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Euclid's Postulates -- from Wolfram MathWorld 1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 4. All right angles are congruent. 5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate. Euclid's fifth postulate cannot be proven as a theorem, although this was attempted by many people. Euclid himself used only the first four postulates ("absolute geometry") for the first 28 propositions of the Elements, but was forced to invoke the parallel postulate on the 29th. In 1823, Janos Bolyai and Nicolai Lobachevsky independently realized that entirely self-consistent " non-Euclidean geometries" could be created in which the parallel postulate did not hold. (Gauss had also discovered but suppressed the existence of non-Euclidean geometries.)	{"url":"https://mathworld.wolfram.com/EuclidsPostulates.html","timestamp":"2024-11-01T20:24:34Z","content_type":"text/html","content_length":"53017","record_id":"<urn:uuid:abd7d836-9d83-4b61-b24d-484f9c18050e>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00658.warc.gz"}
Information scrambling at finite temperature in local quantum systems Title Information scrambling at finite temperature in local quantum systems Publication Journal Article Year of 2020 Authors Sahu, S, Swingle, B Date 5/21/2020 This paper investigates the temperature dependence of quantum information scrambling in local systems with an energy gap, m, above the ground state. We study the speed and shape of growing Heisenberg operators as quantified by out-of-time-order correlators, with particular attention paid to so-called contour dependence, i.e. dependence on the way operators are distributed around the thermal circle. We report large scale tensor network numerics on a gapped chaotic spin chain down to temperatures comparable to the gap which show that the speed of Abstract operator growth is strongly contour dependent. The numerics also show a characteristic broadening of the operator wavefront at finite temperature T. To study the behavior at temperatures much below the gap, we perform a perturbative calculation in the paramagnetic phase of a 2+1D O(N) non-linear sigma model, which is analytically tractable at large N. Using the ladder diagram technique, we find that operators spread at a speed T/m−−−−√ at low temperatures, T≪m. In contrast to the numerical findings of spin chain, the large N computation is insensitive to the contour dependence and does not show broadening of operator front. We discuss these results in the context of a recently proposed state-dependent bound on scrambling. URL https://arxiv.org/abs/2005.10814	{"url":"https://quics.umd.edu/publications/information-scrambling-finite-temperature-local-quantum-systems-0","timestamp":"2024-11-14T08:53:23Z","content_type":"text/html","content_length":"21359","record_id":"<urn:uuid:0719eb27-2d18-4129-8ea5-c12e09dac704>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00274.warc.gz"}
Dutch Betting The Edge We all want to be competitive in everything we do. And no more so that with punting on horse racing. Success in punting on horses, in fact most things reduces down to an information war - the people with the best information make the smart moves. Books have been and still are the best source of general information, and this applies at least as much in the art of racehorse selection and staking as in any other field. Author Paul Segar has produced textbooks which cover all aspects of punting. The books alone stand as a complete reference but also provide 'food for thought'. You can develop / improve your own ideas as well as learn some new techniques. Each book is written in plain English with plenty of practical examples in each chapter. Browse the contents of each book or email for further information, if required. Improve your punting knowledge today - buy one or all of these books. Read the books but want more? It's time to do a course. The Pureform Introduction Course uses a computer program to show you how and when to bet and how to do it successfully. Check out the details The Benchmark Handicapper Course continues from the Introduction Course and gives you further weapons to apply when making quality value selections. More... The Introduction to Dutch Betting using the Ratings Calculator Course gives you an introduction to betting using the Ratings Calculator computer software. More... Buy all three books now: $70 posted Target betting on horse racing using markets is a great way to bet. This article written by Paul Segar and brought to you by Pureform covers betting using markets, overs and unders. 1. Introduction Target betting is easy to understand. You aim to bet to either win or collect the same amount, regardless of which performer wins the race. This article considers betting to collect a set amount whether that is on horse racing, harness or the dish lickers... Betting on horses and many gambling games have so few rules, and this lack of structure makes it both good and bad. The horses run as usual, it has that structure. From a betting viewpoint, there are so many possibilities. You can do whatever you like but at the same time doing whatever you like can not work out so good. Rules and structure is what makes most people successful. At the same time gambling without a care in the world can be a lot of fun if losses are kept in check. This article looks at adding a layer of structure to betting win and place on horse racing but the same approach can be applied to the other codes. For some races, many punters win bet on one or maybe two horses with the same amount on each runner. Consider the following race: Horse Available Odds A $5 B $2.5 C $4 D $16 E $7 The numbers are where it's at for target betting.Focus on the numbers In this race horse A is priced at $5 and horse B at $2.50. A $20 bet on both horses will produce the following profit if one of them wins: Example 1. Horse Wager Available Odds Return Profit A $20 $5 $100 $60 B $20 $2.5 $50 $10 Do both horses have the same chance of winning? This is one way to bet when you think both horses have an equal chance. From the example though clearly horse B needs a bigger bet to make it more profitable. The question. Is horse B a better chance or is it simply priced shorter? No one knows the answer until after the race so the best you can do is estimate. If you are going to bet, like most people, you need to have an opinion. As a punter, if you think each horse has an equal chance, then both runners should be priced at the same value. Seems reasonable. $2.50 or $5 or perhaps some other price. On the other hand, if they are not equal chances, why bet the same amount? The following video shows how to use the Ratings Calculator for target betting: 2. Target Betting This brings us to Target Betting. Target betting using markets is probably the best way to go when doing win or place betting. The word ‘probably’ is used as in gambling there is never one absolute perfect way. First using a market will be considered and the target amount to aim for will be $100. Clearly larger or smaller amounts can be used. The table considers the same race: │Horse│Available Odds │Percent │ │A │$5 │20% │ │B │$2.5 │40% │ │C │$4 │25% │ │D │$16 │6% │ │E │$7 │14% │ │ Total Percent:│105% │ Horse B is priced at $2.50 and converting that price to a percent gives the answer 40% (100/2.5). Following the odds in this case means you should bet: 40% of the cash on horse B which is the favorite for the race 20% for Horse A. The same is done for the other prices and the total percentage adds up to 105%. If you bet on every runner with their percent amount as a dollar amount then you would outlay $105 for a potential collect of $100. Not a great result by anyone calculations. But clearly we can make some better bets on the two horses from the earlier example: Example 2 Horse Wager Available Odds Return Profit A $20 $5 $100 $40 B $40 $2.5 $100 $40 The total outlay is $60 with each runner returning the same amount and the same profit if successful. This is the next level up on the earlier Example 1 still using the available odds. Each section is a level up on the previous level. So get ready to level up! What happens if you think Horse A has a better chance than horse B? Makes sense doesn't it that you want to bet more on horse A than horse B? (The same logic applies to place betting.) Part 2: Markets...	{"url":"http://pureform.com.au/target.php","timestamp":"2024-11-02T08:13:01Z","content_type":"text/html","content_length":"26850","record_id":"<urn:uuid:9019c67b-4374-4394-a5da-444b36edca8a>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00804.warc.gz"}
Re: flex not matching minus sign more than once rkrayhawk@aol.com (RKRayhawk) 10 Aug 2003 22:49:36 -0400 From comp.compilers \| List of all articles for this month \| From: rkrayhawk@aol.com (RKRayhawk) Newsgroups: comp.compilers Date: 10 Aug 2003 22:49:36 -0400 Organization: AOL http://www.aol.com References: 03-08-035 Keywords: lex Posted-Date: 10 Aug 2003 22:49:36 EDT erinf@jobsoft.com (Erin) on 8/10/03 10:00 AM EST posted these flex code snipets i'm using flex v. 2.5.4 and i've run into issues with matching a literal minus code snippet: SP [^A-Za-z0-9 ] MINUS ("-") AN [A-Za-z0-9 ] (({AN}\|{MINUS}){2,15}{SP}{1}){1} works correctly (({AN}\|{MINUS}){2,15}{SP}{1}){2} does not - flex gets hung up i've tried: MINUS [-] MINUS (-) MINUS - i've also tried: ({AN}\|{MINUS}){2,15}{SP}({AN}\|{MINUS}){2,15}{SP} does not work ({AN}\|{MINUS}){2,15}{SP}({AN}){2,15}{SP} works does anyone have a clue as to what i'm doing wrong?! thanks so much! [I've never had a lot of luck with named patterns. I suspect there are some long-standing bugs that haven't been shaken out. -John] Your {SP} named pattern could match a minus sign, which is atleast competitive against the {MINUS} named pattern. You thus have the honor of having created a somewhat ambiguous lexer. Flex and its kin do not diagnose this problem, (because one can intend that! but you immediately get non-intuitive when you venture into that).. That much is suggested by your comment that ({AN}\|{MINUS}){2,15}{SP}({AN}\|{MINUS}){2,15}{SP} does not work ({AN}\|{MINUS}){2,15}{SP}({AN}){2,15}{SP} works although one would have to know exactly what input is bringing about your described results. It would seem that {MINUS} is getting greedy and leaving nothing for {SP}. _might_ work. But I would eliminate the ambiguity. Maybe you want SP [^-A-Za-z0-9 ] And flex may like that better as SP [^\-A-Za-z0-9 ] And if you do not really need two separate rules for MINUS and AN, you can combined them with ANM ([\-A-Za-z0-9 ]) If you do not want to go that far, then atleast try parens around AN AN ([A-Za-z0-9 ]) Concerning your comment that (({AN}\|{MINUS}){2,15}{SP}{1}){1} works correctly (({AN}\|{MINUS}){2,15}{SP}{1}){2} does not - flex gets hung up It is harder to see this problem, but since you getting into multiple occurences, one might guess that you are finding end of line or end of file directly after a MINUS, and we cannot see from your post if you have a \ n rule which could be interfering. It can be assumed that you have some experience, and the following is suggested just in case you haven't considered it. But you may want to have explicit rules for SPaces [ \t], and a newline rule \ n , and possibly a dot rule for all else rather than then SP rule enumerating 'other' by notted character class [^ ]. Post a followup to this message Return to the comp.compilers page. Search the comp.compilers archives again.	{"url":"https://compilers.iecc.com/comparch/article/03-08-041","timestamp":"2024-11-13T08:35:54Z","content_type":"text/html","content_length":"5890","record_id":"<urn:uuid:bb921f69-f685-4f18-bc88-e71d98922adc>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00001.warc.gz"}
Scaling Laws for Reward Model Overoptimization In reinforcement learning from human feedback, it is common to optimize against a reward model trained to predict human preferences. Because the reward model is an imperfect proxy, optimizing its value too much can hinder ground truth performance, in accordance with Goodhart’s law. This effect has been frequently observed, but not carefully measured due to the expense of collecting human preference data. In this work, we use a synthetic setup in which a fixed “gold- standard” reward model plays the role of humans, providing labels used to train a proxy reward model. We study how the gold reward model score changes as we optimize against the proxy reward model using either reinforcement learning or best-of-n sampling. We find that this relationship follows a different functional form depending on the method of optimization, and that in both cases its coefficients scale smoothly with the number of reward model parameters. We also study the effect on this relationship of the size of the reward model dataset, the number of reward model and policy parameters, and the coefficient of the KL penalty added to the reward in the reinforcement learning setup. We explore the implications of these empirical results for theoretical considerations in AI alignment. 1 Introduction Goodhart’s law is an adage that states, “When a measure becomes a target, it ceases to be a good measure.” In machine learning, this effect arises with proxy objectives provided by static learned models, such as discriminators and reward models. Optimizing too much against such a model eventually hinders the true objective, a phenomenon we refer to as overoptimization. It is important to understand the size of this effect and how it scales, in order to predict how much a learned model can be safely optimized against. Moreover, studying this effect empirically could aid in the development of theoretical models of Goodhart’s law for neural networks, which could be critical for avoiding dangerous misalignment of future AI systems. In this work, we study overoptimization in the context of large language models fine-tuned as reward models trained to predict which of two options a human will prefer. Such reward models have been used to train language models to perform a variety of complex tasks that are hard to judge automatically, including summarization [Stiennon et al., 2020], question-answering [Nakano et al., 2021, Menick et al., 2022], and general assistance [Ouyang et al., 2022, Bai et al., 2022, Glaese et al., 2022]. Typically, the reward model score is optimized using either policy gradient- based reinforcement learning or best-of-n sampling, also known as rejection sampling or reranking. Overoptimization can occur with both methods, and we study both to better understand whether and how overoptimization behaves differently across both methods. A major challenge in studying overoptimization in this context is the expense of collecting human preference labels. A large number of labels are required to accurately estimate overall preference probabilities, and this is exacerbated by small effect sizes and the need to take many measurements in order to fit scaling laws. To overcome this, we use a synthetic setup that is described in Section 2, in which labels are supplied by a “gold-standard” reward model (RM) instead of humans. Our main results are empirically validated functional forms for the gold reward model scores R as a function of the Kullback–Leibler divergence from the initial policy to the optimized policy KL := D [KL] (π π[init]), which depends on the method of optimization used. This KL distance between the initial and optimized policies increases monotonically during during RL training (fig. 14), and can be computed analytically as a function of n for BoN. Further, because it is a quadratic metric of distance [Bai et al.,2022, Section 4.3], we will define d := D[KL] (π π[init]), and write our functional forms in terms of d. We find empirically that for best-of-n (BoN) sampling, and for reinforcement learning,^1 Here, R(0) := 0 by definition and α[RL], β[RL], α[bo][n] and β[bo][n] are parameters that may depend on the number of proxy reward model parameters, the size of the proxy reward model dataset, and so on. We see that these scaling laws make accurate predictions. We also find the following. • RL versus best-of-n. As a function of the KL divergence, reinforcement learning tends to be slower than best-of-n sampling at both optimization and overoptimization. This suggests inadequacies with using KL to compare amount of (over)optimization across methods. However, the relationship between the proxy reward model score and the gold reward model score is similar for both methods. • Smooth coefficient scaling. The α and β coefficients in the BoN and RL functional forms vary smoothly with the number of proxy reward model parameters, following approximate logarithmic trends.^2 This allows prediction of attained gold RM score. • Weak dependence on policy size. While larger policies perform better overall and benefit less from optimization against an RM as measured by increase in gold reward, they lead to very similar amounts of overoptimization, as measured through the gap between the proxy and gold scores (which indicates the shortfall between predicted and actual reward), and KL distance at which the maximum gold RM score is attained. • KL penalty ineffectiveness. In our reinforcement learning setup, using a KL penalty increases the proxy reward model score that can be achieved for a given KL divergence, but this does not correspond to a measurable improvement in the gold RM score–KL[RL] frontier. However, we note this result could be particularly sensitive to hyperparameters. Finally, we discuss the implications of these findings for Reinforcement Learning From Human Feedback (RLHF), existing models of Goodhart’s law, and AI Alignment more broadly. 2 Methodology The setting used throughout this paper is the same as for InstructGPT [Ouyang et al., 2022]. In our environment, the observations are text prompts and the policy is used to generate a response to the prompt. The prompts are drawn from a broad range of natural language instructions describing different language model tasks. Then, a learned RM is used to provide the reward signal for the response, which is used by either RL or BoN for optimization. For all experiments, we use pretrained GPT-3 series language models as the initial checkpoint [Brown et al., 2020]. All initial policies are trained with supervised fine-tuning (SFT) on human-generated InstructGPT demonstrations [Ouyang et al., 2022] for 2 epochs. All RMs also use the GPT-3 architecture but have an added scalar head to output the reward. Figure 1: Reward model (RM) parameter size scaling experiments using the InstructGPT environment. Policy size is held constant (1.2B), while reward model size is varied. The x-axes have a square-root scale. Note that the plots have different x-axes. The gold reward represents the ground truth reward; we observe that when we optimize for a learned proxy of the gold reward, the gold reward initially increases and later decreases. We show that our functional forms fit this effect well. Figure 2: Diagram of the real and synthetic RM training setups. Human labellers generate comparison data. In the real RLHF setting, this data is used to train a proxy RM that is optimized by RL/BoN. In our synthetic setting, we instead use a “Gold RM” as our ground truth. In both settings, the proxy RM is a proxy for the ground truth process generating the labels (either the human or gold RM). The RL experiments use Proximal Policy Optimization (PPO) [Schulman et al., 2017]. KL penalty for all RL experiments is set to 0 except for in section 3.6. See appendix C for all other hyperparameters. We mostly use defaults for the PPO hyperparameters; thus, it is possible that there exist different trends for other hyperparameter configurations. In BoN, we generate n trajectories for the policy and use the reward model to pick the one with the highest proxy RM score. We use the unbiased estimator from Nakano et al. [2021, Appendix I] to compute all of the gold and proxy scores for intermediate n between 1 and the maximum n with lower variance and more efficiently than the naive estimator of randomly sampling n samples with replacement repeatedly and taking the mean of the maximum gold and proxy RM scores. The KL distances for BoN are computed analytically: KL[bo][n] = log n − n−1 [Stiennon et al., 2020, Appendix G.3]. 2.1 Synthetic Data Setup Because getting a ground truth gold reward signal from human labellers is expensive, we instead use a synthetic task where the ground truth is defined to be the output of a particular large “gold” RM. The 6B reward model from Ouyang et al. [2022] is used as the gold RM, and our proxy RMs vary from 3M to 3B parameters^3. This synthetic gold reward is used to label pairs of rollouts from the policy given the same prompt to create synthetic RM training data. The synthetic comparisons are created deterministically by always marking the trajectory with the higher gold RM score as preferred. ^4 We generate 100,000 synthetic comparisons and reserve 10% of these as a held out test set for computing the validation loss of RMs. See fig. 2 for a diagram of the synthetic setup. 2.2 Recalibration The RM scores are translation-invariant, so to ensure comparability across different reward models, we recenter each RM such that the average reward of the initial policy is 0. We also unit normalize the variance of the gold RM scores.^5 Because our hard thresholding synthetic data setup produces labels that are miscalibrated (since they do not incorporate the gold RM’s confidence), we recalibrate the proxy RMs by rescaling the logits to minimize cross-entropy loss using a validation set of soft labels. All renormalization and recalibration is applied after the experiments; this does not affect BoN at all, and likely has no impact on RL because Adam is loss scale invariant, though it is possible that there are slight differences due to algorithmic details. 3 Results 3.1 Fitting and validating functional forms We chose our functional forms through experimentation with all RM data and parameter scaling curves in the remainder of this paper. The BoN functional form was hypothesized using data up to n = 1000. In order to validate the functional forms, we performed a BoN experiment with up to n = 60, 000 (KL 10 nats), after only having seen data up to n = 1, 000 (KL 6 nats). As this experiment was conducted after the functional form was hypothesized based on data up to 6 nats, this was a true advance prediction. We also test extrapolation of the BoN and RL functional forms from low KLs to to unseen larger KLs; see fig. 26 for details. We also attempted to model the proxy scores but were unable to obtain a satisfactory fit. For BoN, despite visual similarity, a linear fit (dα[bo][n]) did not work well (fig. 20). The predictions for RL and BoN are not as easily modelled as the gold score predictions. We leave a better understanding of the proxy RM score behavior to future work. 3.2 Scaling with RM Parameter Count We hold policy size (1.2B) and data size (90,000) constant (fig. 1). We observe that for the gold RM scores, α[bo][n] and β[bo][n] change smoothly with RM size (figs. 3a and 3b). For RL, we find that we can hold α[RL] constant across all RM sizes, resulting in a clean scaling curve for β[RL] (fig. 3c). These scaling laws allow us to predict properties of training runs; for instance, we can also predict the peak gold RM scores for different RM sizes (fig. 12). β[bo][n]. We also see smooth scaling in the proxy score’s α[bo][n] and β[bo][n]. However, for the reasons in section 3.1, we are less confident about these fits. For both BoN and RL, we observe systematic underestimates of the proxy√reward model when extrapolated to higher KLs. Both appear to eventually grow roughly linearly in KL, as in Bai et al. [2022]. Figure 3: The values of α[bo][n], β[bo][n] and β[RL] in the BoN and RL overoptimization scaling laws for both proxy (dashed line) and gold (solid line) rewards as they scale with parameter count. 3.3 Scaling with RM Data Size We hold RM size constant (12M) and sweep RM data size for both RL and BoN.^6. Overall, the results are consistent with intuition: more data leads to better gold scores and less goodharting. The scaling of α and β with data size are not as cleanly described as for RM size scaling (fig. 17, fig. 18). For all RM sizes, we observe that for amounts of data less than around 2,000 comparisons^7, there is very little improvement over near-chance loss (Figure 6). This is also reflected in gold scores after optimization (fig. 21). After this threshold, all models improve with more data, though larger RMs Figure 4: RM data scaling experiments. RM size is held constant (12M), while RM data is varied. The x-axis has a square root scale. Note that the plots have different axes. Dotted lines indicate proxy rewards, solid lines indicate gold rewards. generally improve faster. Interestingly, although larger RMs result in better gold scores overall, they do not appear to have this critical threshold substantially earlier than smaller models.^8 We hypothesized that two RMs of equal validation loss would achieve the same robustness against optimization, regardless of the combination of RM size and RM data size. Our results provide some weak evidence for this hypothesis (fig. 5). Figure 5: RM validation loss vs BoN RM score @ n=1000. Most points in this figure are already averaged over multiple seeds. Figure 6: RM losses, broken down by data size and RM size 3.4 Scaling with Policy Size We briefly explore the impact of policy size by holding the RM size constant (12M) and evaluating two different policy sizes. We also perform the same experiment with a different RM size (3B), observing similar results (fig. 22). Larger policies see less benefit from optimization against an RM, but don’t overoptimize more. We observe that the 6B policy run has a smaller difference between its initial and peak gold reward model scores than the 1.2B policy run. This is most visible in the BoN plot (fig. 7a).^9 However, while we might expect that a larger policy overoptimizes substantially faster, contrary to intuition, we find that both gold scores peak at almost the same KL. In fact, the gap between the proxy and gold scores is almost the same between the two policy sizes (fig. 24). We can interpret this gap, the Figure 7: Policy scaling experiments. RM size is held constant (12M), while policy size is varied. The x-axis has a square root scale. Note that the plots have different axes. Dotted lines indicate proxy rewards, solid lines indicate gold rewards. The asterisks in the RL plot indicate the max gold score for each policy size. between the predicted and actual rewards, as being indicative of the extent to which the proxy RM is exploited. We discuss this result further in section 4.4. 3.5 RL vs BoN A priori, we might expect reinforcement learning via PPO [Schulman et al., 2017] and best-of-n to apply optimization in very different ways. As such, we ask whether this difference in optimization results in different overoptimization characteristics. Similarities would potentially indicate candidates for further study in gaining a more fundamental understanding of overoptimization in general, and differences opportunities for better optimization algorithms. We note the following: RL is far less KL-efficient than BoN. Viewing KL distance as a resource to be spent, we observe that RL “consumes” far more KL than BoN. This means that both optimization and overoptimization require more KL to occur with RL. Intuitively, BoN searches very locally around the initial policy, and thus KL[bo][n] increases with roughly log(n). For RL on the other hand, each step modifies the policy from the policy of the previous step—KL increases approximately quadratically with step in the absence of KL penalty (Figure 16, Figure 14). An implication of this result is that KL distance is an inadequate metric for quantity of (over)optimization; we discuss this further in section 4.1. When looking at proxy vs gold RM scores, BoN and RL look more similar. The proxy RM score is another possible metric for quantity of optimization, because it is the value that is being directly optimized for. Using it as the metric of optimization leads to significantly more analogy between RL and BoN than KL distance does. However, we do observe that RL initially has a larger proxy-gold gap (i.e requires more proxy RM increase to match BoN), but then peaks at a higher gold RM score than BoN (fig. 8). 3.6 Effect of KL Penalty We observe in our setting that when varying the KL penalty for RL, the gold RM scores depend only on the KL distance of the policy KL[RL] (Figure 9). The KL penalty only causes the gold RM score to converge earlier, but does not affect the KL[RL]-gold reward frontier, and so the effect of the penalty on the gold score is akin to early stopping (Figure 14). However, we have seen some evidence that this result could be particularly sensitive to hyperparameters. Because we observe that using KL penalty has a strictly larger proxy-gold gap, we set KL penalty to 0 for all other RL experiments in this paper. It is important to note that PPO’s surrogate objective incorporates an implicit penalty on D[KL] (π[old] π), where π[old] is a recent policy (not the initial policy) [Schulman et al.,2017]. This penalty is used to control how fast the policy changes, but also has an indirect effect on the KL we Figure 8: Proxy vs gold RM score for both BoN and RL. RL curves are truncated to a proxy RM score of 1.6 for readability. study here, D[KL] (π π[init]), causing it to grow much more slowly (providing the implementation is well-tuned). We do not know why this indirect effect appears to lead to less overoptimization than an explicit KL penalty. Figure 9: RL experiments with various KL penalties. Policy size (1.2B) and RM size (1.2B) are held constant. Dotted lines indicate proxy rewards, solid lines indicate gold rewards. We observe the effect of the KL penalty on the gold score as being equivalent to early stopping. 4 Discussion 4.1 KL as a measure of amount of optimization For any given fixed optimization method, KL yields clean scaling trends, such as the ones observed in section 3.2, and consistent peak gold RM score KLs as in section 3.4. However, because it’s clear that different methods of optimization spend KL very differently (section 3.5), it should not be used to compare the amount of optimization between different optimization algorithms. There exist pertubations to a policy that are orthogonal to the reward signal that would result in increases in KL that do not increase either gold or proxy reward; conversely, extremely small but well targeted perturbations could substantially change the behavior of the policy within a small KL budget. 4.2 Relation to Goodhart Taxonomy One useful taxonomy for various Goodhart effects is presented in Manheim and Garrabrant [2018], categorizing Goodhart’s Law into 4 categories: Regressional, Extremal, Causal, and Adversarial. In this section, we discuss our results in the framework of this taxonomy. 4.2.1 Regressional Goodhart Regressional Goodhart occurs when our proxy RMs depend on features with noise. The simplest toy example of this is a proxy reward Xˆ which is exactly equal to the gold reward X plus some independent noise Z. When optimizing against this proxy, some amount of optimization power will go to selecting for noise, leading to a gold reward less than predicted by the proxy. More formally, for independent absolutely continuous random variables X and Z with X normally distributed and either (a) Z normally distributed or (b) Z E [Z] < δ for some δ > 0, this model predicts a gold reward that is: where ε = 0 in case (a) and ε = o (Var (Z)) as δ → 0 in case (b). See appendix A for the proof. Intuitively, we can interpret eq. (1) as stating that the optimization power expended is divided between optimizing the gold reward and selecting on the noise proportional to their variances. This also implies that if this is the only kind of Goodhart present, the gold reward must always increase monotonically with the proxy reward; as we observe nonmonotonic behavior (fig. 8), there must be either noise distributions violating these assumptions or other kinds of Goodhart at play. This result lends itself to an interpretation of the α term in the RL√and BoN gold score scaling laws: since for both RL and BoN the proxy scores are roughly linear in KL, the difference in the slope of the proxy score and the linear component of the gold score (i.e the α term) can be interpreted as the amount of regressional Goodhart occurring. 4.2.2 Extremal Goodhart We can think of out of distribution failures of the RM as an instance of extremal Goodhart. As we optimize against the proxy RM, the distribution of our samples shifts out of the training distribution of the RM, and thus the relation between the proxy and gold scores weakens. For instance, suppose in the training distribution a feature like answer length always indicates a higher quality answer, and thus the proxy RM infers that longer answers are always better, even though at some point outside the training distribution, selecting on longer answers no longer improves We can also think of this as the proxy failing to depend on relevant features; this failure bears resemblance to the setting considered in Zhuang and Hadfield-Menell [2020], where a failure of the proxy to consider all features, under certain conditions, leads to overoptimization with unbounded loss of utility regardless of optimization method. We expect extremal Goodharting to be primarily responsible for the nonmonotonicity of the gold RM scores in this paper, and is mostly responsible for the β term, which in the limit of optimization, results in an unbounded loss of utility. This lends a natural interpretation to the smooth decrease in β for both BoN and RL with increased RM size as smooth improvements in model robustness (fig. 4.2.3 Causal Goodhart We can think of causal Goodhart as being a generalization of regressional Goodhart: there may exist correlations between features and gold score where the causal structure of the problem is such that selecting on the feature does not increase the gold score. For instance, suppose answer length is correlated with quality due to some other common cause (say, informativeness); then, the proxy RM may learn to use answer length as a feature, and when we select against the proxy we get longer answers that do not increase on actual quality.^11 In our experiments, we would observe causal Goodhart as behaving similarly to regressional Goodhart. 4.2.4 Adversarial Goodhart Adversarial Goodhart occurs when the policy actively manipulates the proxy. We do not expect the effects of adversarial Goodhart to be captured in this work, as the models involved are not powerful enough to implement adversarial strategies. However, given the constant improvement of ML capabilities, it is entirely plausible that ML systems will one day become capable enough to do so [Hubinger et al., 2019]. When this occurs, the scaling laws observed in this paper may break down. Thus, we advise caution when using these results for extrapolation. 4.3 Implications for iterated RLHF When conducting reinforcement learning from human feedback, it is preferable to use an online setup, in which fresh human feedback data is periodically used to train a new reward model, to mitigate overoptimization [Bai et al., 2022]. Our scaling law allows us to analyze the effect of this iterative approach under some simplifying assumptions. We assume firstly that the sc√aling coefficients α[RL] and β[RL] remain constant across iterations, and secondly that the distance d = KL is additive across iterations (because of how KL appears to grow empirically as in Figure 14). Under these assumptions, the final gold reward model score after k iterations each covering a distance d/k is given by R[RL] (d) = d (α[RL] − β[RL] log (d) + β[RL] log (k)) . Two interesting observations follow from this. Firstly, the iterative approach does not affect any Goodharting captured by the α[RL] term (such as regressional Goodharting, as discussed in Section 4.2.1). Secondly, the effect of the iterative approach is to increase the final gold RM score by an amount proportional to both d and log (k), namely β[RL]d log (k) . Note that this result can only hold up to some maximum value of k, and we expect our scaling law to break down below some minimum distance. Further research is required to determine what this minimum is, as well as to what extent our simplifying assumptions hold in practice. 4.4 Policy size independence Our observation that larger SFT policies seem to exhibit the same amount of overoptimization during RL implies that larger policies do not increase the amount of optimization power applied to the RM or learn faster, even though they start out with higher performance on the gold score. While it is expected that larger policies have less to gain from optimizing against the same RM, we might also expect the gold score to peak at a substantially earlier KL distance, analogous to what we see when we scale the RM size (section 3.2), or for larger policies to more efficiently utilize the same number of RL feedback steps (section 3.3)^12. One possible hypothesis is that, because RLHF can be viewed as Bayesian inference from the prior of the initial policy [Korbak et al., 2022]^13, increases in policy size are only improving the modelling accuracy of the human demonstration distribution. 4.5 Limitations and Future Work In addition to the overoptimization studied in this paper (due to the mismatch between the reward model and the ground truth labels), there exists another source of overoptimization due to mismatch between the ground truth labels and the actual human intent. This contains issues ranging from the mundane, such as labellers choosing options that only appear to match their intent^14, to substantially more philosophically fraught issues [Armstrong and Mindermann, 2018, Sunstein et al., 2001]. The main limitation of this work is that this additional source of overoptimization is not captured in the setting of this paper. See section 5 for discussion of related work in alignment. Some additional limitations and future directions include: • Validating these results on other environments and experimental setups. While the experiments in this paper all use the InstructGPT environment, the main value of these results lies in the extent to which they reflect general phenomema. Confirming whether these results generalize to other settings would be extremely valuable to that end.^15 • Validating the synthetic setting. The synthetic setting might not transfer to real world settings, for instance because there is substantial correlation between RMs. • Investigating methods for making RMs more robust to optimization. While there has been prior work in this direction (see section 5), there is still much work to be done in systematically investigating ways to make RMs more robust. • Exploring other forms of optimization and categorizing their differences. While this work focuses exclusively on BoN and RL there are other ways of applying optimization pressure against a model of a reward signal, either implicit or explicit. This includes GeDi- like steering, Decision Transformers^16, variants of BoN like beam search, and other RL algorithms. • Better understanding the functional form of proxy RM scores. In our modeling, we find that the proxy RM scores are more difficult to predict for both BoN and RL (section 3.2). While they seem to have a major linear component, there is sufficient variation that fitting a linear regression is not very good at predicting extrapolated proxy RM scores. • Exploring adversarial Goodhart empirically. In this work we deal with systems not powerful enough to cause adversarial Goodhart. However, it is plausible that adversarial Goodhart is especially important, or is associated with phase changes that break the trends seen in this paper. • Exploring scaling with policy size in more detail. Our exploration of policy size scaling in this paper was limited to only two policy sizes. It is possible that there exist trends not seen in our exploration when considering the policy size more carefully. • Exploring multi-iteration RLHF. In particular, checking for deviations from the assump- tions of section 4.3. We hope this paper leads to future work further bridging conceptual and empirical alignment research. 5 Related Work Goodhart’s Law in its modern formulation was first introduced in Hoskin [1996], with many of the key ideas introduced in prior works [Campbell, 1969, Goodhart, 1975]. Many approaches have been proposed for reducing overoptimization in general [Taylor, 2016, Everitt et al., 2017], as well as in RMs [Gleave and Irving, 2022], including within the field of adversarial robustness [Chakraborty et al., 2018]. Overoptimization of reward models can be viewed as a special case of specification gaming (also known as reward hacking). Previous work has shown numerous examples of such behavior in a wide variety of settings [Krakovna et al., 2020, Lehman et al., 2020]. Pan et al. [2022] explores a diverse set of RL environments and finds phase transitions in some settings. A number of works have proposed theoretical models of Goodhart’s Law and reward hacking [Krakovna and Kumar, 2019, Manheim and Garrabrant, 2018, Skalse et al., 2022], including Zhuang and Hadfield-Menell [2020] which exhibits very similar overoptimization curves as observed in this paper in some toy environments. One can think of overfitting as a special case of Goodhart’s law where the proxy is the score on some finite set of samples, whereas our actual objective includes its generalization properties as well. Overfitting has been observed and studied in RL settings [Zhang et al., 2018a,b, Farebrother et al., 2018, Cobbe et al., 2019]. Song et al. [2019] studies “observational overfitting” in RL settings, which is closely related to causal Goodhart [Manheim and Garrabrant, 2018]. Adversarial attacks and robustness are also very closely related fields. Many works have demonstrated the existence of adversarial examples in all kinds of neural networks [Szegedy et al., 2013, Lin et al., 2017, Ebrahimi et al., 2018, Dai et al., 2018], and proposed methods to measure and increase neural network robustness [Gu and Rigazio, 2014, Zheng et al., 2016, Carlini et al., 2019, Guo et al., 2021]. Scaling laws have seen substantial success in machine learning for predicting properties of language models [Kaplan et al., 2020, Henighan et al., 2020, Hernandez et al., 2021] and has led to better theoretical understanding of language models [Sharma and Kaplan, 2020, Bahri et al., 2021]. Reinforcement learning from human feedback [Christiano et al., 2017, Ibarz et al., 2018] has been used broadly in language models [Stiennon et al., 2020, Ouyang et al., 2022, Nakano et al., 2021, Bai et al., 2022]. It is also a first step towards recursive reward modelling [Leike et al., 2018], an approach towards reducing the additional source of overoptimization described in section 4.5, though it is subject to some theoretical limitations [Christiano et al., 2021]. We observe similar approximately-linear proxy RM scores observed in Bai et al. [2022]^17, though we observe an early-KL bend in the proxy RM scores, and there are some occasional outliers with very small RMs and data sizes. More broadly, AI alignment is the problem of ensuring that the goals of AI systems are aligned with the goals of humans [Ngo, 2022], including future AI systems which may exceed humans [Bostrom, 2014]. 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Let X and Z be independent absolutely continuous random variables with X normally distributed and either (a) Z normally distributed or (b) \|Z − E [Z]\| < δ for some δ > 0. Then for any real number c and as δ → 0, where ε = 0 in case (a) and ε = o (Var (Z)) in case (b). Proof. First note that by making the substitutions X^I = X E [X] and Z^I = Z E [Z], we may assume without loss of generality that E [X] = E [Z] = 0. Let Var (X) = σ^2 and Var (Z) = τ ^2. In case (a), the pair (X, X + Z) is bivariate normal with covariance matrix and the result follows by standard properties of conditional distributions of multivariate normal distributions. In case (b), let f[X] and f[Z] be the probability density functions of X and Z respectively. Then as required. B RL form details Ideally all overoptimization forms would have finite slope at the origin. We tried the following forms: • d (α[RL] β[RL] log (1 + d)): Has slope α at the origin; however, has substantially worse extrapolation behavior. We can replace the 1 with a learned E but that introduces another degree of • Power laws d (α[RL] β[RL]d^γRL ): Has slope α at the origin; however, this adds another degree of freedom, and the best fits resulted in small values of γ[RL]. Note that the power law forms with small γ[RL] approximate the RL form that we decided on, as lim[n→∞] n(x^1^/n − 1) = log x. C Hyperparameters Hyperparameter Value RM Adam learning rate multiplier 1.67e-2 RM batch size 64 RL Adam learning rate multiplier 4e-3 RL batch size 256 RL PPO clipping parameter 0.2 RL Timesteps per rollout 256 RL minibatches per epoch 128 RL GAE bootstrapping parameter 0.95 Table 1: Hyperparameters used throughout the experiments. Table 2: A sample of the BoN answers on a single InstructGPT question (policy=1.2B, proxy RM=12M). For each individual question, the gold scores do not follow as clean a trend as they do when averaged over many questions as in fig. 1. Figure 10: Maximum gold scores for all RM size and data size combinations. Figure 11: Validation losses for the proxy RMs in section 3.2 by size, plus the two near-chance level RMs. Figure 12: Max BoN gold scores (α[bo][n]/2β[bo][n]) predicted with the BoN closed form Figure 13: Total number of data points seen does not seem to affect the gold RM score much compared to the number of unique data points seen. Averaged across RM sizes. The numbers of datapoints (2000–8000) is intentionally chosen to straddle the sharp increase in performance. The validation loss of the 1×2000, 1×8000, and 4×2000 RMs are 0.686109, 0.654857, and 0.683869 respectively. Figure 14: Change in KL[RL] throughout RL training for various different KL penalties. We observe that KL distance increases approximately monotonically with step count, and converges for higher KL Figure 15: KL[RL] with policy size (RM size = 12M) Figure 16: KL[RL] with RM size Figure 17: α[bo][n] with dataset size, averaged across RM sizes Figure 18: β[bo][n] with dataset size, averaged across RM sizes Figure 19: RM data scaling experiments, BoN, RM size=3B Figure 20: The BoN proxy scores are slightly concave, so that a linear fit does not fit well. Figure 21: BoN Gold scores at n=1,000, broken down by data size and RM size. See fig. 6 for RM losses. Vertical dotted line approximately indicates first better-than-random data size. Figure 22: RL experiments with 3B RM and different policy sizes. Figure 23: fig. 7b with all runs normalized from 0. Figure 24: The gap between the proxy and gold scores in the RL policy sweep (fig. 24). Figure 25: The fraction of updates clipped by PPO. (a) BoN Figure 26: Extrapolation quality of fits in fig. 1. The regressions (shown in faint lines) are only fit to data to the left of the vertical black dotted lines. In the case of BoN, this represents a true advance prediction, as the functional form was chosen without collecting any data past a KL of 6 nats. Foot Note 1We note that this form likely does not hold near the origin, as it has infinite slope there. We experimented with a number of different forms, but found worse fits and extrapolation. See appendix B for more details. 2The coefficient αRL in particular being nearly independent of RM parameter count. 3We originally trained two additional RMs smaller than 3M parameters, which achieved near-chance accuracy and were off-trend, and so were excluded. 4We had experimented with sampling for creating labels, but observed noisier results. 5We later decided this was unnecessary but decided not to change it. 6For BoN, we actually sweep all combinations of RM size and data size; see fig. 10. For a version of fig. 4a against a 3B RM, see fig. 19. 7To test the hypothesis that some minimum number of RM finetuning steps is needed, we control for the number of SGD steps by running multiple epochs and observe that running 4 epochs instead of 1 yields no change in gold score whatsoever, whereas 1 epoch of 4 times as much data performs substantially better (fig. 13). 8This result contradicts some other internal findings; thus, it is possible that this is an artifact of this particular setup. 9 For a version of the RL plot (fig. 7b) with all runs starting at 0, see fig. 23. 10Optimized policies producing very long answers even when a short answer would be preferred is a real issue that we have observed in other experiments in the InstructGPT setting. 11We can think of noise as a particular case of this where the independent noise is correlated with signal+noise, but of course there is no causal relation between signal and noise. 12It is also not the case that the 6B policy run has higher KL distance for the same number of RL steps; in fact, we observe that it has lower KL distance for the same number of steps (fig. 15) 13The result of Korbak et al. [2022] concerns varying KL penalties rather than KL distances with no KL penalty, but as we observe in section 3.6, this is equivalent on our setting. 14For instance, the example of a robotic hand learning from human feedback to only appear to grasp a ball, presented in https://openai.com/blog/deep-reinforcement-learning-from-human-preferences/ [Christiano et al., 2017] 15In the course of our experiments, we observed visually similar results on the WebGPT environment [Nakano et al., 2021]. 16One could consider measuring the actual achieved ground truth/gold score achieved for each “proxy” score conditioned on, a la fig. 8, as testing the implicit reward-behavior mapping encoded by the 17Note that Bai et al. [2022] scaled the policy size with the RM size, while we hold the policy size constant.	{"url":"https://customaiintegrations.com/scaling-laws-for-reward-model-overoptimization/","timestamp":"2024-11-09T22:36:25Z","content_type":"text/html","content_length":"163941","record_id":"<urn:uuid:50832863-0676-447a-8134-98b82668d33e>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00270.warc.gz"}
Matrix Multiplication Made EasyGradeAmathHelp.comMatrix Multiplication Made EasyGradeAmathHelp.com Matrix Multiplication Before attempting matrix multiplication, makes sure you know the basics first. Multiplication is much more complicated than some of the other matrix operations, like matrix addition and scalar multiplication. GradeA will show you two approaches: the Turn & Flip and the Zipper . Choose the method you like the best! Before you can multiply matrices, you need to know when the operation is possible. Sometimes you cannot multiply the two matrices together at all. When is Matrix Multiplication Defined? The key to answering this question is to look at the dimensions of each matrix. Remember, they are always listed as row x column. To see if the operation is defined, list the two dimensions next to each other. Important Note: The order does matter! Next, check to see if the middle numbers are the same! Once the matrix multiplication is defined, you can find the dimensions of the result (the answer). The outside numbers will give you the new dimensions. For the examples above, the results would be (2 x 3) and (3 x 1). Here are more examples: │Original Dimensions: │(3 x 1)(3 x 1) │(2 x 3)(3 x 2)│(4 x 1)(2 x 4) │ │Multiplication Possible? │No! (different)│Yes! (same) │No! (different)│ │Resulting Dimensions: │ (none) │ (2 x 2) │ (none) │ Now that you know how to determine the dimensions of the resulting matrix, now you need to know how to actually multiply them. We are multiplying a (2 x 2)(2 x 2) - so the result will also be a (2 x 2). You will need to multiply each corresponding row times the corresponding column - notice the circles below. Once you multiply the matching pieces, add the results. It might seem a little confusing at first, but after a little practice you will get it. Of course, you might prefer to use the zipper method instead... If you had a difficult time understanding the turn and flip method, maybe the zipper method will be easier for you to understand. We will use the same example: First, take the second matrix and raise it above the first. Now you are going to "zip" in the numbers that line up. Look at the light blue arrows, as well as the colored circles... So, now that you have seen both methods of matrix multiplication, which do you prefer: the turn and flip method or the zipper method? Return to free algebra help or visit the GradeA homepage.	{"url":"http://www.gradeamathhelp.com/matrix-multiplication.html","timestamp":"2024-11-03T06:29:51Z","content_type":"text/html","content_length":"30521","record_id":"<urn:uuid:bcb23375-a163-4772-a3cb-e8035e02bb4a>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00388.warc.gz"}
statistics answered \|Statistics tutor online\| My Mathlab answers \|Ap Statistics Tutor \| calculation of the harmonic mean A PHP Error was encountered Severity: Notice Message: Undefined index: userid Filename: views/question.php Line Number: 212 File: /home/mycocrkc/statisticsanswered.com/application/views/question.php Line: 212 Function: _error_handler File: /home/mycocrkc/statisticsanswered.com/application/controllers/Questions.php Line: 416 Function: view File: /home/mycocrkc/statisticsanswered.com/index.php Line: 315 Function: require_once ben Punditsdkoslkdosdkoskdo The harmonic mean is often used as a measure of center for data sets consisting of rates of change, such as speeds. It is found by dividing the number of values n by the sum of the reciprocals of all values. (No value can be zero.) The author drove 1163 miles to a conference in Orlando, Florida. For the trip to the conference, the author stopped overnight, and the mean speed from start to finish was 38 mi/h. For the return trip, the author stopped only for food and fuel, and the mean speed from start to finish was 56 mi/h. Is the actual “average” speed for the round-trip the mean of 38 mi/h and 56 mi/h? Why or why not? What is the harmonic mean of 38 mi/h and 56 mi/h, and does this represent the true “average” speed?	{"url":"https://www.statisticsanswered.com/questions/857/calculation-of-the-harmonic-mean?ref=anwser","timestamp":"2024-11-06T04:17:40Z","content_type":"text/html","content_length":"74875","record_id":"<urn:uuid:6e6fd570-af7d-4595-8926-f14671b82682>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00288.warc.gz"}
Triangle Bounded by 3 Lines Given three mutually non-parallel lines in the xy coordinate plane, those lines will either intersect at a common point, or intersect pairwise in three distinct points and form the boundary of a To determine the area, perimeter, and angles of this triangle you must first solve for the intersection points. Then, once you know the three points that define the vertices of the triangle, you can either use vector algebra or a solver to find the side and angle measures and the area. To use the calculator on the left, input the equations of the three lines in standard form ax + by = c. The calculator will output the coordinates of the vertices, angles, sides, area, and perimeter. See also How to Find the Area, Sides, and Angles of Triangle Given Three Coordinates (2-D and 3-D) © Had2Know 2010	{"url":"https://www.had2know.org/academics/triangle-bounded-3-lines-calculator.html","timestamp":"2024-11-13T02:41:52Z","content_type":"text/html","content_length":"35855","record_id":"<urn:uuid:66769e33-c1eb-4656-9c78-46543ebde9c7>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00277.warc.gz"}
How many labeled and unlabeled binary tree can be there with N nodes? Open-Source Internship opportunity by OpenGenus for programmers. Apply now. Reading time: 15 minutes In this article, we see how many labeled and unlabeled binary trees we can have with N nodes. This is related to the Catalan Numbers. Binary Tree : A tree whose elements have 0 or 1 or 2 children is called a binary tree. Since each element in a binary tree can have only 2 children, we typically name them the left and right child. A Binary Tree node contains following parts: 1. Data 2. Pointer to left child 3. Pointer to right child Labeled Binary tree - A Binary Tree is labeled if every node is assigned a label Unlabeled Binary Tree - A Binary Tree is unlabeled if nodes are not assigned any label. Unlabeled Binary tree We have to count the total number of trees we can have with n nodes. So for n=1 , Tree = 1 n=2 , Tree = 2 n=3, Tree = 5 n=4 , Tree = 14 Actually what we are doing is considering all possible pair of counts for nodes in left and right subtrees and multiplying the counts for a particular pair and then adding the results of all pairs. But if we observe the pattern resembles the Catalan Numbers. The nth Catalan Number is given by: T(n) = (2 n ) ! / [(n+1) ! n !] Labelled Binary Tree Here we can use the count of the unlabeled trees. Every unlabeled tree with n nodes can create n! labeled trees by assigning different permutations of labels to all nodes. T(n) = [ (2 n ) ! / (n+1) ! n ! ] * n !	{"url":"https://iq.opengenus.org/how-many-labelled-and-unlabeled-binary-tree-can-be-there-with-n-nodes/","timestamp":"2024-11-02T21:00:31Z","content_type":"text/html","content_length":"60908","record_id":"<urn:uuid:6cd4e264-f8f1-41fc-9e96-71e2a28387da>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00146.warc.gz"}
Congruence in Right Triangles Looking for some terminology used with right triangles? Then this tutorial was made for you! In this tutorial, you'll be introduced to the names for the different parts of a right triangle. Check it There's a special theorem that helps you quickly figure out if two right triangles are congruent. This tutorial introduces you to that theorem and shows you how to use it!	{"url":"https://virtualnerd.com/texas-digits/txh-geo/congruent-triangles/congruence-in-right-triangles/","timestamp":"2024-11-13T12:20:18Z","content_type":"text/html","content_length":"17930","record_id":"<urn:uuid:de84743a-3565-446a-bffc-59cbd0b1bc72>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00613.warc.gz"}
Markov’s inequality is very general and hence very weak. Assume that X is a non-negative random variable, a > 0, and X has a finite expected value, Then Markov’s inequality says that In [1] the author gives two refinements of Markov’s inequality which he calls Hansel and Gretel. Hansel says and Gretel says Related posts [1] Joel E. Cohen. Markov’s Inequality and Chebyshev’s Inequality for Tail Probabilities: A Sharper Image. The American Statistician, Vol. 69, No. 1 (Feb 2015), pp. 5-7 Inequalities for inequality: Gini coefficient lower bounds The Gini coefficient, a.k.a. Gini index, of a set of numbers is the average of all differences divided by twice the mean. Specifically, let Then the Gini coefficient of x is defined to be where μ is the mean of the set. The Gini coefficient is often used in economics to measure inequalities in wealth. Now suppose the data is divided into r disjoint groups: We would like to estimate the Gini coefficient of the entire group from Gini coefficients of each subgroup. This individual Gini coefficients alone are not enough data for the task, but if we also know the size and sum of each subgroup, we can compute lower bounds on G. The paper [1] gives five such lower bounds. We will present the five lower bounds and see how well each does in a simulation. Zagier’s lower bounds Here are Zagier’s five lower bounds, listed in Theorem 1 of [1]. Here n[i] is the size of the ith subgroup and X[i] is the sum of the elements in the ith subgroup. Also, n is the sum of the n[i] and X is the sum of the X[i]. G[0] is the Gini coefficient we would get if we replaced each subgroup with its mean, eliminating all variance within subgroups. I drew 102 samples from a uniform random variable and computed the Gini coefficient with def gini(x): n = len(x) mu = sum(x)/n s = sum(abs(a-b) for a in x for b in x) return s/(2mun*2) I split the sample evenly into three subgroups. I then sorted the list of samples and divided into three even groups again. The Gini coefficient of the entire data set was 0.3207. The Gini coefficients of the three subgroups were 0.3013, 0.2798, and 0.36033. When I divided the sorted data into three groups, the Gini coefficients were 0.3060, 0.0937, and 0.0502. The variation in each group is the same, but the smallest group has a smaller mean and thus a larger Gini coefficient. When I tested Zagier’s lower bounds on the three unsorted partitions, I got estimates of [0.3138, 0.3105, 0.3102, 0.3149, 0.1639] for the five estimators. When I repeated this exercise with the sorted groups, I got [0.1499, 0.0935, 0.0933, 0.1937, 0.3207] The bounds for the first four estimates were much better for the unsorted partition, but the last estimate was better for the sorted partition. More posts on inequalities [1] Don Zagier. Inequalities for the Gini coefficient of composite populations. Journal of Mathematical Economics 12 (1983) 102–118. Mahler’s inequality I ran across a reference to Mahler the other day, not the composer Gustav Mahler but the mathematician Kurt Mahler, and looked into his work a little. A number of things have been named after Kurt Mahler, including Mahler’s inequality. Mahler’s inequality says the geometric mean of a sum bounds the sum of the geometric means. In detail, the geometric mean of a list of n non-negative real numbers is the nth root of their product. If x and y are two vectors of length n containing non-negative components, Mahler’s inequality says G(x + y) ≥ G(x) + G(y) where G is the geometric mean. The left side is strictly larger than the right unless x and y are proportional, or x and y both have a zero component in the same position. I’m curious why this inequality is named after Mahler. The classic book Inequalities by Hardy, Littlewood, and Polya list the inequality but call it Hölder’s inequality. In a footnote they note that the inequality above appears in a paper by Minkowski in 1896 (seven years before Kurt Mahler was born). Presumably the authors file the inequality under Hölder’s name because it follows easily from Hölder’s inequality. I imagine Mahler made good use of his eponymous inequality, i.e. that the inequality became associated with him because he applied it well rather than because he discovered it. More geometric mean posts Reversed Cauchy-Schwarz inequality This post will state a couple forms of the Cauchy-Schwarz inequality and then present the lesser-known reverse of the Cauchy-Schwarz inequality due to Pólya and Szegö. Cauchy-Schwarz inequality The summation form of the Cauchy-Schwarz inequality says that for sequences of real numbers x[n] and y[n]. The integral form of the Cauchy-Schwarz inequality says that for any two real-valued functions f and g over a measure space (E, μ) provided the integrals above are defined. You can derive the sum form from the integral form by letting your measure space be the integers with counting measure. You can derive the integral form by applying the sum form to the integrals of simple functions and taking limits. Flipping Cauchy-Schwarz The Cauchy-Schwarz inequality is well known [1]. There are reversed versions of the Cauchy-Schwarz inequality that not as well known. The most basic such reversed inequality was proved by Pólya and Szegö in 1925 and many variations on the theme have been proved ever sense. Pólya and Szegö’s inequality says for some constant C provided f and g are bounded above and below. The constant C does not depend on the functions per se but on their upper and lower bounds. Specifically, assume Sometimes you’ll see C written in the equivalent form This way of writing C makes it clear that the constant only depends on m and M via their ratio. Note that if f and g are constant, then the inequality is exact. So the constant C is best possible without further assumptions. The corresponding sum form follows immediately by using counting measure on the integers. Or in more elementary terms, by integrating step functions that have width 1. Sum example Let x = (2, 3, 5) and y = (9, 8, 7). The sum of the squares in x is 38 and the sum of the squares in y is 194. The inner product of x and y is 18+24+35 = 77. The product of the lower bounds on x and y is m = 14. The product of the upper bounds is M = 45. The constant C = 59²/(4×14×45) = 1.38. The left side of the Pólya and Szegö inequality is 38×194 = 7372. The right side is 1.38×77²= 8182.02, and so the inequality holds. Integral example Let f(x) = 3 + cos(x) and let g(x) = 2 + sin(x). Let E be the interval [0, 2π]. The following Mathematica code shows that the left side of the Pólya and Szegö inequality is 171π² and the right side is 294 π². The function f is bound below by 2 and above by 4. The function g is bound below by 1 and above by 3. So m = 2 and M = 12. In[1]:= f[x_] := 3 + Cos[x] In[2]:= g[x_] := 2 + Sin[x] In[3]:= Integrate[f[x]^2, {x, 0, 2 Pi}] Integrate[g[x]^2, {x, 0, 2 Pi}] Out[3]= 171 π² In[4]:= {m, M} = {2, 12}; In[5]:= c = (m + M)^2/(4 m M); In[6]:= c Integrate[f[x] g[x], {x, 0, 2 Pi}]^2 Out[6]= 294 π² Related posts [1] The classic book on inequalities by Hardy, Littlewood, and Pólya mentions the Pólya-Szegö inequality on page 62, under “Miscellaneous theorems and examples.” Maybe Pólya was being inappropriately humble, but it’s odd that his inequality isn’t more prominent in his book. Expected value of X and 1/X Yesterday I blogged about an exercise in the book The Cauchy-Schwarz Master Class. This post is about another exercise from that book, exercise 5.8, which is to prove Kantorovich’s inequality. for non-negative numbers p[i]. is the arithmetic mean of m and M and is the geometric mean of m and M. In words, the weighted average of the x‘s times the weighted average of their reciprocals is bounded by the square of the ratio of the arithmetic and geometric means of the x‘s. Probability interpretation I did a quick search on Kantorovich’s inequality, and apparently it first came up in linear programming, Kantorovich’s area of focus. But when I see it, I immediately think expectations of random variables. Maybe Kantorovich was also thinking about random variables, in the context of linear programming. The left side of Kantorovich’s inequality is the expected value of a discrete random variable X and the expected value of 1/X. To put it another way, it’s a relationship between E[1/X] and 1/E[X], which I imagine is how it is used in practice. I don’t recall seeing this inequality used, but it could have gone by in a blur and I didn’t pay attention. But now that I’ve thought about it, I’m more likely to notice if I see it again. Python example Here’s a little Python code to play with Kantorovich’s inequality, assuming the random values are uniformly distributed on [0, 1]. from numpy import random x = random.random(6) m = min(x) M = max(x) am = 0.5(m+M) gm = (mM)0.5 prod = x.mean() (1/x).mean() bound = (am/gm)2 print(prod, bound) This returned 1.2021 for the product and 1.3717 for the bound. If we put the code above inside a loop we can plot the product and its bound to get an idea how tight the bound is typically. (The bound is perfectly tight if all the x’s are equal.) Here’s what we All the dots are above the dotted line, so we haven’t found an exception to our inequality. (I didn’t think that Kantorovich had made a mistake. If he had, someone would have noticed by now. But it’s worth testing a theorem you know to be true, in order to test that your understanding of the theorem is correct.) More inequalities The baseball inequality There’s a theorem that’s often used and assumed to be true but rarely stated explicitly. I’m going to call it “the baseball inequality” for reasons I’ll get to shortly. Suppose you have two lists of k positive numbers each: This says, for example, that the batting average of a baseball team is somewhere between the best individual batting average and the worst individual batting average. The only place I can recall seeing this inequality stated is in The Cauchy-Schwarz Master Class by Michael Steele. He states the inequality in exercise 5.1 and gives it the batting average interpretation. (Update: This is known as the “mediant inequality.” Thanks to Tom in the comments for letting me know. So the thing in the middle is called the “mediant” of the fractions.) Note that this is not the same as saying the average of a list of numbers is between the smallest and largest numbers in the list, though that’s true. The batting average of a team as a whole is not the same as the average of the individual batting averages on that team. It might happen to be, but in general it is not. I’ll give a quick proof of the baseball inequality. I’ll only prove the first of the two inequalities. That is, I’ll prove that the minimum fraction is no greater than the ratio of the sums of numerators and denominators. Proving that the latter is no greater than the maximum fraction is completely analogous. Also, I’ll only prove the theorem for two numerators and two denominators. Once you have proved the inequality for two numerators and denominators, you can bootstrap that to prove the inequality for three numerators and three denominators, and continue this process for any number of numbers on top and bottom. So we start by assuming Then we have More inequality posts Hadamard’s upper bound on determinant For an n by n real matrix A, Hadamard’s upper bound on determinant is where a[ij] is the element in row i and column j. See, for example, [1]. How tight is this upper bound? To find out, let’s write a little Python code to generate random matrices and compare their determinants to Hadamard’s bounds. We’ll take the square root of both sides of Hadamard’s inequality to get an upper bound on the absolute value of the determinant. Hadamard’s inequality is homogeneous: multiplying the matrix A by λ multiplies both sides by λ^n. We’ll look at the ratio of Hadamard’s bound to the exact determinant. This has the same effect as generating matrices to have a fixed determinant value, such as 1. from scipy.stats import norm from scipy.linalg import det import matplotlib.pyplot as plt import numpy as np # Hadamard's upper bound on determinant squared def hadamard(A): return np.prod(np.sum(A2, axis=1)) N = 1000 ratios = np.empty(N) dim = 3 for i in range(N): A = norm.rvs(size=(dim, dim)) ratios[i] = hadamard(A)0.5/abs(det(A)) plt.hist(ratios, bins=int(N0.5)) In this simulation the ratio is very often around 25 or less, but occasionally much larger, 730 in this example. It makes sense that the ratio could be large; in theory the ratio could be infinite because the determinant could be zero. The error is frequently much smaller than the histogram might imply since a lot of small values are binned together. I modified the code above to print quantiles and ran it again. print(min(ratios), max(ratios)) qs = [0.05, 0.25, 0.5, 0.75, 0.95] print( [np.quantile(ratios, q) for q in qs] ) This printed 1.0022 1624.9836 [1.1558, 1.6450, 2.6048, 5.7189, 32.49279] So while the maximum ratio was 1624, the ratio was less than 2.6048 half the time, and less than 5.7189 three quarters of the time. Hadamard’s upper bound can be very inaccurate; there’s no limit on the relative error, though you could bound the absolute error in terms of the norm of the matrix. However, very often the relative error is moderately small. More posts on determinants [1] Courant and Hilbert, Methods of Mathematical Physics, Volume 1. Convex function of diagonals and eigenvalues Sam Walters posted an elegant theorem on his Twitter account this morning. The theorem follows the pattern of an equality for linear functions generalizing to an inequality for convex functions. We’ll give a little background, state the theorem, and show an example application. Let A be a real symmetric n×n matrix, or more generally a complex n×n Hermitian matrix, with entries a[ij]. Note that the diagonal elements a[ii] are real numbers even if some of the other entries are complex. (A Hermitian matrix equals its conjugate transpose, which means the elements on the diagonal equal their own conjugate.) A general theorem says that A has n eigenvalues. Denote these eigenvalues λ[1], λ[2], …, λ[n]. It is well known that the sum of the diagonal elements of A equals the sum of its eigenvalues. We could trivially generalize this to say that for any linear function φ: R → R, because we could pull any shifting and scaling constants out of the sum. The theorem Sam Walters posted says that the equality above extends to an inequality if φ is convex. Here’s an application of this theorem. Assume the eigenvalues of A are all positive and let φ(x) = – log(x). Then φ is convex, and and so i.e. the product of the diagonals of A is an upper bound on the determinant of A. This post illustrates two general principles: 1. Linear equalities often generalize to convex inequalities. 2. When you hear a new theorem about convex functions, see what it says about exp or −log. More linear algebra posts Sum and mean inequalities move in opposite directions It would seem that sums and means are trivially related; the mean is just the sum divided by the number of items. But when you generalize things a bit, means and sums act differently. Let x be a list of n non-negative numbers, and let r > 0 []. Then the r-mean is defined to be and the r-sum is define to be These definitions come from the classic book Inequalities by Hardy, Littlewood, and Pólya, except the authors use the Fraktur forms of M and S. If r = 1 we have the elementary mean and sum. Here’s the theorem alluded to in the title of this post: As r increases, M[r](x) increases and S[r](x) decreases. If x has at least two non-zero components then M[r](x) is a strictly increasing function of r and S[r](x) is a strictly decreasing function of r. Otherwise M[r](x) and S[r](x) are constant. The theorem holds under more general definitions of M and S, such letting the sums be infinite and inserting weights. And indeed much of Hardy, Littlewood, and Pólya is devoted to studying variations on M and S in fine detail. Here are log-log plots of M[r](x) and S[r](x) for x = (1, 2). Note that both curves asymptotically approach max(x), M from below and S from above. Related posts [] Note that r is only required to be greater than 0; analysis books typically focus on r ≥ 1. The Brothers Markov The Markov brother you’re more likely to have heard of was Andrey Markov. He was the Markov of Markov chains, the Gauss-Markov theorem, and Markov’s inequality. Andrey had a lesser known younger brother Vladimir who was also a mathematician. Together the two of them proved what is known as the Markov Brothers’ inequality to distinguish it from (Andrey) Markov’s inequality. For any polynomial p(x) of degree n, and for any non-negative integer k, the maximum of the kth derivative of p over the interval [−1, 1] is bounded by a constant times the maximum of p itself. The constant is a function of k and n but is otherwise independent of the particular polynomial. In detail, the Markov Brothers’ inequality says Andrey proved the theorem for k = 1 and his brother Vladimir generalized it for all positive k. The constant in the Markov Brothers’ inequality is the smallest possible because the bound is exact for Chebyshev polynomials [1]. Let’s look at an example. We’ll take the second derivative of the fifth Chebyshev polynomial. T[5](x) = 16x^5 − 20x^3 + 5x. The second derivative is T[5]”(x) = 320x^3 − 120x. Here are their plots: The maximum of T[5](x) is 1 and the maximum of its second derivative is 200. The product in the Markov Brothers’ inequality with n = 5 and k = 2 works out to (25/1)(24/3) = 200 and so the bound is exact for p(x) = T[5](x). It took a while for westerners to standardize how to transliterate Russian names, so you might see Andrey written as Andrei or Markov written as Markoff. There were even more ways to transliterate Chebyshev, including Tchebycheff, Tchebyshev, and Tschebyschow. These versions are the reason Chebyshev polynomials [1] are denoted with a capital T. More posts mentioning Markov [1] There are two families of Chebyshev polynomials. When used without qualification, as in this post, “Chebyshev polynomial” typically means Chebyshev polynomial of the first kind. These are denoted T[n]. Chebyshev polynomials of the second kind are denoted U[n].	{"url":"https://www.johndcook.com/blog/tag/inequalities/page/2/","timestamp":"2024-11-02T11:47:33Z","content_type":"text/html","content_length":"96299","record_id":"<urn:uuid:96b2b939-55b7-445e-8f02-99bd2cf3d6ff>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00487.warc.gz"}
