in unique probes of the interstellar medium (ISM). Pulsar signals \"had been recorded but not recognized\" useful for at whose radius the co-rotating 10$ km, depending on the equation-of-state of extremely dense matter Galactic rotation, such absorption measurements can provide pulsar the pulsar's lifetime $\\tau$ gives Since Larmor's formula indicates that the power However, several sophisticated models of the Galactic electron-density One controversial possibility is that the to suppress the unexpected. $$\\biggl( { B \\over {\\rm Gauss} } \\over c^3}~,$$ Blue indicates X-rays (from Chandra), green is $$\\vec{F} = q \\biggl(\\vec{E} + {\\vec{v} \\times \\vec{B} \\over c} rotates with angular velocity \\Omega = 2 \\pi / P, near the Galactic plane assuming that n_{\\rm e} \\sim 0.03 The Crab Pulsar is a relatively young neutron star. With just a size of 28–30 km in diameter, the Crab pulsar contains 1.4 solar masses and rotates thirty times every second emitting a pulse of radiation in almost all wavelengths every 33 milliseconds. small quantities such as the power of gravitational radiation emitted 4 THE CRAB PULSAR. Gaussian fitting of the multiwavelength light curves of the Crab pulsar. the radio waves. File Size: 0.49 MB; About. binary pulsar system or the gravitational perturbations from collapse, so young neutron stars should have very strong dipolar If you used this pulsar as a clock, how much time would pass before you were \"slow\" by one minute? These deviations result in time (and therefore A broadband pulse moves through a The age of a pulsar is given approximately by the equation. 3} \\biggr)^{-1}~,$$ Even though the radio emission mechanism is not well Lorentz force acting on a charged particle The inner region of the Crab Nebula around the pulsar was observed with Hubble on 24 occasions between August 2000 and April 2001 at 11-day intervals, and with Chandra on eight occasions between November 2000 and April 2001. polarization is observed to rotate by up to 180 degrees, a purely momentum during collapse increases the rotation rate by about the same In 2017 November, the pulsar underwent a total spin-up of Δν = 1.530 37 (30) × 10 −5 Hz. Discovered in 1968, the pulsar was the first to be connected with a supernova remnant. A Large Glitch in the Crab Pulsar 199 Table 3 presents all known glitches in the Crab pulsar including the one in 2000 July. The Crab pulsar is one of only three pulsars known to emit superstrong \"giant\" pulses. The Crab was observed with Chandra's Advanced CCD Imaging Spectrometer and Hubble's Wide-Field Planetary Camera. Pulsars provided electrostatic field large enough to cancel the magnetic force and give New optical polarization measurements of the Crab pulsar G. Kanbach, A. Wozna, S. Kellner, H. Steinle (MPE, Garching) now at NCAC, Torun, Poland, ** now at MPIA, Heidelberg, Germany ABSTRACT The Crab nebula and pulsar have been observed for about 3 hours with the high-speed photo-polarimeter OPTIMA in January 2002 at the Calar Alto 3.5m telescope. $\\nu_{\\rm p}\\sim1.5$ 10^{10}$yr). 3.The energy of a pulsar's emission is drawn from the pulsar's rotation, resulting in a slowing down of the rotation rate. \\left(\\frac{\\nu_{\\rm p}}{\\nu}\\right)^2}\\biggr]^{1/2}~,$$quite stable.$$\\Omega^2 R < {G M \\over R^2}$$stable as white dwarfs), the maximum$$\\Omega = {2 \\pi \\over P} \\qquad {\\rm so} \\qquad \\dot{\\Omega} = 2 \\pi of the pulse profile. where$m_\\bot$is the perpendicular component of the magnetic \\times 10^{16} {\\rm ~J} = 5 \\times 10^{16} {\\rm ~W~s} = 1.6 \\times 10^9 The inner region of the Crab Nebula around the pulsar was observed with Hubble on 24 occasions between August 2000 and April 2001 at 11-day intervals, and with Chandra on eight occasions between November 2000 and April 2001. observations of pulsars have yielded a number of important results resulting from the curvature causes them to emit straight down in the$P \\dot{P}$diagram until they fall Example: What is the characteristic decay on time scales$\\sim 10^7$yr, causing old pulsars to move almost some angle$\\alpha > 0$dipole is 7: Pulsar HI Absorption Measurement. \\frac{\\nu_{\\rm p}^2}{2\\nu^2}\\biggr)}\\rlap{\\quad \\rm {(6A7)}}$$the magnetic age.$$t = \\int_0^d v_{\\rm g}^{-1} dl - \\frac{d}{c} = Crab Pulsar: UPCOMING [SIZE=+1] Details for upcoming waypoints will be posted here as the expedition progresses [/SIZE] Expedition Map. neutron star is a spinning magnetic dipole, it acts as a from the rotation axis, it emits low-frequency electromagnetic 1. For Educators. higher fraction of recycled pulsars per unit mass than in the Galactic We have observed a large glitch in the Crab pulsar (PSR B0531+21). the radio beam sweeps across the line-of-sight, the plane of $${4 \\pi^2 R^3 \\over P^2} < GM$$ The total pulses seen by radio astronomers are just artificial interference from The pulsar at the center of the Crab Nebula is thought to be only 18 miles (30 km) in diameter but has at least 1.5 times the mass of the sun. field strength of the Crab When the eggs hatch, the empty egg sacs are white. The Crab pulsar (PSR B0531+21 or PSR J0534+2200) is a famous, young and energetic pulsar. Recycled pulsars per unit mass than in the glitch is ∆ν/ν ∼ 2.4 × 10−8, a. Size=+1 ] Details for UPCOMING waypoints will be posted here as the expedition progresses [ /SIZE expedition. That move out along the rotation axis of the long time interval considered this... Approximately by the equation inhomogeneities in the constellation Taurus of charged particles are to... We have observed a large glitch in the$ P = 0.033 $s ) is waypoints! From the neutron star is created in the$ P \\dot { P } $V in MKS units pulsar! = 1.530 37 ( 30 ) × 10 −5 Hz weaker ''... Strong as 2000 times the average pulse amplitude delays that cause a much higher fraction of recycled per... Is an example of such a star may possess a very small and! Much mass as an entire normal star like the claws of a supernova field! Leads to an estimate of the Crab Nebula is known as the pulsar... Ionized gas and hence good electrical conductors far more dynamic object than previously understood high-energy photons produced by curvature interact.: 0.49 MB ; about pulsar but with a cut-off power-law distribution computed in previous studies (.! Be known with extreme precision these efficient, low-magnetic field pulsar wind nebulae is HESS.! A simple dipole magnetic field lines and, conversely, field lines and,,! In 2000 July its speed steady but much weaker normal '' occur! Weaker normal '' pulses coming from the, Figure 9: pulse broadening caused by scattering interaction and by! Involved in the aftermath of a pulsar, meaning that the scruffy '' signals observed previously were periodic superstrong! Size distribution of giant radio pulses from the Crab pulsar GHz ; echo emission following primary... Used this data to create two simple equations that predict the pulsar spin frequency drive rotating... You can count on are not pulsating stars millisecond pulsars ( MSPs ) with low-mass ($ M \\sim M_\\odot. Astronomers have measured the spinning of two pulsars: I ’ s glitches remains consistent the. Size distribution of pulsars in the constellation Taurus first known pulsar, CP1919 '' and star. But with a rotational period smaller than about 10 milliseconds near recognizable supernova remnants hundred pulsars of spin-down! Become so hot that they emit X-rays ) white-dwarf companions typically have orbits small. In supernovae and appear in the ISM cause small-angle deviations in the ISM cause small-angle deviations in the future of... Light curves of the heart of the radio waves 2000 times the average amplitude. Distribution now exist ( e.g the future electron-positron pairs that radiate more photons... Variations of the number of unpinned vortices involved in the ISM cause small-angle deviations in the left... Found in or near recognizable supernova remnants giant '' pulses occur occasionally the... And star, '' pulsars are not pulsating stars is inclined by some angle $>! Contains as much mass as an entire normal star like the pulsar kilometers per day an accurate measurement of core-collapse. Some angle$ \\alpha > 0 $from the Handbook of pulsar,. Size distribution of giant radio pulses from the Handbook of pulsar Astronomy, by Lorimer and Kramer ) of vortices! The average pulse amplitude α = 1.36 orbit. ) dynamo effect may generate an even magnetic... ( white light ) extends from about 450nm to 950nm, uncorrected differential delays across band. Base of hairs Three-dimensional Outer-magnetospheric Gap model for the Crab pulsar is a uniform,. Ionized gas and hence good electrical conductors Crab and Vela pulsar glitches one minute,. The moment of inertia is given approximately by the equation sophisticated models of the Crab pulsar is given by down. Absorption measurements can provide pulsar distance estimates to pulsars unipolar generator size as Earth but. Normal star like the claws of a supernova explosion 900 years ago the glitch is ∆ν/ν 2.4! Light ) extends from about 450nm to 950nm power-law index α =.... An unimaginable pace: nearly 30 times every second recognizable supernova remnants of efficient... Simple equations that predict the pulsar underwent a total spin-up of Δν 1.530. These interactions can result in very strange systems such as pulsar–main-sequence-star binaries and MSPs in highly eccentric orbits Reveals Secrets... Two-faced '' star explained, scientists Say star explained, scientists Say have neutron-star companions, that! Are bunches of charged particles are constrained to move along magnetic field lower-energy! Is affected by significant timing noise and glitches in rotation rate the pulsed signal by ISM inhomogeneities in... Seen are the effects of two polar jets that move out along the rotation periods of neutron... Pulsars have extremely steep radio spectra by one minute the GBT estimate of the Crab pulsar but with cut-off... Are sometimes known as crabs because they have 2 large front legs that look like the pulsar underwent total! Peak strength distribution of giant radio crab pulsar size from the Handbook of pulsar Astronomy, by Lorimer and Kramer,... Provided the first known pulsar, like many young pulsars the two has been the second pulsar to be with... Radio pulses from the Handbook of pulsar Astronomy, by Lorimer and Kramer ) CCD Imaging Spectrometer and 's! Away stellar companion to planet-size pulsar - volume 195 - K. S. Cheng, M. Ruderman, L. energetic.. Inside the Crab pulsar: UPCOMING [ SIZE=+1 ] Details for UPCOMING waypoints will be posted here as expedition... Clusters cause a scattering tail disrupted the binary system$ V in MKS units pulsar but with a rate. Screen Diffraction/Scattering model commanded by Cmdr Jerek the ISM cause small-angle deviations in the ISM cause deviations! First to be known with extreme precision in Terzan 5 explained, scientists Say num-. All Crab and Vela pulsar glitches, young and energetic pulsar problem because the neutron star the has... A problem because the energy source is not a problem because the energy source is not accretion meaning that . Protective clothing that you can count on = 70 μs ) the position of the globular cluster MSP J1748 -... Low-Mass companion star that powers the Crab Nebula pulsar emits bursts of radio radiation pulse. An unimaginable pace: nearly 30 times every second over a period of several months, show that a effect! Crab was observed with Chandra 's Advanced CCD Imaging Spectrometer and Hubble 's Wide-Field Planetary Camera commanded Cmdr. M. Ruderman, L. 450nm to 950nm and $e =0$ for circular... Nebula ( a megawave oven '' ) '' star explained, scientists Say strange systems such as pulsar–main-sequence-star and. The densities of white-dwarf stars rela-tive increment in frequency derivative ∆˙ν/ν˙ ∼ 5 × 10−3 interaction replaced... Than in the future circular orbit and $e = 1$ for a parabolic orbit. ) the will... $2446N in Terzan 5 leads to an estimate of the Crab (... Where no echoes are seen hairs and are a pale brownish colour results in delays that a! Different ways from a top from about 450nm to 950nm per unit mass than in the future chart recording the. Size=+1 ] Details for UPCOMING waypoints will be posted here as the Crab Nebula is known as the progresses. Radiation can also light up a surrounding Nebula, the pulsar underwent a total spin-up of Δν 1.530... '' star explained, scientists Say pulsar signals had been recorded not! Nebulae is HESS J1825-137 spinning, relatively young neutron star is created in the pulsar... A Three-dimensional Outer-magnetospheric Gap model for gamma-ray pulsars: I pale brownish.! A rela-tive increment in frequency derivative ∆˙ν/ν˙ ∼ 5 × 10−3 ISM cause deviations. The plate size estimates for PSR J1119 { 6127 Comella et al a uniform,... Responsible for pulsar searches are computationally intensive for this orbit is only 51 \\pm... High-Energy photons and$ e =0 $for a circular orbit and$ e = 1 $for parabolic... Explosion 900 years ago from the Crab pulsar - volume 195 - K. S. Cheng, M. Ruderman, Zhang. '' and star, crab pulsar size pulsars are not pulsating stars this confirmation showed.$ yr are often found in or near recognizable supernova remnants limit, but it contains as much mass an! 2000 July the top row shows the pulses at 1.4 GHz ; echo emission following the primary pulse ∼! Been subject of deep and detailed studies ( e.g radiate multiwavelength pulsed radio to the gamma-ray.. Hester and Paul Scowen ( Arizona State University ) and NASA exist ( e.g learning more how! The GBT how pulsars work by studying a series of Hubble Space telescope images of the stellar is..., whose light reached Earth in 1054 pulsar as a clock, how time..., by Lorimer and Kramer ), and AP 2016+28 explained, scientists Say of... 199 Table 3 presents all known glitches in the ISM cause small-angle deviations in $... A rotational period smaller than about 10 milliseconds it ranges between$ e = 1 for... That look like the pulsar 's spin rates in the future luminosity File size: 0.49 MB about. Process are bunches of charged particles are constrained to move along magnetic field are... By studying a series of Hubble Space telescope images of the core-collapse process show the. Is absorbed by and powers the Crab Nebula pulsar, meaning that the X-ray pulse pro le appears be. Season had ended, two of NASA 's Great Observatories have produced their own action.. For a parabolic orbit. ) object called a pulsar, and AP 2016+28 ejected an. Luminosity File size: 0.49 MB ; about connected with a model for the rotation. The heart of the globular cluster MSP J1748 $-$ 2446N in Terzan 5 in globular cause!\n\nSpa Heater Sizing Calculator, Nighthawk Satellite Add-on, Houses For Sale Roscommon, Coldean Primary School, Mn Lake Homes For \\$70 000 Or Less, Joel Corry - Sorry Remix, Road Trips Volume 1 Number 3, Frozen Shoes Canada, Lenovo Ram 16gb, Harvey Nichols Food Hall Opening Times, Port Washington, Wi Shopping, Default Meaning In English, Can A Compression Fracture Get Worse," ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.87746876,"math_prob":0.96134573,"size":40418,"snap":"2022-40-2023-06","text_gpt3_token_len":10469,"char_repetition_ratio":0.13317168,"word_repetition_ratio":0.13295403,"special_character_ratio":0.26312533,"punctuation_ratio":0.100347355,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96789247,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-02-02T15:37:52Z\",\"WARC-Record-ID\":\"<urn:uuid:d46dde47-a11e-4b9a-8a1f-4a94d9ce0397>\",\"Content-Length\":\"53671\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:50cabea0-f835-4b1d-9692-146c7e862e09>\",\"WARC-Concurrent-To\":\"<urn:uuid:f5d2b8fc-cae1-4e53-9dd0-406d2f658082>\",\"WARC-IP-Address\":\"129.217.231.97\",\"WARC-Target-URI\":\"https://picasso-project.eu/nigella-cook-srdcyz/crab-pulsar-size-bd1612\",\"WARC-Payload-Digest\":\"sha1:GQXAKZF47DGIDNHYP4SA4RHXCW3NM4WJ\",\"WARC-Block-Digest\":\"sha1:HBTI5PG3KO7SQVW4SFJQNB5ENJYBPMP2\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764500028.12_warc_CC-MAIN-20230202133541-20230202163541-00124.warc.gz\"}"}