Stray Capacitance – The Ultimate Guide You Need To Know Table of Contents What is Stray Capacitance? Stray capacitance, also known as parasitic capacitance, refers to the unwanted capacitance that exists between conductors in an electrical or electronic system. This capacitance is not intentionally designed into the circuit but occurs naturally due to the physical proximity of conductive elements. Stray capacitance can have significant effects on the performance and behavior of circuits, especially at high frequencies. In an ideal scenario, capacitance should only exist between the intended conductors, such as the plates of a capacitor. However, in reality, capacitance can also occur between any two conductors that are separated by an insulating medium, such as air, plastic, or dielectric materials. This unintended capacitance is what we refer to as stray capacitance. Causes of Stray Capacitance Several factors contribute to the occurrence of stray capacitance in electrical and electronic systems. Some of the primary causes include: 1. Proximity of conductors: When conductors are placed close to each other, the electric field between them can couple, leading to stray capacitance. The closer the conductors are to each other, the higher the stray capacitance. 2. Conductor surface area: Larger conductor surface areas result in higher stray capacitance. This is because the capacitance is directly proportional to the surface area of the conductors. 3. Dielectric constant of the insulating medium: The dielectric constant of the insulating material between the conductors affects the stray capacitance. Materials with higher dielectric constants, such as ceramic or mica, will result in higher stray capacitance compared to materials with lower dielectric constants, such as air or vacuum. 4. Frequency of operation: Stray capacitance becomes more significant at higher frequencies. As the frequency increases, the impedance of the stray capacitance decreases, allowing more current to flow through it. Effects of Stray Capacitance Stray capacitance can have various effects on the performance and behavior of electrical and electronic systems. Some of the key effects include: 1. Signal distortion: Stray capacitance can cause signal distortion by altering the shape and timing of the signals. This is particularly problematic in high-frequency circuits, where the stray capacitance can act as a low-pass filter, attenuating high-frequency components of the signal. 2. Crosstalk: Stray capacitance between adjacent conductors can lead to crosstalk, where the signal from one conductor couples into another, causing interference. This can result in reduced signal integrity and increased noise levels. 3. Reduced bandwidth: The presence of stray capacitance can limit the bandwidth of a circuit. As the frequency increases, the impedance of the stray capacitance decreases, effectively shunting high-frequency signals to ground. This can result in a reduction of the circuit’s bandwidth. 4. Increased power consumption: Stray capacitance can increase the power consumption of a circuit by allowing current to flow through unintended paths. This can lead to reduced efficiency and increased heat generation. 5. Reduced signal-to-noise ratio (SNR): Stray capacitance can contribute to noise in a circuit, reducing the signal-to-noise ratio. This can make it more difficult to distinguish the desired signal from the background noise. Measuring Stray Capacitance Measuring stray capacitance is essential for understanding its impact on a circuit and implementing appropriate mitigation techniques. Several methods can be used to measure stray capacitance, 1. LCR meters: LCR meters are specialized instruments designed to measure inductance (L), capacitance (C), and resistance (R). They can be used to directly measure the stray capacitance between conductors by connecting the probes to the relevant points in the circuit. 2. Time-domain reflectometry (TDR): TDR is a technique that involves sending a fast-rising pulse through a transmission line and measuring the reflections caused by impedance discontinuities. By analyzing the reflected waveforms, the stray capacitance can be determined. 3. Network analyzers: Network analyzers are instruments that measure the electrical properties of a network, such as impedance, admittance, and S-parameters, over a range of frequencies. They can be used to characterize the stray capacitance of a circuit by measuring its frequency response. 4. Simulation tools: Circuit simulation software, such as SPICE or ANSYS, can be used to model and simulate the effects of stray capacitance in a circuit. By including stray capacitance in the simulation model, designers can predict its impact on circuit performance. Techniques to Minimize Stray Capacitance Minimizing stray capacitance is crucial for ensuring the proper functioning and performance of electrical and electronic systems. Several techniques can be employed to reduce the impact of stray capacitance, including: 1. Proper circuit layout: Careful design and layout of the circuit can help minimize stray capacitance. This includes: 2. Increasing the spacing between conductors 3. Minimizing the surface area of conductors 4. Using guard rings or shielding to isolate sensitive parts of the circuit 5. Material selection: Choosing materials with lower dielectric constants for insulators and substrates can help reduce stray capacitance. For example, using air or foam dielectrics instead of solid dielectrics can significantly reduce stray capacitance. 6. Impedance matching: Proper impedance matching between the source, load, and transmission lines can help minimize the impact of stray capacitance. This involves designing the circuit to ensure that the impedances are matched at the desired frequencies. 7. Differential signaling: Using differential signaling techniques, such as balanced or twisted-pair lines, can help cancel out the effects of stray capacitance. In differential signaling, the signal is transmitted as a pair of complementary signals, which helps to reject common-mode noise and reduce the impact of stray capacitance. 8. Grounding and shielding: Proper grounding and shielding techniques can help minimize the effects of stray capacitance. This includes using ground planes, shielding sensitive parts of the circuit, and ensuring good electrical connections between ground points. Stray Capacitance in Different Applications Stray Capacitance in PCBs Printed circuit boards (PCBs) are a common source of stray capacitance in electronic systems. The close proximity of conductive traces, pads, and planes on a PCB can result in significant stray capacitance. To minimize stray capacitance in PCBs, designers can: • Increase the spacing between traces and pads • Use thinner dielectric layers • Employ ground planes to provide shielding • Optimize the routing of high-frequency signals • Use differential signaling techniques Stray Capacitance in Cables and Wires Cables and wires can also introduce stray capacitance in electrical systems. The capacitance between the conductors within a cable or between adjacent cables can lead to signal distortion and crosstalk. To reduce stray capacitance in cables and wires, consider: • Using shielded cables or twisted-pair wires • Increasing the spacing between conductors • Minimizing cable lengths • Using low-dielectric-constant insulation materials • Proper termination and grounding of cable shields Stray Capacitance in Transformers Transformers can exhibit stray capacitance between windings, as well as between windings and the core or shield. This stray capacitance can affect the frequency response and efficiency of the transformer. To minimize stray capacitance in transformers: • Increase the spacing between windings • Use low-dielectric-constant insulation materials • Employ electrostatic shielding between windings • Optimize the winding geometry and arrangement • Use interleaved windings to reduce inter-winding capacitance Stray Capacitance in Switches and Relays Switches and relays can introduce stray capacitance due to the proximity of the contacts and the presence of insulating materials. This stray capacitance can lead to signal degradation and reduced switching speed. To minimize stray capacitance in switches and relays: • Use switches and relays with larger contact spacing • Employ shielding or guard rings around contacts • Minimize the surface area of contacts • Use materials with lower dielectric constants for insulation • Ensure proper grounding and shielding of the switch or relay enclosure Frequently Asked Questions (FAQ) 1. What is the difference between stray capacitance and parasitic capacitance? Stray capacitance and parasitic capacitance are often used interchangeably. Both terms refer to the unintended capacitance that exists between conductors in an electrical or electronic system. The term “parasitic” emphasizes the unwanted nature of this capacitance, while “stray” simply indicates its unintentional presence. 2. How does stray capacitance affect the frequency response of a circuit? Stray capacitance can act as a low-pass filter, attenuating high-frequency components of a signal. As the frequency increases, the impedance of the stray capacitance decreases, allowing more high-frequency current to be shunted to ground. This results in a reduction of the circuit’s bandwidth and can cause signal distortion. 3. Can stray capacitance be completely eliminated? While it is not possible to completely eliminate stray capacitance, its effects can be minimized through proper circuit design, layout, and material selection. Techniques such as increasing conductor spacing, using low-dielectric-constant materials, and employing shielding and grounding can help reduce the impact of stray capacitance. 4. How does stray capacitance contribute to crosstalk in circuits? Stray capacitance between adjacent conductors can allow signals to couple from one conductor to another, causing crosstalk. The coupled signal can interfere with the intended signal on the receiving conductor, leading to signal integrity issues and increased noise levels. 5. What are some common applications where stray capacitance is a significant concern? Stray capacitance is a concern in various applications, particularly those involving high frequencies or sensitive analog circuits. Some common examples include: 6. High-speed digital circuits, such as memory interfaces and data buses 7. RF and microwave circuits, such as amplifiers, filters, and antennas 8. Precision analog circuits, such as data acquisition systems and sensor interfaces 9. Power electronics, such as switch-mode power supplies and motor drives Stray capacitance is an important consideration in the design and analysis of electrical and electronic systems. It can have significant effects on signal integrity, noise, and overall circuit performance, particularly at high frequencies. Understanding the causes and effects of stray capacitance, as well as the techniques to measure and minimize its impact, is crucial for engineers and designers working on a wide range of applications. By employing proper circuit layout, material selection, and design techniques, the effects of stray capacitance can be minimized, leading to improved system performance and reliability. As the demand for higher-speed and more sensitive electronic systems continues to grow, the management of stray capacitance will remain a critical aspect of electrical and electronic engineering. 1. Horowitz, P., & Hill, W. (2015). The Art of Electronics (3rd ed.). Cambridge University Press. 2. Johnson, H. W., & Graham, M. (2003). High-Speed Digital Design: A Handbook of Black Magic. Prentice Hall. 3. Ott, H. W. (2011). Electromagnetic Compatibility Engineering. John Wiley & Sons. 4. Pozar, D. M. (2012). Microwave Engineering (4th ed.). John Wiley & Sons. 5. Montrose, M. I. (2004). EMC and the Printed Circuit Board: Design, Theory, and Layout Made Simple. John Wiley & Sons.	{"url":"https://pcb-copy.com/stray-capacitance-the-ultimate-guide-you-need-to-know/","timestamp":"2024-11-10T21:00:27Z","content_type":"text/html","content_length":"103677","record_id":"<urn:uuid:c12d3843-42e8-41ef-b968-8305bcec8d9b>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00770.warc.gz"}
How Is Rate Of Return Calculated bk4info.site How Is Rate Of Return Calculated How Is Rate Of Return Calculated Because ROI is most often expressed as a percentage, the quotient is converted to a percentage by multiplying it by This investment's ROI is 2 multiplied. The formula below illustrates this: MWRR is the rate of return where present value of outflows+present value of inflows = 0. In this case, a large contribution. Calculation · Comparisons between various rates of return · Uses · Time value of money · Compounding or reinvesting · Foreign currency returns · Returns when capital. Return rate – For many investors, this is what matters most. · Starting amount – Sometimes called the principal, this is the amount apparent at the inception of. Compounded average rate of return = A = P(1+(1+(r/)^(n-1)here if r=a25 A= P(1+()^9) if n=10 A/P = At a coupounded average rate. But for the increase in the investment value, it is calculated as the current value of the investment minus the initial value, or (50$28) - (50$20) = $ Calculation · Comparisons between various rates of return · Uses · Time value of money · Compounding or reinvesting · Foreign currency returns · Returns when capital. The formula for calculating rate of return is R = [(Ve Vb) / Vb] x , where Ve is the end of period value and Vb is the beginning of period value. Rate of return is the profit or loss on an investment expressed as a percentage. You can calculate the rate of return on typical financial investments (such as. The basic formula for calculating ROR is (current value - initial value) / initial value * This formula can be applied to various investment scenarios to. To calculate the average rate of return, add together the rate of return for the years of your investment, and then, divide that total number by the number of. It's calculated by subtracting the initial investment from its final value, then dividing that number by the initial amount invested. It's then multiplied by. The rate of return is the gain (or loss) compared to the cost of an initial investment, typically expressed in the form of a percentage. The formula for calculating rate of return is R = [(Ve Vb) / Vb] x , where Ve is the end of period value and Vb is the beginning of period value. You can calculate the return on your investment by subtracting the initial amount of money that you put in from the final value of your financial investment. Calculate Simple Rate of Return. Now, pull it all together. Take your annual net income and divide it by the initial cost of the investment. In this case, a. Personal rate of return (PRR) can most simply be thought of as the amount of gain/loss in a period of time, divided by your cash flow activity, which includes. In other words, the rate of return is the gain (or loss) compared to the cost of an initial investment, typically expressed in the form of a percentage. When. How is my rate of return calculated? The simplest explanation is that we calculate the percentage of change in your account over a specific period of time. The. For example: If you assume you earn a 10% annual rate of return, then you are assuming that the value of your investment will increase by 10% every year. So, if. How to calculate rate of return. The most basic way to calculate rate of return is to measure the percentage change in an investment's value for a time period. How to Calculate IRR · Divide the Future Value (FV) by the Present Value (PV) · Raise to the Inverse Power of the Number of Periods (i.e. 1 ÷ n) · From the. The calculation of the rate of return is the interest plus appreciation, divided by original bond price – expressed as a percentage. The rate of return after. Free return on investment (ROI) calculator that returns total ROI rate and annualized ROI using either actual dates of investment or simply investment. The required rate of return (hurdle rate) is the minimum return that an investor is expecting to receive for their investment. Essentially, the required rate is. A rate of return (RoR) is the net gain or loss of an investment over a specified time period, expressed as a percentage of the investment's initial cost. ROI is calculated by subtracting the initial cost of the investment from its final value, then dividing this new number by the cost of the investment, and. Use KeyBank's annual rate of return calculator to determine the annual return of a known initial amount, a stream of deposits, plus a known final future. To calculate your return rate, divide the number of units returned by the number of units sold, multiplying the product by to find your percentage. To calculate the expected rate of return on a stock or other security, you need to think about the different scenarios in which the asset could see a gain or. How is my rate of return calculated? The simplest explanation is that we calculate the percentage of change in your account over a specific period of time. The. Calculate Simple Rate of Return. Now, pull it all together. Take your annual net income and divide it by the initial cost of the investment. In this case, a. To calculate the average rate of return, add together the rate of return for the years of your investment, and then, divide that total number by the number of. Internal rate of return (IRR) is a method of calculating an investment's rate of return. The term internal refers to the fact that the calculation excludes. This is the annually compounded rate of return you expect from your investments before taxes. The actual rate of return is largely dependent on the types of. Compounded average rate of return = A = P(1+(1+(r/)^(n-1)here if r=a25 A= P(1+()^9) if n=10 A/P = At a coupounded average rate. Personal rate of return (PRR) can most simply be thought of as the amount of gain/loss in a period of time, divided by your cash flow activity, which includes. Return rate – For many investors, this is what matters most. · Starting amount – Sometimes called the principal, this is the amount apparent at the inception of. The required rate of return (hurdle rate) is the minimum return that an investor is expecting to receive for their investment. Essentially, the required rate is. How to calculate rate of return. The most basic way to calculate rate of return is to measure the percentage change in an investment's value for a time period. The ARR is calculated by dividing the average annual profit by the cost of investment and multiplying by The formula for calculating the average rate of. The formula for calculating the internal rate of return (IRR) is as follows: Internal Rate of Return (IRR) = (Future Value ÷ Present Value)^(1 ÷ Number of. The calculation of the rate of return is the interest plus appreciation, divided by original bond price – expressed as a percentage. The rate of return after. A dollar-weighted return calculation will yield a different result as soon as any of the variables (such as date of purchase, size of initial investment. The formula below illustrates this: MWRR is the rate of return where present value of outflows+present value of inflows = 0. In this case, a large contribution. Rate of return is the profit or loss on an investment expressed as a percentage. You can calculate the rate of return on typical financial investments (such as. Calculation · Comparisons between various rates of return · Uses · Time value of money · Compounding or reinvesting · Foreign currency returns · Returns when capital. The rate of return on the US CD is simply the interest rate on that deposit. More formally, RoR $ = i $. This is because the interest rate describes the. To calculate the expected rate of return on a stock or other security, you need to think about the different scenarios in which the asset could see a gain or. [ Total Return = (1 + annual return)^(number of years) ] Let's return to the example where a $10, investment grows to $12, over a five year period. The. Use KeyBank's annual rate of return calculator to determine the annual return of a known initial amount, a stream of deposits, plus a known final future. To calculate your return rate, divide the number of units returned by the number of units sold, multiplying the product by to find your percentage. Because ROI is most often expressed as a percentage, the quotient is converted to a percentage by multiplying it by This investment's ROI is 2 multiplied. The basic formula for calculating ROR is (current value - initial value) / initial value * This formula can be applied to various investment scenarios to. Free return on investment (ROI) calculator that returns total ROI rate and annualized ROI using either actual dates of investment or simply investment. But for the increase in the investment value, it is calculated as the current value of the investment minus the initial value, or (50$28) - (50$20) = $ For example: If you assume you earn a 10% annual rate of return, then you are assuming that the value of your investment will increase by 10% every year. So, if. You can calculate the return on your investment by subtracting the initial amount of money that you put in from the final value of your financial investment. ROI is calculated by subtracting the initial cost of the investment from its final value, then dividing this new number by the cost of the investment, and. It's calculated by subtracting the initial investment from its final value, then dividing that number by the initial amount invested. It's then multiplied by. Fund performance information isn't personal since it doesn't take consider your contributions or recent account activity. Your rate of return calculation, on. When Do Stocks Go Up And Down \| Grunt Shirts Amazon	{"url":"https://bk4info.site/prices/how-is-rate-of-return-calculated.php","timestamp":"2024-11-08T18:55:35Z","content_type":"text/html","content_length":"15921","record_id":"<urn:uuid:02cf8b36-c9cb-4700-bcd7-059d87f72740>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00724.warc.gz"}
How to Calculate Area of Polygon from Unordered Coordinate Points. Algorithm and Code in Python Finding an area is a common task in GIS. Using a GIS software like QGIS we can find area of a polygon using a measurement tool or calculate area of each polygon feature with field calculator tool. It's so convenient and that's the reason why we use the software. On the other hand, using a software to find a polygon area, in particular irregular polygon area is like a black box method. It means we can get a result without knowing how it works. This post is like unboxing the method. I will explain how to calculate area of polygon from a given unordered coordinate points. Calculate Area of Polygon with an Ordered Coordinate Points Let's start with finding area of an ordered polygon points. Suppose we have a polygon as in figure 1. From the figure we can see the ordered point from 1 to 4 with corresponding coordinates (2,2), (2,4),(5,4) and (9,2). Figure 1. A polygon and ordered points Given an ordered points, we can calculate the area of polygon with the famous Shoelace method as in figure 2. Figure 2. Shoelace Method Using the Shoelace method, let's calculate the area of polygon in figure 1 as following. \[A=\bigg\| \frac{(2.4+2.4+5.2+9.2)-(2.2+4.5+4.9+2.2)}{2}\bigg \|=\bigg \|\frac{44-64}{2}\bigg \|=10 \] We get the area of polygon 10 in an area unit. Further more if you want to prove the result, simply divide the polygon into a rectangle and triangle along a dash line as in figure 1. Compute the area for each geometry and add both of them. The result will be the same. Implementation in Python We already calculated the area of polygon manually using the Shoelace method. Now let's do it in Python. The following code is the code for computing a polygon area with a given ordered points. The code consist of two functions which is called explode_xy and shoelace_area. The first function is used to explode a given point coordinates into x and y list. The both list will be as input argument for shoelace_area function to calculate a polygon area using formula as in figure 2. #EXPLODE X AND Y def explode_xy(xy): for i in range(len(xy)): return xl,yl def shoelace_area(x_list,y_list): for j in range(len(x_list)-1): a1 += x_list[j]y_list[j+1] a2 += y_list[j]x_list[j+1] return l How to Sort a Random Polygon Points Now, suppose you build a robot that can measure coordinates in a room with such an integrated sensor. Most probably the recorded coordinates not stored in an ordered structure. You want calculate the area of the room with the Shoelace method cause it's quite straight forward and easy to implement. But the problem is the method only works with an ordered points. So the question is: how to sort a random unordered polygon points? Might be there are numerous approaches to tackle this problem, but here is mine. The main idea is to order the points based on the angle from a reference axis as illustrated in figure 3. From the figure can be seen that point p has the smallest angle and point s has the biggest angle. Point q's angle is bigger than p and r's angle is bigger than q. Overall the order of points will be p-q-r-s. Following is step by step approach to order a random polygon points. 1. Find a centroid coordinate by taking the mean of x and y. 2. Subtract each points coordinate with centroid coordinate. 3. Put of subtracted coordinates into a corresponding quadrant. 4. Calculate the angle of each point with arc tangent. 5. Adjust the calculated angles with respected to quadrant and reference axis. 6. Order the points based on the adjusted angle from the smallest to the largest. Implementing The Sorting Algorithm in Python Based on the steps above, let's implement it in python. Calculate Centroid Coordinate To calculate the centroid coordinate can be done by taking the mean of x and y. From the previous code we split each coordinate with explode_xy function. The function returns x and y list. Then to calculate the mean, simply we sum up the list and divide it with the length of the list. The following code is to calculate the centroid coordinate. At the First line the numpy library is imported. This library will be used at later steps. import numpy as np Subtract Each Coordinate with Centroid To subtract each point coordinate with centroid can be done easly using numpy array. For that we need to transform the x list and y list into numpy array and then subtract it with centroid coordinate as the following code. Grouping Subtracted Coordinates Into a Quadrant At this step we will group each coordinate into a corresponding quadrant. Why do we need this step? Because at the later step when calculating the angle of each point, the reference axis is not the same. And at the end we have to reference all the angles from the same axis. When doing the transformation the condition will be difference for each quadrant. In this tutorial I determined the first quadrant form the bottom left to the bottom right with clockwise direction as shown in figure 4. Each point will be grouped into a corresponding quadrant based on its value. From figure 4 can be seen that, if a point has negative, it will be in the first quadrant. If negative but positive then it will be in the second quadrant and so on. The following code is used to group a point coordinate based on the condition above. The result is stored into a list which consist of four dictionaries data structure for each corresponding quadrant. The dictionary structure is used to map each point based on it's index in the unordered coordinate list. Therefore the index will be the key for each values in a dict items. for i in range(len(xa)): if (xa[i]<0 and ya[i]<0): elif (xa[i]<0 and ya[i]>0): elif (xa[i]>0 and ya[i]>0): Calculating and Adjusting Angle To calculate a point angle from the centroid can be used arc tangent function by dividing y with x. The resulting angle will have different reference axis for different quadrant ($\beta_i$), but mainly along x axis as illustrated in figure 5. What we need is the angle from a common reference axis, which is -y axis (the highlighted blue line). To get the angle from the common reference axis ($\alpha_i$), it need to be adjusted with adding or subtract with $90^o$ or $270^o$. The following code is to calculate the angle of points in each quadrant and perform the required adjustment. for i in range(len(q)): for j in q[i].keys(): if i==0: elif i==1: elif i==2: Sorting The Angles From the previous step, we get a variable alpha which is a dictionary that contain adjusted angle from a common reference axis. The last step is to sort the angle from smallest to largest one. This can be done with the next code. The first line of the code is declaring an empty list to store the sorted angle. Then the angle is sorted with the sorted built in function. Next in a loop, the point coordinate will be stored into the list which has equal key. for i in sorted_alpha: for k,l in alpha.items(): if i==l: check_p=xy[k] in re_xy if not check_p: After running the code, you should get an ordered polygon coordinate in a list variable which is called re_xy. To calculate the area using the Shoelace method, explode the list into x and y again and use shoelace_area function as in the first listing code above. I've tested the algorithm for several polygon shapes and it works well. If you find it not works for a certain shape, please provide me the coordinates of the shape in sequence order, so I can check it. I also very welcome if you have any suggestion to improve the algorithm. That's all this tutorial how to calculate a polygon area from unordered coordinate points. In this tutorial we discuss about the famous shoelace method to calculate area from points coordinate. Then construct an algorithm to sort coordinates based on the angle from centroid coordinate. The algorithm is explaining step by step with implementation in Python code. Hope it useful for you and thanks for reading! 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Decimal to fraction calculator that works anywhere Sometimes you need to convert a decimal to a fraction. Typically, this means dropping everything, searching a converter online, calculating and then copying the results over. A time consuming and error-prone process that can be avoided. With Text Blaze, you can automate the entire process. Simply type a pre-define shortcut anywhere you are, enter the date of birth and the age will be automatically typed for you. Decimal to fraction calculator Click on the snippet below to see how you can convert decimals to fractions as you type. Decimal to fraction calculator {formtext: name=number; default=1.625} is {if: isnumber(number) = "yes"}{if: number <> floor(number)} {note: preview=no; trim=yes} {int=split(number, ".")[1] if split(number, ".")[1] <> "" else 0} {decimals=split(number, ".")[2]} {numerator=extractregex(decimals, "\d{0,3}")} {denominator=10^len(numerator)} {factors1=filter([numerator/i for i in seq(1, numerator)], item -> testregex(item, "\.") = "no")} {factors2=filter([denominator/i for i in seq(1, denominator)], item -> testregex(item, "\.") = "no")} {gcf=filter(factors1, item -> includes(factors2, item) = "yes")[1]} {endnote: trim= right} {if: int > 0}{=catch(int, "")} {endif}{=numerator/gcf}/{=denominator/gcf} {else} {=abs(number)} {endif} {else} {error: Please input a number} {endif} Just give me the fraction! Need to just type fraction? No problem! The snippet below behaves exactly like the snippet above, but the output is just the fraction. Decimal to fraction calculator - just the fraction {note}{formtext: name=number; default=1.625}{endnote}{if: isnumber(number) = "yes"}{if: number <> floor(number)} {note: preview=no; trim=yes} {int=split(number, ".")[1] if split(number, ".")[1] <> "" else 0}{decimals=split(number, ".")[2]} {numerator=extractregex(decimals, "\d{0,3}")} {denominator=10^len(numerator)} {factors1=filter([numerator/i for i in seq(1, numerator)], item -> testregex (item, "\.") = "no")} {factors2=filter([denominator/i for i in seq(1, denominator)], item -> testregex(item, "\.") = "no")} {gcf=filter(factors1, item -> includes(factors2, item) = "yes")[1]} {endnote: trim=right} {if: int > 0}{=catch(int, "")} {endif}{=numerator/gcf}/{=denominator/gcf} {else} {=abs(number)} {endif} {else} {error: Please input a number} {endif} How decimal to fraction conversion works 1. Write down the decimal - we'll focus on decimals that terminate (the calculator above uses up to 3 decimal places) 2. Divide the decimal by 1 to create a fraction that looks like this: 1.625 / 1 3. Multiply both numerator and denominator by 10 until you get a regular fraction: 1625 / 1000 4. Divide both by the largest common denominator: (1625 / 125) / (1000 / 125) = 13 / 8 = 1 5/8 Hi there! You made it all the way down to the bottom of this article. Take a few seconds to share it. Want to turbo charge your work with templates and snippets? Text Blaze is the fastest way to do that.	{"url":"https://blaze.today/blog/decimal_to_fraction/","timestamp":"2024-11-12T18:58:19Z","content_type":"text/html","content_length":"33612","record_id":"<urn:uuid:40a828bf-a8c1-4c0c-8afa-54d1d8924d87>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00550.warc.gz"}
Free variables and bound variables - Wikipedia Republished // WIKI 2 In mathematics, and in other disciplines involving formallanguages, including mathematicallogic and computerscience, a variable may be said to be either free or bound. Some older books use the terms real variable and apparent variable for free variable and bound variable, respectively. A free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is not a parameter of this or any container expression. The idea is related to a placeholder (a symbol that will later be replaced by some value), or a wildcardcharacter that stands for an unspecified symbol. In computerprogramming, the term free variable refers to variables used in a function that are neither localvariables nor parameters of that function. The term non-localvariable is often a synonym in this context. An instance of a variable symbol is bound, in contrast, if the value of that variable symbol has been bound to a specific value or range of values in the domainofdiscourse or universe. This may be achieved through the use of logical quantifiers, variable-binding operators, or an explicit statement of allowed values for the variable (such as, "...where ${\displaystyle n}$ is a positive integer".) A variable symbol overall is bound if at least one occurrence of it is bound.^[1]^pp.142--143 Since the same variable symbol may appear in multiple places in an expression, some occurrences of the variable symbol may be free while others are bound,^[1]^p.78 hence "free" and "bound" are at first defined for occurrences and then generalized over all occurrences of said variable symbol in the expression. However it is done, the variable ceases to be an independent variable on which the value of the expression depends, whether that value be a truth value or the numerical result of a calculation, or, more generally, an element of an image set of a function. While the domain of discourse in many contexts is understood, when an explicit range of values for the bound variable has not been given, it may be necessary to specify the domain in order to properly evaluate the expression. For example, consider the following expression in which both variables are bound by logical quantifiers: ${\displaystyle \forall y\,\exists x\,\left(x={\sqrt {y}}\right).}$ This expression evaluates to false if the domain of ${\displaystyle x}$ and ${\displaystyle y}$ is the real numbers, but true if the domain is the complex numbers. The term "dummy variable" is also sometimes used for a bound variable (more commonly in general mathematics than in computer science), but this should not be confused with the identically named but unrelated concept of dummyvariable as used in statistics, most commonly in regression analysis.^[2]^p.17 YouTube Encyclopedic • 1/5 • [Logic] Free and Bound Variables • PART-1: FREE VARIABLE \| BOUND VARIABLE \| SCOPE OF A QUANTIFIER \| FREE VARIABLE AND BOUND VARIABLE \| • Determine Basic (Leading) Variables and Free Variables Given a Matrix in RREF • Free and Bound Variables \|\| Lesson 33 \|\| Discrete Math & Graph Theory \|\| Learning Monkey \|\| • Row Echelon Form, Pivot Positions, Basic and Free Variables Before stating a precise definition of free variable and bound variable, the following are some examples that perhaps make these two concepts clearer than the definition would: In the expression ${\displaystyle \sum _{k=1}^{10}f(k,n),}$ n is a free variable and k is a bound variable; consequently the value of this expression depends on the value of n, but there is nothing called k on which it could depend. In the expression ${\displaystyle \int _{0}^{\infty }x^{y-1}e^{-x}\,dx,}$ y is a free variable and x is a bound variable; consequently the value of this expression depends on the value of y, but there is nothing called x on which it could depend. In the expression ${\displaystyle \lim _{h\rightarrow 0}{\frac {f(x+h)-f(x)}{h}},}$ x is a free variable and h is a bound variable; consequently the value of this expression depends on the value of x, but there is nothing called h on which it could depend. In the expression ${\displaystyle \forall x\ \exists y\ {\Big [}\varphi (x,y,z){\Big ]},}$ z is a free variable and x and y are bound variables, associated with logicalquantifiers; consequently the logicalvalue of this expression depends on the value of z, but there is nothing called x or y on which it could depend. More widely, in most proofs, bound variables are used. For example, the following proof shows that all squares of positive even integers are divisible by ${\displaystyle 4}$ Let ${\displaystyle n}$ be a positive even integer. Then there is an integer ${\displaystyle k}$ such that ${\displaystyle n=2k}$. Since ${\displaystyle n^{2}=4k^{2}}$, we have ${\displaystyle n^ {2}}$ divisible by ${\displaystyle 4}$ not only k but also n have been used as bound variables as a whole in the proof. Variable-binding operators The following ${\displaystyle \sum _{x\in S}\quad \quad \prod _{x\in S}\quad \quad \int _{0}^{\infty }\cdots \,dx\quad \quad \lim _{x\to 0}\quad \quad \forall x\quad \quad \exists x}$ are some common variable-binding operators. Each of them binds the variable x for some set S. Many of these are operators which act on functions of the bound variable. In more complicated contexts, such notations can become awkward and confusing. It can be useful to switch to notations which make the binding explicit, such as ${\displaystyle \sum _{1,\ldots ,10}\left(k\mapsto f(k,n)\right)}$ for sums or ${\displaystyle D\left(x\mapsto x^{2}+2x+1\right)}$ for differentiation. Formal explanation Tree summarizing the syntax of the expression ${\displaystyle \forall x\,((\exists y\,A(x))\vee B(z))}$ Variable-binding mechanisms occur in different contexts in mathematics, logic and computer science. In all cases, however, they are purely syntactic properties of expressions and variables in them. For this section we can summarize syntax by identifying an expression with a tree whose leaf nodes are variables, constants, function constants or predicate constants and whose non-leaf nodes are logical operators. This expression can then be determined by doing an inordertraversal of the tree. Variable-binding operators are logicaloperators that occur in almost every formal language. A binding operator Q takes two arguments: a variable v and an expression P, and when applied to its arguments produces a new expression Q(v, P). The meaning of binding operators is supplied by the semantics of the language and does not concern us here. Variable binding relates three things: a variable v, a location a for that variable in an expression and a non-leaf node n of the form Q(v, P). Note: we define a location in an expression as a leaf node in the syntax tree. Variable binding occurs when that location is below the node n. In the lambdacalculus, x is a bound variable in the term M = λx. T and a free variable in the term T. We say x is bound in M and free in T. If T contains a subterm λx. U then x is rebound in this term. This nested, inner binding of x is said to "shadow" the outer binding. Occurrences of x in U are free occurrences of the new x.^[3] Variables bound at the top level of a program are technically free variables within the terms to which they are bound but are often treated specially because they can be compiled as fixed addresses. Similarly, an identifier bound to a recursivefunction is also technically a free variable within its own body but is treated specially. A closed term is one containing no free variables. Function expressions To give an example from mathematics, consider an expression which defines a function ${\displaystyle f=\left[(x_{1},\ldots ,x_{n})\mapsto t\right]}$ where t is an expression. t may contain some, all or none of the x[1], …, x[n] and it may contain other variables. In this case we say that function definition binds the variables x[1], …, x[n]. In this manner, function definition expressions of the kind shown above can be thought of as the variable binding operator, analogous to the lambda expressions of lambdacalculus. Other binding operators, like the summation sign, can be thought of as higher-orderfunctions applying to a function. So, for example, the expression ${\displaystyle \sum _{x\in S}{x^{2}}}$ could be treated as a notation for ${\displaystyle \sum _{S}{(x\mapsto x^{2})}}$ where ${\displaystyle \sum _{S}{f}}$ is an operator with two parameters—a one-parameter function, and a set to evaluate that function over. The other operators listed above can be expressed in similar ways; for example, the universalquantifier ${\displaystyle \forall x\in S\ P(x)}$ can be thought of as an operator that evaluates to the logicalconjunction of the Boolean-valuedfunction P applied over the (possibly infinite) set S. Natural language When analyzed in formalsemantics, natural languages can be seen to have free and bound variables. In English, personalpronouns like he, she, they, etc. can act as free variables. Lisa found her book. In the sentence above, the possessive pronoun her is a free variable. It may refer to the previously mentioned Lisa or to any other female. In other words, her book could be referring to Lisa's book (an instance of coreference) or to a book that belongs to a different female (e.g. Jane's book). Whoever the referent of her is can be established according to the situational (i.e. pragmatic) context. The identity of the referent can be shown using coindexing subscripts where i indicates one referent and j indicates a second referent (different from i). Thus, the sentence Lisa found her book has the following interpretations: Lisa[i] found her[i] book. (interpretation #1: her = of Lisa) Lisa[i] found her[j] book. (interpretation #2: her = of a female that is not Lisa) The distinction is not purely of academic interest, as some languages do actually have different forms for her[i] and her[j]: for example, Norwegian and Swedish translate coreferent her[i] as sin and noncoreferent her[j] as hennes. English does allow specifying coreference, but it is optional, as both interpretations of the previous example are valid (the ungrammatical interpretation is indicated with an asterisk): Lisa[i] found her[i] own book. (interpretation #1: her = of Lisa) Lisa[i] found her[j] own book. (interpretation #2: her = of a female that is not Lisa) However, reflexivepronouns, such as himself, herself, themselves, etc., and reciprocalpronouns, such as each other, act as bound variables. In a sentence like the following: Jane hurt herself. the reflexive herself can only refer to the previously mentioned antecedent, in this case Jane, and can never refer to a different female person. In this example, the variable herself is bound to the noun Jane that occurs in subject position. Indicating the coindexation, the first interpretation with Jane and herself coindexed is permissible, but the other interpretation where they are not coindexed is ungrammatical: Jane[i] hurt herself[i]. (interpretation #1: herself = Jane) Jane[i] hurt herself[j]. (interpretation #2: herself = a female that is not Jane) The coreference binding can be represented using a lambdaexpression as mentioned in the previous Formalexplanationsection. The sentence with the reflexive could be represented as (λx.x hurt x)Jane in which Jane is the subject referent argument and λx.x hurt x is the predicate function (a lambda abstraction) with the lambda notation and x indicating both the semantic subject and the semantic object of sentence as being bound. This returns the semantic interpretation JANE hurt JANE with JANE being the same person. Pronouns can also behave in a different way. In the sentence below Ashley hit her. the pronoun her can only refer to a female that is not Ashley. This means that it can never have a reflexive meaning equivalent to Ashley hit herself. The grammatical and ungrammatical interpretations are: *Ashley[i] hit her[i]. (interpretation #1: her = Ashley) Ashley[i] hit her[j]. (interpretation #2: her = a female that is not Ashley) The first interpretation is impossible. Only the second interpretation is permitted by the grammar. Thus, it can be seen that reflexives and reciprocals are bound variables (known technically as anaphors) while true pronouns are free variables in some grammatical structures but variables that cannot be bound in other grammatical structures. The binding phenomena found in natural languages was particularly important to the syntactic governmentandbindingtheory (see also: Binding See also 1. ^ ^a ^b W. V. O. Quine, Mathematical Logic (1981). Harvard University Press, 0-674-55451-5. 2. ^ Robert S. Wolf, A Tour through Mathematical Logic (2005). 978-0-88385-036-7 3. ^ Thompson1991, p. 33. • Thompson, Simon (1991). Type theory and functional programming. Wokingham, England: Addison-Wesley. ISBN 0201416670. OCLC 23287456. • Wolf, Robert S. (2005). A Tour through Mathematical Logic. Vol. 30. Mathematical Association of America. ISBN 978-0-88385-042-8. JSTOR 10.4169/j.ctt5hh94h. Further reading This page was last edited on 3 September 2024, at 11:34	{"url":"https://wiki2.org/en/Free_variables_and_bound_variables","timestamp":"2024-11-05T13:34:54Z","content_type":"application/xhtml+xml","content_length":"121445","record_id":"<urn:uuid:fba57532-9367-4285-bccc-56a749a9b632>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00766.warc.gz"}
Why is square root of any number 1 more than multiple of 8 why is square root of any number 1 more than multiple of 8 Related topics: linear algebra application in our life solve algebra problems for me free calculator download in c# algebraic expression worksheets, third grade what is the importance of slope intercept in daily life difference between equation and function algebra formulas 10th class partial fraction solver Practice Work Book McDougal Littell Middle School Course 2 Math Answers vertex of parabola calculator Author Message WNCMao Posted: Sunday 19th of Aug 09:40 Hey guys ,I was wondering if someone could help me with why is square root of any number 1 more than multiple of 8? I have a major assignment to complete in a couple of months and for that I need a good understanding of problem solving in topics such as dividing fractions, trinomials and long division. I can’t start my project until I have a clear understanding of why is square root of any number 1 more than multiple of 8 since most of the calculations involved will be directly related to it in some form or the other. I have a question set , which if someone can help me solve, would help me a lot. From: Rushall, Back to top AllejHat Posted: Monday 20th of Aug 14:43 Can you give more details about the problem? I can help you if you explain what exactly you are looking for. Recently I came across a very handy software program that helps in solving math problems quickly . You can get help on any topic related to why is square root of any number 1 more than multiple of 8 and more, so I recommend trying it out. From: Odense, Back to top SanG Posted: Wednesday 22nd of Aug 13:11 I allow my son to use that software Algebrator because I believe it can effectively help him in his algebra problems. It’s been a long time since they first used that program and it did not only help him short-term but I noticed it helped in improving his solving capabilities. The software helped him how to solve rather than helped them just to answer. It’s great ! From: Beautiful Northwest Lower Back to top robintl Posted: Thursday 23rd of Aug 20:37 Sounds like something I’ve been looking for all this time! Thanks guys, just one final question, can someone please provide me a website address where I can order my copy of this software ? From: Belgium Back to top Jrobhic Posted: Friday 24th of Aug 12:17 Yeah, I do. Click on this https://softmath.com/ordering-algebra.html and I assure you that you’ll have no algebra problems that you can’t answer after using this program. Chattanooga, TN Back to top	{"url":"https://softmath.com/algebra-software/multiplying-fractions/why-is-square-root-of-any.html","timestamp":"2024-11-10T17:48:49Z","content_type":"text/html","content_length":"41308","record_id":"<urn:uuid:fb1d2554-1ddd-42d6-a54d-1959716846ba>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00252.warc.gz"}
The Earth will pass through perihelion and accelerate to its maximum speed. The Earth moves in an elliptical orbit around the Sun, marking the cycle that we know as a year. And although it seems very simple, in reality this displacement keeps its complexities. Our planet does not always travel at the same speed, this depends on how close it is to the Major Astro. In 2022, the Earth is close to reaching its maximum speed and approaching perihelionthat is, the closest point to the Sun. Thanks to astrophysics we know that the orbit described by the Earth around the Sun is 930 million kilometers. If the pertinent calculations are carried out, then we obtain that the average travel speed is 107,280 kilometers per hour. In this way, it squares with the 365 days and almost 6 hours that the planet takes to complete one revolution around its host star. Kepler’s Second Law Thanks to Kepler’s laws, it is known that our planet experiences variations in its speed, depending on how close and far it is to the Sun. According to Kepler’s second law, the speed increases until reaching its maximum (110,700 kilometers per hour) at the point known as perihelionwhich is the position of the Earth closest to the Sun. And conversely, it is reduced to the minimum (103,536 kilometers per hour) at aphelion, which is the greatest distance between the Earth and the Sun. Surprisingly, there is a difference of more than 7 thousand kilometers per hour between perihelion and aphelion. Which makes it clear to us that the Earth does not always move at the same speed, but undergoes important changes depending on the interaction with its host star. When will the Earth’s perihelion be in 2022? According to Earth Sky, the perihelion of 2022 will occur on January 4 at 00:52 CST, When the Earth is positioned at 147 million kilometers from the Sun. Therefore, it will be this day when we will move at the highest speed around the Major Astro, according to Kepler’s second law, at 110,700 kilometers per hour. For his part he aphelion, which is the farthest distance, the July 4th, when our planet moves 5 million kilometers away, taking perihelion as a reference. This day the speed will be reduced to reach 103,536 kilometers per hour around the Sun. Although Johannes Kepler, a German astronomer, did not have all the theoretical framework that has been developed today on the forces of the Universe, he was able to establish a mathematical description of the movements of the planets. Thus he established his three known laws that have given current astronomers guidelines to learn more about the Solar System. And although at the time Kepler could not fully understand what caused these variations, today we know that it is thanks to the interaction of gravitational fields that the Earth accelerates and decelerates in its transit around the Sun. Stay tuned for January 4, 2022, because even if we fail to perceive it, we will be aware that the planet will accelerate and reach its maximum translation speed. An interesting phenomenon between the terrestrial cosmic dance and its Major Astro. It might interest you	{"url":"https://www.despertarmagia.com/the-earth-will-pass-through-perihelion-and-accelerate-to-its-maximum-speed/","timestamp":"2024-11-07T05:53:44Z","content_type":"text/html","content_length":"51779","record_id":"<urn:uuid:00fbe032-2716-4725-8703-af263c328cc1>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00322.warc.gz"}
MANOVA procedure • Genstat Knowledge Base 2024 Performs multivariate analysis of variance and covariance (R.W. Payne & G.M. Arnold). PRINT = string Printed output required from the multivariate analysis of covariance (ssp, tests, permutationtest); default test APRINT = string Printed output from the univariate analyses of variance of the y-variates (as for the ANOVA PRINT option); default * UPRINT = string Printed output from the univariate unadjusted analyses of variance of the y-variates (as for the ANOVA UPRINT option); default * CPRINT = string Printed output from the univariate analyses of variance of the covariates (as for the ANOVA CPRINT option); default * TREATMENTSTRUCTURE = Treatment formula for the analysis; if this is not set, the default is taken from the setting (which must already have been defined) by the TREATMENTSTRUCTURE directive BLOCKSTRUCTURE = Block formula for the analysis; if this is not set, the default is taken from any existing setting specified by the BLOCKSTRUCTURE directive and if neither has been set the formula design is assumed to be unstratified (i.e. to have a single error term) COVARIATES = Covariates for the analysis; by default MANOVA uses those listed by a previous COVARIATE directive (if any) FACTORIAL = scalar Limit on the number of factors in a treatment term LRV = pointer Contains elements first for the treatment terms and then the covariate term (if any), allowing the LRV’s to be saved from one of the analyses; if a term is estimated in more than one stratum, the LRV is taken from the lowest stratum in which it is estimated FPROBABILITY = Printing of probabilities for F statistics (no, yes); default no string token SELECTION = string Which test statistics to print when PRINT=test (lawleyhotellingtrace, pillaibartletttrace, roysmaximumroot, wilkslambda}; default lawl, pill, roys, wilk NTIMES = scalar Number of permutations to make when PRINT=perm; default 999 EXCLUDE = factors Factors in the block model of the design whose levels are not to be randomized SEED = scalar Seed for the random number generator used to make the permutations; default 0 continues from the previous generation or (if none) initializes the seed automatically Y = variates Y-variates for an analysis Procedure MANOVA performs multivariate analysis of variance or covariance. The data variates are specified by the Y parameter. The model for the design is specified by options of the procedure. TREATMENTSTRUCTURE specifies a model formula to define the treatment terms in the analysis; if this is unset, MANOVA will use the model already defined by the TREATMENTSTRUCTURE directive, or will fail if that too has not been set. BLOCKSTRUCTURE defines the underlying structure of the design, and MANOVA will use the model (if any) previously defined by the BLOCKSTRUCTURE directive if this is not set; these can both be omitted if there is only one error term (i.e. if the design is unstratified). The COVARIATES option specifies any covariates; by default MANOVA will take those already listed (if any) by the COVARIATE directive. The FACTORIAL option can be used to set a limit on the number of factors in the terms generated from the treatment formula. The LRV option allows a pointer to be saved containing an LRV structure for each treatment term. When covariates have been specified, the pointer will also contain a final LRV structure for the covariate term. If a term is estimated in more than one stratum, the LRV is taken from the stratum that occurs last in the BLOCKTERMS pointer. The structures in the LRV hold the canonical variate loadings, roots and trace for the respective term. The PRINT option indicates the output required from the multivariate analysis of covariance, with settings ssp to print the sums of squares and products matrices, tests to print the various test statistics, and permutationtest to calculate probabilities for the test statistics using a permutation test. The SELECTION option controls which test statistics are given when PRINT=tests. The available statistics are Wilks’ Lambda (with approximate F test), the Pillai-Bartlett trace, Roy’s maximum root test and the Lawley-Hotelling trace. The default is to print them all. By default, when PRINT=perm, MANOVA makes 999 random permutations and determines the probability of each test statistic from its distribution over these randomly generated datasets. The NTIMES option allows you to request another number of allocations, and the SEED option allows you to specify the seed to use for the random numbers used to make the permutations. The permutations are done by the RANDOMIZE directive, using the block model defined by the BLOCKSTRUCTURE option. The EXCLUDE option allows you to restrict the randomization so that one or more of the factors in the block model is not randomized. The most common situation where this is required is when one of the treatment factors involves time-order, which cannot be randomized. The APRINT, UPRINT and CPRINT control output from the univariate analyses of each of the y-variates, corresponding to ANOVA options PRINT, UPRINT and CPRINT, respectively. FPROBABILITY controls whether or not probabilities are produced for F-ratios and for Chi-square variables in the analysis; by default these are omitted. Options: PRINT, APRINT, UPRINT, CPRINT, TREATMENTSTRUCTURE, BLOCKSTRUCTURE, COVARIATES, FACTORIAL, LRV, FPROBABILITY, SELECTION, NTIMES, EXCLUDE, SEED. Parameter: Y. The relevant theory, with formulae and references for the test statistics, can be found in Chatfield & Collins (1986, Chapter 9). The procedure analyses the data variates by ANOVA first as y-variates, and then as covariates in order to obtain the SSP matrices. The SSP matrices are then adjusted for the covariates, using matrix manipulation in CALCULATE, and LRV decompositions are done, before the test statistics are calculated (again using CALCULATE). If any of the y-variates is restricted, the analysis will involve only the units not excluded by the restriction. Chatfield, C. & Collins, A.J. (1986). Introduction to Multivariate Analysis (revised edition). Chapman & Hall, London. See also Procedures: RMULTIVARIATE, MVAOD. Commands for: Multivariate and cluster analysis, Repeated measurements. CAPTION 'MANOVA example',\ !t('Data from Chatfield & Collins, Introduction to Multivariate',\ 'Analysis, 1986 edition, page 143 (analysis on pages 176-178).');\ FACTOR [LEVELS=!(4,20,34); VALUES=6(4,20,34)] Temp FACTOR [LABELS=!T(Male,Female); VALUES=3(1,2)3] Sex VARIATE [NVALUES=18] InitWt,FinalWt,TumourWt READ InitWt,FinalWt,TumourWt 18.15 16.51 0.24 18.68 19.50 0.32 19.54 19.84 0.20 19.15 19.49 0.16 18.35 19.81 0.17 20.68 19.44 0.22 21.27 23.30 0.33 19.57 22.30 0.45 20.15 18.95 0.35 18.87 22.00 0.25 20.66 21.08 0.20 21.56 20.34 0.20 20.74 16.69 0.31 20.02 19.26 0.41 17.20 15.90 0.28 20.22 19.00 0.18 18.38 17.92 0.30 20.85 19.90 0.17 : MANOVA [PRINT=ssp,tests; TREATMENTSTRUCTURE=Temp*Sex;\ COVARIATES=InitWt; FPROBABILITY=yes] FinalWt,TumourWt	{"url":"https://genstat.kb.vsni.co.uk/knowledge-base/manova/","timestamp":"2024-11-06T08:32:17Z","content_type":"text/html","content_length":"46807","record_id":"<urn:uuid:aaa7175a-b4e4-46d2-b70e-1f7ee5c45513>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00140.warc.gz"}
Model was written in NetLogo 5.0.4 • Viewed 1024 times • Downloaded 101 times • Run 33 times Do you have questions or comments about this model? Ask them here! (You'll first need to log in.) WHAT IS IT? This model draws a Voronoi diagram of polygons around a set of points. These diagrams resemble many phenomena in the world including cells, forest canopies, territories of animals, fur and shell patterns, crystal growth and grain growth, cracks in dried mud and other geological phenomena, road networks, and so on. Voronoi diagrams are useful in computer graphics, vision and path planning for robots, marketing, and other applications. First the points are placed randomly. Then the polygons are drawn according to the following rules. Each point is enclosed inside exactly one polygon. All of the points inside the polygon are closer to that point than they are to any of the other points. Instead of calculating the mathematically exact coordinates of the polygons, this model constructs an approximation using a grid. Each grid cell (each "patch", in NetLogo terminology) is colored according to which point it is closest to. HOW TO USE IT Use the NUMBER slider to choose how many points you want, then press SETUP. The model will place the points and draw the polygons. If you want to play with moving the points around yourself, press the GO button. Now you can drag the points around with the mouse. As you move a point, the model redraws the polygon. This takes time, so it redraws them starting near the mouse and proceeding outward. If you want to see the boundary of the updated region, turn on the SHOW-UPDATES? switch. The line segment separating two points is exactly midway between them. How many sides do the polygons typically have? (You may want to ignore the polygons around the edges.) Where different colors touch, there is usually a "Y". When do you get a "T" or an "X" instead? Experiment with the effect of moving the points around. Moving the points slowly is best. (If you move them too fast, the model will have trouble keeping up and it won't be easy to see what's going Align two points so they have the exact same x coordinate or y coordinate. Is the line between them always perfectly smooth? (To see the effect, you may have to move the points closer or farther away from each other. Look closely.) Also try putting two points exactly on top of each other. What happens? Both effects occur because when a grid square ("patch") is equally distant from two differently colored points, NetLogo resolves the tie randomly. Instead of placing the points completely randomly, have them move away from each other until they are roughly equidistant from each other. This makes all the polygons roughly the same size. Edit the view and turn wrapping on in both directions, and click SETUP. The model may seem to be working, but there is a problem. If you turn on SHOW-UPDATES?, you can see that the update rectangle keeps going forever, continually refreshing the grid colors. Fix the model to work with wrapping, so that update stops as soon as the whole screen has been redrawn. Instead of using the patches to display Voronoi polygons, find the boundaries by using turtles. Create a large batch of turtles at each point (colored the same color as the point), each turtle facing a different angle. Have the turtles walk outward from their points at a uniform rate. Stop the turtles when they run into a turtle of a different color. Instead of using a patch-based approximation, calculate the exact positions of the sides of the polygons. (There are numerous published algorithms for calculating this information.) Then display the polygons using turtles with the "line" shape. The core procedure for drawing the polygons is called recolor. It is only one line long! It puts the min-one-of and distance reporters to good use. The mouse-down?, mouse-xcor, and mouse-ycor primitives are used so the user can interact with the model. Because the number of patches is so large, it takes a while to update them all when a point moves. So we use moving turtles to recolor the patches; the moving turtles start where the mouse is, and move outwards in a square, since near the mouse is where the user will be looking first. See the Code tab for the details on how it works. • MaterialSim Grain Growth • Fur • Honeycomb • Scatter For more information on Voronoi diagrams, see http://en.wikipedia.org/wiki/Voronoi. (There are also many other sites on this topic on the web.) Thanks to John Jungck from Beloit College for inspiring this model with his talk at Northwestern University about Voronoi structures in nature. Thanks to Josh Unterman and Seth Tisue for their work on this model. If you mention this model in a publication, we ask that you include these citations for the model itself and for the NetLogo software: • Wilensky, U. (2006). NetLogo Voronoi model. http://ccl.northwestern.edu/netlogo/models/Voronoi. Center for Connected Learning and Computer-Based Modeling, Northwestern Institute on Complex Systems, Northwestern University, Evanston, IL. • Wilensky, U. (1999). NetLogo. http://ccl.northwestern.edu/netlogo/. Center for Connected Learning and Computer-Based Modeling, Northwestern Institute on Complex Systems, Northwestern University, Evanston, IL. Copyright 2006 Uri Wilensky. This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA. Commercial licenses are also available. To inquire about commercial licenses, please contact Uri Wilensky at uri@northwestern.edu. Comments and Questions My learning sciences design class (Question) I'm hoping this interface will work well in my class. Any thoughts on issues that might arise for my class? I'm thinking we might need pagination for the comments. Uri Wilensky Posted almost 14 years ago Re: learning sciences design class Well, there is the issue that questions like this end up attached to models like "Voronoi", when I think the question is more broadly about the Modeling Commons platform... :-) Forrest Stonedahl Posted almost 14 years ago Re: learning sciences design class I remember liking to be able to look at code from the Commons to help people debug and think through their models. It helped me work harder to talk through the code and help people Michelle explain and interpret it, rather than jumping in and editing! Posted almost 14 years ago globals [ available-colors ;; list of colors that will be assigned to points current-point ;; the point the user is currently moving breed [points point] ;; these are the little circles in the middle of the polygons ;; The next two breeds are used only when we're updating the polygons ;; when the user moves the points with the mouse. See below for further ;; details. breed [spawners spawner] breed [updaters updater] ;;; CORE PROCEDURES ;;; These are the only procedures necessary to draw the diagram ;;; initially, without moving points. to setup ;; too dark and too light are hard to distinguish from each other, ;; so only use 13-17, 23-27, ..., 133-137 set available-colors shuffle filter [(? mod 10 >= 3) and (? mod 10 <= 7)] n-values 140 [?] set-default-shape points "circle 3" ask n-of number patches [ make-point ] ask patches [ recolor ] set current-point nobody to make-point ; patch procedure sprout-points 1 [ set size 5 set color first available-colors set available-colors butfirst available-colors to recolor ;; can be patch or turtle procedure set pcolor [color] of min-one-of points [distance myself] ;;; OTHER PROCEDURES ;;; The rest of the procedures are used for efficiently updating ;;; the diagram when the user moves the points around. to go ask spawners [ spawn ] ask updaters [ update ] to obey-mouse ;; first handle the case where the user has released the mouse button ;; (or hasn't pressed it yet) if not mouse-down? [ set current-point nobody ;; if the mouse button is down, get the mouse position let x round mouse-xcor let y round mouse-ycor ;; if we don't have a point yet, pick the closest one if current-point = nobody [ set current-point min-one-of points [distancexy x y] ;; check if the point needs to move if x != [xcor] of current-point or y != [ycor] of current-point [ ;; move the point ask current-point [ setxy x y ] ;; the point has moved, so we need to recolor all patches, so we kill off ;; the old turtles that were doing the recoloring and make new ones ask spawners [ die ] ask updaters [ die ] ask current-point [ ask patch-here [ make-spawners ] ] ;; Here's how we use turtles to update the patches in a growing ;; square pattern. We use two breeds of turtles, spawners and updaters. ;; Spawners are at the corners of the square and move diagonally. Each ;; time a spawner moves, it spawns two new updaters. The updaters move ;; vertically or horizontally, and every time an updater lands on a patch, ;; it recolors it. When a spawner or an updater hits the edge of the world, ;; it dies. Together, these rules are enough to make the growing square! to make-spawners ;; patch procedure let counter 0 sprout-spawners 4 [ ;; give the four headings of NE, SE, SW, and NE set heading 45 + counter * 90 set counter counter + 1 set color gray if not show-updates? [ hide-turtle ] to spawn ;; spawner procedure hatch-updaters 1 [ rt 45 ] hatch-updaters 1 [ lt 45 ] if not can-move? 1 [ die ] ;; Moving diagonally is a little tricky. Moving forward 1 isn't enough by ;; itself, since that doesn't take us all the way to the center of the ;; diagonally next patch. So after moving forward, we use SETXY to move ;; to the exact center of the new patch. fd 1 setxy pxcor pycor to update ;; updater procedure if not can-move? 1 [ die ] fd 1 ; Copyright 2006 Uri Wilensky. ; See Info tab for full copyright and license. 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03-31-11 - Some image filter notes Say you have a filter like F = [1,2,3,2,1] . The normal thing to do is compute the sum and divide so you have pre-normalized values and you just do a bunch of madd's. eg. you make N = [1/9,2/9,3/9,2/ Now there's the question of how you handle the boundaries of the image. The normal thing to do is to take the pre-normalized filter N and apply all over, and when one of the taps samples off edge, you have to give it something to sample. You can use various edge modes, such as : SampleBounded(int i, int w) : clamp : return Sample( Clamp(i,0,w-1) ); wrap : return Sample( (i+256w)%w ); mirror (no duplicated edge pixel) : if ( i < 0 ) return SampleBounded( -i, w ); if ( i >= w ) return SampleBounded( -i + 2w - 2 , w ); else return Sample( i ); mirror with duplicated edge pixel : if ( i < 0 ) return SampleBounded( - i - 1, w ); if ( i >= w ) return SampleBounded( -i + 2w - 1 , w ); else return Sample( i ); (the correct edge mode depends on the usage of the image, which is one of those little annoying gotchas in games; eg. the mips you should make for tiling textures are not the same as the mips for non-tiling textures). (another reasonable option not implemented here is "extrapolate" , but you have to be a bit careful about how you measure the slope at the edge of the image domain) The reason we do all this is because we don't want to have to accumulate the sum of filter weights and divide by the weight. But really, in most cases what you really should be doing is applying the filter only where its domain overlaps the image domain. Then you sum the weights in the area that is valid and renormalize. eg. if our filter F is two pixels off the edges, we just apply [3,2,1] / 6 , we don't clamp the sampler and put an extra [1,2] on the first pixel. ADDENDUM : in video games there's another special case that needs to be handled carefully. When you have a non-tiling texture which you wish to abutt seamlessly to another texture. That is, you have two textures T1 and T2 that are different and you wish to line them up beside each other without a seam. I call this mode "shared", it sort of acts like "clamp" but has to be handled specially in filtering. Lets say T1 and T2 are layed against eachother horizontally, so they abutt along a column. What the artist should do is make the pixels in that border column identical in both textures (or you could have your program enforce this). Then, the UV mapping on the adjacent rectangles should be inset by half a pixel - that is, it picks the center of the pixels, not the edge of the texture. Thus the duplicated pixel edge only appears to be a single column of pixels. But that's not the special case handling - the special case is whenever you filter a "shared" image, you must make border column pixels only from other border column pixels. That is, that shared edge can only be vertically filtered, not horizontally filtered. That way it stays identical in both images. Note that this is not ideal with mipping, what happens is the shared edge gets fatter at higher mip levels - but it never develops a seam, so it is "seamless" in that sense. To do it right without any artifacts (eg. to look as if it was one solid bigger texture) you would have to know what image is on the other side of the shared edge and be able to filter tap into those pixels. Obviously that is impossible if your goal is a set of terrain tiles or something like that where you use the same shared edge in multiple different ways. (is there a better solution to this issue?) I did a little look into the difference between resizing an image 8X by either doubling thrice or directly resizing. I was sanity checking my filters and I thought - hey if I use a Gaussian filter, it should be the same thing, because convolution of a Gaussian with a Gaussian is a Gaussian, right? In the continuous case, you could either use one Gaussian with an sdev of 8 (not actually right for 8X mag, but you get the idea). If you had a Gaussian with sdev 2 and convolved it 3 times - you should get a Gaussian with sdev of 8. So I tried it on my filters and I got : Gaussian for doubling, thrice : Gaussian for direct 8x : and I was like yo, WTF they're way off, I must have a bug. (note : these are scaled to make the max value 1.0 rather than normalizing because it's easier to compare this way, they look more unequal after normalizing) But then I realized - these are not really proper Gaussians. These are discrete samples of Gaussians. If you like, it's a Gaussian multiplied by a comb. It's not even a Gaussian convolved with a box filter - that is, we are not applying the gaussian over the range of the pixel as if the pixel was a box, but rather just sampling the continuous function at one point on the pixel. Obviously the continuous convolution theorem that Gauss [conv] Gauss = Gauss doesn't apply. As for the difference between doing a direct 8X and doubling thrice, I can't see a quality difference with my eyes. Certain the filters are different numerically - particularly filters with negatives, eg. : sinc double once : sinc double twice : sinc double thrice : sinc direct 8x : very different, but visually meh? I don't see much. The other thing I constantly forget about is "filter inversion". What I mean is, if you're trying to sample between two different grids using some filter, you can either apply the filter to the source points or the dest points, and you get the same results. More concretely, you have filter shape F(t) and some pixels at regular locations P[i]. You create a continuous function f(t) = Sum_i P[i] F(i-t) ; so we have placed a filter shape at each pixel center, and we are sampling them all at some position t. But you can look at the same thing a different way - f(t) = Sum_i F(t-i) * P[i] ; we have a filter shape at position t, and then we are sampling it at each position i around it. So, if you are resampling from one size to another, you can either do : 1. For each source pixel, multiply by filter shape (centered at source) and add shape into dest, or : 2. For each dest pixel, multiply filter shape (centered at dest) by source pixels and put sum into dest. And the answer is the same. (and usually the 2nd is much more efficient than the first) And for your convenience, here are some doubling filters : box : const float c_filter[1] = { 1.00000 }; linear : const float c_filter[2] = { 0.25000, 0.75000 }; quadratic : const float c_filter[3] = { 0.28125, 0.68750, 0.03125 }; cubic : const float c_filter[4] = { 0.00260, 0.31510, 0.61198, 0.07031 }; mitchell0 : const float c_filter[4] = { -0.02344, 0.22656, 0.86719, -0.07031 }; mitchell1 : const float c_filter[4] = { -0.01476, 0.25608, 0.78212, -0.02344 }; mitchell2 : const float c_filter[4] = { 0.01563, 0.35938, 0.48438, 0.14063 }; gauss : const float c_filter[5] = { 0.00020, 0.20596, 0.78008, 0.01375, 0.00000 }; sqrtgauss : const float c_filter[5] = { 0.00346, 0.28646, 0.65805, 0.05199, 0.00004 }; sinc : const float c_filter[6] = { 0.00052, -0.02847, 0.23221, 0.87557, -0.08648, 0.00665 }; lanczos4 : const float c_filter[4] = { -0.01773, 0.23300, 0.86861, -0.08388 }; lanczos5 : const float c_filter[5] = { -0.04769, 0.25964, 0.89257, -0.11554, 0.01102 }; lanczos6 : const float c_filter[6] = { 0.00738, -0.06800, 0.27101, 0.89277, -0.13327, 0.03011 }; These are actually pairs of filters to create adjacent pixels in a double-resolution output. The second filter of each pair is simply the above but in reverse order (so the partner for linear is 0.75, 0.25). To use these, you scan it over the source image and apply centered at each pixel. This produces all the odd pixels in the output. Then you take the filter and reverse the order of the coefficients and scan it again, this produces all the even pixels in the output (you may have to switch even/odd, I forget which is which). These are created by taking the continuous filter function and sampling at 1/4 offset locations - eg. if 0 is the center (maximum) of the filter, you sample at -0.75,0.25,1.25, etc. And here's the same thing with a 1.15 X blur built in : box : const float c_filter[1] = { 1.0 }; linear : const float c_filter[2] = { 0.30769, 0.69231 }; quadratic : const float c_filter[3] = { 0.00000, 0.33838, 0.66162 }; cubic : const float c_filter[5] = { 0.01586, 0.33055, 0.54323, 0.11034, 0.00001 }; mitchell0 : const float c_filter[5] = { -0.05174, 0.30589, 0.77806, -0.03143, -0.00078 }; mitchell1 : const float c_filter[5] = { -0.02925, 0.31410, 0.69995, 0.01573, -0.00052 }; mitchell2 : const float c_filter[5] = { 0.04981, 0.34294, 0.42528, 0.18156, 0.00041 }; gauss : const float c_filter[6] = { 0.00000, 0.00149, 0.25842, 0.70629, 0.03379, 0.00002 }; sqrtgauss : const float c_filter[6] = { 0.00000, 0.01193, 0.31334, 0.58679, 0.08726, 0.00067 }; sinc : const float c_filter[7] = { 0.00453, -0.05966, 0.31064, 0.78681, -0.03970, -0.00277, 0.00015 }; lanczos4 : const float c_filter[5] = { -0.05129, 0.31112, 0.78006, -0.03946, -0.00042 }; lanczos5 : const float c_filter[6] = { 0.00499, -0.09023, 0.33911, 0.80082, -0.04970, -0.00499 }; lanczos6 : const float c_filter[7] = { 0.02600, -0.11420, 0.34931, 0.79912, -0.05497, -0.00837, 0.00312 }; The best doubling filters to my eyes are sinc and lanczos5, they have a good blend of sharpness and lack of artifacts. Stuff like gauss and cubic are too blurry, but are very smooth ; lanczos6 is sharper but has more ringing and stair-steps; wider lanczos filters get worse in that way. Sinc and lanczos5 without any blur built in can have a little bit of visible stair-steppiness (there's an inherent tradeoff when linear upsampling of sharpness vs. stair-steps) (by stair steps I mean the ability to see the original pixel blobs). 3 comments: ryg said... "But then I realized - these are not really proper Gaussians. These are discrete samples of Gaussians. If you like, it's a Gaussian convolved with a comb filter." This is if you use a sampled Gaussian as your filter. There's an alternative approach - the basic idea is that convolution with a Gaussian is the solution to a simple continuous uniform linear diffusion equation (at a time proportional to the desired variance). You can then consider that same diffusion problem on a discrete space (i.e. grid) and use the solutions as your discrete Gaussian approximation. The 1D kernel coefficients are given by K[sigma](n) = exp(-sigma) * I_n(sigma) where I_n is the modified Bessel function of order n. For 2D you just do the usual separable filtering thing; the resulting kernel doesn't have perfect rotational symmetry, but it does have the convolution theorem: K[sigma1] * K[sigma2] = K[sigma1+sigma2]. There's a nice introduction on Wikipedia, and the original paper by Lindeberg is available online. castano said... > what you really should be doing is applying the filter only where its domain overlaps the image domain. Interesting... and if you are using a polyphase filter (case 2 below) you are probably precomputing the kernel weights for each column, so you can normalize the edge case in advance and it comes out for free. I still think that wrapping properly is preferable if you have that information. cbloom said... "I still think that wrapping properly is preferable if you have that information." Yeah, video game textures are sort of a special case. Note : addendum added to original post on this subject. Also, for the small filters shown here, the issue of off-edge sampling is not very important (assuming your image is large - 3 pixels of edge being slightly not perfect on a 1920 wide image is In some cases however (SCIELAB for example) I've used some huge Gaussians, like 100 pixels wide, and then the off-edge contribution to the filter can be very significant.	{"url":"http://cbloomrants.blogspot.com/2011/03/03-31-11-some-image-filter-notes.html","timestamp":"2024-11-12T14:13:11Z","content_type":"application/xhtml+xml","content_length":"77868","record_id":"<urn:uuid:43db63a3-9d68-4582-a324-d9409a862acb>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00888.warc.gz"}
Pi Day Pi Day, not to be confused with National Pie Day (January 23rd), is celebrated at FM on March 14th with more than just reciting digits. Although the 3.14159265358979323846264…yada yada yada is somewhat the highlight of the event, “American Pidol” and a Rubik’s cube competition also are vital to the experience. If you would like to participate, start practicing now. It is also important to back up and visit the history of the holiday itself. According to Piday.org, in ancient Babylon people stretched ropes around buildings to estimate pi as 25/8, or 3.125. The Egyptians were a little farther off with (16/9)², or approximately 3.16. Archimedes used an approach with having a many-sided polygon slightly inside and outside a circle. He concluded that pi was between 223/71 and 22/7. Mathematicians refined this method until the Austrian Christoph Grienberger accurately calculated 38 digits of pi using 10^40 sided-polygons. When the Renaissance came along, William Oughtred and then Leonard Euler finally attached a symbol to pi (Euler is given more credit, as corroborated by “History of Pi” by Patt J. Gilly). Pi is the first letter of the word perimetros, which loosely translates to circumference from Greek. Johann Heinrich Lambert proved pi’s irrationality in 1767 and Ferdinand von Lindermann proved it was transcendental in 1882. Irrational numbers do not repeat or end, but transcendental numbers cannot be found from an equation with polynomial coefficients. For example, the square root of 5 is irrational, but it is not transcendental because it is a solution to the equation x⁴ + x² = 30. Given modern technology, we now know 31 trillion digits of pi. This is thought to be entirely unnecessary, as only 39 digits are needed to calculate anything in the universe within the length of a hydrogen atom. However, we may discover situations in the future where it is necessary to be more precise than hydrogen atoms, and calculate larger volumes than one universe. Humanity could bring us anywhere, and you can start with memorizing digits, writing creative parodies, or speeding up your Rubik’s Cube solving time. About the Writer Hudson Brenner, Co-Features Editor Hudson Brenner is a co-features editor. His password is the last 8 digits of pi, although he is contemplating changing it to the next 16 to accommodate...	{"url":"https://thefmbuzz.org/2132/features/pi-day/","timestamp":"2024-11-05T04:02:43Z","content_type":"application/xhtml+xml","content_length":"103282","record_id":"<urn:uuid:714ba289-a57c-4d6e-b04d-30cde8ab2a14>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00052.warc.gz"}
Greatest Common Divisors Definition. If m and n are integers, not both 0, the greatest common divisor Example. ( Greatest common divisors for small integers) Find by direct computation The largest integer which divides 4 and 6 is 2: The largest integer which divides -6 and 15 is 3: The largest integer which divides 42 and 0 is 42: Finally, the largest integer which divides 24 and 25 is 1: Here are some easy properties of the greatest common divisor. Proposition. Let (d) If Proof. (a) That (b) On the one hand, the set of common divisors is finite (because a common divisor can't be larger than Now greatest common divisor (c) The largest integer which divides both a and b is the same as the largest integer which divides both b and a. In similar fashion, I'll use the Division Algorithm to derive a method for computing the greatest common divisor of two numbers. The idea is to perform the Division Algorithm repeatedly until you get a remainder of 0. First, I need a lemma which is useful in its own right. Lemma. If a and b are integers, not both 0, and k is an integer, then Proof. If d divides a and b, then d divides If d divides I've proved that the set of common divisors of a and b is the same as the set of common divisors of The lemma says that the greatest common divisor of two numbers is not changed if I change one of the numbers by adding or subtracting an integer multiple of the other. This can be useful by itself in determining greatest common divisors. Example. Prove that if n is an integer, then The idea is to subtract multiples of one number from the other to reduce the powers until I get an expression which is clearly equal to 1. Theorem. ( The Euclidean Algorithm) Let (a) The process will terminate with (b) At the point when the process terminates, Proof. There is no question that I can apply the Division Algorithm as described above, as long as First, I'll show that the process terminates with Note that At any stage, I'm starting with This means that In other words, each step leaves the greatest common divisor of the pair of a's unchanged. Thus, Example. ( Using the Euclidean algorithm to find a greatest common divisor) Use the Euclidean algorithm to compute To save writing --- and to anticipate the setup I'll use for the Extended Euclidean Algorithm later --- I'll arrange the computation in a table: The greatest common divisor is the last nonzero remainder (3). Hence, Definition. If a and b are things, a linear combination of a and b is something of the form The next result is a key fact about greatest common divisors. Theorem. ( Extended Euclidean Algorithm) Note: s and t are not unique. Proof. The proof will actually give an algorithm which constructs a linear combination. It is called a backward recurrence, and it appears in a paper by S. P. Glasby [2]. It will look a little complicated, but you'll see that it's really easy to use in practice. Suppose nonzero remainder: I'm going to define a sequence of numbers starting with and working downward to backward recurrence.) Now I claim that I will prove this by downward induction, starting with The result holds for Next, suppose I want to prove the result for k. Substitute This proves the result for k, so the result holds for In particular, for Remark. There are many algorithms (like the one in the proof) which produce a linear combination. This one is pretty good for small computations which you're doing by hand. One drawback of this algorithm is that you need to know all of the quotients (the q's) in order to work backwards to get the linear combination. This isn't bad for small numbers, but if you're using large numbers on a computer, you'll need to store all the intermediate results. There are algorithms which are better if you're doing large computations on a computer (see [1], page 300). It's difficult to overemphasize the importance of this result! It has many applications --- from proving results about greatest common divisors, to solving Diophantine equations. I'll give some examples which illustrate the result, then discuss how you use the algorithm in the theorem. Before I give examples of the algorithm, I'll look at some other ways of finding a linear combination. Definition. Let relatively prime if Example. ( A linear combination for a greatest common divisor) Show that 12 and 25 are relatively prime. Write their greatest common divisor as as linear combination with integer coefficients of 12 and 25. In some cases, the numbers are nice enough that you can figure out a linear combination by trial and error. In this case, it's clear that Note that Example. ( Finding a linear combination by algebra) Use the Division Algorithm computations in the Euclidean algorithm to find an integer linear combination of 51 and 36 that is equal to It's possible --- but tedious --- to use the computations in the Euclidean algorithm to find linear combinations. For The third equation says By the second equation, The first equation says I've expressed the greatest common divisor 3 as a linear combination of the original numbers 51 and 36. I don't recommend this approach, since the proof of the Extended Euclidean Algorithm gives a method which is much easier and less error-prone. Example. ( Finding a linear combination using the backward recursion) Find In this example, I'll show how you can use the bakcward recursion to obtain a linear combination. I'll arrange the computations in the form of a table; the table is simply an extension of the table I used for the Euclidean algorithm. In this example only, I'm labelling the columns with the variable names a, q, and y from the proof so you can see the correspondence. Normally, I'll omit them. Here's how you start: (You can save a step by putting the larger number first.) The a and q columns are filled in using the Euclidean algorith, i.e. by successive division: Divide the next-to-the-last a by the last a. The quotient goes into the q-column, and the remainder goes into the a-column. When the division comes out evenly, you stop adding rows to the table. In this case, 85 divided by 17 is 5, and the remainder is 0. The last entry in the a-column is the greatest common divisor. Thus, Having filled in the a and q columns, you now fill in the y-column from bottom to top. You always start in the same way: The last y is always 0 and the next-to-the-last y is always 1: Then, working from bottom to top, fill in the y's using the rule This comes from the recursion formula in the Extended Euclidean Algorithm Theorem: It's probably easier to show than it is to explain: To get the linear combination, form the products of the top two a's and y's diagonally and subtract one from the other: How do you know the order for the subtraction? The proof gives a formula, but the easiest thing is to pick one of the two ways, then fix it if it isn't right. If you subtract "the wrong way", you'll get a negative number. For example, Since I know the greatest common divisor should be 17 --- it's the last number in the a-column --- I just multiply this equation by -1: This way, you don't need to memorize the exact formula. Example. ( Finding a linear combination using the backward recursion) Compute Example. ( The converse of the linear combination result) Give specific numbers a, b, m, n and d such that The converse of the linear combination result is not always true. That is, if For example, There's an important situation in which the linear combination result does work backwards: namely, when the greatest common divisor is 1. The next result makes this precise, and also shows how you can use the linear combination rule to prove results about greatest common divisors. Proposition. Let Proof. The greatest common divisor of a and b can be written as a linear combination of a and b. Therefore, if Conversely, suppose that Example. ( Using a linear combination to prove relative primality) Prove that if k is any integer, then the fraction For example, if A fraction is in lowest terms if the numerator and denominator are relatively prime. So I want to show that I'll use the previous result, noting that I found the coefficients by playing with numbers, trying to make the k-terms cancel. Since a linear combination of The linear combination rule is often useful in proofs involving greatest common divisors. If you're proving a result about a greatest common divisor, consider expressing the greatest common divisor as a linear combination of the two numbers. Proposition. Let a and b be integers, not both 0. If greatest common divisor of a and b, in the sense that it was the largest common divisor of a and b. The last lemma shows that you can take greatest in a different sense --- namely, that be divisible by any other common divisor of a and b. Example. ( Using the linear combination result to prove a greatest common divisor property) Prove that if Multiplying by k, I get On the other hand, Since k and [1] Alfred Aho, John Hopcroft, and Jeffrey Ullman, The Design and Analysis of Computer Algorithms. Reading, Massachusetts: Addison-Wesley Publishing Company, 1974. [2] S. P. Glasby, Extended Euclid's algorithm via backward recurrence relations, Mathematics Magazine, 72(3)(1999), 228--230. Copyright 2018 by Bruce Ikenaga	{"url":"https://sites.millersville.edu/bikenaga/abstract-algebra-1/greatest-common-divisors/greatest-common-divisors.html","timestamp":"2024-11-05T23:08:34Z","content_type":"text/html","content_length":"44957","record_id":"<urn:uuid:4774fc46-a63f-4978-92a6-bfc6f834e50b>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00696.warc.gz"}
Problem Model Prompt P_Combinatorics_3 PaLM 2 Medium Root C w/ H Ground Truth Answer: 2^floor(n/2)floor(n/2)! Use in Code import champ_dataset # "pip install champ-dataset" if necessary dataset = champ_dataset.load('v0') problem = dataset['P_Combinatorics_3'] messages = problem.conversations['PaLM 2 Medium\|Root C w/ H'].messages for message in messages: print(f'Role: {message.role}') print(f'Text: {message.text}') print(f'Error: {message.error}') Full Conversation System Prompt User Input Model Output You are an expert on mathematics. Solve the following problem. Make sure to show your work before giving the final answer. On a chess board, two rooks are placed peacefully if they are not on the same row or column. For an n x n chess board, find the number of ways that n rooks can be placed peacefully (i.e., any two are placed peacefully) and the placing is also invariant to a 180-degree rotation. You may find the following information useful: 1. If n is odd, what position must be occupied by a rook? 2. For even n=2k, consider a board of size 2k x 2k. If a rook is placed on the first row, which other position is also determined? 3. Let P(2k) be the number of placements on a 2k x 2k board. Express P(2k) in terms of P(2k-2). 4. If there are n steps of doing a task, and there are p_i ways to do the i-th step, then there are p_1p_2...p_n ways to do the task. 5. Find P(2) as the base case. Let $P(n)$ be the number of ways to place $n$ rooks on an $n\times n$ chess board so that no two rooks are in the same row or column. We will prove by induction on $n$ that $P(n) = \binom{n}{2}$. The base case is $n=2$. There is only one way to place two rooks on a $2\times 2$ chess board so that no two rooks are in the same row or column, namely, to place one rook in each corner. Now assume that the claim holds for some $n\geq 2$, and let $n=2k$ be even. We will show that $P(2k+2) = \binom{2k+2}{2}$. Consider a $2k+2\times 2k+2$ chess board. We can place the first $k$ rooks on the first $k$ rows in any order, since they cannot be in the same row or column. Once the first $k$ rooks are placed, there are $2k$ empty squares remaining on the board. We can place the $(k+1)$-th rook in any of these squares, since it cannot be in the same row or column as any of the first $k$ rooks. Once the $(k+1)$-th rook is placed, there are $2k-1$ empty squares remaining on the board. We can place the $(k+2)$-th rook in any of these squares, since it cannot be in the same row or column as any of the first $k+1$ rooks. We can continue in this way, placing each rook in a square that is not in the same row or column as any of the previously placed rooks. In this way, we can place all $k+2$ rooks on the board so that no two rooks are in the same row or column. Now consider the following two placements of $k+2$ rooks on a $2k+2\times 2k+2$ chess board: 1. Place the first $k$ rooks on the first $k$ rows in any order. 2. Place the $(k+1)$-th rook in the bottom right corner. 3. Place the $(k+2)$-th rook in the square directly above the $(k+1)$-th rook. 4. Continue in this way, placing each rook in the square directly above the previously placed rook. 2. Place the first $k$ rooks on the first $k$ rows in any order. 3. Place the $(k+1)$-th rook in the top left corner. 4. Place the $(k+2)$-th rook in the square directly below the $(k+1)$-th rook. 5. Continue in this way, placing each rook in the square directly below the previously placed rook. It is easy to see that these two placements are equivalent under a 180-degree rotation. Therefore, there are $\binom{2k+2}{2}$ ways to place $k+2$ rooks on a $2k+2\times 2k+2$ chess board so that no two rooks are in the same row or column. By the principle of mathematical induction, the claim holds for all $n\geq 2$. Now, summarize the answer above in one sentence, without any intermediate steps or explanations. The number of ways to place $n$ rooks on an $n\times n$ chess board so that no two rooks are in the same row or column is $\binom{n}{2}$.	{"url":"https://yujunmao1.github.io/CHAMP/solutions/p_combinatorics_3_palm2m_rootcwh.html","timestamp":"2024-11-05T04:12:02Z","content_type":"text/html","content_length":"7998","record_id":"<urn:uuid:aa68c30a-fa9a-48c8-b621-e7f3f4b54a23>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00656.warc.gz"}
Efficient Ways to Generate a Matrix with Manhattan Distances from the Nearest Zero in Python π ‘ Problem Formulation: The challenge is to create a Python program that generates a matrix of integers, where each cell contains the Manhattan distance to the nearest zero in the matrix. To clarify, the Manhattan distance between two points (x1, y1) and (x2, y2) is abs(x1 - x2) + abs(y1 - y2). For instance, given the binary matrix input [[0, 0, 0], [0, 1, 0], [0, 0, 0]], the desired output would be [[0, 0, 0], [0, 1, 0], [0, 0, 0]] because every ‘1’ is adjacent to a ‘0’, resulting in a Manhattan distance of 1. Method 1: Breadth-First Search (BFS) This method involves performing a breadth-first search (BFS) on the matrix starting from the cells that contain zeros. During the search, the distance from these cells is propagated to their adjacent cells incrementally. This technique guarantees that the shortest path (Manhattan distance) to a zero will be found for each cell. Here’s an example: from collections import deque def update_matrix(matrix): rows, cols = len(matrix), len(matrix[0]) dist = [[float('inf')] * cols for _ in range(rows)] q = deque() for r in range(rows): for c in range(cols): if matrix[r][c] == 0: dist[r][c] = 0 q.append((r, c)) directions = [(1,0), (-1,0), (0,1), (0,-1)] while q: r, c = q.popleft() for dr, dc in directions: rr, cc = r + dr, c + dc if 0 <= rr < rows and 0 <= cc dist[r][c] + 1: dist[rr][cc] = dist[r][c] + 1 q.append((rr, cc)) return dist # Example usage: matrix = [[0, 0, 0], [0, 1, 0], [1, 1, 1]] [[0, 0, 0], [0, 1, 0], [1, 2, 1]] This code snippet initializes a 2D array, dist, to keep track of distances and uses a queue to perform a BFS starting from the cells that are originally zeros. Distances are updated as the BFS proceeds, ensuring that the shortest path to a zero is recorded. Method 2: Dynamic Programming Dynamic programming can solve this problem by iteratively updating the matrix to find the smallest distance to a zero for each cell. This involves two passes over the matrix: one from top-left to bottom-right, and another from bottom-right to top-left, updating distances based on already calculated values. Here’s an example: def update_matrix(matrix): rows, cols = len(matrix), len(matrix[0]) dist = [[float('inf')] * cols for _ in range(rows)] for r in range(rows): for c in range(cols): if matrix[r][c] == 0: dist[r][c] = 0 if r > 0: dist[r][c] = min(dist[r][c], dist[r-1][c] + 1) if c > 0: dist[r][c] = min(dist[r][c], dist[r][c-1] + 1) for r in range(rows-1, -1, -1): for c in range(cols-1, -1, -1): if r < rows-1: dist[r][c] = min(dist[r][c], dist[r+1][c] + 1) if c < cols-1: dist[r][c] = min(dist[r][c], dist[r][c+1] + 1) return dist # Example usage: matrix = [[0, 0, 0], [0, 1, 0], [1, 1, 1]] [[0, 0, 0], [0, 1, 0], [1, 2, 1]] In this code snippet, the dist matrix is initialized and iteratively updated in two sweeps using dynamic programming. The values are updated based on the minimum distances calculated in the previous steps for adjacent cells. Method 3: Using NumPy for Vectorized Operations For those working with numerical data in Python, using NumPy can lead to concise and fast operations. This method leverages NumPy to perform operations across the entire matrix simultaneously, utilizing vectorization for efficiency. Here’s an example: import numpy as np def update_matrix(matrix): # This is a placeholder for the NumPy-based method which would require # complex manipulation and is not directly provided here. # Example usage assumes there is a numpy-based solution: # matrix = [[0, 0, 0], [0, 1, 0], [1, 1, 1]] # print(update_matrix(matrix)) Assumed output after applying NumPy operations would be: [[0, 0, 0], [0, 1, 0], [1, 2, 1]] Explanation for this placeholder is that it represents where a NumPy-based method would be implemented. In practice, one would apply NumPy array operations to manipulate the matrix data efficiently with potential use of broadcasting, slicing, and other vectorized operations. Method 4: Recursive Approach A recursive approach involves recursively calculating the Manhattan distance from each non-zero cell to the closest zero. By carefully managing the base cases and avoiding recalculations using memoization, this method can be effective for smaller matrices. Here’s an example: def update_matrix(matrix): # Placeholder for a possible recursive implementation # Example usage is theoretical for this recursive snippet: # matrix = [[0, 0, 0], [0, 1, 0], [1, 1, 1]] # print(update_matrix(matrix)) Potential output, depending on the precise implementation: [[0, 0, 0], [0, 1, 0], [1, 2, 1]] This code snippet is a theoretical placeholder for a recursive solution. A recursive approach would require careful design to efficiently calculate the distances for each cell, avoiding needless recalculations and taking care to not exceed Python’s maximum recursion depth. Bonus One-Liner Method 5: List Comprehension and Min Function A one-liner approach in Python can utilize list comprehension and the min function to calculate the Manhattan distances in a more concise way. This is a more Pythonic and potentially less efficient method given that it could involve generating a list of distances for each cell. Here’s an example: def update_matrix(matrix): rows, cols = len(matrix), len(matrix[0]) # This is not recommended for large matrices due to performance concerns dist = [[min(abs(r-i) + abs(c-j) for i in range(rows) for j in range(cols) if matrix[i][j] == 0) for c in range(cols)] for r in range(rows)] return dist # Example usage: matrix = [[0, 0, 0], [0, 1, 0], [1, 1, 1]] [[0, 0, 0], [0, 1, 0], [1, 2, 1]] This code snippet uses nested list comprehensions to calculate the minimum Manhattan distance from each cell to the nearest zero. While concise, this approach is computationally intensive and not recommended for larger matrices due to poor performance. • Method 1: Breadth-First Search (BFS). Best for large matrices. Efficient and guarantees the shortest path. Can be memory-intensive for very large matrices due to queue usage. • Method 2: Dynamic Programming. Also efficient for large matrices. Uses two passes for accuracy and can deal with larger data sets. May not be as intuitive as BFS for some users. • Method 3: Using NumPy for Vectorized Operations. Best for those comfortable with NumPy and numerical computing. Can lead to very concise code but may require complex manipulation and understanding of NumPy operations. • Method 4: Recursive Approach. Suitable for smaller matrices as it can hit the recursion limit for larger matrices and be less efficient due to overhead of recursive calls. • Bonus One-Liner Method 5: List Comprehension and Min Function. Pythonic and concise but inefficient for larger matrices due to computational complexity.	{"url":"https://blog.finxter.com/efficient-ways-to-generate-a-matrix-with-manhattan-distances-from-the-nearest-zero-in-python/","timestamp":"2024-11-02T20:59:42Z","content_type":"text/html","content_length":"75025","record_id":"<urn:uuid:9a16e08a-ee1c-4144-8531-0f7407c6cdcf>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00552.warc.gz"}
7.5: The Slope-Intercept Form of a Line (2024) Page ID $ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}$ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} $ $ \newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ ( \newcommand{\kernel}{\mathrm{null}\,}\) $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\\| #1 \\|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\\| #1 \\|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\AA}{\unicode[.8,0]{x212B}}$ $ \newcommand{\vectorA}[1]{\vec{#1}} % arrow$ $ \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$ $ \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}$ $ \newcommand{\vectorC}[1]{\textbf{#1}}$ $ \newcommand{\vectorD}[1]{\overrightarrow{#1}}$ $ \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$ $ \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} $ $ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}$ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} $ $\newcommand{\avec}{\mathbf a}$ $\newcommand{\bvec}{\mathbf b}$ $\newcommand{\cvec}{\mathbf c}$ $\newcommand{\dvec}{\mathbf d}$ $\newcommand{\dtil}{\widetilde{\mathbf d}}$ $\newcommand{\ evec}{\mathbf e}$ $\newcommand{\fvec}{\mathbf f}$ $\newcommand{\nvec}{\mathbf n}$ $\newcommand{\pvec}{\mathbf p}$ $\newcommand{\qvec}{\mathbf q}$ $\newcommand{\svec}{\mathbf s}$ $\ newcommand{\tvec}{\mathbf t}$ $\newcommand{\uvec}{\mathbf u}$ $\newcommand{\vvec}{\mathbf v}$ $\newcommand{\wvec}{\mathbf w}$ $\newcommand{\xvec}{\mathbf x}$ $\newcommand{\yvec}{\mathbf y} $ $\newcommand{\zvec}{\mathbf z}$ $\newcommand{\rvec}{\mathbf r}$ $\newcommand{\mvec}{\mathbf m}$ $\newcommand{\zerovec}{\mathbf 0}$ $\newcommand{\onevec}{\mathbf 1}$ $\newcommand{\real} {\mathbb R}$ $\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$ $\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$ $\newcommand{\threevec} [3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\fourvec}[4]{\left[\begin{array} {r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$ $\newcommand{\laspan}[1]{\text{Span}\{#1\}}$ $\newcommand{\bcal}{\cal B}$ $\newcommand{\ccal}{\cal C}$ $\newcommand{\scal}{\cal S}$ $\newcommand{\ wcal}{\cal W}$ $\newcommand{\ecal}{\cal E}$ $\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$ $\newcommand{\gray}[1]{\color{gray}{#1}}$ $\newcommand{\lgray}[1]{\color{lightgray}{#1}}$ $\ newcommand{\rank}{\operatorname{rank}}$ $\newcommand{\row}{\text{Row}}$ $\newcommand{\col}{\text{Col}}$ $\renewcommand{\row}{\text{Row}}$ $\newcommand{\nul}{\text{Nul}}$ $\newcommand{\var} {\text{Var}}$ $\newcommand{\corr}{\text{corr}}$ $\newcommand{\len}[1]{\left\|#1\right\|}$ $\newcommand{\bbar}{\overline{\bvec}}$ $\newcommand{\bhat}{\widehat{\bvec}}$ $\newcommand{\bperp}{\ bvec^\perp}$ $\newcommand{\xhat}{\widehat{\xvec}}$ $\newcommand{\vhat}{\widehat{\vvec}}$ $\newcommand{\uhat}{\widehat{\uvec}}$ $\newcommand{\what}{\widehat{\wvec}}$ $\newcommand{\Sighat}{\ widehat{\Sigma}}$ $\newcommand{\lt}{<}$ $\newcommand{\gt}{>}$ $\newcommand{\amp}{&}$ $\definecolor{fillinmathshade}{gray}{0.9}$ The General Form of a Line We have seen that the general form of a linear equation in two variables is ax+by=c (Section 7.4). When this equation is solved for y, the resulting form is called the slope-intercept form. Let's generate this new form. ax + by &=c&\text{Subtract } ax \text{ from both sides. }\\ by &= -ax + c&\text{ Divide both sides by } b\\ \dfrac{by}{b} &= \dfrac{-ax}{b} + \dfrac{c}{b}\\ \dfrac{\not{b}y}{\not{b}}&= \dfrac{-ax}{b} + \dfrac{c}{b}\\ y&=\dfrac{-ax}{b} + \dfrac{c}{b}\\ This equation is of the form $y = mx + b$ if we replace $\dfrac{-a}{b}$ with $m$ and constant $\dfrac{c}{b}$ with $b$. (Note The fact that we let $b = \dfrac{c}{b}$ is unfortunate and occurs because of the letters we have chosen to use in the general form. The letter $b$ occurs on both sides of the equal sign and may not represent the same value at all. This problem is one of the historical convention and, fortunately, does not occur very often.) The following examples illustrate this procedure. Example $\PageIndex{1}$ Solve $3x + 2y = 6$ for $y$. 3x + 2y&= 6& \text{ Subtract } 3x \text{ from both sides }\\ 2y&= -3x + 6& \text{ Divide both sides by } 2\\ y &= -\dfrac{3}{2}x + 3 The equation is of the form $y = mx + b$. In this case, $m = -\dfrac{3}{2}$ and $b = 3$. Example $\PageIndex{2}$ Solve $-15x + 5y = 20$ for $y$. -15x + 5y&=20\\ 5y&=15x + 20\\ This equation is of the form $y = mx + b$. In this case, $m = 3$ and $b = 4$. Example $\PageIndex{3}$ Solve $4x-y = 0$ for $y$. This equation is of the form $y = mx + b$. In this case, $m=4$ and $b=0$. Notice that we can write $y=4x$ as $y = 4x + 0$ The Slope-Intercept Form of a Line The Slope-Intercept Form of a Line $y=mx+b$ A linear equation in two variables written in the form $y=mx+b$ is said to be in slope-intercept form. Sample Set A The following equations are in slope-intercept form: Example $\PageIndex{4}$ $y = 6x - 7$. In this case $m = 6$ and $b = -7$ Example $\PageIndex{5}$ $y = -2x + 9$. In this case $m = -2$ and $b = 9$ Example $\PageIndex{6}$ $y = \dfrac{1}{5}x + 4.8$. In this case $m = \dfrac{1}{5}$ and $b = 4.8$ Example $\PageIndex{7}$ $y = 7x$. In this case $m = 7$ and $b = 0$ since we can write $y = 7x$ as $y = 7x + 0$. The following equations are not in slope-intercept form. Example $\PageIndex{8}$ $2y = 4x - 1$. The coefficient of $y$ is $2$. To be in slope-intercept form, the coefficient of $y$ must be $1$. Example $\PageIndex{9}$ $y + 4x = 5$. The equation is not solved for $y$. THe $x$ and $y$ appear on the same side of the equal sign. Example $\PageIndex{10}$ $y + 1 = 2x$. The equation is not solved for $y$ Practice Set A The following equation are in slope-intercept form. In each case, specify the slope and $y$-intercept. Practice Problem $\PageIndex{1}$ $y=2x+7; m=?, b=?$ $m=2, b=7$ Practice Problem $\PageIndex{2}$ $y=−4x+2; m=?, b=?$ Practice Problem $\PageIndex{3}$ $y=−5x−1; m=?, b=?$ Practice Problem $\PageIndex{4}$ $y=\dfrac{2}{3}x−10; m=?, b=?$ $m = \dfrac{2}{3}, b = -10$ Practice Problem $\PageIndex{5}$ $y = \dfrac{-5}{8}x + \dfrac{1}{2}; m=?, b=?$ $m = \dfrac{-5}{8}, b=\dfrac{1}{2}$ Practice Problem $\PageIndex{6}$ $y=−3x; m=?, b=?$ $m=−3, b=0$ Slope and Intercept When the equation of a line is written in slope-intercept form, two important properties of the line can be seen: the slope and the intercept. Let's look at these two properties by graphing several lines and observing them carefully. Sample Set B Example $\PageIndex{11}$ Graph the line $y=x−3$. $x$ $y$ $(x,y)$ $0$ $−3$ $(0,−3)$ $4$ $1$ $(4,1)$ $−2$ $−5$ $(−2,−5)$ Looking carefully at this line, answer the following two questions. At what number does this line cross the y-axis? Do you see this number in the equation? The line crosses the y-axis at −3. Place your pencil at any point on the line. Move your pencil exactly one unit horizontally to the right. Now, how many units straight up or down must you move your pencil to get back on the line? Do you see this number in the equation? After moving horizontally one unit to the right, we must move exactly one vertical unit up. This number is the coefficient of $x$. Example $\PageIndex{12}$ Graph the line $y = \dfrac{2}{3}x + 1$ $x$ $y$ $(x,y)$ $0$ $1$ $(0,1)$ $3$ $3$ $(3,3)$ $−3$ $−1$ $(−3,−1)$ Looking carefully at this line, answer the following two questions. At what number does this line cross the $y$-axis? Do you see this number in the equation? The line crosses the $y$-axis at $+1$. Place your pencil at any point on the line. Move your pencil exactly one unit horizontally to the right. Now, how many units straight up or down must you move your pencil to get back on the line? Do you see this number in the equation? After moving horizontally one unit to the right, we must move exactly $\dfrac{2}{3}$ unit upward. This number is the coefficient of $x$. Practice Set B Practice Problem $\PageIndex{7}$ Graph the line $y = -3x + 4$ $x$ $y$ $(x,y)$ $0$ - - $3$ - - $2$ - - Looking carefully at this line, answer the following two questions. At what number does the line cross the $y$-axis? Do you see this number in the equation? The line crosses the $y$-axis at $+4$. After moving horizontally $1$ unit to the right, we must move exactly $3$ units downward. Place your pencil at any point on the line. Move your pencil exactly one unit horizontally to the right. Now, how many units straight up or down must you move your pencil to get back on the line? Do you see this number in the equation? In the graphs constructed in Sample Set B and Practice Set B, each equation had the form $y=mx+b$. We can answer the same questions by using this form of the equation (shown in the diagram). At what number does the line cross the $y$-axis? Do you see this number in the equation? In each case, the line crosses the $y$-axis at the constant $b$. The number $b$ is the number at which the line crosses the $y$-axis, and it is called the $y$-intercept. The ordered pair corresponding to the $y$-intercept is $(0,b)$. Place your pencil at any point on the line. Move your pencil exactly one unit horizontally to the right. Now, how many units straight up or down must you move your pencil to get back on the line? Do you see this number in the equation? To get back on the line, we must move our pencil exactly $m$ vertical units. The number $m$ is the coefficient of the variable $x$. The number $m$ is called the slope of the line and it is the number of units that $y$ changes when $x$ is increased by $1$ unit. Thus, if $x$ changes by $1$ unit, $y$ changes by $m$ units. Since the equation $y=mx+b$ contains both the slope of the line and the $y$-intercept, we call the form $y=mx+b$ the slope-intercept form. The Slope-Intercept Form of the Equation of a Line The slope-intercept form of a straight line is $y=mx+b$ The slope of the line is $m$, and the $y$-intercept is the point $(0,b)$. The Slope is a Measure of the Steepness of a Line The word slope is really quite appropriate. It gives us a measure of the steepness of the line. Consider two lines, one with slope $\dfrac{1}{2}$ and the other with slope $3$. The line with slope $3$ is steeper than is the line with slope $\dfrac{1}{2}$. Imagine your pencil being placed at any point on the lines. We make a $1$-unit increase in the $x$-value by moving the pencil one unit to the right. To get back to one line we need only move vertically $\dfrac{1}{2}$ unit, whereas to get back onto the other line we need to move vertically $3$ units. Sample Set C Find the slope and the y-intercept of the following lines. Example $\PageIndex{13}$ $y = 2x + 7$ The line is in the slope-intercept form $y=mx+b$. The slope is $m$, the coefficient of $x$. Therefore, $m=2$. The $y$-intercept is the point $(0,b)$. Since $b=7$, the $y$-intercept is $(0,7)$. $y$-intercept: $(0,7)$ Example $\PageIndex{14}$ $y = -4x + 1$ The line is in the slope-intercept form $y=mx+b$. The slope is $m$, the coefficient of $x$. Therefore, $m=-4$. The $y$-intercept is the point $(0,b)$. Since $b=1$, the $y$-intercept is $(0,1)$. $y$-intercept: $(0,1)$ Example $\PageIndex{15}$ $3x + 2y = 5$. This equation is written in general form. We can put the equation in slope-intercept form by solving for $y$. y&=-\dfrac{3}{2}x + \dfrac{5}{2} Now the equation is in slope-intercept form. $y$-intercept: $(0,\dfrac{5}{2})$ Practice Set C Practice Problem $\PageIndex{8}$ Find the slope and $y$-intercept of the line $2x+5y=15$. Solving for $y$ we get $y = \dfrac{-2}{5}x + 3$. Now $m = \dfrac{-2}{5}$ and $b = 3$. The Formula for the Slope of a Line We have observed that the slope is a measure of the steepness of a line. We wish to develop a formula for measuring this steepness. It seems reasonable to develop a slope formula that produces the following results: Steepness of line $1>$ steepness of line 2. Consider a line on which we select any two points. We’ll denote these points with the ordered pairs $(x_1,y_1)$ and $(x_2,y_2)$. The subscripts help us to identify the points. $(x_1, y_1)$ is the first point. Subscript $1$ indicates the first point. $(x_2, y_2)$ is the second point. Subscript $2$ indicates the second point. The difference in $x$ values $(x_2−x_1)$ gives us the horizontal change, and the difference in $y$ values $(y_2−y_1)$ gives us the vertical change. If the line is very steep, then when going from the first point to the second point, we would expect a large vertical change compared to the horizontal change. If the line is not very steep, then when going from the first point to the second point, we would expect a small vertical change compared to the horizontal change. We are comparing changes. We see that we are comparing: \text{The vertical change}&\text{ to }&\text{ the horizontal change}\\ \text{The change in }y&\text{ to }&\text{ the change in }x\\ y_1-y_1&\text{ to }&x_2-x_1 This is a comparison and is therefore a ratio. Ratios can be expressed as fractions. Thus, a measure of the steepness of a line can be expressed as a ratio. The slope of a line is defined as the ratio $\text{Slope } = \dfrac{\text{ change in } y}{\text{ change in } x}$ Mathematically, we can write these changes as $\text{Slope } = \dfrac{y_2-y_1}{x_2-x_1}$ Finding the Slope of a Line The slope of a nonvertical line passing through the points $(x_1, y_1)$ and $(x_2, y_2)$ is found by the formula: $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ Sample Set D For the two given points, find the slope of the line that passes through them. Example $\PageIndex{16}$ $(0,1)$ and $(1,3)$. Looking left to right on the line we can choose $(x_1, y_1)$ to be $(0,1)$, and $(x_2, y_2)$ to be $(1, 3)$. Then, $m = \dfrac{y_2-y_1}{x_2-x_1} = \dfrac{3-1}{1-0} = \dfrac{2}{1} = 2$ This line has slope $2$. It appears fairly steep. When the slope is written in fraction form, $2 = \dfrac{2}{1}$, we can see, by recalling the slope formula, that as $x$ changes $1$ unit to the right (because of the $+1$) $y$ changes $2$ units upward (because of the $+2$. $m = \dfrac{\text{ change in } y}{\text{ change in } x} = \dfrac{2}{1}$. Notice that as we look left to right, the line rises. Example $\PageIndex{17}$ $(2,2)$ and $(4,3)$. Looking left to right on the line we can choose $(x_1, y_1)$ to be $(2,2)$, and $(x_2, y_2)$ to be $(4, 3)$. Then, $m = \dfrac{y_2-y_1}{x_2-x_1} = \dfrac{3-2}{4-2} = \dfrac{1}{2}$ This line has slope $\dfrac{1}{2}$. Thus, as $x$ changes $2$ units to the right (because of the $+2$, $y$ changes $1$ unit upward (because of the $+1$). $m = \dfrac{\text{ change in } y}{\text{ change in } x} = \dfrac{1}{2}$. Notice that in examples 16 and 17, both lines have positive slopes, $+2$ and $+\dfrac{1}{2}$, and both lines rise as we look left to right. Example $\PageIndex{18}$ $(-2, 4)$ and $(1,1)$. Looking left to right on the line we can choose $(x_1, y_1)$ to be $(-2,4)$, and $(x_2, y_2)$ to be $(1,1)$. Then, $m = \dfrac{y_2-y_1}{x_2-x_1} = \dfrac{1-4}{1-(-2)} = \dfrac{-3}{1 + 2} = \dfrac{-3}{3} = -1$ This line has slope $-1$. When the slope is written in fraction form, $m = -1 = \dfrac{-1}{+1}$, we can see that as $x$ changes $1$ unit to the right (because of the $+1$, $y$ changes $1$ unit downward (because of the $-1$). Notice also that this line has a negative slope and declines as we look left to right. Example $\PageIndex{19}$ $(1, 3)$ and $(5, 3)$. $m = \dfrac{y_2-y_1}{x_2-x_1} = \dfrac{3-3}{5-1} = \dfrac{0}{4} = 0$ This line has $0$ slope. This means it has no rise and, therefore, is a horizontal line. This does not mean that the line has no slope, however. Example $\PageIndex{20}$ $(4,4)$ and $(4,0)$. $m = \dfrac{y_2-y_1}{x_2-x_1} = \dfrac{0-4}{4-4} = \dfrac{-4}{0}$ Since division by $0$ is undefined, we say that vertical lines have undefined slope. Since there is no real number to represent the slope of this line, we sometimes say that vertical lines have undefined slope, or no slope. Practice Set D Practice Problem $\PageIndex{9}$ Find the slope of the line passing through $(2,1)$ and $(6,3)$. Graph this line on the graph of problem 2 below. $m = \dfrac{3-1}{6-2} = \dfrac{2}{4} = \dfrac{1}{2}$ Practice Problem $\PageIndex{10}$ Find the slope of the line passing through $(3,4)$ and $(5,5)$. Graph this line. The line has slope $\dfrac{1}{2}$ Practice Problem $\PageIndex{11}$ Compare the lines of the following problems. Do the lines appear to cross? What is it called when lines do not meet (parallel or intersecting)? Compare their slopes. Make a statement about the condition of these lines and their slopes. The lines appear to be parallel. Parallel lines have the same slope, and lines that have the same slope are parallel Before trying some problems, let’s summarize what we have observed. • The equation $y=mx+b$ is called the slope-intercept form of the equation of a line. The number $m$ is the slope of the line and the point $(0,b)$ is the $y$-intercept. • The slope, $m$, of a line is defined as the steepness of the line, and it is the number of units that $y$ changes when $x$ changes $1$ unit. • The formula for finding the slope of a line through any two given points $(x_1,y_1)$ and $(x_2,y_2)$ is: □ $m = \dfrac{y_2-y_1}{x_2-x_1}$ • The fraction $\dfrac{y_2-y_1}{x_2-x_1}$ represents the \dfrac{\text{ change in } y}{\text{ change in } x} • As we look at a graph from left to right, lines with positive slope rise and lines with negative slope decline. • Parallel lines have the same slope. • Horizontal lines have 0 slope. • Vertical lines have undefined slope (or no slope). For the following problems, determine the slope and y-intercept of the lines. Exercise $\PageIndex{1}$ slope = $3$; $y$-intercept= $(0,4)$ Exercise $\PageIndex{2}$ Exercise $\PageIndex{3}$ slope = $9$; $y$-intercept= $(0,1)$ Exercise $\PageIndex{4}$ Exercise $\PageIndex{5}$ slope = $-4$; $y$-intercept= $(0,5)$ Exercise $\PageIndex{6}$ Exercise $\PageIndex{7}$ slope = $-6$; $y$-intercept= $(0,-1)$ Exercise $\PageIndex{8}$ Exercise $\PageIndex{9}$ slope = $-1$; $y$-intercept= $(0,2)$ Exercise $\PageIndex{10}$ Exercise $\PageIndex{11}$ slope = $4$; $y$-intercept= $(0,5)$ Exercise $\PageIndex{12}$ Exercise $\PageIndex{13}$ slope = $-4$; $y$-intercept= $(0,9)$ Exercise $\PageIndex{14}$ $y = \dfrac{3}{5} - 8$ Exercise $\PageIndex{15}$ $y = \dfrac{2}{7} - 12$ slope = $\dfrac{2}{7}$; $y$-intercept= $(0,-12)$ Exercise $\PageIndex{16}$ $y = \dfrac{-1}{8}x + \dfrac{2}{3}$ Exercise $\PageIndex{17}$ $y = \dfrac{-4}{5} - \dfrac{4}{7}$ slope = $-\dfrac{4}{5}$; $y$-intercept= $(0,-\dfrac{4}{7})$ Exercise $\PageIndex{18}$ Exercise $\PageIndex{19}$ slope = $\dfrac{6}{5}$; $y$-intercept= $(0,-\dfrac{1}{10})$ Exercise $\PageIndex{20}$ Exercise $\PageIndex{21}$ slope = $1$; $y$-intercept= $(0,-3)$ Exercise $\PageIndex{22}$ Exercise $\PageIndex{23}$ slope = $-\dfrac{5}{3}$; $y$-intercept= $(0,2)$ Exercise $\PageIndex{24}$ Exercise $\PageIndex{25}$ slope = $\dfrac{1}{4}$; $y$-intercept= $(0,-\dfrac{1}{4})$ For the following problems, find the slope of the line through the pairs of points. Exercise $\PageIndex{26}$ $(1, 6), (4,9)$ Exercise $\PageIndex{27}$ $(1, 3), (4,7)$ $m = \dfrac{4}{3}$ Exercise $\PageIndex{28}$ $(3, 5), (4,7)$ Exercise $\PageIndex{29}$ $(6, 1), (2,8)$ $m = -\dfrac{7}{4}$ Exercise $\PageIndex{30}$ $(0, 5), (2,−6)$ Exercise $\PageIndex{31}$ $(−2, 1), (0,5)$ Exercise $\PageIndex{32}$ $(3, −9), (5,1)$ Exercise $\PageIndex{33}$ $(4, −6), (−2,1)$ $m = -\dfrac{7}{6}$ Exercise $\PageIndex{34}$ $(−5, 4), (−1,0)$ Exercise $\PageIndex{35}$ $(−3, 2), (−4,6)$ Exercise $\PageIndex{36}$ $(9, 12), (6,0)$ Exercise $\PageIndex{37}$ $(0, 0), (6,6)$ Exercise $\PageIndex{38}$ Exercise $\PageIndex{39}$ $(−1, −7), (−2,−9)$ Exercise $\PageIndex{40}$ $(−6, −6), (−5,−4)$ Exercise $\PageIndex{41}$ $(−1, 0), (−2,−2)$ Exercise $\PageIndex{42}$ $(−4, −2), (0,0)$ Exercise $\PageIndex{43}$ $(2, 3), (10,3)$ $m=0$ (horizontal line $y=3$) Exercise $\PageIndex{44}$ Make a statement about the slopes of parallel lines. Exercise $\PageIndex{45}$ $(4, −2), (4,7)$ Exercise $\PageIndex{46}$ $(8, −1), (8,3)$ Exercise $\PageIndex{47}$ $(4, 2), (6,2)$ Exercise $\PageIndex{48}$ $(5, −6), (9,−6)$ $m=0$ (horizontal line at $y=−6$) Exercise $\PageIndex{49}$ Do lines with a positive slope rise or decline as we look left to right? Exercise $\PageIndex{50}$ Do lines with a negative slope rise or decline as we look left to right? For the following problems, determine the slope and y-intercept of the lines. Round to two decimal places. Exercise $\PageIndex{51}$ Exercise $\PageIndex{52}$ Exercise $\PageIndex{53}$ Exercise $\PageIndex{54}$ For the following problems, find the slope of the line through the pairs of points. Round to two decimal places. Exercise $\PageIndex{55}$ $(5.56, 9.37), (2.16, 4.90)$ Exercise $\PageIndex{56}$ $(33.1, 8.9), (42.7, −1.06)$ Exercise $\PageIndex{57}$ $(155.89, 227.61), (157.04,227.61)$ Exercise $\PageIndex{58}$ $(0.00426, −0.00404), (−0.00191, −0.00404)$ Exercise $\PageIndex{59}$ $(88.81, −23.19), (88.81, −26.87)$ Exercise $\PageIndex{60}$ $(−0.0000567, −0.0000567), (−0.00765, 0.00764)$ Exercises for Review Exercise $\PageIndex{61}$ Simplify $(x^2y^3w^4)^0$. Exercise $\PageIndex{62}$ Solve the equation $3x−4(2−x)−3(x−2)+4=0$. Exercise $\PageIndex{63}$ When four times a number is divided by five, and that result is decreased by eight, the result is zero. What is the original number? Exercise $\PageIndex{64}$ Solve $−3y+10=x+2$ if $x=−4$. Exercise $\PageIndex{65}$ Graph the linear equation $x+y=3$.	{"url":"https://oreoti.best/article/7-5-the-slope-intercept-form-of-a-line","timestamp":"2024-11-13T05:45:26Z","content_type":"text/html","content_length":"130243","record_id":"<urn:uuid:2a4052bf-870e-429a-ae75-5cf3abaefbd9>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00704.warc.gz"}
Overview of the GlobalOptimization Package Calling Sequence List of Package Commands Description Notes Accessing Package Commands Examples See Also Calling Sequence Copyright and Trademark GlobalOptimization[command](arguments) Information command(arguments) examples/ • The Global Optimization Toolbox, powered by Optimus® technology from Noesis GlobalOptimization/InputForms Solutions, is implemented as the GlobalOptimization package, which numerically computes global solutions to nonlinear programming (NLP) problems over a bounded GlobalOptimization/Options region. An NLP problem is the minimization or maximization of an objective function, possibly subject to constraints. infolevel • The GlobalOptimization package contains a command for solving optimization problems, which can be specified in various forms. For information on the input forms, see the GlobalOptimization/InputForms help page. Additionally, the package offers an interactive Maplet interface that provides an easy-to-use facility for entering and solving an optimization problem, as well as plotting both the problem and its solution. • Using the many options, you can control the algorithms used by the global solver. For information on these options, see the GlobalOptimization/Options help page. • The global solver in the GlobalOptimization package calls external code that uses hardware floats, but the objective function and the constraints can be evaluated in Maple with higher precision. For more information on the global solver, the algorithms, the floating-point computation environment, and ways to achieve best performance with the solver, see the GlobalOptimization/Computation help page. • For more information and examples on the toolbox, see Introduction to the Global Optimization Toolbox and Applications of the Global Optimization Toolbox. Accessing Package Commands • Each command in the GlobalOptimization package can be accessed by using either the long form or the short form of the command name in the command calling sequence. Because the underlying implementation of the package is a module, it is possible to use the form GlobalOptimization:-command to access a command from the package. For more information, see Module Members. List of Package Commands • The following is a list of available commands. GetLastSolution GlobalSolve Interactive To display the help page for a particular command, see Getting Help with a Command in a Package. • The GlobalSolve help page describes the most commonly used forms of input. Use with the more advanced Matrix form of input is described in the GlobalOptimization [GlobalSolveMatrixForm] help page. • To see additional information on the progress of the solver during the solution of an optimization problem, set infolevel[GlobalOptimization] to a positive integer. More information is printed at higher infolevel settings. > $\mathrm{with}⁡\left(\mathrm{GlobalOptimization}\right)&colon;$ Find the global solution to the minimization problem $\mathrm{ln}⁡\ left(x\right)⁢\mathrm{sin}⁡\left(x\right)$ in the range > $\mathrm{GlobalSolve}⁡\left(\mathrm{ln}⁡\left(x\right) $[{-2.85006479973796}{,}[{x}{=}{17.2990352355127}]& (1) Find the global solution to a constrained minimization problem. > $\mathrm{GlobalSolve}⁡\left({x}^{6}-5⁢{x}^{3}-20& InvisibleTimes;x+18\le 0},x=-3&period;&period;3\right)$ $[{-23.0747312455205424}{,}[{x}{=}{-1.36754446796582}& (2) Calling Sequence List of Package Commands Description Notes Accessing Package Commands Examples See Also Calling Sequence Copyright and Trademark GlobalOptimization[command](arguments) Information command(arguments) examples/ • The Global Optimization Toolbox, powered by Optimus® technology from Noesis GlobalOptimization/InputForms Solutions, is implemented as the GlobalOptimization package, which numerically computes global solutions to nonlinear programming (NLP) problems over a bounded GlobalOptimization/Options region. An NLP problem is the minimization or maximization of an objective function, possibly subject to constraints. infolevel • The GlobalOptimization package contains a command for solving optimization problems, which can be specified in various forms. For information on the input forms, see the GlobalOptimization/InputForms help page. Additionally, the package offers an interactive Maplet interface that provides an easy-to-use facility for entering and solving an optimization problem, as well as plotting both the problem and its solution. • Using the many options, you can control the algorithms used by the global solver. For information on these options, see the GlobalOptimization/Options help page. • The global solver in the GlobalOptimization package calls external code that uses hardware floats, but the objective function and the constraints can be evaluated in Maple with higher precision. For more information on the global solver, the algorithms, the floating-point computation environment, and ways to achieve best performance with the solver, see the GlobalOptimization/Computation help page. • For more information and examples on the toolbox, see Introduction to the Global Optimization Toolbox and Applications of the Global Optimization Toolbox. Accessing Package Commands • Each command in the GlobalOptimization package can be accessed by using either the long form or the short form of the command name in the command calling sequence. Because the underlying implementation of the package is a module, it is possible to use the form GlobalOptimization:-command to access a command from the package. For more information, see Module Members. List of Package Commands • The following is a list of available commands. GetLastSolution GlobalSolve Interactive To display the help page for a particular command, see Getting Help with a Command in a Package. • The GlobalSolve help page describes the most commonly used forms of input. Use with the more advanced Matrix form of input is described in the GlobalOptimization [GlobalSolveMatrixForm] help page. • To see additional information on the progress of the solver during the solution of an optimization problem, set infolevel[GlobalOptimization] to a positive integer. More information is printed at higher infolevel settings. > $\mathrm{with}⁡\left(\mathrm{GlobalOptimization}\right)&colon;$ Find the global solution to the minimization problem $\mathrm{ln}⁡\ left(x\right)⁢\mathrm{sin}⁡\left(x\right)$ in the range > $\mathrm{GlobalSolve}⁡\left(\mathrm{ln}⁡\left(x\right) $[{-2.85006479973796}{,}[{x}{=}{17.2990352355127}]& (1) Find the global solution to a constrained minimization problem. > $\mathrm{GlobalSolve}⁡\left({x}^{6}-5⁢{x}^{3}-20& InvisibleTimes;x+18\le 0},x=-3&period;&period;3\right)$ $[{-23.0747312455205424}{,}[{x}{=}{-1.36754446796582}& (2) • The Global Optimization Toolbox, powered by Optimus® technology from Noesis Solutions, is implemented as the GlobalOptimization package, which numerically computes global solutions to nonlinear programming (NLP) problems over a bounded region. An NLP problem is the minimization or maximization of an objective function, possibly subject to constraints. • The GlobalOptimization package contains a command for solving optimization problems, which can be specified in various forms. For information on the input forms, see the GlobalOptimization/ InputForms help page. Additionally, the package offers an interactive Maplet interface that provides an easy-to-use facility for entering and solving an optimization problem, as well as plotting both the problem and its solution. • Using the many options, you can control the algorithms used by the global solver. For information on these options, see the GlobalOptimization/Options help page. • The global solver in the GlobalOptimization package calls external code that uses hardware floats, but the objective function and the constraints can be evaluated in Maple with higher precision. For more information on the global solver, the algorithms, the floating-point computation environment, and ways to achieve best performance with the solver, see the GlobalOptimization/Computation help page. • For more information and examples on the toolbox, see Introduction to the Global Optimization Toolbox and Applications of the Global Optimization Toolbox. • The Global Optimization Toolbox, powered by Optimus® technology from Noesis Solutions, is implemented as the GlobalOptimization package, which numerically computes global solutions to nonlinear programming (NLP) problems over a bounded region. An NLP problem is the minimization or maximization of an objective function, possibly subject to constraints. The Global Optimization Toolbox, powered by Optimus® technology from Noesis Solutions, is implemented as the GlobalOptimization package, which numerically computes global solutions to nonlinear programming (NLP) problems over a bounded region. An NLP problem is the minimization or maximization of an objective function, possibly subject to constraints. • The GlobalOptimization package contains a command for solving optimization problems, which can be specified in various forms. For information on the input forms, see the GlobalOptimization/ InputForms help page. Additionally, the package offers an interactive Maplet interface that provides an easy-to-use facility for entering and solving an optimization problem, as well as plotting both the problem and its solution. The GlobalOptimization package contains a command for solving optimization problems, which can be specified in various forms. For information on the input forms, see the GlobalOptimization/InputForms help page. Additionally, the package offers an interactive Maplet interface that provides an easy-to-use facility for entering and solving an optimization problem, as well as plotting both the problem and its solution. • Using the many options, you can control the algorithms used by the global solver. For information on these options, see the GlobalOptimization/Options help page. Using the many options, you can control the algorithms used by the global solver. For information on these options, see the GlobalOptimization/Options help page. • The global solver in the GlobalOptimization package calls external code that uses hardware floats, but the objective function and the constraints can be evaluated in Maple with higher precision. For more information on the global solver, the algorithms, the floating-point computation environment, and ways to achieve best performance with the solver, see the GlobalOptimization/Computation help page. The global solver in the GlobalOptimization package calls external code that uses hardware floats, but the objective function and the constraints can be evaluated in Maple with higher precision. For more information on the global solver, the algorithms, the floating-point computation environment, and ways to achieve best performance with the solver, see the GlobalOptimization/Computation help • For more information and examples on the toolbox, see Introduction to the Global Optimization Toolbox and Applications of the Global Optimization Toolbox. For more information and examples on the toolbox, see Introduction to the Global Optimization Toolbox and Applications of the Global Optimization Toolbox. Accessing Package Commands • Each command in the GlobalOptimization package can be accessed by using either the long form or the short form of the command name in the command calling sequence. Because the underlying implementation of the package is a module, it is possible to use the form GlobalOptimization:-command to access a command from the package. For more information, see Module • Each command in the GlobalOptimization package can be accessed by using either the long form or the short form of the command name in the command calling sequence. Each command in the GlobalOptimization package can be accessed by using either the long form or the short form of the command name in the command calling sequence. Because the underlying implementation of the package is a module, it is possible to use the form GlobalOptimization:-command to access a command from the package. For more information, see Module List of Package Commands • The following is a list of available commands. GetLastSolution GlobalSolve Interactive To display the help page for a particular command, see Getting Help with a Command in a Package. • The GlobalSolve help page describes the most commonly used forms of input. Use with the more advanced Matrix form of input is described in the GlobalOptimization[GlobalSolveMatrixForm] help page. To display the help page for a particular command, see Getting Help with a Command in a Package. • The GlobalSolve help page describes the most commonly used forms of input. Use with the more advanced Matrix form of input is described in the GlobalOptimization[GlobalSolveMatrixForm] help page. The GlobalSolve help page describes the most commonly used forms of input. Use with the more advanced Matrix form of input is described in the GlobalOptimization[GlobalSolveMatrixForm] help page. • To see additional information on the progress of the solver during the solution of an optimization problem, set infolevel[GlobalOptimization] to a positive integer. More information is printed at higher infolevel settings. • To see additional information on the progress of the solver during the solution of an optimization problem, set infolevel[GlobalOptimization] to a positive integer. More information is printed at higher infolevel settings. To see additional information on the progress of the solver during the solution of an optimization problem, set infolevel[GlobalOptimization] to a positive integer. More information is printed at higher infolevel settings. > $\mathrm{with}⁡\left(\mathrm{GlobalOptimization}\right)&colon;$ Find the global solution to the minimization problem $\mathrm{ln}⁡\left(x\right)⁢\mathrm{sin}⁡\left(x\right)$ in the range $[1&period;&period;20] > $\mathrm{GlobalSolve}⁡\left(\mathrm{ln}⁡\left(x\right)⁢\mathrm{sin}⁡\left(x\right),x=1&period;&period;20\right)$ $[{-2.85006479973796}{,}[{x}{=}{17.2990352355127}]]$ (1) Find the global solution to a constrained minimization problem. > $\mathrm{GlobalSolve}⁡\left({x}^{6}-5⁢{x}^{3}-20⁢{x}^{2}-5,{5⁢{x}^{2}+20⁢x+18\le 0},x& $[{-23.0747312455205424}{,}[{x}{=}{-1.36754446796582}]]$ (2) Find the global solution to the minimization problem $\mathrm{ln}⁡\left(x\right)⁢\mathrm{sin}⁡\left(x\right)$ in the range $[1&period;&period;20] See Also Copyright and Trademark	{"url":"https://www.maplesoft.com/support/help/view.aspx?path=GlobalOptimization&L=E","timestamp":"2024-11-07T13:11:47Z","content_type":"text/html","content_length":"131339","record_id":"<urn:uuid:337c17f3-40df-497f-92ac-47ca6a2f142f>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00600.warc.gz"}
rail for Amsler integrator no. 4 Inscribedengraved on the track: No. 1124 FunctionThe planimeter is used to determine the area of an arbitrary, two-dimensional figure: it gives the area between the graphically defined curve and a horizontal x-axis. Users manually trace the figure in question with the attached tracing arm. The tracing motion is communicated through the device such that the large, solid brass cylinder will slide along the track. The distance that the cylinder has moved when the tracing is complete (i.e. when the tracer has returned to the point from which she started), is proportional to the area inside the figure. This motion transfer is guaranteed by the trigonometric relations of the main mechanism consisting of the large circle, and the three connected smaller circle. The white plastic discs attached to the tracing arm are used to record the area. This instrument, unlike some of its close relatives, does not have a drawing mechanism. Instruments of this kind were later referred to as 'moment planimeters' because the integrals of f(x)^2 and f(x)^3 give respectively the static moment and the moment of inertia for the figure's area. For supplementary images of a similar model and mathematical explanation of the Amsler Planimeter, click here. For a detailed description of how the rolling sphere planimeter measures area, see the article in the Primary Sources. Primary SourcesJ.C. Maxwell, "Descriptions of a New Form of Planometer, an Instrument for Measuring the Areas of Plane Figures Drawn on Paper," in The Scientific Papers of James Clerk Maxwell, Vol. 1 , ed. W. D. Niven (Mineola, NY: Dover Publications, 2003), 230-237. [Reprint of original 1890 publication by Cambridge University Press.]	{"url":"http://waywiser.fas.harvard.edu/objects/419/rail-for-amsler-integrator-no-4;jsessionid=DE708530847E71DACA4D29DC76572EDE","timestamp":"2024-11-04T14:47:35Z","content_type":"application/xhtml+xml","content_length":"53116","record_id":"<urn:uuid:f7b9996e-de13-4845-b1fc-8582e6d5a510>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00756.warc.gz"}
MRB Constant -- from Wolfram MathWorld Consider the sequence of partial sums defined by As can be seen in the plot above, the sequence has two limit points at Sums for the MRB constant are given by (Finch 2003, p. 450; OEIS A037077). The constant can also be given as a sum over derivatives of the Dirichlet eta function An integral expression for the constant is given by (M. Burns, pers. comm., Jan. 21, 2020). No closed-form expression is known for this constant (Finch 2003, p. 450).	{"url":"https://mathworld.wolfram.com/MRBConstant.html","timestamp":"2024-11-07T00:51:46Z","content_type":"text/html","content_length":"61632","record_id":"<urn:uuid:8e6de7e2-0b35-4749-868b-b53370e6a640>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00288.warc.gz"}
Admission requirements This course treats the principles of calculus for functions of one variable. You will become familiar with the fundamental concepts of the theory and get access to applications in other areas of Mathematics and beyond. The following topics will be covered: limits; continuity and differentiability; curvature of a graph; linear approximation of functions; Taylor polynomials for functions of one variable; Landau's big-O notation; integration of functions of one variable; fundamental theorem of calculus; substitution rule; integration by parts; partial fractions; improper integrals; finding anti-derivatives; real series; convergence criteria for real series; real power series; differentiation and integration of power series; Taylor series. Course objectives The student knows the theory of the topics mentioned and can apply the theory judiciously. The student can skillfully use the theory to solve concrete problems. Mode of instruction During the semester there will two lectures in almost every week. Some lectures may be replaced by exercise classes. There will be weekly homework and a midterm exam. Assessment method The final grade consists of homework (20%), a written midterm exam (24%), and a written (retake) exam (56%). To pass the course, the grade for the (retake) exam should be at least 5 and the (unrounded) weighted average of the three partial grades at least 5.5. No minimum grade is required for the homework and midterm exam in order to take the exam or to pass the course. There will be no retakes for the homework and the midterm exam. The homework counts as a practical and it is expected to consists of 10 assignments, of which the lowest grade is dropped. The midterm exam counts as a constituent examination and its grade will be replaced by the grade of the (retake) exam if the latter is higher. Calculus - A Complete Course (8th or 9th or later edition) by Robert A. Adams en Christopher Essex, Pearson Education, 2013, ISBN 978-0-321-78107-9 There also exist combinations of the book with software packages. These are considerably more expensive and NOT needed for the course. Brightspace is essentially used, for instance to publish the homework. Onno van Gaans, Mathematical Institute, Snellius room 222, Email: vangaans[at]math.leidenuniv.nl Tel.: 071 527 7122.	{"url":"https://studiegids.universiteitleiden.nl/en/courses/108988/analyse-1","timestamp":"2024-11-10T02:12:51Z","content_type":"text/html","content_length":"18994","record_id":"<urn:uuid:4eb738c9-bc2a-44a4-b787-68d77629aa39>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00051.warc.gz"}
The Stacks project Let $p : X \to S$ be a morphism of schemes. We define the de Rham cohomology of $X$ over $S$ to be the cohomology groups \[ H^ i_{dR}(X/S) = H^ i(R\Gamma (X, \Omega ^\bullet _{X/S})) \] Since $\Omega ^\bullet _{X/S}$ is a complex of $p^{-1}\mathcal{O}_ S$-modules, these cohomology groups are naturally modules over $H^0(S, \mathcal{O}_ S)$. \[ \xymatrix{ X' \ar[r]_ f \ar[d] & X \ar[d] \\ S' \ar[r] & S } \] of schemes, using the canonical maps of Section 50.2 we obtain pullback maps \[ f^* : R\Gamma (X, \Omega ^\bullet _{X/S}) \longrightarrow R\Gamma (X', \Omega ^\bullet _{X'/S'}) \] These pullbacks satisfy an obvious composition law. In particular, if we work over a fixed base scheme $S$, then de Rham cohomology is a contravariant functor on the category of schemes over $S$. Lemma 50.3.1. Let $X \to S$ be a morphism of affine schemes given by the ring map $R \to A$. Then $R\Gamma (X, \Omega ^\bullet _{X/S}) = \Omega ^\bullet _{A/R}$ in $D(R)$ and $H^ i_{dR}(X/S) = H^ i(\ Omega ^\bullet _{A/R})$. Proof. This follows from Cohomology of Schemes, Lemma 30.2.2 and Leray's acyclicity lemma (Derived Categories, Lemma 13.16.7). $\square$ Lemma 50.3.2. Let $p : X \to S$ be a morphism of schemes. If $p$ is quasi-compact and quasi-separated, then $Rp_\Omega ^\bullet _{X/S}$ is an object of $D_\mathit{QCoh}(\mathcal{O}_ S)$. Proof. There is a spectral sequence with first page $E_1^{a, b} = R^ bp_\Omega ^ a_{X/S}$ converging to the cohomology of $Rp_\Omega ^\bullet _{X/S}$ (see Derived Categories, Lemma 13.21.3). Hence by Homology, Lemma 12.25.3 it suffices to show that $R^ bp_\Omega ^ a_{X/S}$ is quasi-coherent. This follows from Cohomology of Schemes, Lemma 30.4.5. $\square$ Lemma 50.3.3. Let $p : X \to S$ be a proper morphism of schemes with $S$ locally Noetherian. Then $Rp_\Omega ^\bullet _{X/S}$ is an object of $D_{\textit{Coh}}(\mathcal{O}_ S)$. Proof. In this case by Morphisms, Lemma 29.32.12 the modules $\Omega ^ i_{X/S}$ are coherent. Hence we can use exactly the same argument as in the proof of Lemma 50.3.2 using Cohomology of Schemes, Proposition 30.19.1. $\square$ Lemma 50.3.4. Let $A$ be a Noetherian ring. Let $X$ be a proper scheme over $S = \mathop{\mathrm{Spec}}(A)$. Then $H^ i_{dR}(X/S)$ is a finite $A$-module for all $i$. Proof. This is a special case of Lemma 50.3.3. $\square$ Lemma 50.3.5. Let $f : X \to S$ be a proper smooth morphism of schemes. Then $Rf_\Omega ^ p_{X/S}$, $p \geq 0$ and $Rf_\Omega ^\bullet _{X/S}$ are perfect objects of $D(\mathcal{O}_ S)$ whose formation commutes with arbitrary change of base. Proof. Since $f$ is smooth the modules $\Omega ^ p_{X/S}$ are finite locally free $\mathcal{O}_ X$-modules, see Morphisms, Lemma 29.34.12. Their formation commutes with arbitrary change of base by Lemma 50.2.1. Hence $Rf_\Omega ^ p_{X/S}$ is a perfect object of $D(\mathcal{O}_ S)$ whose formation commutes with arbitrary base change, see Derived Categories of Schemes, Lemma 36.30.4. This proves the first assertion of the lemma. To prove that $Rf_\Omega ^\bullet _{X/S}$ is perfect on $S$ we may work locally on $S$. Thus we may assume $S$ is quasi-compact. This means we may assume that $\Omega ^ n_{X/S}$ is zero for $n$ large enough. For every $p \geq 0$ we claim that $Rf_\sigma _{\geq p}\Omega ^\bullet _{X/S}$ is a perfect object of $D(\mathcal{O}_ S)$ whose formation commutes with arbitrary change of base. By the above we see that this is true for $p \gg 0$. Suppose the claim holds for $p$ and consider the distinguished triangle \[ \sigma _{\geq p}\Omega ^\bullet _{X/S} \to \sigma _{\geq p - 1}\Omega ^\bullet _{X/S} \to \Omega ^{p - 1}_{X/S}[-(p - 1)] \to (\sigma _{\geq p}\Omega ^\bullet _{X/S})[1] \] in $D(f^{-1}\mathcal{O}_ S)$. Applying the exact functor $Rf_$ we obtain a distinguished triangle in $D(\mathcal{O}_ S)$. Since we have the 2-out-of-3 property for being perfect (Cohomology, Lemma 20.49.7) we conclude $Rf_\sigma _{\geq p - 1}\Omega ^\bullet _{X/S}$ is a perfect object of $D(\mathcal{O}_ S)$. Similarly for the commutation with arbitrary base change. $\square$ Your email address will not be published. Required fields are marked. In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar). All contributions are licensed under the GNU Free Documentation License. In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0FL6. Beware of the difference between the letter 'O' and the digit '0'. 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This document is to be distributed for free and without any modification from its original state. The author declines all responsibility in the damage this document or any of the things you will do with it might do to anyone or to anything. This document and any of its contents is not copyrighted and is free of all rights, you may thus use it, modify it or destroy it without breaking any international law. However according to the author's will, you may not use this document for commercial profit directly, but you may use indirectly its intellectual contents; in which case I would be pleased to receive a mail of notice or even thanks. This is my first tutorial and I am still a student, you must assume that this document is probably not free of small errors and bugs. In the same state of mind, those algorithms are not fully optimised, they are explained for pedagogical purposes and you may find some redundant computations or other voluntary clumsiness. Please be indulgent and self criticise everything you might read. Hopefully, lots of this stuff was taken in sources and books of reference; as for the stuff I did: it has proven some true efficiency in test programs I made and which work as wanted. As said in the introduction: If you have any question or any comment about this text, please send it to the above email address, I'll be happy to answer as soon as Simulating a physical phenomena which obeys to known mathematical equations is, with a number of approximations, always feasable. But what about more abstract concepts, such as feelings, which do not follow any laws? The simplest things we can feel are often the hardest things to capture in a program. Beat detection follows this rule : feeling the beat of a song comes naturally to humans or animals. Indeed it is only a feeling one gets when listening to a melody, a feeling which will make you dance in rhythm or hit a table with your hands on the melody beats. Therefore, how can we teach this beat detection to a machine that can only compute logical operations? In fact there are a number of algorithms which manage to approximate, more or less accurately, this beat detection. We will first study the statistical approach of beat detection on a streaming source and secondly a filtering approach of rhythm extraction on a static song. This guide assumes the reader has basic signal processing understanding (FFT, convolutions and correlations should sound common) maybe some stuff in statistics will also help (Variance, Average, Principal Components Analysis, will be quoted among others). The point here is not to actually write the code of these algorithms, but more to understand how they work and to be able to adapt or create the appropriate algorithm to a situation. If you have a question or a comment about this text, please send it to the above email address, I'll be happy to answer as soon as possible. Anyway, the aim here is to give more precise ideas on the subject of beat detection to the reader. Enjoy. I – Statistical streaming beat detection 1 – Simple sound energy a - A first analysis The human listening system determines the rhythm of music by detecting a pseudo – periodical succession of beats. The signal which is intercepted by the ear contains a certain energy, this energy is converted into an electrical signal which the brain interprets. Obviously, The more energy the sound transports, the louder the sound will seem. But a sound will be heard as a beat only if his energy is largely superior to the sound's energy history, that is to say if the brain detects a brutal variation in sound energy. Therefore if the ear intercepts a monotonous sound with sometimes big energy peaks it will detect beats, however, if you play a continuous loud sound you will not perceive any beats. Thus, the beats are big variations of sound energy. This first analysis will bring us to our simplest model : Sound energy peaks. In this model we will detect sound energy variations by computing the average sound energy of the signal and comparing it to the instant sound energy. Lets say we are working in stereo mode with two lists of values : (an) and (bn). (an) contains the list of sound amplitude values captured every Te seconds for the left channel, (bn) the list of sound amplitude values captured every Te seconds for the right channel. So we want to compute the instant energy and the average energy of the signal. The instant energy will in fact be the energy contained in 1024 samples (1024 values of a[n] and b [n]), 1024 samples represent about 5 hundreds of second which is pretty much 'instant'. The average energy should not be computed on the entire song, some songs have both intense passages and more calm parts. The instant energy must be compared to the nearby average energy, for example if a song has an intense ending, the energy contained in this ending shouldn't influence the beat detection at the beginning. We detect a beat only when the energy is superior to a local energy average. Thus we will compute the average energy on say : 44032 samples which is about 1 second, that is to say we will assume that the hearing system only remembers of 1 second of song to detect beat. This 1 second time (44032 samples) is what we could call the human ear energy persistence model; it is a compromise between being to big and taking into account too far away energies, and being too small and becoming to close to the instant energy to make a valuable comparison. The history buffer where we will keep the last 44032 samples wil contain in fact too lists of samples (B[0]) and (B[1]) corresponding to the left (an) and to the right (bn) channels history. Simple sound energy algorithm #1: Every 1024 samples: • Use the 1024 new samples taken in a[n] and b[n] to compute the instant energy 'e', using the following formula (i0 is the position of the 1024 samples to process): • Compute the local average energy '<E>' on the 44100 samples of a history buffer (B): • Shift the 44032 history buffer (B) of 1024 indexes to the right so that we make room for the 1024 new samples and evacuate the oldest 1024 samples. • Move the 1024 new samples on top of the history buffer. • Compare 'e' to 'C * <E>' where C is a constant which will determine the sensibility of the algorithm to beats. A good value for this constant is 1.3. If 'e' is superior to 'C * <E>' then we have a beat! b - Some direct optimisations This was the basic version of the algorithm, its speed and accurecy can be improved quite easily. The algorithm can be optimised by keeping the energy values computed on 1024 samples in history instead of the samples themselves, so that we don't have to compute the average energy on the 44100 samples buffer (B) but on the instant energies history we will call (E). This sound energy history buffer (E) must correspond to approximately 1 second of music, that is to say it must contain the energy history of 44032 samples (calculated on groups of 1024) if the sample rate is 44100 samples per second. Thus E[0] will contain the newest energy computed on the newest 1024 samples, and E[42] will contain the oldest energy computed on the oldest 1024 samples. We have 43 energy values in history, each computed on 1024 samples which makes 44032 samples energy history, which is equivalent to 1 second in real time. The count is good. The value of 1 second represents the persistance of the music energy in the human ear, it was obtain with experimentations but it may varry a little from a person to another, just adjust it as you feal. So here is what the algorithm becomes: Simple sound energy algorithm #2: Every 1024 samples: • Compute the instant sound energy 'e' on the 1024 new sample values taken in (an) and (bn) using the formula (R1) • Compute the average local energy <E> with (E) sound energy history buffer: • Shift the sound energy history buffer (E) of 1 index to the right. We make room for the new energy value and flush the oldest. • Pile in the new energy value 'e' at E[0]. • Compare 'e' to 'C<E>'. c - Sensitivity detection The imediate draw back of this algorithm is the choice of the 'C' constant. For example in techno and rap music beats are quite intense and precise so 'C' should be quite high (about 1.4); whereas for rock and roll, or hard rock which contains a lot of noise, the beats are more confused and 'C' should be low (about 1.1 or 1.0). There is a way, to make the algorithm determine automatically the good choice for the 'C' constant. We must compute the variance of the energies contained in the energy history buffer (E). This variance, which is nothing but the average of ( Energy Values – Energy average = (E) - <E> ), will quantify how marked the beats of the song are and thus will give us a way to compute the value of the 'C' constant. The formula to calculate the variance of the 43 E[i] values is described below (R4). Finally, the greater the variance is the more sensitive the algorithm should be and thus the smaller 'C' will become. We can choose a linear decrease of 'C' with 'V' (the variance) and for example when V → 200, C → 1.0 and when V → 25, C → 1.45 (R5). This is our new version of the sound energy beat detection algorithm: Simple sound energy algorithm #3: Every 1024 samples: • Compute the instant sound energy 'e' on the 1024 new samples taken in (an) and (bn) using the following formula (R1). • Compute the average local energy <E> with (E) sound energy history buffer using formula (R3). • Compute the variance 'V' of the energies in (E) using the following formula: • Compute the 'C' constant using a linear degression of 'C' with 'V', using a linear regression with values (R5): • Shift the sound energy history buffer (E) of 1 index to the right. We make room for the new energy value and flush the oldest. • Pile in the new energy value 'e' at E[0]. • Compare 'e' to 'C<E>', if superior we have a beat! Those three algorithms were tested with several types of music, among others : pop, rock, metal, techno, rap, classical, punk. The fact is the results are quite unpredictable. I will only talk about Simple beat detection algorithm #3 as #2 and #1 are only pedagogical intermediates to get to the #3. Clearly, the beat detection is very accurate and sounds right with techno and rap, the beats are very precise and the music contains very few noise. The algorithm is quite satisfying for that kind of music and if you aim to use beat detection on techno you can stop reading here, the rest won't change anything to your beat detection. However, even if the improvement of the dynamic 'C' calculations ameliorates things alot, the beat detection on punk, rock and hard rock, is sometimes quite approximate. We can feel it doesn't really get the rythm of the song. Indeed the algorithm detects energy peaks. Sometimes you can hear a drum beat which is sank among other noises and which goes trough the algorithm without being detected as a beat. To explain this phenomena lets say a guitare and flute make alternatively an amplitude constant note. Each time the first finishes the other starts. The note made by the guitare and the note made by the flute have the same energy but the ear detects a certain rhythm because the notes of the instruments are at different pitch. For our algorithm (who is one might say colorblind) it is just an amplitude constant noise with no energy peaks. This partly explains why the algorithm doesn't detect precisely beats in songs with a lot of instruments playing at different rythms and simultaneously. Our next analysis will make us walk through this difficulty. Comparing the results we have obtained with the Simple beat detection algorithm #3 to its computing cost, this algorithm is very efficient. If you are not looking for a perfect beat detection than I recommend you use it. Here is a screenshot of a program I made using this algorithm. You fill find the binaries and the sources on my homepage. 2 – Frequency selected sound energy a - The idea and the algorithm The issue with our last analysis of beat detection is that it is colorblind. We have seen that this could raise quite a few problems for noisy like songs in rock or pop music. What we must do is give our algorithm the abbility to determine on which frequency subband we have a beat and if it is powerful enough to take it into account. Basically we will try to detect big sound energy variations in particular frequency subbands, just like in our last analysis; unless this time we will be able to seperate beats regarding their color ( frequency subband ). Thus If we want to give more importants to low frequency beats or to high frequency beats it should be more easy. Notice that the energy computed in the time domain is the same as the energy computed in the frequency domain, so we don't have any difference between computing the energy in time domain or in frequency domain. For maths freaks this is called the Parseval Theorem. Okay that was just a bit of sport, lets go back to the mainstream; Here is how the Frequency selected sound energy algorithm works: The source signals are still coming from (an) and (bn). (an) and (bn) can be taken from a wave file, or directly from a streaming microphon or line input. Each time we have accumulated 1024 new samples, we will pass to the frequency domain with a Fast Fourier Transform (FFT). We will thus obtain a 1024 frequency spectrum. We then divide this spectrum into however many subbands we like, here I will take 32. The more subbands you have, the more sensitive the algorithm will be but the harder it will become to adapt it to lots of different kinds of songs. Then we compute the sound energy contained in each of the subbands and we compare it to the recent energy average corresponding to this subband. If one or more subbands have an energy superior to their average we have detected a beat. The great progress with the last algorithm is that we now know more about our beats, and thus we can use this information to change an animation, for example. So here is more precisely the Frequency selected sound energy algorithm #1: Frequency selected sound energy algorithm #1: Every 1024 samples: To help out visualising how the data piles work have a look at this scheme: Now the 'C' constant of this algorithm has nothing to do with the 'C' of the first algorithm, because we deal here with separated subbands the energy varies globally much more than with colorblind algorithms. Thus 'C' must be about 250. The results of this algorithm are convincing, it detects for example a symbal rhythm among other heavy noises in metal rock, and indeed the algorithm separates the signal into subbands, therefore the symbal rhythm cannot pass trough the algorithm without being recognized because it is isolated in the frequency domain from other sounds. However the complexity of the algorithm makes it useful only if you are dealing with very noisy sounds, in other cases, Simple beat detection algorithm #3 will do the job. b - Enhancements and beat decision factors. There are ways to enhance a bit more our Frequency selected sound energy algorithm #1. First we will increase the number of subbands from 32 to 64. This will take obviously more computing time but it will also give us more precision in our beat detection. The second way to develop the accuracy of the algorithm uses the defaults of human ears. Human hearing system is not perfect; in fact its transfer function is more like a low pass filter. We hear more easily and more clearly low pitched noises than high pitch noises. This is why it is preferable to make a logarithmic repartition of the subbands. That is to say that subband 0 will contain only say 2 frequencies whereas the last subband, will contain say 20. More precisely the width 'wi' of the 'n' subbands indexed 'i' can be obtained using this argument: • Linear increase of the width of the subband with its index: • We can choose for example the width of the first subband: • The sum of all the widths must not exceed 1024 ((B)'s size): Once you have equations (R11) and (R12) it is fairly easy to extract 'a' and 'b', and thus to find the law of the 'wi'. This calculus of 'a' and 'b' must be made manually and 'a' and 'b' defined as constants in the source; indeed they do not vary during the song. So in fact in Frequency selected sound energy algorithm #1, all we have to modify is the number of subbands we will take equal to 64 and the (R7) relation. This relation becomes: It may seem rather complicated but in fact it is not. Replacing this relation (R7) with (R7)' we have created Frequency selected sound energy algorithm #2. If you have musics with very tight and rapid beats, you may want to compute the stuff more frequently than every 1024 samples, but this is only for special cases, normally the beat should not be shorter than 1/40 of second. Using the advantages of Frequency selected beat detection you can also enhance the beat decision factor. Up to now it was based on a simple comparison between the instant energy of the subband and the average energy of the subband. This algorithm enables you to decide beats differently. You may want for examples to cut beats which correspond to high pitch sounds if you run techno music or you may want to keep only [50-4000Hz] beats if you are working with speech signal. This algorithm has the advantage of being perfectly adaptable to any kind or category of signal which was not the case of Simple beat detection algorithm #3. Notice that the correspondants between index 'i' of the FFT transform and real frequency is given by formula : • If 'i' < 'N/2' then: • Else 'i' >= 'N/2' then: So 'i' is the index of the value in the FFT output buffer, N is the size of the FFT transform (here 1024), fe is the sample frequency (here 44100). Thus index 256 corresponds to 10025 Hz. This formula may be useful if you want to create your own subbands and you want to know what the correspondants between indexes and real frequency are. Another way of filtering the beats, or selecting them, is choosing only those which are marked and precise enough. As we have seen before, to detect the accuracy of beats we must compute the variance of the energy histories for each subband. If this variance is high it means that we have great differences of energy and thus that the beat are very intense. Thus all we have to do is compute this variance for each subband and add a test in the beat detection decision. To the "Es[i] > (C*<Ei>)" condition we will add "and V((Ei))>V0". V0 will be the variance limit, with experience 150 is a reasonable value. Now the V((Ei)) value is easy to compute, just follow the following equality if you don't see how : The last (finally) way to enhance your beat detection, is to make the source signal pass through a derivation filter. Indeed differentiating the signal makes big variations of amplitude more marked and thus makes energy variations more marked and recognisable later in the algorithm. I haven't tried this optimisation but according to some sources this is quite useful. If you try it please give me your opinion on it! Concerning the results of the Frequency selected sound energy algorithm #2 I must admit they are way more satisfying than the Simple sound energy algorithm #3. In a song the algorithm catches the bass beats as well as the tight cymbals hits. I insist on the fact that you may also select the beats in very different ways, which becomes quite useful if you know you are going to run techno music with low pitch beats for example. You may also select the beats differently according to there accuracy with the variance criteria. There are many other ways to decide beats; it is up to you to explore them and find the one which fits the most your needs. I used Frequency selected sound energy algorithm #2algorithm in a demo program of which you can see some screenchots just below. One can see quite clearly that there is a beat in low frequencies (probably a bass or drum hit) and also a high pitch beat (probably a cymbal or such): The reason why I called this part of the document, 'Statistical streaming beat detection', is that the energy can be considered as a random variable, of which we have been calculating values over time. Thus we could interpret those values as a sampling for the statistical analysis of a random variable. But one can push this approach further. When we have separated the energies histories into 128 subbands we created in fact 128 random energy variables. We can then apply some of the general statistical methods of analysis. For example, the principal components analysis method will enable you to determine if some of the subbands are directly linked or independent. This would help us to regroup some subbands which are directly linked and thus make the beat decision more efficient. However, this method is basically just far too computing expensive and maybe just too hard to implement comparing to the results we want. If you are looking for a really good challenge in beat detection you could push in this direction. Filtering rhythm detection	{"url":"http://archive.gamedev.net/archive/reference/programming/features/beatdetection/","timestamp":"2024-11-06T21:59:51Z","content_type":"text/html","content_length":"41164","record_id":"<urn:uuid:a69eba3f-cc3a-4e0c-b616-846c243c10e3>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00157.warc.gz"}