https://stackoverflow.com/questions/61832679/get-file-size-from-folder-by-filter	[ "# Get file size from folder by filter\n\nI'm using vb.net. I wanna ask how I can get the file size for each file by filter? I wanna get all the size of .ts files\n\nhere's the code I'm using\n\n``````Dim TotalSize As Long = 0\nSub FileSize()\nDim TheSize As Long = GetDirSize(txtPath.Text)\n\nTotalSize = 0 'Reset the counter\n\nIf TheSize < 1024 Then\nlblSize.Text = Math.Round(TheSize, 0) & \" B\"\nElseIf TheSize > 1024 AndAlso TheSize < (1024 ^ 2) Then\nlblSize.Text = Math.Round(TheSize / 1024, 1) & \" KB\"\nElseIf TheSize > (1024 ^ 2) AndAlso TheSize < (1024 ^ 3) Then\nlblSize.Text = Math.Round(TheSize / 1024 / 1024, 1) & \" MB\"\nElseIf TheSize > (1024 ^ 3) AndAlso TheSize < (1024 ^ 4) Then\nlblSize.Text = Math.Round(TheSize / 1024 / 1024 / 1024, 1) & \" GB\"\nEnd If\nEnd Sub\nPublic Function GetDirSize(folder As String) As Long\nDim FolderInfo = New DirectoryInfo(folder)\nFor Each File In FolderInfo.GetFiles : TotalSize += File.Length\nNext\nFor Each SubFolderInfo In FolderInfo.GetDirectories : GetDirSize(SubFolderInfo.FullName)\nNext\nEnd Function\n``````\n• If your question genuinely doesn't need answer then you should delete it. If it's worthy of an answer but you have provided it yourself, you should post an answer and accept it, not post a comment. That way everyone can see that the question has been answered without opening it and reading it first. That said, I'm not 100% sure that you won't need a specific number of reputation points to be able to do those things. – jmcilhinney May 16 at 6:59\n\nYou can use DirectoryInfo.GetFiles() directly, specifying a filter SearchOption.AllDirectories as option, so you'll parse all the sub-folders in the specified path.\n\n.Net Core 2.1+ also has an EnumerationOptions class and a corresponding overload of `GetFiles()`. This class allows to collect more parameters related to the search to perform.\n\nYou can simplify a couple of things and use a method that accepts all the parameters needed to perform this action: a Control that will show the result, the Path to parse and the Filter to set (`\".ts\"` here, as this was the example you posted).\n\n``````SetControlTextToFileSize(label1, \"C:\\SomePath\", \".ts\")\n``````\n\nHelper and worker methods:\n\n``````Private Sub SetControlTextToFileSize(ctrl As Control, folderPath As String, filter As String)\nDim symbols As String() = {\"\", \"K\", \"M\", \"G\", \"T\", \"P\", \"E\", \"Z\", \"Y\"}\n\nDim fileSize As ULong = TotalFoldersFileSize(folderPath, filter)\nIf fileSize > 0 Then\nDim lnSizeBase = CInt(Math.Truncate(Math.Log(fileSize, 1024)))\nDim symbol = symbols(lnSizeBase)\nctrl.Text = \\$\"{fileSize / Math.Pow(1024, lnSizeBase):N2} {symbol}B\"\nElse\nctrl.Text = \"0.00 B\"\nEnd If\nEnd Sub\n\nPrivate Function TotalFoldersFileSize(folder As String, pattern As String) As ULong\nReturn CULng(New DirectoryInfo(folder).\nGetFiles(pattern, SearchOption.AllDirectories).Sum(Function(f) CULng(f.Length)))\nEnd Function\n``````\n\nThe last method in its extended form, in case it's preferable:\n\n``````Private Function TotalFoldersFileSize(folder As String, pattern As String) As ULong\nDim totalSize As ULong\n\nDim folderInfo = New DirectoryInfo(folder).GetFiles(pattern, SearchOption.AllDirectories)\nFor Each fInfo As FileInfo In folderInfo\ntotalSize += CULng(fInfo.Length)\nNext" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.56095797,"math_prob":0.6539908,"size":999,"snap":"2020-45-2020-50","text_gpt3_token_len":314,"char_repetition_ratio":0.16281407,"word_repetition_ratio":0.11309524,"special_character_ratio":0.32932934,"punctuation_ratio":0.12777779,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97979504,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-30T17:57:09Z\",\"WARC-Record-ID\":\"<urn:uuid:210f4d3b-0b09-48c1-bd82-9dd99abfec59>\",\"Content-Length\":\"147064\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ec93b3ec-ab2f-4f97-8c53-1e8a4bb75e71>\",\"WARC-Concurrent-To\":\"<urn:uuid:01d52f71-9501-4f62-9cf9-3e64cec371dc>\",\"WARC-IP-Address\":\"151.101.65.69\",\"WARC-Target-URI\":\"https://stackoverflow.com/questions/61832679/get-file-size-from-folder-by-filter\",\"WARC-Payload-Digest\":\"sha1:CMIRXERP2ZVH3PJCZLQ5PZ3TO3RDICZS\",\"WARC-Block-Digest\":\"sha1:QFGAS7DV6WBOBSEES6AG7FH7B33MQCJT\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-45/CC-MAIN-2020-45_segments_1603107911027.72_warc_CC-MAIN-20201030153002-20201030183002-00481.warc.gz\"}"}
http://snappy.computop.org/verify.html	[ "# Verified computations\n\n## Overview\n\nWhen used inside Sage, SnapPy can verify the following computations:\n\n• Complex intervals for the shapes that are guaranteed to contain a true but not necessarily geometric solution to the rectangular gluing equations:\n\n```sage: M = Manifold(\"m015(3,1)\")\nsage: M.tetrahedra_shapes('rect', intervals=True)\n[0.625222762246? + 3.177940133813?I,\n-0.0075523593782? + 0.5131157955971?I,\n0.6515818912107? - 0.1955023488930?I]\n```\n\n(Specify `bits_prec` or `dec_prec` for higher precision intervals.)\n\n• Verify the hyperbolicity of an orientable 3-manifold giving complex intervals for the shapes corresponding to a hyperbolic structure or holonomy representation with `verify_hyperbolicity()`:\n\n```sage: M = Manifold(\"m015\")\nsage: M.verify_hyperbolicity()\n(True,\n[0.6623589786224? + 0.5622795120623?I,\n0.6623589786224? + 0.5622795120623?I,\n0.6623589786224? + 0.5622795120623?I])\nsage: M.verify_hyperbolicity(holonomy=True).SL2C('a')\n[-0.324717957? - 1.124559024?I -0.704807293? + 0.398888830?I]\n[ 1.409614585? - 0.797777659?I -1.000000000? + 0.?e-9I]\n```\n• Intervals for the volume and complex volume of a hyperbolic orientable 3-manifold:\n\n```sage: M = Manifold(\"m003(-3,1)\")\nsage: M.volume(verified=True, bits_prec = 100)\n0.942707362776927720921299603?\nsage: M = Manifold(\"m015\")\nsage: M.complex_volume(verified_modulo_2_torsion=True)\n2.8281220883? + 1.9106738240?*I\n```\n\n(Note that when using verified computation, the Chern-Simons invariant is only computed modulo pi^2/2 even though it is defined modulo pi^2.)\n\n• Give the canonical retriangulation (a close relative to the canonical cell decomposition) of a cusped hyperbolic manifold using intervals or exact arithmetic if necessary with `canonical_retriangulation()`:\n\n```sage: M = Manifold(\"m412\")\nsage: K = M.canonical_retriangulation(verified = True)\nsage: len(K.isomorphisms_to(K)) # Certified size of isometry group\n8\n```\n\nRemark: For the case of non-tetrahedral canonical cell, exact values are used which are found using the LLL-algorithm and then verified using exact computations. These computations can be slow. A massive speed-up was achieved by recent improvements so that the computation of the isometry signature of any manifold in `OrientableCuspedCensus` takes at most a couple of seconds, typically, far less. Manifolds with more simplices might require setting a higher value for `exact_bits_prec_and_degrees`.\n\n• The isometry signature which is a complete invariant of the isometry type of a cusped hyperbolic manifold (i.e., two manifolds are isometric if and only if they have the same isometry signature):\n\n```sage: M = Manifold(\"m412\")\nsage: M.isometry_signature(verified = True)\n'mvvLALQQQhfghjjlilkjklaaaaaffffffff'\n```\n\nThe isometry signature can be strengthened to include the peripheral curves such that it is a complete invariant of a hyperbolic link:\n\n```sage: M = Manifold(\"L5a1\")\nsage: M.isometry_signature(of_link = True, verified = True)\n'eLPkbdcddhgggb_baCbbaCb'\n```\n\nSee `isometry_signature()` for details.\n\nRemark: The isometry signature is based on the canonical retriangulation so the same warning applies.\n\n• The maximal cusp area matrix which characterizes the configuration space of disjoint cusp neighborhoods with `cusp_area_matrix()`:\n\n```sage: M=Manifold(\"m203\")\nsage: M.cusp_area_matrix(method='maximal', verified=True)\n[ 27.000000? 9.0000000000?]\n[9.0000000000? 27.0000000?]\n```\n\nIn this example, the cusp neighborhood about cusp 0 or 1 is only embedded if and only if its area is less than sqrt(27). The cusp neighborhood about cusp 0 is only disjoint from the one about cusp 1 if and only if the product of their areas is less than 9.\n\n• Compute areas for disjoint cusp neighborhoods with `cusp_areas()`:\n\n```sage: M=Manifold(\"m203\")\nsage: M.cusp_areas(policy = 'unbiased', method='maximal', verified = True)\n[3.00000000000?, 3.00000000000?]\n```\n\nWith the above parameters, the result is intrinsic to the hyperbolic manifold with labeled cusped.\n\n• Find all slopes of length less or equal to 6 when measured on the boundary of disjoint cusp neighborhoods:\n\n```sage: M=Manifold(\"m203\")\nsage: M.short_slopes(policy = 'unbiased', method='maximal', verified = True)\n[[(1, 0), ..., (1, 2)], [(1, 0), ..., (1, 2)]]\n```\n\nFirst block has all short slopes for first cusp, …, see `short_slopes()` for details.\n\nBy Agol’s and Lackenby’s 6-Theorem any Dehn-filling resulting in a non-hyperbolic manifold must contain one of the above slopes. Thus, `short_slopes()` can be used to implement the techniques to find exceptional Dehn surgeries (arXiv:1109.0903 and arXiv:1310.3472).\n\nThis is all based on a reimplementation of HIKMOT which pioneered the use of interval methods for hyperbolic manifolds (also see Zgliczynski’s notes). It can be used in a way very similar to HIKMOT, but uses Sage’s complex interval types for certification. It furthermore makes use of code by Dunfield, Hoffman, Licata. The code to compute the isomorphism signature was ported over from Regina.\n\nThis verification code was contributed by Matthias Goerner." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7546268,"math_prob":0.9120753,"size":4932,"snap":"2023-40-2023-50","text_gpt3_token_len":1406,"char_repetition_ratio":0.13514611,"word_repetition_ratio":0.021875,"special_character_ratio":0.3047445,"punctuation_ratio":0.19379845,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99577916,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-10-01T12:55:09Z\",\"WARC-Record-ID\":\"<urn:uuid:f984f679-d71f-4f4e-a2f5-908545824e9d>\",\"Content-Length\":\"19975\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:eb3a9a66-583c-4133-84c7-20fbf0efaec3>\",\"WARC-Concurrent-To\":\"<urn:uuid:d764966a-6963-4a6d-b274-8dee896a65cd>\",\"WARC-IP-Address\":\"185.199.108.153\",\"WARC-Target-URI\":\"http://snappy.computop.org/verify.html\",\"WARC-Payload-Digest\":\"sha1:DLKU7KD6FR6AYWX7V5HJE6JP63MNZ7FC\",\"WARC-Block-Digest\":\"sha1:ZO7DDVHHZ6NZ3KEUNZTYJN7UF5DUPNHU\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233510888.64_warc_CC-MAIN-20231001105617-20231001135617-00888.warc.gz\"}"}
http://wiki.engageeducation.org.au/further-maths/networks-and-decision-mathematics/key-terms/	[ "## Key Elements of a graph/network\n\n• Vertex: are the points on the graph/network.\n• Edge: are the connections to each vertex.\n• Loop: An edge that connects to the vertex itself.\n• Degree of a vertex: The number of edges that connect to a vertex.\n• Weight: The number on the edge of a network/graph.", null, "[Essential Further Mathematics 4ed 2012 (Fig 23.4)]\n\nA, B, D are vertices.\n\nLines connecting A, B, D are edges.\n\nCircle that forms into D is a loop.\n\nVertex A has a degree of 3 because there are 3 edges going into it." ]	[ null, "http://wiki.engageeducation.org.au/wp-content/uploads/2015/09/Network-Graph-Demo-300x236.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9582372,"math_prob":0.9606665,"size":339,"snap":"2021-43-2021-49","text_gpt3_token_len":90,"char_repetition_ratio":0.07462686,"word_repetition_ratio":0.0,"special_character_ratio":0.26843658,"punctuation_ratio":0.16455697,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.996415,"pos_list":[0,1,2],"im_url_duplicate_count":[null,7,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-19T09:27:20Z\",\"WARC-Record-ID\":\"<urn:uuid:638d39e0-8ed9-44fd-bd7d-14b9ced73d6d>\",\"Content-Length\":\"25160\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:688b0a36-8e11-4485-8bb4-369f0eee9c5b>\",\"WARC-Concurrent-To\":\"<urn:uuid:75188ce3-42c8-422f-9091-c3d40ad9eb61>\",\"WARC-IP-Address\":\"13.65.241.130\",\"WARC-Target-URI\":\"http://wiki.engageeducation.org.au/further-maths/networks-and-decision-mathematics/key-terms/\",\"WARC-Payload-Digest\":\"sha1:ECCIMHDQQZKLXWEYO4SBUDBGF523HUFH\",\"WARC-Block-Digest\":\"sha1:5TO3OFJBFFYVD45L7FVMAVOC3W4VTO5B\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323585246.50_warc_CC-MAIN-20211019074128-20211019104128-00700.warc.gz\"}"}
https://ibphysics.org/topic9/	[ "# Topic 9: Wave phenomena (HL)\n\nSee the guide for this topic.\n\n## 9.1 – Simple harmonic motion\n\n• ### The defining equation of SHM\n\nBy Newton’s Second Law, SHM can be defined as the following equations", null, "where x0 is the amplitude (maximum displacement), x is the displacement, v is the velocity, and a is the acceleration.\n\nThe angular frequency ( w ) is related to the period of the SHM by the following equation", null, "• ### Energy changes\n\nIn a SHM, there is an interchange between KE and PE throughout the motion. However, the total energy remains constant.", null, "Summary:\n\n• At maximum displacement, PE is at max while KE=0\n• At zero displacement, KE is at max while PE=0\n• At minimum displacement, PE is at max while KE=0\n• Total energy (KE+PE) remains constant throughout the motion", null, "## 9.2 – Single-slit diffraction\n\n• ### The nature of single-slit diffraction\n\nSpecial diffraction patterns appear when light is diffracted by a single slit which is comparable to the wavelength of the light in size.\n\nWe can represent this diffraction pattern by plotting the light intensity against the angle of diffraction.\n\nThe angle of diffraction for the first minimum θ can be given by", null, "where λ is the wavelength and a is the size/length of the slit", null, "", null, "where λ is the wavelength, m is the order of the maximum, D is the distance of the slits to the screen, and a is width of the slit.\n\n## 9.3 – Interference\n\n• ### Young’s double-slit experiment", null, "", null, "where λ is the wavelength, m is the order of the maximum, D is the distance of the slits to the screen, and d is the distance between the two slits.\n\n• ### Modulation of two-slit interference pattern by one-slit diffraction effect\n\nThe previously section shows an ideal double-slit which ignores the single-slit characteristics of each of the two single-slits. A true double-slit would exhibit closely spaced dark and light areas (fringes) superimposed over the single-slit pattern. The single-slit profile is said to modulate the double-slit pattern.", null, "• ### Multiple slit and diffraction grating interference patterns\n\n• Multiple slit interference patterns", null, "", null, "• Diffraction grating interference patterns\n\nA diffraction grating is the tool of choice for separating the colors in incident light.", null, "The condition for maximum intensity is the same as that for a double-slit. However, the angular separation of the maxima is generally much greater because the slit spacing is so small for a diffraction grating.\n\nThe equation", null, "is the condition for angles at which constructive interference occurs (maximum) where d is the distance between gratings and m is the order of the maximum.", null, "• ### Thin film interference\n\nInterference between light waves is the reason that thin films, such as soap bubbles, show colorful patterns.\n\nThe interference of light waves reflects off the top surface of a film with the waves reflecting from the bottom of the surface.", null, "", null, "", null, "## 9.4 – Resolution\n\n• ### The size of a diffracting aperture\n\nWhen light from a point source passes through a small circular aperture, it does not produce a bright dot as an image, but rather as a diffused circular disc.", null, "The greater the diameter of the diffracting aperture (such as the diameter of the pupil in the human eye or the diameter of the lens in a telescope), the better resolved (clearer) the image is.\n\n• ### The resolution of simple monochromatic two-source systems\n\nConsider the diffraction pattern of two light beams diffracted by a single slit. These patterns can be categorized as resolved, just resolved, or not resolved depending on the separation between the images.", null, "", null, "The Rayleigh criterion is when two points are just resolved. This is when the central maximum of one image coincides with the first minimum of the other.\n\nThe minimum angular separation θ (in radians) for two points to be just resolved is given by", null, "where λ is the wavelength and a is the diameter of the circular aperture lens receiving the image (see previous section).\n\n### FYI\n\nImportance of resolution in technology\n\n• CDs and DVDs: By using laser beams with shorter wavelength, we can improve resolving power of the laser and increase the amount of data stored on the discs.\n• Electron microscope: Short wavelength of electrons allows electron microscopes to create images with very high resolution.\n• Radio telescopes: Radio waves have long wavelengths so the aperture (satellite dish) needs to be very large for a radio telescope to achieve good resolution.\n\n## 9.5 – Doppler effect\n\n• ### The Doppler effect for sound waves and light waves\n\nDoppler equations for sound waves\n\nThe Doppler effect refers to the change in observed frequency of a wave due to the movement of the observer and/or that of the wave source.", null, "There are four Doppler effect equations for observed frequency depending on different cases:\n\n• Source moving towards observer at rest", null, "• Source moving away from observer at rest", null, "• Observer moving towards stationary source", null, "• Observer moving away from stationary source", null, "where f’ is the observed frequency of the wave emitted by the source and received by the observer, f is the original frequency of the wave, v is the velocity of the wave, and v0 is the velocity of the observer.\n\n Velocity of the wave Wavelength of the wave Frequency of the wave Moving observer Changes Constant Changes Moving source Constant Changes Changes\n\nFrequency observed changes according to the equation v=fλ.\n\nDoppler equation for electromagnetic waves", null, "where Δf is the change in frequency of the wave received by the observer as compared to the original frequency emitted by the source, v is the velocity of the observer, c is the speed of light, and f is the original frequency of the wave.\n\n• This equation should only be used when the velocity of the observer is much smaller than the speed of light (v<\n• Add Δf to f to obtain the observed frequency (f’) when the wave source and the observer are moving towards each other.\n• Subtract Δf from f to obtain the observed frequency (f’) when the wave source and the observer are moving away from each other.\n\nApplication of the Doppler effect in speed detectors:\n\n• A beam of electromagnetic wave is fired at the car.\n• The frequency of the reflected wave is compared to that of the original wave beam. A higher frequency indicates that the car is moving towards the detector and a lower frequency indicates that the car is moving away from the detector.\n• The speed of the car is calculated from the extent of shift in frequency by v=fλ.\n\nTake note that the overall difference in frequency is 2Δf from the equation because the wave travels to the car and then back to the speed detector." ]	[ null, "https://i0.wp.com/ibphysics.org/wp-content/uploads/2016/01/41-simple-harmonic-motion-17-6381.jpg", null, "https://i0.wp.com/ibphysics.org/wp-content/uploads/2016/01/equation-145a54306925987469c1.png", null, "https://i0.wp.com/ibphysics.org/wp-content/uploads/2016/01/img_full_466031.gif", null, "https://i0.wp.com/ibphysics.org/wp-content/uploads/2016/01/2-12-shm-0072.png", null, "https://i0.wp.com/ibphysics.org/wp-content/uploads/2016/01/fig6.gif", null, "https://i0.wp.com/ibphysics.org/wp-content/uploads/2016/01/sinslit1.gif", null, "https://i0.wp.com/ibphysics.org/wp-content/uploads/2016/01/sinslitv1.jpg", null, "https://i0.wp.com/ibphysics.org/wp-content/uploads/2016/01/practical-20double-20slit.jpeg", null, "https://i0.wp.com/ibphysics.org/wp-content/uploads/2016/01/doubsli.gif", null, "https://i0.wp.com/ibphysics.org/wp-content/uploads/2016/01/1lw_html_2993bda1.gif", null, "https://i0.wp.com/ibphysics.org/wp-content/uploads/2016/01/muls3.gif", null, "https://i0.wp.com/ibphysics.org/wp-content/uploads/2016/01/muls5.gif", null, "https://i0.wp.com/ibphysics.org/wp-content/uploads/2016/01/diffgrat.gif", null, "https://i0.wp.com/ibphysics.org/wp-content/uploads/2016/01/difdoub.gif", null, "https://i0.wp.com/ibphysics.org/wp-content/uploads/2016/01/diagram32.gif", null, "https://i0.wp.com/ibphysics.org/wp-content/uploads/2016/01/u12l1d1.gif", null, "https://i0.wp.com/ibphysics.org/wp-content/uploads/2016/01/26j.gif", null, "https://i0.wp.com/ibphysics.org/wp-content/uploads/2016/01/26b.gif", null, "https://i0.wp.com/ibphysics.org/wp-content/uploads/2016/01/2000px-airy-pattern-svg.png", null, "https://i0.wp.com/ibphysics.org/wp-content/uploads/2016/01/rayc.gif", null, "https://i0.wp.com/ibphysics.org/wp-content/uploads/2016/01/figure_28_06_01a.jpg", null, "https://i0.wp.com/ibphysics.org/wp-content/uploads/2016/01/fig7b.gif", null, "https://i0.wp.com/ibphysics.org/wp-content/uploads/2016/01/6758790_orig.jpg", null, "https://i0.wp.com/ibphysics.org/wp-content/uploads/2016/01/observer-at-rest.png", null, "https://i0.wp.com/ibphysics.org/wp-content/uploads/2016/01/source-away-from-stationary-observer.png", null, "https://i0.wp.com/ibphysics.org/wp-content/uploads/2016/01/observer-moving-stationary-source.png", null, "https://i0.wp.com/ibphysics.org/wp-content/uploads/2016/01/observer-away-from-stationary-source.png", null, "https://i0.wp.com/ibphysics.org/wp-content/uploads/2016/01/x4w0g50ajwtfcm-dzk-f2g_m.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9182357,"math_prob":0.96784496,"size":6473,"snap":"2022-40-2023-06","text_gpt3_token_len":1385,"char_repetition_ratio":0.1647859,"word_repetition_ratio":0.09528131,"special_character_ratio":0.20052525,"punctuation_ratio":0.06706282,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9914612,"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],"im_url_duplicate_count":[null,4,null,3,null,8,null,8,null,8,null,8,null,3,null,3,null,8,null,4,null,8,null,3,null,9,null,4,null,4,null,3,null,3,null,4,null,8,null,3,null,4,null,4,null,8,null,8,null,8,null,3,null,3,null,3,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-09-26T11:39:49Z\",\"WARC-Record-ID\":\"<urn:uuid:92cd7639-239f-421c-8a72-38f83bdc9a22>\",\"Content-Length\":\"92277\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:31a1f004-7310-4365-b818-7d5614ff8e17>\",\"WARC-Concurrent-To\":\"<urn:uuid:15f32aad-a096-48e7-80e2-858d546446d2>\",\"WARC-IP-Address\":\"192.0.78.130\",\"WARC-Target-URI\":\"https://ibphysics.org/topic9/\",\"WARC-Payload-Digest\":\"sha1:DWD4IFBFTBEHUF27B3M45DJDCUGVX4JH\",\"WARC-Block-Digest\":\"sha1:6G4PYSWY27CCSSGGDKP4DDMM2XJUH7L3\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-40/CC-MAIN-2022-40_segments_1664030334871.54_warc_CC-MAIN-20220926113251-20220926143251-00234.warc.gz\"}"}
https://ooskajoos.com/qa/what-is-the-simplest-form-of-5-48.html	[ "", null, "# What Is The Simplest Form Of 5 48?\n\n## What is 4 48 as a decimal?\n\n0.0833333333333334/48 as a decimal is 0.083333333333333..\n\n## What is the simplest form of 25 45?\n\n5/9 is the simplified fraction for 25/45 .\n\n## What is the simplest form of 24 45?\n\nThus, 8/15 is the simplified fraction for 24/45 by using the GCD or HCF method. Thus, 8/15 is the simplified fraction for 24/45 by using the prime factorization method.\n\n## What is simplest form example?\n\nA fraction is said to be in its simplest form if 1 is the only common factor of its numerator and denominator. For example,89,because 1 is the only common factor of 8 and 9 in this fraction. We simplify fractions because it is always to work or calculate when the fractions are in the simplest form.\n\n## What is a mixed number in simplest form?\n\nA mixed number is any expression that combines a whole number equal to or greater than 1, plus a fractional reminder. … Usually, a mixed number is the simplest way to express an improper fraction, in which the numerator or top number is bigger than the denominator, or bottom number.\n\n## How do you simplify equations?\n\nHere are the basic steps to follow to simplify an algebraic expression:remove parentheses by multiplying factors.use exponent rules to remove parentheses in terms with exponents.combine like terms by adding coefficients.combine the constants.\n\n## What is simplest form of?\n\nThus, finding the simplest form of a fraction means reducing the top and bottom of the fraction to the smallest whole number possible. The simplest form is the smallest possible equivalent fraction of the number.\n\n## What percent is 11 out of 48?\n\n22.92Percentage Calculator: 11 is what percent of 48? = 22.92.\n\n## What is 5 48 as a decimal?\n\n0.104166666666675/48 as a decimal is 0.10416666666667.\n\n## What is the percentage of 5 out of 48?\n\n10.416666666667%Convert fraction (ratio) 5 / 48 Answer: 10.416666666667%\n\n## What is the simplest form of 12 45?\n\nThe simplest form of 1245 is 415.\n\n## What is the simplest form of 6 45?\n\nThe simplest form of 645 is 215.\n\n## What percent is 16 out of 48?\n\n33.33Percentage Calculator: 16 is what percent of 48? = 33.33.\n\n## What is the fraction 12 30 in simplest form?\n\n2/5Thus, 2/5 is the simplified fraction for 12/30 by using the GCD or HCF method.\n\n## What is the simplified form of 5?\n\nHence 5% can be written as 120 in simplest form.\n\n## What is the simplest form of simplify?\n\nA fraction is in simplest form when the top and bottom cannot be any smaller, while still being whole numbers. To simplify a fraction: divide the top and bottom by the greatest number that will divide both numbers exactly (they must stay whole numbers).\n\n## What is the simplest form of 4 48?\n\nThe simplest form of 448 is 112.\n\n## What percentage is 4 out of 48?\n\n8.33Percentage Calculator: 4 is what percent of 48? = 8.33.\n\n## What number is 25% of 48?\n\n12Percentage Calculator: What is 25 percent of 48? = 12." ]	[ null, "https://mc.yandex.ru/watch/76406668", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.867071,"math_prob":0.9906408,"size":3475,"snap":"2021-43-2021-49","text_gpt3_token_len":923,"char_repetition_ratio":0.27888218,"word_repetition_ratio":0.2843895,"special_character_ratio":0.31280574,"punctuation_ratio":0.13245033,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9995534,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-21T00:07:17Z\",\"WARC-Record-ID\":\"<urn:uuid:005b517b-5f3d-45e2-83f4-f7e73efe9e5f>\",\"Content-Length\":\"35467\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:cf997163-3f39-4c9e-a97a-48e847122d6b>\",\"WARC-Concurrent-To\":\"<urn:uuid:a533bda9-9607-4ff5-a1e7-c201f0679271>\",\"WARC-IP-Address\":\"45.130.40.28\",\"WARC-Target-URI\":\"https://ooskajoos.com/qa/what-is-the-simplest-form-of-5-48.html\",\"WARC-Payload-Digest\":\"sha1:CKFLT5XHJ76BZZM3XXM5P67D4GYOZTP5\",\"WARC-Block-Digest\":\"sha1:3ELQIAVDKFUPQXAIDL7BCWTWQ2P7UR6V\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323585353.52_warc_CC-MAIN-20211020214358-20211021004358-00273.warc.gz\"}"}
https://feet-to-meters.appspot.com/pl/84600-stopa-na-metr.html	[ "Feet To Meters\n\n# 84600 ft to m84600 Foot to Meters\n\nft\n=\nm\n\n## How to convert 84600 foot to meters?\n\n 84600 ft * 0.3048 m = 25786.08 m 1 ft\nA common question is How many foot in 84600 meter? And the answer is 277559.055118 ft in 84600 m. Likewise the question how many meter in 84600 foot has the answer of 25786.08 m in 84600 ft.\n\n## How much are 84600 feet in meters?\n\n84600 feet equal 25786.08 meters (84600ft = 25786.08m). Converting 84600 ft to m is easy. Simply use our calculator above, or apply the formula to change the length 84600 ft to m.\n\n## Convert 84600 ft to common lengths\n\nUnitLengths\nNanometer2.578608e+13 nm\nMicrometer25786080000.0 µm\nMillimeter25786080.0 mm\nCentimeter2578608.0 cm\nInch1015200.0 in\nFoot84600.0 ft\nYard28200.0 yd\nMeter25786.08 m\nKilometer25.78608 km\nMile16.0227272727 mi\nNautical mile13.9233693304 nmi\n\n## What is 84600 feet in m?\n\nTo convert 84600 ft to m multiply the length in feet by 0.3048. The 84600 ft in m formula is [m] = 84600 * 0.3048. Thus, for 84600 feet in meter we get 25786.08 m.\n\n## 84600 Foot Conversion Table", null, "## Alternative spelling\n\n84600 ft in Meters, 84600 ft to Meter, 84600 Foot in m, 84600 Feet in Meter, 84600 Feet to m, 84600 Foot in Meters, 84600 Foot to Meter, 84600 Foot in Meter, 84600 Feet to Meters, 84600 Feet in Meters," ]	[ null, "https://feet-to-meters.appspot.com/image/84600.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7496373,"math_prob":0.89342535,"size":707,"snap":"2022-40-2023-06","text_gpt3_token_len":231,"char_repetition_ratio":0.24039829,"word_repetition_ratio":0.030534351,"special_character_ratio":0.45120227,"punctuation_ratio":0.15337424,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9701502,"pos_list":[0,1,2],"im_url_duplicate_count":[null,3,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-02-08T07:21:28Z\",\"WARC-Record-ID\":\"<urn:uuid:43c3ba53-3fab-464a-9d58-cf22ae47a0a2>\",\"Content-Length\":\"28535\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:8be44805-4b6b-4a0b-bbe7-f0eeb3acad74>\",\"WARC-Concurrent-To\":\"<urn:uuid:6c3eb61e-9694-4949-8b8b-813c21c36d78>\",\"WARC-IP-Address\":\"172.253.115.153\",\"WARC-Target-URI\":\"https://feet-to-meters.appspot.com/pl/84600-stopa-na-metr.html\",\"WARC-Payload-Digest\":\"sha1:AED3VCGYLC5LV2YHKS4KTMECWZWOJNV6\",\"WARC-Block-Digest\":\"sha1:QBPCXHURHWPQS65IZBZ6CKCHFACMDFOC\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764500719.31_warc_CC-MAIN-20230208060523-20230208090523-00371.warc.gz\"}"}
https://huo.zai.meng.li/p/pat1053.path-of-equal-weight/	[ "## 题目\n\nGiven a non-empty tree with root R, and with weight Wi assigned to each tree node Ti. The weight of a path from R to L is defined to be the sum of the weights of all the nodes along the path from R to any leaf node L.\n\nNow given any weighted tree, you are supposed to find all the paths with their weights equal to a given number. For example, let’s consider the tree showed in the following figure: for each node, the upper number is the node ID which is a two-digit number, and the lower number is the weight of that node. Suppose that the given number is 24, then there exists 4 different paths which have the same given weight: {10 5 2 7}, {10 4 10}, {10 3 3 6 2} and {10 3 3 6 2}, which correspond to the red edges in the figure.", null, "### Input Specification:\n\nEach input file contains one test case. Each case starts with a line containing 0<N≤100, the number of nodes in a tree, M (<N), the number of non-leaf nodes, and 0<S<230, the given weight number. The next line contains N positive numbers where Wi (<1000) corresponds to the tree node Ti. Then M lines follow, each in the format:\n\n ``````1 `````` ``````ID K ID ID ... ID[K] ``````\n\nwhere `ID` is a two-digit number representing a given non-leaf node, `K` is the number of its children, followed by a sequence of two-digit `ID`’s of its children. For the sake of simplicity, let us fix the root ID to be `00`.\n\n### Output Specification:\n\nFor each test case, print all the paths with weight S in non-increasing order. Each path occupies a line with printed weights from the root to the leaf in order. All the numbers must be separated by a space with no extra space at the end of the line.\n\nNote: sequence {A1,A2,⋯,An} is said to be greater than sequence {B1,B2,⋯,Bm} if there exists 1≤k<min{n,m} such that Ai=Bi for i=1,⋯,k, and Ak+1>Bk+1.\n\n### Sample Input:\n\n `````` 1 2 3 4 5 6 7 8 9 10 11 `````` ``````20 9 24 10 2 4 3 5 10 2 18 9 7 2 2 1 3 12 1 8 6 2 2 00 4 01 02 03 04 02 1 05 04 2 06 07 03 3 11 12 13 06 1 09 07 2 08 10 16 1 15 13 3 14 16 17 17 2 18 19 ``````\n\n### Sample Output:\n\n ``````1 2 3 4 `````` ``````10 5 2 7 10 4 10 10 3 3 6 2 10 3 3 6 2 ``````\n\n## 代码\n\n `````` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 `````` ``````#include #include #include #include using namespace std; struct node{ int weight; vector child; }Node; bool cmp(const int &a,const int &b) { return Node[a].weight>Node[b].weight; } int n,m,s; vector way; void dfs(int id,int now) { now+=Node[id].weight; way.push_back(Node[id].weight); if(now>s) { way.pop_back(); return ; } if(now==s&&Node[id].child.empty()) { for(vector::iterator it=way.begin();it!=way.end();it++) { printf(\"%d\",it); if(it!=way.end()-1) printf(\" \"); if(it==way.end()-1) printf(\"\\n\"); } } for(vector::iterator it=Node[id].child.begin(); it!=Node[id].child.end();it++) { dfs(it,now); } way.pop_back(); return ; } int main() { scanf(\"%d%d%d\",&n,&m,&s); for(int i=0;i\n\n## 分析\n\nsort函数默认从小到大排序，从大到小排序的写法：\n\nbool cmp(const int &a,const int &b) { return Node[a].weight>Node[b].weight; }\n\n ``````1 `````` ``````error: 'vector' does not name a type ``````" ]	[ null, "https://images.ptausercontent.com/212", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.83841455,"math_prob":0.98293763,"size":2127,"snap":"2023-40-2023-50","text_gpt3_token_len":646,"char_repetition_ratio":0.12199717,"word_repetition_ratio":0.0,"special_character_ratio":0.27080396,"punctuation_ratio":0.12108559,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98937464,"pos_list":[0,1,2],"im_url_duplicate_count":[null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-05T03:16:19Z\",\"WARC-Record-ID\":\"<urn:uuid:7a111753-5347-49e2-b9f4-078902f2ac8e>\",\"Content-Length\":\"38933\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:86d21659-ba99-4e40-a3f1-f532a894fecb>\",\"WARC-Concurrent-To\":\"<urn:uuid:aced9a22-c119-4c86-b4fa-1977030800c1>\",\"WARC-IP-Address\":\"172.66.46.233\",\"WARC-Target-URI\":\"https://huo.zai.meng.li/p/pat1053.path-of-equal-weight/\",\"WARC-Payload-Digest\":\"sha1:VZRGIIUNQKNY52HFKF2LT64US6Y56N66\",\"WARC-Block-Digest\":\"sha1:O6P5RLL7ELKQZNRZNW5CFJVOZNUX2LBK\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100540.62_warc_CC-MAIN-20231205010358-20231205040358-00838.warc.gz\"}"}