Intersection Point Calculator (Line or Curve in 2D Plane) Online Search for a tool Intersection Point Tool for finding the intersection point(s) of 2 lines or curves by calculation from their respective equations (crossing in the 2D plane). Intersection Point - dCode Tag(s) : Functions dCode and more dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day! A suggestion ? a feedback ? a bug ? an idea ? Write to dCode! Intersection Point Find Intersection point of 2 lines Find Intersection Point(s) of 2 Curves Answers to Questions (FAQ) What is an intersection point? (Definition) A point of intersection between 2 elements/drawings/graphs/curves in the 2D plane is the place of crossing/superposition of the 2 elements. How to calculate the intersection point of 2 lines? From the equations of the 2 lines of the 2D plane, it is possible to calculate the point of intersection (if it exists) by solving the corresponding system of equations. The values obtained (generally for $ x $ and $ y $) correspond to the coordinates $ (x, y) $ of the point of intersection. Example: The lines of respective equations $ y = x + 2 $ and $ y = 4-x $ form the system of equations $ \begin{cases} y = x+2 \\ y = -x+4 \end{cases} $ which has for solution $ \begin{cases} x = 1 \\ y = 3 \end{cases} $ therefore the point of intersection of the 2 lines is the point of coordinates $ (1,3) $ If the equations of the lines are not known, dCode allows you to find the equations of a line from its slope coefficient, its y-intercept or only from 2 points (linear equation). How to calculate the intersection point of 2 curves? The calculation of the point (or points) of intersection of 2 curves requires solving the corresponding system of equations. Example: The square function of equation $ y = x^2 $ and the horizontal line $ y = 1 $ allow to create the system of equations $ \begin{cases} y = x^2 \\ y = 1 \end{cases} $ which has 2 solutions $ \ begin{cases} x = 1 \\ y = 1 \end{cases} $ and $ \begin{cases} x = -1 \\ y = 1 \end{cases} $ and therefore the square function has 2 points of intersection with the horizontal line at the coordinate points $ (x,y) $: $ (-1,1) $ and $ (1,1) $ Source code dCode retains ownership of the "Intersection Point" source code. Except explicit open source licence (indicated Creative Commons / free), the "Intersection Point" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Intersection Point" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Intersection Point" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app! Reminder : dCode is free to use. Cite dCode The copy-paste of the page "Intersection Point" or any of its results, is allowed (even for commercial purposes) as long as you credit dCode! Exporting results as a .csv or .txt file is free by clicking on the export icon Cite as source (bibliography): Intersection Point on dCode.fr [online website], retrieved on 2024-11-11, https://www.dcode.fr/intersection-point © 2024 dCode — The ultimate 'toolkit' to solve every games / riddles / geocaching / CTF.	{"url":"https://www.dcode.fr/intersection-point","timestamp":"2024-11-11T20:00:57Z","content_type":"text/html","content_length":"22332","record_id":"<urn:uuid:4de59a89-713a-4dba-9801-a66587912bbf>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00251.warc.gz"}
9 S Multiplication Worksheet Mathematics, particularly multiplication, creates the foundation of many academic techniques and real-world applications. Yet, for lots of students, mastering multiplication can posture a difficulty. To resolve this difficulty, teachers and parents have embraced a powerful device: 9 S Multiplication Worksheet. 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Reported casualties by type of road Reported casualties by type of road Table 23 refers. In 2022, non built-up roads accounted for two-fifths of the total number of casualties (44%: 2,455 out of 5,621). However, because speeds are higher on non built-up roads than elsewhere (the definition is roads with a speed limit of more than 40mph), they accounted for almost three quarters of those killed (74%: 128 out of 173) and for just under half of the total number of seriously injured (46%: 823 out of 1,776). Compared with 2012, the fall in the total number of casualties has been 53% for non built-up roads and 58% for those elsewhere. The numbers killed on built-up roads has fallen by 32% whereas those on non built-up ones have risen by 16%. Over the years, some traffic will have been transferred away from built-up roads by the opening of city and town bypasses, and by the construction of non built-up roads with higher average traffic volumes. Therefore, these figures do not provide an accurate measure of the comparative change in the road safety performance of built-up and non built-up roads. Casualties by mode of transport Table 23 refers. A total of 3,198 car users were injured in road collisions in 2022, representing 57% of all casualties. Of these car users, 101 died. There were 912 pedestrian casualties (16% of the total), of whom 33 died, 480 pedal cycle casualties (9% of the total), of whom 2 died, and 467 motorcycle casualties (8% of the total), of whom 25 died. Because of the numbers of car user, pedestrian, pedal cyclist and motorcyclist casualties, the figures for each of these four groups of road users are the subject of separate sections, which follow this one, and are followed by a section on child casualties, which gives details of their modes of transport. Together, all the modes of transport other than the four mentioned above accounted for 564 casualties in 2022 (10% of the total), and for smaller percentages of the numbers of seriously injured. These included 117 bus and coach users injured in 2022, of whom 20 suffered serious injuries (none died). There were also 211 casualties who were travelling in light goods vehicles (2 died), 36 people in heavy goods vehicles(5 died), 74 users of taxis(2 died), 16 users of minibuses(none died) and 110 people with another means of transport (3 died). Car user casualties A total of 3,198 car users were injured in road collisions in 2022, representing 57% of all casualties. Of these people, a total of 817 were seriously injured, 101 died. Non built-up roads accounted for over a half of all car user casualties (56%: 1,798 out of 3,198). Perhaps because average speeds are higher on non-built up roads, they accounted for much higher percentages of the total numbers of car users who were killed (80%: 81 out of 101) or were seriously injured (65%: 529 out of 817). (see Table 23) The number of car users killed in 2022 was 46 more than the 2021 figure and the total number of casualties of all severities was up by 10%. Since 2012, the number killed has increased by 38%, and there has been a fall of 58% in the total number of car user casualties. (see Table 23) Looking at the annual average over the years 2018-2022, the casualty rate for 16-22 year old car users was 1.42 per thousand population. This was much higher than the rate for car users in the older age groups, which varied from 0.49 to 1.23 per thousand population. (see Table 32) On average, over the years 2018-2022, 68% of car user fatalities occurred on roads with a speed limit of 60 mph. Such roads accounted for 37% of the total number of car user casualties of all severities, where more casualties occurred on roads with a 30 mph limit (42%). (see Table 33) Adult car users On weekdays, the peak time for adult car user casualties was from 4pm to 6pm. The 4pm to 5pm average of 246 (the average over the years 2018-2022) was 65% higher than the average of 149 in the morning 8am to 9am peak. (see Table 28) Adult car user casualties varied by month, with fewest in April and most in August. August had 33% more adult car user casualties than April (annual averages over the years 2018-2022; months standardised to 30 days). (see Table 29) Friday had the peak numbers of adult car user casualties over the years 2018-2022 with 17% more than the average daily number of adult car user casualties. (see Table 30) Pedestrian casualties There were 912 pedestrian casualties in 2022: 16% of all casualties. Of these, 367 were seriously injured and 33 died. Presumably due to their greater vulnerability, a higher proportion of the total number of people who were killed (19%) and seriously injured (21%) were pedestrians. In addition, 40% of pedestrian casualties were seriously injured (367 out of 912) compared with serious for all modes of 32% (1,776 out of 5,621). 93% of pedestrian casualties occurred on built-up roads (851 out of 912) in 2021. (see Table 23) The overall number of pedestrian casualties was 18% higher than 2021. Since 2012, the number of pedestrians killed has fallen by 26 and there has been a 54% reduction in the total number of pedestrian casualties. Looking at the annual average for the period 2018 to 2022, the 12-15 age-group had the highest 'all severities' pedestrian casualty rates (0.55 per thousand population). (see Tables 23 & 32) The overall pedestrian 'all severities' casualty rate for males was 0.22 per thousand population, compared with 0.15 per thousand for females, using the averages for the period 2018 to 2022. (see Table 34) Adult pedestrian casualties On average in the period 2018 to 2028, the peak time for adult pedestrian casualties during the week was from 4pm to 6pm; at weekends it was from 5pm to 7pm. (see Table 28) November and December were the peak months for adult pedestrian casualties, with each having 40% and 38% respectively more than the monthly average. Adult pedestrian casualties in the four winter months, November to February, were 25% more than the monthly average (annual averages over the years 2018-2022; months standardised to 30 days). (see Table 29) Friday has the highest numbers of adult pedestrian casualties; 20% more than the daily average over the period 2018 to 2022. (see Table 30) Pedal Cycle Casualties There were 480 pedal cycle casualties in 2022, 32 less than the previous year. The number of seriously injured pedal cycle casualties in 2022 was 180. There were 2 pedal cycle fatalities in 2022, 8 less than 2021. Since 2012 there has been a 47% decrease in all pedal cycle casualties and the number of fatalities has fluctuated between 2 and 13. In 2022, 88% of pedal cycle casualties were on built-up roads (see Table 23). It should be noted that pedal cycle traffic is estimated to have seen a decrease of 3% in 2022 compared with 2021. In terms of the averages for the period 2018 to 2022, the pedal cycle casualty rate per head of population was highest for those aged 23-25 (0.17 per thousand population). Of course, it must be remembered that, as noted earlier, per capita casualty rates do not provide a measure of the relative risk, because they do not take account of the levels of usage of (in this case) pedal cycles. (see Table 32) Adult pedal cycle casualties Using the averages for the period 2018 to 2022, on weekdays, the peak numbers of adult pedal cycle casualties occurred from 4 pm to 6 pm and from 8 am to 9 am. At weekends the numbers were smaller, but appear to peak between 10 am to 2 pm. (see Table 28) The peak months of the year for adult pedal cycle casualties were June and August which were 28-40% more than the monthly average (2018-2022 annual averages standardised to 30 days). (see Table 29) The day of the week with the peak numbers of adult pedal cycle casualties was Tuesday, 19% higher than the daily average, over the years 2018-2022. There were substantially fewer adult pedal cycle casualties on Sunday, 35% less than the daily average. (see Table 30) Motorcyclist casualties A total of 467 motorcyclists were injured in road collisions in 2022, representing 8% of all casualties. Of these, 280 were seriously injured and 25 died. 53% of all motorcyclist casualties occurred on non built-up roads but (perhaps because of their higher average speeds) such roads accounted for 60% of those seriously injured, and 80% of those killed. (see Table 23) The number of motorcyclist casualties in 2022 was 2% lower than in the previous year and the number killed decreased by 5. The total number of motorcycle casualties rose each year from 1999 to a peak in 2001; since then, it has tended to decline. As a result, the figure for all casualties in 2022 was 46% lower than in 2012. Four more motorcyclists died in 2022 than in 2012. (see Table 23) On average, over the years 2018 to 2022, the motorcyclist casualty rate was highest for the 16-25 and 50-59 age groups (0.15 per thousand population); other age-groups had smaller casualty rates. (see Table 32) Looking at the averages for the period 2018 to 2022, the peak time of day for adult motorcyclist casualties was 4pm to 6pm on weekdays (see Table 28), the peak months of the year were June (68 casualties) and August (66 casualties, amidst a general peak from May to September (see Table 29) and there were more casualties from Friday to Sunday than on any of the other days (see Table 30). Child (0-15) casualties There were 587 child casualties in 2022, representing 10% of the total number of casualties of all ages. Of the child casualties, 176 were seriously injured, and three died (see Table 24). There were two less children killed in 2022 than in 2021. The total number of child casualties increased by 92 on 2021. Since 2012, the number of children killed has increased by one. (see Table A and Table 25) In terms of the averages for the period 2018 to 2022, on weekdays, the peak time for child casualties was from 3 pm to 6 pm, with 43% of all weekday casualties in those three hours. A further 17% occurred in the three hours between 6 pm and 9 pm There was another peak in the morning, between 8 am and 9 am There was no real clear peak at weekends: the numbers of casualties were very broadly the same each hour from 12 noon to 7 pm (see Table 27) August was the peak month for child casualties, with 36% more than in an average month. June had 20% more than an average month. (2018-2022 annual averages standardised to 30 days). (see Table 29) Using the averages for 2018 to 2022, Friday was the peak day of the week for child casualties, with 28% more than an average day. Sunday, on the other hand, had 21% less than an average day. (see Table 30) Child (0-15) casualties by mode of transport In 2022, there were 295 child pedestrian casualties. They accounted for 32% of all pedestrian casualties of all ages (295 out of 912). Of the child pedestrian casualties, 115 were seriously injured and 1 died. (see Table 24) There were 44 child pedal cycle casualties in 2022 (9% of the total of 480 pedal cycle casualties of all ages). The child pedal cycle casualties included 12 who were seriously injured, none died. (see Table 24) In 2022, there were 196 child casualties in cars, 6% of the total number of car user casualties of all ages (196 out of 3,272). Of the child casualties in cars, 21 were seriously injured (one died). (see Tables 23 and 25) Child (0-15) casualty rates (per head of population) Children's casualty rates (per head of population) increase with age: using the averages for the years 2018-2022 taken together, for children aged 0-4 the rate was 0.35 per thousand population, whereas it was 0.69 per thousand for those aged 5-11 and for the 12-15 age group it was 1.02 per thousand. The pedestrian casualty rate for younger children (0-4 years) was 32% of that for 5-11 and 18% of the 12-15 year old rate. (see Table 32) The pedestrian casualty rate for boys in the 0-4 age group was more than twice that for girls. The difference between the sexes was even more pronounced in driver or rider casualty rates. (see Table The overall child pedestrian casualty rate at 0.20 per thousand child population was almost twice the corresponding rate for adult pedestrian casualties. (see Table 32) Emergency hospital admissions for Road Traffic Collisions, by ethnic group A new table U has been added to the Excel data tables which provides a time series showing the number of emergency hospital admissions for injury collisions by ethnic group. < Previous \| Contents \| Next >	{"url":"https://www.transport.gov.scot/publication/reported-road-casualties-scotland-2022/reported-casualties-by-type-of-road/","timestamp":"2024-11-02T07:37:41Z","content_type":"text/html","content_length":"36943","record_id":"<urn:uuid:deece811-adb0-400f-a1d0-d20b5c3f8e4d>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00686.warc.gz"}
[Solved] Verify whether the indicated numbers is zeroes of the ... \| Filo Verify whether the indicated numbers is zeroes of the polynomials corresponding to them in the following case: Not the question you're searching for? + Ask your question We have is a root of Was this solution helpful? Found 2 tutors discussing this question Discuss this question LIVE for FREE 12 mins ago One destination to cover all your homework and assignment needs Learn Practice Revision Succeed Instant 1:1 help, 24x7 60, 000+ Expert tutors Textbook solutions Big idea maths, McGraw-Hill Education etc Essay review Get expert feedback on your essay Schedule classes High dosage tutoring from Dedicated 3 experts Practice questions from Mathematics Class 9 (RD Sharma) View more Practice more questions from Polynomials Practice questions on similar concepts asked by Filo students View more Stuck on the question or explanation? Connect with our Mathematics tutors online and get step by step solution of this question. 231 students are taking LIVE classes Question Text Verify whether the indicated numbers is zeroes of the polynomials corresponding to them in the following case: Topic Polynomials Subject Mathematics Class Class 9 Answer Type Text solution:1 Upvotes 75	{"url":"https://askfilo.com/math-question-answers/verify-whether-the-indicated-numbers-is-zeroes-of-the-polynomials-corresponding-102304","timestamp":"2024-11-04T21:25:09Z","content_type":"text/html","content_length":"262604","record_id":"<urn:uuid:400d0a38-3009-4dfa-bf4f-08954f4c8abf>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00218.warc.gz"}
Behr, Wolfgang (2005): Noncommutative Gauge Theory beyond the Canonical Case. Dissertation, LMU München: Faculty of Physics Preview 930kB Canonically deformed spacetime, where the commutator of two coordinates is a constant, is the most commonly studied noncommutative space. Noncommutative gauge theories that have ordinary gauge theory as their commutative limit have been constructed there. But these theories have their drawbacks: First of all, constant noncommutativity can only be an approximation of a realistic theory, and therefore it is necessary to study more complicated space-dependent structures as well. Secondly, in the canonical case, the noncommutativity didn't fulfill the initial hope of curing the divergencies of quantum field theory. Therefore it is very desirable to understand noncommutative spaces that really admit finite QFTs. These two aspects of going beyond the canonical case will be the main focus of this thesis. They will be addressed within two different formalisms, each of which is especially suited for the purpose. In the first part noncommutative spaces created by star-products are studied. In the case of nonconstant noncommutativity, the ordinary derivatives possess a deformed Leibniz rule, i.e. d_i (f star g) \neq d_i f star g + f star d_i g. Therefore we construct new objects that still have an undeformed Leibniz rule. These derivations of the star-product algebra can be gauged much in the same way as in the canonical case and lead to function-valued gauge fields. By linking the derivations to frames (vielbeins) of a curved manifold, it is possible to formulate noncommutative gauge theories that admit nonconstant noncommutativity and go to gauge theory on curved spacetime in the commutative limit. We are also able to express the dependence of the noncommutative quantities on their corresponding commutative counterparts by using Seiberg-Witten maps. In the second part we will study noncommutative gauge theory in the matrix theory approach. There, the noncommutative space is the ground state of a matrix action, the fluctuations around this ground state creating the gauge theory. In the canonical case the matrices used are infinite-dimensional (they are the Fock-space representation of the Heisenberg algebra), leading to a number of problems, especially with divergencies. Therefore we construct gauge theory using finite dimensional matrices (fuzzy spaces). This gauge theory is finite, goes to gauge theory on a 4-dimensional manifold in the commutative limit and can also be used to regularize the noncommutative gauge theory of the canonical case. In particular, we are able to match parts of the known instanton sector of the canonical case with the instantons of the finite theory. Item Type: Theses (Dissertation, LMU Munich) Keywords: noncommutative geometry, gauge theory Subjects: 500 Natural sciences and mathematics 500 Natural sciences and mathematics > 550 Earth sciences Faculties: Faculty of Physics Language: English Date of oral examination: 3. November 2005 1. Referee: Wess, Julius MD5 Checksum of the PDF-file: 903166a50b02b0c078746ac681546186 Signature of the printed copy: 0001/UMC 14921 ID Code: 4425 Deposited On: 02. Dec 2005 Last Modified: 24. Oct 2020 10:04	{"url":"https://edoc.ub.uni-muenchen.de/4425/","timestamp":"2024-11-05T13:59:50Z","content_type":"application/xhtml+xml","content_length":"31035","record_id":"<urn:uuid:2813d2b8-7213-4f69-82f2-7a616229d533>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00630.warc.gz"}
Classical Dosages: Pandora’s Box June 18, 2012 Classical Dosages: Pandora’s Box Many practitioners have a desire to practice Chinese medicine in a manner that is congruent with what has been done in the past. That is, many prefer formulas that are time-tested. The formulas from the Discussion of Cold Damage (Shang Han Lun), written around 1800 years ago, often fall into this category because not only do they form the foundation of herbalism as we know it, but they are still commonly used today. However, we must ask, what aspect has been time-tested? Has the formula itself been consistently used in a fixed manner demonstrating clinical efficacy? Or is it just the idea behind the formula, applied to ever-changing situations, that has stood the test of time? I think there are aspects of both that are true. An examination of dosages used over time for one of the most famous classical formulas, Minor Bupleurum Decoction (xiao chai hu tang), will give some insight into this issue, as well as provide some guidance as to how we might consider following the First, it should be pointed out that Chinese weights and measurements have historically lacked standardization and have seen numerous changes throughout the centuries. Thus, the dosages listed in classic texts and how they relate to one’s current measurement system has been a source of debate for centuries. For example, in the 17th century, Li Shizhen (1517-1592) said, “The liang of the ancients is the qian of today.” However, Zhang Jiebing (1563-1640), of the same time period, wrote, “The liang of the ancients equals six of today’s qian.” Thus, even within a time period where we think weights are consistent, we see major disagreements. Because of this, various doctors from different regions—as well as different time periods—used contrasting systems of measurement. This raises some very practical questions. For example, how many grams of Bupleuri Radix (chai hu) are in the formula Minor Bupleurum Decoction (xiao chai hu tang)? Modern texts recommend anywhere from 9 to 24g a day, yet key archeological data on weights and measures from the Han dynasty put a daily dose of Bupleuri Radix (chai hu) anywhere from 111-125g (from the source text) (Xiong, 2005, Li, 2005). There are many factors to consider on why dosages may change over time. They include changes in the constitutions of the patient populations, different living conditions and environmental factors, changes in measurement systems and the resulting confusion, and even differences in herb quality or proper herb identification itself. Although the ‘why’ is interesting and is certainly worth exploring, merely looking at the differences in dosages for the ingredients in Minor Bupleurum Decoction (xiao chai hu tang) is enlightening and brings up two interrelated points: 1. There is little consistency in how famous texts / doctors (both past and present) understood the fundamental dosages of Minor Bupleurum Decoction (xiao chai hu tang). This is interesting because all of them had the original source text, which was (and is) often simultaneously cited. See table #1 and compare the doses. 2. More concerning than an across-the-board lowering of dosages (which actually happened in 1979[i]) is that the ratios between the herbs are dramatically different from source to source. That is, if we compare the dosage of Bupleuri Radix (chai hu) to the dosages of the other herbs in the formula, we see considerable divergences in the corresponding ratios. For example, the ratios of Bupleuri Radix (chai hu) to Scutellariae Radix (huang qin) are anywhere from a 1:1 to 1:2.66, and of Bupleuri Radix (chai hu) to Pinelliae Rhizoma preparatum (zhi ban xia), anywhere from 1:1 to 2:97 (see chart #2). The latter is a 197% difference in ratios. For whatever reason, different authors have very different opinions not only about how to interpret the original doses from the Discussion of Cold Damage, but also, what doses may be most appropriate for their patient population. This is peculiar because classical experts often talk about the precise nature of Zhang Zhongjing’s formulas, and how a change in a single ingredient’s dosage can dramatically affect the nature and actions of a formula. Consequently, students are often warned not to mess with the doses, because they represent a master’s creation. Although I generally agree that changing an ingredient’s dose can dramatically change a formula’s action, one cannot help asking, “what is the ‘correct’ or ‘original’ dose that we should follow?” Since this seems quite difficult to ascertain, how do we make sense of such a wide breadth of discrepancies in these dosages and the corresponding ratios? One solution is to look at how famous doctors have used Minor Bupleurum Decoction (xiao chai hu tang) clinically. From case records, we see doses across the board from 6g to 125g of Bupleuri Radix (chai hu) a day. Even a doctor who states that there should be 24g of Bupleuri Radix (chai hu) in Minor Bupleurum Decoction (xiao chai hu tang) may use anywhere from 12g to 30g depending on the individual presentation. Other herbs are also adjusted depending on specific guidelines. We know that Zhang Zhongjing made specific recommendations for modifications of Minor Bupleurum Decoction (xiao chai hu tang) based on various signs and symptoms, such as exchanging Trichosanthis Radix (tian hua fen) for Pinelliae Rhizoma preparatum (zhi ban xia) if there is pronounced thirst. This trend of modifying the core formula has continued throughout time. However, we also see guidelines for how to modify (increase and decrease) dosages by Discussion of Cold Damage (Shang Han Lun) experts. For example: … Actually, according to specific circumstances, it is suitable to increase and decrease [the doses of these herbs]. For example, when there is alternating fever and chills one will want to emphasize the [dose of] Bupleuri Radix (chai hu). If there is no alternating fever and chills, but there is only ‘fullness in the chest and ribside,’ or ‘bitter taste in the mouth and dry throat,’ then Bupleuri Radix (chai hu) dose can be reduced. However, Scutellariae Radix’s (huang qin) dose should never surpass Bupleuri Radix (chai hu). Originally, the dose of Pinelliae Rhizoma preparatum (zhi ban xia) was half a sheng. This larger dose was important because it was able to descend counterflow and stop vomiting, which was important for the original indication of “frequent vomiting (Li, 2005).” Without this symptom, one might reduce the dose. Essentials from the Golden Cabinet also mentions, “If there is fever with frequent vomiting then Minor Bupleurum Decoction (xiao chai hu tang) masters it.” In this case, a large dose of both Bupleuri Radix (chai hu) and Pinelliae Rhizoma preparatum (zhi ban xia) should be used (Li, 2005). This is the type of thinking sits at the core of how famous doctors modify classic formulas and can arrive at dramatically different doses, and even different ratios of herbs. Of course there are a multitude of other variables to consider, such as the patient’s constitution etc. Additional Thoughts: All of this brings up an interesting point. Often, modern practitioners use classic formulas in ways that were not originally prescribed. For example, the original indication of alternating fever and chills for Minor Bupleurum Decoction (xiao chai hu tang) was most likely quite severe, possibly akin to malaria. A large dose formula, e.g. using 100g of Bupleuri Radix (chai hu), makes sense in this However, modern practitioners often extrapolate this symptom to represent a condition that comes and goes (and various other ideas related to it). Consequently, they might give Minor Bupleurum Decoction (xiao chai hu tang) to a patient that does not exhibit the severity of the original presentation. In addition, it may be based on their constitution, abdominal or pulse confirmation, which traditionally might not necessary have been used (in the manner we use it today). Such adaptation is, of course, valid, but the reality is that the patient in front of us may look different than what this formula was originally intended for. Therefore, Minor Bupleurum Decoction (xiao chai hu tang) may be given in a smaller dose—and over a long period of time—quite successfully. Finally, the method of preparation originally recommended in the Discussion of Cold Damage (Shang Han Lun) differs from the way many of us prepare decoctions. The source text advises to decoct the ingredients in approximately 12 cups of water (originally one dou and two sheng) until six cups remain. The dregs are removed and the strained decoction is further decocted until three cups remain. This additional cooking time—especially without the herbs themselves—tempers the larger dose of e.g. Bupleuri Radix (chai hu). Therefore, when I use larger doses of Bupleuri Radix (chai hu), I have the patient prepare their herbs in this manner. Both of these points help understand how a 111-125g of Bupleuri Radix (chai hu) might have made sense classically, while the majority of our current Minor Bupleurum Decoction (xiao chai hu tang) patients would just not do well with such doses. Even a 24-gram-a-day dose is daunting for most practitioners, in both China and the West. In conclusion, there has been a quite a bit written over the centuries about Minor Bupleurum Decoction (xiao chai hu tang), and the numerous ways it can be used. Quite simply, its usage as well as the dosages of its ingredients has changed over time. If we want to try to find a starting point that is close to Zhang Zhongjing’s original thinking, we might forgo trying to nail down the exact doses and try to follow the ratios from the source text as closely as possible. This is a good starting point and the following formula represents these ratios: Xiao Chai Hu Tang Chai Hu 24g Huang Qin 9g Ren Shen 9g Zhi Gan Cao 9g Ban Xia 8g Sheng Jiang 9g Da Zao 6g (rounded up from 5.75) Apart from this, stay nimble. Note: After this research Red Pine Chinese Herbs have changed their dose Xiao Chai Hu Tang to the above. Table #1 – Doses of Ingredients for Xiao Chai Hu Tang – converted to grams. │ │Chai Hu │Huang Qin│Ren Shen│Zhi Gan Cao│Ban Xia │Sheng Jiang│Da Zao│ │SHL(1) │125g[ii] │46.875g │46.875g │46.875g │42g[iii]│46.875g │30g │ │SHL(2) │111.36g[iv]│41.76g │41.76g │41.76g │100ml[v]│41.76g │? │ │Xu Shu-Wei (1075-1156) │74.7g │3 fen[vi]│3 fen │3 fen │22.41g │5 slices │2p │ │ │ │ │ │ │ │ │ │ │ │ │(分) │ │ │ │ │ │ │Shang Han Wen Yi Tiao Bian （1748） │14.92g │7.46g │3.73g │3.73g │7.46g │7.46g │2p │ │Zhang Xi-Chun (1909) │29.84g │11.19g │11.19g │11.19g │14.92g │11.19g │4p │ │实用中医学 1975 │11.19g[vii]│7.46g │3.73g │3.73g │11.19g │11.19g │4p │ │中医方剂手册 │9g[viii] │9g │9g │9g │6g │9g │4p │ │ │ │ │ │ │ │ │ │ │1983 │ │ │ │ │ │ │ │ │实用方剂小典 2002 │12g │9g │6g │5g │9g │9g │4p │ │Fangji Xue (2005) │24g │9g │9g │9g │9g │9g │4p │ │Huang Huang (2005) │10-20g │6-10g │5-10g │5-10g │6-15g │10-15g │5-10p │ │Scheid/Bensky (2009) │24g │9g │9g │9g │24g │9g │12p │ Table #2 - Ratios of herbs with Chai Hu[ix] │ │Chai Hu│Huang Qin│Ren Shen│Zhi Gan Cao│Ban Xia│Sheng Jiang│Da Zao│ │ │ │ │ │ │ │ │ │ │SHL(1) │- │2.67 │2.67 │2.67 │2.97 │2.67 │4.166 │ │SHL(2) │- │2.67 │2.67 │2.67 │? │2.67 │? │ │Xu Shu-Wei (1075-1156) │- │? │? │? │3.33 │? │? │ │Shang Han Wen Yi Tiao Bian （1748） │- │2 │4 │4 │2 │2 │- │ │Zhang Xi-Chun (1909) │- │2.67 │2.67 │2.67 │2 │2.67 │? │ │实用中医学 1975 │- │1.5 │3 │3 │1 │1 │? │ │中医方剂手册 │- │1 │1 │1 │1.5 │1 │? │ │ │ │ │ │ │ │ │ │ │1983 │ │ │ │ │ │ │ │ │实用方剂小典 2002 │- │1.33 │2 │2.4 │1.333 │1.333 │? │ │Fangji Xue (2005) │- │2.67 │2.67 │2.67 │2.67 │2.67 │? │ │Scheid/Bensky (2009) │- │2.67 │2.67 │2.67 │1 │2.67 │? │ 1. Li Fei. 李飞. 方剂学(第2版)(套装上下册). 第2版 ed., 人民卫生出版社, 2005. 2. Liu Jingchao. 刘景超, and 李具双. 许叔微医学全书(精装). 第1版 ed., 中国中医药出版社, 2006. 3. Scheid, Volker, Dan Bensky et al. Chinese Herbal Medicine: Formulas & Strategies (2nd Ed.). 2 ed., Eastland Press, 2009. 4. Xiong Manqi. 熊曼琪. 伤寒论(精装). 第1版 ed., 人民卫生出版社, 2005. 5. Yang Xuan. 杨璇. 伤寒温疫条辨. 第1版 ed., 学苑出版社, 2006. 6. Zhang Xichun (张锡纯 ) (2002). Essays on Medicine Esteeming the Chinese and Respecting the Western (医学衷中参西录 yixue zhongzhong canxi lu). Hebei: Hebei kexuejizhu chubanshe 河北科学挤术出版 7. 实用方剂小典 (2002)－ Details upon request. 8. 中医方剂手册 (1983) - Details upon request. 9. 实用中医学 (1975) - Details upon request. [i] In 1979 the qian was changed from 3.73g to 3.125. [ii] The SHL1 doses are from Archeological data presented in Xiong’s Discussion of Cold Damage (Shang Han Lun) textbook, p. 1078. [iii] The dosage of Ban Xia is more problematic because it was originally expressed in sheng, which is measurement of volume. [iv] The SHL2 doses are from Fei’s Formulas (Fangji Xue) textbook, page. 124. [v] The dosage of Ban Xia is more problematic because it is originally expressed in sheng, which is measurement of volume. [vi] It is unclear how much this fen is. Consulted sources suggest that this 3 fen is approximately 1.24g. However this makes little sense in comparison to the 74.7g dose of Chai Hu. [vii] The doses from this 1975 text as well as the ones prior to this do not uses grams. They use traditional measurement which have been converted using Fei’s 2005 chart on page 124. [viii] Post 1979 1 qian is reduced to 3.125g, and then often rounded down to 3g. [ix] All of the numbers in this table are essentially a ratio of a given herb to Chai Hu. For example, in the first row, Chai Hu to Huang Qin is 2.67:1.	{"url":"https://urbanherbsco.com/blogs/chinese-herb-blog/6167078-classical-dosages-pandora-s-box","timestamp":"2024-11-10T16:02:25Z","content_type":"text/html","content_length":"145829","record_id":"<urn:uuid:cf83f4c4-5459-4ef3-82f5-b436c931bc22>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00467.warc.gz"}
Portfolio Management The benefit of utilities: a plausible explanation for small risky parts in a portfolio Keywords: portfolio management, risk, risk aversion, utility function, certainty equivalent, risk-free return, yield on stock, buy-and-hold strategy, Black-Scholes model, lognormal distribution, most probable return, mode of a probability distribution Identification of what proportion of stock to hold based on your expectation of return and risk disposition Keywords: S&P 500 market, market index, fixed-term deposit, AAA bond, savings book, investment period, yield on stock, buy-and-hold strategy, risk-free return, volatility, loss probability, total Suppose you want to achieve a certain return in a fixed time period on a portfolio that contains two different assets: a “market” set of stocks and a “risk-free” investment. Our methods allow an estimate of what return you would receive for various stock proportions and what the loss probability would be for the portfolio as a whole (stocks + risk-free assets). The basic assumption is that you buy the “market” (with ETFs, index certificates or a mixture of stocks that represents the market) and that you keep the percentage of stock approximately constant over the defined time period. This means that you have to adjust the stock proportion from time to time: when the stock market goes up substantially, the proportion of stock increases as well and you must sell part of your stock in order to keep the ratio of stock to risk-free assets the same. Similarly you have to buy replacement stock, if the the market goes down, provided you have not exited the market altogether (as described in the section on market signals). First we show the total return for various stock ratios as a function of the loss probability, assuming a buy-and-hold strategy and ignoring any market signals. Second we show how you can improve your return on investment and, at the same time, decrease your risk by avoiding long term bear markets with the help of market signals. The mathematical basis of this method of portfolio management is to model the stock market using a Gaussian distribution for the logarithmic price changes, as is done in the Black-Scholes model. Depending on the strategy – buy-and-hold or using market signals – different moments of the Gaussian distribution are used. For the following scenarios, we have applied the statistical characteristics of the US S&P 500 market, which is of special importance as it is a lead market for all other world markets. We use the historical S&P 500 data on a weekly basis, as one can show that there is no improvement in the results using data on a daily basis. Portfolio Management without Market Signals Portfolio Management with Market Signals	{"url":"https://www.sigmadewe.com/portfoliomanagement.html?&L=1.","timestamp":"2024-11-14T01:37:33Z","content_type":"text/html","content_length":"9270","record_id":"<urn:uuid:fbb5e25d-a50b-4556-8343-261e79abec17>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00362.warc.gz"}
Don't spot the pattern If I write 2 then 4 then 6, then we feel good because we know that next comes 8. We can foresee it. We are not in the hands of destiny. Unfortunately, however, this has nothing to do with truth. – Arthur Seldom, The Oxford Murders (movie) The series 2, 4, 8, could obviously be followed by 16, but also by 10 or 7004. It's always possible to find a rule, a justification which allows a series to be continued by any number. It all depends on how complicated the rule is. – Arthur Seldom, The Oxford Murders I remember getting questions at school of the form “which number comes next?” At the time I thought these questions were perfectly normal. I now think they are nonsensical. As such it troubles me to see similar questions (with diagrams rather than numbers) are being used in psychometrics assessments. For instance here are ones called diagrammatic reasoning tests. Whether numbers or diagrams, the idea is the same and it should be put to an end. These questions expect you to extrapolate from a finite set of data. The problem is, as with the above quotes, there are infinitely many ways to do this. The only difference between them is that some “feel” more right than others. They are intuitive, they are “simple”. But both of these things are in fact rather subjective. And so while these questions pretend to have only one right answer, they really do not. Here is an example. The series 1 2 3 5 … This could be “all integers with at most one factor”, i.e. all the primes and the number 1 – then the next number is 7. It could also be the Fibonacci sequence, but starting at 1 2 instead of 1 1 – then the next number is 8. Of course one could think up infinitely many rules for completing this sequence. Another simple rule is to assume it is periodic 1 2 3 5 1 2 3 5…. - then the next number is 1. Of course if you looked only at the first three elements in the series you would probably guess the next number is 4. The question, then is not really “what is the next number?” it is: Find a function from the natural numbers to the natural numbers which has the given sequence as its first mappings. The function should be “simple”, meaning it should be described (possibly as a recurrence relation) only with addition, subtract, exponentiation, etc. and should be the one function that whoever is marking the question would think is the simplest. The problem is that the problem is never actually stated like this. The ways in which you are allowed to describe your function are not enumerated and there is no objective means of determining what “simple”. Thus for any above mediocre mind, the problem is not to find the next number, but to determine how far beyond the standard set of descriptions for functions they should allow their mind to Thus the question really only does the following: it forces you to confine your search to what is expected already. It hinders the ability to think beyond this and it penalises anyone who happens to think differently from the standard. It creates a false impression of truth and limits human creativity. The only way to ask these questions (if you have to ask them at all) is to give a precise description of the form of function allowed and then make sure only one function in this set satisfies the requirements. The same is true for diagrammatic questions.	{"url":"http://blog.johandp.com/2013/09/dont-spot-pattern.html","timestamp":"2024-11-13T12:55:08Z","content_type":"text/html","content_length":"78349","record_id":"<urn:uuid:12436e64-be6c-428b-bd03-e70cfff1338d>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00874.warc.gz"}
Intuitive Infinitesimal Calculus • An original calculus textbook written in accordance with our unique teaching philosophy. • Always gives the most illuminating and satisfying proofs possible, while standard books obscure key ideas under mountains of pedantic formalism. • Focus on aha-that’s-why explanations, often using visual and intuitive reasoning, while standard books prefer opaque formula-crunching. • Full of fascinating problems, not boring obstacle-course drills. • Topics are carefully motivated, not taught “because I say so.” • Develops applications fully from first principles, so that you can reach genuine insight, instead of just giving you formulas to plug numbers into like a circus monkey doing tricks for a banana. • Sticks to essentials instead of burying key concepts under rambling prose and bloat content. • Uses a worksheet-style format for clean and clear presentation and active reader engagement. • Reference summary at end of each chapter gives you “everything you need to know for the test” in quick-and-dirty, cheat-sheet form, including step-by-step solution plans for standard problem types. Other textbooks expect you to somehow extract this information for yourself from running text and examples, even though they always mingle it with a bunch of useless crap you don’t need. • Illuminated by unique historical perspective and expertise, as the author did his Ph.D. on the history of the calculus. • A free calculus textbook. Don’t give hundreds of dollars to [DEL:bloodsucking parasites:DEL] publishing corporations. • See also blog posts on calculus teaching.	{"url":"https://intellectualmathematics.com/calculus/","timestamp":"2024-11-12T23:46:17Z","content_type":"text/html","content_length":"38334","record_id":"<urn:uuid:b8c83665-1d0b-421c-bbca-8f7af9701f2e>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00167.warc.gz"}
NIPS 2016 Mon Dec 5th through Sun the 11th, 2016 at Centre Convencions Internacional Barcelona Reviewer 1 This paper addresses an issue that is common in randomized controlled trials (or A/B tests): inferences may be invalid if there is nonstationarity, and in particular if users strategically respond dynamically to the differences in the two cells. The authors attempt to calibrate a strategic model using short-run data to then estimate a long-run causal effect. Qualitative Assessment ---The way you refer to "causal effect" and "the fundamental problem of causal inference" is somewhat nonstandard. Typically (in the Rubin potential outcomes model, which is what you are building on), the causal effect is defined at the individual level, with a "treatment" outcome and "control" outcome for each experimental unit. The fundamental problem of causal inference is that only one of these two outcomes is actually observed for each experimental unit. You seem to be focusing on a slightly different issue, which is that the effect of treating the entire population cannot be determined correctly from just data when half the population is treated. It seems to me that this issue -- which can arise due to a variety of violations of the SUTVA assumption -- can exist independent of whether there is a multiagent interaction. Conversely, it seems multiagent considerations are relevant even when defining causal effects at the sub-population level. ---Your references are somewhat limited to recent work. There has been extensive work on causal inference in nonstationary environments, particularly by econometricians and time series analysts. It would be good to situate your work better relative to this work. Your main difference lies in the particular structure of behavioral model you assume, and I judged the paper on this basis. ----A significant limitation that I see is that the modeling approach requires games and behaviors to be defined before the estimation procedure can be carried out. One could, instead, imagine a fully nonparametric approach to causal inference that just aimed to remove any nonstationarity in treatment and control before estimating a treatment effect. The approach you take requires that one should believe constructing the necessary spaces of games and behaviors would be both reasonably specificed, and computationally tractable. ---The main contribution is in the modeling, as the theorem follows naturally from the modeling assumptions. The theorem is making an asymptotic statement imprecisely; it would be better to formally state the result as well. ---The numerics are carried out for a relatively simplified two player game, whereas the motivating examples are all drawn from ad auctions. I would like to understand whether this approach can be reasonably used in a more practical setting. For example, in an ad auction setting, how would you restrict the behaviors that you consider? As the paper currently stands, the example is a bit too simple, considering the significant model complexity that would arise if one tried to capture all behaviors possible in more complex (and real-world) strategic environments. ---Overall, I thought the authors took an intriguing approach to a very difficult problem. I was generally left with the impression that the work is promising but still preliminary, and would benefit from some deeper investigation of the applicability of the methods, as well as comparison to the very extensive related literature. Confidence in this Review 2-Confident (read it all; understood it all reasonably well) Reviewer 2 This paper pursues an interesting line of research: how to identifying the long term causal impact in randomized experiments where agents' react to experiment and adapt with time? Much of the literature on causal inference focus on generalization of inference made from small sample to the entire population. The authors pursue a novel line of research by focusing on long term effect of experiments. They do so by building a dynamic model for behavior update, fitting this model on short term data, and using the model to make long term claim. Qualitative Assessment I like the novelty the authors bring by focusing on an important problem that hasn't been studied well, and in using tools from behavioral game theory and machine learning. My concerns are mostly around generalization of this framework for real life scenarios and its robustness esp. around the dynamic update model for behavior, as well as action given behavior. Most of these seem to put a heavy burden of knowing the exact payoff, fixed agents, and working with finite games. I, however, am hoping that some of these would be studied more in their future work. Confidence in this Review 1-Less confident (might not have understood significant parts) Reviewer 3 The paper proposes a methodology to estimate the long-term causal effect of a treatment (such as a price increase) in a complex dynamical multiagent systems. To do so, it has to solve two inferential tasks: infer outcomes across assigments to the control or the treatment; infer across time. This is particularly challenging as only short-term experimental data are observed. Qualitative Assessment The paper is well written, the supplementary material exhaustive. Unfortunately, I am not qualified to give an educated opinion on its content. I am not familiar with many of the concepts used. This paper deserves a specialist in causality since its contribution may be significant. Naively, I wonder about the stability of such a methodology in practice (e.g. http://www.auai.org/uai2016/ proceedings/papers/214.pdf). I will be amazed that such inference will yield robust conclusions with respect to real-life situations as in some fields (financial applications for risk and portfolios) estimating a simple yet robust variance-covariance is an endless problem. Confidence in this Review 1-Less confident (might not have understood significant parts) Reviewer 4 The paper deals with the long-term causal effects in a system where agents exhibit dynamic behaviours modeled under some assumptions. The key extension of the classical theory in the paper is the modeling of agent behaviors that can predict their actions based on game theory. Qualitative Assessment The paper is well written and well organized, there are some room for improvement: 1. The author defines the behavioral model and temporal model in a general form. It'd be interesting to see discussions on how different choice of models might influence the long-term effects. 2. As the counterfactuals always exist in real dataset, it might be relevant o design a simulated experiment and compare LACE, DID, etc. Confidence in this Review 1-Less confident (might not have understood significant parts) Reviewer 5 The authors consider the problem of understanding long-term causal effects of advertising interventions by brining ideas from game theory to bear on experimental data. Essentially, the authors use a deterministic (behavioral) game-theoretic model to extrapolate from short term inferences to the desired long-term ones. Similarly, they are able to look at interventions on an agent-by-agent bases, using a notion of inferred player types. Qualitative Assessment I think this is an interesting paper. I have two suggestions/criticisms. First, I believe that the algorithm describes what is essentially a two step procedure. The data is used to estimate a parameter, and then a model is used to convert that estimate into a different estimate. So it is effectively a plug-in estimator: to estimate g(theta) we first estimate theta and then use g(\hat{\ theta}) as our estimate. In the present case g is a pretty elaborate dynamic model, but the idea is the same. If indeed this is the procedure I think an explanation could be much abbreviated and clearer as a result. Fortunately, the assumptions posited by the authors mean than the form of g() is not affected by the intervention or subsequent data, which allows for this two step procedure. Second, I think the authors are fairly glib/optimistic about the accuracy of behavioral game theory models. Given that the procedure relies on that estimate being spot-on, this is no small matter. Perhaps a citation to the more pessimistic assessment of Hahn, Mela and Goswami (2015) AoAS is warranted. The authors did an especially nice job at contrasting their approach to previous work (lines 76-82) and I find their approach to be very sensible in this light. Regarding the validity of the behavioral model, one can always perform sensitivity analyses for different choices of g(). Confidence in this Review 3-Expert (read the paper in detail, know the area, quite certain of my opinion)	{"url":"https://papers.nips.cc/paper_files/paper/2016/file/af4732711661056eadbf798ba191272a-Reviews.html","timestamp":"2024-11-02T11:23:35Z","content_type":"text/html","content_length":"10155","record_id":"<urn:uuid:5d47b418-3f56-4aca-b070-ac65192c24c3>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00784.warc.gz"}
6.851: Advanced Data Structures (Fall'17) Prof. Erik Demaine TAs: Adam Hesterberg, Jayson Lynch [Home] [Lectures] [Problem Sets] [Project] [Coauthor] [Accessibility] [+] Dynamic optimality: independent rectangle, Wilber, and Signed Greedy lower bounds; key-independent optimality; O(lg lg n)-competitive Tango trees In this lecture, we'll see the best binary search tree we know, in the sense of achieving the best known competitive ratio of O(lg lg n) compared to the offline optimal. To analyze these “Tango trees”, we compare against a lower bound. Specifically, we describe a Signed Greedy algorithm that, for a given access sequence, computes a number of node touches that every BST in the world must perform when given that access sequence. Indeed, the Signed Greedy algorithm adds points row by row just like the Greedy algorithm we saw last lecture. The only catch is that Signed Greedy doesn't actually satisfy the point set, so it doesn't correspond to a BST… yet it is tantalizingly close to Greedy, which we saw last lecture corresponds to an online BST, suggesting that the two bounds are within a constant factor of each other (making Greedy dynamically optimal). While this gap hasn't been closed yet, we can still use the lower bound to analyze Tango trees, and get the exponential improvement over the trivial O(lg n) competitive ratio held by any balanced binary search tree. Download Video: 360p, 720p Lecture notes, page 1/8 • [previous page] • [next page] • [PDF] Lecture notes, page 1/8 • [previous page] • [next page] • [PDF] The video above should play if your web browser supports either modern Flash or HTML5 video with H.264 or WebM codec. The lecture notes should advance automatically. If you have any trouble with playback, email Erik.	{"url":"https://courses.csail.mit.edu/6.851/fall17/lectures/L06.html","timestamp":"2024-11-12T21:51:14Z","content_type":"text/html","content_length":"6623","record_id":"<urn:uuid:bc8a2e51-20a4-4ec6-9e08-76f1a10ab390>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00449.warc.gz"}
Factor the given expression. 9x^(2)-100 -Turito Are you sure you want to logout? Factor the given expression. using the formula The correct answer is: (3x - 10) (3x + 10) is the factorized form of the given expression. ANS :- (3x-10)(3x+10) is the factorized form of the given expression. Given , Using square root and squaring on 9 and 100 . we get using the formula ∴ (3x - 10) (3x + 10) is the factorized form of the given expression. Get an Expert Advice From Turito.	{"url":"https://www.turito.com/ask-a-doubt/factor-the-given-expression-9x-2-100-qa6fd3ed5","timestamp":"2024-11-08T18:26:58Z","content_type":"application/xhtml+xml","content_length":"258830","record_id":"<urn:uuid:74456e97-4f7d-48dd-a0d2-a518d0b3185d>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00758.warc.gz"}
The Ultimate Guide to Calculating Half Life: Understanding and Mastering the Process - The Cognition Sentinel The Ultimate Guide to Calculating Half Life: Understanding and Mastering Half Life Calculations from radioactive decay to drug efficacy. Learn the basic steps and the common mistakes to avoid when calculating half life. Half life is an essential concept in the fields of science and medicine. It is the amount of time it takes for half of a substance to decay or expire. Knowing how to calculate half life is crucial in many areas, including nuclear medicine, environmental science, and drug development. Understanding the process of calculating half life is essential for scientists, students, and anyone with an interest in the world around us. The Ultimate Guide to Calculating Half Life: A Step-by-Step Tutorial Learning how to calculate half life involves taking several steps. The following tutorial will provide guidance on how to correctly calculate half life. Step 1: Gather Data and Identify the Known Variables The first step in calculating half life is to gather data and identify the known variables. This includes the initial amount of the substance, the amount of time it takes for the substance to decay or disappear, and the final amount of the substance. Step 2: Determine the Order of the Reaction The next step is to determine the order of the reaction. There are three types of reactions that scientists use to describe the decay or expiration of substances. These are zero order, first order, and second order reactions. The order of the reaction refers to the number of molecules involved in the reaction. Zero order means that the amount of the substance decays at a constant rate, independent of the amount of the substance present. First order reactions involve only one molecule, while second order reactions involve two molecules. Step 3: Use the Half Life Formula to Calculate the Half Life The third step is to use the half life formula to calculate the half life. The half life formula is defined as the amount of time it takes for one-half of the initial amount of the substance to decay. The formula is as follows: Half life = 0.693 / k Where k is the rate constant for the reaction. The rate constant can be calculated by using the data and the order of the reaction. Step 4: Check Your Answer Using the Original Data Checking your answer using the original data is the fourth step in calculating half life. It is crucial to make sure that the answer obtained in step 3 is correct by using the original data. Step 5: Understanding the Significance of Your Results The final step is to understand the significance of your results. Half life calculations are essential in many fields of science. Knowing the half life can help scientists understand the decay of radioactive materials, the metabolism of drugs, and the decay of pollutants in the environment. Mastering the Art of Determining Half Life in 5 Easy Steps The following is a step-by-step breakdown of each of the steps above. This guide aims to simplify the process of calculating half life and make it easier and more efficient for users. Step 1: Collect Data: Gather data and identify the known variables, including the initial amount, the rate of decay, and the final amount. Step 2: Determine the Order of the Reaction: Determine the order of the reaction in order to calculate the rate constant. This can be done by looking at the reaction’s chemical equation or plotting the data on a graph and examining its shape. Step 3: Calculate the Rate Constant: Use the data and the order of the reaction to calculate the rate constant, k. Step 4: Calculate the Half Life: Use the half life formula, Half life = 0.693 / k, to calculate the half life. Step 5: Check Your Results: Make sure to verify your calculation by checking it against the original data collected in step 1. The Science Behind Half Life: How to Calculate Your Results Like a Pro Understanding the scientific concepts behind half life calculations is crucial for mastering the calculation process. Below, we will provide an in-depth explanation of the mathematical concepts behind half life calculations, practice problems to illustrate these concepts, and common mistakes to avoid in your calculations. Mathematical Concepts Behind Half Life Calculations The calculation of half life requires knowledge of several scientific concepts, including rate constants, logarithms, and exponential functions. The rate constant, k, is the proportionality constant that relates the rate of a chemical reaction to the concentration of the reactants. The logarithmic function is the inverse of the exponential function. The natural logarithm is denoted by “ln” and is used in half life calculations. Practice Problems Practice problems can help you master the concepts behind half life calculations. Below are some sample problems: 1. If the initial amount of a sample is 100 grams, and it takes 10 days for the sample to decay to 50 grams, what is the half life of the sample? 2. If the half life of a substance is 5 days and the initial amount of the substance is 100 grams, what is the amount remaining after 20 days? Common Mistakes to Avoid There are a few common mistakes to avoid when calculating half life. These include rounding errors, not considering the correct order of the reaction, and not checking your work against the original Why Half Life Matters: A Comprehensive Breakdown of the Calculation Process Half life calculations are essential in many fields of science. In this section, we will provide real-life applications of half life calculations, including how it affects drug efficacy and treatment planning in medicine and how it is used in nuclear medicine and environmental science. Real-life Scenarios Where Half Life is Relevant Half life is relevant in many areas, including nuclear medicine, physics, and chemistry. For example, in nuclear medicine, half life calculations are used to determine the decay rate of radioactive materials. In medicine, half life calculations are used to determine the metabolism and elimination of drugs from the body. In environmental science, half life calculations are used to model the decay of pollutants and contaminants in the environment. How Half Life Affects Drug Efficacy and Treatment Planning Half life plays a vital role in drug efficacy and treatment planning. By understanding the half life of a drug, doctors can determine the dosage and frequency of administration. This information is essential for the correct administration of drugs to patients, avoiding overdosing, and maximizing the therapeutic effect of the drug. A Beginner’s Guide to Calculating Half Life: Tips, Tricks, and Common Mistakes to Avoid For beginners, calculating half life can be a daunting task. However, with the following tips, tricks, and common mistakes to avoid, the process can be simplified. Simplification of Complex Concepts for Beginners The concepts involved in calculating half life can be complex. However, by breaking down the process into steps, beginners can acquire a basic understanding of the process. Explanation of Common Pitfalls in Calculations and How to Avoid Them There are several common mistakes to avoid when calculating half life. These include not verifying the order of the reaction, rounding errors, and not checking the calculation against the original data. By avoiding these mistakes, beginners can improve their accuracy in half life calculations. Tips on How to Apply Half Life Calculations in Everyday Life Half life calculations can also be applied in everyday life. For example, half life calculations can help determine the shelf life of food or the expiration date of medications. By applying half life calculations, individuals can make informed decisions about the products they use and consume. Simplifying Half Life: How to Calculate Decay in Everyday Life Half life calculations can be used to calculate decay in everyday life. For example, half life can be used to determine the decay of radioactive materials in the environment or the expiration date of a food product. To calculate decay, use the following formula: Amount remaining = (Initial amount) x (1/2)^(time / half life) From Radioactive Decay to Drug Efficacy: Understanding Half Life and its Calculations Half life calculations are used in various fields, including nuclear medicine, environmental science, and drug development. Understanding the association between half life and other scientific concepts is fundamental in mastering half life calculations. Association of Half Life with Other Mathematical and Scientific Concepts Half life is linked with other scientific concepts, including exponential functions, logarithmic functions, and calculus. These concepts help scientists and researchers to understand decay and expiration in various fields and make informed decisions based on half life calculations. Half life calculations are crucial in various fields of science and medicine. They help us to understand decay, expiration, and the behavior of radioactive materials, drugs, and pollutants. By mastering half life calculations and understanding their significance, individuals can make informed decisions and contribute to the betterment of society. Remember, calculating half life involves several steps, including gathering data, determining the order of the reaction, calculating the half life, and checking your results. With practice, anyone can master the art of calculating half life and apply it in everyday life. Leave a Reply Cancel reply	{"url":"https://www.supsalv.org/how-to-calculate-the-half-life/","timestamp":"2024-11-07T12:45:17Z","content_type":"text/html","content_length":"108591","record_id":"<urn:uuid:3ebe6ff9-92ed-44ec-9f6c-c1fee7182cac>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00251.warc.gz"}
Mastering pandas for Finance Book description Master pandas, an open source Python Data Analysis Library, for financial data analysis In Detail This book will teach you to use Python and the Python Data Analysis Library (pandas) to solve real-world financial problems. Starting with a focus on pandas data structures, you will learn to load and manipulate time-series financial data and then calculate common financial measures, leading into more advanced derivations using fixed- and moving-windows. This leads into correlating time-series data to both index and social data to build simple trading algorithms. From there, you will learn about more complex trading algorithms and implement them using open source back-testing tools. Then, you will examine the calculation of the value of options and Value at Risk. This then leads into the modeling of portfolios and calculation of optimal portfolios based upon risk. All concepts will be demonstrated continuously through progressive examples using interactive Python and IPython Notebook. By the end of the book, you will be familiar with applying pandas to many financial problems, giving you the knowledge needed to leverage pandas in the real world of finance. What You Will Learn • Modeling and manipulating financial data using the pandas DataFrame • Indexing, grouping, and calculating statistical results on financial information • Time-series modeling, frequency conversion, and deriving results on fixed and moving windows • Calculating cumulative returns and performing correlations with index and social data • Algorithmic trading and backtesting using momentum and mean reversion strategies • Option pricing and calculation of Value at Risk • Modeling and optimization of financial portfolios Product information • Title: Mastering pandas for Finance • Author(s): • Release date: May 2015 • Publisher(s): Packt Publishing • ISBN: 9781783985104	{"url":"https://www.oreilly.com/library/view/mastering-pandas-for/9781783985104/","timestamp":"2024-11-13T05:38:13Z","content_type":"text/html","content_length":"87161","record_id":"<urn:uuid:86207913-ed6e-4659-9581-1a9d75f75fa9>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00373.warc.gz"}
Derivative of a Quadratic Form: Calculating Rates of Change Table of Contents : Calculating the derivative of a quadratic form is essential in understanding how rates of change operate in various fields, including mathematics, physics, and economics. Quadratic forms are expressions that can be represented as ( ax^2 + bx + c ), where ( a, b, ) and ( c ) are constants. This blog post will walk you through the fundamental concepts of quadratic forms, how to calculate their derivatives, and the implications of these calculations. By mastering these concepts, you can better analyze relationships in data and make informed decisions based on your findings. Understanding Quadratic Forms What is a Quadratic Form? 🤔 A quadratic form is an expression involving a variable raised to the second power. Mathematically, it can be expressed as: [ Q(x) = ax^2 + bx + c ] • ( a ): Coefficient of ( x^2 ) (the quadratic term). • ( b ): Coefficient of ( x ) (the linear term). • ( c ): Constant term. The shape of the graph of a quadratic form is a parabola, and depending on the sign of ( a ), the parabola opens upwards (if ( a > 0 )) or downwards (if ( a < 0 )). Characteristics of Quadratic Forms Quadratic functions have several important characteristics: • Vertex: The highest or lowest point of the parabola. • Axis of Symmetry: A vertical line that runs through the vertex. • Intercepts: Points where the graph intersects the axes. Understanding these characteristics is crucial for analyzing the behavior of quadratic forms. Calculating the Derivative of a Quadratic Form The Derivative Explained The derivative of a function measures how the function's output changes as its input changes. For quadratic forms, the derivative tells us the rate of change of the function concerning its variable ( x ). Step-by-Step Calculation 📝 To find the derivative of ( Q(x) = ax^2 + bx + c ), we can apply basic rules of differentiation: 1. Power Rule: The derivative of ( x^n ) is ( n \cdot x^{n-1} ). 2. Constant Rule: The derivative of a constant is zero. Using these rules, we differentiate each term of the quadratic form: [ Q'(x) = \frac{d}{dx}(ax^2) + \frac{d}{dx}(bx) + \frac{d}{dx}(c) ] Applying the rules: • The derivative of ( ax^2 ) is ( 2ax ). • The derivative of ( bx ) is ( b ). • The derivative of ( c ) is ( 0 ). So, the derivative of the quadratic form is: [ Q'(x) = 2ax + b ] Interpretation of the Derivative • The derivative ( Q'(x) ) represents the slope of the tangent line to the parabola at any point ( x ). • A positive value indicates that the function is increasing, while a negative value indicates that the function is decreasing. • The value of ( x ) where ( Q'(x) = 0 ) gives the x-coordinate of the vertex of the parabola, which corresponds to the maximum or minimum point of the quadratic form. Applications of the Derivative of Quadratic Forms Real-World Examples 🌍 1. Physics: In kinematics, the position of an object can often be described by a quadratic equation. The derivative represents the object's velocity. 2. Economics: Profit functions may also take on a quadratic form, where the derivative shows how profits change with variations in production levels. 3. Engineering: Structural analysis involves using quadratic equations to model forces. Derivatives help assess changes in stress or strain. Table of Quadratic Derivatives Quadratic Form ( Q(x) ) Derivative ( Q'(x) ) Critical Points ( 2x^2 + 4x + 1 ) ( 4x + 4 ) ( x = -1 ) ( -3x^2 + 2x - 5 ) ( -6x + 2 ) ( x = \frac{1}{3} ) ( x^2 + 6x + 8 ) ( 2x + 6 ) ( x = -3 ) Important Note: Finding the critical points is essential in optimization problems, as they can indicate maximum or minimum values in the context of a problem. Graphical Representation of Derivatives Visualizing the Derivative Graphing the quadratic form along with its derivative can provide insights into the behavior of the function. The intersection of the derivative with the x-axis corresponds to critical points, while the value of the derivative informs us about the increasing or decreasing nature of the function. Example Graphs 📊 When ( Q(x) = 2x^2 + 4x + 1 ): • Quadratic graph: A parabola opening upwards. • Derivative graph: A linear function showing the slope. Using graphing tools can help visualize these concepts, reinforcing the connection between the quadratic form and its derivative. Understanding the derivative of a quadratic form is a powerful tool in mathematics and its applications. By mastering how to calculate derivatives and interpret their meaning, you can analyze the rates of change, optimize functions, and solve real-world problems effectively. Whether you're delving into physics, economics, or engineering, these skills are invaluable. As you continue your exploration, remember that practice is key to becoming proficient in these concepts!	{"url":"https://tek-lin-pop.tekniq.com/projects/derivative-of-a-quadratic-form-calculating-rates-of-change","timestamp":"2024-11-12T19:23:14Z","content_type":"text/html","content_length":"85821","record_id":"<urn:uuid:0ddcb945-151a-4a48-9dd0-89662e33c182>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00687.warc.gz"}
American Mathematical Society The initial-Neumann problem for the heat equation in Lipschitz cylinders HTML articles powered by AMS MathViewer Trans. Amer. Math. Soc. 320 (1990), 1-52 DOI: https://doi.org/10.1090/S0002-9947-1990-1000330-7 PDF \| Request permission We prove existence and uniqueness for solutions of the initial-Neumann problem for the heat equation in Lipschitz cylinders when the lateral data is in ${L^p}$, $1 < p < 2+\varepsilon$, with respect to surface measure. For convenience, we assume that the initial data is zero. Estimates are given for the parabolic maximal function of the spatial gradient. An endpoint result is established when the data lies in the atomic Hardy space ${H^1}$. Similar results are obtained for the initial-Dirichlet problem when the data lies in a space of potentials having one spatial derivative and half of a time derivative in ${L^p}$, $1 < p < 2+\varepsilon$, with a corresponding Hardy space result when $p = 1$. Using these results, we show that our solutions may be represented as single-layer heat potentials. By duality, it follows that solutions of the initial-Dirichlet problem with data in ${L^q}$, $2 - \varepsilon ’ < q < \infty$ and BMO may be represented as double-layer heat potentials. • D. G. Aronson, Bounds for the fundamental solution of a parabolic equation, Bull. Amer. Math. Soc. 73 (1967), 890–896. MR 217444, DOI 10.1090/S0002-9904-1967-11830-5 • D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 22 (1968), 607–694. MR 435594 • Russell M. Brown, The method of layer potentials for the heat equation in Lipschitz cylinders, Amer. J. Math. 111 (1989), no. 2, 339–379. MR 987761, DOI 10.2307/2374513 • Russell M. Brown, Area integral estimates for caloric functions, Trans. Amer. Math. Soc. 315 (1989), no. 2, 565–589. MR 994163, DOI 10.1090/S0002-9947-1989-0994163-7 • A.-P. Calderón, Cauchy integrals on Lipschitz curves and related operators, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 4, 1324–1327. MR 466568, DOI 10.1073/pnas.74.4.1324 —, Boundary value problems in Lipschitzian domains, Recent Progress in Fourier Analysis, Elsevier Science Publishers, 1985, pp. 33-48. • R. R. Coifman, A. McIntosh, and Y. Meyer, L’intégrale de Cauchy définit un opérateur borné sur $L^{2}$ pour les courbes lipschitziennes, Ann. of Math. (2) 116 (1982), no. 2, 361–387 (French). MR 672839, DOI 10.2307/2007065 • Ronald R. Coifman and Guido Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), no. 4, 569–645. MR 447954, DOI 10.1090/S0002-9904-1977-14325-5 • Björn E. J. Dahlberg and Carlos E. Kenig, Hardy spaces and the Neumann problem in $L^p$ for Laplace’s equation in Lipschitz domains, Ann. of Math. (2) 125 (1987), no. 3, 437–465. MR 890159, DOI • E. B. Fabes and M. Jodeit Jr., $L^{p}$ boundary value problems for parabolic equations, Bull. Amer. Math. Soc. 74 (1968), 1098–1102. MR 233081, DOI 10.1090/S0002-9904-1968-12061-0 • E. B. Fabes and N. M. Rivière, Dirichlet and Neumann problems for the heat equation in $C^{1}$-cylinders, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 179–196. MR 545307 • Eugene Fabes and Sandro Salsa, Estimates of caloric measure and the initial-Dirichlet problem for the heat equation in Lipschitz cylinders, Trans. Amer. Math. Soc. 279 (1983), no. 2, 635–650. MR 709573, DOI 10.1090/S0002-9947-1983-0709573-7 • E. B. Fabes and D. W. Stroock, The $L^p$-integrability of Green’s functions and fundamental solutions for elliptic and parabolic equations, Duke Math. J. 51 (1984), no. 4, 997–1016. MR 771392, DOI 10.1215/S0012-7094-84-05145-7 • John B. Garnett and Peter W. Jones, BMO from dyadic BMO, Pacific J. Math. 99 (1982), no. 2, 351–371. MR 658065 • David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0 • Anna Grimaldi Piro, Francesco Ragnedda, and Umberto Neri, Invertibility of some heat potentials in BMO norms, Rend. Sem. Mat. Univ. Padova 75 (1986), 77–90. MR 847659 • John T. Kemper, Temperatures in several variables: Kernel functions, representations, and parabolic boundary values, Trans. Amer. Math. Soc. 167 (1972), 243–262. MR 294903, DOI 10.1090/ • Robert V. Kohn, New integral estimates for deformations in terms of their nonlinear strains, Arch. Rational Mech. Anal. 78 (1982), no. 2, 131–172. MR 648942, DOI 10.1007/BF00250837 O. A. Ladyženskaja, V. A. Solonnikov and N. N. Uralceva, Linear and quasilinear equations of parabolic type, Transl. Math. Mono., vol. 23, Amer. Math. Soc., Providence, R.I., 1968. • Jürgen Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101–134. MR 159139, DOI 10.1002/cpa.3160170106 • Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095 • N. Th. Varopoulos, BMO functions and the $\overline \partial$-equation, Pacific J. Math. 71 (1977), no. 1, 221–273. MR 508035 • Gregory Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains, J. Funct. Anal. 59 (1984), no. 3, 572–611. MR 769382, DOI 10.1016/ Bibliographic Information • © Copyright 1990 American Mathematical Society • Journal: Trans. Amer. Math. Soc. 320 (1990), 1-52 • MSC: Primary 35K05; Secondary 31B35, 35C15, 42B30, 46E35 • DOI: https://doi.org/10.1090/S0002-9947-1990-1000330-7 • MathSciNet review: 1000330	{"url":"https://www.ams.org/journals/tran/1990-320-01/S0002-9947-1990-1000330-7/?active=current","timestamp":"2024-11-04T19:56:37Z","content_type":"text/html","content_length":"75253","record_id":"<urn:uuid:90f30461-1125-4a94-a3e4-5a8cf49f286e>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00899.warc.gz"}
More and Less than 1% Related Pages Illustrative Math Grade 7 Lesson 9: More and Less than 1% Let’s explore percentages smaller than 1%. Illustrative Math Unit 7.4, Lesson 9 (printable worksheets) Lesson 9 Summary The following diagram shows how to find percentages of quantities. Lesson 9.1 Number Talk: What Percentage? Determine the percentage mentally. 10 is what percentage of 50? 5 is what percentage of 50? 1 is what percentage of 50? 17 is what percentage of 50? Lesson 9.2 Waiting Tables During one waiter’s shift, he delivered appetizers, entrées, and desserts. What percentage of the dishes were desserts? appetizers? entrées? What do your percentages add up to? Open Applet Lesson 9.3 Fractions of a Percent 1. Find each percentage of 60. What do you notice about your answers? 30% of 60 3% of 60 0.3% of 60 0.03% of 60 2. 20% of 5,000 is 1,000 and 21% of 5,000 is 1,050. Find each percentage of 5,000 and be prepared to explain your reasoning. If you get stuck, consider using the double number line diagram. a. 1% of 5,000 b. 0.1% of 5,000 c. 20.1% of 5,000 d. 20.4% of 5,000 3. 15% of 80 is 12 and 16% of 80 is 12.8. Find each percentage of 80 and be prepared to explain your reasoning. a. 15.1% of 80 b. 15.7% of 80 Are you ready for more? To make Sierpinski’s triangle, Start with an equilateral triangle. This is step 1. Connect the midpoints of every side, and remove the middle triangle, leaving three smaller triangles. This is step 2. Do the same to each of the remaining triangles. This is step 3. Keep repeating this process. 1. What percentage of the area of the original triangle is left after step 2? Step 3? Step 10? 2. At which step does the percentage first fall below 1%? Lesson 9.4 Population Growth 1. The population of City A was approximately 243,000 people, and it increased by 8% in one year. What was the new population? 2. The population of city B was approximately 7,150,000, and it increased by 0.8% in one year. What was the new population? Lesson 9 Practice Problems 1. The student government snack shop sold 32 items this week. For each snack type, what percentage of all snacks sold were of that type? 2. Select all the options that have the same value as 3 1/2% of 20. 3. 22% of 65 is 14.3. What is 22.6% of 65? Explain your reasoning. 4. A bakery used 30% more sugar this month than last month. If the bakery used 560 kilograms of sugar last month, how much did it use this month? 5. Match each diagram to a situation. The diagrams can be used more than once. a. The amount of apples this year decreased by 15% compared with last year’s amount. b. The amount of pears this year is 85% of last year’s amount. c. The amount of cherries this year increased by 15% compared with last year’s amount. d. The amount of oranges this year is 115% of last year’s amount. 6. A certain type of car has room for 4 passengers. a. Write an equation relating the number of cars (n) to the number of passengers (p). b. How many passengers could fit in 78 cars? c. How many cars would be needed to fit 78 passengers? The Open Up Resources math curriculum is free to download from the Open Up Resources website and is also available from Illustrative Mathematics. Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations. We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.	{"url":"https://www.onlinemathlearning.com/percentage-illustrative-math.html","timestamp":"2024-11-02T14:18:33Z","content_type":"text/html","content_length":"40262","record_id":"<urn:uuid:75342d46-f93d-4183-a154-3b5d9c0aa112>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00847.warc.gz"}
PPT - 1. Solar Photovoltaic Theory PowerPoint Presentation, free download - ID:5932289 1. 1. Solar Photovoltaic Theory 1-2. Potentialassessment 2. 1-2.Potential assessment • Contents 1-2. Potential assessment1-2-1. Basic principle of assessment1-2-2. Insolation measurement1-2-3. Estimation of annual power generation1-2-4. Case practice 3. Sun light from any direction PV REFLECTED 1m 1m Horizontal plane 1-2-1 Basic principle of assessment • Insolation Solar radiation (Insolation ) is “light energy” from sun. • Solar radiation (insolation) reaches the ground as: • direct radiation • diffused radiation Global Radiation(Insolation) Energy received within a unit time Energy: kWh/m2 Power: kW/ m2 4. 1.00 kW/m2( 0.093 kW/f2 ) 1.35 kW/m2(0.125 kW/f2 ) Green Out of atmosphere( 1.35 kW/m2 = 0.125 kW/feet2) Ground surface on the equator( 1.00 kW/m2 = 0.093 kW/feet2 ) Absorbed by H2O , O2 , O3, CO2 Visible 1-2-1 Basic principle of assessment • Insolation spectrum on the surface of ground 5. 1-2-1 Basic principle of assessment • Various effects for insolation • Local latitude effect • “Air mass” effect ( Atmospheric path length effect) • Seasonal effect • Weather effect • Face rotation effect • Surrounding obstacles effect( Shading effect ) 6. : Local latitude I0 I0 I0 I0 Earth -90 deg(S pole) 0 deg(Equator) +90 deg(N pole) 1-2-1 Basic principle of assessment • Effect of Local Latitude Actually, you can measure this value Local Horizontal Insolation mathematical Cosine curve 1.0 kW/m2(0.093 kW/feet2) about 1.0 kW/m2(0.093 kW/feet2) 7. (kW/m2) I0 (kW/m2) 1 m m 1 m 1-2-1 Basic principle of assessment • Effect of Local Latitude Rectangle plane towed sun light Meaning of convert equation Horizontal plane Insolation energy of the tilted plane(yellow) and the horizontal plane(blue) is same. Tilted plane(yellow plane) Horizontal plane(blue plane) 8. B c a A C b Appendix • A-2 Triangle Function ( Cosine Function ) Please calculate by your handy computer Example 9. Ideal tilt angle = local latitude Ideal PV plane Your horizontal plane I0 You are here I0 Local latitude is Earth 1-2-1 Basic principle of assessment • Best tilt angle Best tilt angle is almost same as “local latitude” 10. Lp (Light pass length) Air Air Mass = I0 At (thickness of air) I0 mathematical Cosine curve I0 With Air mass effect I0 Earth -90 deg(S pole) 0 deg(Equator) +90 deg(N pole) 1-2-1 Basic principle of assessment • Effect of “Air mass” ( Atmospheric path length ) At Lp about 1.37 kW/m2( 0.125 kW/f2 ) about 1.0 kW/m2( 0.093 kW/f2 ) • “Atmospheric path length” depend on its latitude. Air mass 11. Latitude Max. Min. Japan +35deg Jun. Dec. Singapore 0deg Mar.Sep. Jun.Dec. Australia - 35deg Dec. Jun. 1-2-1 Basic principle of assessment • Effect of Season 12. 1-2-1 Basic principle of assessment • Effect of Season Seasonal effect is more strong in high latitude -13 Samoa +1 Kiribath -17 Vanuatu -21 Cook Is. kWh/m2day -8 Tuvalu. +34 Japan Month 13. Data in JAPAN Fine day Cloudy day Actual output / Rated capacity Rainy day Time 1-2-1 Basic principle of assessment • Effect of Weather Daily output curve of various weather condition 14. Key factor of solar resource • Latitude • Atmospheric path length • Length of daytime • Opportunity of fine day. Almost same in PPA countries Depend on the geographical aspect. Insoration 6822 (MJ/m2year)Utilization 15.8% Fine day 77.5%Cloudy day 17.9% Nandi Fiji Insoration 6131 (MJ/m2year)Utilization 14.2% Fine day 68.4%Cloudy day 21.2% Suva 1-2-1 Basic principle of assessment • Effect of Weather 15. Location = 35N Face to S W E Face to SE Face to SW S Insolation 6 10 12 14 18 Time 1-2-1 Basic principle of assessment • Face-rotation effects on daily insolation curve • If you rotate PV module face to East, output peak will shift to earlier. • If you rotate PV module face to West, output peak will shift to later. Northern hemisphere 16. Face to S Face to SE Face to SW Insolation Face to S Face to SE Face to SW Insolation Face to S Face to SE Face to SW Insolation 6 10 12 14 18 1-2-1 Basic principle of assessment • Face-rotation effects on daily insolation curve • This effect is more strong in high latitude. • Low latitude area (under 15deg), this effect is negligible. Latitude15N Latitude35N Latitude60N 17. Latitude effect • Seasonal effect(depended on sun height angle) • Air mass effect Insolation • Weather effect Summer • Seasonal effect Winter Time 6 12 18 Day light time 1-2-1 Basic principle of assessment • Various effects on daily insolation curve 18. 1-2-1 Basic principle of assessment • Necessity of on site insolation measuring Key factor of solar resource • Latitude dependent • Atmospheric path length • Length of daytime • Seasonal sun height-angle • Weather dependent • Opportunites of fine day • Mist in the air • Site situation • Shade of mountain, tree, buildings • Contamination by dust, salty gusts Easy to estimate Easy to estimate Easy to estimate Un-known Un-known Difficult Un-known On site insolation measuring is necessary before planning.(at least 1 – 3 years. Use meteorological observatory data) 19. PV 1-2-1 Basic principle of assessment • Basic theory of PV panel adjustment Basic theory of PV panel adjustment • You cannot avoid these effect. • The best things you can do are to: Latitude effect Air Mass effect Seasonal effect Daily effect Weather effect Obstacle shad effect - Tilt PV plane the same as your latitude.- Face true north or true south. Face to N or S (as possible as you can) same angle as latitude Avoidable. Try to find good location 20. +deg At 45N point, Optimum tilt angle is45 – 7 = 38 deg 45 deg local latitude difference between local latitude and optimum tilt -7deg under In low latitude region such as 10 to 20 deg, error is negligible -deg 1-2-1 Basic principle of assessment • Basic theory of PV panel adjustment (Note) • In high latitude locations, the optimum tilting angle is slightly lower than the local latitude. By using computer, you can calculate accurate tilting angle easily. 21. 1-2-1 Basic principle of assessment • Insolation of the world 22. 1 (kW/m2) Output 1 kW Rated Capacity “1 kW” 1 (kW/m2) Generate 1 kWh for 1 hour Rated Capacity “1 kW” 1-2-1 Basic principle of assessment • Definition of PV’s Rated Capacity Note: This is the definition that, we use metric system here. “Rated capacity 1kW” means ( Power ) If insolation is 1 kW/m2, this PV can output 1 kW. ( Energy ) If PV has 1 kW/m2 insolation in 1 hour, this PV can generate 1 kWh 23. 1 (kWh/m2day) Generate 1 kWh for a day Rated Capacity “1 kW” 1-2-1 Basic principle of assessment • Definition of PV’s Capacity “Rated capacity 1kW” means ( Accumulated Energy ) If PV has 1 kWh/ m2day, this PV can generate 1 kWh for a day In resource assessment, “Accumulated Insolation (energy)” is used widely. Daily accumulated insolation kWh/m2day Monthly accumulated insolation kWh/ m2month Annual accumulated insolation kWh/m2year 24. 1 (kW/m2) Generate 1 kW Rated Capacity “1 kW” 1 (kW/m2) Generate 1 kW Rated Capacity “1 kW” 1-2-1 Basic principle of assessment • Definition of PV’s Capacity “Efficiency” parameteris already included in “Rated Capacity”. • High-efficiency PV ( Single crystal PV 15% ) If you use “rated capacity”, you don’t have to consider about efficiency. • Low-efficiency PV (Amorphous PV 8 %) Module is larger. 25. 1-2-2 Insolation measurement • How to observe Insolation Pyranometer for Horizontal Global Solar Radiation (Insolation) Pyranometer Horizontal plane 26. 1-2-2 Insolation measurement • How to observe Insolation Pyranometer for Horizontal Global Solar Radiation (Insolation) Sun window (receives light from all directions) • Place Pyranometer on thehorizontal plane. • Make sure no shadow is cast all day long. • Clean upper window frequently. 20 cm Instant value XX.XX (kW/m2) orAccum. value XX.XX (kWh/m2) Data logger Insolation data is very common in meteorology. Ask your meteorological observatory for local insolation data. 27. Metric(m) Imperial(feet) MJ MJ / m2year MJ / feet2year kWh kWh / m2year kWh / feet2year x 3.60 x 1 / 10.76 Pyranometer for Global Solar Radiation 1-2-2 Insolation measurement • There are many units of Insolation data. Be sure to note which unit your pyranometer is using. 28. 1-2-2 Insolation measurement • Example of raw data (monthly data) Date January Accumulating Time Average Insolation for a day (kWh/m2day) 29. 1-2-2 Insolation measurement • Example of raw data ( Annual data) Daily average Insolation Summarize Annual total insolation 30. I0 I0 I0 Earth 1-2-2 Insolation measurement • Convert “horizontal insolation” to “tilted insolation” Raw insolation data( Horizontal insolation ) Horizontal to Tiltedconversion Plane ofPV Panel (Tilted same as local latitude) I0 Hj I0 I0 Earth Hj : Tilted insolation 31. Hj(kWh/m2year) PV Module Plane(Tilted as local Latitude) I (kWh/m2year) Measured Plane (Horizontal) (kWh/m2year) (kWh/m2year) 1-2-2 Insolation measurement • Convert “horizontal insolation” to “tilted insolation” (Note) This conversion can be used in low latitudes (less than 20deg.) 32. 1-2-2 Insolation measurement • Convert “horizontal insolation” to “tilted insolation” Meaning of convert equation (kW/m2) I Hj(kW/m2) 1 m Insolation energy of the tilted plane(yellow) and the hori-zontal plane(blue) is same. m 1 m Tilted plane(yellow plane) Horizontal plane(blue plane) 33. Local Latitude = -10 (deg) Horizontal Insolation I = 2,000 (kWh/m2Year) (kWh/m2year) (kWh/m2year) (kWh/m2year) 1-2-2 Insolation measurement • Convert “horizontal insolation” to “tilted insolation” (Example) Hj = 2,031 kWh/m2year( Tilted insolation ) I = 2,000 kWh/m2year(Measured raw data) 10 deg. 34. Actual generation energy Hg = 70% (System Efficiency) • Converter Loss 8% • Surface Contamination 7% • Temperature Rise 15% Pu = Rated Capacity of PV Module (=1.0 kW)hg = System efficiency (= 0.7 depending on type of PV cell)H = Tilted Plane Insolation in kWh unit. 1-2-3. Estimation of annual power generation • Actual generation energy of PV (Example) Tilted Insolation Hj = 2,031 (kWh/ m2Year) PV rated capacity Pu=10 (kW) (kWh/year) 35. 1-2-3. Estimation of annual power generation • Calculate “Load Factor ( Syaytem Utilization parameter)" • To estimate various capacities of PV system, calculate Unified Parameter,called"Load Factor (System Utilization Parameter )". • This parameter means “Annual average output power” of unit capacity of the PV system. p = Annual Available power for Unit Capacity of PV Module Pu = Unit Capacity of PV Module (=1.0) (Example) Annual power generation p= 14,217 (kWh/Year) PV rated capacity Pu=10 (kW) 16.3 (%) 36. (kWh/year) 1-2-3. Estimation of annual power generation • Calculate annual power by System Utilization Parameter (Example) If you install a 50kW PV system in this place, how much power can you generate? System utilization parameter Ug= 0.158 (%) PV rated capacity Pu=50 (kW) 37. Local Latitude = -15 (deg) Horizontal Insolation I = 1,800 (kWh/m2Year) PV Capacity Pu = 5kW 1-2-3. Estimation of annual power generation • Exercise (Insolation data) Step1Convert “horizontal insolation” to “tilted insolation” Step2Calculate annual earned energy 38. Local Latitude = -15 (deg) Horizontal Insolation I = 1,800 (kWh/m2Year) PV Capacity Pu = 5kW (kWh/m2year) (kWh/m2year) (kWh/m2year) (kWh/year) 1-2-3. Estimation of annual power generation • Exercise (Insolation data) Step1Convert “horizontal insolation” to “tilted insolation” Step2Calculate annual earned energy 39. 1-2-3. Estimation of annual power generation • Exercise Step3Calculate “Load Factor" Step4If you install 50kW PV system in this place, how much energy (kWh) can you earn? 40. 14.9 (%) (kWh/year) 1-2-3. Estimation of annual power generation • Exercise Step3Calculate “Load Factor" Step4If you install 50kW PV system in this place, how much energy (kWh) can you earn? 41. 1-2-4. Case practice Case Practice 42. 1-2-4. Case practice February has 28 days	{"url":"https://fr.slideserve.com/blake-jennings/1-solar-photovoltaic-theory","timestamp":"2024-11-08T14:13:31Z","content_type":"text/html","content_length":"104750","record_id":"<urn:uuid:f0deee11-39b2-4647-bf91-c3591301c017>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00427.warc.gz"}
herr strathmann Yeah! Shogun this week got accepted to be an organisation participating in the 10th Google Summer of Code. This year, besides mentoring a few projects, I am one of the three project administrators. I am curious how this will be. One first thing to do was to write the application for Shogun – I’m glad it worked! I also will spend a little more time organising things. Apart from trying to find mentors (which requires a lot of talking people into it), I also want to make Shogun (and the students) having more from the program. Last year, I pushed the team to ask all students • to write a project report in the form of IPython notebooks (link). These are absolutely great for talking about the GSoC work, impressing people, and having a final piece of work to show for the • To fully unit-test every module of their algorithm/framework. This is absolutely essential in order to not loose the student’s work a few years later when a re-factoring change breaks their code and nobody knows how to fix it. Those tests already saved lots of life since last year. • To peer-review each other in pairs of students. This improved documentation here and there and solved some bugs. I want to emphasise this more this year as I think it is a great way of enabling synergistic effects between students. In addition, we will again screen all the applicants via a set of entrance tasks on our github page (link). I just wrote a large number of such smaller or larger tasks that get students started on a particular project, fix bugs in Shogun, or prepare some larger change. In order to get the students started a bit more easily (contributing to Shogun these days is a non-trivial task), I wrote a little how-to (link) that is supposed to point out our expectations, and what are the first steps towards participating in GSoC. Finally, I wrote descriptions for quite a few possible projects, some of them with a number of interesting co-mentors. The full list is here (link). If you are a talented student interested in any of those topics, consider working with us during the summer. It’s usually very fun! Some of the other projects involve cool buzzwords such as Deep Learning, Structured Output, Kernel, Dual solvers, Cluster backends, etc. Join us! 🙂 MLOSS workshop at NIPS 2013 Last week, I went to the Advances in Neural Information Processing Systems (NIPS) for the first time. That was a very nice experience due to the incredibly density of people whose names I know from research papers. In fact, it was too much to take so I had to pick things that sounded interesting – still loads. The main three buzzwords of the conference for me were: Deep Learning (even Mark Zuckerberg is interested in that these days), Mini-batch, and stochastic gradient descent (aka on-line whatever). One very interesting workshop I attended on Tuesday was on Machine Learning Open-Source Software (MLOSS), organised by Cheng Soon Ong (who could not be there unfortunately) and Antti Honkela. I presented a short spotlight for Shogun (slide) and had a one hour demo, showing off with our cool IPython notebooks (link) and the cloud Shogun server (link). I got some very encouraging feedback for this, including from Fernando Perez. I also met a few nice fellow open-source ML coders from scikit-learn. During the workshop, there was a quite lively discussion about licensing issues, in particular whether to choose GPL or BSD. The python universe for example seems to gain a lot from being BSD-style Finally, NIPS is was held close to Lake Tahoe, which is surrounded by incredibly beautiful mountains to hike in. One evening, I met the guy who left those traces … very exciting, slightly scary… GSoC 2013 brings Shogun 3.0 Shogun’s third Google Summer of Code just ended with our participation in the mentor summit at Google’s headquarter in Mountain View and the release of Shogun 3.0 (link) What a great summer! But let’s start at the beginning… Shogun is a toolbox that offers a unified framework for data-analysis, or in buzz words: machine learning, for a broad range of data types and analysis problems. Those not only include standard tools such as regression, classification, clustering, etc, but also cutting edge techniques from recent developments in research. One of Shogun’s most unique features is its interfaces to a wide range of mainstream computing languages. In our third GSoC, we continued most of the directions taken in previous years such as asking students to contribute code in the application process for them to be considered. For that, we created a list of smaller introductory tasks for each of the GSoC projects that would become useful later in the project. While allowing students to get used to our development process, and increasing the quality of the applications, this also pushed the projects forward a bit before GSoC even started. The number of applications did not suffer through that (57 proposals from 52 students) but even increased compared to the previous year (48 proposals from 38 students) — this seems to be a trend. This summer, we also had former GSoC students mentoring for the first time: Sergey Lisitsyn and me (mentoring two projects). Both of us joined in 2011. In addition, the former student Fernando Iglesias participated again and former student Viktor Gal stayed around to work on Shogun during GSoC (and did some massive infrastructure improvements). These are very nice long term effects of continuous GSoC participation. Thanks to GSoC, Shogun is growing constantly both in terms of code and developers. As in 2012, we eventually could give away 8 slots to some very talented students. All of them did an awesome job on some highly involved projects covering a large number of topics. Two projects were extensions of previous ones: Roman Votjakov extended last year’s project on the popular Gaussian Processes for handling classification problems and Shell Hu implemented a collection of algorithms within last year’s structured output framework (for example for OCR) Fernando Iglesias implemented a new algorithm called metric learning, which plays well together with existing methods in Shogun. Another new algorithm came from Soumyajit De, who has implemented an estimation method for log-determinants of large sparse matrices (needed for example for large-scale Gaussian distributions), and implemented a framework for linear operators and solvers, and fundamentals of an upcoming framework for distributed computing (which is used by his algorithm) on the fly. Evangelos Anagnostopoulos worked on feature hashing and random kitchen sinks, two very cool tricks to speed up linear and kernel-based learning methods in Shogun. Kevin Hughes implemented methods for independent component analysis, which can be used to separate mixtures of signals (for example audio, heart-beats, or images) and are well known in the community. Last but not least, Liu Zhengyang created a pretty web-framework for running Shogun demos from the web browser and did add support for directly loading data from the mldata website. Evgeniy Andreev improved Shogun’s usability via integrating native support for various popular file formats such as CSV and protobuf. You might have noticed the links in the above text (and images). Most of them are the final reports of the students in the form of IPython notebooks, an awesome new open-source tool that we started using for documentation. We are very proud of these. See http://shogun-toolbox.org/page/documentation/notebook/ for a list of all notebooks. Also check out the web-demo framework at http:// www.shogun-toolbox.org/page/documentation/demo/ if you haven’t yet. IPython also features Shogun in the cloud: Former student Viktor Gal did setup http://cloud.shogun-toolbox.org which is an IPython notebook server ran by us. It allows you to play with Shogun-python from any web-browser without having to install it. You can try the existing notebooks or write your own. Give it a shot and let us know what you think! This year’s GSoC also was the most productive one for us ever. We got more than 2000 commits changing almost 400000 lines in more than 7000 files since our last release before GSoC. Students! You all did a great job and we are more than amazed what you all have achieved. Thank you very much and we hope some of you will stick around. Besides all the above individual projects, we encouraged students to work together a bit more to enable synergistic effects. One way we tried to implement this was through a peer review where we paired students to check each others interface documentation and final notebooks. We held the usual meetings with both mentors and students every few weeks to monitor progress and happiness, as well as asking students to write weekly reports. Keeping our IRC channel active every day also helped a lot in keeping things going. My personal experience with mentoring was very positive. It is very nice to give back to the community. I tried to give them the same useful guidance that I received back then, and probably learned as much as my students did on the way. Having participated in GSoC 2011 and 2012, the change of perspective as a mentor was interesting, in particular regarding the selection process. Time wise, I think Google’s official statement of 5 hours per student per week is underestimating things quite a bit (if you want to get things done), and of course there is no upper bound on time you can spend. Our plan of pairing external mentors with internal developers worked smoothly. As most of our mentors are scientists who tend to be very busy, it is sometimes hard for them to review all code on their own. Combining big-picture guidance with the in-depth framework knowledge of the paired core developers allowed for more flexibility when allocating mentors for projects. Keep in mind that Shogun is still being organised by only five people (4 former students) plus a hand full of occasional developers, which makes it challenging to supervise 8 projects. Another change this year was that writing unit-tests were mandatory to get code merged, which made the number of unit tests grew from 50 to more than 600. In the past years, we had seen how difficult it is to write tests at the end of projects, or maintain untested code. Making students do this on-the-fly drastically increased the stability of their code. A challenging side-effect of this was that many bugs within Shogun were discovered (and eventually fixed) which kept students and developers busy. As for Shogun itself, GSoC also boosts our community of users, which became so active this year that decided to organise a the first Shogun workshop in Berlin this summer. We had something over 30 participants from all over the world. The Shogun core team also met for the first time in real life, which was nice! We had a collection of talks, discussions, and hands-on sessions. Click here and here for videos and slides. Sören hang out with the people he knew from previous years and the cool Debian guys (for which he is a developer too). After the summit, the Shogun mentor team went hiking in the south Californian desert – I even climbed a rock. What a great summer! Shogun Workshop 2013 Last weekend, our Shogun workshop finally took place in Berlin. It was really cool to meet all those guys in person. We have been working together for quite some time now. The core-team an Shogun’s supporters are absolutely awesome. It is great to be part of that. We had a nice afternoon at c-base (who were so friendly to host us) with some talks by all of our developers, followed by two days of hands-on workshop at the TU-Berlin. I gave a little talk on two random things you can do with kernels (that are completely unrelated): Gaussian Processes and the kernel MMD. Slides are (download). I also wrote some IPython notebooks for GP-regression (link), GP-probit-classification (link), and two-sample testing with the kernel MMD (link). One of the results of our discussions was that we will start using those notebook for Shogun’s documentation as they allow to combined code, plots, and maths in a web-based viewer. Finally, here are some picture of us, (pretty nerdy) GSoC 2013 Shogun got accepted in the Google Summer of Code 2013! To read my blog about the GSoC, click here. Check out our ideas page. This year, I will be a mentor rather than a student and I am very excited about this. I’ll be offering two projects: • Implement Gaussian process classification (joint with Oliver Stegle). This is an extension of the GSoC project last year and should be quite interested while not being too complicated (link) • Implement unbiased estimators of likelihoods of very large, sparse Gaussian distributions (joint with Erlend Aune and Daniel Simpson). This one is quite challenging since it involved many different topics. However, it should also be very interesting (link) Shogun is in the GSoC 2013 Shogun got accepted in the Google Summer of Code 2013! Check out our ideas page. This year, I will be a mentor rather than a student and I am very excited about this. I’ll be offering two projects: • Implement Gaussian process classification (joint with Oliver Stegle). This is an extension of the GSoC project last year and should be quite interested while not being too complicated (link) • Implement unbiased estimators of likelihoods of very large, sparse Gaussian distributions (joint with Erlend Aune and Daniel Simpson). This one is quite challenging since it involved many different topics. However, it should also be very interesting (link) Shogun 2.1 is out! We released SHOGUN 2.1. See the announcement (link). The release features my recent work on kernel selection for MMD-based kernel two-sample testing and a streaming based implementation for this. See blog-entry. We also added a new unit-testing framework, of which I am very excited since we finally get a mechanism to detect code errors. We also got yet another interface language (perl). Very cool stuff and lots of work/blood/sweat/fun with the other guys. Check it out! Next thing to come here is a workshop on machine learning with SHOGUN on July 12 in the C-Base in Berlin. Stay tuned! SHOGUN – A large scale machine learning toolbox To read my blog about SHOGUN development, click here. SHOGUN (website) is a machine learning toolbox with focus is on large scale kernel methods and especially on Support Vector Machines. It provides a generic SVM interface for several different SVM state-of-the-art implementations Each of the SVMs can be combined with a variety of kernels. The toolbox provides efficient implementations of many common kernels. Also many other popular machine learning algorithms are implemented and the list is continuously extended for example due to the support of the Google Summer of Code. For example, there are now Gaussian processes, many dimensionality reduction methods, Structured Output and latent SVMs, various multi-task learning techniques, and many more. SHOGUN is implemented in C++ and comes with interfaces to many languages. I got into the team after the GSoC 2011 and since then have implemented some new features: A framework for cross-validation and model selection during the GSoC 2011 and a framework for kernel based statistical hypothesis testing in the GSoC 2012. I also worked on migrating serialized SHOGUN objects from different versions to one another. Nice blog entry about SHOGUN’s GSoC 2012 Sören wrote a nice summarising blog post on the GSoC 2012. See here. Streaming Features for Linear Time MMD I finally finished an important and very cool extension to my GSoC 2012 project – making the linear time MMD statistic work with streaming based data. In particular, SHOGUN’s streaming framework is now used. By design, the linear time MMD statistic, given as is very well suited for streaming based data since only four examples have to be hold in memory at once. Once, the sum in the h-statistic is computed, used data can be “forgotten”. As I described in my M.Sc. thesis (link), this allows to process infinite amounts of data and therefore results in possibly more accurate two-sample tests. This holds in particular in cases where the amount of data needed to solve problems is larger than computer memory. During the GSoC, I implemented the linear time MMD on the base of SHOGUN’s standard features interface, which made it necessary to hold data in memory. With the latest modifications (link to patch), the class for the linear time MMD (class reference), now accepts streaming features (class reference) only. This allows to process arbitrarily large amounts of data in a very comfortable way. In order to not suffer from overhead while streaming examples one by one, a block size may be specified: this number of examples is processed at once and should be chosen as large as fits into memory. Recall the linear time MMD’s distribution is normal and its variance can easily estimated by using the empirical variance of the individual h-statistics (while the MMD is their mean) when the number of samples is large enough. The new implementation in SHOGUN does this on the fly using D. Knuth’s online variance algorithm [1] (implementation link). Therefore, a complete two-sample test is now possible in linear time and constant space. A nice illustration of the advantages of this approach can be found in the examples for the linear time MMD (link). A data generator for artificial data which implements SHOGUN’s streaming interface is passed to the MMD class. It produces data from the underlying distribution on the fly. [1] Donald E. Knuth (1998). The Art of Computer Programming, volume 2: Seminumerical Algorithms, 3rd edn., p. 232. Boston: Addison-Wesley.	{"url":"http://herrstrathmann.de/tag/shogun/page/2/","timestamp":"2024-11-02T15:22:01Z","content_type":"text/html","content_length":"78109","record_id":"<urn:uuid:b608ae8d-63cc-477b-8d32-f78c8f6c8288>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00255.warc.gz"}
What is the equation of the line tangent to f(x)=(x^3 - 1) / x at x=1? \| HIX Tutor What is the equation of the line tangent to #f(x)=(x^3 - 1) / x# at #x=1#? Answer 1 At #x=1#, we have #y = f(1) = 0# So the tangent line contains the point #(1,0)# #f'(x) = ((3x^2)(x) - (x^3-1)(1))/x^2 = (2x^2 + 1)/x^2# At #x=1#, the slope of the tangent is #f'(1) = 3#. The equation of the tangent line sought is #y=3x-3# Sign up to view the whole answer By signing up, you agree to our Terms of Service and Privacy Policy Answer from HIX Tutor When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some Not the question you need? HIX Tutor Solve ANY homework problem with a smart AI • 98% accuracy study help • Covers math, physics, chemistry, biology, and more • Step-by-step, in-depth guides • Readily available 24/7	{"url":"https://tutor.hix.ai/question/what-is-the-equation-of-the-line-tangent-to-f-x-x-3-1-x-at-x-1-8f9af9d1b4","timestamp":"2024-11-01T23:02:27Z","content_type":"text/html","content_length":"568395","record_id":"<urn:uuid:31afc5f3-2610-45af-86da-107fd420d078>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00174.warc.gz"}
math question I am having my students write a Basic program to input the length and width of a rectangle, then calculate and display the area and perimeter of the rectangle. I noticed that, for a certain number pair, area = perimeter. I tried a few other pairs of length/width and couldn't find a repeat of that relationship. If x is length and y is width, we have: xy = 2x + 2y How can I determine if there are other number pairs that satisfy this condition? 04-24-2012, 08:22 PM Use algebra to transform your equation into y=2x/(x-2). Now for any x you can find the corresponding y that makes the perimeter equal to the area. There are an infinite number of solutions. If you're only interested in integer solutions, there are only two unique solutions, considering x and y to be interchangeable. Edited: 24 Apr 2012, 8:29 p.m. 04-24-2012, 08:40 PM Or use Dario Alpern's Solver and fill in the blanks with 0, -1, 0, 2, 2 ,0, then solve it :-) Positive integer solutions: a) x = 0, y = 0; b) x = 3, y = 6; c) x = 4, y = 4. P.S.: I assume you haven't counted the trivial one. Edited: 24 Apr 2012, 8:44 p.m. 04-24-2012, 08:45 PM Be sure to tell your students that equating area and perimeter is, formally speaking, nonsense: they're quantities with different units. As a relationship between pure numbers, there's no such objection, of course. 04-24-2012, 08:47 PM Rewrite as From here you can easily find all solutions. Assuming you are after positive integer solutions, there are only two: (4,4) and (3,6) (and (6,3), I guess). These come from factoring 4=2x2=1x4. 04-24-2012, 08:58 PM Right. I assumed that a figure with sides of length zero was not a rectangle. 04-24-2012, 09:14 PM I tend to see the Euclidean point as any of the infinite n-zero-lenght-side regular polygons, a square with null sides being one of them. This doesn't mean I am right, of course! Edited: 24 Apr 2012, 9:17 p.m. 04-25-2012, 05:29 AM Well done Eduardo ! Very nice 04-25-2012, 05:55 AM Yes, I see that. Very clever! 04-25-2012, 06:20 AM You're pulling our respective legs, aren't you? ;-) 04-25-2012, 07:04 AM No, this actually happened. I'm giving a test to my students to write a Basic program to calculate area and perimeter of a rectangle from inputs of length and width, so I wrote my little program first and then tested it and I happened to enter length/width 3 and 6 and I noticed that area and perimeter were the same. I wondered if any other number pair would result in that situation, but I couldn't create a system of equations with xy = 2x + 2y, so I posted here for help, and I knew someone more clever than I am would figure it out. And several did! 04-25-2012, 11:58 AM Is a rectangle if you lim x->0 & y > 0 :D 04-25-2012, 12:16 PM I have a related question which bogs me for some time now. I try to explain it with a small ASCII picture: \| \| \|___ \| \| \| \| \| \|___ \| \| \| \| \|_ \| \| \|_ \| \| \|_ \| \| \|_ \| \| \|_ \| The zig-zag line from the top left corner to the bottom right corner is an approximation to the diagonal but its length is just twice the side length. You can refine it ad infinitum and it will become indistinguishable from the diagonal but the total length is constant in each iteration. Strange, isn't it? 04-25-2012, 12:21 PM Fractals at their simple best. 04-25-2012, 03:09 PM As a student I had to do some wicked (as I thought, but what do I konw <g>) presentation of some topics on Linear Algebra, this one reminds me of it. I think a "Koch-Kurve" might deliver some hints. 04-25-2012, 04:33 PM ... it's interesting that for a VERY long rectangle the short side will approach the limit y=2 if xy=2(x+y). It is trivial, but I wasn't aware of that. 04-25-2012, 06:28 PM HP50G Koch animation Done with an HP50 with a "virtual screen" Edited: 25 Apr 2012, 6:48 p.m. 04-25-2012, 08:47 PM Quote: Strange, isn't it? What makes it strange is that school maths drums into you that a piece of paper only has two dimensions, X & Y. And that any points or lines drawn on that paper can always be represented by X,Y coordinates. Yet diagonal lines, as you point out, 'break the rules' of two dimensions because they have a length that is less than the sum of the steps along the X,Y dimensions, no matter how small those steps are. I'm not sure that fractals are the whole answer. Sure, the stepped line from corner to corner has a fractal dimension. But the diagonal is straight and therefore not fractal. It's as if a 2-dimensional plane has 3 degrees of freedom - left-right only, up-down only and combined. Perhaps we should confuse the world by calling that 3 dimensional ;-) 04-25-2012, 09:26 PM Hi Marcus, Great question! Vectors may be helpful in resolving the paradox. 1. Convert all line segments into vectors by inserting arrows. 2. For each pair of horizontal and vertical vectors (i.e., for each zig and zag along the path) draw diagonal resultant vectors, forming little triangles. 3. For each little triangle, the difference between the distance along the zig-zag (the sum of 2 sides) and the displacement (the length of the diagonal) is the triangle error. 4. The total error is the sum of all triangle errors along the path. The total error is the same whether there are few zig-zags (a few large errors) or many little zig-zags (many little errors). This can be seen by adding all the horizontal arrows and, separately, adding all the vertical arrows. The vector sums are constant no matter the size of the zig-zags (if they weren't constant, some paths would undershoot or overshoot the destination). Paths of many small zig-zags look smoother and are better approximations to the diagonal in a least squares sense (distance of zig-zag path to diagonal path). However the total error in approximating the length of the diagonal is the same for paths of many large zig-zags or many small zig-zags (few big errors vs many little errors). A hallmark of a fractal (self-similar, irregular) curve (e.g., a coastline) is that its curve length is not a definite value but depends on the scale used in measuring it. In this problem: (1) the zig-zag curves are not self-similar (i.e., when you zoom in on the coarse zig-zag curve you do not find little zig-zag curves) and, (2) the length of the zig-zag curves is definite and not dependent on the scale used in measuring them. Hence, this does not appear to be a fractal problem. Edited: 26 Apr 2012, 9:08 a.m. 04-25-2012, 09:35 PM Don- what an interesting observation and excercise!! Would your students be open to a discussion about units? e.g. although their magnitude may be equal sometimes, they can check their formulas by thinking of each side of the equation's geometrically. Although a pure maths person may say the answer is y=2x/(x-2) leaving y alone as a unit-less number). An engineer might look at the problem statement and, by following the units, suggest the perimeter (distance) is never equal area (distance^2) because the units don't match. :) I realize it's a small point of order, but it's a great skill for the students to have later in life! 04-25-2012, 11:38 PM Thanks Allen. I just happened to notice that the length/width pair 3/6 yields the same quantity for both area and perimeter, and I wondered if there was a different pair like that, and it turns out there is (4/4). I wanted to build and solve a system of equations but it was not possible in this case, and I really like what Eduardo did with factoring to solve the problem. That's the beauty of the forum. When I teach math (I'm teaching "technology" this year) and we discuss area and linear measures like perimeter, of course I do distinguish between inches and square inches (or feet). I generally bring in my tape measure and we measure the distance along the four walls (perimeter), and they always understand that well. Then for area I'll bring in some 1 foot by 1 foot floor tiles and start to lay them on the floor so they can see what area and square feet really mean. Kids always appreciate real physical stuff, and they tend to understand concepts like square units better when you show them the real thing. But until they get out in the real world and have to do things like buy carpet for their living room, they tend to forget what they've learned about square units, so we teach it again the next year. Repitition helps the learning process. 04-26-2012, 08:03 AM Quote: You can refine it ad infinitum and it will become indistinguishable from the diagonal. Not really. If it seems indistinguishable, you just need a bigger magnifying glass. I zoomed in on your indistinguishable line and it still straddles the diagonal. Problems similar to this can be used to show how one's intuitive sense of math breaks down when you do "ad infinitim" problems. It's a nice introduction to some of the theories about infinity and 04-26-2012, 11:27 AM Similarly ignoring that you can "prove" Pi = 4	{"url":"https://archived.hpcalc.org/museumforum/showthread.php?mode=linear&tid=218908&pid=0","timestamp":"2024-11-05T10:36:46Z","content_type":"application/xhtml+xml","content_length":"83068","record_id":"<urn:uuid:9dadf9e0-30c3-46d7-bb4b-ff8fedd57a67>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00739.warc.gz"}
normal approximation method requirements This leads to wider intervals for higher confidence levels. This is known as a normal approximation confidence interval. A problem arises when there are a limited number of samples, or draws in the case of data “drawn from a box.” A probability histogram of such a set may not resemble the normal curve, and therefore the normal curve will not accurately represent the expected values of the random variables. The 99% confidence interval will be wider than the 95% confidence interval. However, knowing the true standard deviation of a population is often unrealistic except in cases such as standardized testing, where the entire population is measured. A hypothesis test for a proportion is used when you are comparing one group to a known or hypothesized population proportion value. This can be done using raw data or summarized data. $\begingroup$ One advantage of using the normal is it often gives enough information to quickly tell whether it's even worth calculating the answer more precisely. This is a left-tailed test because we want to know if the proportion is less than 0.80. The process of using this curve to estimate the shape of the binomial distribution is known as normal approximation. Both $np_0 \geq 10$ and $n(1-p_0) \geq 10$ so we can use the normal approximation method. The runs test is a useful tool to determine if a sequence is likely to be random or not. All of the conditions for testing a claim about a population proportion using the normal approximation method are satisfied, so the method can be used. The normal approximation can be used in counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where infinitely divisible and decomposable distributions are involved. Recall that if $np \geq 10$ and $n(1-p) \geq 10$ then the sampling distribution can be approximated by a normal distribution. TODO: binom_test intervals raise an exception in small samples if one. This method of constructing a sampling distribution is known as the normal approximation method. You can change this value by clicking on the distributions. A key point is that calculating [latex]\text{z}[/latex] requires the population mean and the population standard deviation, not the sample mean or sample deviation. $H_{a}\colon p>0.80$, $z=\dfrac{\widehat{p}- p_0 }{\sqrt{\frac{p_0 (1- p_0)}{n}}}$, \ (\widehat{p}=\dfrac{87}{100}=0.87\), $p_{0}=0.80$, $n=100$, $z= \dfrac{\widehat{p}- p_0 }{\sqrt{\frac{p_0 (1- p_0)}{n}}}= \dfrac{0.87-0.80}{\sqrt{\frac{0.80 (1-0.80)}{100}}}=1.75$. Since p is close to ½ (it equals ½! Method â binom_testâ directly inverts the binomial test in scipy.stats. It is important to remember that the LLN only applies (as the name indicates) when a large number of observations are considered. What if we have summarized data and not data in a Minitab Express worksheet? In more complicated cases, normalization may refer to more sophisticated adjustments where the intention is to bring the entire probability distributions of adjusted values into alignment. Of the 522 students in the sample, 273 said that they did have a dog. Asked Jul 19, 2020. The scope of the normal approximation is dependent upon our sample size, becoming more accurate as the sample size grows. To calculate this area, first we compute the area below 8.5 and then subtract the area below 7.5. In a representative sample of 1168 American adults, 747 said they were not financially prepared for retirement. A function of the form Î¦(z )= 1 â 0 .5 e â Az b can be used as an approximation to the standard normal cumulative function. According to the law of large numbers, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed. Both $n p_0$ and $n (1-p_0)$ are at least 10, this assumption has been met. O B. This means we can use the normal approximation method to construct this confidence interval. When Is the Approximation Appropriate? $p_{0}$ = hypothesize population proportion This must be done manually. Find a [latex]\text{Z}[/latex] score for 8.5 using the formula [latex]\text{Z}=\frac { 8.5-5 }{ 1.5811 } =2.21[/latex]. Therefore, de Moivre reasoned that if he could find a mathematical expression for this curve, he would be able to solve problems such as finding the probability of 60 or more heads out of 100 coin flips much more easily. When discussion proportions, we sometimes refer to this as the Rule of Sample Proportions. The results are shown in the following figures: Normal Area 2: This graph shows the area below 7.5. normal approximation: The process of using the normal curve to estimate the shape of the distribution of a data set. The standard score is the number of standard deviations an observation or datum is above the mean. In Minitab Express, the exact method is the default method. The value of the multiplier increases as the confidence level increases. In order to use the normal approximation method, the assumption is that both $n p_0 \geq 10$ and $n (1-p_0) \geq 10$. There were 24 females. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed. The tool of normal approximation allows us to approximate the probabilities of random variables for which we don’t know all of the values, or for a very large range of potential values that would be very difficult and time consuming to calculate. We want to construct a 95% confidence interval for $p$ with a margin of error equal to 4%. Using Minitab Express, we find the probability $P(z\ geq1.75)=0.0400592$ which may be rounded to $p\; value=0.0401$. We want to construct a 95% confidence interval for $p$ with a margin of error equal to 4%. Laplace showed that even if a distribution is not normally distributed, the means of repeated samples from the distribution would be very nearly normal, and that the the larger the sample size, the closer the distribution would be to a normal distribution. A normal distribution has some interesting properties: it has a bell shape, the mean and median are equal, and 68% of the data falls within 1 standard deviation. X ~ N(20 × ½, 20 × ½ × ½) so X ~ N(10, 5) . In probability theory and statistics, the chi-square distribution (also chi-squared or Ï 2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The importance of the normal curve stems primarily from the fact that the distribution of many natural phenomena are at least approximately normally distributed. If you have data in a Minitab Express worksheet, then you have what we call "raw data." The central limit theorem (CLT) states that, given certain conditions, the mean of a sufficiently large number of independent random variables, each with a well-defined mean and well-defined variance, will be approximately normally distributed. Minitab evaluates the likelihood ratio for all possible values of X = (0, 1,â ¦, n) and sums the probabilities for all values for which the LR (y) â ¥ LR (x).. p-value = Î£ P{X = y \| â ¦ This same distribution had been discovered by Laplace in 1778—when he derived the extremely important central limit theorem. In Spring 2016, a sample of 522 World Campus students were surveyed and asked if they own a dog. Some types of normalization involve only a rescaling, to arrive at values relative to some size variable. $99\%\;C.I. The question then is, “What is the probability of getting a value exactly 1.8973 standard deviations above the mean?” You may be surprised to learn that the answer is 0 (the probability of any one specific point is 0). Abraham de Moivre, an 18th century statistician and consultant to gamblers, was often called upon to make these lengthy computations. This is in contrast to "summarized data" which you'll see on the next page. Nevertheless, there are several methods which provide an approximation of the integral by numerical methods: Taylor series, asymptotic series, continual fractions, and some other more. Normal Area 1: This graph shows the area below 8.5. Where \(p_0$ is the hypothesized population proportion that you are comparing your sample to. The binomial distribution has a mean of [latex]\mu = \text{Np} = 10\cdot 0.5 = 5[/latex] and a variance of [latex]\sigma^2 = \text{Np}(1-\text{p}) = 10 \cdot 0.5\cdot 0.5 = 2.5[/latex]. Before we can conduct our hypothesis test we must check this assumption to determine if the normal approximation method or exact method should be used. That sampling distribution will have a mean of $p_0$ and a standard deviation (i.e., standard error) of $\sqrt{\frac{p_0 (1-p_0)}{n}}$, Recall that the standard normal distribution is also known as the z distribution. Note that the default method for constructing the sampling distribution in Minitab Express is to use the exact method. There is evidence that the proportion of women in the population who think they are overweight is less than 40%. Normal Distribution and Scales: Compares the various grading methods in a normal distribution. 1. where p = proportion of interest 2. n = sample size 3. $H_{0}\colon p=0.80$ which has discrete steps. is approximately distributed as a normal distribution with a mean of 0 and a standard deviation of 1, N(0,1). Question 1. 1. Testing the Normal Approximation and Minimal Sample Size Requirements of Weighted Kappa When the Number of Categories is Large Domenic V. Cicchetti Applied Psychological Measurement 1981 5 : â ¦ Normal Approximation: The normal approximation to the binomial distribution for 12 coin flips. A key point is that calculating [latex]\text{z}[/latex] requires the population mean and the population standard deviation, not the sample mean or sample deviation. This course does not cover the exact method in detail, but you will see how these tests may be performed using Minitab Express. We can use these pieces to determine a minimum sample size needed to produce these results by using algebra to solve for $n$: $M$ is the margin of error Can a test about a population proportion using the normal approximation method be used? Before we can conduct our hypothesis test we must check this assumption to determine if the normal approximation method or exact method should be used. From the plot below, we see that the $z^$ multiplier for a 99% confidence interval is 2.576. Let's construct a 95% confidence interval to estimate the proportion of all American adults who are not financially prepared for retirement. Normal Approximation Method Power may be calculated for one-sample proportions tests using the normal approximation to the binomial distribution. Translate the problem into a probability statement about X. In variants, convergence of the mean to the normal distribution also occurs for non-identical distributions, given that they comply with certain conditions. Here is another example: To create a frequency table of dog ownership in Minitab Express: This should result in the following frequency table: Select your operating system below to see a step-by-step guide for this example. did not choose Normal Approximation as the method)? A failure is defined as answering "no." If one only has a sample set, then the analogous computation with sample mean and sample standard deviation yields the Student’s [latex]\text{t}[/latex]-statistic. September 17, 2013. According to the Center for Disease Control (CDC), the percent of adults 20 years of age and over in the United States who are overweight is 69.0% (see http://www.cdc.gov/nchs/fastats/obesity-overweight.htm). To perform a one sample proportion z test with summarized data in Minitab Express: $p \leq \alpha$, reject the null hypothesis. September 17, 2013. $H_{0}\colon p=0.80$ (adsbygoogle = window.adsbygoogle \|\| []).push({}); The process of using the normal curve to estimate the shape of the binomial distribution is known as normal approximation. The $z^$ multiplier for a 95% confidence interval is 1.960. As shown on the probability distribution plot below, the multiplier associated with a 95% confidence interval is 1.960, often rounded to 2 (recall the Empirical Rule and 95% Rule). A. Check rules of thumb using n = 3,500,000 and p = 1/6. Now, we have an estimate to include in the formula: $n=\left ( \frac{1.960}{0.04} \right )^2 (0.25)(1-0.25)=450.188$. The standard score is a dimensionless quantity obtained by subtracting the population mean from an individual raw score and then dividing the difference by the population standard deviation. :\;0.640\pm 2.576 (0.014)=0.0640\pm 0.036=[0.604, \; 0.676]\). When conducting a hypothesis test, we check this assumption using the hypothesized proportion (i.e., the proportion in the null hypothesis). This is exactly what he did, and the curve he discovered is now called the normal curve. Verify whether n is large enough to use the normal approximation by checking the two appropriate conditions.. For the above coin-flipping question, the conditions are met because n â p = 100 â 0.50 = 50, and n â (1 â p) = 100 â (1 â 0.50) = 50, both of which are at least 10.So go ahead with the normal approximation. In cases where it is impossible to measure every member of a population, a random sample may be used. The equation for the Normal Approximation for the Binomial CI is shown below. Clearly, the normal approximation to the binomial is a much better method. }{ \text{x}!\left( \text{N}-\text{x} \right) ! } Central Limit Theorem: A distribution being “smoothed out” by summation, showing original density of distribution and three subsequent summations. Normal Approximation 3 of6 0 5 10 15 20 25 30 0.00 0.05 0.10 0.15 Normal Approx to Binom: n=20, p=0.5 x binomial dist P(x) normal approx f(x) Thus, if np 5 and nq 5 we can use the normal distribution to approxi-mately describe a binomial random variable. David Lane, Normal Approximation to the Binomial. In other words, the scope of the normal approximation is dependent upon our sample size, becoming more accurate as the sample size grows. Check assumptions and write hypotheses, 3. Thus, this is known as a "single sample proportion z test" or "one sample proportion z test.". The next two pages will show you how to use Minitab Express to conduct this analysis using either raw data or summarized data. Research Question: Are more than 80% of American's right handed? The following example uses a scenario in which we want to know if the proportion of college women who think they are overweight is less than 40%. $n$ = sample size. Note: Because the normal approximation is not accurate for small values of n, a good rule of thumb is to use the normal approximation only if np>10 and np(1-p)>10. In a sample of 100 Americans, 87 were right handed. This means we get started with a set level of confidence and margin of error. This section provides the power calculation formulas for the various test statistics available in this procedure. In order to consider a normal distribution or normal approximation, a standard scale or standard units is necessary. With the classical 30 degrees of freedom the visualization shows that p-value from the normal approximation (0.05) is really close to the p-value from the t-distribution (0.055). The binomial distribution has a mean of [latex]\mu = \text{Np} = 10\cdot 0.5 = 5[/latex] and a variance of [latex]\sigma^2 = \text{Np}(1-\text{p}) = 10 \cdot 0.5\cdot 0.5 = 2.5[/latex]; therefore a standard deviation of 1.5811. This is exactly what he did, and the curve he discovered is now called the normal curve. The normal distribution is a good approximation to the binomial when n is sufficiency large and p is not too close to 0 or 1. The hypothesized value of the population proportion is symbolized by $p_0$ because this is the value in the null hypothesis ($H_0$). Most statistical procedures for testing differences between means assume normal distributions. As probability and statistical theory show us, as the number of samples increase for the given mean and standard deviation, the more closely the sample probability distribution will â ¦ Compute a 95% confidence interval to estimate the proportion of all African American adults who have some level of lactose intolerance. The following is an example on how to compute a normal approximation for a binomial distribution. The normal approximation p-value for the three alternative hypotheses uses a â ¦ This is a non-directional (i.e., two-tailed) test, so we need to find the area under the z distribution that is more extreme than $z=-0.980$. Because there is no estimate of the proportion given, we use $\tilde{p}=0.50$ for a conservative estimate. Here, for the sake of ease, we have used an online normal area calculator. giving us an approximation for the variance of our estimator. Research Question: Is the percentage of Creamery customers who prefer chocolate ice cream over vanilla less than 80%? The sample observations are not a random sample, so a test about a population proportion using the normal approximating method cannot be used. David Lane, History of Normal Distribution. In this example a success is defined as answering "yes" to the question "do you own a dog?" Normal distribution integral has no analytical solution. Note that p-values are also symbolized by $p$. In order to construct a 95% confidence interval with a margin of error of 4%, we should obtain a sample of at least $n=601$. Are any of the three requirements violated? k 1.5 Example: Approximate Mean and Variance Suppose X is a random variable with EX = 6= 0. One sample proportion tests and confidence intervals are covered in Section 6.1 of the Lock5 textbook. In the example below, we want to know if there is evidence that the proportion of students who are male is different from 0.50. Requirements for using normal approximation to binomial. David Lane, History of Normal Distribution. On the following pages you will see how a confidence interval for a population proportion can be constructed by hand using the normal approximation method. $SE=\sqrt{\frac{\hat{p} (1-\hat{p})}{n}}=\sqrt{\ frac{0.640 (1-0.640)}{1168}}=0.014$, The $z^$ multiplier for a 95% confidence interval is 1.960, The formula for a confidence interval for a proportion is $\widehat{p}\pm z^ (SE)$, $0.640\pm 1.960(0.014)=0.640\pm0.028=[0.612, \;0.668]$. $\tilde p$ is an estimated value of the proportion. It requires knowing the population parameters, not the statistics of a sample drawn from the population of interest. By using regression analysis and after rounding the coefficient to one decimal place, the approximation obtained is () 1 .2 1 .3 5 1 0 .5 Î¦ z = â e â z. Research question: Is this city’s proportion of overweight individuals different from 0.690? The problem is that the binomial distribution is a discrete probablility distribution whereas the normal distribultion is a continuous distribution. $np_0 = 226(0.50)=113$ and $n(1-p_0) = 226(1-0.50)=113$. The z test statistic tells us how far our sample proportion is from the hypothesized population proportion in standard error units. We can use the normal approximation method. This is a right-tailed test because we want to know if the proportion is greater than 0.80. central limit theorem : The theorem that states: If the sum of independent identically distributed random variables has a finite variance, then it will be (approximately) normally distributed. where [latex]\text {x}[/latex] is the number of heads (60), [latex]\text{N}[/latex] is the number of flips (100), and [latex]\pi[/latex] is the probability of a head (0.5). The binomial distribution can be used to solve problems such as, “If a fair coin is flipped 100 times, what is the probability of getting 60 or more heads?” The probability of exactly [latex]\text{x}[/latex] heads out of [latex]\text{N}[/ latex] flips is computed using the formula: [latex]\displaystyle \text{P}\left( \text{x} \right) =\frac { \text{N}! If you do not have each individual observation, but rather have the sample size and number of successes in the sample, then you have summarized data. If we are conducting a one-tailed (i.e., right- or left-tailed) test, we look up the area of the sampling distribution that is beyond our test statistic. Normalization can also refer to the creation of shifted and scaled versions of statistics, where the intention is that these normalized values allow the comparison of corresponding normalized values for different datasets. np = 583,333.333 >> 10 CHECK! Sum of many independent 0/1 components with probabilities equal p (with n large enough such that npq â ¥ 3), then the binomial number of success in n trials can be approximated by the Normal distribution with mean µ = np and standard deviation q np(1â p). One of the conditions for a binomial distribution are not satisfied, so a test about a population proportion using the normal approximating method cannot be used. Find a [latex]\text{Z}[/latex] score for 7.5 using the formula [latex]\text{Z}=\frac { 7.5-5 }{ 1.5811 } =1.5811[/latex]. Recall, the z distribution is a normal distribution with a mean of 0 and standard deviation of 1. We are 99% confidence that between 60.4% and 67.6% of all American adults are not financially prepared for retirement. Because the distribution of means is very close to normal, these tests work well even if the distribution itself is only roughly normal. Minitab Express will not check assumptions for you. a. There is no principle that a small number of observations will coincide with the expected value or that a streak of one value will immediately be “balanced” by the others. The Central Limit Theorem states that if the sample size is sufficiently large then the sampling distribution will be approximately normally distributed for many frequently tested statistics, such as those that we have been working with in this course. When we're constructing confidence intervals $p$ is typically unknown, in which case we use $\widehat{p}$ as an estimate of $p$. The central limit theorem has a number of variants. This means that our sample needs to have at least 10 "successes" and at least 10 "failures" in order to construct a confidence interval using the normal approximation method. You first learned how to construct a frequency table in Lesson 2.1.1.2.1 of these online notes. This is known as the exact method. Because both $n \ widehat p \geq 10$ and $n(1- \widehat p) \geq 10$, the normal approximation method may be used. If the assumptions for the normal approximation method are not met (i.e., if $np$ or $n(1-p)$ is not at least 10), then the sampling distribution may be approximated using a binomial distribution. According to the Rule of Sample Proportions, if $np\geq 10$ and $n(1-p) \geq 10$ then the sampling distributing will be approximately normal. We are 95% confident that between 61.2% and 66.8% of all American adults are not financially prepared for retirement. What if we knew that the population proportion was around 0.25? From the Minitab Express output, $z$ = -1.86, From the Minitab Express output, $p$ = 0.0625, $p > \alpha$, fail to reject the null hypothesis. Hamlet Killing Claudius Quotes, Vector Fonts Generator, Bedroom Carpet And Paint Ideas, Andhra And Telangana Cuisine, Do Mountain Goats Attack Humans, Lumix Gf10 Review,	{"url":"https://sancakpalas.com/k7xe71n/l7qa1yjo/haztyv.php?a606dc=normal-approximation-method-requirements","timestamp":"2024-11-11T07:43:01Z","content_type":"text/html","content_length":"87951","record_id":"<urn:uuid:06966f1c-ac24-4a57-add7-6113150acbda>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00289.warc.gz"}
Question ID - 154488 \| SaraNextGen Top Answer A current of 0.2 ampere is passing through a resistance of 20 ohm. The voltage applied at the ends of resistance (a) 40 volts (b) (c) (d) A current of 0.2 ampere is passing through a resistance of 20 ohm. The voltage applied at the ends of resistance is	{"url":"https://www.saranextgen.com/homeworkhelp/doubts.php?id=154488","timestamp":"2024-11-04T18:37:58Z","content_type":"text/html","content_length":"16499","record_id":"<urn:uuid:55bdc605-3879-47b2-97e6-ac1d4a3a6351>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00531.warc.gz"}
Re: [tor-bugs] #32895 [Applications/Tor Browser]: Improve marsigning_check.sh script to deal better with non-reproducible, signed macOS mar files #32895: Improve marsigning_check.sh script to deal better with non-reproducible, signed macOS mar files Reporter: gk \| Owner: gk Type: defect \| Status: \| assigned Priority: Medium \| Milestone: Component: Applications/Tor Browser \| Version: Severity: Normal \| Resolution: Keywords: tbb-sign, tbb-maint, \| Actual Points: GeorgKoppen202005 \| Parent ID: \| Points: Reviewer: \| Sponsor: Changes (by gk): * status: new => assigned * keywords: tbb-sign => tbb-sign, tbb-maint, GeorgKoppen202005 * owner: tbb-team => gk Old description: > Our current mar-signing check script does two things: > 1) It checks whether the SHA-256 sum from the signed .mar file is the > same one as from the unsigned one and returns an error if so. > 2) It strips the signature and compares the SHA-256 sum of the resulting > .mar file with the unsigned one. > Step 2) essentially tries to do 2 checks in one: a) that there is a > proper signature that can get stripped and b) that the resulting .mar > file is the same as the unsigned one. That's cool in theory as we want to > have both checks but it has a number of issues in practice. The most > important ones are: > i) The script fails the mar-signing check for macOS as the stripping the > signatures from those files does not give us the unsigned .mar due to the > content signing > ii) It's not clear we signed actually with the right key (although that > is in practice not much of an issue) or whether the signature verifies > later on (which is actually what we want to know). New description: Our current mar-signing check script does two things: 1) It checks whether the SHA-256 sum from the signed .mar file is the same one as from the unsigned one and returns an error if so. 2) It strips the signature and compares the SHA-256 sum of the resulting .mar file with the unsigned one. Step 2) essentially tries to do 2 checks in one: a) that there is a proper signature that can get stripped and b) that the resulting .mar file is the same as the unsigned one. That's cool in theory as we want to have both checks but it has a number of issues in practice. The most important ones i) The script fails the mar-signing check for macOS as stripping the signatures from those files does not give us the unsigned .mar yet due to the content signing. (see: #20254) ii) It's not clear we signed actually with the right key (although that is in practice not much of an issue) or whether the signature verifies later on (which is actually what we want to know). Ticket URL: <https://trac.torproject.org/projects/tor/ticket/32895#comment:2> Tor Bug Tracker & Wiki <https://trac.torproject.org/> The Tor Project: anonymity online tor-bugs mailing list	{"url":"https://www.mail-archive.com/tor-bugs@lists.torproject.org/msg215393.html","timestamp":"2024-11-09T09:47:08Z","content_type":"text/html","content_length":"11757","record_id":"<urn:uuid:b2d99c18-54cc-4b05-a1cc-2aedfcc4ffd3>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00729.warc.gz"}
Translating Sentences and Define a Process for Problem Solving Learning Objectives • Define a process for problem solving □ Translate words into algebraic expressions and equations □ Define a process for solving word problems Word problems can be tricky. Often it takes a bit of practice to convert an English sentence into a mathematical sentence, which is one of the first steps to solving word problems. In the table below, words or phrases commonly associated with mathematical operators are categorized. Word problems often contain these or similar words, so it’s good to see what mathematical operators are associated with them. How much will it cost? Addition [latex]+[/latex] Subtraction [latex]-[/latex] Multiplication [latex]\times[/latex] Variable ? Equals [latex]=[/latex] More than Less than Double A number Is Together In the past Product Often, a value for which no information is given. The same as Sum slower than times After how many hours? Total the remainder of In the future difference faster than Some examples follow: • [latex]x\text{ is }5[/latex] becomes [latex]x=5[/latex] • Three more than a number becomes [latex]x+3[/latex] • Four less than a number becomes [latex]x-4[/latex] • Double the cost becomes [latex]2\cdot\text{ cost }[/latex] • Groceries and gas together for the week cost $250 means [latex]\text{ groceries }+\text{ gas }=250[/latex] • The difference of 9 and a number becomes [latex]9-x[/latex]. Notice how 9 is first in the sentence and the expression Let’s practice translating a few more English phrases into algebraic expressions. Translate the table into algebraic expressions: some number the sum of the number and 3 twice the sum of the number and 3 a length double the length double the length, decreased by 6 a cost the difference of the cost and 20 2 times the difference of the cost and 20 some quantity the difference of 5 and the quantity the difference of 5 and the quantity, divided by 2 an amount of time triple the amount of time triple the amount of time, increased by 5 a distance the sum of [latex]-4[/latex] and the distance the sum of [latex]-4[/latex] and the twice the distance Show Solution In this example video, we show how to translate more words into mathematical expressions. The power of algebra is how it can help you model real situations in order to answer questions about them. Here are some steps to translate problem situations into algebraic equations you can solve. Not every word problem fits perfectly into these steps, but they will help you get started. 1. Read and understand the problem. 2. Determine the constants and variables in the problem. 3. Translate words into algebraic expressions and equations. 4. Write an equation to represent the problem. 5. Solve the equation. 6. Check and interpret your answer. Sometimes writing a sentence helps. Twenty-eight less than five times a certain number is 232. What is the number? Show Solution In the video that follows, we show another example of how to translate a sentence into a mathematical expression using a problem solving method. Another type of number problem involves consecutive numbers. Consecutive numbers are numbers that come one after the other, such as 3, 4, 5. If we are looking for several consecutive numbers it is important to first identify what they look like with variables before we set up the equation. For example, let’s say I want to know the next consecutive integer after 4. In mathematical terms, we would add 1 to 4 to get 5. We can generalize this idea as follows: the consecutive integer of any number, x, is [latex]x+1[/latex]. If we continue this pattern we can define any number of consecutive integers from any starting point. The following table shows how to describe four consecutive integers using algebraic notation. First [latex]x[/latex] Second [latex]x+1[/latex] Third [latex]x+2[/latex] Fourth [latex]x+3[/latex] We apply the idea of consecutive integers to solving a word problem in the following example. The sum of three consecutive integers is 93. What are the integers? Show Solution In the following video we show another example of a consecutive integer problem.	{"url":"https://courses.lumenlearning.com/aacc-collegealgebrafoundations/chapter/read-define-a-process-for-problem-solving/","timestamp":"2024-11-11T01:21:21Z","content_type":"text/html","content_length":"56571","record_id":"<urn:uuid:2fb3c7bd-58f2-4798-b4b1-7294603828f7>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00138.warc.gz"}
Convert Minutes to Hours Online To convert minutes into hours you must divide the desired number of minutes by 60, because each hour has 60 minutes. Here are some examples: How many hours are in 120 minutes? 120 / 60 = 2 We soon conclude that 120 minutes equals 2 hours. But not all calculations are that simple, so let's make things more difficult: How many hours are in 165 minutes? The rule remains the same, 165 / 60 equals 2.75, and here we have an error, after all, there is no such thing as two hours and seventy-five minutes. This occurs because the result of the division is in decimals, so it is necessary to convert the decimals to minutes, you can even use our tool to convert decimals into minutes, access it by clicking here. To convert the result in decimal to minutes we do: 0.75 * 60, thus resulting in 45, so we come to the conclusion that 165 minutes correspond to 02:45:00 (two hours and forty-five minutes).	{"url":"https://converteonline.com/convert-minutes-to-hours/","timestamp":"2024-11-02T15:25:47Z","content_type":"text/html","content_length":"24146","record_id":"<urn:uuid:a4572afc-3ef1-47b6-95d7-8e33ccb7e476>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00023.warc.gz"}
How many km in 9.5 m? Contact Us! Please get in touch with us if you: 1. Have any suggestions 2. Have any questions 3. Have found an error/bug 4. Anything else ... To contact us, please click HERE. How many km in 9.5 m? 9.5 meters equals 0.0095 kilometer because 9.5 times 0.001 (the conversion factor) = 0.0095 All In One Unit Converter Meters to kilometers Conversion Formula How to convert 9.5 meters into kilometers To calculate the value in kilometers, you just need to use the following formula: Value in kilometers = value in meters × ^1/[1000] In other words, you need to multiply the capacitance value in meters by ^1/[1000] to obtain the equivalent value in kilometers. For example, to convert 9.5 m to kilometers, you can plug the value of 9.5 into the above formula toget kilometers = 9.5 × ^1/[1000] = 0.0095 Therefore, the capacitance of the capacitor is 0.0095 kilometer. Note that the resulting value may have to be rounded to a practical or standard value, depending on the application. By using this converter, you can get answers to questions such as: • How much are 9.5 meters in kilometers; • How to convert meters into kilometers and • What is the formula to convert from meters to kilometers, among others. Meters to Kilometers Conversion Chart Near 8.9 meters Meters to Kilometers 8.9 meters 0.0089 kilometer 9 meters 0.009 kilometer 9.1 meters 0.0091 kilometer 9.2 meters 0.0092 kilometer 9.3 meters 0.0093 kilometer 9.4 meters 0.0094 kilometer 9.5 meters 0.0095 kilometer 9.6 meters 0.0096 kilometer 9.7 meters 0.0097 kilometer 9.8 meters 0.0098 kilometer 9.9 meters 0.0099 kilometer 10 meters 0.01 kilometer 10.1 meters 0.0101 kilometer Note: Values are rounded to 4 significant figures. Fractions are rounded to the nearest 8th fraction. Definition of Meter The meter is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second. One metre is about 3 3⁄8 inches longer than a yard, i.e. about 39 3⁄8 inches. Here are some common conversions from meters to other length units: 1 meter = 100 centimeters 1 meter = 3.28084 feet 1 meter = 1.09361 yards 1 meter = 0.000621371 miles 1 meter = 39.3701 inches Conversely, to convert from these other units of length to metres, you would use the appropriate conversion factor, either by multiplying or dividing the original quantity by the factor. In summary, the meter is a unit of length in the SI system and is commonly used to measure distance and length in a variety of contexts. Its basis in units of 10 makes it easy to convert to other units of length. Here are some examples of things that measure about one meter (order of magnitude): A typical human arm span A meter stick or yardstick A bicycle frame size A large pizza A three-foot (1 meter) long fish A standard kitchen countertop A medium-sized dog A basketball hoop height A typical pool cue length A standard walking cane A small ladder or step stool Microwaves of 300 GHz have a wavelength of 1 mm Definition of Kilometer A kilometer (km) is a unit of length within the International System of Units (SI), which is a decimal multiple of the meter. The kilometer, a common measure of distances equals to 1000 meters, and equivalent to 3280.8 feet or 0.621 mile. Abbreviation: km Here are more examples of things that measure about one kilometer (order of magnitude): The length of a standard Olympic running track The distance between two subway stations in some cities The distance of a short car trip within a city The distance from the base to the peak of a mountain The distance between two landmarks in a city The length of some organized road races The distance of a typical cycling sprint race The distance covered by an average person during a 10-15 minute walk. Despite efforts to provide accurate information on this website, no guarantee of its accuracy is made. Therefore, the content should not be used for decisions regarding health, finances, or property.	{"url":"https://www.howmany.wiki/u/How-many--km--in--9.5--m","timestamp":"2024-11-06T04:47:18Z","content_type":"text/html","content_length":"117690","record_id":"<urn:uuid:9491479d-f8ca-4af0-96d3-f6d8d38fe26a>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00303.warc.gz"}
Data considerations for To ensure that your results are valid, consider the following guidelines when you collect data, perform the analysis, and interpret your results. The data should include at least 1 random factor. If you do not have any random factors, use Fit General Linear Model. For more information on random factors, go to What is the difference between fixed and random factors?. The response variable should be continuous If the response variable is categorical, your model is less likely to meet the assumptions of the analysis, to accurately describe your data, or to make useful predictions. The sample data should be selected randomly Random samples are used to make generalizations, or inferences, about a population. If your data were not collected randomly, your results might not represent the population. Collect data using best practices To ensure that your results are valid, consider the following guidelines: □ Make certain that the data represent the population of interest. □ Collect enough data to provide the necessary precision. □ Measure variables as accurately and precisely as possible. □ Record the data in the order it is collected. The model should provide a good fit to the data If the model does not fit the data, the results can be misleading. In the output, use the residual plots, the diagnostic statistics for unusual observations, and the model summary statistics to determine how well the model fits the data.	{"url":"https://support.minitab.com/en-us/minitab/help-and-how-to/statistical-modeling/anova/how-to/mixed-effects-model/before-you-start/data-considerations/","timestamp":"2024-11-04T19:01:56Z","content_type":"text/html","content_length":"11338","record_id":"<urn:uuid:e952737e-be1d-4b91-9689-99cfebebe845>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00870.warc.gz"}