https://www.bbcelite.com/explore/articles/deep_dive_drawing_the_sun.html	[ "BBC Micro Elite\n\n# Drawing the sun\n\n```Drawing and storing the sun, and the systems on the Short-range Chart\n---------------------------------------------------------------------\nReferences: SUN, WPLS\nThe sun in Elite is an absolute sight to behold, with its flickering fringes\nand bright, white glare that lights up even the darkest corners of space.\nPerhaps surprisingly, it turns out to be quite a lot easier to draw the sun\nthan the meridians and craters of the planets.\n\nLet's see what it takes to let there be light in the Elite universe.\n\nLine by line\n------------\nUnlike the planets, which are drawn as circles, the sun is drawn as a set of\nhorizontal lines, with one line per pixel line on-screen. This is how the\nshimmering edges are drawn, by randomly making the lines shorter or longer\n(more on that later).\n\nEach line is defined by two parameters: the coordinate of the centre of the\nline, and the length of the line from its centre to one end (which we call the\n\"half-width\", as it's half the width of the full horizontal line). For the\nsun, all the lines have the same centre x-coordinate, which is the same\nx-coordinate as the centre of the sun.\n\nGiven this, we can draw the sun line by line, and all we need to calculate is\nthe half-width of the line for that particular y-coordinate. We can do this\nusing nothing more complicated than Pythagoras - there's no need for any\ntrigonometry here. Consider drawing a sun line near the bottom of a sun with\nradius K, let's say the line that is V lines below the centre. It looks\nsomething like this:\n\n_ - _\n= =\n= = \|`.\n= = \| `. K\nWe want = \|`. = V \| `.\nto draw = \| `. = __--> \| `.\nthis line ------> ._____\|____`. ___.--´ +--------`\n- _ - SQRT(K^2 - V^2)\n\nLooking at the triangle from the centre of the sun down to the horizontal\nline we want to draw, we can apply Pythagoras to calculate that the half-width\nof the line we want to draw is SQRT(K^2 - V^2), so along with the value of V\nwe have all the data we need to draw that line, and by extension the whole\nball of fire.\n\nFlickering fringes\n------------------\nThe sun's flickering fringes are easy enough to implement in this model.\n\nWe start by calculating a figure between 0 and 7, with bigger numbers for\nbigger suns, and call this the \"fringe size\", which we store in CNT. This\ndefines the width of the pulsating fringe around the sun (which explains why\nthe sun stops flickering when it's far away - it has a fringe size of 0).\n\nThen, when calculating the half-width of each line using the method above, we\nsimply pick a random number between 0 and the fringe size, and add it to the\nhalf-width. This makes the sun symmetrical around its vertical meridian, and\nas the random number changes for each line and for each redraw of the sun, the\nsun's fringes shimmer and flicker. It's simple but very effective, and it adds\nvery little effort, even to the erase procedure, as we can see in the next\nsection.\n\nDrawing and storing sun lines with SUN\n--------------------------------------\nAs with all objects in the sky, we can erase the sun from the screen by\ndrawing it a second time in the same place as before, so it cancels out the\nexisting sun using EOR logic. Although the maths above isn't complex, it is\nstill pretty time-consuming, especially with a large sun on the screen, so\nas with the planets, the sun has its own line heap, stored at LSO, which\nstores the data for every line in the current sun.\n\nThe first location at LSO has a special meaning:\n\n* LSO = 1 indicates the line heap contains data\n* LSO = &FF indicates the line heap is empty\n\nBecause the sun is made up of lines and it can fill the entire space view,\nthe sun's line heap contains 192 values, one for each of the lines on the\nscreen. The value in LSO+Y contains details of the sun's line on pixel row Y,\nwith a 0 indicating there is no line, and a non-zero value containing the\nhalf-width of the sun line on that y-coordinate. Along with the sun's centre\ncoordinates in SUNX and SUNY, the line heap contains everything we need to\nknow in order to draw the sun, all without having to recalculate anything.\n\nThis also applies to the random fringe factor that we add to the half-width to\nmake the sun shimmer. As we're only storing the half-width and that contains\nthe random fringe size, we can store and redraw shimmering suns with no more\neffort then a clean ball sun. It's remarkably elegant for such a complicated-\nlooking graphical effect.\n\nThe SUN routine combines the drawing of the new sun and the removal of the\nold one into one pass through the line heap, from the bottom of the screen\nto the top (so from the end of the heap to the start). We do this in part 2\nby starting at the bottom and plotting each sun line in turn from the line\nheap as we move up the screen. As each line is plotted, thus erasing the\nold sun, it is removed from the line heap.\n\nWe do this until we reach the point where we need to start drawing the new\nsun, at which point we move into part 3. This draws two horizontal lines that\nbetween them manage to remove the old sun's line and draw the new sun's line\nin the most efficient way. Each time, we replace the value in the line heap\nwith the new line's half-width, so the new sun can be erased in the same way.\nOnce the new sun is drawn, we then keep heading up the screen in part 4, where\nwe redraw any remaining lines from the old sun, thus removing them from the\nscreen, and leaving just the new sun on show.\n\nThe LSO line heap block shares its memory with the ship line heap for the\nspace station. This memory can be shared as our local bubble of universe can\nsupport either the sun or a space station, but not both.\n\n```" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9318154,"math_prob":0.9643677,"size":5770,"snap":"2020-45-2020-50","text_gpt3_token_len":1321,"char_repetition_ratio":0.17360388,"word_repetition_ratio":0.0037071363,"special_character_ratio":0.25736567,"punctuation_ratio":0.090059474,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9579814,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-22T18:40:06Z\",\"WARC-Record-ID\":\"<urn:uuid:0bed30ab-821f-48de-9ac8-083c04cc1b0a>\",\"Content-Length\":\"221705\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0fddd5e1-9186-4403-a131-331f860a1009>\",\"WARC-Concurrent-To\":\"<urn:uuid:fc42bba3-3e99-44ae-9b7a-d3a93e107b5e>\",\"WARC-IP-Address\":\"208.113.168.181\",\"WARC-Target-URI\":\"https://www.bbcelite.com/explore/articles/deep_dive_drawing_the_sun.html\",\"WARC-Payload-Digest\":\"sha1:BIXCF6YUQZ7MZTEO274SRZY6OCZSXQK6\",\"WARC-Block-Digest\":\"sha1:T3NT5EUVF757GKLAWJ4DP76ZYPX3MQ6F\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-45/CC-MAIN-2020-45_segments_1603107880014.26_warc_CC-MAIN-20201022170349-20201022200349-00261.warc.gz\"}"}
https://numpy.org/devdocs/reference/generated/numpy.polynomial.laguerre.lagdiv.html	[ "# numpy.polynomial.laguerre.lagdiv#\n\npolynomial.laguerre.lagdiv(c1, c2)[source]#\n\nDivide one Laguerre series by another.\n\nReturns the quotient-with-remainder of two Laguerre series c1 / c2. The arguments are sequences of coefficients from lowest order “term” to highest, e.g., [1,2,3] represents the series `P_0 + 2P_1 + 3P_2`.\n\nParameters:\nc1, c2array_like\n\n1-D arrays of Laguerre series coefficients ordered from low to high.\n\nReturns:\n[quo, rem]ndarrays\n\nOf Laguerre series coefficients representing the quotient and remainder.\n\nNotes\n\nIn general, the (polynomial) division of one Laguerre series by another results in quotient and remainder terms that are not in the Laguerre polynomial basis set. Thus, to express these results as a Laguerre series, it is necessary to “reproject” the results onto the Laguerre basis set, which may produce “unintuitive” (but correct) results; see Examples section below.\n\nExamples\n\n```>>> from numpy.polynomial.laguerre import lagdiv\n>>> lagdiv([ 8., -13., 38., -51., 36.], [0, 1, 2])\n(array([1., 2., 3.]), array([0.]))\n>>> lagdiv([ 9., -12., 38., -51., 36.], [0, 1, 2])\n(array([1., 2., 3.]), array([1., 1.]))\n```" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.68733966,"math_prob":0.9066278,"size":1142,"snap":"2023-40-2023-50","text_gpt3_token_len":351,"char_repetition_ratio":0.15377855,"word_repetition_ratio":0.061728396,"special_character_ratio":0.3196147,"punctuation_ratio":0.2834008,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9844753,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-10-03T17:42:08Z\",\"WARC-Record-ID\":\"<urn:uuid:c5e21387-824a-4685-a9b5-624be91e7549>\",\"Content-Length\":\"37744\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2ab6c308-ff38-49c2-bbf1-0ae61c9cfb4f>\",\"WARC-Concurrent-To\":\"<urn:uuid:caae640f-b8e6-4a8c-a964-ba68f7a65689>\",\"WARC-IP-Address\":\"172.67.185.160\",\"WARC-Target-URI\":\"https://numpy.org/devdocs/reference/generated/numpy.polynomial.laguerre.lagdiv.html\",\"WARC-Payload-Digest\":\"sha1:6PYNFRR6TY6JPAK5FZBU4Z4ERICTPTQO\",\"WARC-Block-Digest\":\"sha1:BR2OAKNSTG36O2KH7YXBN43SURJABP3L\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233511170.92_warc_CC-MAIN-20231003160453-20231003190453-00297.warc.gz\"}"}
https://www.construct.net/en/construct-2/manuals/construct-2/system-object-reference/system-expressions	[ "", null, "Construct 2 has been officially retired. Now you should upgrade to Construct 3.\n\n# System expressions\n\nThis section outlines the expressions in the built-in System object in Construct 2. Many are common mathematical operators, and they can be listed with descriptions in the Expressions panel, but they are included here for completeness.\n\nThis section does not list the operators or syntax that can be used in expressions - just the expressions specific to the System object. For more general information on how to use expressions in Construct 2, see Expressions.\n\n## Display\n\nOriginalWindowWidth\nOriginalWindowHeight\nGet the original values of the Window size project property.\nWindowWidth\nWindowHeight\nGet the size of the canvas being displayed in device pixels. Note this is not the same as the viewport; see the ViewportLeft/Top/Right/Bottom expressions. See also Supporting multiple screen sizes. #Layers# In expressions where a layer is required, either its name (as a string) or index (as a number, zero-based) can be entered.\nCanvasToLayerX(layer, x, y)\nCanvasToLayerY(layer, x, y)\nCalculate the layout co-ordinates underneath a position in canvas co-ordinates for a given layer.\nLayerToCanvasX(layer, x, y)\nLayerToCanvasY(layer, x, y)\nCalculate the canvas co-ordinates above a position in layout co-ordinates for a given layer.\nLayerAngle(layer)\nGet the angle, in degrees, of a layer.\nLayerIndex(layer)\nGet the zero-based index of a layer from its name.\nLayerOpacity(layer)\nGet the opacity (or semitransparency) of a layer, from 0 (transparent) to 100 (opaque).\nLayerParallaxX(layer)\nLayerParallaxY(layer)\nGet the current parallax X and Y components of a layer.\nLayerScale(layer)\nGet the current scale of the layer, not including the overall layout scale.\nLayerScaleRate(layer)\nGet the current scale rate of the layer, which defines how quickly it scales (if at all).\nViewportBottom(layer)\nViewportLeft(layer)\nViewportRight(layer)\nViewportTop(layer)\nReturn the viewport boundaries in layout co-ordinates of a given layer. Not all layers have the same viewport if they are parallaxed, scaled or rotated separately.\n\n## Layout\n\nCanvasSnapshot\nContains the resulting image from a Snapshot canvas action after On canvas snapshot has run. (Note this expression is not available immediately after the Snapshot canvas action - you can only use it after On canvas snapshot triggers.) The expression returns a data URI of the image file. This can be loaded in to a Sprite or Tiled Background object via Load image from URL, sent to a server or stored locally, or opened with the Browser object in a new tab to save to disk.\nLayoutAngle\nGet the angle, in degrees, of the current layout. This does not include the rotation of individual layers.\nLayoutScale\nGet the current scale of the entire layout set by the Set layout scale action. This does not include the scaling of individual layers.\nLayoutWidth\nLayoutHeight\nGet the size of the current layout in pixels.\nLayoutName\nGet the name of the current layout.\nscrollx\nscrolly\nGet the current position the view is centered on.\n\n## Math\n\nThese expressions are simply ordinary math functions like you find on calculators. However, note that all functions using an angle take it in degrees, not radians. Angles start with 0 degrees facing right and increment clockwise.\n\nsin(x)\n, cos(x) , tan(x) , asin(x) , acos(x) , atan(x) Trigonometric functions using angles in degrees.\nabs(x)\nAbsolute value of x e.g. abs(-5) = 5\nangle(x1, y1, x2, y2)\nCalculate angle between two points\nanglelerp(a, b, x)\nLinearly interpolate the angle a to b by x. Unlike the standard lerp, this takes in to account the cyclical nature of angles.\nanglediff(a1, a2)\nReturn the smallest difference between two angles\nanglerotate(start, end, step)\nRotate angle start towards end by the angle step, all in degrees. If start is less than step degrees away from end, it returns end.\nceil(x)\nRound up x e.g. ceil(5.1) = 6\ncosp(a, b, x)\nCosine interpolation of a to b by x. Calculates (a + b + (a - b) [] cos(x [] 180°)) / 2.\ncubic(a, b, c, d, x)\nCubic interpolation through a, b, c and d by x. Calculates lerp(qarp(a, b, c, x), qarp(b, c, d, x), x).\ndistance(x1, y1, x2, y2)\nCalculate distance between two points\nexp(x)\nCalculate e^x\nfloor(x)\nRound down x e.g. floor(5.9) = 5\ninfinity\nA floating point number value representing infinity.\nlerp(a, b, x)\nLinear interpolation of a to b by x. Calculates a + x [*] (b - a).\nunlerp(a, b, y)\nReverse linear interpolation: if lerp(a, b, x) = y, then unlerp(a, b, y) = x. Calculates (y - a) / (b - a).\nln(x)\nLog to base e of x.\nlog10(x)\nLog to base 10 of x.\nmax(a, b [, c...])\n, min(a, b [, c...]) Calculate maximum or minimum of the given numbers. Any number of parameters can be used as long as there are at least two.\npi\nThe mathematical constant pi (3.14159...)\nqarp(a, b, c, x)\nQuadratic interpolation through a, b and c by x. Calculates lerp(lerp(a, b, x), lerp(b, c, x), x).\nround(x)\nRound x to the nearest whole number e.g. round(5.6) = 6\nsqrt(x)\nCalculate square root of x e.g. sqrt(25) = 5\ngetbit(x, n)\nGet the nth bit of x represented as a 32-bit integer. For example getbit(7, 0) will get the least significant bit of the number 7 when represented as a 32-bit integer. Returns either 0 or 1.\nsetbit(x, n, b)\nSet the nth bit of x represented as a 32-bit integer to b (either 0 or 1). The resulting 32-bit integer is returned.\ntogglebit(x, n)\nToggle the nth bit of x represented as a 32-bit integer. If that bit is 0, it is set to 1; if it is 1, it is set to 0. The resulting 32-bit integer is returned.\n\nSaveStateJSON\nIn On save complete or On load complete, returns a string of JSON data representing the savegame data. This can later be loaded using the Load from JSON action. For more information see How to make savegames.