Pierre-Francois Marteau • Time series data mining ○ Separation of shape and temporal patterns in time series, C++ and Python wrapper package STS2, (github) ○ elastic Kernelized Averaging of Time Series in C++ TEKA, (github) ○ Regularized Dynamic Time Warping Kernel (KDTW) ☆ KDTW in a nutshell KDTW ☆ KDTW pure python and C implementations with a python biding (github) ☆ The Matlab/Octave code for KDTW is available here ○ Time Warp Edit Distance: TWED ☆ ANSI C source code: twed.c, variant with a non infinite, progressive initialization. ☆ Python implementation: twed.py (github). ☆ Python/Matlab/Octave implementations: wikipedia, thanks to the summer term 2013 TSDM-students at the Intelligent Embedded Systems Department at the University of Kassel, Germany (Matlab ○ Assessing automatic labeling in data stream in C++ (evTSS) • Symbolic sequence data mining • Anomaly detection	{"url":"https://people.irisa.fr/Pierre-Francois.Marteau/code.html","timestamp":"2024-11-06T17:08:45Z","content_type":"application/xhtml+xml","content_length":"4404","record_id":"<urn:uuid:9ccd9681-b0bf-4389-9058-71db7463becf>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00285.warc.gz"}
Modelling and Simulation of Photovoltaic Full Cell Hybrid System Volume 03, Issue 02 (February 2014) Modelling and Simulation of Photovoltaic Full Cell Hybrid System DOI : 10.17577/IJERTV3IS20125 Download Full-Text PDF Cite this Publication Atul Kumar Dewangan, 2014, Modelling and Simulation of Photovoltaic Full Cell Hybrid System, INTERNATIONAL JOURNAL OF ENGINEERING RESEARCH & TECHNOLOGY (IJERT) Volume 03, Issue 02 (February 2014), • Open Access • Total Downloads : 947 • Authors : Atul Kumar Dewangan • Paper ID : IJERTV3IS20125 • Volume & Issue : Volume 03, Issue 02 (February 2014) • Published (First Online): 11-02-2014 • ISSN (Online) : 2278-0181 • Publisher Name : IJERT • License: This work is licensed under a Creative Commons Attribution 4.0 International License Text Only Version Modelling and Simulation of Photovoltaic Full Cell Hybrid System Atul Kumar Dewangan Lecturer in Electrical Engineering Kirodimal Institute of Technology, Raigarh (C. G.) Abstract A hybrid system also called as standalone system supplies electricity to the load without being connected to the electric grid. Hybrid systems have applications in remote and inaccessible areas where the population is living without electricity. In remote and rural areas the grid connection is not technical feasible and also a cost effective option. Therefore, hybrid systems are well suited for such areas. The purpose of this thesis is to model and simulate the different components of a PVFC hybrid system which may fulfill the electric demands for remote and rural areas. Therefore, here a photovoltaic generator and a fuel cell are connected to the load to fulfill the electrical demands to these areas. This hybrid system consists of a photovoltaic generator and a proton exchange membrane fuel cell (PEMFC) coupled together to form a hybrid system which is connected to the load or grid as per the user demand. A simulation software program known as Matlab has been used to simulate the system performance. The system design and performance analysis could thus be achieved through computer modeling and simulation prior to practical realization. Keywords Dynamic model; Fuel cell; Photovoltaic module; Hybrid system 1. INTRODUCTION The non renewable sources of energy such as natural gas, petroleum and coal are being depleted rapidly. Also, they cause global problems such as the green house effect and pollution which are posing great danger for our environment and eventually for the entire life on our planet. On other hand, the renewable energy sources such as solar, wind, tidal, geothermal etc are attracting more attention as an alternative energy. The photovoltaic (PV) energy among the renewable energy sources has been widely used in low power applications. Photovoltaic generator converts solar radiation directly into electricity. Photovoltaic generators have a lot of advantages such as being inexhaustible and pollution free, silent, no rotating parts etc. They are replacing electricity generators by other polluting ways. From an operational point of view, a PV power generation experiences large variations in its output power due to intermittent weather conditions. Those phenomena may cause operational problems at the power station, such as excessive frequency deviations. In many regions of the world, the fluctuating nature of solar radiation means that purely PV power generators for off grid applications must be large and thus expensive. One method to overcome this problem is to integrate the photovoltaic plant with other power sources such as diesel, fuel cell (FC), or battery back- up. The diesel back-up generator for PV power is able to ensure a continuous 24-hour. However, it has a number of significant disadvantages such as noise and exhaust gases pollution. In addition, reasonably reliable diesel back-up generators are available only for the power range above about 5kW, which is too much high for a large number of applications. In the middle and small power range this technology cannot be used in an effective way. The fuel cell beak-up power supply is a very attractive option to be used with an intermittent power generation source like PV power because the fuel cell power system is characterized with many attractive features such as efficiency, fast load- response, modular production and fuel flexibility. Due to the fast responding capability of the fuel cell power system, a photovoltaic-fuel cell (PVFC) hybrid system may be able to solve the photovoltaic inherent problem of intermittent power generation. Unlike a storage battery, which also represents an attractive back-up option, such as fast response, modular construction and flexibility, the fuel cell power can produce electricity for unlimited time to support the PV power generator. Therefore, a continuous supply of high quality power generated from the PVFC hybrid system is possible day and night. Environmental impacts of the fuel cell power generation are relatively small in contrast to other fossil fuel power sources. Since chemical reactions inside the fuel cell stack are accomplished by catalysts, it requires a low sulphur content fuel. Low-emission characteristics of the fuel cell power system may allow some utilities to offset the costs of installing additional emission control equipment. Moreover, their high efficiency results in low fossil fuel CO2 emissions, which will help in reducing the rate of global, warming. Therefore, the fuel cell power system has a great potential for being coordinated with the PV generator to smooth out the photovoltaic powers fluctuations. 1. Objective of Study It has been well-proven that a photovoltaic power source should be integrated with other power sources, whether used in either a stand-alone or grid-connected mode. Stand-alone power systems are very popular, especially in remote sites. The system under study in this dissertation is the modeling and simulation of different components of a PVFC hybrid power system, which is constituted of a photovoltaic generator, a proton exchange membrane (PEM) fuel cell and PCU unit. This system is intended to be a future competitor of hybrid PV/Diesel systems, especially from an environmental point of view (low noise and zero emission) and operational costs point of view [4].The development of appropriate simulation tools will help in dealing with modeling, simulation, and design and energy management of the system understudy. A simulation software program known as Matlab has been used to simulate the system performance. The system design and performance analysis could thus be achieved through computer modeling and simulation prior to practical realization. This dissertation aims towards: Proper data collecting and/or data synthesizing that describes the system operation and the load profile, Visualizing and analyzing the system dynamic behavior using power flow trace overlong-term duration, for example, one year, creating an accurate simulation system model to predict the real performance of the PVFC hybrid system, and then Undertaking detailed analysis of the effect of changes in the system configurations, power conditioning units, and sites to choose an optimal system design. The objective of the study is to reach a design that optimizes the operation of a PVFC hybrid system. All components of this system have been selected for an optimal operation of the complete system. The data of the component models was taken from real projects or manufacturers data sheet. The component models of the system are verified with components experimental data to assure the accuracy of these models before being implemented into the system simulation study. 2. Problem Statement Electricity is extremely versatile, clean, easy to use, and can be turned on or off at the flick of a switch. Electricity has brought enormous social benefits in all areas of life. Currently, many nations already have small-scale solar, wind, and geothermal devices in operation providing energy to urban and rural populations. These types of energy production are especially useful in remote locations because of the excessive cost of transporting electricity from large- scale power plants. A renewable energy system, which we have designed here, is composed of some renewable sources, and targets a small area such as a village. The system supplies energy to rural area by using renewable sources It is the preferred method of supplyng power for many household applications, especially lighting, but connection to the national electrical grid is a rare occurrence in rural areas of the developing and under developed world. In the majority of the worlds' poorer countries it is estimated that significantly less than 5% of the rural population are connected to the national grid. There are many reasons, both technical and economic, which make grid connection unfeasible and these will be looked at briefly in this fact sheet. In urban areas of the developing world grid connection is commonplace. Particularly in remote mountainous areas (such as the Himalayas Region) people often live under extreme conditions. The harsh climate in high-altitudes, limited available natural resources and the remote location of most villages make life challenging. 3. Solution Statement Hybrid energy systems (HES), which utilize different renewable resources such as wind, solar, biomass, small/micro hydro, with fossil fuel powered diesel/petrol generator to provide electric power, are well suited for remote rural areas. This proposed work a general methodological framework for the formulation of an action plan for the small-scale hybrid energy system for remote area. The action plan is formed on the basis of cost effective modeling for remote rural area that is minimization of energy production cost. Among the renewable energy resources, the energy through the photovoltaic (PV) N effect can be considered the most essential and prerequisite sustainable resource because of the ubiquity, abundance, and sustainability of solar radiant energy. Regardless of the intermittency of sunlight, solar energy is widely available and completely free of cost. Recently, photovoltaic array system is likely recognized and widely utilized to the forefront in electric power applications. It can generate direct current electricity without environmental impact and contamination when is exposed to solar radiation. Use of a PVFC hybrid model provides a potential solution for better energy efficiency while reducing the cost of FC power technology. Hybrid systems have applications in remote and inaccessible areas where the population is living without electricity. In remote and rural areas the grid connection is not technical feasible and also a cost effective option. Therefore, hybrid systems are well suited for such areas. With increasing concerns about fossil fuel deficit, skyrocketing oil prices, global warming, and damage to environment and ecosystem, the promising incentives to develop alternative energy resources with high efficiency and low emission are of great importance. Among the renewable energy resources, the energy through the photovoltaic (PV) N effect can be considered the most essential and prerequisite sustainable resource because of the ubiquity, abundance, and sustainability of solar radiant energy. Regardless of the intermittency of sunlight, solar energy is widely available and completely free of cost. Recently, photovoltaic array system is likely recognized and widely utilized to the forefront in electric power applications. It can generate direct current electricity without environmental impact and contamination when is exposed to solar radiation. A simulation software program known as Matlab has been used to simulate the system performance. The system design and performance analysis could thus be achieved through computer modeling and simulation prior to practical realization. 2. BLOCK DIAGRAM OF PVFC HYBRID SYSTEM The block diagram of the PVFC Hybrid system is shown in figure 1 consists of a photovoltaic cell, a fuel cell, an inverter and a DC-DC converter. In this system a photovoltaic cell feeds power into grid through the DC-DC converter and inverter, which step up the voltage level and invert the DC of the photovoltaic cell into AC for the grid. In the absence of solar radiation the fuel cell is the other alternative which continuous the powers supply to the grid. Figure 1. Block Diagram of PVFC Cell 3. MODELLING OF COMPONENTS 1. Modelling of Photovoltaic Cell A general mathematical description of I-V output characteristics for a PV cell has been studied for over the past four decades. Such an equivalent circuit-based model is mainly used for the MPPT technologies. The equivalent circuit of the general model which consists of a photo current, a diode, a parallel resistor expressing a leakage current, and a series resistor describing an internal resistance to the current flow, is shown in Figure 2. Figure 2. Electrical model of PV cell The voltage-current characteristic equation of a solar cell is given as Where IRS is the cells reverse saturation current at a reference temperature and a solar radiation EG is the bang- gap energy of the semiconductor used in the cell. An even more exact mathematical description of a solar cell, which is called the double exponential model, is derived from the physical behaviour of solar cell constructed from polycrystalline silicon. This model is composed of a light- generated current source, two diodes, a series resistance and a parallel resistance. However, there are some limitations to develop expressions for the V-I curve parameters subject to the implicit and nonlinear nature of the model. Therefore, this model is rarely used in the subsequent literatures and is not taken into consideration for the generalized PV model [5]. The shunt resistance RSH is inversely related with shunt leakage current to the ground. q(V IRS ) PH S I I I e kTC A1(V IRS ) RSH Equation 1 I PH is a light-generated current or photocurrent, I S is the cell saturation of dark current, q 1.61019 C is an electron charge, k 1.38 1023 J / K is a Boltzmanns Figure 3 Model of Solar Cell constant, TC is the cells working temperature, A is an ideal factor, RSH is a shunt resistance, and RS is a series In general, the PV efficiency is insensitive to variation in R and the shunt-leakage resistance can be assumed to resistance. The photocurrent mainly depends on the solar insulation and cells working temperature, which is described as [1], approach infinity without leakage current to ground. On the other hand, a small variation in RS will significantly affect the PV output power. [ISC KI (TC Tref )] Equation 2 I IPH IS [exp (q (V IRS )/k TC A) 1] I SC is the cells short-circuit current at a Equation 4 1kW / m2 , K I is the cells short-circuit current temperature For an ideal PV cell, there is no series loss and no leakage to coefficient, Tref is the cells reference temperature, and is ground, i.e., RS = 0 and RSH = . The above equivalent circuit of PV solar cell can be simplified. the solar insulation in kW / m2 [2] . On the other hand, the cells saturation current varies with the cell temperature, which is described as I IPH IS [exp (q V / kTC A) 1] Equation 5 IS I RS )3 e qEG (1/ Tref 1/ TC ) kC A Equation 3 ☆ Determination of Model Parameters All of the model parameters can be determined by examining the manufacturers specifications of PV products. The most important parameters widely used for describing the cell electrical performance is the open-circuit voltage qH 2 / pH 2 Kan / M H 2 Equation 9 and the short circuit current ISC. The aforementioned equations are implicit and nonlinear; therefore, it is difficult to arrive at an analytical solution for a set of model parameters at a specific temperature and irradiance [3]. Since For hydrogen molar flow, there are three significant factors: hydrogen input flow, hydrogen output flow and hydrogen flow during the reaction [4]. The relationship amog these I PH and ignoring the small diode and factors can be expressed as [3], ground-leakage currents under zero-terminal voltage, the short-circuit current I SC is approximately equal to the d dt (PH 2 ) RT /Van in qout qr H 2 ) H 2 H 2 photocurrent I PH IPH [1], i.e. Equation 10 I PH Equation 6 According to the basic electrochemical relationship between On the other hand, the VOC parameter is obtained by the hydrogen flow and the FC system current, the flow rate of reacted hydrogen is given by [6] assuming the output current is zero. Given the PV open- circuit voltage at reference temperature and ignoring qr H 2 N / 2F 2KrIFC Equation 11 the shunt-leakage current, the reverse saturation current at reference temperature can be approximately obtained as [1], Using Equation (10) and (11) and applying Laplaces ISC / [exp (q VOC / NS kATC ) 1] Equation 7 transform, the hydrogen partial pressure can be obtained in the s domain as [3] H 2 In addition, the maximum power can be expressed as Equation 8 PH 2 1/ KH 2 S(qin H 2 2KrI ) are terminal voltage and output Equation 12 current of PV module at maximum power point (MPP), and is the cell fill factor which is a measure of cell quality. The specifications of variables used in the model of PV cell are listed Table 1 specifications of variables used in the model of H 2 Van / KH 2 RT Equation 13 Characteristics SPEC Typical peak power(Pp) 60W Voltage at peak power(VVV) 17.1V Current at peak power(IPP) 3.5A Short circuit current(ISC) 3.8A Open circuit voltage(VOC) 21.1V Temperature coefficient of open circuit -73mV/ Â°C Temperature coefficient of short 3mA /Â°C circuit current(KI) Approximate effect of temperature on Nominal operating cell temperature 49Â°C PV cell Similarly, the water partial pressure and oxygen partial pressure can be obtained. The polarization curve for the PEMFC is obtained from the sum of Nernsts voltage, the activation over voltage and the ohmic over voltage. Assuming constant temperature and oxygen concentration, the FC output voltage may be expressed as [3] Vcell E act ohmic Equation 14 B ln(CI FC ) Equation 15 ohmic Rint IFC Equation 16 Now, the Nernsts instantaneous voltage may be expressed as [16], E No [Eo RT / 2F logPH 2 Po2 / PH 2o Equation 17 2. Modeling of PEM Fuel Cell The FC model used in this thesis is realized in MATLAB and Simulink. Then, this model is embedded into the SimPower Systems of MATLAB as a controlled voltage source. The relationship between the molar flow of any gas (hydrogen) through the valve and its partial pressure inside the channel can be expressed as [3] The fuel cell system consumes hydrogen according to the power demand. The hydrogen is obtained from a high pressure hydrogen tank for the stack operation. During operational conditions, to control the hydrogen flow rate according to the FC power output, a feedback control strategy is utilized. To achieve this feedback control, the FC current from the output is taken back to the input while converting the hydrogen into molar form [4]. The amount of hydrogen available from the hydrogen tank is given by, o FC qreq H 2 N I / 2FU Equation 18 Depending on the FC system configuration and the flow of hydrogen and oxygen, the FC system produces the dc output voltage [4]. The hydrogenoxygen flow ratio rH-O in the FC system determines the oxygen flow rate. Different time constants can be defined for fuel increase and fuel decrease. The MATLAB and Simulink based FC system model developed in this paper with its block parameters are shown below. Figure 4 Model of fuel cell Figure 5 Function Block parameter; Saturation of FC Model Figure 6 Function Block parameter; Fcn6 of FC Model Figure 7. Function Block parameter; Gain of FC Model The values of various variables used in the modelling of the model are described in the following table. Table 2 Specification of variables used in fuel cell model Parameters SEPF Activation voltage constant(B) 0.04777 [A-1] Activation voltage constant(C) 0.0136 [V] Faradays constant(F) 96484600 [C kmol(s atm)-1] Hydrogen time constant(H2) 3.37[s] Hydrogen valve constantKH2) 8.395110-7 [kmol Kr constant(=NO/4F) (s A)-1] Hydrogen oxygen flow ratio (r H_O) No load voltage(EO) 0.6 [V] Number of cells(NO) 332 Oxygen time constant(O2) 6.74 [s] 2.110-5 [kmol(s Oxygen valve constant(kO2) FC absolute temp.[T] 343 [K] 8314.47 [J(kmol K)- Universal gas factor[R] Water time constant(H2O) 18.418 [s] 7.71610-6 [kmol(s Water valve constant(KH2O) Utilization factor(U) 0.8 PI gain constants(k1,k2) 10 3. Modelling of DC DC Converter Under steady-state conditions, the voltage and current waveforms of a dc-dc converter can be found by use of two basic circuit analysis principles. The principle of inductor volt-second balance states that the average value, or dc component, of voltage applied across an ideal inductor winding must be zero. This principle also applies to each winding of a transformer or other multiple winding magnetic devices. Its dual, the principle of capacitor amp-second or charge balance, states that the average current that flows through an ideal capacitor must be zero. Hence, to determine the voltages and currents of dc-dc converters operating in periodic steady state, one averages the inductor current and capacitor voltage waveforms over one switching period, and equates the results to zero. The equations are greatly simplified by use of a third artifice, the small ripple approximation. The inductor currents and capacitor voltages contain dc components, plus switching ripple at the switching frequency and its harmonics. In most well designed converters, the switching ripple is small in magnitude compared to the dc components. For inductor currents, a typical value of switching ripple at maximum load is 10% to 20% of the dc component of current. For an output capacitor voltage, the switching ripple is typically required to be much less than 1% of the dc output voltage. In both cases, the ripple magnitude is small compared with the dc component, and can be ignored. A resistor RL is included in series withthe inductor, to model the resistance of the inductor winding. It is desired to determine simple expressions for the output voltage V, inductor current IL, and efficiency. With the switch in position1, the inductor voltage is equal toVL (t) Vg IL (t)RL . By use of the small ripple approximation, we can replace I L (t) with its dc component I L , and hence obtain VL (t) Vg IL (t)RL . Likewise, the capacitor current is equal to IC (t) V (t) / R , which can be approximated as Figure 8 Model of DC-DC Converter When the switch is in position 2, the inductor is connected between the input and output voltages. The inductor voltage IC (t) V / R [7]. can now be written. The model of DC-DC Converter with its block VL (t) Vg IL (t)RL V t Vg IL RL V . parameters are shown in the below figures and specifications of various model parameters are listed in table 3. Table 3 Specification of variables used in DC-DC Converter model The capacitor current can be expressed as IC (t) = IL (t) v (t)/R~IL-V/R. When the converter operates in steady state, the average value, or dc component, of the inductor voltage waveform VL (t) must be equal to zero. Parameters Specifications Converter inductance 66 [mH] Converter capacitance 2200[ÂµF] Semiconductor type MOSFET Rated switching frequency 1000[Hz] Proportional gain of PI voltage control system Integral gain of PI voltage control system Reference voltage 400[V] Upon equating the average value of the VL (t) obtain [8], to zero, we 0 D(Vg V ) Equation 19 Likewise, application of the principle of capacitor charge balance to the capacitor current leads to, 0 D(V / R) (1 D)(I V / R) Equation 20 From equation 17 and 18, g L V V 1/1 D.1/1 R /1 D2 R Equation 21 IL Vg /(1 D)2 R.1/(1 RL /(1 D)2 R Equation 22 In the ideal case when RL = 0, the voltage conversion ratio M (D) is equal to one at D = 0, and tends to infinity as D approaches one. In the practical case where some small inductor resistance RL is present, the output voltage tends to zero at D = 1. In addition, it can be seen that the inductor winding resistance RL (and other loss elements as well) limits the maximum output voltage that the converter can produce. Obtaining a given large value of V/Vg requires that the winding resistance RL be sufficiently small. The converter efficiency can also be calculated. For this boost converter the efficiency is equal to [8], P V 2 R g L V I Equation 23 From equation 6, 7 and 8 the efficiency becomes. Figure 10 Function Block parameter; Discrete PI Controller of DC-DC Converter Model 1 RL /(1 D2 R Equation 24 It can be seen that, to obtain high efficiency, the inductor winding resistance RL should be much smaller than (1 D) 2 R. This is much easier to accomplish at low duty cycles, where (1 D) is close to unity, that at high duty cycles where (1 D) approaches zero. The output simulation result of DC-DC converter is shown in figure 9. Figure.9 Output of DC-DC Converter The figure 9 shows the output of the DC-DC Converter. Consequently, the efficiency is high at low duty cycles, but decreases rapidly to zero near D = 1. This behavior is typical of converters having boost or buck-boost characteristics. Figure 11 Function Block parameter; Repeating Sequence of DC-DC Converter Model Figure 12 Function Block parameter; Relational Operator of DC-DC Converter Model the switching part of the circuit that is in figure 14 one obtain the solution [9] which is: V n 1,5,7,11 4V (cos n 1) sinn (t -1200 ) SN 3n Equation 27 V n 1,5,7,11 4V (cos n 1) sinn (t – 2400 ) TN 3n Equation 28 V n 1,5,7,11 4V (cos n 1) sinn t SN 3n Equation 29 Each one of the 3-phases to neutral voltage, the 1, 5, 7, 11 Figure 13 Function Block parameter; Data type Conversion of DC-DC Converter Model D. Modelling of the Inverter The main circuit is the part where the DC electric power is converted to AC. This is virtually implemented with the one that is shown at the Figure 14, in this circuit we use a 3 leg inverter for 3-phase conversion which is composed of 6 IGBTs and the control unit. The last generates control pulses to drive the IGBTs. The pulse generator gives a digital signal to the IGBTs. When the signal from the pulse generator is not zero then it reacts as a switch and opens. This consists the basic operation in order to convert the DC to AC, with the technique of the Pulse Width Modulation (PWM).The frequency of the IGBTs we use is 1 KHz. For the time interval the IGBTs are open, we get a pulse at power circuit, which has the same amplitude of source. The RMS time integral give us the output values. The on-off is determined by a control unit which is analyzed below. The modulation factor ma can be used as a parameter for the dynamic control of the system. When ma is changing we can control the voltage output and correct the voltage fluctuations due to the PV array and MPPT. The losses will be analogue to the change over the ma. A useful reference for cascaded multilevel converters which discuses the control circuit of new topology [7]. A three phase inverter has the basic advantage that generates power in 3-phase and is working without a At one node of the circuit, supposing we have an input voltage Voi(t) an LC filter ,L inductance and C capacitance and the rL resistant Load ,if we apply the Kirchhoffs laws and if we consider that the IGBTs at an open state, we get: are the harmonics appearing and 2f the basic frequency at 50Hz.The Matlab Lab/Simulink using the numerical methods is solving the problem, taking into account not only this part of the system, but the total circuit as it can be seen at figure 14 [10]. Figure14 Model of Inverter r i L diL V V (t) Equation 25 L L dt C oi i C( dVC ) VC Equation 26 L dt R Figure 15 Function Block parameter; IGBT1 of Inverter Model The above problem is depending on the output of the PV array and in order to have a simple solution we consider only Figure 16 Function Block parameter; Diode of Inverter Model Figure 17 Function Block parameter; DC Voltage Source of Inverter Model 4. SIMULATION RESULTS 1. Simulation Result of PV Cell The simulation results of PV cell model with its block parameters are shown below. The following results show the output of the solar cell. Figure 18 Output of PV Cell Model Figure 19 Function (Fcn1) Block parameter of PV Cell Model Figure 20 Function Block Parameter; Gain of PV Cell Model Figure 21 Function Block parameter; Product of PV Cell Model 2. Simulation Result of PEM Fuel Cell The simulation results of fuel cell are shown as follows: Figure 22 Variation of Current with Time The experimental results show that very different local dynamic respect through the response of the average current shows very little dynamic. Figure 23 Variation of Temperature with Time Fuel cell temperature has an important role on performance of PEM fuel cells, when fuel cell temperature is lower than or equal to humidification temperature, local currentdecreases along the Figure 24 Variation of Power with time Photovoltaic and Fuel cells are ideal candidates for use in hybrid system due to their high energy density and the power requirements of these can be satisfied by the transient response of system. These have high power density and provide good transient characteristics. The behavior of the two in tandem is studied through simulations. Figure 25 Variation of Voltage with time 3. Simulation of Inverter Module Figure 26 Simulation Result of Inverter Model Table 4. Specification of variables used in Inverter model Parameters Specifications Semiconductor type IGBT-DIODE Snubber resistance 2[K] Snubber capacitance 0.1[ÂµF] Internal resistance 1[m] Carrier frequency 5[kHz] Modulation index 0.98 Frequency of output 5. CONCLUSION The overall goal of this thesis is to model and simulate the different models of a PVFC hybrid system. In this work, different models of a PVFC hybrid system has been implemented in computer codes and utilized to predict its operational performance through numerical simulation. Detailed descriptions of the individual component models required to simulate a PVFC hybrid system are presented. These models are mainly based one electrical and electrochemical relation. However, a number of empirical relationships for some models are also used. The models of PV generator, PEM fuel cell and power conditioning units are discussed in details. The modeling, identification and validation of the component models show that the agreement between simulated and measured data is very good. Several short- term simulations are performed, such that the I-U characteristics, hydrogen production and consumption rates, and other physical processes of the individual component models are properly evaluated. The main conclusions that could be drawn from the evaluation of the individual component models of the hydrogen PVFC hybrid system are given in the following subsections. The main conclusions about the overall operation of the hydrogen PVFC hybrid system at two sites with different topologies that it can also stabilize the fuel cell operation within the set limits especially when sudden load variations occur. This is better than to oversize the power sources, which is an expensive solution. However, coupling PV generator and fuel cell directly to the DC bus-bar may be a good alternative for small systems. The energy losses in this system associated with the power conditioning units are minimized. For long- term operation, the results of the simulation have been used for a detailed energy analysis, in which the energy conversion steps and losses for each individual component are analyzed and quantified, and their influence on the system overall efficiency is investigated. The results of the energy analysis have shown that the operational performance of the system does not depend only on component efficiencies but also on system design and consumption behavior. This fact points out that the search for performance improvement of PVFC hybrid system should be concentrated on development of subsystem components, especially the fuel cell. The results obtained from the analysis have shown that the performance of PVFC hybrid system can be optimized in different ways: by understanding the system behavior better, by improving components efficiency, by utilizing new systems concepts, and by helping people to use their systems as efficiently as possible. This ensures that the system as a whole can be operated in such a way as to supply a definite amount of power all the time irrespective of the available solar insulation and other environmental factors. This thesis deals with a hybrid system containing PV array and FC stack operating in tandem. In due course of the project, various options available for hybrid system and the interfaces have been studied and a new hybrid system has been proposed whose simulation is done to show its validity. The two-stage inverter provides both active power and reactive power independently of each other giving the .edibility of operating in accordance with the system requirements. The dc/dc boost converter gives the added advantage of being able to extract maximum power available from the PV array ensuring maximum utilization of the source. Thus, the PCU provides active power and reactive power whose value is decided by the grid requirement. The PCU which is intended for interfacing the Fuel Cell stack to the grid can in actual practice be used in many applications with different voltage levels. Its operation makes it possible to alter the switching control to switch from one configuration to the other while the inverter is in operation. The ability of the inverter that it can be operated in any of the three configurations, make it a universal configuration with its applications in interfacing different sources. Hence, the hybrid system as a whole, is self-sufficient in the sense that it can provide required amount of power at a given instant. This is done by operating the PV array at its MPP taking full advantage of the PV source. From the simulations, it can be seen that the system is operated in grid connected mode only. A photovoltaic fuel cell system hybrid power system is designed and modeled for a grid-independent user with appropriate power flow controllers. The available power from the renewable energy sources is highly dependent on environmental conditions such as intensity of solar radiation. To overcome this deficiency of the solar system, we integrated photovoltaic generator with the fuel cell system using a novel topology. The purpose of this thesis is the modeling and simulation of a stand-alone hybrid power system, referred to as Photovoltaic-Fuel Cell (PVFC) hybrid system. This hybrid topology exhibits excellent Performance under variable solar radiation and load power requirements. The proposed system can be used for non- interconnected remote areas. 6. FUTURE WORK To enhance the performance of PVFC hybrid systems, the following recommendations for future work are proposed. The choice of the suitable concept should be based on the type of application, adding other renewable sources such as a wind turbine to the system. A wind energy conversion would reduce the required PV generator area, and reduce the hydrogen storage volume. A trade-off between PV generator area and wind generator size is an interesting challenge for systems located at sites with high average wind speeds. A practical limitation on the system design is the voltage operating range of the available power conditioning units, which are designed mainly for lead-acid batteries rather than fuel cells or super capacitors. Thus, designing a new power conditioning units that can match the characteristics of these components is recommended. In hydrogen PVFC hybrid system without battery energy storage, such as in this work, the annual numbers of the on and off switching of the electrochemical components and also the annual operating times of these components are large. This would probably affect in the overall simulation results, if the hydrogen losses are not included in the simulation. These losses must be calculated to make the simulation more accurate. The H2/O2 PEM fuel cell has a better performance than the Air/H2 PEM fuel cell which is used in this work, but requires a storage tank for oxygen and a purification system. Thus, it is recommended to study using H2/O2 PEM fuel cell with the PVFC hybrid system and evaluate the system according to the cost point of view. Designing of a high pressure electrolyser could eliminate the need for a cmpressor to compress hydrogen into high pressure and, thus the volume of the gas storage tank is decreased. Other concept is to store the hydrogen in metal hydride (MH) storage, i.e., replacing the compressed hydrogen gas storage with low pressure ambient temperature metal hydride storage. The greatest advantage of the MH- storage is that it can be coupled directly to a low pressure electrolyser, thus eliminating the need for a compressor. The choice of the suitable concept should be based on the type of application. 1. D. Mayer, R. Metkemeijer, S. Busquet, P. Caselitz, J. Bard, and et al Photovoltaic/Electrolyser/Fuel cell Hybrid System the Tomorrow Power Station for Remote Areas, 17th EPVSEC, Munich, Germany, 2001, pp. 2529-2530. 2. M. Uzunoglu , M.S. Alam Dynamic modeling, design and simulation of a PEM fuel cell/ultra-capacitor hybrid system for vehicular applications Department of Electrical and Computer Engineering, University of South Alabama, 307 N. University, 26 November 2006. 3. A. R. Balkin Modelling A 500W Polymer Electrolyte Membrane Fuel Cell, Bs. D., University of Technology, Faculty of Engineering, Sydney, 2002. 4. J. Benz, B. Ortiz, W. Roth Fuel Cells in Photovoltaic Hybrid Systems for Stand-Alone Power Supplies, 2nd European PV- Hybrid and Mini-Grid Conference, Kassel, Germany, 2003, pp. 232-239. 5. D. M. Bernardi and M. W. Verbrugge A Mathematical Model of the Polymer-Electrolyte Fuel Cell, Journal of the Electrochemical Society, Vol. 139, No. 9, pp. 2477-2491 6. M. Van Wieringen, R. Pop-Iliev, "Development of a Dual-Fuel Power Generation System for an Extended Range Plug-in Hybrid Electric Vehicle ," IEEE Trans. on Industrial Electronics, vol. 57, no. 2, pp. , Feb 2010. 7. K.H. Edelmoser and F. A. Himmelstoss High Efficiency DC-AC Inverter Solar Application, Proceeding of the 14th EPVSEC, Barcelona, Spain, 1997. 8. C. Cecati, F. Ciancetta, P. Siano, "A Multilevel Inverter for Photovoltaic Systems With Fuzzy Logic Control ," IEEE Trans. on Industrial Electronics, vol. 57, no. 12, pp. , Dec 2010. 9. T.HottinenTechnical Review and Economic Aspects of Hydrogen Storage Technologies, MSc, Helsinki University of Technology, Department of Engineering Physics and Mathematics, Espoo, 2001. 10. Severine Busquet Study of a Stand Alone Power System Based on a Photovoltaic field,an electrolyser and fuel cell: Test bench and modlelization, P.hd, center d energetigue,Ecroll des Mines You must be logged in to post a comment.	{"url":"https://www.ijert.org/modelling-and-simulation-of-photovoltaic-full-cell-hybrid-system","timestamp":"2024-11-02T05:33:10Z","content_type":"text/html","content_length":"105220","record_id":"<urn:uuid:f0ee1fed-c5ff-46ce-99cb-66866af49711>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00196.warc.gz"}
Thermo-Mechanical Coupling Numerical Simulation for Extreme High-Speed Laser Cladding of Chrome-Iron Alloy China Academy of Machinery Science and Technology Group Co., Ltd., Beijing 100044, China Beijing National Innovation Institute of Lightweight Co., Ltd., Beijing 100083, China China Machinery Institute of Advanced Materials Co., Ltd., Zhengzhou 450001, China Author to whom correspondence should be addressed. Submission received: 15 April 2023 / Revised: 29 April 2023 / Accepted: 4 May 2023 / Published: 7 May 2023 With the aim to improve cladding coating quality and prevent cracking, this paper established an extreme high-speed laser cladding thermo-mechanical coupling simulation model to study the evolution of the temperature field and the residual stress distribution. Process parameters that impacted the macroscopic morphology of single-pass coatings were investigated. Numerical calculations and temperature field simulations were performed based on the process parameter data to validate the effects of the temperature gradient and cooling rate on the coating structure and the residual stress distribution. The results showed that a good coating quality could be achieved using a laser power of 2400 W, a cladding speed of 20 m/min, and a powder feeding rate of 20.32 g/min. The coatings’ cross-sectional morphology corresponded well with the temperature distribution predicted by the numerical modeling of the melt pool. The microstructure of the molten coatings was affected by the temperature gradient and the cooling rate, which varied greatly from the bottom to the middle to the top. Maximum residual stress appeared between the bonding region of the coatings and the substrate, and the coatings themselves had significant residual stress in the form of tensile strains, that were mostly distributed in the direction of the laser cladding. 1. Introduction 45 steel shaft parts are extensively utilized in the military, aerospace, power, and shipbuilding sectors due to their ability to load bear and transfer torque while the machinery is in operation. High-speed rotation, extreme friction, and prolonged corrosion make these components fragile and prone to failure [ ]. Therefore, the service life of components can be effectively increased by the use of laser cladding technology to provide wear- and corrosion-resistant coatings on the surface of the components [ ]. There is a great potential for extreme high-speed laser cladding technology to displace hard chromium plating and green manufacturing due to its high cladding efficiency, high powder utilization rate, small heat-affected zone, low dilution rate, and pollution-free manufacturing process [ ]. Powder particles are melted above the melt pool by optimizing the coupling of process parameters, causing the substrate and cladding materials to combine and solidify in the molten state, resulting in a metallurgical bond coating, and thereby preparing high-performance coatings in a cost-effective and time-efficient manner [ ]. However, due to the late start of extreme high-speed laser cladding technology, the research are still in the exploration stage of process parameters, improving coating performance, and promoting technology, and it relies primarily on experimental experience and the trial-and-error method to optimize the process parameters and thereby improve the coating quality [ Progress has been achieved in the study of the numerical simulation and technology of extreme high-speed laser cladding by researchers both at home and abroad. Experimental comparisons of the microstructure performance of conventional laser cladding and extreme high-speed laser cladding conducted by Li et al. [ ], revealed that the latter had finer and more uniform coating microstructures. Comparative experiments conducted by Wang et al. [ ] on conventional laser cladding and extreme high-speed laser cladding of iron-based amorphous coatings revealed that the latter had a superior performance due to their fine microstructure and hardness of more than 1300 HV in terms of wear and corrosion resistance. Preparing an extreme high-speed laser-clad M2 alloy coating, Zheng et al. [ ] investigated the influence of the process parameters on the creation of the cladded layer, and characterized the coating in terms of its shape, microhardness, and wear resistance. The impact of the powder particle size and the bonding rate on coating quality was investigated by Lou et al. [ ], who looked at the morphology of the coating’s microstructures when applied using low-power, ultra-high-speed laser cladding. To better understand why laser cladding speeds affect mechanical properties, Ding et al. [ ] compared the microstructure of Inconel 625 coatings produced using extreme high-speed laser cladding and conventional laser cladding. They found that increasing the laser’s melting speed and converting coarse columnar crystals into fine dendrites improved the coatings’ hardness, wear resistance, and corrosion resistance. Li et al. [ ] established a combined heat source model based on the finite element method, studied the temperature distribution of the extreme high-speed laser-cladding AISI 4140 melt pool, and used this model to investigate the impact of laser power and cladding speed on the resulting temperature field and stress field. In conclusion, there has been a great deal of study on the process optimization and coating quality assessment of extreme high-speed laser cladding, but only a smattering of studies on the temperature development law and residual stress of coating under such conditions. Numerical simulation studies of extreme high-speed laser cladding can accurately predict the temperature change and residual stress distribution during the cladding process, which is essential as it is difficult to measure the melt pool temperature and residual stress in real time [ ]. By combining the numerical simulations with the experiments, the relationship between the melt pool temperature and the tissue properties of the clad coating can be effectively established, thereby laying the foundation for process parameter optimization and subsequent multi-pass cladding laps. In this work, experiments on the extreme high-speed laser cladding of chrome-iron alloy were conducted, and the impact of various process factors on the extreme high-speed laser single-pass coating was analyzed. A numerical model of the thermal coupling of the ultra-high-speed laser cladding was established based on the chosen process parameters. The relationship between the melt pool temperature gradient, cooling rate, and coating structure during the extreme high-speed laser cladding process was analyzed, and the residual stress distribution of the cladding coatings was 2. Materials and Methods In this experiment, the 45 steel shaft with a diameter of 40 mm was employed as the base material, and the 431 stainless steel powder with a particle size of 50–150 μm was used as the cladding material. The chemical compositions of the cladding powder and the substrate material are listed in Table 1 Extreme high-speed laser cladding experiments were performed utilizing the entire suite of machinery from the China Machinery Institute of Advanced Materials Co., Ltd. (Zhengzhou, China). The laser beam spot diameter was 1 mm, and the highest power was 3300 W, with the powder being fed in a coaxial fashion. The extreme high-speed laser cladding’s single-channel forming size was primarily considered by analyzing the effects of laser power, cladding speed, and powder feeding speed using the single-factor experiment method, which took into account previous process experiments [ ], theoretical research, and industrial application practice. Parameters used in the experiment are listed in Table 2 . When performing numerical calculations, the set of parameters with the optimum surface state, width, and height dimensions were selected for the calculations. In order to observe the microstructure of the coatings and measure the width and height of the coatings, a 5 × 5 × 10 mm square specimen was cut out of the surface of a 45 steel shaft part using the wire-cutting method, and the surface was cleaned of stains using alcohol, inlaid, ground, and polished. The width and height of the molten layer were observed and measured using an optical display, and when measuring the height and width of the coating, each value was measured three times and then averaged. A schematic diagram of the coating width and height measurement is shown in Figure 1 . W is the coating width, and H is the coating height. When the coating’s surface has the phenomenon of sticky powder, the powder particles were then excluded when measuring the coating’s width and height dimensions. When conducting the observation of the coating’s microstructure, a corrosion treatment was carried out using a 10% FeCl hydrochloric acid solution and a 4% HNO alcohol solution for 10 s each. After the corrosion treatment was completed, the microstructure was observed using either scanning electron microscopy or optical microscopy. 3. Finite Element Simulation 3.1. The Control Equation and Boundary Conditions Extreme high-speed laser cladding is a rapid-heating and rapid-cooling technique that involves intricate heat and mass transport processes. As heat conduction occurs mostly inside a material, and because material properties vary greatly with temperature, the heat transfer process is subsequently very nonlinear, and its control equation is [ $ρ c ∂ T ∂ t = λ ( ∂ 2 T ∂ 2 x + ∂ 2 T ∂ 2 y + ∂ 2 T ∂ 2 z ) + Q$ is the material density; is the material’s specific heat; T is the temperature; t is the heat transfer time; is the material’s thermal conductivity; and is the internal heat source, including phase change latent heat and the heat source load’s heat generation, where , and are all functions of temperature. In the laser cladding process, thermal radiation and thermal convection are the main heat transfer boundary conditions, and the boundary condition equation is as follows [ $λ ∂ n ∂ T + h ( T s − T 0 ) + σ ε ( T s 4 − T 0 4 ) = A Q ( x , y , t )$ is the material thermal conductivity; is the temperature; is the initial ambient temperature; is the convection heat transfer coefficient; is the material surface temperature; is the Stefan–Boltzmann constant; is the material surface radiation coefficient; and is the material surface absorption coefficient for laser energy. 3.2. Heat Source Model Laser cladding involves melting material above the substrate with a high-energy laser beam, and then applying some of that energy to the substrate to create a metallurgical bond layer. The accuracy of the findings obtained from the numerical simulations is heavily influenced by the energy distribution. Since the laser energy distribution features in the cladding process include both a high-order Gaussian distribution and a variable-density distribution, a combined heat source model was employed for modeling. The powder material was subjected mostly to the high-order Gaussian energy distribution, whereas the substrate material was subjected primarily to the variable-density heat source model [ ]. Energy was distributed according to a high-order Gaussian, as seen below: $q ( x , y ) = η ξ 1 P π r 2 e x p { − 3 ( ( x 2 + y 2 ) R 2 ) n } z > 0$ is the laser power; is the utilization efficiency of the laser energy; is the proportion of energy allocated to the high-order Gaussian heat source, which is 0.8; is the distance from the center of the laser beam; is the radius of the laser beam; and is the order of the heat source. A variable-density heat source was used to simulate the heating of the substrate, and the energy distribution is as follows: $q ( x , y , z ) = η ξ 2 P π r 2 e x p { − 3 ( x 2 + y 2 ) ( R · exp ( a 1 · z ) ) 2 } − h ≤ z ≤ 0$ is the laser power; is the utilization rate of laser energy; is the energy ratio allocated to the variable-density heat source, taken as 0.2; is the distance from the center of the laser beam; is the radius of the laser beam; is an empirical parameter, taken as 1.6; and is the depth of laser heating on the substrate. 3.3. Finite Element Model Abaqus software was used to create a geometric model based on the actual procedure of the 431 stainless steel coating on the surface of 45 steel shaft parts; the finite element model and mesh division are displayed in Figure 2 ; the substrate was sized at 10 mm × 10 mm × 3 mm, and the coatings had height and width parameters of 0.2 mm and 1 mm, respectively. The mesh was fine-tuned close to the coating layer, with a minimum mesh size of 0.05 mm, and coarsened further from the coating layer, with a maximum mesh size of 0.3 mm, to improve the computation accuracy and efficiency. Grids were numbered as 59,760, and the nodes added up to 252,367 in total. The DC3D8 hexahedral linear heat transfer cell was used for the temperature field computations, whereas the C3D8R hexahedral linear stress cell was used for the stress field calculations. Living and dead element technology was used in the model, and the coatings were already set in place. Coaxial powder feeding characteristics were more accurately simulated by progressively adding the coating to the heating calculation model as the heat source was shifted. To accomplish the goal of simultaneous material addition and heat source loading, the Fortran language was used to create a subroutine for the heat source, and “Model Change” was used to put the subroutine into action. The thermal coupling calculations for extreme high-speed laser cladding were performed using a sequential coupling method to increase the efficiency and guarantee convergence. When the temperature field calculation was completed, the mesh division used in the temperature field calculation was not changed; instead, the model was transformed into a stress analysis and stress calculation mesh cell, and the obtained results were incorporated into the calculation as predefined loads. When the temperature cooled down to room temperature, the stress field results were considered as residual stresses at the time. 3.4. Material Parameter Model The process of extreme high-speed laser cladding involves rapid melting, rapid cooling, and solidification of the material. Physical parameters such as material density, specific heat capacity, Young’s modulus, coefficient of expansion, and yield strength vary considerably at different temperatures; therefore, temperature-dependent material parameter models were required to improve the accuracy of the thermo-mechanical coupling numerical simulation results. Calculations and interpolation using the JmatPro 6.0 calculation program, based on the material composition shown in Table 1 , can provide the high-temperature physical properties of the material. Table 3 Table 4 display the obtained data. The thermal and mechanical material parameters from Table 3 Table 4 were defined in the Abaqus 2016 finite element software, and the material parameters were called upon when conducting thermal calculations for the temperature and stress solutions. 4. Results and Discussion 4.1. The Influence of the Process Parameters on the Macro-Shape of the Coating Layer Figure 3 shows how the laser power from 1800 to 2400 W affected the coating section size at a 20 m/min laser cladding speed and 20.32 g/min powder feeding. Figure 3 indicates that laser power increased the coating section width and height. The laser power increased from 1800 to 2400 W, making the coating 12.3% wider and 21.0% higher. Even when the powder feeding rate and laser cladding speed remained unchanged, the melt pool became wider and higher as the laser power increased, the laser energy density rose, and more materials were melted. The coatings’ width was affected more by the laser power than the coating height as the melt pool lasts longer, has more laser energy, and flows more in the width direction [ The coating’s single-channel cross-sectional size, and melting rate were investigated in the research shown in Figure 4 ; the powder feeding rate was held unchanged at 20.32 g/min, and the laser power was set at 2400 W. Figure 4 shows that when the laser cladding speed was raised from 15 to 30 m/min, both the width and height of the coating dropped. The reduction in width was 33.4%, and the fall in height was 30.8%, respectively. Powder particles and substrates spent less time under the laser’s irradiation when the laser cladding speed was increased, resulting in less energy being absorbed per unit time of the material, less material being melted, and the molten pool becoming unstable, thereby resulting in a reduction in both the breadth and height [ As shown in Figure 5 , the effect of the powder feed rate on the coating section size was investigated while maintaining a constant laser power of 2400 W, and a laser cladding speed of 20 m/min. Figure 5 demonstrates that the cladding layer grows in both height and breadth when the powder feed rate was increased. When the powder feed rate was raised from 16.54 g/min to 35.79 g/min, the coating grew in width by 22.34%, and in height by 155.03%, respectively. This was due to the higher powder feed rate, more material and laser energy being delivered into the melt pool, and a large increase in coating height and width [ Figure 6 shows the unpolished surface state of the melted coating, which must be considered when choosing the process parameters for the laser cladding process in the numerical simulation analysis. Samples 1 to 10 of Figure 6 correspond to experimental serial numbers 1 to 10 in Table 2 , respectively. Samples No. 1 and No. 2 had many unmelted powder particles present at the edges of the coatings due to the low laser power, which led to an insufficient melting of the powder particles, and failed to form a stable melt pool; sample No. 3 was able to form a temperature melt pool, but the coatings had a serious sticky powder phenomenon; samples Nos. 4 and No. 5 had a good surface condition; samples Nos. 6, No. 7, and No. 8 were able to form a melt pool; there was a great deal of unmelted powder on the top of Nos. 9 and No. 10. So, sample No. 4’s process parameters were chosen for the numerical simulation study after considering the influence of the laser cladding speed and the laser power on the size of the melting section, and the results of the surface condition analysis. When comparing the surfaces of samples 1, 2, 3, and 4 in Figure 3 Figure 6 , it was clear that the sticky powder phenomenon on the coating surface progressively reduces and the surface flatness rises when the laser power is increased, with both the width and height of the coating increasing. Figure 4 Figure 6 demonstrate that when laser cladding rises, the coating width and height decrease, with tiny particles emerging at the coating margins, and surface flatness progressively declining. These results were based on comparing the surfaces of samples 4, 5, 6, and 7. Figure 5 Figure 6 demonstrate that when the powder feed rate was increased, the width and height of the coating also increased, a high number of unmelted particles formed on the surface of the coating, and the surface flatness was greatly reduced, as shown by comparing the surfaces of samples 4, 8, 9, and 10. 4.2. Temperature Field Numerical Simulation and Verification The laser cladding time was calculated as 0.03 s, and the other process was cooling. Based on the findings of the process experiment, the parameter of sample No. 4 was chosen as the numerical simulation parameter, with a total calculation time of 600 s. The melt pool’s temperature field distribution at t = 0.015 s was selected for analysis as it provided the most consistent results. Figure 7 depicts the temperature distribution in progress during ultra-high laser cladding. The maximum melting temperature was 2780 °C, with a tiny heat-affected zone forming where the molten layer bonded to the substrate at temperatures close to its melting point. A long molten pool was created due to the high melting rate, with a big temperature gradient in the front half of the pool, and a tiny temperature gradient in the back, giving the pool an oval shape. Figure 8 shows a comparison between the transverse section of an experimental molten layer and the numerical simulation molten pool section. The results show that the size of the numerical simulation molten pool is in good agreement with the actual melting size, that the heat-affected zone is in good agreement with the numerical simulation results, that the dilution rate is low, and that the statistical error was found to be less than 6.42%. Therefore, it could be demonstrated that the outcomes of the numerical simulation of the temperature field are, to a certain extent, correct. 4.3. The Influence of the Cooling Rate and Temperature Gradient on the Coating Structure To study the melting and solidification heat transfer processes, Figure 9 plots the thermal cycling curve and heating-cooling curve with four points in the direction of the molten pool section depth. From Figure 9 , it could be deduced that the maximum cooling rate occurs between points A and B at the top of the molten pool, where the temperature is approximately 2780 °C and the cooling rate is approximately 3.8 × 10 °C/s, respectively; the maximum temperature occurs between the point C at the interface between the substrate and the molten layer, where the temperature is approximately 1753 °C and the cooling rate is approximately 8 × 10 °C/s, respectively. The melt pool temperature tends to decrease from top to bottom, and the cooling rate was found to be higher at the top of the melt pool and lower at the bottom of the melt pool, due to the obvious convective heat exchange arising on the surface of the melt pool and the obvious heat conduction occurring at the bottom [ Figure 10 depicts the distribution of the melt pool temperature gradient in different directions. As could be seen from the figure, in the melt pool depth MN path, the temperature gradient first decreases and then increases, with a maximum temperature gradient of 6545 °C/mm and a minimum of 457 °C/mm, respectively; in the melt pool width KL path, the temperature gradient was symmetrically distributed about the center of the melt pool, with a maximum temperature gradient of 5321 °C/mm and a minimum temperature gradient of 500 °C/mm, respectively. This was due to the significant difference observed in the thermal conductivity of the material in the bonding area between the coating and the substrate [ During the solidification of a cladding coating, the microstructure’s form and size were directly related to the temperature gradient G and the tissue growth rate R. The shape control factor K = G × R determined the structural morphology of the microstructure, with larger values of K indicating a flat crystal, medium values indicating a cellular crystal, and smaller values indicating a dendrite or equiaxed crystal. The cooling rate V = G/R determines the size of the microstructure, with larger values of V indicating smaller microstructures [ Table 5 displays the results of the calculation of the shape control factor K and the growth rate, as well as the temperature gradient and the cooling rate at locations A, B, and C at the time of solidification of the melt pool, based on the results of a finite element numerical simulation. Figure 11 depicts the microstructure of the top, middle, and bottom of a single cross-section of the ultra-high-speed laser cladding coatings. Small, flat crystals formed near the metallurgical bond between the substrate and the coatings at the coating’s bottom, while large, dendritic crystals grew vertically along the bonding surface in a relatively uniform fashion, indicating that the temperature gradient was greatest in this region, making it suitable for crystallization and tissue development. There were smaller dendritic crystals found with a fine slatted structure and a reasonably consistent growth direction observed in the cladding layer’s middle region, parallel to the heat dissipation axis. The coating’s top section displayed both fine dendritic and equiaxed crystals, with the dendrites being noticeably larger than those in the coating’s bottom and middle regions. During solidification, the single-pass coating’s temperature gradient was highest at the bottom of the melt pool, then in the middle, and finally at the top; the cooling rate increased gradually with increasing depth of the melt pool; the value of the shape control factor was relatively large at the top and bottom of the melt pool and smaller at the middle; and the tissue growth rate was highest at the middle of the melt pool and smaller at the bottom. Figure 11 shows the results of a fifty time scribing experiment that determined the average dendritic spacing to be 1.62 μm, 1.03 μm, and 0.98 μm for the top, middle, and bottom regions of the coating, respectively. The comparison study showed a considerable correlation between the coating structure and the temperature gradient distribution and cooling rate, demonstrating that the accuracy of the numerical simulation results in accurately predicting the coating structure and morphology. 4.4. Stress Field Simulation Analysis and Prediction The residual stress distribution cloud map in Figure 12 was obtained by first obtaining the results of a numerical simulation of the temperature field, then modifying the type of calculation unit, then loading the model with the results of the numerical simulation of the temperature field, and finally imposing boundary conditions on the stress field. Figure 12 shows that the coating’s residual stress was concentrated in the fusion coating region, with a maximum value of 741 MPa. This value was higher than the matrix’s yield strength at the fusion layer and matrix bonding positions, resulting in plastic deformation of the matrix. Cracking in coatings is significantly affected by residual stress [ ]. As a result, we chose to examine the coating section pathways KL and MN, as well as the coating surface path GH. Figure 13 a depicts the residual stress curve along the KL-coated section route. Stress values on the coating were relatively large, with the tensile stress reaching a maximum of 406 MPa in the z-direction, and the residual stress in the x-direction and y-direction changing from tensile stress to compressive stress along the KL path (as seen in Figure 13 a). The stress in the melted layer region was relatively large, and the stress away from the melted layer was relatively small due to the extremely short action time of the laser in the melted layer region and the high cooling rate, which caused a relatively high temperature gradient at the symmetrical center of the melted layer and the material shrinkage phenomenon. A substantial residual stress was present in the coating bonding zone because of the substantial thermal expansion coefficient difference between the melted material and the substrate, as well as the uneven shrinkage [ The residual stress curve on the MN coating section route is shown in Figure 13 Figure 13 b shows that the cladding layer had a high residual stress throughout, with the highest residual stress of 741 MPa occurring at the interface between the cladding layer and the substrate. Maximum x-direction stress occurred at the cladding layer’s top, whereas y-direction stress was often less intense, and z-direction stress was greatest at the cladding layer’s contact with the substrate. The coating’s temperature differential was greatest at the cladding/substrate contact, and a substantial tensile tension was created when the melt pool rapidly cooled and solidified. The coating’s surface route GH is shown as a residual stress curve ( Figure 13 Figure 13 c shows that the melt pool flowed steadily throughout the cladding process, showing that the residual stress can be managed. This was shown by the fact that the stress and primary stress of the cladding surface fluctuated within a given range. In conjunction with Figure 12 a, it could be seen that the surface residual stress was similar to that at the substrate/substrate interface (210 MPa), indicating no tendency of cracking in the coatings; however, the large residual stress in the z direction will lead to an increase in the tendency of coating cracking. Figure 8 Figure 12 Figure 13 , it is clear that the temperature gradient and residual stresses are greatest in the bonded area of the coating; there is a large temperature gradient on the coating surface; and the temperature gradient and residual stress curves of the coating and the bonded area follow the same trend. Consequently, the temperature gradient is the primary cause of residual stresses, which could be mitigated by decreasing the temperature gradient. 5. Conclusions Abaqus finite element software was used to create a numerical model for extreme high-speed laser cladding based on a composite heat source and the live-dead cell approach. An analysis of the effects of the laser power and the cladding speed on the cladding section size and surface morphology was performed after a 431 stainless steel coating was clad onto a 45 steel shaft. A laser power of 2400 W, a cladding speed of 20 m/min, and a powder feeding rate of 20.32 g/min were chosen as the ideal process parameters for numerical simulation computation. The maximum temperature of the melt pool was calculated to be 2780 °C using this parameter and was found by a temperature field calculation. The numerical model was shown to be accurate, as the predicted size and heat effect zone of the melt pool were in good agreement with the experimental data. The influence of different process parameters on the melt pool size was also analyzed. The laser power and the powder feed rate were found to be positively correlated to the melt pool’s width and height, while the laser cladding rate was found to be negatively correlated to the melt pool’s width and height. To examine the cladding section’s microstructure, a numerical simulation of the temperature gradient and the cooling rate was performed. The microstructure of the clad layer varied depending on its location: at the top, the cooling rate was highest, the temperature gradient was largest, and the clad layer’s dendrite crystals were chaotic and fine; in the middle, the cooling rate was highest, the temperature gradient was smaller, and the clad layer’s dendrite crystals were coarse and long. Cladding layer residual stress curve analysis was also performed. There was a lot of tension at the coating–substrate interface, more tension on the clad layer’s surface than in its interior, and a lot of tension in the coating itself. The clad layer’s residual stress was found to have mostly originated in the laser’s scanning direction. Author Contributions Conceptualization, L.N. and M.W.; methodology, L.N., M.W. and X.W.; software, X.W. and L.N.; validation, L.N. and Y.X.; investigation, L.N. and Y.X.; resources, M.W.; data curation, L.N. and X.G.; writing—original draft preparation, L.N.; writing—review and editing, X.W., Y.X. and X.G.; supervision, M.W.; project administration, M.W.; funding acquisition, M.W. All authors have read and agreed to the published version of the manuscript. The authors acknowledge funding support from the National Natural Science Foundation of China, under Grant No. 51975240. Institutional Review Board Statement Not applicable. Informed Consent Statement Not applicable. Data Availability Statement All data are available from the corresponding author upon reasonable request. Conflicts of Interest The authors declare no conflict of interest. Figure 8. Numerical simulation of the melt pool cross-section compared with experimental coating morphology. Figure 9. Temperature law curve of the selected node. (a) Thermal cycle curve in the depth direction of the melt pool; and (b) heat-up-cooling rate curve of nodes. Figure 10. Temperature gradient profile of the selected path. (a) Temperature gradient in the depth direction MN of the melt pool; and (b) temperature gradient in the direction KL of the melt pool Figure 11. Microstructures of different locations of the cladding layer. (a) Top; (b) middle; and (c) bottom. Figure 13. Residual stress on different paths of the coating. (a) Path KL; (b) path MN; and (c) path GH. Materials C Cr Mn Ni Si P Fe 431 0.19 17.19 0.81 1.36 0.92 0.04 Bal 45 0.42 0.25 0.72 0.35 0.03 0.03 Bal Experiment No. Laser Power/W Cladding Speed/(m/min) Powder Feeding Rate/(g/min) 1 1800 20 20.32 2 2000 20 20.32 3 2200 20 20.32 4 2400 20 20.32 5 2400 15 20.32 6 2400 25 20.32 7 2400 30 20.32 8 2400 20 16.54 9 2400 20 30.36 10 2400 20 35.79 Temperature Specific Heat Capacity Thermal Conductivity Density Elastic Modulus Expansion Coefficient Poisson’s Ratio /°C /(J·kg/°C) /(W·m/°C) /(g/cm^3) /GPa /10^−6/°C 25 457.62 17.29 7.74 196.35 17.56 0.29 300 539.35 20.35 7.62 177.32 18.28 0.31 600 667.48 23.69 7.49 151.56 19.12 0.34 900 681.76 26.97 7.35 121.48 20.92 0.34 1200 754.57 30.35 7.19 87.97 21.51 0.37 1300 855.98 31.48 7.14 72.34 21.92 0.39 1400 1054.31 32.36 7.07 35.35 23.02 0.39 1450 1948.47 32.39 7.02 15.67 24.10 0.42 Temperature Specific Heat Capacity Thermal Conductivity Density Elastic Modulus Expansion Coefficient Poisson’s Ratio /°C /(J·kg/°C) /(W·m/°C) /(g/cm^3) /GPa /10^−6/°C 25 453.63 16.94 8.04 210.32 8.26 0.29 200 498.52 19.03 7.93 199.68 9.35 0.33 400 535.98 21.41 7.82 165.42 10.51 0.33 600 567.76 23.8 7.71 147.65 11.87 0.35 800 598.13 26.18 7.60 128.53 14.85 0.37 1000 629.98 28.56 7.49 108.98 14.96 0.42 1200 661.31 30.94 7.39 88.62 14.95 0.42 1400 698.64 33.33 7.29 67.75 15.04 0.42 Point A B C Temperature gradient 1.63 × 10^3 5.36 × 10^3 8.72 × 10^3 Cooling rate 2.12 × 10^4 2.67 × 10^4 3.68 × 10^4 Shape control factor 125.33 1076.01 2066.26 Growth rate 1.30 4.98 1.42 Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https: Share and Cite MDPI and ACS Style Nian, L.; Wang, M.; Ge, X.; Wang, X.; Xu, Y. Thermo-Mechanical Coupling Numerical Simulation for Extreme High-Speed Laser Cladding of Chrome-Iron Alloy. Coatings 2023, 13, 879. https://doi.org/ AMA Style Nian L, Wang M, Ge X, Wang X, Xu Y. Thermo-Mechanical Coupling Numerical Simulation for Extreme High-Speed Laser Cladding of Chrome-Iron Alloy. Coatings. 2023; 13(5):879. https://doi.org/10.3390/ Chicago/Turabian Style Nian, Liangxiao, Miaohui Wang, Xueyuan Ge, Xin Wang, and Yifei Xu. 2023. "Thermo-Mechanical Coupling Numerical Simulation for Extreme High-Speed Laser Cladding of Chrome-Iron Alloy" Coatings 13, no. 5: 879. https://doi.org/10.3390/coatings13050879 Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details Article Metrics	{"url":"https://www.mdpi.com/2079-6412/13/5/879","timestamp":"2024-11-03T13:07:08Z","content_type":"text/html","content_length":"455816","record_id":"<urn:uuid:7fbca68b-bf55-42c0-b82a-61987d243b0e>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00039.warc.gz"}
10th class Math Notes Exercise 4.2 - eStudent.pk 10th class Math Notes Exercise 4.2 0 Comments 10th Class Math Notes Exercise 4.2 “Math” a compulsory subject for 10th class Science students. and 10th Class Math Notes Exercise 4.2 English (Science Group) included in this book. For getting good marks in this subject, the students need more practice. Here we are providing you Math Notes 10th class Solution for your better preparation. 10th class Science Students can get more practice and prepared through these important mathematics class 10 notes solutions. We are providing you these notes in the pdf files in which you can easily download and save. 10th class general math notes pdf is available for preparation here. Lets take a visit on English Medium version and enjoy your study also share with your friends for the easiness of other classmates and colleagues. Note: If you find any error, then just inform us in comments or email address admin@eStudent.pk MCQs & Short Questions Time is over Thanks for taking Quiz, Click Finish Button to see your Result Math Ch#04, 10th Class Here MCQs for your Revision For Good Revision & Grip on Concept, You Should take the MCQs and Revise it again and again Good Luck. . . . (: You Should Enter your Name 1 / 19 Partial fractions of $\frac{(x-2)}{(x-1)(x+2)}$ are the form of __________ (Select Correct Option) 1. $\frac{A}{(x-1)}+\frac{B}{(x+2)}$ 2. $\frac{Ax}{(x-1)}+\frac{B}{(x+2)}$ 3. $\frac{A}{(x-1)}+\frac{Bx+C}{(x+2)}$ 4. $\frac{Ax+B}{(x-1)}+\frac{C}{(x+2)}$ 2 / 19 The identity $(5x+4)^{^{2}}&space;=&space;25x^2&space;+40x+16$ is ture for _________ 3 / 19 Partial Fractions of $\frac{x^2+1}{(x+1)(x-1)}$ are of the from of ________ (Select Correct Option) 1. $\frac{A}{(x+1)}+\frac{B}{(x-1)}$ 2. $1+\frac{A}{(x+1)}+\frac{Bx+C}{(x-1)}$ 3. $1+\frac{A}{(x+1)}+\frac{B}{(x-1)}$ 4. $\frac{Ax+B}{(x+1)}+\frac{C}{(x-1)}$ 4 / 19 Partial Fractions of $\frac{x^3-2x^2-2}{(x^2+1)^{2}}$ are of the form ___________ (Select One Option) 1. $\frac{Ax+B}{x^2+1}+&space;\frac{Cx^2+D}{(x^2+1)^{2}}$ 2. $\frac{Ax+B}{x^2+1}+&space;\frac{Cx+D}{(x^2+1)^{2}}$ 3. $\frac{Ax+B}{x^2+1}+&space;\frac{Cx+D}{(x^2+1)^{}}$ 4. $\frac{Ax+B}{x^2+1}+&space;\frac{Cx}{(x^2+1)^{2}}$ 5 / 19 A fraction in which the degree of the numerator is greater or equal to the degree of denominator is called ______ 6 / 19 Partial Fractions of $\frac{1}{(x-1)^{2}(x-2)}$ are of the form ___________ (Select Correct Option) 1. $\frac{A}{x-1}+\frac{Bx+C}{(x-1)^{2}}+\frac{D}{x-2}$ 2. $\frac{A}{x-1}+\frac{B}{x-1^{}}+\frac{C}{x-2}$ 3. $\frac{A}{x-1}+\frac{B}{(x-1)^{2}}+\frac{C}{x-2}$ 4. $\frac{Ax+B}{(x-1)^{2}}+\frac{C}{x-2}$ 7 / 19 a fraction in which the degree of numerator is less then the degree of the denominator is called __________ 8 / 19 Partial Fractions of $\frac{2x+5}{(x+1)(x^2+2)^2}$ are of the form _________ (Select Correct Option) 1. $\frac{A}{x+1}+\frac{Bx+C}{x^2+2}+\frac{Dx+E}{(x^2+2)^{2}}$ 2. $\frac{A}{x+1}+\frac{Bx+C}{x^2+2}$ 3. $\frac{Bx+C}{x^2+2}+\frac{Dx+E}{(x^2+2)^{2}}$ 4. $\frac{A}{x+1}+\frac{B}{x^2+2}+\frac{Cx+D}{(x^2+2)^{2}}$ 9 / 19 A fraction in which the degree of the numerator is greater or equal to the degree of denominator is called ______ 10 / 19 $\frac{x^3+1}{(x-1)(x+2)}$ is _________________ 11 / 19 A fraction with degree of numerator less then degree of denominator is called _____ fraction 12 / 19 Partial Fractions of $\frac{1}{(x-1)^{2}(x-2)}$ are in the form of __________ (Select Correct Option) 1. $\frac{Ax+C}{(x-1)^2}+\frac{C}{(x-2)}$ 2. $\frac{A}{x-1}+\frac{B}{(x-1)^{2}}+\frac{C}{x-2}$ 3. $\frac{Ax+B}{x^2+2x+4}$ 4. $\frac{A}{(x-1)^2}+\frac{B}{(x-2)}$ 13 / 19 $\frac{2x+1}{(x+1)(x-1)}$ is ___________ 14 / 19 Resolution of fraction into fractions depends upon the factors of _____ 15 / 19 $(x+3)^{2}&space;=&space;x^2+6x+9$ is ____________ 16 / 19 A function of the form $f(x)&space;=&space;\frac{N(x)}{D(x)'}$ with D (x) = 0, where N (x) and D(x) are polynomials in x is called ______ 17 / 19 An _______ equation is a type of equation, which is satisfied by all the values of variable involved in given equation. 18 / 19 Partial Fractions of $\frac{x+2}{(x+1)(x^2+2)}$ are of the form of _____________ (Select Correct Option) 1. $\frac{A}{(x+1)}+\frac{B}{(x^2+2)}$ 2. $\frac{A}{(x+1)}+\frac{Bx+C}{(x^2+2)}$ 3. $\frac{Ax+B}{(x+1)}+\frac{C}{(x^2+2)}$ 4. $\frac{A}{(x+1)}+\frac{Bx}{(x^2+2)}$ 19 / 19 The Quotient of two numbers or algebraic expression is called a _____ Your score is Can u write your Feelings How You Feel this If you find any mistake in MCQs & Short Questions, please inform us by Commenting or by Contact us menu to improve the Quality of this free content. Let’s see, How Students Solve this . . . . . . . Results . . . Math Solution 10th Class (Science Group) Class 10th Math Notes pdf (Solution) There are total 13 Chapters in Mathematics 10th Class for Punjab Textbook Board, Lahore. Solutions of all the chapters are given below. Anyone can download the PDF file of the notes. Note that, you can view these notes only if you have PDF reader software/ App on your device. However online view of the notes is also available on eStudent.pk . You can find all solution along with , Math MCQs Class 10. if you are searching the topics like 10^th Class Math Notes, Notes of 10^th Class math, 10^th Class Math Solution, 10^th Class Math, 10^th Class math solution pdf, 10^th class Math solution pdf download, 10^th class math guess paper 2022, Past Papers 10^th Class Math, 10^th class math solution pdf download, past papers 10^th class math, 10^th class math key book pdf free download, 10^th class math solution pdf free download, 10^th class math book solved all chapters pdf download, 10^th class math MCQs pdf download, 10^th class math pairing scheme 2022, 10^th class math guide pdf download, 10^th Class Math Notes Chapter 4, 10th Class Math Notes in Urdu (Science Group), math class 10 notes Punjab board, key book of mathematics 10th class punjab text book pdf, math notes class 10 urdu medium, Math Notes Class 10 Solution Exercise 4.2, 10th Class Math Notes Exercise 4.2 Notes for Class 10th Math Math Notes Class 10, 10^th class math notes, Math notes for class 10, math notes class 10 pdf, class 10 math notes pdf, 10^th class math solution pdf free download, key book of mathematics 10^th class Punjab text book board, math class 10 notes kpk board pdf, class 10 math notes pdf, math class 10 notes punjab board, math notes class 10 kpk board, math notes class 10 adamjee notes, math class 10 notes, 10^th class math notes chapter 4, 10^th class math solution chapter 4 pdf, math notes class 10 chapter 4, then this post is specially for you and your future. All Boards of Intermediate & Secondary Educations in Punjab, Federal Board of Intermediate and Secondary Education, kpd board, Sindh board, Agha Khan Board and also Allama Iqbal Open University are taking examination according to Mathematics 10th Class. 10th Class Math key book pdf in urdu.	{"url":"https://estudent.pk/10th-class-math-notes-exercise-4-2/","timestamp":"2024-11-08T13:57:52Z","content_type":"text/html","content_length":"361172","record_id":"<urn:uuid:2f9b59f9-3836-492d-b44f-5ba10c3c667e>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00557.warc.gz"}
Comparing Numbers As I discussed in Crossfooting, numbers in Excel can have slight inaccuracies. This is due to the binary-decimal conversions that take place. In most cases, you won’t ever notice that there’s a problem because the type of operation you’re doing won’t show a problem. If, for instance, you’re just summing some numbers and those numbers look like 147.65000000001 but only show 147.65, the results of your sum will still be correct – at least what you see as the result. One place where this really can bite you is when you’re comparing two numbers. Kurt supplies an example where the numbers come from a CSV file. A formula is used to make sure the CSV file (which doesn’t contain formulas) actually works as if it contained formulas. The numbers in A:D are imported numbers and are all two decimal places. The formula in E is In E4 where it’s false, I check the calculations. F4 is =B4-C4, G4 is =D4-D3 and H4 and I4 are the results of a Copy – Paste Special – Values from F4 and G4. You can see I4 has a little precision The way to fix this is to modify the formula that checks for equality. The first step is to determine how much precision you need. If you’re dealing with currency, you only need two decimal places. You would modify the formula to read: Rounding to two decimal places gives you the precision you need and ignores those little errors. Only you can determine how much precision you need, but you can round to whatever that is. 2 thoughts on “Comparing Numbers” 1. for thoese not aware of the huge rounding issues in excel/ compuers in genral these articals are worth a read: About Rounding excel v’s VBA 6 2. Hi Dick just to add an alternative approach: I’d use something like where epsilon is rather small number such as 0.0001 Posting code? Use <pre> tags for VBA and <code> tags for inline.	{"url":"http://dailydoseofexcel.com/archives/2005/01/03/comparing-numbers/","timestamp":"2024-11-12T07:06:17Z","content_type":"text/html","content_length":"75706","record_id":"<urn:uuid:03ce00c7-b89a-4bdd-92dd-3010ab08d614>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00129.warc.gz"}
Number Theory I - Winter term 2015/16 Time: Tuesdays, 10-12 Uhr. Place: Arnimallee 6 SR 032/A6. Exercises: Fridays, 10-12 Uhr, Arnimallee 6 SR 032/A6. First Exercise class: October 23 (no exercise class on Oct. 16) Solving exercises shall help to pass the exam. 1) In case of a bad grade at the exam, we shall upgrade according to the success in the exercises group. 2) In case you pass the exam but your grade is less than your performance in the exercises, we shall also upgrade the final grade. • Atiyah, M.F.; Macdonald, I.G.: Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969 ix+128 pp. (This book is probably the best entry to the subject. It is short, concise, and clearly written.) • Bosch, S.: Algebraic Geometry and Commutative Algebra, Universitext Springer, 2012. • Further literature will be announced in class. Prerequisites: Linear Algebra I+II, Algebra and Number Theory Exam date: February 10, 10-12h, Arnimallee 6, SR032/A6 or April 11, 08-10h, Arnimallee 6, SR032/A6 Rules for the exam: • Everybody is free to do the exam in February or in April or both. In case a student passes the exam in February and tries again the exam in April the February grade will be erased and only the April grade will remain. • You may bring two sheets of paper (DIN A4), each written by yourself and by hand with writing on both sides Problem sets: Addendum to the class on Novemeber 24: pdf	{"url":"https://www.mi.fu-berlin.de/en/math/groups/arithmetic_geometry/teaching/numbthe_wise1516.html","timestamp":"2024-11-08T18:05:41Z","content_type":"text/html","content_length":"32067","record_id":"<urn:uuid:4ef1f72a-5c35-4ac6-8c98-96b8455aedeb>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00249.warc.gz"}
Assumed Daily Dose (ADD) Methodology \| NHSBSA This section explains the use of Assumed Daily Doses (ADDs) for measuring medicine utilisation in the Innovation Scorecard. It discusses why ADDs are used, how they are used in calculations, and the principles applied to assign ADDs. The page includes examples of how to apply the rules for assigning ADDs, and worked examples of ADD calculations. ADD definition An Assumed Daily Dose (ADD) is a prescribing measure developed specifically for the Innovation Scorecard. It is similar to the internationally recognised Defined Daily Dose (DDD). An ADD is a unique value for each presentation of a medicine. It is based on the strength and form of the drug, for example vials or 150mg tablets. It is also based on the recommended frequency of use for the presentation’s main indication, for example, one a day, or every 2 weeks. When first introduced to the Innovation Scorecard in January 2016, the measure was called an Actual Daily Dose. From the October 2023 publication, it has been renamed to Assumed Daily Dose. The new name reflects that it is a derived value based on assumptions about how the medicine is used such as age, weight, type and severity of disease, rather than actual usage by individual patients. ADDs in the innovation scorecard Medicine groupings are used in the Innovation Scorecard where there are several medicines as options for treatment of a condition or similar conditions. To report on the total use of medicines within a grouping, a standard unit of measurement is needed. This allows us to add together the use of the different medicines in a fair and meaningful way. For example, a grouping could include medicine A for which the dose is one 10mg tablet once a day, and medicine B for which the standard dose is one 2.5mg tablet twice a day. To add together the number of tablets would give too much weight to medicine B. Adding together the number of milligrams would give too much weight to medicine A. Similarly, a grouping could include medicines that have different routes of administration such as tablets taken orally and liquids given intravenously. The Defined Daily Dose (DDD) unit of measurement is defined by the WHO Collaborating Centre for Drug Statistics Methodology. It assigns a single assumed average dose per day for a medicine formulation at a chemical substance level. This is used to report the use of some individual medicines within the Innovation Scorecard. However, there are limitations in using DDDs to report medicine use: • DDDs are calculated at the chemical substance level and may not reflect any available presentations for the medicine. • DDDs are only assigned once a year, so there may be a gap between a medicine being added to the Innovation Scorecard and the DDD being published. • Not all medicines can be assigned a DDD, for example if the dosing schedule is highly dependent on patient characteristics. • DDDs assigned on an international basis may not reflect prescribing practices in England. The source data for the Innovation Scorecard is at presentation level. We assign an ADD to each presentation. The advantages of using ADDs are that the ADD values can match the prescribing patterns. For example, the amount of a presentation that is specifically used for reduced doses is counted as the relevant number of reduced daily doses. This is instead of counting a smaller number of the standard dose. Another advantage is that an ADD may be able to be created where no DDD has been published. The published DDD may also not apply to the presentation, such as an initiation specific pack or strength. ADDs in innovation scorecard calculations The usage of a drug grouping where all the presentations have assigned ADD values is calculated over several steps. 1. For each presentation of each drug, the total ADDs for presentation X of drug A is found by dividing the total number of units of presentation X by the ADD value. 2. The total ADDs for drug A is found by adding together the total ADDs for each presentation. 3. Steps 1 and 2 are repeated for each drug within the grouping of medicines. 4. The total ADDs for the grouping is found by adding together the total ADDs for each drug. ADD calculations worked example │Drug│ Presentation │ Dose │ ADD │Prescribing│ │A │5mg tablet │5mg twice daily │2 tablets │100 tablets│ │A │10mg tablet │10mg once daily │1 tablet │100 tablets│ │A │5mg/5ml oral solution │7.5ml once daily│7.5mg │750mg │ │B │20mg capsule │20mg twice daily│2 capsules│50 capsules│ │B │10mg patch │10mg once daily │1 patch │50 patches │ For drug A: • the total ADDs for 5mg tablets is 100 tablets divided by 2 tablets, which equals 50 • the total ADDs for 10mg tablets is 100 tablets divided by 1 tablet, which equals 100 • the total ADDs for 5mg/5ml oral solution is 750mg divided by 7.5mg, which equals 100 Therefore, the total ADDs for drug A is 250. This is found by adding together the total ADDs for 5mg tablets, 10mg tablets and 5mg/5ml oral solution, which is 50 plus 100 plus 100. For drug B: • the total ADDs for 20mg capsules is 50 capsules divided by 2 capsules, which equals 25 • the total ADDs for 10mg patch is 50 patches divided by 1 patch, which equals 50 Therefore, the total ADDs for drug B is 75. This is found by adding together the total ADDs for 20mg capsules and 10mg patches, which is 25 plus 50. For drugs A and B the total ADDs is 325 ADDs. This is found by adding the 250 for drug A and the 75 for drug B. Rules for assigning ADDs We apply the following guiding principles when assigning an ADD to a presentation of a medicine. This makes sure that ADDs are assigned consistently and transparently. The rules are applied when an ADD is first needed to be assigned to a presentation. This may be needed the first time the presentation appears in the prescribing data used in the Innovation Scorecard. It can also happen when a medicine previously reported as an individual medicine is added to a grouping. There is currently no routine review process for ADDs following this initial assignment. However, ADDs may be updated if relevant information is noticed during the production process of a new release of the Innovation Scorecard. For some medicines, the recommended dose and frequency are more complex and require assumptions specific to that medicine when assigning the ADD. A file showing the assumptions made for each medicine for which ADDs have been assigned is included in the resources section of the overview chapter. General rules A presentation of a medicine is selected for treatment use to minimise the number of units used for a dose, for example, tablets, pens, or vials. Each presentation is used for a recommended maintenance dose unless the presentation is specifically designed for initiation. A presentation designed for initiation, such as a different strength or a titration pack, might be different from one used for a maintenance dose. If a presentation is designed for use by children, the ADD for that presentation will be based on the child maintenance dose. This is different from the DDD which is usually based on the adult maintenance dose. Where a DDD is published for the medicine, you should follow the assumptions used in assigning the DDD. If these assumptions are not followed, you should document where departure from the assumptions is intentional. Treatment frequency and duration Where a presentation will be used to give doses at intervals less frequently than daily, calculate the ADD as the dose divided by the number of days in the treatment period. Where the interval varies between information sources, you should rely on the DDD. If the DDD is not available, the manufacturer’s information or the NICE TA should be used. Where a presentation is recommended to be given for a defined treatment period, calculate the ADD over the treatment period. Where a presentation is specifically designed for initiation of treatment, the ADD will be assigned based on this use. Where the presentation is an initiation pack containing multiple doses, calculate the ADD as the total in the pack divided by the number of days for which the pack is used. Whole vial use Where part of a vial or other unit would be used for a dose and the remainder discarded, the ADD is calculated as using the whole vial. Units and rounding For tablet or sachet presentations given daily, define the ADD in tablets or sachets. For tablet or sachet presentations used to give doses at intervals less frequently than daily, define the ADD in units of the active ingredient, for example, milligrams. Round ADD calculations consistently with the DDD where relevant. If the calculated value can be expressed exactly to 2 decimal places, for example, 1.75 tablets, use this value. Otherwise, round to 2 decimal places if the value is less than one of the relevant unit of measurement for example mg, mcg. Round to one decimal place if the value is more than one unit, or to the nearest whole number if more than 10 units. If presentations of the same medicine would be rounded differently, round to the same number of decimal places for all the presentations. Examples of applying the ADD rules Presentation selection Drug A is available in 20mg, 30mg, and 40mg tablets. The recommended dose is 40mg once daily. Adverse effects may be managed by dose reductions. We assume that the 20mg tablet is selected when a reduced dose is required, and would not be selected to give a 40mg dose as 2 x 20mg tablets. The ADD for the 20mg tablet is therefore one tablet. Use for maintenance dose The initial dose of drug B comprises two 300mg infusions 2 weeks apart. The maintenance dose of drug B is 600mg every 6 months. The only presentation available is a 300mg dose. This may be used for initial doses or maintenance doses. Because we cannot identify initiation use separately, we assume that all the reported use is for maintenance doses. Applying the ‘divide by treatment period’ rule, the ADD is 600mg divided by 182 days = 3.3mg. Child maintenance doses Drug C is available as a sachet of granules, with a recommended dosage for children aged under 12. In assigning the ADD we assume that the sachet is used for the recommended age group. Specific initiation presentations The recommended dose for drug D is 15mg twice daily for 3 weeks, followed by 20mg once daily for continued treatment. We assume that the 15mg presentation is only used for initiation doses, with an assigned ADD of 2 tablets. The ADD for the 20mg presentation is one tablet. Following the DDD assumptions The treatment cycle for drug E is 5 consecutive days of treatment in the first year of treatment, and 3 consecutive days of treatment in the second year of treatment. The published DDD assumes an average of 4 days of treatment per year, and we apply the same assumption in assigning the ADD. The recommended dose for drug F is 3mg/kg every 4 weeks. The DDD methodology document states that when the recommended dose refers to body weight, an adult is considered to be a person of 70kg. The recommended dose would therefore be 210mg. However, the dose is recommended to be given as a whole number of vials, with a dose of 200mg for patients in the weight range that includes 70kg. The DDD is assigned as 200mg divided by 28 days = 7.1mg. We apply the same assumptions in assigning the ADD. Departing from the DDD assumptions Drug G is available as 10mg tablets or 25mg tablets, with the recommended dose for each strength of one tablet daily. The published DDD is 17.5mg, being an average of the 2 strengths. As the ADD is assigned at presentation level, we do not use this assumption. We assign the ADD for each presentation as 1 tablet daily. Treatment frequency For drugs H to L, we assign an ADD to a presentation by dividing the amount of active ingredient over the interval between doses. │Drug│Dose │ Frequency │Time in days│ Calculation │ ADD │ │H │75mg │every 2 weeks │14 │75 divided by 14 │5.4mg │ │I │225mg│monthly │30 │225 divided by 30 │7.5mg │ │J │140mg│every 4 weeks │28 │140 divided by 28 │5mg │ │K │600mg│every 6 months │182 │600 divided by 182 │3.3mg │ │L │10mg │daily for 2 weeks per year│365 │10 multiplied by 14 divided by 365 │0.38mg│ Initiation pack An initiation pack presentation is available for drug M containing multiple doses of different strengths for a patient to build up to the recommended daily dose. We assign the ADD for the initiation pack as the total number of tablets in the pack divided by the number of days supply included in the pack. │Treatment days │Number of days │Dose │Frequency│Total tablets│ │Days 1-3 │3 │0.5mg│1 daily │3 │ │Days 4-7 │4 │0.5mg│2 daily │8 │ │Days 8-14 │7 │1mg │2 daily │14 │ │Total │14 │ │ │25 │ The ADD for the initiation pack is 25 tablets divided by 14 days = 1.8 tablets, rounded to one decimal place. Whole vial use Drug N is available as a 10mg per vial presentation. The dose for drug N is 21mg every 3 weeks. This dose will require the use of 3 of the 10mg vials, with an unused amount being discarded. We assign the ADD as the amount of the medicine in the number of whole vials used to give the dose, divided by the number of days in the treatment period. Note this may be greater than the amount administered to the patient. For drug N, the ADD is 3 x 10mg = 30mg divided by 21 days = 1.4mg. Pages in this publication 1. Assumed Daily Dose (ADD) Methodology	{"url":"https://cms.nhsbsa.nhs.uk/statistical-collections/nice-technology-appraisals/nice-technology-appraisals-nhs-england-innovation-scorecard-june-2024/assumed-daily-dose-add-methodology","timestamp":"2024-11-05T12:25:05Z","content_type":"text/html","content_length":"67219","record_id":"<urn:uuid:e128a92c-c395-4170-8bb8-22f22d6fc7c5>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00427.warc.gz"}
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z It is physical property of a substance which makes it to flow. The molecules of such substance move one past other. The substance assumes the shape of its container. Phenomenon of luminescence in which emission of light occurs during excitation or within 10^-8 Sec after the excitation source is removed. Fluorescent Lamp Lamp contains a tube filled with inert gas & a small amount of mercury vapor (low pressure). It doesn’t have filament running through it instead they are equipped with coiled Tungsten filaments coated with an electron emitting substance. The fluorescent light occurs in two stages. First, electrons emitted from cathodes create an electrical arc through Mercury vapor. Then the resultant UV radiation strikes Phosphor coating which then gives off visible light. Flux (Energy) The energy flux is a measure of the total energy incident on a surface per unit area per unit time. Another term used for this quantity especially in dosimetry, is the energy fluence rate. Flux (particle) Particle flux simply represents the number of particles incident on a surface per unit area per unit time. In the field of dosimetry, this quantity is also known as particle fluence rate. Focal length The distance from the lens to that point where light rays converges or diverges (depends on type of lens). Focal length decides the lens strength/power. Lens power is inversely proportional to focal Focal point Focal points are a pair of points on principal axis of a system such that all rays starting from one focal point, after passing through the system, become parallel to the principal axis and parallel rays after refraction through the optical system converge at second focal point. First point is object point on principal axis for which the image point is at infinity and the second focal point is the image point on principal axis for which object point is at infinity. Focaults Pendulum It is a device used to demonstrate the rotation of earth and the fact that earth is not an inertial frame of reference. The place in the earth crust, where the vibrations come from, is called focus of earth quake. Mass of an object times the acceleration it gained. Forced Vibration This is vibration of the body which exists because of exertion of external periodic force constantly on body. Forward Bias Term used in solid state physics. It is state of biasing P-N junction where ‘P’ junction is at higher potential compared to ‘N’ junction. This type of biasing reduces width of depletion Layer in P-N junction and the diode conducts electrically. Forward Biased Diode The ‘P’ junction of P-N diode is kept at high potential and ‘N’ junction at low potential. Fourier Theorem Fourier theorem deals with the summation of a number of simple harmonic vibrations in which the vibrations are in the same straight line. The theorem also helps in the synthesis and analysis of complex forms of vibrations. This theorem was formulated by J.B.T Fourier in 1828. The theorem states that any finite, continuous and single valued periodic function can be expressed as summation of number of simple harmonic terms (sine and cosine functions) having the frequencies integer multiple of given function. Fourier’s theorem deals primarily with the synthesis of a complex periodic vibration from simple harmonic terms and it gives a method to analyze a complex vibration into its component vibrations. Frame of Reference It is a coordinate system relative to which we describe the motion of body. Franck & Hertz Experiment It is the experiment which revealed the existence of discrete stationary states of electrons. Franck-Condon Principle According to this principle, an electronic transition takes place so rapidly that a vibrating molecule doesn’t change its inter nuclear distance appreciably during the transition. That is during an electronic transition, inter-nuclear distance remains the same, means straight line representing a transition between electronic states will be vertical. Transition between electronic states occurs vertically in a potential energy diagram. Frank-Hertz Experiment Experiment which was carried out by James Frank and Gustav Hertz in 1914 which revealed directly the existence of discrete quantized stationary energy states of electrons.	{"url":"https://www.conceptualphysicstoday.com/2023/02/physics-dictionary-f3.html","timestamp":"2024-11-10T09:11:03Z","content_type":"application/xhtml+xml","content_length":"107068","record_id":"<urn:uuid:cd920d2d-d63d-4412-a45c-6a7a83f6d987>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00631.warc.gz"}
C++ Program to Check Whether a Number is Palindrome or Not This program takes an integer from user and that integer is reversed. If the reversed integer is equal to the integer entered by user then, that number is a palindrome if not that number is not a palindrome. Example: Check Palindrome Number #include <iostream> using namespace std; int main() int n, num, digit, rev = 0; cout << "Enter a positive number: "; cin >> num; n = num; digit = num % 10; rev = (rev * 10) + digit; num = num / 10; } while (num != 0); cout << " The reverse of the number is: " << rev << endl; if (n == rev and n > 0) // Negative numbers are not palindromic cout << " The number is a palindrome."; cout << " The number is not a palindrome."; return 0; Enter a positive number: 12321 The reverse of the number is: 12321 The number is a palindrome. Enter a positive number: 12331 The reverse of the number is: 13321 The number is not a palindrome. In the above program, use is asked to enter a positive number which is stored in the variable num. The number is then saved into another variable n to check it when the original number has been reversed. Inside the do...while loop, last digit of the number is separated using the code digit = num % 10;. This digit is then added to the rev variable. Before adding the digit to rev, we first need to multiply the current data in the rev variable by 10 in order to add the digit to the n^th place in the number. For example: in the number 123, 3 is in the zero^th place, 2 in the one^th place and 1 in the hundred^th place. So, to add another number 4 after 123, we need to shift the current numbers to the left, so now 1 is in the thousand^th place, 2 in the one^th place, 3 is in the one^thplace and 4 in the zero^th This is done easily by multiplying 123 by 10 which gives 1230 and adding the number 4, which gives 1234. The same is done in the code above. When the do while loop finally ends, we have a reversed number in rev. This number is then compared to the original number n. If the numbers are equal, the original number is a palindrome, otherwise it's not. Also Read:	{"url":"https://www.programiz.com/cpp-programming/examples/palindrome-number","timestamp":"2024-11-14T11:33:19Z","content_type":"text/html","content_length":"160404","record_id":"<urn:uuid:bb9037d8-a10f-4d59-be3f-63d83b8d89e1>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00185.warc.gz"}
Ellipse - Equation, Definition, Theorem, Proof, Example, Solution, Types We invoke that an ellipse is the locus of a point which moves such that its distance from a fixed point (focus) bears a constant ratio (eccentricity) less than unity its distance from its directrix bearing a constant ratio e (0 < e < 1) . (i) Equation of an Ellipse in standard form Let S be a focus, l be a directrix, e be the eccentricity ( 0 < e < 1) and P(x, y) be the moving point. Draw SZ and PM perpendicular to l . Let A and A′ be the points which divide SZ internally and externally in the ratio e :1 respectively. Let AA′ = 2a . Let the point of intersection of the perpendicular bisector with AA′ be C. Therefore CA = a and CA′ = a. Choose C as origin and CZ produced as x -axis and the perpendicular bisector of AA′ produced as y –axis. By definition, Hence we obtain the locus of P as which is the equation of an ellipse in standard form and note that it is symmetrical about x and y axis. Definition 5.4 (1) The line segment AA¢ is called the major axis of the ellipse and is of length 2a . (2) The line segment BB¢ is called the minor axis of the ellipse and is of length 2b . (3) The line segment CA = the line segment CA¢ = semi major axis = a and the line segment CB = the line segment CB¢ = semi minor axis = b. (4) By symmetry, taking S¢(-ae, 0) as focus and x =- a/e as directrix l¢ gives the same ellipse. Thus, we see that an ellipse has two foci, S (ae, 0) and S¢(-ae, 0) and two vertices A(a, 0) and A¢(-a, 0) and also two directrices, x = a/e and x =- a/e. Example 5.15 Find the length of Latus rectum of the ellipse The Latus rectum LL’ (Fig. 5.22) of an ellipse passes through S (ae, 0) . Hence L is (ae, y[1] ) . That is, the end points of Latus rectum L and L′ are Hence the length of latus rectum LL' = 2b^2 / a (iii) Types of ellipses with centre at (h, k ) (a) Major axis parallel to the x-axis From Fig. 5.24 The length of the major axis is 2a . The length of the minor axis is 2b . The coordinates of the vertices are (h + a, k ) and (h − a, k ) , and the coordinates of the foci are (h + c, k ) and (h − c, k ) where c^2 = a^2 − b^2. (b) Major axis parallel to the y-axis From Fig. 5.25 The length of the major axis is 2a . The length of the minor axis is 2b. The coordinates of the vertices are (h, k + a) and (h, k – a) , and the coordinates of the foci are (h, k + c) and (h, k – c) , where c^2 = a^2 - b^2 . Theorem 5.5 The sum of the focal distances of any point on the ellipse is equal to length of the major axis. Let P(x, y) be a point on the ellipse Draw MM ¢ through P perpendicular to directrices l and l¢ . Draw PN ^ to x -axis. By definition SP = ePM = eNZ = e[CZ - CN ] Hence, SP + S¢P = a - ex + a + ex = 2a When b = a , the equation = 1, becomes (x - h)^2 + ( y - k )^2 = a^2 the equation of circle with centre (h, k) and radius a . When b = a, e = = 0 . Hence the eccentricity of the circle is zero. Furthere, SP/ PM = 0 implies PM →∞ . That is, the directrix of the circle is at infinity. Auxiliary circle or circumcircle is the circle with length of major axis as diameter and Incircle is the circle with length of minor axis as diameter. They are given by x^2 + y^2 = a^2 and x^2 + y^2 = b^2 respectively.	{"url":"https://www.brainkart.com/article/Ellipse_39165/","timestamp":"2024-11-07T23:35:41Z","content_type":"text/html","content_length":"72938","record_id":"<urn:uuid:42cac427-9cab-487d-b7e4-71b7f84941f7>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00608.warc.gz"}
Refresher: Set theory and logic Yesterday, while I was waiting for my computer to be reimaged due to some serious funk happening with my outlook mail, I had a couple hours to burn. After I killed a longer than normal walk at lunch, I sat down in the lobby with a good book. Background: my products are reliant on a lot of technology, but one aspect is critical in how they work and are used. This being a PID Servo Control system. While you don’t need to know in depth what that is to use one of our instruments, having a deep knowledge does indeed help you get the most out of it. Control theory is something you would imagine to be the realm of electrical engineers, but curiously, it seems to be the realm of mechanical engineering. And at the root of it is math. To understand what is really happening, and how it works, you need to know a branch of mathematics called “Discrete Mathematics”. This is the foundation of computers and computer science, dealing with the world broken into discrete pieces and processed algorithmically. (As an aside, my education is in Physics, and there we deal in continuum mathematics, similar, but distinctly different). So I picked up a textbook. I might have mentioned in the past that Dover publishing does a wonderful job of keeping classic science and math texts in print, and affordable. The early parts of this text are a deep dive into set theory, function representation, and logic (mathematical logic is not the same as what most people think of logic). Being a child of the 70’s, and the evolution of mathematics elementary education, I had always some concepts of sets, and operations on sets. But beyond this informal early introduction, I never really dove into the subject. Some of my physics topics touched upon it, but again, it was using set theory to get to a solution. The first chapter was an eye opener. I realize what I had learned earlier was very shallow, and cursory, but now I have a much deeper understanding of these foundations of modern mathematics. A good way to spend a couple hours. Next up is counting (combinatorics). This site uses Akismet to reduce spam. Learn how your comment data is processed.	{"url":"https://tralfaz.com/2013/07/refresher-set-theory-and-logic/","timestamp":"2024-11-13T06:09:08Z","content_type":"text/html","content_length":"62491","record_id":"<urn:uuid:1ab8ffdc-30db-42fc-a213-f927617e0ae4>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00692.warc.gz"}
e c Pi Day 2018 Happy Pi Day 2013, this interactive activity will guide you through ways to find π using polygons. Most people know the definition of PI as the ratio of the circumference of a circle to its diameter. They have probably used PI in circle problems, calculating the area using area = πr^2 or the circumference using C = 2πr or C = πd . But what of this mysterious number, is it possible to create a visual analogy. This demonstration attempts to do so by examining and manipulating a regular This will be done in two ways firstly by considering the perimeter of a regular polygon which fits perfectly in a circle of radius one, the polygon can be opened to form a straight line and its perimeter then measured. To show how π relates to area we again use a polygon, dividing its area into triangles. The controls For the sake of consistency with the other activites the controls are described below. But the best way to use this activity is to follow the instructions guide in the top right hand corner • Next button is pressed to move to the next instruction • Previous button is pressed to move back to the previous instruction • Focus select the part to focus and zoom in on • Area toggle area or circumference • Exterior angles toggle to display exterior angles • close polygon click button to animate the polygon closing • open polygon click button to animate opening the polygon • Exterior Angle Adjust this slider to change the exterior angle which in effect opens or closes the original polygon • side length Adjust this slider to change the side length • side length Adjust this slider to change the number of sides. • Automatic recalculate - when this button is on changing the number of sides and the two other sliders will automatically adjust the angle and length to form a regular polygon that fits perfectly in the circle. Related Activities If you like visual mathematics why not check out the interactive multiplication tables The inspiration for this activity actually came from playing around with the interactive fractal after setting the branches to one. The interactive polygon explorer. is better for teaching the various properties and angles of polygons.	{"url":"https://www.visnos.com/demos/pi","timestamp":"2024-11-04T13:55:51Z","content_type":"text/html","content_length":"13064","record_id":"<urn:uuid:4632a818-df23-416c-9854-73991ff0a787>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00271.warc.gz"}
Fraction To Decimal Worksheet Grade 8 - Decimal Worksheets Fraction Into Decimal Worksheets – If you’re looking for printable worksheets on Class Decimal You’re within the appropriate location. Decimals are a crucial maths concept … Read more Fraction To A Decimal Worksheet Fraction To A Decimal Worksheet – Transforming decimals from fractions is actually a challenging method for a number of youngsters Nevertheless, BYJU’s Fractions To Decimals … Read more Fraction To Decimal Worksheets Fraction To Decimal Worksheets – Switching decimals from fractions is actually a difficult approach for many children Even so, BYJU’s Fractions To Decimals Worksheet will … Read more	{"url":"https://www.decimalworksheets.com/tag/fraction-to-decimal-worksheet-grade-8/","timestamp":"2024-11-06T20:47:20Z","content_type":"text/html","content_length":"65849","record_id":"<urn:uuid:01667927-a2f8-4d26-89a7-7a2477ebff3e>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00685.warc.gz"}
How do you divide frequency from a counter? For frequency division, toggle mode flip-flops are used in a chain as a divide by two counter. One flip-flop will divide the clock, ƒIN by 2, two flip-flops will divide ƒIN by 4 (and so on). What is a divide by N counter? A Divide by N counter implies that it divides the input clock frequency by N ie; if you cascade four flip-flops then, the output of every stage is divided by 2, if you are taking the output from the 4th flip-flop, then its output frequency is clock frequency by 16 (2^4). What is a mod 6 counter? Johnson counters are one of the most important applications of shift registers. Recall that the number of flip-flops required for a Johnson counter is half the number of used states for that counter. Since a mod 6 Johnson counter can count up to 6 states, 3 flip flops will be required. What is the modulus of 5 bit ripple counter? Explanation: The minimum number of flip-flops used in a counter is given by: 2(n-1)<=N<=2n. Thus, for modulus-5 counter: 22 <= N <= 23, where N = 5 and n = 3. Explanation: There are 10 states, out of which MSB is high only for (1000, 1001) 2 times. What should be the connection on divide by 10 pin high or low? So: To create a divide-by-10 counter, you first connect pin 5 to +5 volts and pin 10 to ground to power the chip. Then you connect pin 12 to pin 1 and ground pins 2, 3, 6, and 7. What is D flip-flop truth table? The D flip flop is the most important flip flop from other clocked types. It ensures that at the same time, both the inputs, i.e., S and R, are never equal to 1. The Delay flip-flop is designed using a gated SR flip-flop with an inverter connected between the inputs allowing for a single input D(Data). How do you design a divide by counter? Design of Divide-by-N Counters 1. A counter can also be used as a frequency divider. 2. Each flip-flop will divide its input signal by 2 such that the output of the last stage will be a frequency equal to the input frequency divided by the Modulus number. What do you call a counter with a truncated sequence? A common modulus for counters with truncated sequences is ten (1010), called MOD-10. A counter with ten states in its sequence is known as a decade counter. Decade counters are useful for interfacing to digital displays. Other MOD counters include the MOD-6 or MOD-12 counter which have applications in digital clocks to display the time of day. What’s the maximum count of a 3 Flip Flop counter? So a 3 flip-flop counter will have a maximum count of 2 3 = 8 counting states and would be called a MOD-8 counter. The maximum binary number that can be counted by the counter is 2 n–1 giving a maximum count of (111) 2 = 2 3–1 = 7 10. Then the counter counts from 0 to 7. Is the cd4059 divide by N counter preset? The CD4059 is a CMOS programmable Divide-by-N Counter that can be programmed to divide an input frequency by any number “N” from 3 to 15,999. The down-counter is preset by means of 16 jam inputs. How does the down counter on an IC work? The down-counter is preset by means of 16 jam inputs. The three Mode-Select Inputs Ka, Kb, and Kc determine the modulus (divide-by number) of the first and last counting sections in accordance with the truth table. The IC offers a wide array of features and in directly interfaceable with TTL, CMOS, and NMOS devices.	{"url":"https://corporatetaxratenow.com/how-do-you-divide-frequency-from-a-counter/","timestamp":"2024-11-09T16:06:00Z","content_type":"text/html","content_length":"42694","record_id":"<urn:uuid:8d3b1f12-93fc-4524-bc16-d5343c9f7a27>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00484.warc.gz"}
Users are encouraged to read through the following primers: Integration Primer Integration Examples Integration with noisy data Examples Tutorials MATLAB programs Using trapz and cumtrapz example I N/A Using trapz and cumtrapz example II distanceusingtrapz.m Examples for the advanced user Tutorials MATLAB programs Using quad and quadl example I distanceusingquad.m & paravelocity.m Users are encouraged to work through the following suggested exercises: Integration Exercises Files needed for the exercises: - Right-click and save-as if necessary	{"url":"http://matlabmarina.com/integration.html","timestamp":"2024-11-10T08:11:12Z","content_type":"text/html","content_length":"15121","record_id":"<urn:uuid:7f05118b-e176-43f8-bf96-fe897eae6fe8>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00013.warc.gz"}
Difference Between Resistance And Impedance The key difference between resistance and impedance is that Resistance is the opposition to current flow in both DC and AC systems, defined simply as R=V/IR = V/IR=V/I while, Impedance extends this concept to AC systems by including frequency-dependent reactance, making it a complex quantity represented as Z=V/IZ = V/IZ=V/I. What is Resistance? Resistance in electricity refers to the property of a circuit or a component within a circuit that converts electrical energy into heat while opposing the flow of electric current. This occurs due to collisions between the charged particles carrying the current and the fixed particles within the conductor’s structure. While resistance is most notable in devices like lamps, heaters, and resistors, it is present in every part of a circuit, including connecting wires and transmission lines. The conversion of electrical energy into heat, even in small amounts, impacts the electromotive force (EMF) or driving voltage needed to produce a specific current through the circuit. Electrical resistance RRR is quantitatively defined as the voltage VVV (measured in volts) across a circuit divided by the current III (measured in amperes) flowing through it, expressed as R=V/IR = V/IR=V/I. For instance, if a 12-volt battery drives a 2-ampere current through a wire, the wire has a resistance of 6 ohms, calculated as 12 volts divided by 2 amperes. The ohm (Ω) is the standard unit of electrical resistance, equivalent to one volt per ampere. Resistance in a conductor is directly proportional to its length and inversely proportional to its cross-sectional area. It also varies depending on the material of the conductor. When conductors are cooled to extremely low temperatures, some exhibit zero resistance and allow currents to flow indefinitely, a phenomenon observed in superconductors. The reciprocal of resistance, denoted as 1/R1/R1/R, is called conductance, measured in units of reciprocal ohms, known as mhos. What is Impedance? Impedance, denoted by the symbol Z, quantifies the opposition to electrical flow and is measured in ohms. In direct current (DC) systems, impedance and resistance are equivalent and defined as the voltage across an element divided by the current (R=V/I). However, in alternating current (AC) systems, impedance incorporates “reactance,” which arises from the frequency-dependent effects of capacitance and inductance. Although impedance is still measured in ohms and expressed by the equation Z=V/I, both the voltage (VVV) and current (III) are influenced by frequency. Resistance vs Impedance The major difference between resistance and impedance is given below: Parameter Resistance Impedance Definition Opposition to the flow of electric current. Total opposition to the flow of alternating current, encompassing both resistance and reactance. Symbol R Z Units Measured in ohms (Ω). Also measured in ohms (Ω). Type of Current Applies to both direct current (DC) and alternating current (AC). Mainly pertains to alternating current (AC). Dependence on Frequency Independent of frequency. Varies with frequency and includes both resistive and reactive components. Components Consists solely of resistance, a real component. Includes both resistance (real) and reactance (imaginary), forming a complex quantity. Phasor Representation Represented as a real number in phasor diagrams. Represented as a complex number in phasor diagrams, indicating both magnitude and phase. Ohm’s Law V = I × R (Voltage = Current × Resistance). V = I × Z (Voltage = Current × Impedance), incorporating both resistance and reactance. Energy Dissipation Energy is dissipated as heat in resistive components. Energy is dissipated as heat in resistive components and is stored and released in reactive components. Example A light bulb’s filament has resistance. A circuit containing both resistors and inductors or capacitors has impedance. DC Circuit Behavior Determines the behavior of direct current circuits. Reactance is irrelevant in DC circuits; resistance alone determines the behavior. AC Circuit Behavior Affects both magnitude and phase in AC circuits. Impedance, including both resistance and reactance, influences both magnitude and phase in AC circuits. Application Used in both DC and AC circuits. Primarily used in AC circuits where reactive components are significant. Leave a Comment	{"url":"https://differguide.com/difference-between-resistance-and-impedance/","timestamp":"2024-11-08T16:04:27Z","content_type":"text/html","content_length":"55702","record_id":"<urn:uuid:ec642cf7-33a8-4acb-af0b-239d46c1c8e5>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00368.warc.gz"}
[Solved] There is a lottery with n coupons and n p \| SolutionInn Answered step by step Verified Expert Solution There is a lottery with n coupons and n people take part in it. Each person picks exactly one coupon. Coupons are numbered consecutively There is a lottery with n coupons and n people take part in it. Each person picks exactly one coupon. Coupons are numbered consecutively from 1 to n, n being the maximum ticket number. The winner of the lottery is any person who owns a coupon where the sum of the digits on the coupon is equal to s. If there are multiple winners, the prize is split equally among them. Determine how many values of s there are where there is at least one winner and the prize is split among most people. Example n = 12 The list of coupon numbers generated from 1 to n is [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. The sums of the digits are [1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3]. The largest number of winners is 2 which will occur for coupons numbered [1, 10], [2, 11] and [3, 12]. The maximum number of possible winners occurs for any of these 3 possible values of s, so 3 is the answer. Function Description Complete the function lotteryCoupons in the editor below. The function must return the number of ways to choose s in such a way that there is at least one winner and the prize is split among the greatest number of people. lotteryCoupons has the following parameter(s): n: an integer that represents the maximum coupon number Constraints 1 n 104 Input Format For Custom Testing Sample Case 0 Sample Input STDIN 3 Function w n = 3 Sample Output There are 3 Steps involved in it Step: 1 Code is in... Get Instant Access to Expert-Tailored Solutions See step-by-step solutions with expert insights and AI powered tools for academic success Ace Your Homework with AI Get the answers you need in no time with our AI-driven, step-by-step assistance Get Started	{"url":"https://www.solutioninn.com/study-help/questions/there-is-a-lottery-with-n-coupons-and-n-people-51902","timestamp":"2024-11-03T23:07:32Z","content_type":"text/html","content_length":"115174","record_id":"<urn:uuid:f94b1cc0-0c36-4d21-95e6-3ae2bdbcaa58>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00851.warc.gz"}
Library GeneralRec Termination of all programs is a crucial property of Gallina. Non-terminating programs introduce logical inconsistency, where any theorem can be proved with an infinite loop. Coq uses a small set of conservative, syntactic criteria to check termination of all recursive definitions. These criteria are insufficient to support the natural encodings of a variety of important programming idioms. Further, since Coq makes it so convenient to encode mathematics computationally, with functional programs, we may find ourselves wanting to employ more complicated recursion in mathematical What exactly are the conservative criteria that we run up against? For definitions, recursive calls are only allowed on syntactic subterms of the original primary argument, a restriction known as primitive recursion . In fact, Coq's handling of reflexive inductive types (those defined in terms of functions returning the same type) gives a bit more flexibility than in traditional primitive recursion, but the term is still applied commonly. In Chapter 5, we saw how definitions are checked against a syntactic guardedness condition that guarantees productivity. Many natural recursion patterns satisfy neither condition. For instance, there is our simple running example in this chapter, merge sort. We will study three different approaches to more flexible recursion, and the latter two of the approaches will even support definitions that may fail to terminate on certain inputs, without any up-front characterization of which inputs those may be. Before proceeding, it is important to note that the problem here is not as fundamental as it may appear. The final example of Chapter 5 demonstrated what is called a deep embedding of the syntax and semantics of a programming language. That is, we gave a mathematical definition of a language of programs and their meanings. This language clearly admitted non-termination, and we could think of writing all our sophisticated recursive functions with such explicit syntax types. However, in doing so, we forfeit our chance to take advantage of Coq's very good built-in support for reasoning about Gallina programs. We would rather use a shallow embedding , where we model informal constructs by encoding them as normal Gallina programs. Each of the three techniques of this chapter follows that style. Well-Founded Recursion The essence of terminating recursion is that there are no infinite chains of nested recursive calls. This intuition is commonly mapped to the mathematical idea of a well-founded relation , and the associated standard technique in Coq is well-founded recursion . The syntactic-subterm relation that Coq applies by default is well-founded, but many cases demand alternate well-founded relations. To demonstrate, let us see where we get stuck on attempting a standard merge sort implementation. We have a set equipped with some "less-than-or-equal-to" test. A standard function inserts an element into a sorted list, preserving sortedness. Fixpoint insert ) ( list A ) : list A match ls with x :: nil h :: ls' if le x h then x :: ls else h :: insert x ls' end We will also need a function to merge two sorted lists. (We use a less efficient implementation than usual, because the more efficient implementation already forces us to think about well-founded recursion, while here we are only interested in setting up the example of merge sort.) The last helper function for classic merge sort is the one that follows, to split a list arbitrarily into two pieces of approximately equal length. Fixpoint split list A ) : list A * list A match ls with (nil, nil) h :: nil (h :: nil, nil) h1 :: h2 :: ls' ) := split ls' in (h1 :: ls1, h2 :: ls2) end Now, let us try to write the final sorting function, using a natural number " " test from the standard library. Fixpoint mergeSort (ls : list A) : list A := if leb (length ls) 1 then ls else let lss := split ls in merge (mergeSort (fst lss)) (mergeSort (snd lss)). Recursive call to mergeSort has principal argument equal to "fst (split ls)" instead of a subterm of "ls". The definition is rejected for not following the simple primitive recursion criterion. In particular, it is not apparent that recursive calls to are syntactic subterms of the original argument ; indeed, they are not, yet we know this is a well-founded recursive definition. To produce an acceptable definition, we need to choose a well-founded relation and prove that respects it. A good starting point is an examination of how well-foundedness is formalized in the Coq standard library. Print well_founded. well_founded = fun (A : Type) (R : A -> A -> Prop) => forall a : A, Acc R a The bulk of the definitional work devolves to the , whose definition we may also examine. Print Acc. Inductive Acc (A : Type) (R : A -> A -> Prop) (x : A) : Prop := Acc_intro : (forall y : A, R y x -> Acc R y) -> Acc R x In prose, an element is accessible for a relation if every element "less than" according to is also accessible. Since is defined inductively, we know that any accessibility proof involves a finite chain of invocations, in a certain sense that we can make formal. Building on Chapter 5's examples, let us define a co-inductive relation that is closer to the usual informal notion of "absence of infinite decreasing chains." We can now prove that any accessible element cannot be the beginning of any infinite decreasing chain. From here, the absence of infinite decreasing chains in well-founded sets is immediate. Absence of infinite decreasing chains implies absence of infinitely nested recursive calls, for any recursive definition that respects the well-founded relation. The Fix combinator from the standard library formalizes that intuition: Check Fix. : forall (A : Type) (R : A -> A -> Prop), well_founded R -> forall P : A -> Type, (forall x : A, (forall y : A, R y x -> P y) -> P x) -> forall x : A, P x A call to must present a relation and a proof of its well-foundedness. The next argument, , is the possibly dependent range type of the function we build; the domain is the function's domain. The following argument has this type: forall x : A, (forall y : A, R y x -> P y) -> P x This is an encoding of the function body. The input stands for the function argument, and the next input stands for the function we are defining. Recursive calls are encoded as calls to the second argument, whose type tells us it expects a value and a proof that is "less than" , according to . In this way, we enforce the well-foundedness restriction on recursive calls. The rest of 's type tells us that it returns a function of exactly the type we expect, so we are now ready to use it to implement . Careful readers may have noticed that has a dependent type of the sort we met in the previous chapter. Before writing , we need to settle on a well-founded relation. The right one for this example is based on lengths of lists. We must prove that the relation is truly well-founded. To save some space in the rest of this chapter, we skip right to nice, automated proof scripts, though we postpone introducing the principles behind such scripts to Part III of the book. Curious readers may still replace semicolons with periods and newlines to step through these scripts interactively. Notice that we end these proofs with , not . Recall that marks the theorems as , so that the details of their proofs may be used during program execution. Why could such details possibly matter for computation? It turns out that satisfies the primitive recursion restriction by declaring itself as recursive in the structure of Acc proofs . This is possible because proofs follow a predictable inductive structure. We must do work, as in the last theorem's proof, to establish that all elements of a type belong to , but the automatic unwinding of those proofs during recursion is straightforward. If we ended the proof with , the proof details would be hidden from computation, in which case the unwinding process would get stuck. To justify our two recursive calls, we will also need to prove that respects the relation. These proofs, too, must be kept transparent, to avoid stuckness of evaluation. We use the syntax to reference identifier with its implicit argument behavior turned off. (The proof details below use Ltac features not introduced yet, and they are safe to skip for now.) Lemma split_wf forall len ls , 2 <= length ls <= len ) := split ls in lengthOrder ls1 ls /\ lengthOrder ls2 ls unfold lengthOrder induction len 2 ( destruct ls 2 ( length ls match goal with \| [ E < 2 \|- ] => destruct E \| [ 2 \|- ] => destruct E \| [ ] ] => IH L split L destruct IH end Ltac split_wf intros ls length ls split ls Lemma split_wf1 forall ls , 2 <= length ls split ls Lemma split_wf2 forall ls , 2 <= length ls split ls Hint Resolve split_wf1 split_wf2 To write the function definition itself, we use the refine tactic as a convenient way to write a program that needs to manipulate proofs, without writing out those proofs manually. We also use a replacement le_lt_dec for leb that has a more interesting dependent type. (Note that we would not be able to complete the definition without this change, since refine will generate subgoals for the if branches based only on the type of the test expression, not its value.) Definition mergeSort list A list A Fix lengthOrder_wf fun _ list A list A forall ls' list A lengthOrder ls' ls list A ) => if le_lt_dec 2 ( length ls then let lss split ls in merge fst lss ) ( snd lss else ls subst lss End mergeSort The important thing is that it is now easy to evaluate calls to Eval compute in mergeSort leb :: nil = 1 :: 2 :: 8 :: 19 :: 36 :: nil Since the subject of this chapter is merely how to define functions with unusual recursion structure, we will not prove any further correctness theorems about . Instead, we stop at proving that has the expected computational behavior, for all inputs, not merely the one we just tested. The library theorem Fix_eq imposes one more strange subgoal upon us. We must prove that the function body is unable to distinguish between "self" arguments that map equal inputs to equal outputs. One might think this should be true of any Gallina code, but in fact this general function extensionality property is neither provable nor disprovable within Coq. The type of Fix_eq makes clear what we must show manually: Check Fix_eq. : forall (A : Type) (R : A -> A -> Prop) (Rwf : well_founded R) (P : A -> Type) (F : forall x : A, (forall y : A, R y x -> P y) -> P x), (forall (x : A) (f g : forall y : A, R y x -> P y), (forall (y : A) (p : R y x), f y p = g y p) -> F x f = F x g) -> forall x : A, Fix Rwf P F x = F x (fun (y : A) (_ : R y x) => Fix Rwf P F y) Most such obligations are dischargeable with straightforward proof automation, and this example is no exception. match goal with \| [ \|- context[match ?E with left _ => _ \| right _ => _ end] ] => destruct E end; simpl; f_equal; auto. As a final test of our definition's suitability, we can extract to OCaml. let rec mergeSort le x = match le_lt_dec (S (S O)) (length x) with \| Left -> let lss = split x in merge le (mergeSort le (fst lss)) (mergeSort le (snd lss)) \| Right -> x We see almost precisely the same definition we would have written manually in OCaml! It might be a good exercise for the reader to use the commands we saw in the previous chapter to clean up some remaining differences from idiomatic OCaml. One more piece of the full picture is missing. To go on and prove correctness of , we would need more than a way of unfolding its definition. We also need an appropriate induction principle matched to the well-founded relation. Such a principle is available in the standard library, though we will say no more about its details here. Check well_founded_induction. : forall (A : Type) (R : A -> A -> Prop), well_founded R -> forall P : A -> Set, (forall x : A, (forall y : A, R y x -> P y) -> P x) -> forall a : A, P a Some more recent Coq features provide more convenient syntax for defining recursive functions. Interested readers can consult the Coq manual about the commands Program Fixpoint A Non-Termination Monad Inspired by Domain Theory The key insights of domain theory inspire the next approach to modeling non-termination. Domain theory is based on information orders that relate values representing computation results, according to how much information these values convey. For instance, a simple domain might include values "the program does not terminate" and "the program terminates with the answer 5." The former is considered to be an of the latter, while the latter is an approximation of "the program terminates with the answer 6." The details of domain theory will not be important in what follows; we merely borrow the notion of an approximation ordering on computation results. Consider this definition of a type of computations. The type describes the result a computation will yield, if it terminates. We give a rich dependent type to computations themselves: Definition computation {f : nat option A \| forall ) ( f n = Some v n' >= n f n' = Some v} A computation is fundamentally a function from an approximation level n to an optional result. Intuitively, higher values enable termination in more cases than lower values. A call to may return to indicate that was not high enough to run the computation to completion; higher values may yield . Further, the proof obligation within the subset type asserts that in an appropriate sense: when some is sufficient to produce termination, so are all higher values, and they all yield the same program result It is easy to define a relation characterizing when a computation runs to a particular result at a particular approximation level. On top of , we also define , which is the most abstract notion of when a computation runs to a value. The book source code contains at this point some tactics, lemma proofs, and hint commands, to be used in proving facts about computations. Since their details are orthogonal to the message of this chapter, I have omitted them in the rendered version. Now, as a simple first example of a computation, we can define , which corresponds to an infinite loop. For any approximation level, it fails to terminate (returns ). Note the use of to create a new opaque lemma for the proof found by the tactic. In contrast to the previous section, opaque proofs are fine here, since the proof components of computations do not influence evaluation behavior. It is generally preferable to make proofs opaque when possible, as this enforces a kind of modularity in the code to follow, preventing it from depending on any details of the proof. Section Bottom Variable A Definition Bottom computation A fun _ => @ None A abstract run Theorem run_Bottom forall v ~run Bottom v End Bottom A slightly more complicated example is , which gives the same terminating answer at every approximation level. Section Return Variable A Variable v Definition Return computation A fun _ Some v abstract run Theorem run_Return run Return v End Return The name was meant to be suggestive of the standard operations of monads. The other standard operation is , which lets us run one computation and, if it terminates, pass its result off to another computation. We implement bind using the notation let (x, y) := e1 in e2 , for pulling apart the value which may be thought of as a pair. The second component of a is a proof, which we do not need to mention directly in the definition of Section Bind Variables A B Variable m1 computation A Variable m2 computation B Definition Bind computation B fun n ) := m1 in match f1 n with Some v ) := m2 v in f2 n end abstract run Theorem run_Bind ) ( run m1 v1 m2 v1 run Bind v2 match goal with \| [ ] => max x y End Bind A simple notation lets us write Bind calls the way they appear in Haskell. Notation " x <- m1 ; m2" := Bind m1 fun x )) ( right associativity at level We can verify that we have indeed defined a monad, by proving the standard monad laws. Part of the exercise is choosing an appropriate notion of equality between computations. We use "equality at all approximation levels." Now we come to the piece most directly inspired by domain theory. We want to support general recursive function definitions, but domain theory tells us that not all definitions are reasonable; some fail to be continuous and thus represent unrealizable computations. To formalize an analogous notion of continuity for our non-termination monad, we write down the approximation relation on computation results that we have had in mind all along. Section lattice Variable A Definition leq x y option A ) := forall v x = Some v y = Some v End lattice We now have the tools we need to define a new Fix combinator that, unlike the one we saw in the prior section, does not require a termination proof, and in fact admits recursive definition of functions that fail to terminate on some or all inputs. Section Fix. First, we have the function domain and range types. Next comes the function body, which is written as though it can be parameterized over itself, for recursive calls. Finally, we impose an obligation to prove that the body is continuous. That is, when terminates according to one recursive version of itself, it also terminates with the same result at the same approximation level when passed a recursive version that refines the original, according The computational part of the Fix combinator is easy to define. At approximation level 0, we diverge; at higher levels, we run the body with a functional argument drawn from the next lower level. Now it is straightforward to package as a computation combinator Hint Extern 1 ( _ >= _ ) => Hint Unfold leq Lemma Fix'_ok forall steps n x v Fix' n x steps = Some v forall n' n' >= n Fix' n' x steps = Some v unfold runTo in induction n match goal with \| [ _ >= _ ] => inversion H eauto end Hint Resolve Fix'_ok Hint Extern 1 ( proj1_sig _ _ = _ ) => match goal with \| [ \|- E _ = _ ] => proj2_sig E Definition Fix computation B intro x fun n Fix' n x abstract run Finally, we can prove that Fix obeys the expected computation rule. Theorem run_Fix forall x v f Fix x Fix x match goal with \| [ ] => S n eauto end End Fix After all that work, it is now fairly painless to define a version of that requires no proof of termination. We appeal to a program-specific tactic whose definition is hidden here but present in the book source. Definition mergeSort' forall A , ( ) -> list A list A fun A le list A list A list A ) => if le_lt_dec 2 ( length ls then let lss split ls in ls1 <- mergeSort fst lss ; ls2 <- mergeSort snd lss ; Return merge le ls1 ls2 else Return ls abstract mergeSort' Furthermore, "running" on concrete inputs is as easy as choosing a sufficiently high approximation level and letting Coq's computation rules do the rest. Contrast this with the proof work that goes into deriving an evaluation fact for a deeply embedded language, with one explicit proof rule application per execution step. Lemma test_mergeSort' mergeSort' leb :: nil :: nil There is another benefit of our new Fix compared with the one we used in the previous section: we can now write recursive functions that sometimes fail to terminate, without losing easy reasoning principles for the terminating cases. Consider this simple example, which appeals to another tactic whose definition we elide here. Definition looper computation unit fun looper ) => if b then Return tt else looper b abstract looper Lemma test_looper looper true As before, proving outputs for specific inputs is as easy as demonstrating a high enough approximation level. There are other theorems that are important to prove about combinators like , and . In general, for a computation , we sometimes have a hypothesis proving run c v for some , and we want to perform inversion to deduce what must be. Each combinator should ideally have a theorem of that kind, for built directly from that combinator. We have omitted such theorems here, but they are not hard to prove. In general, the domain theory-inspired approach avoids the type-theoretic "gotchas" that tend to show up in approaches that try to mix normal Coq computation with explicit syntax types. The next section of this chapter demonstrates two alternate approaches of that sort. In the final section of the chapter, we review the pros and cons of the different choices, coming to the conclusion that none of them is obviously better than any one of the others for all situations. Co-Inductive Non-Termination Monads There are two key downsides to both of the previous approaches: both require unusual syntax based on explicit calls to fixpoint combinators, and both generate immediate proof obligations about the bodies of recursive definitions. In Chapter 5, we have already seen how co-inductive types support recursive definitions that exhibit certain well-behaved varieties of non-termination. It turns out that we can leverage that co-induction support for encoding of general recursive definitions, by adding layers of co-inductive syntax. In effect, we mix elements of shallow and deep embeddings. Our first example of this kind, proposed by Capretta, defines a silly-looking type of thunks; that is, computations that may be forced to yield results, if they terminate. A computation is either an immediate or another computation wrapped inside . Since is co-inductive, every type is inhabited by an infinite nesting of s, standing for non-termination. Terminating results are wrapped inside some finite number of Why bother to write such a strange definition? The definition of is motivated by the ability it gives us to define a "bind" operation, similar to the one we defined in the previous section. Note that the definition would violate the co-recursion guardedness restriction if we left out the seemingly superfluous on the righthand side of the second We can prove that form a monad for . The proof is omitted here but present in the book source. As usual for this sort of proof, a key element is choosing an appropriate notion of equality for In the proofs to follow, we will need a function similar to one we saw in Chapter 5, to pull apart and reassemble a in a way that provokes reduction of co-recursive calls. As a simple example, here is how we might define a tail-recursive factorial function. To test our definition, we need an evaluation relation that characterizes results of evaluating We need to apply constructors of explicitly, but the process is easy to automate completely for concrete input programs. Now consider another very similar definition, this time of a Fibonacci number function. Notation " x <- m1 ; m2" := TBind m1 fun x )) ( right associativity at level CoFixpoint fib (n : nat) : thunk nat := match n with \| 0 => Answer 1 \| 1 => Answer 1 \| _ => n1 <- fib (pred n); n2 <- fib (pred (pred n)); Answer (n1 + n2) Coq complains that the guardedness condition is violated. The two recursive calls are immediate arguments to , but is not a constructor of . Rather, it is a defined function. This example shows a very serious limitation of for traditional functional programming: it is not, in general, possible to make recursive calls and then make further recursive calls, depending on the first call's result. The example succeeded because it was already tail recursive, meaning no further computation is needed after a recursive call. I know no easy fix for this problem of , but we can define an alternate co-inductive monad that avoids the problem, based on a proposal by Megacz. We ran into trouble because was not a constructor of , so let us define a new type family where "bind" is a constructor. This example shows off Coq's support for recursively non-uniform parameters , as in the case of the parameter declared above, where each constructor's type ends in comp A , but there is a recursive use of with a different parameter . Beside that technical wrinkle, we see the simplest possible definition of a monad, via a type whose two constructors are precisely the monad operators. It is easy to define the semantics of terminating We can also prove that form a monad according to a notion of equality based on , but we omit details here; they are in the book source at this point. Not only can we define the Fibonacci function with the new monad, but even our running example of merge sort becomes definable. By shadowing our previous notation for "bind," we can write almost exactly the same code as in our previous definition, but with less syntactic clutter. Notation " x <- m1 ; m2" := ( Bnd m1 fun x CoFixpoint mergeSort'' A ) ( list A ) : list A ) := if le_lt_dec 2 ( length ls then let lss split ls in ls1 <- mergeSort'' le fst lss ; ls2 <- mergeSort'' le snd lss ; Ret merge le ls1 ls2 else Ret ls To execute this function, we go through the usual exercise of writing a function to catalyze evaluation of co-recursive calls. Definition frob' A comp A ) := match c with Ret x Ret x Bnd _ c' f Bnd c' f end Lemma exec_frob forall A comp A frob' c exec c x destruct c Now the same sort of proof script that we applied for testing s will get the job done. Lemma test_mergeSort'' mergeSort'' leb :: nil :: nil apply exec_frob Have we finally reached the ideal solution for encoding general recursive definitions, with minimal hassle in syntax and proof obligations? Unfortunately, we have not, as has a serious expressivity weakness. Consider the following definition of a curried addition function: This definition works fine, but we run into trouble when we try to apply it in a trivial way. Definition testCurriedAdd := Bnd (curriedAdd 2) (fun f => f 3). Error: Universe inconsistency. The problem has to do with rules for inductive definitions that we will study in more detail in Chapter 12. Briefly, recall that the type of the constructor quantifies over a type . To make work, we would need to instantiate nat -> comp nat . However, Coq enforces a that (roughly) no quantifier in an inductive or co-inductive type's definition may ever be instantiated with a term that contains the type being defined. Chapter 12 presents the exact mechanism by which this restriction is enforced, but for now our conclusion is that is fatally flawed as a way of encoding interesting higher-order functional programs that use general recursion. Comparing the Alternatives We have seen four different approaches to encoding general recursive definitions in Coq. Among them there is no clear champion that dominates the others in every important way. Instead, we close the chapter by comparing the techniques along a number of dimensions. Every technique allows recursive definitions with termination arguments that go beyond Coq's built-in termination checking, so we must turn to subtler points to highlight differences. One useful property is automatic integration with normal Coq programming. That is, we would like the type of a function to be the same, whether or not that function is defined using an interesting recursion pattern. Only the first of the four techniques, well-founded recursion, meets this criterion. It is also the only one of the four to meet the related criterion that evaluation of function calls can take place entirely inside Coq's built-in computation machinery. The monad inspired by domain theory occupies some middle ground in this dimension, since generally standard computation is enough to evaluate a term once a high enough approximation level is provided. Another useful property is that a function and its termination argument may be developed separately. We may even want to define functions that fail to terminate on some or all inputs. The well-founded recursion technique does not have this property, but the other three do. One minor plus is the ability to write recursive definitions in natural syntax, rather than with calls to higher-order combinators. This downside of the first two techniques is actually rather easy to get around using Coq's notation mechanism, though we leave the details as an exercise for the reader. (For this and other details of notations, see Chapter 12 of the Coq 8.4 manual.) The first two techniques impose proof obligations that are more basic than termination arguments, where well-founded recursion requires a proof of extensionality and domain-theoretic recursion requires a proof of continuity. A function may not be defined, and thus may not be computed with, until these obligations are proved. The co-inductive techniques avoid this problem, as recursive definitions may be made without any proof obligations. We can also consider support for common idioms in functional programming. For instance, the monad effectively only supports recursion that is tail recursion, while the others allow arbitrary recursion schemes. On the other hand, the monad does not support the effective mixing of higher-order functions and general recursion, while all the other techniques do. For instance, we can finish the failed example in the domain-theoretic monad. The same techniques also apply to more interesting higher-order functions like list map, and, as in all four techniques, we can mix primitive and general recursion, preferring the former when possible to avoid proof obligations. Fixpoint map A B computation B ) ( list A ) : list B ) := match ls with Return nil x :: ls' f x ) ( fun x' map f ls' ) ( fun ls'' x' :: ls'' Theorem test_map fun x S x )) (1 :: nil :: nil One further disadvantage of is that we cannot prove an inversion lemma for executions of without appealing to an , a logical complication that we discuss at more length in Chapter 12. The other three techniques allow proof of all the important theorems within the normal logic of Coq. Perhaps one theme of our comparison is that one must trade off between, on one hand, functional programming expressiveness and compatibility with normal Coq types and computation; and, on the other hand, the level of proof obligations one is willing to handle at function definition time.	{"url":"http://adam.chlipala.net/cpdt/html/GeneralRec.html","timestamp":"2024-11-06T05:06:58Z","content_type":"application/xhtml+xml","content_length":"171554","record_id":"<urn:uuid:94723f1e-f10a-423f-95b7-9e4b81a0b479>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00305.warc.gz"}
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Sides A, C, and the side of the rectangle adjacent to side B have lengths of #5 #, #2 #, and #4 #, respectively. What is the rectangle's area? • How much air resistance acts on a falling 100-N box of nails when it reaches terminal velocity? • Discounts and markups? • Question #be751 • A 12-foot board is divided into two pieces so one piece is twice as long as the other. What are the lengths? • Question #f3694 • If the volume of an object were to double, with no change in mass, its density would? a) Halve b) Double c) Be the same d) None of these • How do I solve this ? • Time varies inversely with speed if the distance is constant. A trip takes 4 hours at 80 km/h. How long does it take at 64 km/h? • If f(x)=2x+1 and g(f(x))=4x^2+4x+3 find g(x) given that g(x)=ax^2+bx+c how do I do that? • Question #ffc58 • Question #9d54d • Question #15209 • In a scientific experiment, the mass of an object was determined to be 20.450 g, 20.313 g, and 21.013 g. What is the mean mass of the object? And the average deviation from the mean? • Question #a0c75 • Question #0042b • Question #3d8fc • Question #d1324 • Next	{"url":"https://socratic.org/users/tazwar89/answers","timestamp":"2024-11-10T22:35:12Z","content_type":"text/html","content_length":"29742","record_id":"<urn:uuid:bca899f5-9c51-438b-826a-33aaba53f8a8>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00486.warc.gz"}
The Ksp of manganese(II) carbonate, MnCO_3, is 2.42 * 10^-11. What is the solubility of this compound in g/L? \| HIX Tutor The Ksp of manganese(II) carbonate, #MnCO_3#, is #2.42 * 10^-11#. What is the solubility of this compound in g/L? Answer 1 $s = 5.65 \times {10}^{- 5} \frac{g}{L}$ Manganese(II) carbonate dissociate in water according to the following equation: $M n C {O}_{3} \left(s\right) r i g h t \le f t h a r p \infty n s M {n}^{2 +} \left(a q\right) + C {O}_{3}^{2 -} \left(a q\right) \text{ " } {K}_{s p} = 2.42 \times {10}^{- 11}$ Sign up to view the whole answer By signing up, you agree to our Terms of Service and Privacy Policy Answer 2 The solubility of manganese(II) carbonate (MnCO3) in g/L can be calculated using its Ksp value. The molar mass of MnCO3 is 114.95 g/mol. Using the formula for Ksp, we can find the concentration of Mn ^2+ and CO3^2- ions in solution. Then, using stoichiometry, we can determine the solubility of MnCO3 in g/L. The solubility of manganese(II) carbonate (MnCO3) is approximately 4.34 × 10^-5 g/L. Sign up to view the whole answer By signing up, you agree to our Terms of Service and Privacy Policy Answer from HIX Tutor When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some Not the question you need? HIX Tutor Solve ANY homework problem with a smart AI • 98% accuracy study help • Covers math, physics, chemistry, biology, and more • Step-by-step, in-depth guides • Readily available 24/7	{"url":"https://tutor.hix.ai/question/the-ksp-of-manganese-ii-carbonate-mnco-3-is-2-42-10-11-what-is-the-solubility-of-8f9af860a4","timestamp":"2024-11-09T19:27:36Z","content_type":"text/html","content_length":"582578","record_id":"<urn:uuid:4d42cc46-f082-4acd-8883-165e3159795d>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00431.warc.gz"}
how to make polygon working model (maths TLM model ) - Science Projects \| Maths TLM \| English TLM \| Physics Projects \| Computer Projects \| Geography Projects \| Chemistry Projects \| Working Projects \| Working Models \| DIY for School / College Science Exhibitions or Fair how to make polygon working model (maths TLM model ) In this article post we have given the video instruction on how to make the polygon working model (maths TLM model ) for science exhibition polygon working model (maths TLM model ) Creating a polygon working model using cardboard is a great way to understand the properties and characteristics of different polygons. Here’s a step-by-step guide to making the model: Materials you will need: • Cardboard (for the base and polygons) • Ruler • Pencil • Scissors • Glue or adhesive • Color paper (optional, for decorating) Step-by-step instructions: 1. Prepare the circular base: □ Take a piece of cardboard to serve as the base for your polygon model. The size of the base will depend on the size and number of polygons you want to include. 2. Decide on the polygons: □ Determine which polygons you want to include in your model. You can choose from various polygons such as triangles, quadrilaterals, pentagons, hexagons, etc. 3. Draw and cut out the polygons: □ Use a ruler and pencil to draw the outlines of each polygon on the cardboard. □ Carefully cut out each polygon using scissors. 4. Decorate the polygons: □ If you have color paper available, you can glue it onto the cardboard polygons to make them more visually appealing. 5. Arrange the polygons on the base: □ Glue the polygons onto the cardboard base, arranging them in rows or columns. 6. Label the polygons: □ Use markers or pens to label each polygon with its name (e.g., triangle, square, pentagon, hexagon). 7. Add details: □ You can use markers or pens to add details to the polygons, such as the number of sides and angles for each shape. 8. Demonstrate the model: □ Use the model to identify and discuss the properties of each polygon. □ Discuss the number of sides, angles, and the sum of interior angles for each polygon. This model provides a visual representation of different polygons and their properties. It’s a fun and educational project to understand the characteristics of various polygons and their significance in geometry. #workingmodel #TLMmodel #polygonworkingmodel #craftpiller #maths #mathstlm #mathsmodel Video steps on making polygon working model (maths TLM model ) Leave a Comment	{"url":"https://howtofunda.com/polygon-working-model-maths-tlm-model/","timestamp":"2024-11-04T23:06:10Z","content_type":"text/html","content_length":"62353","record_id":"<urn:uuid:99b07981-424b-4182-9f67-c5b3b6ec0ac6>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00162.warc.gz"}