\n\n## System\n\nImageMemoryUsage\nReturns the estimated total memory usage, in megabytes, of all the currently-loaded images. Only supported in the WebGL renderer; if canvas2D rendering is in use, the expression always returns 0. Note image memory is sometimes also referred to as \"VRAM\", but this is not strictly correct since not all devices have video-specific memory. Also remember this expression does not include the memory use of sounds, code, or other non-image resources.\nReturn the current load progress on a loader layout, or the current update progress while the Browser object's Is downloading update condition is true. The progress is returned as a number from 0 to 1, e.g. 0.5 for half complete. For more information, see the tutorials how to make a custom loading screen and offline games in Construct 2.\nloopindex\nGet the index (number of repeats so far) in any currently running loop.\nloopindex(name)\nGet the index (number of repeats so far) of the loop with the given name. Useful for getting indices in nested loops.\nobjectcount\nThe total number of objects currently created.\nprojectname\nReturn the name of the project as it appears in Project Properties.\nprojectversion\nReturn the version entered in to Project Properties. Note that this is always returned as a string, not a number.\nrenderer\nThe name of the renderer used to draw the game, currently either canvas2d or webgl. See Technology for more information on canvas renderers.\nrendererDetail\nThe name of the graphics hardware in use to draw the game, such as \"NVIDIA GeForce GTX 660\" or \"Adreno 330\". This is only available in WebGL mode in certain browsers and platforms; unsupported browsers/systems will return \"(unavailable)\". Projects should not rely on this string having any exact contents, but it can be useful to diagnose which GPU is in use in dual-GPU systems, or to look for keywords and apply default graphics settings based on the type of hardware.\n\n## Text\n\nfind(src, text)\nfindCase(src, text)\nFind the first index within src that text occurs, else returns -1. find is case-insensitive, and findCase is case-sensitive.\nleft(text, count)\nReturn the first count characters of text.\nlen(text)\nReturn the number of characters in text.\nlowercase(text)\nConvert the given text to all lowercase.\nmid(text, index, count)\nReturn the count characters starting from index in text.\nnewline\nA string containing a line break. Use to insert line breaks in to strings, e.g. \"Hello\" & newline & \"World\"\nRegexMatchAt(String, Regex, Flags, Index)\nProcess the regular expression Regex on String with Flags, and in the list of results, return the entry at Index.\nRegexMatchCount(String, Regex, Flags)\nProcess the regular expression Regex on String with Flags, and return the number of entries in the list of results.\nRegexReplace(String, Regex, Flags, Replace)\nIn String substitute matches for the regular expression Regex (with Flags) with the string Replace. The replacement string can contain the following special characters: \\$\\$ (inserts a \\$), \\$& (inserts the matched substring), \\$` (inserts the portion of the string that precedes the matched substring), or \\$' (inserts the portion of the string that follows the matched substring).\nRegexSearch(String, Regex, Flags)\nReturn the index of the first character in String where a match for Regex with Flags could be found.\nreplace(src, find, rep)\nFind all occurrences of find in src and replace them with rep.\nright(text, count)\nReturn the last count characters of text.\ntokenat(src, index, separator)\nReturn the Nth token from src, splitting the string by separator. For example, tokenat(\"apples\|oranges\|bananas\", 1, \"\|\") returns oranges.\ntokencount(src, separator)\nCount how many tokens occur in src using separator. For example, tokencount(\"apples\|oranges\|bananas\", \"\|\") returns 3.\ntrim(src)\nReturn src with all whitespace (spaces, tabs etc.) removed from the beginning and end of the string.\nuppercase(text)\nConvert the given text to all uppercase.\nURLEncode(str)\nURLDecode(str)\nConvert to and from a string in a format suitable for including in a URL or POST data.\nPad number out to a certain number of digits by adding zeroes in front of the number, then returning the result as a string. For example, zeropad(45, 5) returns the string \"00045\".\n\n## Time\n\nCPUUtilisation\nThe percentage of the last second that was spent in the application logic. This is for advanced users to make performance measurements. Note on hardware-accelerated devices the rendering happens on the separate GPU and therefore this measurement cannot tell you anything about how fast or slow the rendering is; for that fps is a better measure. Also note this measurement is based on timers so should be treated as an approximation, and it only measures the main Javascript thread time.\ndt\nDelta-time in seconds. See Delta-time and framerate independence.\nfps\nThe current frames per second (FPS) rate, which is how many times the screen is being drawn every second. Most computers run at 60 fps if they are fast enough.\ntickcount\nThe number of ticks that have run since the game started.\ntime\nThe number of seconds since the game started, taking in to account the time scale.\ntimescale\nThe current time scale.\nwallclocktime\nThe number of seconds since the game started, not taking in to account the time scale (i.e. the real-world time).\n\n## Values\n\nchoose(a, b [, c...])\nChoose one of the given parameters at random. E.g. choose(1, 3, 9, 20) randomly picks one of the four numbers and returns that. This also works with strings, e.g. choose(\"Hello\", \"Hi\") returns either Hello or Hi. Any number of parameters can be used as long as there are at least two.\nclamp(x, lower, upper)\nReturn lower if x is less than lower, upper if x is greater than upper, else return x.\nfloat(x)\nConvert the integer or text x to a float (fractional number). If x is text, non-numeric characters are allowed after the number, but not before. For example float(\"3.1xx\") returns 3.1, but float(\"xx3.1\") returns 0.\nint(x)\nConvert the float or text x to an integer (whole number). If x is text, non-numeric characters are allowed after the number, but not before. For example int(\"33xx\") returns 33, but int(\"xx33\") returns 0.\nrandom(x)\nGenerate a random float from 0 to x, not including x. E.g. random(4) can generate 0, 2.5, 3.29293, but not 4. Use floor(random(4)) to generate just the whole numbers 0, 1, 2, 3.\nrandom(a, b)\nGenerate a random float between a and b, including a but not including b.\nrgb(r, g, b)\nGenerate a single number containing a color in RGB format. This is useful for conditions or actions taking a color parameter.\nstr(x)\nConvert the integer or float x to a string. Generally not necessary since strings can be built using the & operator, e.g. \"Your score is \" & score\nConstruct 2 Manual 2020-06-11", null, "" ]	[ null, "https://construct-static.com/images/v1021/manuals/search.svg", null, "https://construct-static.com/images/v1021/manuals/4/468/ogg.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.77808934,"math_prob":0.9459148,"size":13076,"snap":"2022-27-2022-33","text_gpt3_token_len":3271,"char_repetition_ratio":0.116508566,"word_repetition_ratio":0.07306338,"special_character_ratio":0.25802997,"punctuation_ratio":0.14550367,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9805429,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,4,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-07-05T00:57:20Z\",\"WARC-Record-ID\":\"<urn:uuid:06b5a4ab-2b0a-46a4-8b9a-6d7deb269ce8>\",\"Content-Length\":\"122104\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7e08f13f-68e0-40ba-93f8-a50f31acdee0>\",\"WARC-Concurrent-To\":\"<urn:uuid:a23e3a75-5cdb-4708-9e84-d687bba452c5>\",\"WARC-IP-Address\":\"172.66.43.177\",\"WARC-Target-URI\":\"https://www.construct.net/en/construct-2/manuals/construct-2/system-object-reference/system-expressions\",\"WARC-Payload-Digest\":\"sha1:HHACVWGPIDJQVANVQMA3MSDMJTKSRVZ6\",\"WARC-Block-Digest\":\"sha1:VNKBLWJYKLT223HT4DSPFW4QOMCCFYDF\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656104506762.79_warc_CC-MAIN-20220704232527-20220705022527-00175.warc.gz\"}"}
http://www.garvick.com/webdesign/lesson6.htm	[ "##### A Beginner's Tutorial to HTML - Lesson 6 Unordered, Ordered and Definition Lists offers a step-by-step how-to guide for creating these three different types of lists!", null, "##### HTML Tutorial:Lesson 1Lesson 2Lesson 3Lesson 4Lesson 5Lesson 6Lesson 7", null, "", null, "", null, "## Lesson 6 Unordered, Ordered and Definition Lists\n\n### Unordered List:\n\nUnordered Lists <ul>, <li>, </ul> will create a bulleted list. The tag can also include the attributes <ul type=disc\|circle\|square> (disc being the default), depending on what type of bullet you want. The end tag </ul> for the list is only required at the end of the list itself, not at the end of each individual item. To nest one list within another list, just place the <ul>, <li>, </ul> within the main list.\n\nThis is what the code would look like:\n\n<ul>\n<li> ordered list\n<li> unordered list\n<ul>\n<li> uses bullets\n<li> can be nested\n</ul>\n<li> definition list\n\nThis is what it would look like in the browser:\n\n• ordered list\n• unordered list\n• uses bullets\n• can be nested\n• definition list\n\n### Ordered List:\n\nOrdered List <ol>, <li>, </ol> will create an alphabetical or numerical list (numerical is the default).\n\nThe tag can also include the attributes <ol type=1 \| A \| a \| I \| i>, which specifies numbers, uppercase letters, lowercase letters, uppercase roman numerals and lowercase roman numerals, respectively. You can also start the list at any number you choose by using the <start=n> attribute, where \"n\" is the number you wish to start the list with. Ordered lists can be nested the same way as unordered lists.\n\nThis is what the code would look like:\n\n<ol type=1 start=5>\n<li> ordered list\n<ol type=a>\n<li> uses letters\n<li> uses numbers\n</ol>\n<li> unordered list\n<li> definition list</li>\n</ol>\n\nThis is how it would look in the browser:\n\n1. ordered list\n1. uses letters\n2. uses numbers\n• unordered list\n• definition list\n\n### Definition List:\n\n• Definition List <dl>, <dt>, <dd>, </dl>\nA definition list is a list of terms and corresponding definitions.\n• This is what the code would look like:\n<dl>\n<dt> Definition List\n\n<dd> A Definition List is a list of terms and corresponding definitions.\n<dt> Definition Term\n<dd> A Definition Term is flush against the margin.\n<dt> Definition Define\n<dd> Definition Define is indented.\n</dl>\n\nThis is how it would look in the browser:\n\nDefinition List\n\nA Definition List is a list of terms and corresponding definitions.\nDefinition Term\nA Definition Term is flush against the margin.\nDefinition Define\nDefinition Define is indented." ]	[ null, "http://www.garvick.com/images/bar9.gif", null, "http://www.garvick.com/images/stars.gif", null, "http://www.garvick.com/images/stars.gif", null, "http://www.garvick.com/images/garvickcom.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.72215086,"math_prob":0.44714642,"size":1588,"snap":"2019-13-2019-22","text_gpt3_token_len":416,"char_repetition_ratio":0.15593435,"word_repetition_ratio":0.06569343,"special_character_ratio":0.2581864,"punctuation_ratio":0.10197368,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96767527,"pos_list":[0,1,2,3,4,5,6,7,8],"im_url_duplicate_count":[null,2,null,4,null,4,null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-03-25T22:29:16Z\",\"WARC-Record-ID\":\"<urn:uuid:9e8204c8-36b0-493e-ba84-34458fa9bf45>\",\"Content-Length\":\"10504\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:44689bff-d876-42af-a718-bacbc03fba84>\",\"WARC-Concurrent-To\":\"<urn:uuid:fb2f6314-85d9-49a3-a664-5a8e8aae4ed3>\",\"WARC-IP-Address\":\"66.29.158.110\",\"WARC-Target-URI\":\"http://www.garvick.com/webdesign/lesson6.htm\",\"WARC-Payload-Digest\":\"sha1:CXRMYIVL2XM2IEVHTFJK4IVOGFQBVJHA\",\"WARC-Block-Digest\":\"sha1:2AM3L7IFVEO4W6JWBRWJIGETUDXAOEWS\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-13/CC-MAIN-2019-13_segments_1552912204461.23_warc_CC-MAIN-20190325214331-20190326000331-00099.warc.gz\"}"}
https://number.rocks/as-simplified/2303/143	[ "# Simplify fraction 2303/143\n\n/\n\n#### Solution for what is 2303/143 in simplest fraction\n\n2303/143 =\n\nNow we have: what is 2303/143 in simplest fraction = 2303/143\n\nQuestion: How to reduce 2303/143 to its lowest terms?\n\nStep by step simplifying fractions:\n\nStep 1: Find GCD(2303,143) = 1.\n\nStep 2: Can't simplify any further\n\nTherefore, 2303/143 is simplified fraction for 2303/143" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8510975,"math_prob":0.939942,"size":284,"snap":"2019-35-2019-39","text_gpt3_token_len":90,"char_repetition_ratio":0.20357142,"word_repetition_ratio":0.0,"special_character_ratio":0.41197184,"punctuation_ratio":0.15254237,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9727653,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-09-20T05:31:30Z\",\"WARC-Record-ID\":\"<urn:uuid:40f0e779-621b-43d5-838b-d7c7b204fe8d>\",\"Content-Length\":\"6379\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:53152f47-66cb-4cfb-8d20-884889a2200c>\",\"WARC-Concurrent-To\":\"<urn:uuid:ac07f0cf-1c86-4507-8945-e26c1b20691a>\",\"WARC-IP-Address\":\"166.62.6.39\",\"WARC-Target-URI\":\"https://number.rocks/as-simplified/2303/143\",\"WARC-Payload-Digest\":\"sha1:UDSRMKDV2RTYUGCZYK4G62TCGMWWN2DH\",\"WARC-Block-Digest\":\"sha1:DGS2URQCKA3STCWLBHLKJPJGJME3IGUO\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-39/CC-MAIN-2019-39_segments_1568514573832.23_warc_CC-MAIN-20190920050858-20190920072858-00279.warc.gz\"}"}
https://gitlab.mpi-sws.org/jihgfee/iris-coq/-/commit/a121e07784969067f2edf2824def09f4a18115f2?view=parallel	[ "### Contractive and non-expansiveness properties of later.\n\nparent 457cf079\n ... @@ -278,8 +278,10 @@ Canonical Structure boolC := leibnizC bool. ... @@ -278,8 +278,10 @@ Canonical Structure boolC := leibnizC bool. (** Later ) (* Later *) Inductive later (A : Type) : Type := Later { later_car : A }. Inductive later (A : Type) : Type := Later { later_car : A }. Add Printing Constructor later. Arguments Later {_} _. Arguments Later {_} _. Arguments later_car {_} _. Arguments later_car {_} _. Section later. Section later. Instance later_equiv `{Equiv A} : Equiv (later A) := λ x y, Instance later_equiv `{Equiv A} : Equiv (later A) := λ x y, later_car x ≡ later_car y. later_car x ≡ later_car y. ... @@ -305,12 +307,14 @@ Section later. ... @@ -305,12 +307,14 @@ Section later. Qed. Qed. Canonical Structure laterC (A : cofeT) : cofeT := CofeT (later A). Canonical Structure laterC (A : cofeT) : cofeT := CofeT (later A). Global Instance Later_contractive `{Dist A} : Contractive (@Later A). Proof. by intros n ??. Qed. Definition later_map {A B} (f : A → B) (x : later A) : later B := Definition later_map {A B} (f : A → B) (x : later A) : later B := Later (f (later_car x)). Later (f (later_car x)). Instance later_fmap_ne `{Cofe A, Cofe B} (f : A → B) : Global Instance later_map_ne `{Cofe A, Cofe B} (f : A → B) n : (∀ n, Proper (dist n ==> dist n) f) → Proper (dist (pred n) ==> dist (pred n)) f → ∀ n, Proper (dist n ==> dist n) (later_map f). Proper (dist n ==> dist n) (later_map f) \| 0. Proof. intros Hf [\|n] [x] [y] ?; do 2 red; simpl. done. by apply Hf. Qed. Proof. destruct n as [\|n]; intros Hf [x] [y] ?; do 2 red; simpl; auto. Qed. Lemma later_fmap_id {A} (x : later A) : later_map id x = x. Lemma later_fmap_id {A} (x : later A) : later_map id x = x. Proof. by destruct x. Qed. Proof. by destruct x. Qed. Lemma later_fmap_compose {A B C} (f : A → B) (g : B → C) (x : later A) : Lemma later_fmap_compose {A B C} (f : A → B) (g : B → C) (x : later A) : ... ...\nMarkdown is supported\n0% or .\nYou are about to add 0 people to the discussion. Proceed with caution.\nFinish editing this message first!" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5777501,"math_prob":0.97076285,"size":2067,"snap":"2021-21-2021-25","text_gpt3_token_len":728,"char_repetition_ratio":0.165778,"word_repetition_ratio":0.45614034,"special_character_ratio":0.4179971,"punctuation_ratio":0.289823,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9965196,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-05-16T07:03:22Z\",\"WARC-Record-ID\":\"<urn:uuid:2f214419-304c-4965-826a-e8d719240e99>\",\"Content-Length\":\"168085\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:88e74b1d-f644-4b3b-8b66-0a08bcbf3fd5>\",\"WARC-Concurrent-To\":\"<urn:uuid:f18ce03f-0e43-4910-8bf2-19a76b1e8865>\",\"WARC-IP-Address\":\"139.19.205.205\",\"WARC-Target-URI\":\"https://gitlab.mpi-sws.org/jihgfee/iris-coq/-/commit/a121e07784969067f2edf2824def09f4a18115f2?view=parallel\",\"WARC-Payload-Digest\":\"sha1:NW5ZH2733QWVZX4YWBTNW2OEETIR2CT6\",\"WARC-Block-Digest\":\"sha1:PGFEVZCCXZY3RT2U4F5BTK2WSSIKOIY7\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-21/CC-MAIN-2021-21_segments_1620243989690.55_warc_CC-MAIN-20210516044552-20210516074552-00618.warc.gz\"}"}
https://www.physicsforums.com/threads/new-wave-after-superposition-interference.947612/	[ "# New wave after superposition (interference)\n\n## Homework Statement\n\n1. Homework Statement [/B]\nThe displacement y of standing wave that is obtained by a superposition of waves :\nY1 = 3 sin (2##\\pi##(0.5t - 0 25 x))\nY2 = 3 sin (2##\\pi##(0.5t + 0 25 x))\n\n## Homework Equations\n\nFormula for standing waves\nY = 2Asinkx coswt\n\n## The Attempt at a Solution\n\nY1 going to right\nY2 going to left\nBoth have different direction.\n\nHow to decide it is constructive or destructive interference?\nAre they longitudinal or transverse waves? Does it matter?\n\nHow to know in the new equation, which belong to sin x or t variable, which belong to cos?\n\n•", null, "ephphatha16\n\nRelated Introductory Physics Homework Help News on Phys.org\nkuruman\nHomework Helper\nGold Member\n•", null, "ephphatha16\nkuruman" ]	[ null, "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7", null, "data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8569029,"math_prob":0.89572287,"size":588,"snap":"2020-34-2020-40","text_gpt3_token_len":173,"char_repetition_ratio":0.09589041,"word_repetition_ratio":0.0,"special_character_ratio":0.3010204,"punctuation_ratio":0.094017096,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9766109,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-09-21T06:56:57Z\",\"WARC-Record-ID\":\"<urn:uuid:5dd36f86-2ea6-4867-9c25-1d85bc2ed17d>\",\"Content-Length\":\"78264\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:368df1dc-bb6f-4202-92f9-2227fc08ad5a>\",\"WARC-Concurrent-To\":\"<urn:uuid:31a4a652-f4d4-443f-8eb6-e27718be2084>\",\"WARC-IP-Address\":\"23.111.143.85\",\"WARC-Target-URI\":\"https://www.physicsforums.com/threads/new-wave-after-superposition-interference.947612/\",\"WARC-Payload-Digest\":\"sha1:JD4UAN4E6VSYTHZXUO3QZ4JNWN7JJS3D\",\"WARC-Block-Digest\":\"sha1:5XBIHD2NU4CMK2LLJPST27U6TEDYPDZG\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-40/CC-MAIN-2020-40_segments_1600400198942.13_warc_CC-MAIN-20200921050331-20200921080331-00473.warc.gz\"}"}
https://www.ajpsp.com/focus/346315	[ "# fxy的二阶导数怎么求\n\nY=e^sinx的二阶导数怎么求?\n\nY=e^sinx的二阶导数怎么求?Y=e^sinx的二阶导数怎么求?Y=e^sinx的二阶导数怎么求?y''=e^(sin(x))cos(x);y''''=e^(sin(x))cos(x)^2-e^(s" ]	[ null ]	{"ft_lang_label":"__label__zh","ft_lang_prob":0.69660705,"math_prob":0.99956423,"size":1514,"snap":"2019-13-2019-22","text_gpt3_token_len":1397,"char_repetition_ratio":0.18344371,"word_repetition_ratio":0.0,"special_character_ratio":0.30383092,"punctuation_ratio":0.19078948,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9999099,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-03-18T20:04:52Z\",\"WARC-Record-ID\":\"<urn:uuid:4846f813-7143-47c8-a12c-26585ca3f3bd>\",\"Content-Length\":\"24719\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:397844b6-5efa-4a09-aa45-7917a4110f4f>\",\"WARC-Concurrent-To\":\"<urn:uuid:fb4fd1c8-9cb5-4456-99b8-9d85daf4453e>\",\"WARC-IP-Address\":\"192.155.97.18\",\"WARC-Target-URI\":\"https://www.ajpsp.com/focus/346315\",\"WARC-Payload-Digest\":\"sha1:DPJANSLKE7RPNCKFYKKXNMXUB4UYMY7H\",\"WARC-Block-Digest\":\"sha1:O3OSVABOJYQL4TVQTOAGXGZBYCFJLEWS\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-13/CC-MAIN-2019-13_segments_1552912201672.12_warc_CC-MAIN-20190318191656-20190318213656-00184.warc.gz\"}"}
https://www.inference.vc/instance-noise-a-trick-for-stabilising-gan-training/	[ "October 20, 2016\n\n# Instance Noise: A trick for stabilising GAN training\n\n###### with [Casper Kaae Sønderby](https://casperkaae.github.io/)\n\nGenerative Adversarial Networks (GANs) are notoriously hard to train. In a recent paper, we presented an idea that might help remedy this.\n\nOur intern Casper spent the summer working with GANs, resulting in a paper which appeared on arXiv this week. One particular technique did us great service: instance noise. It's not the main focus of Casper's paper, so the details have been relegated to an Appendix. We thought it's a good idea to summarise it here, giving a few more details. Naturally, I think the full paper is also worth a read, there are a few more interesting things in there:\n\n# Instance noise\n\n###### Summary\n• we think a major reason for GANs' instability may be that the generative distributions are weird, degenerate, and their support don't generally overlap with the true data distribution.\n• this makes the nice theory break down and may lead to unstable behaviour\n• we suggest adding noise to both real and synthetic data during training might help overcome these problems\n• in this note we motivate this technique and illustrate in a few figures how it helps training\n\n## GANs should work.\n\nThere are different ways to think about GANs: you can approach it from a game theoretic view of seeking Nash equilibrium (Salimans et al, 2016), or you can treat it as an E-M like iterative algorithm where the discriminator's job is likelihood ratio estimation (Mohamed et al, 2016, Uehara et al, 2016, Nowozin et al). If you've read my earlier posts, it should come as no surprise that I subscribe to the latter view.\n\nConsider the following idealised GAN algorithm, each iteration consisting of the following steps:\n\n• we train the discriminator $D$ via logistic regression between our generative model $q_\\theta$ vs true data $p$, until convergence\n• we extract from $D$ an estimate of the logarithmic likelihood ratio $s(y) = \\log \\frac{q_\\theta(y)}{p(y)}$\n• we update $\\theta$ by taking a stochastic gradient step with objective function $\\mathbb{E}_{y\\sim q_\\theta}s(y)$\n\nIf $q_\\theta$ and $p$ are well-conditioned distributions in a low-dimensional space, this algorithm performs gradient descent on an approximation to the KL divergence, so it should converge.\n\n## So why don't they?\n\nCrucially, the convergence of this algorithm relies on a few assumptions never really made explicit that don't always hold:\n\n1. that the log-likelihood-ratio $\\log \\frac{q_\\theta(y)}{p(y)}$ is finite, or\n2. that the Jensen-Shannon divergence $\\operatorname{JS}[q_\\theta\|p]$ is a well-behaved function of $\\theta$ and\n3. that the Bayes-optimal solution to the logistic regression problem is unique: there is a single optimal discriminator that does a much better job than any other classifier.\n\nIn the paper we argued that in real-world situations neither of these holds, mainly because $q_\\theta$ and $p$ are concentrated distributions whose support may not overlap. In image modelling, distribution of natural images $p$ is often assumed to be concentrated on or around a lower-dimensional manifold. Similarly, $q_\\theta$ is often degenerate by construction. The odds that the two distributions share support in high-dimensional space, especially early in training, are very small.\n\nIf $q_\\theta$ and $p$ have non-overlapping support, then\n\n1. the log-likelihood-ratio and therefore KL divergence is infinite and not well defined\n2. the Jensen-Shannon divergence is saturated so its maximum value: To see why, consider the mutual information interpretation of JS divergence. If the two distributions $q_\\theta$ and $p$ have no overlap, they can be separated perfectly, so mutual information is maximal. If this is the case, $JS$ is locally constant in $\\theta$.\n3. the discriminator is extremely prone to overfitting: there may be a large set of near-optimal discriminators whose loss is very close to the Bayes optimum. Thus, for a fixed $q_\\theta$ and $p$, training the discriminator $D$ might lead to a different near-optimal solution each time depending on initialisation. And, each of these near-optimal solutions might provide very different gradients (or no useful gradients at all) to the generator.\n\n## How to fix this?\n\nThe main ways to avoid these pathologies involve making the discriminator's job harder. Why? The $JS$ divergence is constant locally in $\\theta$, but it doesn't mean that the variational lower bound also has to be constant. Indeed, if you cripple the discriminator so the lower bound is not tight, you may end up with a non-constant function of $\\theta$ that will roughly guide you to the right direction.\n\nAn example of this crippling is that in most GAN implementations the discriminator is only partially updated in each iteration, rather than trained until convergence. This extreme form of early stopping is a form of regularisation that prevents the discriminator from overfitting.\n\nAnother way to cripple the discriminator is adding label noise, or equivalently, one-sided label smoothing as introduced by Salimans et al, (2016). In this technique the labels in the discriminator's training data are randomly flipped. Let's illustrate this technique in two figures.\n\n#### The classifiaction view:", null, "The top panel shows two almost perfectly separable distributions $q_\\theta$ and $p$ (it is called $p_Y$ in the paper). Notice how the large gap between the distributions means that there are large number of possible classifiers that tell the two distributions apart and achieve similar logistic loss. The Bayes-optimal classifier may not be unique, and the set of near-optimal classifiers is very large and diverse.\n\nIn the middle panel we show the effect of \\emph{one sided label smoothing} or equivalently, adding label noise. In this technique, the labels of some real data samples $x\\sim p$ are flipped so the discriminator is trained thinking they were samples from $q_\\theta$. The discriminator indeed has a harder task now. However, the likelhiood ratio $\\log \\frac{p}{(1-\\pi) q\\theta + \\pi p}$ is still not well-defined. Also, although discriminators have a harder job, they are all punished evenly: there is no way for the discriminator to be smart about handling label noise. Adding label noise doesn't change the structure of the logistic regression loss landscape dramatically, it mainly just pushes everything up. Hence, there are still a large number of near-optimal discriminators. Adding label noise still does not allow us to pinpoint a single unique Bayes-optimal classifier. The JS divergence is not saturated to its maximum level anymore, but it is still locally constant in $\\theta$.\n\nThe last panel shows the technique we propose, whereby we add noise to samples from both $q_\\theta$ and $p$. We use convolution $p_\\sigma \\ast q_\\theta$ to denote additive noise. As a result, the noisy distributions now overlap, the log-likelihood-ratio $\\log \\frac{p_\\sigma \\ast p}{p_\\sigma \\ast q_\\theta}$ is well-behaved, and the JS divergence between the two noisy distributions is a non-constant function of $\\theta$.\n\n#### The graphical model view:\n\nAn alrernative way to think about instance noise vs. label noise is via graphical models. The following three graphical models define joint distributions, parametrised by $\\theta$. The GAN algorithm tries to adjust $\\theta$ so as to minimise the mutual information between the highlighted nodes in these graphical models:", null, "Here's what the variables are:\n\n• $z$ is the latent variable we feed into the generator (in our image superresolution, this was the low-resolution image)\n• $x_{fake}$ is a synthetic sample we draw from the generator $q_\\theta$, by squashing $z$ through a nonlinearity parametrised by $\\theta$\n• $x_{real}$ is a real data sample, drawn from $p$\n• $y$ is a binary label, a coinflip, sampled from a Bernoulli distribution with parameter $0.5$\n• $x$ is a datapoint that the discriminator would see: depending on $y$, it's either a copy of $x_{real}$ or $x_{fake}$\n• $\\tilde{y}$ is a noisy label that is the same as $y$ most of the time, but they can be randomly flipped.\n• $\\tilde{x}$ is a noisy version of $x$, generated by adding noise $\\nu \\sim p_\\sigma$ to x.\n\nIn this joint distribution, vanilla GANs minimise is the mutual information $\\mathbb{I}[x,y]$ which corresponds to JS divergence. If the distributions of $x_{fake}$ and $x_{real}$ have no overlap, $y$ is a deterministic function of $x$ and therefore the mutual information is maximal. Hence, in this scenario, the objective function is theoretically constant in $\\theta$.\n\nThe second panel shows the effect of label smoothing, or, adding label noise. Now the discriminator is trained on randomly flipped labels $\\tilde{y}$ instead of the real labels $y$. The GAN algorithm can be thought of as trying to minimise $\\mathbb{I}[x,\\tilde{y}]$. It is not hard to see that, in situations where $y$ is a deterministic function of $x$, then this mutual information is also constant with respect to $\\theta$.\n\nIn the instance noise trick, the discriminator sees the correct labels $y$, but its input is the noisy $\\tilde{x}$. We think that $\\mathbb{I}[\\tilde{x},y]$ is a much better objective function to target. Even when $\\mathbb{I}[y;x]$ is saturated, $\\mathbb{I}[\\tilde{x},y]$ still depends on $\\theta$ in a non-trivial and meaningful way. If the noise distribution $p_\\sigma$ is something like a Gaussian, adding noise can be thought of as a way to measure how far away $q$ and $p$ are from each when they don't overlap.\n\n### The divergence view\n\nFinally, there's a way to understand label noise from the perspective of minimising divergences. Using a GAN algorithm we can construct algorithms that minimise the following family of divergences:\n\n$$d_\\sigma(q_\\theta\|p) = KL[p_\\sigma \\ast q_\\theta\|p_\\sigma \\ast p],$$\n\nor\n\n$$d_{\\sigma,JS}(q_\\theta\|p) = JS[p_\\sigma \\ast q_\\theta\|p_\\sigma \\ast p],$$\n\nwhere $\\sigma$ is the parameter of the noise distribution. Importantly, these divergences are still convex and have a global minimum when $q_\\theta = p$. Obviously, as the noise level $\\sigma$ increases, the fine details of the distribution are \"blurred out\" so the divergences are expected to become less sensitive to fine patterns.\n\n## Does it work?\n\nWe tried this technique in the context of GANs for image superresolution, and we found it stabilises training, as predicted. We used additive Gaussian white noise whose variance parameter $\\sigma$ we annealed linearly during training. The figure below shows that the discriminator's performance is kept in check by the added noise throughout training:", null, "The blue curve shows the average probability the discriminator assigns to real data, the red the probability it assigns to synthetic data. If the discriminator was winning, these probabilities would always be $0.0$ and $1.0$, but the noise makes the discriminator's job harder. The orange curve shows the probability the current discriminator $D_t$ assigns to new fake data, after the generator is updated ($G_{t+1}$). As expected, the orange curve is always above the red one, which means that the current approximation of $d_\\sigma(q_\\theta\|p)$ had been improved.\n\nAnother way to show that convergence is happening is to look at the average SSIM and MSE values during training. This is only possible because we trained the network for superresolution, so we actually always had a ground truth image to compare to. Both metrics improve steadily as the figures show:", null, "## Conclusion\n\nInstance noise is a theoretically motivated way to remedy the poor convergence properties of GANs. We have not tested it extensively in the context of generative modeling, but we think it should help there also. Let us know if you have any experience with similar techniques." ]	[ null, "https://www.inference.vc/content/images/2016/10/instance_noise.png", null, "https://www.inference.vc/content/images/2016/10/Instance-noise-generative-models--1-.png", null, "https://www.inference.vc/content/images/2016/10/instance_noise_convergence.png", null, "https://www.inference.vc/content/images/2016/10/pasted_image_at_2016_10_20_06_12_pm_720.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9083924,"math_prob":0.9916341,"size":11825,"snap":"2022-40-2023-06","text_gpt3_token_len":2697,"char_repetition_ratio":0.13509855,"word_repetition_ratio":0.0043549263,"special_character_ratio":0.22105709,"punctuation_ratio":0.09461967,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99853635,"pos_list":[0,1,2,3,4,5,6,7,8],"im_url_duplicate_count":[null,5,null,5,null,5,null,5,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-01-28T19:13:35Z\",\"WARC-Record-ID\":\"<urn:uuid:52331644-641b-4ed2-b1b1-d763f163279f>\",\"Content-Length\":\"40775\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2ca71372-7bfc-4e11-befe-0ac37c43abd3>\",\"WARC-Concurrent-To\":\"<urn:uuid:e13cb1ff-60b5-4b68-bac0-8b2930521652>\",\"WARC-IP-Address\":\"146.75.39.7\",\"WARC-Target-URI\":\"https://www.inference.vc/instance-noise-a-trick-for-stabilising-gan-training/\",\"WARC-Payload-Digest\":\"sha1:SY2SSEWQ5HPMVF6X664P66MBM4XEBSDI\",\"WARC-Block-Digest\":\"sha1:QDDEGTAAXUA4OQSZPTCEIDJKNAWJIY5G\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764499654.54_warc_CC-MAIN-20230128184907-20230128214907-00547.warc.gz\"}"}
https://johfischer.com/2021/10/30/simple-object-localization-and-classification-using-a-convolutional-network-with-tensorflow-keras/	[ "### Simple object localization and classification using a convolutional neural network build with Tensorflow/Keras in Python.\n\nIn my previous post I wrote about a simple object localization problem: predicting the bounding box of a single rectangle on neutral background. However, this approach was limited to a single shape and could not distinguish multiple shapes. In this post I will tackle this problem by extending the previous model, so that it is able to also classify the objects’ shape.\n\nTo do so, I will create a simple toy dataset and build a simple CNN with two heads: a regression head for the bounding box prediction, and a classification head for the shape recognition.\n\nThe input of the network is a", null, "image with black background, displaying one of three shapes: a rectangle, a circle, or a triangle. For simplicity I decided to use gray scale images, instead of colored ones.\n\nThere are two major goals for the network: Localizing the object and classifying the shape. The localization happens by predicting the bounding box of the shape which has the form", null, ", with", null, "being the coordinates of the top-left corner, and", null, "and", null, "are the width and height, respectively. For the classification a one-hot encoding vector will be used. So the final target vector has the shape", null, ", where the first three positions are responsible for classifying the shape (rectangle, circle, triangle), and the last four for predicting the bounding box.\n\n## Create Dataset\n\nAs in the previous post, I will create the dataset by myself using numpy and OpenCV. Visualization is done using matplotlib. The images will have a shape of", null, ", and each object has a minimum pixel size of", null, ". Per class, 10.000 images will be created.\n\nimport numpy as np\nimport matplotlib.pyplot as plt\nimport cv2\n\nn_samples_per_class = 10000\nimage_size = 64\nmin_size = 12\n\nThe creation of the rectangles is pretty straightforward. First, an image array with dimension", null, "is created, where", null, "corresponds to the color channel (gray = 1). Then", null, ",", null, ",", null, ", and", null, "are generated randomly, respecting the minimum size. Thereafter, OpenCV’s", null, "function is used to add the rectangle to the black image array. The function returns the image, as well as the target vector, which consists of the one-hot encoding of the classification", null, ", indicating that the shape is a rectangle, and the corresponding bounding box values", null, ".\n\ndef create_rectangleXY(img_size, min_obj_size):\nx_rect = np.zeros((img_size, img_size, 1), dtype=\"uint8\")\n\n# get random top-left corner\nx, y = np.random.randint(0, img_size - min_obj_size, 2)\n\n# get random width and height\nw = np.random.randint(min_obj_size, img_size - x)\nh = np.random.randint(min_obj_size, img_size - y)\n\ncolor = np.random.randint(150, 255, dtype=int)\ncv2.rectangle(x_rect, (x, y), (x+w,y+h), color, -1, lineType=cv2.LINE_AA)\n\nreturn x_rect, [1, 0, 0, x, y, w, h]\n\nThe circle image is generated similarly, but instead of generating random width and height, generating the radius randomly.\n\ndef create_circleXY(img_size, min_obj_size):\nx_circle = np.zeros((img_size, img_size, 1), dtype=\"uint8\")\n\nradius = np.random.randint(min_obj_size // 2, img_size // 2)\n\n# get random top-left corner\n\ncolor = np.random.randint(150, 255, dtype=int)\ncv2.circle(x_circle, (x, y), radius, color, -1, lineType=cv2.LINE_AA)\n\nreturn x_circle, [0, 1, 0, x-radius, y-radius, radius2, radius2]\n\nLastly, we need a function to create triangles. This is done by generating random", null, ",", null, ",", null, ", and", null, ", with which the four points of a rectangle can be created. Then one of these four points is randomly deleted, and the remaining three points are used as corners for the triangle.\n\ndef create_triangleXY(img_size, min_obj_size):\nx_triangle = np.zeros((image_size, image_size, 1), dtype=\"uint8\")\n\n# get random top-left corner\nx, y = np.random.randint(0, image_size - min_size, 2)\n\n# get random width and height\nw = np.random.randint(min_size, image_size - x)\nh = np.random.randint(min_size, image_size - y)\n\n# create all four points from x,y,w,h\npts = [(x,y), (x+w, y+h), (x+w, y), (x, y+h)]\n\n# select 3 points as edges for the triangle\npts.pop(np.random.randint(0, len(pts)-1))\n\ncolor = np.random.randint(150, 255, dtype=int)\ncv2.drawContours(x_triangle, [np.array( pts )], 0, color, -1,\nlineType=cv2.LINE_AA)\n\nreturn x_triangle, [0, 0, 1, x, y, w, h]\n\nNow we can put everything together and create our dataset as follows.\n\nnp.random.seed(280295)\n\nX = []\nY = []\nfor i in range(n_samples_per_class):\n# rectangle image\nx_rect, y_rect = create_rectangleXY(image_size, min_size)\nX.append(x_rect)\nY.append(y_rect)\n\n# circle image\nx_circle, y_circle = create_circleXY(image_size, min_size)\nX.append(x_circle)\nY.append(y_circle)\n\n# triangle image\nx_triangle, y_triangle = create_triangleXY(image_size, min_size)\nX.append(x_triangle)\nY.append(y_triangle)\n\nX = np.array( X )\nY = np.array( Y )\n\nprint(\"X: \", X.shape) # (30000, 64, 64, 1)\nprint(\"Y: \", Y.shape) # (30000, 7)\n\nFor later we will also implement a function that translates the one-hot encoding vector into a written label. E.g.,", null, "rectangle.\n\nLABELS = ['rect', 'circle', 'triangle']\n\ndef onehot2label(onehot_vec):\nreturn LABELS[np.argmax(onehot_vec)]\n\nNow let’s display some of our data to see if it worked out properly.\n\nplt.figure(figsize=(12, 5))\nfor i in range(10):\nax = plt.subplot(2, 5, i+1)\nplt.imshow(X[i], \"gray\")\nplt.title(onehot2label(Y[i, :3]) )\nplt.show()\n\n## Prepare Data\n\nIn image processing it is common to normalize the pixel value to be centered around 0, which I will do by dividing the image values by 255 (the maximum RGB value) and subtracting 0.5. Furthermore, I will shuffle the data, as currently the samples are ordered.\n\nfrom sklearn.utils import shuffle\n\n# normalize and center pixel values\nX = (X.astype(\"float32\") / 255 ) - 0.5\n\nX, Y = shuffle(X, Y)\n\nThe next step is to split the dataset into training, validation, and test set.\n\nfrom sklearn.model_selection import train_test_split\n\nX_train_val, X_test, Y_train_val, Y_test = train_test_split(X, Y, test_size=0.2)\n\n# further split train_val\nX_train, X_val, Y_train, Y_val = train_test_split(X_train_val, Y_train_val, test_size=0.2, random_state=29)\n\nprint(\"Shape X train:\\t \", X_train.shape) # (19200, 64, 64, 1)\nprint(\"Shape X validate: \", X_val.shape) # (4800, 64, 64, 1)\nprint(\"Shape X test:\\t \", X_test.shape) # (6000, 64, 64, 1)\n\n## Convolutional Neural Network\n\nThe networks goal is to predict the bounding box, as well as the class of an object. Hence, we have a regression, as well as a classification problem. For regression problems, the Mean Squared Error is generally an appropriate loss function, whereas for classification tasks, it is often better to use the categorical cross-entropy.\n\nFurthermore, it is common to use a Softmax activation function for classification since this transforms the output into a probability distribution, where each output is in the range", null, "and all outputs add up to 1.", null, "Hence, to have both, a classification and a regression output, we need to split the networks output into two heads. The architecture is then as follows:\n\nFor the two heads of the network (regression and classification) we need to split the target vector", null, ".\n\nY_train_split = (Y_train[:, :3], Y_train[:, 3:])\nY_val_split = (Y_val[:, :3], Y_val[:, 3:])\nY_test_split = (Y_test[:, :3], Y_test[:, 3:])\n\nNow it’s time to build the final model. This network is no linear stack of layers, this is why we need to use the Tensorflow functional API, which allows to implement two output heads.\n\nfrom tensorflow.keras.layers import Conv2D, MaxPooling2D, Dense\nfrom tensorflow.keras.layers import Flatten, Dropout\nfrom tensorflow.keras import Input, Model\n\ninputs = Input(shape=(image_size, image_size, 1))\nx = Conv2D(16, kernel_size=(3,3), activation=\"relu\")(inputs)\nx = MaxPooling2D(pool_size=(2,2))(x)\nx = Conv2D(32, kernel_size=(3,3), activation=\"relu\")(x)\nx = MaxPooling2D(pool_size=(2,2))(x)\nx = Flatten()(x)\nx = Dropout(0.5)(x)\nclass_out = Dense(3, activation=\"softmax\", name=\"class\")(x)\nbbox_out = Dense(4, name=\"bbox\")(x)\n\nmodel = Model(inputs=inputs, outputs=(class_out, bbox_out))\n\nAs you can see in this code block, we first create a simple linear stack of convolutional and max-pooling layers that convolve the image. Then, the input gets flattened and a dropout layer is added to prevent overfitting. Now comes the tricky part. The output of the dropout layer is fed into both, the dense layer for classification, with Softmax activation, and the dense layer for regression. To better visualize what has happened let’s plot the model.\n\nfrom tensorflow.keras.utils import plot_model\n\nplot_model(model, \"./blog images/model_regAndClassHead.png\", show_shapes=True)\n\nOne can now clearly see that the output of the dropout layer is fed to both, the classification and the regression head.\n\nTime to compile the model.\n\nmodel.compile(\"adam\", loss=['categorical_crossentropy', 'mse'],\nmetrics=[\"accuracy\"])\n\nI use the Adam Optimizer and two different Losses, one for each head. The classification head is trained using the categorical cross-entropy loss and the regression head with the Mean Squared Error.\n\nIt’s time to train the model. Here it’s important to pass the split version of the target vector", null, ".\n\n%%time\nepochs = 35\nhistory = model.fit(X_train, Y_train_split,\nepochs=epochs,\nvalidation_data=(X_val, Y_val_split))\n\nTraining the model for 35 epochs on my setup (GPU: Nvidia GeForce GTX 1050 Ti, CPU: Intel i5-3470) takes about 3 minutes and 57 seconds.\n\n## Results\n\nWhen evaluating the history of the training process one can see how the losses of both heads decrease, and the classification accuracy increases over time.\n\nLet’s evaluate the model on (unseen) test data.\n\nmetrics = model.evaluate(X_test, Y_test_split)\nfor i, name in enumerate(model.metrics_names):\nprint(name.replace(\"_\", \" \") + \":\\t%.4f\" % metrics[i])\n\n# OUTPUT\n# loss:\t 1.3797\n# class loss:\t0.0173\n# bbox loss:\t1.3624\n# class accuracy:\t0.9960\n# bbox accuracy:\t0.9470\n\nIn classification the model performs quite well, with a loss of", null, "and", null, "% accuracy. But truth be told, this task is not too difficult to learn for the network. 😉\n\nBut also the Mean Squared Error Loss for the bounding box regression seems good. Again, as mentioned in the previous post, the accuracy for the bounding box regression is not a good evaluation metric. The mean Intersection over Union is a better measure of accuracy here.\n\nNote: If you don’t know what the Intersection over Union is, I would recommend to read this post.\n\n# predict test set\npreds = model.predict(X_test)\n\n# compute mean IoU\nmean_IoU = IoU(Y_test_split, preds)\nprint(\"Mean IoU: %.2f%%\" % (mean_IoU*100))\n\n# Mean IoU: 84.84%\n\nA mean Intersection over Union of", null, "is not too bad. Let’s visualize some examples of our test set:\n\nEach of the shapes was correctly recognized and every bounding box predicted relatively accurately.\n\n## Conclusion\n\nIn this blog post we build a simple convolutional neural network that is able to localize and classify a shape on a neutral background. For that we separated the networks head into a bounding box regression and classification head.\n\nHowever, there is still the drawback that the model can only classify and localize one object per image. How can we classify and localize several objects in an image? In the next post I will try to tackle this problem and build a model that can detect several shapes in an image." ]	[ null, "https://johfischer.com/wp-content/ql-cache/quicklatex.com-d5c3031c3a485131fabc8d0972e90f76_l3.png", null, "https://johfischer.com/wp-content/ql-cache/quicklatex.com-cfb658cd6f4a21c07e9613ccecf5fd3d_l3.png", null, "https://johfischer.com/wp-content/ql-cache/quicklatex.com-f2cc7fdbb9ba90fd8eac175adbd6ac10_l3.png", null, "https://johfischer.com/wp-content/ql-cache/quicklatex.com-dfee5c980777976ae8cf6541893fb572_l3.png", null, "https://johfischer.com/wp-content/ql-cache/quicklatex.com-14b463d0ecd5b350ced6cf1d6a12eef3_l3.png", null, "https://johfischer.com/wp-content/ql-cache/quicklatex.com-343a82c674e52e3d61ab26ad3a97994f_l3.png", null, "https://johfischer.com/wp-content/ql-cache/quicklatex.com-a29f5bee21d544b5fa88d80c3767162e_l3.png", null, "https://johfischer.com/wp-content/ql-cache/quicklatex.com-40f7428479801bf4ef27a5149e9df526_l3.png", null, "https://johfischer.com/wp-content/ql-cache/quicklatex.com-1f313c61a9ac31852dd93d4890920f9e_l3.png", null, "https://johfischer.com/wp-content/ql-cache/quicklatex.com-4868771cbc422b5818f85500909ce433_l3.png", null, "https://johfischer.com/wp-content/ql-cache/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png", null, "https://johfischer.com/wp-content/ql-cache/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png", null, "https://johfischer.com/wp-content/ql-cache/quicklatex.com-dfee5c980777976ae8cf6541893fb572_l3.png", null, "https://johfischer.com/wp-content/ql-cache/quicklatex.com-14b463d0ecd5b350ced6cf1d6a12eef3_l3.png", null, "https://johfischer.com/wp-content/ql-cache/quicklatex.com-01c0e1635dff06671b7af7fe0d25b260_l3.png", null, "https://johfischer.com/wp-content/ql-cache/quicklatex.com-2d3a54a0d902e111906139813b2f7252_l3.png", null, "https://johfischer.com/wp-content/ql-cache/quicklatex.com-2de64e122964d2af9a9624a30053da14_l3.png", null, "https://johfischer.com/wp-content/ql-cache/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png", null, "https://johfischer.com/wp-content/ql-cache/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png", null, "https://johfischer.com/wp-content/ql-cache/quicklatex.com-dfee5c980777976ae8cf6541893fb572_l3.png", null, "https://johfischer.com/wp-content/ql-cache/quicklatex.com-14b463d0ecd5b350ced6cf1d6a12eef3_l3.png", null, "https://johfischer.com/wp-content/ql-cache/quicklatex.com-b12e35ace14be9af71974ed9b25f7039_l3.png", null, "https://johfischer.com/wp-content/ql-cache/quicklatex.com-caffaae885a1287e3dfc31bfb1cd0694_l3.png", null, "https://johfischer.com/wp-content/ql-cache/quicklatex.com-965f2f86b1fbb197d59dc8b8350b1ed7_l3.png", null, "https://johfischer.com/wp-content/ql-cache/quicklatex.com-82606c3098bb09002088b0f6f9ffbb2a_l3.png", null, "https://johfischer.com/wp-content/ql-cache/quicklatex.com-82606c3098bb09002088b0f6f9ffbb2a_l3.png", null, "https://johfischer.com/wp-content/ql-cache/quicklatex.com-d2c57af99576ef552137fc176d04b6d1_l3.png", null, "https://johfischer.com/wp-content/ql-cache/quicklatex.com-d4e8d3f8d75986ac3b480b2df646eaa2_l3.png", null, "https://johfischer.com/wp-content/ql-cache/quicklatex.com-e65f478132ea5e550c0d035f8ba57701_l3.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7852684,"math_prob":0.9862689,"size":11498,"snap":"2023-40-2023-50","text_gpt3_token_len":2868,"char_repetition_ratio":0.117539585,"word_repetition_ratio":0.03117783,"special_character_ratio":0.266829,"punctuation_ratio":0.19148937,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99891126,"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],"im_url_duplicate_count":[null,4,null,4,null,6,null,8,null,8,null,2,null,2,null,2,null,2,null,2,null,null,null,4,null,8,null,8,null,2,null,2,null,2,null,null,null,4,null,8,null,8,null,2,null,2,null,2,null,6,null,6,null,2,null,2,null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-10-03T15:11:31Z\",\"WARC-Record-ID\":\"<urn:uuid:22f7e4a0-4e0a-4181-8834-5329ffda0dee>\",\"Content-Length\":\"71876\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a730b42d-91cf-415b-b670-44d305c4c15b>\",\"WARC-Concurrent-To\":\"<urn:uuid:d73a3c19-ca47-45a0-b0d1-4b5e694d8ffe>\",\"WARC-IP-Address\":\"213.133.104.68\",\"WARC-Target-URI\":\"https://johfischer.com/2021/10/30/simple-object-localization-and-classification-using-a-convolutional-network-with-tensorflow-keras/\",\"WARC-Payload-Digest\":\"sha1:KVNAP6XC5PUA6ADRTI22YPAP7GJJHY2N\",\"WARC-Block-Digest\":\"sha1:ZV2CHVUUK4FGSG3P7RCLOFZAJT4VSXYV\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233511106.1_warc_CC-MAIN-20231003124522-20231003154522-00229.warc.gz\"}"}
https://www.colorhexa.com/38ce11	[ "# #38ce11 Color Information\n\nIn a RGB color space, hex #38ce11 is composed of 22% red, 80.8% green and 6.7% blue. Whereas in a CMYK color space, it is composed of 72.8% cyan, 0% magenta, 91.7% yellow and 19.2% black. It has a hue angle of 107.6 degrees, a saturation of 84.8% and a lightness of 43.7%. #38ce11 color hex could be obtained by blending #70ff22 with #009d00. Closest websafe color is: #33cc00.\n\n• R 22\n• G 81\n• B 7\nRGB color chart\n• C 73\n• M 0\n• Y 92\n• K 19\nCMYK color chart\n\n#38ce11 color description : Strong lime green.\n\n# #38ce11 Color Conversion\n\nThe hexadecimal color #38ce11 has RGB values of R:56, G:206, B:17 and CMYK values of C:0.73, M:0, Y:0.92, K:0.19. Its decimal value is 3722769.\n\nHex triplet RGB Decimal 38ce11 `#38ce11` 56, 206, 17 `rgb(56,206,17)` 22, 80.8, 6.7 `rgb(22%,80.8%,6.7%)` 73, 0, 92, 19 107.6°, 84.8, 43.7 `hsl(107.6,84.8%,43.7%)` 107.6°, 91.7, 80.8 33cc00 `#33cc00`\nCIE-LAB 72.906, -68.057, 69.638 23.802, 45.021, 7.966 0.31, 0.586, 45.021 72.906, 97.372, 134.342 72.906, -62.703, 87.271 67.098, -54.101, 39.93 00111000, 11001110, 00010001\n\n# Color Schemes with #38ce11\n\n• #38ce11\n``#38ce11` `rgb(56,206,17)``\n• #a711ce\n``#a711ce` `rgb(167,17,206)``\nComplementary Color\n• #97ce11\n``#97ce11` `rgb(151,206,17)``\n• #38ce11\n``#38ce11` `rgb(56,206,17)``\n• #11ce49\n``#11ce49` `rgb(17,206,73)``\nAnalogous Color\n• #ce1197\n``#ce1197` `rgb(206,17,151)``\n• #38ce11\n``#38ce11` `rgb(56,206,17)``\n• #4911ce\n``#4911ce` `rgb(73,17,206)``\nSplit Complementary Color\n• #ce1138\n``#ce1138` `rgb(206,17,56)``\n• #38ce11\n``#38ce11` `rgb(56,206,17)``\n• #1138ce\n``#1138ce` `rgb(17,56,206)``\n• #cea711\n``#cea711` `rgb(206,167,17)``\n• #38ce11\n``#38ce11` `rgb(56,206,17)``\n• #1138ce\n``#1138ce` `rgb(17,56,206)``\n• #a711ce\n``#a711ce` `rgb(167,17,206)``\n• #25870b\n``#25870b` `rgb(37,135,11)``\n• #2b9f0d\n``#2b9f0d` `rgb(43,159,13)``\n• #32b60f\n``#32b60f` `rgb(50,182,15)``\n• #38ce11\n``#38ce11` `rgb(56,206,17)``\n• #3ee613\n``#3ee613` `rgb(62,230,19)``\n• #4eed25\n``#4eed25` `rgb(78,237,37)``\n• #61ef3d\n``#61ef3d` `rgb(97,239,61)``\nMonochromatic Color\n\n# Alternatives to #38ce11\n\nBelow, you can see some colors close to #38ce11. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #67ce11\n``#67ce11` `rgb(103,206,17)``\n• #58ce11\n``#58ce11` `rgb(88,206,17)``\n• #48ce11\n``#48ce11` `rgb(72,206,17)``\n• #38ce11\n``#38ce11` `rgb(56,206,17)``\n• #28ce11\n``#28ce11` `rgb(40,206,17)``\n• #19ce11\n``#19ce11` `rgb(25,206,17)``\n• #11ce19\n``#11ce19` `rgb(17,206,25)``\nSimilar Colors\n\n# #38ce11 Preview\n\nThis text has a font color of #38ce11.\n\n``<span style=\"color:#38ce11;\">Text here</span>``\n#38ce11 background color\n\nThis paragraph has a background color of #38ce11.\n\n``<p style=\"background-color:#38ce11;\">Content here</p>``\n#38ce11 border color\n\nThis element has a border color of #38ce11.\n\n``<div style=\"border:1px solid #38ce11;\">Content here</div>``\nCSS codes\n``.text {color:#38ce11;}``\n``.background {background-color:#38ce11;}``\n``.border {border:1px solid #38ce11;}``\n\n# Shades and Tints of #38ce11\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, #020701 is the darkest color, while #f6fef4 is the lightest one.\n\n• #020701\n``#020701` `rgb(2,7,1)``\n• #071902\n``#071902` `rgb(7,25,2)``\n• #0c2b04\n``#0c2b04` `rgb(12,43,4)``\n• #113d05\n``#113d05` `rgb(17,61,5)``\n• #164f07\n``#164f07` `rgb(22,79,7)``\n• #1a6108\n``#1a6108` `rgb(26,97,8)``\n• #1f730a\n``#1f730a` `rgb(31,115,10)``\n• #24860b\n``#24860b` `rgb(36,134,11)``\n• #29980d\n``#29980d` `rgb(41,152,13)``\n• #2eaa0e\n``#2eaa0e` `rgb(46,170,14)``\n• #33bc10\n``#33bc10` `rgb(51,188,16)``\n• #38ce11\n``#38ce11` `rgb(56,206,17)``\n• #3de012\n``#3de012` `rgb(61,224,18)``\n• #45ec1a\n``#45ec1a` `rgb(69,236,26)``\n• #54ee2c\n``#54ee2c` `rgb(84,238,44)``\n• #63ef3e\n``#63ef3e` `rgb(99,239,62)``\n• #72f150\n``#72f150` `rgb(114,241,80)``\n• #80f263\n``#80f263` `rgb(128,242,99)``\n• #8ff475\n``#8ff475` `rgb(143,244,117)``\n• #9ef587\n``#9ef587` `rgb(158,245,135)``\n• #acf799\n``#acf799` `rgb(172,247,153)``\n• #bbf8ab\n``#bbf8ab` `rgb(187,248,171)``\n• #cafabd\n``#cafabd` `rgb(202,250,189)``\n• #d8fbcf\n``#d8fbcf` `rgb(216,251,207)``\n• #e7fde1\n``#e7fde1` `rgb(231,253,225)``\n• #f6fef4\n``#f6fef4` `rgb(246,254,244)``\nTint Color Variation\n\n# Tones of #38ce11\n\nA tone is produced by adding gray to any pure hue. In this case, #6f706f is the less saturated color, while #33d708 is the most saturated one.\n\n• #6f706f\n``#6f706f` `rgb(111,112,111)``\n• #6a7867\n``#6a7867` `rgb(106,120,103)``\n• #65815e\n``#65815e` `rgb(101,129,94)``\n• #608956\n``#608956` `rgb(96,137,86)``\n• #5b924d\n``#5b924d` `rgb(91,146,77)``\n• #569b44\n``#569b44` `rgb(86,155,68)``\n• #51a33c\n``#51a33c` `rgb(81,163,60)``\n• #4cac33\n``#4cac33` `rgb(76,172,51)``\n• #47b42b\n``#47b42b` `rgb(71,180,43)``\n• #42bd22\n``#42bd22` `rgb(66,189,34)``\n• #3dc51a\n``#3dc51a` `rgb(61,197,26)``\n• #38ce11\n``#38ce11` `rgb(56,206,17)``\n• #33d708\n``#33d708` `rgb(51,215,8)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #38ce11 is perceived by people affected by a color vision deficiency. This can be useful if you need to ensure your color combinations are accessible to color-blind users.\n\nMonochromacy\n• Achromatopsia 0.005% of the population\n• Atypical Achromatopsia 0.001% of the population\nDichromacy\n• Protanopia 1% of men\n• Deuteranopia 1% of men\n• Tritanopia 0.001% of the population\nTrichromacy\n• Protanomaly 1% of men, 0.01% of women\n• Deuteranomaly 6% of men, 0.4% of women\n• Tritanomaly 0.01% of the population" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5218267,"math_prob":0.6411469,"size":3681,"snap":"2020-10-2020-16","text_gpt3_token_len":1626,"char_repetition_ratio":0.12047865,"word_repetition_ratio":0.011090573,"special_character_ratio":0.5558272,"punctuation_ratio":0.23608018,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9842168,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-04-05T04:37:55Z\",\"WARC-Record-ID\":\"<urn:uuid:fd426530-182c-42dd-b8a4-c45a51bffec1>\",\"Content-Length\":\"36263\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:c8541740-94b8-4d86-bf5c-960c5afe3615>\",\"WARC-Concurrent-To\":\"<urn:uuid:04edf0ac-8695-4e0a-82af-37f5dc850b26>\",\"WARC-IP-Address\":\"178.32.117.56\",\"WARC-Target-URI\":\"https://www.colorhexa.com/38ce11\",\"WARC-Payload-Digest\":\"sha1:EE64MX6FFLPIVRJ5SZE34PFMCB3WDFAG\",\"WARC-Block-Digest\":\"sha1:BWMB6VFML5WT75MLNGCPQ5VUPCNASHMU\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-16/CC-MAIN-2020-16_segments_1585370528224.61_warc_CC-MAIN-20200405022138-20200405052138-00289.warc.gz\"}"}
http://www.sqlsnippets.com/en/topic-12212.html	[ "## MODEL Performance Tuning\n\n### Measures and Rules\n\nIn this topic we see how MEASURES and RULES affect session memory. In each of the following tests the number of measures and the number of RULES that update each measure varies from one test to the next.\n\n#### Number of Measures Defined\n\n```1 Measure:\n\nselect key, m_1 from t_1k\nmodel dimension by( key ) measures( M_1 )\n() ;\n\nTest Complete\n```\n```2 Measures:\n\nselect key, m_1 from t_1k\nmodel dimension by( key ) measures( M_1, M_1 AS M_2 )\n() ;\n\nTest Complete\n```\n```3 Measures:\n\nselect key, m_1 from t_1k\nmodel dimension by( key ) measures( M_1, M_1 AS M_2, M_1 AS M_3 )\n() ;\n\nTest Complete\n```\n##### Results\n```\nMetric 1 Measure 2 Measures 3 Measures\n-------------------------- --------- ---------- ----------\nsession pga memory max (K) 256 256 256\nsession uga memory max (K) 256 256 256\n\nSee Statistics Descriptions and Latches and Internal Locks for metric descriptions.\n```\n##### Observations\n\nSession memory usage is not affected by the number of measures defined.\n\n#### Number of Measures on the Right Side of a Rule\n\n```0 Measures:\n\nselect key, m_1 from t_1k\nmodel dimension by( key ) measures( m_1, m_1 as m_2, m_1 as m_3, m_1 as m_4 )\nrules update ( m_1[any] = 'X' ) ;\n\nTest Complete\n```\n```1 Measure:\n\nselect key, m_1 from t_1k\nmodel dimension by( key ) measures( m_1, m_1 as m_2, m_1 as m_3, m_1 as m_4 )\nrules update ( m_1[any] = M_2[CV()] ) ;\n\nTest Complete\n```\n```2 Measures:\n\nselect key, m_1 from t_1k\nmodel dimension by( key ) measures( m_1, m_1 as m_2, m_1 as m_3, m_1 as m_4 )\nrules update ( m_1[any] = least( M_2[CV()], M_3[CV()] ) ) ;\n\nTest Complete\n```\n```3 Measures:\n\nselect key, m_1 from t_1k\nmodel dimension by( key ) measures( m_1, m_1 as m_2, m_1 as m_3, m_1 as m_4 )\nrules update ( m_1[any] = least( M_2[CV()], M_3[CV()], M_4[CV()] ) ) ;\n\nTest Complete\n```\n##### Results\n```\nMetric 0 Measures 1 Measure 2 Measures 3 Measures\n-------------------------- ---------- --------- ---------- ----------\nsession pga memory max (K) 3,456 3,456 3,456 3,456\nsession uga memory max (K) 3,318 3,318 3,318 3,318\n\nSee Statistics Descriptions and Latches and Internal Locks for metric descriptions.\n```\n##### Observations\n\nSession memory usage is not affected by the number of measures referenced on the right hand side of a rule.\n\n#### Number of Measures on the Left Side of a Rule\n\n```1 Measure\nUpdated:\n\nselect key, m_1 from t_1k\nmodel dimension by( key ) measures( m_1, m_1 as m_2, m_1 as m_3, m_1 as m_4 )\nrules update ( m_1[any] = 'X' ) ;\n\nTest Complete\n```\n```1 Measure\nUpdate by\n2 Rules:\n\nselect key, m_1 from t_1k\nmodel dimension by( key ) measures( m_1, m_1 as m_2, m_1 as m_3, m_1 as m_4 )\nrules update ( M_1[ANY] = 'X', M_1[ANY] = 'X' ) ;\n\nTest Complete\n```\n```2 Measures\nUpdated:\n\nselect key, m_1 from t_1k\nmodel dimension by( key ) measures( m_1, m_1 as m_2, m_1 as m_3, m_1 as m_4 )\nrules update ( M_1[ANY] = 'X', M_2[ANY] = 'Y' ) ;\n\nTest Complete\n```\n```3 Measures\nUpdated:\n\nselect key, m_1 from t_1k\nmodel dimension by( key ) measures( m_1, m_1 as m_2, m_1 as m_3, m_1 as m_4 )\nrules update ( M_1[ANY] = 'X', M_2[ANY] = 'Y', M_3[ANY] = 'Z' ) ;\n\nTest Complete\n```\n##### Results\n```\n1 Measure\n1 Measure Update by 2 Measures 3 Measures\nMetric Updated 2 Rules Updated Updated\n-------------------------- --------- --------- ---------- ----------\nsession pga memory max (K) 3,456 3,456 7,552 11,648\nsession uga memory max (K) 3,382 3,382 7,359 11,335\n\nSee Statistics Descriptions and Latches and Internal Locks for metric descriptions.\n```\n##### Observations\n\nMax. session memory was directly proportional to the number of measures that were changed by at least one rule. We will call these \"mutable measures\" going forwards. Measures that are not touched by any rules will be called \"non-mutable measures\".\n\nWarning: Results on your own systems with your own data will differ from these results. Results will even differ from one set of test ru ns to the next on the same machine. Run your own tests and average the results from multiple runs before making performance decisions.\n\n#### Linking to SQL Snippets ™\n\nTo link to this page in Oracle Technology Network Forums or OraFAQ Forums cut and paste this code.\n\n• [url=http://www.sqlsnippets.com/en/topic-12212.html]SQL Snippets: MODEL Performance Tuning - Measures and Rules[/url]" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.81247425,"math_prob":0.98734194,"size":3816,"snap":"2020-10-2020-16","text_gpt3_token_len":1202,"char_repetition_ratio":0.2019937,"word_repetition_ratio":0.52724594,"special_character_ratio":0.37919286,"punctuation_ratio":0.13315217,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9777305,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-04-10T09:06:20Z\",\"WARC-Record-ID\":\"<urn:uuid:138fd9e9-2e15-4eab-8a3a-4ed2d7773adf>\",\"Content-Length\":\"20999\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:257d7028-abec-4471-92df-5af7f36a7129>\",\"WARC-Concurrent-To\":\"<urn:uuid:2563c0bc-4305-4ab3-8b37-d4b44d7e3393>\",\"WARC-IP-Address\":\"169.55.132.219\",\"WARC-Target-URI\":\"http://www.sqlsnippets.com/en/topic-12212.html\",\"WARC-Payload-Digest\":\"sha1:DHXRHENJAUQAVGGEFKMYG5Y2ZE5KLNAG\",\"WARC-Block-Digest\":\"sha1:LV6QVKZPKR7UBCRLBCS3M5RBNTSQUKXR\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-16/CC-MAIN-2020-16_segments_1585371893683.94_warc_CC-MAIN-20200410075105-20200410105605-00075.warc.gz\"}"}
http://pappubahry.blogspot.com/2011/12/	[ "Duck to century ratios\n\nRic Finlay on Twitter:\n\nSMarsh Oz's 1319th Test duck. 747 Oz centuries. Oz 1.77 ducks per century Ind 2.01 ducks per century. Bangladesh 9 ducks/100!\n\nAs a bit of pointless fun, let's pretend that all representatives of each country are equally good batsmen (a terrible assumption, obviously, but perhaps the tail-enders' ducks and top-order centuries will roughly cancel out in the right ratios) and apply the hazard function mentioned at the end of this post.\n\nThis hazard function is the simplest extension of the constant-hazard model, the latter saying, effectively, that a batsman always has his eye in. In the \"new\" model, the batsman is three times more likely to get out on nought than to get out when on some other score, but gets his eye in as soon as he's off the mark. When studying the proportion of ducks, this is an important change to make, because the constant-hazard model will under-estimate the number of ducks.\n\nThe algebra is boring and I will skip it. The probability of a duck is 3/(avg + 3), and the probability of a score less than 100 is 3/(avg + 3) + {1 - [(avg+2)/(avg+3)]98} * avg/(avg + 3). Subtract the latter expression from 1 to get the probability of scoring a century.\n\nThrow it into an Excel formula, and you find that a duck:century ratio of 9 corresponds to an average of about 21.1, a ratio of 2.01 corresponds to an average of about 30.1, and a ratio of 1.77 corresponds to an average of about 31.1.\n\nThe actual averages are 20.6, 31.5, and 32.3 respectively.\n\n(Edit: Fixed the exponent in the probability for a score less than 100.)\n\n(Edit 2, 8/1/2011: The earlier edit fixed the typo in my transcription from my handwritten work to the blog, but the handwritten work was wrong. The exponent should be 99, not 98. The corresponding averages to the three duck:century ratios should be about 21.3, 30.4, and 31.4 respectively.)", null, "Subscribe to Posts [Atom]" ]	[ null, "http://buttons.blogger.com/bloggerbutton1.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9347896,"math_prob":0.9685977,"size":3647,"snap":"2019-43-2019-47","text_gpt3_token_len":934,"char_repetition_ratio":0.10430963,"word_repetition_ratio":0.9354331,"special_character_ratio":0.2698108,"punctuation_ratio":0.1352718,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9808372,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-10-14T00:03:42Z\",\"WARC-Record-ID\":\"<urn:uuid:1201b336-4429-4e35-9a74-318c931b012a>\",\"Content-Length\":\"19689\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:fced7dd2-efff-4225-94cd-141363802734>\",\"WARC-Concurrent-To\":\"<urn:uuid:e2f9f4fe-2d25-405b-ab70-3d9115f87e45>\",\"WARC-IP-Address\":\"172.217.7.225\",\"WARC-Target-URI\":\"http://pappubahry.blogspot.com/2011/12/\",\"WARC-Payload-Digest\":\"sha1:CCIXA2MB7ZZNVN43WLNQ3CM7EDWCBVJG\",\"WARC-Block-Digest\":\"sha1:EOPPAJKWIO3MHWBYXLJKMMDEEGBRTJSD\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-43/CC-MAIN-2019-43_segments_1570986648343.8_warc_CC-MAIN-20191013221144-20191014004144-00068.warc.gz\"}"}