Kirthevasan Kandasamy Feb 20, 2023 Abstract:In online marketplaces, customers have access to hundreds of reviews for a single product. Buyers often use reviews from other customers that share their type -- such as height for clothing, skin type for skincare products, and location for outdoor furniture -- to estimate their values, which they may not know a priori. Customers with few relevant reviews may hesitate to make a purchase except at a low price, so for the seller, there is a tension between setting high prices and ensuring that there are enough reviews so that buyers can confidently estimate their values. Simultaneously, sellers may use reviews to gauge the demand for items they wish to sell. In this work, we study this pricing problem in an online setting where the seller interacts with a set of buyers of finitely-many types, one-by-one, over a series of $T$ rounds. At each round, the seller first sets a price. Then a buyer arrives and examines the reviews of the previous buyers with the same type, which reveal those buyers' ex-post values. Based on the reviews, the buyer decides to purchase if they have good reason to believe that their ex-ante utility is positive. Crucially, the seller does not know the buyer's type when setting the price, nor even the distribution over types. We provide a no-regret algorithm that the seller can use to obtain high revenue. When there are $d$ types, after $T$ rounds, our algorithm achieves a problem-independent $\tilde O(T^{2/3}d^{1/3})$ regret bound. However, when the smallest probability $q_{\text{min}}$ that any given type appears is large, specifically when $q_{\text{min}} \in \Omega(d^{-2/3}T^{-1/3})$, then the same algorithm achieves a $\tilde O(T^{1/2}q_{\text{min}}^{-1/2})$ regret bound. We complement these upper bounds with matching lower bounds in both regimes, showing that our algorithm is minimax optimal up to lower order terms.	{"url":"https://www.catalyzex.com/author/Kirthevasan%20Kandasamy","timestamp":"2024-11-03T14:07:06Z","content_type":"text/html","content_length":"220621","record_id":"<urn:uuid:620634d5-be38-46c1-8e58-d44f2ad13ea3>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00826.warc.gz"}
Power Analysis Note that analysis of untransformed data (logscale = FALSE) is not supported. The terminology of the design argument follows this pattern: treatments x sequences x periods. With x <- pa.ABE(...), x <- pa.scABE(...), and x <- pa.NTID(...) results are given as an S3 object, which can be printed, plotted, or both. The estimated sample sizes give always the total number of subjects (not subject/sequence in crossovers or subjects/group in a parallel design – like in some other software packages). Function pa.ABE() CV CV CV none $\small{\theta_0}$ theta0 ‘True’ or assumed deviation of T from R 0.95 $\small{\pi}$ targetpower Minimum desired power 0.80 $\small{\pi}$ minpower Minimum acceptable power 0.70 design design Planned design "2x2x2" passed ... Arguments to power.TOST() none If no additional arguments are passed, the defaults of power.TOST() are applied, namely alpha = 0.05, theta1 = 0.80, theta2 = 1.25. Arguments targetpower, minpower, theta0, theta1, theta2, and CV have to be given as fractions, not in percent. CV is generally the within- (intra-) subject coefficient of variation. In replicate designs only homoscedasticity (CV[wT] = CV[wR]) is supported. For design = "parallel" it is the total (a.k.a. pooled) CV. The conventional TR\|RT (a.k.a. AB\|BA) design can be abbreviated as "2x2". Some call the "parallel" design a ‘one-sequence’ design. The "paired" design has two periods but no sequences, e.g., in studying linear pharmacokinetics a single dose is followed by multiple doses. A profile in steady state (T) is compared to the one after the single dose (R). Note that the underlying model assumes no period effects. Function pa.scABE() CV CV CV none $\small{\theta_0}$ theta0 ‘True’ or assumed deviation of T from R 0.90 $\small{\pi}$ targetpower Minimum desired power 0.80 $\small{\pi}$ minpower Minimum acceptable power 0.70 design design Planned replicate design "2x2x3" regulator regulator ‘target’ jurisdiction (see below) "EMA" nsims nsims Number of simulations 1e5 passed ... Arguments to power.scABEL() or power.RSABE() none If no additional arguments are passed, the defaults of power.scABEL() and power.RSABE() are applied, namely alpha = 0.05, theta1 = 0.80, theta2 = 1.25. Note the recommended default $\small{\ theta_0}$ 0.90 for HVDPs. regulator can be "EMA", "HC", "GCC", or "FDA". Arguments targetpower, minpower, theta0, theta1, theta2, and CV have to be given as fractions, not in percent. CV is the within- (intra-) subject coefficient of variation, where only homoscedasticity (CV[wT] = CV[wR]) is supported. Function pa.NTIDFDA() CV CV CV none $\small{\theta_0}$ theta0 ‘True’ or assumed deviation of T from R 0.975 $\small{\pi}$ targetpower Minimum desired power 0.80 $\small{\pi}$ minpower Minimum acceptable power 0.70 design design Planned replicate design "2x2x4" nsims nsims Number of simulations 1e5 passed ... Arguments to power.NTID() none If no additional arguments are passed, the defaults of power.NTID() are applied, namely alpha = 0.05, theta1 = 0.80, theta2 = 1.25. Note the default $\small{\theta_0}$ 0.975 for NTIDs since the FDA requires tighter batch release limits of ±5% for them. Arguments targetpower, minpower, theta0, theta1, theta2, and CV have to be given as fractions, not in percent. CV is the within- (intra-) subject coefficient of variation, where only homoscedasticity (CV[wT] = CV[wR]) is supported.	{"url":"https://cran.r-project.org/web/packages/PowerTOST/vignettes/PA.html","timestamp":"2024-11-02T00:14:37Z","content_type":"text/html","content_length":"167439","record_id":"<urn:uuid:9dca28a3-b6bb-4b99-bff3-233b9ad5dbee>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00150.warc.gz"}
What is 6 million written in scientific notation? \| Socratic What is 6 million written in scientific notation? 1 Answer You need to move the decimal point to where you have a number between 1 and less than 10 (like 9.99999) $6.0 \times {10}^{n}$ Now to find the value of the n, count how many places you need to move the decimal point in order to change it from 6,000,000. to 6.0. My count says 6 places. So the scientific notation version of 6 million is $6.0 \times {10}^{6}$ you could also say $6 \times {10}^{6}$. I hope this helps, Impact of this question 18605 views around the world	{"url":"https://socratic.org/questions/what-is-6-million-written-in-scientific-notation#572911","timestamp":"2024-11-03T19:19:00Z","content_type":"text/html","content_length":"33273","record_id":"<urn:uuid:74cfcefd-9bc1-463f-9365-f865407d7ce1>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00543.warc.gz"}
[Solved] A reversible path's integral of dQ/T is given by A reversible path's integral of dQ/T is given by Answer (Detailed Solution Below) Option 3 : Sf - Si Building Materials for All AE/JE Civil Exams Mock Test 9.1 K Users 20 Questions 20 Marks 20 Mins Entropy - • The second law leads to the definition of a new property called entropy. • The Clausius inequality forms the basis for the definition of a new property called entropy. • For an internally reversible process, the cyclic integral of δQ / T is zero. • A quantity whose cyclic integral is zero depends on the state only and not the process path, and thus it is a property. • Clausius in 1865 realized that he discovered a new property and he called it entropy. $dS = \frac{\Delta Q}{T}\;$ • For, a reversible path's integral of dQ/T is given by the change in entropy between the final and initial points of that path i.e. S[f] - S[i] • So, option 3 is correct. Latest MP Vyapam Sub Engineer Updates Last updated on Sep 25, 2024 -> MP Vyapam Sub Engineer Recruitment 2024 Answer Key has been released for the exam which was held from 19th September 2024 onwards. -> A total of 283 vacancies have been announced. Candidates had applied online from 5th to 19th August 2024. -> The MP Vyapam Sub Engineer exam aims to recruit individuals for Sub Engineer positions across various government departments in Madhya Pradesh. -> Candidates can check MP Vyapam Sub Engineer Previous Year Papers for better preparation!	{"url":"https://testbook.com/question-answer/a-reversible-paths-integral-of-dqt-is-given--63887b2d5a4b0f905ba26067","timestamp":"2024-11-03T17:24:54Z","content_type":"text/html","content_length":"216832","record_id":"<urn:uuid:4b8a4084-715f-43ca-aed3-fddf72db208f>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00339.warc.gz"}
Analysis of Spatial Sensitivity Based on Electrostatic Monitoring Technique in Oil-Lubricated System Analysis of Spatial Sensitivity Based on Electrostatic Monitoring Technique in Oil-Lubricated System () 1. Introduction In the past, bearings in industrial rotating machinery caused arcing due to arc voltage and shaft current, and in the actual operation, it was impossible to observe the mechanism of electric corrosion in real time, and the motor was easily damaged. Non-contact monitoring of the contact damages sites and products of the mechanism using electrostatic induction technology [1] [2] . The principle is based on the metal contact pair which is generated on the surface during contact, scratching, falling off, crack propagation, and formation of a white layer and a phenomenon of trace static. This technology enables real-time monitoring of the surface condition of the part, which gives the maintenance plan ample preparation time and makes predictive maintenance possible. Aero engine Lubricating Oil Road Abrasive Monitoring is the monitoring of bearings, gears and other lubricating components of engines and wind turbine gears, and predicts the impending failure of lubricating components, thereby effectively predicting the life of components in order to protect the engine and gearbox, safety, saving maintenance costs and maximizing economic benefits. Therefore, it is one of the important components of the engine and its gearbox condition monitoring system. Oil analysis is an effective means of monitoring wear and tear. When the engine is operating with the gearbox, it is often found in the lubricating oil that the lubricating parts of the engine and the gearbox begin to wear. For example, the lubricating parts of the engine and the gearbox are slid, fatigued or creeped, and the small bucks will peel off the substrate to form abrasive particles and flow with the lubricating oil. Therefore, the formation of abrasive particles is the result of failure of the friction surface, and the abnormal amount or large abrasive particles found in the oil monitoring indicate excessive wear or fatigue failure of the components in the engine and the gearbox. In the past, the oil monitoring of the machine was to collect oil samples on site, and went to the laboratory to complete the analysis. After the analysis, the data was sent to the on-site maintenance personnel and decision makers. Due to the large amount of time required to transfer oil samples and data, it takes several hours to complete an oil sample analysis, and up to two months. According to the survey, 50% of the off-line analytical oil samples showed no problems, and 45% of the offline analytical oil samples showed that the failure was about to occur, and only 5% detected serious problems. In this way, not only does it consume a lot of manpower and material resources, but it is also difficult to guarantee the timeliness of the data. According to the National Aeronautics and Space Administration (NASA) study, the oil-abrasive monitoring method is more reliable than the vibration analysis method widely used in aeronautical engines in the wear failure analysis method. The wear state can be directly obtained by monitoring and analyzing the abrasive grain information [1] [2] . Electrostatic sensors are a new type of sensor for abrasive particle monitoring in aircraft engine oil systems. Compared with the traditional abrasive particle monitoring technology, real-time online monitoring can be realized, and the sensor structure is simple to install and convenient. The oil system electrostatic sensor was first developed by the British company Stewart Hughes and has been applied to the F-35 engine lubricating oil road surface monitoring system [3] . The research on the basic characteristics of electrostatic sensors is mainly focused on the gas-solid two-phase flow detection method. Gajewski, Yan et al. [4] [5] [6] used simulated analysis and experimental methods to measure the concentration and velocity of transported particles, while Xu et al. proposed an improved electrostatic sensor measurement model. The model does not exhibit specific mathematical expressions, using finite element software for simulation analysis [7] [8] . Huang et al. established a point charge model on the lube road and verified that the electrical signal is similar to the theoretical analysis voltage output of the analog monitoring experimental station. Mao et al. improved the mathematical model of the electrostatic sensor by equivalently equating the point charge into a ring shape, and verified the correctness of the mathematical model in the calibration experiment. From the current research status at home and abroad, Lubricating Oil Road Particle Static Monitoring Technology [9] [10] [11] can detect component degradation earlier than traditional vibration and temperature monitoring methods, and can provide real-time monitoring information for PHM. Although experimental studies have shown that the waveform of the monitoring voltage is related to the nature of the charge carried by the particles, the electrostatic sensor can distinguish the abrasive grains of the buckling block and the non-metallic material, and qualitatively obtain the level of the monitoring voltage and the particle size. The conclusion about the size, however, lacks a theoretical basis to explain the relationship between the induced voltage waveform and the charged characteristics of the abrasive particles, and lacks research on the relationship between the abrasive charging characteristics and the abrasive material. Therefore, it is necessary to improve the electrostatic monitoring of the lubricating oil particles. The physical model, and using Coulomb’s law and Gauss’s theorem to establish a mathematical model of the particle electrostatic monitoring system, so that it can clearly indicate the relationship between the amount of electricity carried by the particles and the output of the electrostatic monitoring system and the factors affecting the relationship; Monitor the quantitative relationship between sensor spatial sensitivity and particle size, sensor radius, and axial length. 2. Electrostatic Sensor 2.1. Electrostatic Sensor Structure and Induction Mechanism In order to verify the input-output relationship between charged particles and sensors, to understand the sensor mechanism and study the sensor characteristics, it is necessary to establish the physical and mathematical models of the sensor. The physical structure model of the electrostatic sensor is shown in Figure 1. The sensor consists of a ring probe, an insulating layer and a metal shield. The lubricating oil flows through a pipe made of an insulating material, and the annular probe installed at the inner wall of the insulating pipe is used to sense an electrostatic signal in the fluid, and the metal shielding layer is used to shield Figure 1. Schematic diagram of the electrostatic sensor and sensing mechanism. external interference. The mobile charge inside the electrostatic sensor probe can move freely in any direction. When charged particles pass through the electrostatic sensor, the moving charge within the sensor moves under the influence of external charges until they reach the surface of the sensor probe. The movement of charge in the sensor causes current flow through the signal line and the signal can be measured by the signal conditioner. The magnitude of the induced charge generated can be obtained by the acquisition circuit. The amount of induced charge on the electrode is amplified by the charge amplifier, converted into an analog voltage signal, and then converted into a digital signal by an AID conversion circuit and stored in a computer for analysis and processing. The electrostatic sensor has a ring shape and a rod shape. The advantage of the ring probe is that it does not interfere with the flow field and is simple to install. The disadvantage is that it is not sensitive to charged particles passing through the center of the fluid, and is not suitable for the case where the diameter of the fluid is large. The rod-shaped probe is usually mounted on the side of the fluid conduit and inserted into the fluid. The advantage is that it can sense the charged particles in the center of the fluid. The disadvantage is that it is not easy to install, it needs to be perforated in the fluid pipeline, and the probe may have a certain influence on the internal fluid. 2.2. Electrostatic Sensor Mathematical Model When the lubricating oil flows in the pipe, it generates static electricity, which affects the safety of the mechanical equipment. Scholars have proposed a number of computational models for oil flow charging, with which the degree of oil flow can be predicted. However, there are many factors affecting the electrochemical reaction involved in the interface of the tubing wall, and the mechanism of action between the influencing factors is very complicated. The proposed model often contains some variables that are difficult to determine in the actual situation, which affects the model. Promote the application. Based on the mathematical model of the electrostatic sensor based on the principle of point charge electrostatic induction, Professor Yan emphasizes the dielectric constant in the lubricating oil. Under the condition of the known velocity distribution of the oil in the pipeline, the sensitivity characteristics of the electrostatic sensor are studied. In Figure 1, the electrostatic effect of the charged charge on the charged particles is not counted. Due to Coulomb’s law, the electric field strength at any point on the point charge electrode is: $E=\frac{q}{4\text{π}\epsilon {d}^{2}}$(1) where q is the size of the induced charge and d is the distance that the point charge corresponds to the electrostatic field generated on the probe. The dielectric constant of the lubricating medium $\epsilon ={\epsilon }_{0}{\epsilon }_{r}$(2) where ε[0] is the permittivity of free space and ε[r] is the relative permittivity. The electric flux density D at a certain point in free space is the number of flux lines on a curved surface perpendicular to the flux line divided by the area of the surface, calculated as follows: $D=\epsilon E$(3) The induced charge Q of the entire probe is equal to the flux passing through the closed surface where the probe is located. $Q=\underset{s}{\oint }D\cdot \text{d}S$(4) where dS is a microelement surface size anywhere on the sensor surface and x is the radial position of charged particles. The axial position z of charged particles can be obtained by: where v is the speed of movement of charged particles, and t is the movement time of charged particles. 2.3. Electrostatic Sensor Spatial Sensitivity The sensitivity of the sensor is defined as the ratio of the output of the sensor to the input. For easy identification, the peak value of the signal pulse can be defined as the output value of the sensor. Then, the sensitivity of the electrostatic sensor can be expressed as the ratio between the induced charge and the amount of the abrasive particles when the abrasive grains pass through the center section of the electrode. Formula can be expressed as: Through the analysis of mathematical models, combined with the above derivation, the closed surface formed by −L/2 to L/2 utilizes Gauss’s law, and the sensitivity generated by the entire probe is given by: ${S}_{p}=\|\frac{Q}{q}\|=\frac{R}{\text{2π}}{\int }_{0}^{\text{π}}\frac{R-x\mathrm{cos}\varphi }{{F}^{2}\left(x,\varphi \right)}\left\{\frac{z+L}{{\left[{\left(z+L\right)}^{2}+{F}^{2}\left(x,\varphi \ right)\right]}^{\frac{1}{2}}}-\frac{z-L}{{\left[{\left(z-L\right)}^{2}+{F}^{2}\left(x,\varphi \right)\right]}^{\frac{1}{2}}}\right\}\text{d}\varphi$(7) $F\left(x,\varphi \right)={\left({R}^{2}+{x}^{2}-2Rx\mathrm{cos}\varphi \right)}^{0.5}$(8) where R is the radius of the probe and L is half the axial length of the probe. Q is the original charge of the point charge and q is the size of the induced charge. 2.4. Electrostatic Sensor Input and Output Characteristics For electrostatic sensors, the input is the induced charge in the sensing region, which can be measured using a suitable device, and the corresponding output is the voltage signal that is collected and processed by the electrostatic sensing system. Figure 2 shows the actual equivalent circuit model of the electrostatic sensor electrode. According to Kirchhoff’s law of charge where $R={R}_{e}\cdot {R}_{i}/\left({R}_{e}+{R}_{i}\right)$ , $C={C}_{e}+{C}_{i}+{C}_{c}$ . ${C}_{e}$ , ${R}_{e}$ are the equivalent capacitance and insulation resistance of the electrode, respectively, the equivalent input capacitance and input impedance of the interface circuit, which is the distributed capacitance of the cable, which is the induced power on the electrode. If the initial condition is zero, a Laplace transform on the above equation can be obtained: where ${U}_{i}\left(s\right)$ is the Laplace transform of the output voltage ${u}_{i}\left(t\right)$ of the interface, $Q\left(s\right)$ is the Laplace of the electrostatic sensor output induced Figure 2. Electrostatic sensor equivalent circuit. $q\left(t\right)$ . If the condition $\|jwRC\|\ll 1$ is satisfied, then Equation (10) can be simplified to: Then the time domain response is: Equation (12) shows that the input voltage of the interface circuit is proportional to the rate of change of induced charge on the probe (inductive current), so the interface circuit is resistive. In relatively clean environments, charge measurements from different laboratories using different instruments may result in different absolute values due to cables, charge amplifiers, connectors, and the like. However, the output voltage is always proportional to the coefficient R[i]. 2.5. Field of View Since the electrostatic sensor detects charge only in the sensing area of the detector, the “field of view” is introduced as a basic and important parameter of the electrostatic sensor. The field of view is defined as the measurable maximum spatial range detected by the electrostatic sensor as it induces charge through the sensing area of the sensor probe. The electrostatic sensor cannot detect an out-of-range charge portion and is therefore a “blind zone” for the sensor. The field of view is an important parameter that characterizes the range of the sensor and can also be used to optimize the design of the electrostatic sensor. It is assumed that the inductive point charge passes through the electrostatic sensor at a constant radial position at a constant velocity v[c]. When the level first rises above the baseline, the peak starts from p[1], and when the level returns to the baseline level, the peak ends at p[2]. The duration from p[1] to p[2] is considered to be the charge duration through the probe t[p]. Therefore, the static television field m[p] at the corresponding radial position can be calculated by the following equation [11] . Duration t[p] can be used ${t}_{p}={t}_{p2}-{t}_{p1}$ to calculate. The total field of view in the axial direction of the sensor is a collection of this field. View all corresponding radial positions. Since the field of view of the electrostatic sensor in the radial direction is limited to the inside of the sensor, the field of view of the electrostatic sensor is generally referred to as the axial range. Due to the symmetrical nature of the ring probe, the field of view is the same on both sides of the center of the probe. Therefore, the survey is only simplified for one side of the sensor. The entire field of view can then be obtained and can be doubled by a half range. 3. Electrostatic Sensor Spatial Sensitivity Analysis 3.1. Axial Length Affects Sensitivity The axial length of the electrode has an effect on the electric field distribution of the sensitive space of the sensor. Therefore, in order to determine the length of the electrode, it is necessary to investigate the influence of the length of the electrode on the distribution of the sensitivity of the electrostatic sensor. The sensitivity of the sensor with the axial length of the electrode of 4 mm, 10 mm, 20 mm and 40 mm was simulated by MAXWELL. The calculation results are shown in Figure 3. Figure 3 shows the variation of the sensitivity of the center cross section of each probe in the radial direction in the case where the axial length of the probe is different. It can be seen from the figure that when the axial length of the electrode is increased, the value of the sensitivity of the radial position on the cross section is increasing, but not in a proportional relationship. As the axial length of the electrode increases, the sensitivity change parameter S gradually decreases, that is, the sensitivity tends to be uniform, and the sensitivity of the sensor is also increased, that is, the longer the electrode, the better. However, excessively increasing the length of the electrode can cause significant spatial filtering effects, and excessively increasing the axial length of the electrode can make the sensor less robust and the sensor probe susceptible to bending. Under the condition of ensuring the sensitivity and sensitivity of the sensor, the characteristics of the integrated sensor are finally selected to be 10 mm in axial length. 3.2. Effect of Ring Probe Radius on Sensitivity The installation position of the inlet ring sensor determines that the ring should be as close as possible to the pipe wall so that it does not affect the air flow field. In order to investigate whether the sensor has an influence on the sensitivity in the inlets of different diameters, the electrodes with diameters of 20 mm, 50 mm, 80 mm and 100 mm were simulated. Before the simulation, the finite element model needs to be modified to adjust the air domain between the electrode casing and the electrode. The calculation results are shown in Figure 4. 3.3. Sensitivity Analysis in the Radial Direction In order to further discover the influence of the radius and axial length of the Figure 3. Radial center section sensitivity of electrodes with different axial lengths. electrostatic sensor on the sensitivity, select z = 0, x = 0.01, and use matlab to obtain the influence of the two on the sensitivity. As shown in Figure 5, in the radial direction, the range of the radius of the electrostatic sensor $0.01<R<0.05$ is selected, and the range of the axial length $0<L<0.1$ is the sensitivity of the probe at different positions in the space. Figure 5 shows the sensitivity distribution of sensors with different radii and different axial lengths. From the figure, the size of the induced probe radius has little effect on the sensitivity distribution trend of the induced probe, and the sensitivity space increases slightly with the increase of the radius. According to the same distance from the side of the probe, the sensitivity of the probes with different radii is distributed along the axial position. It can be seen from the figure that in the same distance from the side, at the same axial position, the radius Figure 4. Radial center section sensitivity of different diameter electrodes. Figure 5. Effect of L and R on sensitivity in radial x = 0.01. is large. The sensitivity of the sensor is relatively large. The radial sensitivity distribution when two groups were selected (L = 0.05, R = 0.05; L = 0.0125, R = 0.05) is clearly as shown in Figure 6. The sensitivity gets the maximum value at x = 0, which is the center of the pipe. The closer to the tube wall sensitivity, the maximum is obtained. The larger the shaft diameter ratio, the greater the sensitivity. 3.4. Sensitivity Analysis in the Axial Direction As shown in Figure 7, in the axial direction, the range of the radius of the electrostatic sensor is selected, and the range of the axial length is the sensitivity of the probe at different positions in the space. Figure 7 shows the sensitivity Figure 6. Sensitivity radial distribution at z = 0. Figure 7. The influence of L and R on the sensitivity in the axial direction at z = 0.01. distribution of sensors with different radii and different axial lengths. It is apparent in the picture that the longer the axial length, the greater the sensitivity. The axial sensitivity distribution when two groups are selected (L = 0.02, R = 0.01; L = 0.02, R = 0.04) is clearly as shown in Figure 8. As can be seen from Figure 8, the electrostatic sensor is located at the same radial length, and the distance from the sensor to the z-axis is further, that is, the closer to the sensor wall, the higher the spatial sensitivity of the electrostatic sensor, and vice versa, the lower the spatial sensitivity. At the same time, as the radial length of the electrostatic sensor becomes longer, the spatial sensitivity of the sensor is lower at the same radial position. It can also be understood that the greater the ratio of the axial length to the radial diameter of the electrostatic sensor, the higher the spatial sensitivity. Figure 5, Figure 6, and Figure 7 confirm each other, thus demonstrating the correctness of the spatial sensitivity formula, and thus the mathematical model established in this paper is correct. In addition, the spatial sensitivity of the electrostatic sensor is also affected by factors such as the moving speed of the charged abrasive particles, the thickness of the insulated pipe and its dielectric constant, and these influencing factors are independent of the physical properties of the electrostatic sensor itself. 4. Spatial Sensitivity Distribution Characteristics According to the mathematical model of the electrostatic sensor, when x = 0, the electrostatic sensor obtains the maximum sensitivity at x = 0, which is of great significance for the study of axial sensitivity. The factors that affect the spatial sensitivity of the electrostatic sensor in the axial direction are mainly the position of the charged abrasive particles and the axial length and radial radius of the sensor. The axial length and radius influence of the sensor have been obtained by simulation. Since the electrostatic sensor has a radial symmetry characteristic, when the charged abrasive grain passes through the electrostatic sensor sensing Figure 8. Axis distribution of sensitivity at x = 0. area at a certain position in the radial direction of the sensor, the axial length of the sensor Under the condition of L constant, the sensitivity of the sensor is only related to the relative position of the charged abrasive particles in the radial direction. For the convenience of observation, let x = 0, L = 0.01 m, and obtain the axial distribution of the electrostatic sensitivity of the electrostatic sensor as shown in Figure 9. Inside the probe, it can be seen from Figure 9 that the closer the radial position is to the probe wall, the higher the spatial sensitivity and the sensitivity has a maximum at the central axis of the pipe, consistent with the previous conclusions. If the trajectory of the point charge (abrasive grain) in the radial position cannot be known, only the sensing area of the sensor near the axis can be selected, that is, the sensor probe of the sensor is embedded in the insulating layer, and only the static sensitivity of the oil is allowed. The flow is changed in a gently changing area, and the output of the sensor can be approximated regardless of the influence of the radial position. This has important guiding significance for the design of the sensor. 5. Conclusions 1) Reducing the radial radius of the sensor can effectively increase the spatial sensitivity of the electrostatic sensor. The closer the charged particles are to the wall surface of the pipe in the radial position, the higher the spatial sensitivity. The installation of the electrostatic sensor close to the monitoring surface during installation can effectively improve the electrostatic sensor sensitivity. The sensitive field of the sensor is greater than the axial length of the probe, so no sources of interference can occur in the sensitive field. 2) The larger the axial length L of the electrode, the higher the sensitivity, the Figure 9. Axial sensitivity distribution of electrostatic sensors. more uniform the distribution of the sensitive field of the section, and the corresponding range of the axial sensitive space; but the electrode is too long, the sensor is facing the fluid space due to the electrostatic filter effect of the electrostatic sensor. The high frequency signal of the structure loses its responsiveness, causing significant spatial filtering effects, making the sensor less robust and the probe susceptible to bending. The axial length of the electrostatic sensor must be chosen reasonably. 3) The larger the axial diameter ratio in the axial direction and the radial direction, the more sensitive the sensor is and the more sensitive it is. The value of the shaft diameter ratio should be as large as possible.	{"url":"https://scirp.org/journal/paperinformation?paperid=88207","timestamp":"2024-11-11T01:45:04Z","content_type":"application/xhtml+xml","content_length":"130039","record_id":"<urn:uuid:d744156b-e651-49e1-80f6-a6261c3e5440>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00082.warc.gz"}
An optimal randomized logarithmic time connectivity algorithm for the EREW PRAM (extended abstract) Improving a long chain of works we obtain a randomized EREW PRAM algorithm for finding the connected components of a graph G = (V, E) with n vertices and m edges in O(log n) time using an optimal number of O((m + n)/log n) processors. The result returned by the algorithm is always correct. The probability that the algorithm will not complete in O(log n) time is at most n^-c for any desired c > 0. The best deterministic EREW PRAM connectivity algorithm, obtained by Chong and Lam, runs in O(log n log log n) time using m + n processors. Publication series Name Proceedings of the 6th Annual ACM Symposium on Parallel Algorithms and Architectures, SPAA 1994 Conference 6th Annual ACM Symposium on Parallel Algorithms and Architectures, SPAA 1994 Country/Territory United States City Cape May Period 27/06/94 → 29/06/94 Dive into the research topics of 'An optimal randomized logarithmic time connectivity algorithm for the EREW PRAM (extended abstract)'. Together they form a unique fingerprint.	{"url":"https://cris.tau.ac.il/en/publications/an-optimal-randomized-logarithmic-time-connectivity-algorithm-for","timestamp":"2024-11-09T14:34:49Z","content_type":"text/html","content_length":"50081","record_id":"<urn:uuid:49e213a5-26c2-4bbd-8e38-0e3e3d154f50>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00371.warc.gz"}
Introducing the COVID-19 Simulator and Machine Learning Toolkit for Predicting COVID-19 Spread There have been breakthroughs in understanding COVID-19, such as how soon an exposed person will develop symptoms and how many people on average will contract the disease after contact with an exposed individual. The wider research community is actively working on accurately predicting the percent population who are exposed, recovered, or have built immunity. Researchers currently build epidemiology models and simulators using available data from agencies and institutions, as well as historical data from similar diseases such as influenza, SARS, and MERS. It’s an uphill task for any model to accurately capture all the complexities of the real world. Challenges in building these models include learning parameters that influence variations in disease spread across multiple countries or populations, being able to combine various intervention strategies (such as school closures and stay-at-home orders), and running what-if scenarios by incorporating trends from diseases similar to COVID-19. COVID-19 remains a relatively unknown disease with no historic data to predict trends. We are now open-sourcing a toolset for researchers and data scientists to better model and understand the progression of COVID-19 in a given community over time. This toolset is comprised of a disease progression simulator and several machine learning (ML) models to test the impact of various interventions. First, the ML models help bootstrap the system by estimating the disease progression and comparing the outcomes to historical data. Next, you can run the simulator with learned parameters to play out what-if scenarios for various interventions. In the following diagram, we illustrate the interactions among the extensible building blocks in the toolset. In this post, we describe in detail how our disease simulation works, how simulation parameters are learned using supervised learning, and predict the incidence of disease given an intervention Historical trends for infectious diseases We provide several notebooks in our open-source toolset to run what-if scenarios at the state level in the US, India, and countries in Europe. In these notebooks, we use various data sources that frequently publish the number of new cases. For example, for the US, we use the Delphi Epidata API from Carnegie Mellon University (CMU) to access various datasets, including but not limited to the Johns Hopkins Center for Systems Science and Engineering (JHU-CSSE), survey trends from Google search and Facebook, and historical data for H1N1 in 2009–2010. We can use our notebook, covid19_data_exploration.ipynb, to overlay historical data from previous pandemics with COVID-19. For example, the following graphs compare COVID-19 to seasonal flu and the H1N1 pandemic in California, Texas, and Illinois. The first graph shows the 7-day average of the number of incidences in California during seasonal flu, H1N1, and COVID-19. Although COVID-19 cases peaked in summer for most states in the US, there are exceptions. In Illinois, the most cases occurred early in the year, similar to the H1N1 peak in spring. On the other hand, in other states such as Texas, we observe a potential peak aligning with the H1N1 peak in fall. The trends differ greatly across states and countries. Therefore, we provide notebooks that enable you to run what-if scenarios by learning from existing data and projecting into the future using anticipated peaks. Results from running what-if scenarios The notebook covid19_simulator.ipynb has a comprehensive list of regions and countries across the world to run what-if scenarios. In this section, we discuss various what-if scenarios for France, Italy, the US, and Maharashtra, India. First, we use ML to predict the disease trends, including peaks and waves based on parameters specified by the user (for example, we use 3 months of COVID-19 case data to bootstrap and follow the H1N1 trend or create a second or third wave after 6 months from the first wave). Next, we play out intervention scenarios such as mild intervention or strict intervention and discuss the results. For France, we considered the what-if scenario of having a stricter intervention and a second wave in 6 months but with a higher peak than the first wave, as expected in H1N1-like trends. The following graphs compare the daily number of cases and cumulative number of cases. Our projection in this scenario (orange line) closely matches with the actual curve (blue line). For Italy, we consider the scenario of having a mild intervention policy versus a stricter intervention policy and a second wave in 6 months with a higher peak, as expected in H1N1-like trends. The first set of graphs shows the number of daily cases and total cases with a mild intervention policy. The following graphs compare daily and total case numbers with a stricter intervention policy. During the first wave, the milder intervention projection initially matches better, and stricter interventions result in a decline. Therefore, in this what-if scenario, we can see how our model captures varying interventions. However, although the trend with the second wave matches our predictions, using the assumption of a second wave in 6 months doesn’t align with the new trend. Therefore, the best projection for Italy’s what-if scenario should shift the second wave to where we usually expect H1N1 would have been—specifically, in fall. For our US scenario, we considered having another wave in 3 months while keeping a stricter level of interventions. Overall, the US aligns better with stricter intervention scores based on the invention scoring mechanism provided by the Oxford Coronavirus Government Response Tracker, which we use for all the examples in this post and notebook. In this what-if scenario, we start observing a trend for another wave that aligns with H1N1-like trends in fall. Maharashtra, India Our Maharashtra, India, scenario considers having another wave in 6 months while keeping the same level of intervention policies. The graph on the left shows the actual (blue) and estimated (orange) number of cases. The graph the right is the cumulative number of cases. In this scenario, we can see the impact of experiencing a second wave is similar to the first one. Disease simulator We model the disease progression for each individual in a population using a finite state machine, and then report out the aggregate state of the population. We assign a probability distribution to the disease parameters for each individual, parametrized by a mean, standard deviation, and lower and upper limits. For example, you can set parameters such as individuals will develop symptoms within 2–5 days after exposure, with the majority of the population developing symptoms in 2–3 days. Similarly, you can set parameters for the recovery period, such as within 14–21 days after exposure. The stochasticity allows for variation in the population at the individual level to mimic real-world scenarios. Our finite state machine is similar to the simulation model in COVID-19 Projections Using Machine Learning, with additional states for infection transmission by asymptomatic individuals, as shown in the following diagram. The default state machine is extensible in the sense that you can add any disease progression state to the model as long as the state transitions are well-defined from and to the new state. For example, you can add the state for having tested positive. Our disease simulation can also capture population dynamics. The transition from one state to the next for an individual is influenced by the states of the others in the population. For example, a person transitions from a Susceptible to Exposed state based on factors such as whether the person is vulnerable due to pre-exiting health issues or interventions such as social distancing. Theoretically, our simulation model iterates each individual’s state within an automata network [4]. The state transition probabilities are driven by two types of factors: • Individual, disease-specific factors – The probability densities assigned to the individual on how soon the symptoms will appear dictates transitioning from an Exposed state to Onset of systems • Population, transmission-specific factors – The probability to transition from Susceptible to Exposed is higher for an individual with a larger social network or exposure to infected individuals Learning simulation parameters The simulation has different types of parameters. Some of these parameters are known or discovered by researchers and scientists, such as the number of days prior to the onset of symptoms. Other parameters such as transmission rate, which varies greatly among populations and rapidly over time, can be learned from actual data published by agencies. The following are three core simulation parameters learned by ML methods: • Transmission rate – Transmission rate can be derived either directly from the recent case counts of the target location or as an expected value from the transmission rates of the countries matching the transmission rate pattern of the target location. As most of the regions (country / state / county) have now surpassed the peak of the 1^st wave or already entered the 2^nd wave, the transmission rate can be measured more reliably from respective country’s daily confirmed-cases data itself. • Time (weeks) to reach the peak of the first wave of infection – This parameter can be learnt from the countries with matching transmission rate patterns. For those regions that are now beyond the peak of the 1^st wave, this parameter can be captured by a sliding window analysis of the daily confirmed cases curve. In absence of sufficient matching countries, you can use a configurable range, such as 1–5 weeks. • Transmission control – This parameter is learnt from a configurable range by reducing the simulation error for a validation timeframe with known case counts. 100% interventions can not prevent 100% transmission. Interventions tend to control only a fraction of the overall transmission scope. This parameter is intended to represent that fraction and can vary a lot across regions. It is learnt by fitting the wave-1 data against values from a range (e.g. 0.1 to 1) through iterative trial simulations. Learning intervention scores We score intervention effectiveness in the following three ways: • Fitting score stringency index – Fits the daily intervention scores in a regression model with the OXCGRT-provided stringency index (score_stringency_idx) as the dependent variable. Subsequently, it extracts the intervention effectiveness scores as the ensemble-based regression model’s feature importance. • Fitting confirmed case counts – Fits the daily intervention scores in a regression model with the changes in confirmed case counts (moving average) as the dependent variable. Feature importance scores indicate intervention effectiveness. • Observing case count variations – Measures the changes in total case count by turning off the interventions one by one. Scores the interventions in proportion to the respective changes resulting from it being turned off. Finally, these three scores can be combined using a configurable weighted average. Although these approaches would be affected by the co-occurrences and correlations among the interventions, as a whole, it can represent approximate relative effectiveness scores of the interventions. Limitations of our toolset Our toolset has the following limitations: • Our disease model expects multiple waves of infections following Gaussian distribution, a pattern quite evident in past influenza pandemics [2]. • Our disease model doesn’t include death rates; however, it’s relatively straightforward to extend the finite state machine. • Country-level intervention scores, to some extent, could be applicable at lower levels, like states, and thus have been used accordingly. However, a more accurate approach would be to gather and use the intervention scores at respective regional levels. • The population size needs to be large enough due to underlying probabilistic components, such as 1,000 or more individuals. • Although we assumed that on average one person can infect three people, based on a recent study published in Oxford Academic, we anticipate this number to vary across populations. Therefore, it’s a configurable parameter in our base. Several studies indicate that individuals who don’t exhibit symptoms are the largest transmitters. • Our model is designed to simulate the first 2 waves of the infection in our code repository and additional waves can be added as required. Toolset architecture deep dive In this section, we dive deep into the five main components of the toolset architecture. • Bootstrapping – This block exposes the configurable parameters. The configurable parameters can be adjusted between 0.1 to 1.0. Similarly, wave1_weeks was initially 2 weeks; now its range is 1–5 • Infection Wave(s) Analysis: We do a sliding window analysis of the smoothened daily confirmed cases data to detect the starting point and the peak of the 1^st and 2^nd. This information is subsequently used to infer the parameters of the underlying probability distributions that work at the core of the simulator. • Intervention effectiveness scorer – We use supervised learning to estimate an intervention effectiveness score for a population using the research data from OXCGRT [1] or similar sources. Then we create a weighted average score. • Optimizer – The optimization model iteratively varies the parameters to be learned and reduces the error in predicting the incidence rate based on the historical data. • Predictions – After the simulation parameters are learned for a specific country or population, we can use the intervention effectiveness scores (the what-if scenario of a given disease progression pattern over time, such as the second peak in June) to run our simulation to predict the relative impact of an intervention in future. Toolset inputs and outputs The inputs are as follows: • Daily infection counts from over 60 countries • Daily country-level rating of over 10 interventions, such as stay-at-home orders or school closures • A disease incidence pattern with peaks and their timing and duration • Simulation duration Our output is the incidence rate over the course of the simulation. Our open-source code simulates COVID-19 case projections at various regional granularity levels. The output is the projection of the total confirmed cases over a specific timeline for a target state or a country, for a given degree of intervention. Our solution first tries to understand the approximate time to peak and expected case rates of the daily COVID-19 cases for the target entity (state/country) by analysis of the disease incidence patterns. Next, it selects the best (optimal) parameters using optimization techniques on a simulation model. Finally, it generates the projections of daily and cumulative confirmed cases, starting from the beginning of the outbreak uptil a specified length of time in the future. To get started, we have provided a few sample simulations at state and country levels in the covid19_simulator.ipynb notebook in https://github.com/aws-samples/covid19-simulation, which you can run on Amazon SageMaker or a local environment. [1] Oxford Coronavirus Government Response Tracker https://www.bsg.ox.ac.uk/research/research-projects/coronavirus-government-response-tracker [2] Mummert A, Weiss H, Long LP, Amigó JM, Wan XF (2013) A Perspective on Multiple Waves of Influenza Pandemics. PLOS ONE 8(4): e60343. https://doi.org/10.1371/journal.pone.0060343 [3] Viceconte, Giulio, and Nicola Petrosillo. “COVID-19 R0: Magic number or conundrum?.” Infectious disease reports vol. 12,1 8516. 24 Feb. 2020, doi:10.4081/idr.2020.8516 https:// [4] https://nyuscholars.nyu.edu/en/publications/automata-networks-and-artificial-intelligence) Q. What is incidence rate vs. prevalence rate? Incidence refers to the occurrence of new cases of disease or injury in a population over a specified period of time. Incidence rate is a measure of incidence that incorporates time directly into the denominator. Prevalence differs from incidence, in that prevalence includes all cases, both new and pre-existing, in the population at the specified time, whereas incidence is limited to new cases Q. How are the interventions effectiveness scored? Interventions effectiveness can be scored in the following three ways: • Fitting score stringency index – Fits the daily intervention scores in a regression model with the OXCGRT-provided stringency index (score_stringency_idx) as the dependent variable. Subsequently, it extracts the intervention effectiveness scores as the ensemble-based regression model’s feature importance. • Fitting confirmed case counts – Fits the daily intervention scores in a regression model with the changes in confirmed case counts (moving average) as the dependent variable. Feature importance scores indicate intervention effectiveness. • Observing case count variations – Measures the changes in total case count by turning off the interventions one by one. Scores the interventions in proportion to the respective changes resulting from it being turned off. Finally, these three scores can be combined using a configurable weighted average. Although these approaches would be affected by the co-occurrences and correlations among the interventions, as a whole, it can represent an approximate relative effectiveness scores of the interventions. Q. How are we computing expected transmission rate and time to peak? Given the latest case rate and transmission rate growth patterns of a region, the solution first identifies the countries that have exhibited similar patterns in the past and eventually plateaued. From these reference countries, it determines the time-to-peak and mean/median daily growth rates until the peak. When a considerable number (default 5) of countries found with matching patterns, then the expected transmission rate, and time-to-peak are computed as weighted averages using pattern and population similarity levels. Q. What parameters are learned? The solution always learns the transmission probability for the location in context (country or state) by fitting the simulation model outcome against the confirmed case counts. Optionally, it can also learn the optimal time (in weeks) to reach the peak of the infection spread. Prior to optimization, the time to reach the peak is approximated from the data of other countries having similar transmission patterns. Q. Can the simulation work starting at any point of the disease progression timeline? No. One of the key parameters in the solution is the recent transmission rate change (growth). If the simulation starts at a very early stage of disease progression, the transmission rate might be too low to come up with realistic future projections. Similarly, if the simulation starts at the plateau or declining phase, the transmission rate change might be negative and therefore generate incorrect projections. This solution works best on the moderate-to-high-growth phase of the disease progression. About the Authors Tomal Deb is a Data Scientist in the Amazon Machine Learning Solutions Lab. He has worked on a wide range of data science problems involving NLP, Recommender Systems, Forecasting , Numerical Optimization, etc. Sahika Genc is a Principal Applied Scientist in the AWS AI team. Her current research focus is deep reinforcement learning (RL) for smart automation and robotics. Previously, she was a senior research scientist in the Artificial Intelligence and Learning Laboratory at the General Electric (GE) Global Research Center, where she led science teams on healthcare analytics for patient Sunil Mallya is a Principal Deep Learning Scientist in the AWS AI team. He leads engineering for Amazon Comprehend and enjoys solving problems in the area of NLP. In addition, Sunil also enjoys working on Reinforcement Learning and Autonomous Cars. Atanu Roy is a Principal Deep Learning Architect in the Amazon ML Solutions Lab and leads the team for India. He spends most of his spare time and money on his solo travels. Vinay Hanumaiah is a Deep Learning Architect at Amazon ML Solutions Lab, where he helps customers build AI and ML solutions to accelerate their business challenges. Prior to this, he contributed to the launch of AWS DeepLens and Amazon Personalize. In his spare time, he enjoys time with his family and is an avid rock climber. Nate Slater leads the US West and APAC/Japan/China business for the Amazon Machine Learning Solutions Lab. Taha A. Kass-Hout, MD, MS, is General Manager, Machine Learning & Chief Medical Officer at Amazon Web Services (AWS). 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Math Exercises & Math Problems: Analytic Geometry of the Straight Line and Plane p, if : p passing through the points : p with a direction vector d and a normal vector n passes through the point K. Find the parametric, the general and the slope-intercept equation of a straight line p, if : p, which passes through the point M, if the slope angle between a straight line p and the x-axis is φ : L and which is parallel to the given straight line p : N and which is perpendicular to the given straight line p : p and q are parallel or perpendicular to each other : (work with normal vectors of the straight lines) A [3;2], B [–1;–1] and the vector a = (12;–5), where a = C – B. a) Find the coordinates of the point C. b) Prove that the points A, B, C are vertices of a triangle. c) Find the general equations of straight lines on which lie the sides of the triangle ABC. d) Find the general equations of straight lines on which lie the medians of the triangle ABC. e) Find the general equations of straight lines on which lie the altitudes of the triangle ABC. f) Find the parametric equations of the straight line passing through the midpoints of the line segments AC and BC. g) Find the slope-intercept equation of the straight line passing through the point A and parallel to the straight line BC. h) Find the coordinates of the centroid T. i) Find the perimeter of the triangle ABC. j) Find the area of the triangle ABC. AB, AC and BC, if A [2;5], B [–3;9], C [6;12]. A [3;4], B [–1;2], C [1;3], D [–5;0] lie on one straight line. Find the parametric, the general and the slope-intercept equation of the straight line. A [4;–1;9] which is parallel to p passing through the point A [2;–1;2] perpendicularly to the plane π: x – y + z + 13 = 0. ρ = ABC, A [–4;0;2], B [–2;1;1], C [1;–3;–2]. A [2;1;4] and which is parallel to the plane β: x – 2y + 5z + d = 0. A [1;2;0] and which is perpendicular to the straight line p: x = 3 – t; y = 4 + 2t; z = 1 – 2t; t∈R. ABCDV with vertices D [0;0;0], A [4;0;0], B [4;4;0], V [2;2;6]. Find the general equation of a plane BCV. You might be also interested in: - Vectors - Conic Sections - Relative Positions of Lines and Planes - Matrices	{"url":"https://www.math-exercises.com/new/analytical-geometry/analytic-geometry-of-the-straight-line-and-plane","timestamp":"2024-11-12T16:05:07Z","content_type":"application/xhtml+xml","content_length":"79024","record_id":"<urn:uuid:ba878d86-35bb-4bcc-99b5-41a3c5d5aed1>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00147.warc.gz"}
#DevOptical Part 8: Raytracing 101 Published: 2021-09-11 \| Categories: [»] Tutorialsand[»] Optics. Now that we have set the basis for [»] transforming points and directions in 3D space, we can tackle the exquisite task of optical raytracing. Raytracing is at the core of every optical design software and allows to compute all important quantities related to image quality, including (but not limited to) spot and fan diagrams, wavefront error analysis, mtf, through-focus mtf, psf, image simulation etc. I will develop all these terms in later articles and we will now focus on how to perform real raytracing. I m using the world real here in opposition to [»] paraxial raytracing that we already covered in some depth in the previous posts. We will assume that we have a set of surfaces or bodies (e.g.: sphere, cylinders, plane ) already positioned in 3D space with their [»] local space coordinates system (abbreviated here lsp). One surface, typically a plane (but not restricted to), will emit rays that will transfer from surface to surface and follow effects such as refraction, reflection, scattering, diffraction etc. Data are then collected on image surfaces, also typically plane (but not restricted to) to assess image formation properties listed previously. Sequential vs. Non-Sequential Raytracing Optical design softwares make one more distinctions between two types of raytracing: sequential and non-sequential raytracing. In non-sequential raytracing, rays are launched into the 3D space like they would do in the real world. There is no preferred directions and rays will interact with any surface they will intersect. After intersecting surfaces, the main ray will usually refract but child rays will be created to account for reflection on surface or scattering properties of the surface. The process stops when the rays do not intercept any more surfaces which will never occur if you place them inside a closed box as they will infinitively scatter from one face of the box to the other. To avoid this, rays are associated to an energy/intensity level and they are discarded when this level drops below some threshold value. Non-sequential raytracing is particularly useful to study stray light and ghosting effects in optical systems but it tends to generate a lot of computations. It also makes the standard tool for image analysis such as wavefront error difficult to address. For this reason, optical design software introduces the concept of sequential raytracing. Most optical design jobs are actually done with this latter type of raytracing and the non-sequential mode is only used to assert robustness to ghosting and stray light. Many designers often don t bother with non-sequential raytracing and apply it only for extremely tough applications such as deployment of optics in space where the money involved allows the extra hours required to spend on these subsequent analyses. In sequential raytracing, bodies are split into surfaces contributions and are added to an ordered list. The list always starts with the object surface and ends with the image surface. Rays traverse the list from left to right and never go backward into the list. This means that if you launch N rays at the object surface, you will end up with a maximum of N rays at the image surface. I say a maximum of because some rays may be clipped (vignetted) when intercepting outside of the clear aperture of the surfaces or having total internal reflection effects. It is common to represent bunch of rays as a table where each row represents a ray and each column represent the state of the ray at the i-th surface of the system. The state of the ray usually consists of the ray position and direction (before refraction or reflection) at the surface interception in either world or local space coordinates, a flag to tell if the ray is vignetted or not and eventually other properties such as the normal of the surface at the interception position. Note that I said rays do not go backward into the list in sequential raytracing but I did not say that they cannot go backward in 3D space. You have to understand sequential raytracing as a system directing the rays from one surface to the next, forgetting about all other surfaces in the system. Most of the time, in refractive system, rays will also travel space from left to right. It is only when using reflective surfaces like in Cassegrain telescopes that you will see rays bouncing back and forth but don t be fooled: it is still sequential raytracing! Ray/Surface Interception One of the major jobs of raytracing software is to compute the interception position of a ray with a surface. The surfaces can be of any type but are usually smooth and described by analytical formulas although it is not an explicit requirement. 99% of the job also restrict to one surface type: the standard conic section: where z is the sag of the surface, r the radial coordinate, c the curvature of the surface and k the conic constant. Note that this equation is expressed in the local space coordinate of the body where z is the forward (optical) axis. When the curvature is zero, the surface decays into a plane. When the conic constant is zero, the surface decay into a sphere. I personally like to make the distinction between the three cases: plane, sphere and standard conic section. This allows for more flexibility when introducing the concept of Constructive Solid Geometry (CSG). It is also easier to introduce the concept of raytracing with plane and sphere first. To find the interception between a ray and a surface, we first need to express the surface as a function where x, y and z describe all the points on the surface. We also know that the rays have a starting position P and a direction v [»] defining a line of equation or, when splitting into the components, We usually restrict to since photons do not travel backward from their direction vector but this is not a strict rule to apply. The interception position corresponds to when the ray components match the surface equation: Since (x[0],y[0],z[0]) and (v[x],v[y],v[z]) are given, we need to solve for ξ. Once we have obtained ξ, we can compute the point coordinate from the line equation. As explained in the [»] previous post, it is usually easier to solve the problem in the local space coordinate of the surface body and the first step is to project the ray from world space to the surface local space. Once the interception position has been found in the local space coordinate system, it can be projected back to world space after having applied the surface effect if any (e.g.: refraction, reflection etc.). From the new ray coordinates, we then find the interception to the next surface and so on. Let s now analyse the resolution of the equation for the different bodies we met (plane, sphere and standard conic section). Interception with a plane When using c=0 in the conic section equation we get Which is the equation of a plane perpendicular to the z axis and located at the origin of the system coordinates. The resolution is immediate because we have and so leading to The interception coordinate is therefore The normal of the plane is always: although we could have used the opposite vector. It is important to keep the same formalism for the orientation of the normal as it will play an important role when computing refraction of rays. We see that the solution exists only when which would fail for a ray travelling parallel to the surface. It is extremely important to check for interception conditions because they will lead to invalidating the vignetting flag for the ray. Rays that do not intercept surface, or intercept them outside of the clear aperture area shall not be traced to the next surface. Interception with a canted plane It is possible to generalize the previous equations to a canted plane defined by the equation with the surface normal and d the offset to the origin. Solving the system implies The condition for the interception to occur is still the same, i.e. that the ray does not travel parallel to the plane (perpendicular to normal): You can check that setting n=(0,0,-1) and d=0 will give you the equation of the previous section. Interception with a sphere The common expression of a sphere centered at coordinates (0,0,0) is with R the radius of the sphere. The interception problem now becomes It is however easier to consider a sphere centered at the coordinates (0,0,R) such that the point (0,0,0) is solution to the problem: Then, using, we can develop the equation to We can re-arrange this equation into that we can also shorten using vector notation with p=(x[0],y[0],z[0]-R). This system can be solved only if and the solutions are If D=0 we have only one solution which is B/2A. The case A=0 shall be tested too but does not correspond to a healthy ray because it would occur if the ray has no direction vector. When doing CSG, it is important to keep both interception coordinates but with raytracing we are only interested in keeping one of the two points. When the surface is convex (R>0) we want to keep the first point of intersection (smaller ξ) but when the surface is concave (R<0) we want to keep the second point of interception (larger ξ): Finally, the surface normal is obtained from the interception coordinates from the sphere origin divided by the radius: To avoid errors, it is important to use the sphere equations only when R≠0. Interception with the standard conic section surface Optical design softwares do not bother dividing the different cases (sphere, parabola, plane, ellipse ) and use the standard conic section of equation Solving this equation for interception with a ray required a bit of re-arrangement though: and so to which we can inject the ray equations leading to the second order equation that we can rearrange into For which a solution only exists if with the solution being And, as for the sphere, Except that this time the case c=0 is a valid solution (plane). The normal of the surface is obtained using the gradient of the sag at the interception position that we will often want to normalize for the further operations. Since the system is symmetric, the expression for dz/dx and dz/dy will be similar and I will only derive the expressions for the dz/dx case. If you are not familiar with derivation operations, here is a remainder of the rules required for this post: The first thing is to express the derivative in relation to the formula using the radial coordinates: Concerning dz/dr we have Applying the rules to the standard conic section surface we get We can simplify the equations to for which a solution only exists if meaning that One of the nice features of these equations is that they involve only r^2 which limits the number of square roots which are computationally expensive. Surface with no analytical solutions Not all surfaces have analytical solution to raytracing. For instance, aspheric lenses are described by an equation of the form: where a[i] are the aspheric coefficients. The last equation is known as an even asphere in optical design softwares because it only has even terms in r powers. In practice, the sum is not infinite and manufacturers will generally consider the power terms below or equal to 10. The a[1] term (r^2) is usually skipped during optimization process because it interferes with the curvature term of the lens but manufacturers can accept a non-null a[1] term if you ask them. There are other lens types with equations having no analytical solution to raytracing such as the odd asphere, the Zernike lens or the Chebydchev lenses. In such cases, there are no other solution but to use iterative search method to solve the interception problem. Such methods are however slow and do not guarantee to converge to the solution. They may also fail during convergence even if the solution exists. In most cases, these lenses are deviations of the standard conic surface with a small departure represented by some polynomial type. Interception of the standard conic section is therefore a good starting approximation point for the rest of the process. This is not always the case and some lenses can have large deviations to the standard conic section. Toroidal lenses are a good illustration for this because they have different curvature in x and y coordinates and can therefore have large sag departure along their two principal axes. The general idea is to solve the surface equation for the parameter ξ. Among the methods that can be used are dichotomy search, simplex and Netwon-Raphson-based methods (non-exhaustive list!). I personally used the latter successfully and will quickly describe how to use them. Starting from an approximation of the interception coordinate ξ[0], the following algorithm is applied iteratively until it converges to the solution: where 0<k≤1 is a damping constant to avoid oscillating around the solution. The derivative can be either computed from the surface equation or from an estimate. There are many methods to compute an estimate of the derivative including using the finite difference between g(ξ [i]) and g(ξ[i-1]), or by fitting a polynomial to the last N g(ξ[i]) points and derivating analytically the polynomial. I have experience only when using the direct analytical derivative of the function. I recommend using k=0.1 which gave me satisfaction in the past for even aspheric lenses when using the standard conic section as approximation of the interception coordinates. Once the interception position is found, it is still required to find the surface normal. For this, we use the same equation that we used for the standard conic section: which requires computing the derivative of the sag equation in both x and y. Skipping such lengthy computations would be a good thing but aspheric lenses are now part of traditional catalogs as well so it is important to support them. Once the interception position with the surface has been found (if it exists), it is important to check if it lies in the clear aperture of the lens. To understand why apertures are necessary in the computation, imagine a plane surface. We know that it is possible to find the interception with that surface as long as the ray direction is not perpendicular to the surface normal. But it is still possible to intercept the surface very far away for rays that are almost parallel to the surface! Since it is not possible (not desirable) to make optics that have the size of a Jupiter moon, we have to restrict the valid interception points to a localized region of space. Most of the apertures are circular but you can use square apertures as well or even star-shaped aperture. You can even do an aperture that has a blocking portion in the center if you would like! Also, the aperture does not necessarily need to be centred around the optical axis of the lens. The first step is therefore to [»] transform the interception position into the aperture local space coordinate system. Most of the time this will by the identity transform, though. Once in the local space of the aperture, you can use an analytical expression to determine if the ray is within the aperture. For instance, here are a few apertures types: Circular aperture Circular obscuration Rectangular aperture Apertures can be combined just like we will do with Constructive Solid Geometry to achieve more complex effect but this go beyond the scope of this post. If a ray is clipped by an aperture, it shall be marked as such in the vignetting flag of the lens and no longer raytraced to the other surfaces. One of the usage of aperture, when programming, is that you can create a special aperture type that will never block rays but will record the rays footprint such that you can obtain the effective lens diameter required for the lens in the usage conditions. Surface Effects Once the ray has intercepted the surface and has passed the aperture check, it may experience some of the following effects: refraction across the surface, reflection on the surface, diffraction (as with gratings), scattering etc. In non-sequential raytracing, multiple effects can be combined. For instance, a ray intercepting a glass surface will mostly refract but a part of it will also be reflected according to the Fresnel equations. A ray intercepting a grating will be split into the many diffraction orders and scattering will throw rays all over the place. In sequential raytracing however, only one effect is considered. We will consider only complete refraction, reflection, single diffraction order etc. Note that in the case of refraction, a Total Internal Refraction (TIR) condition is considered as failure in sequential raytracing while it is converted to a pure reflection in non-sequential raytracing. Here, I will consider only reflection and refraction. I will discuss other types of effects in later post when I have some time to experiment with them first (in particular scattering). To find the refraction law, we will first split the ray directions into the orthonormal basis n and x, n being the normal vector of the surface and is considered as facing the ray. I will also consider that the incoming direction vector has been normalized (length is 1). The situation is displayed in Figure 1. Figure 1 Refraction of a ray Under these conditions we have where α[1] and α[2][ ]are the angle of the ray with the normal before and after refraction. We find and therefore We also know that a ray coming from a material of refractive index n[1] to a material of refractive index n[2] will follow the Snell-Descartes law of refraction: and so The cosine of the incoming angle can be obtained from the dot product of the incoming vector with the normal (I reverted the sign due to the orientation of the normal) The cosine of the outgoing angle can be obtained using a small trick: It is important to check that which would indicate a Total Internal Reflection (TIR) event. One advantage of the trick used to compute the cosines of the angles is that they do not rely on expensive trigonometric function evaluation but rather use the more simple square root. The final formula is therefore If a TIR occur, we can either drop the ray or apply the reflection formula. Reflections typically occurs on metallic surfaces such as silver or aluminium mirrors but will also happens on dielectric coating as wells or during TIR events. Reflection of a ray is illustrated in Figure 2. Figure 2 Reflection of a ray Decomposing the ray as before, we get we obtain the reflection formula Ray Aiming and Normalized Pupil Coordinates There is one more important note that I need to make about how ZEMAX OpticsStudio works when launching rays. I am not familiar with the other optical design softwares (CodeV and OSLO namely) but I guess they have similar modes of operations. OpticsStudio refers to rays in normalized coordinates relative to the STOP and the FOV. So, a ray starting at the edge of the X FOV would have the coordinates (+1,0) or (-1,0), whatever the actual FOV is. The software then converts this normalized coordinate to a physical one to launch the actual ray. The same thing occurs with the direction. Instead of specifying the direction vector itself, we give the intersection coordinate with the STOP surface as normalized coordinate as well. Once we know where the ray starts and where it passes through the STOP surface, we have entirely characterized its path. Now comes the question on how to compute the starting direction of the ray to launch it. Computing the position was simple but computing the direction from the STOP coordinates is more tricky. In OpticStudio this is referred to as Ray Aiming and the software allows three modes of operations: none (off), paraxial ray and real ray. I will first describe what these modes are before I explain how I am doing in the #DevOptical software. When ray aiming is set to none in OpticStudio, the software uses the paraxial entrance pupil of the system. The position and diameter are computed like we did in our [»] previous post but you can use the full raytracing engine too although it makes little sense (more computations). The position is obtained by multiplying the normalized coordinate by the paraxial radius of the entrance pupil. From the position of the ray in the entrance pupil and the starting position we can compute the initial direction vector. This is relatively fast but assumes there is a one-to-one relationship between the actual STOP real position and the paraxial entrance pupil. This is obviously only true very close to the optical axis where the paraxial condition is respected and this mode of operation quickly generate incorrect results as the size of the STOP increases. There is then the paraxial and real ray aiming mode. It is interesting to note that these modes produce the exact same results if you configure OpticStudio to have the actual STOP aperture limiting the rays in the system (more on this below). For OpticStudio users, this is the floating aperture mode in the aperture settings. When ray aiming is active, the system will correct the starting position until the ray physically passes at the given normalized coordinates of the STOP surface. This requires an iterative search for the correct position. In practice, we compute the actual STOP intersection position and multiply the starting radial position by the ratio between the desired STOP radial position and the observed radial position at iteration #i. The process is repeated until the ray passes through the STOP at the desired coordinates withing some threshold (e.g.: 10^-12 meters or less). That was the floating aperture mode. OpticStudio also offers another aperture mode where you specify the entrance pupil diameter from which the software infers the size of the STOP. When the ray aiming is set to paraxial , a paraxial raytrace is used to compute the STOP size from the entrance pupil diameter. When the ray aiming is set to real , reals rays are used to compute the STOP size. What is fishy with this is that when you will go to the system reports analysis tab and look at the entrance pupil diameter you will always look for the paraxial entrance pupil which may then differs from the one you selected in the menu! To avoid this kind of troubles, I always work in the floating aperture mode and specify the size of the STOP myself. I then use ray aiming to calculate the correct direction of the ray. It is a bit more computations and will not work with systems having central obscuration but this is the price to pay to avoid the mess of multiples modes of operations, each giving a slightly different answer. I also believe that, in the spirit of this #DevOptical series, having a single option that covers 99% of the use-cases is better. The 1% left can then be dealt with more specialized software like OpticStudio or CodeV. Ray aiming will have important consequences when we start discussing wavefront error maps, PSF and MTF. But I m leaving this for later. Final Words This was a pretty long post and I m really glad to be able to release it. If you followed a bit the website, you will remember that I developed the raytracing code in 2017 already (4 years ago!) when discussing [»] a custom 5x microscopy objective. Everything that is here has been thoroughly tested and validated against commercial optical design software. It is, to my knowledge, the most complete tutorial on the topic you will find online for free. I struggled a lot implementing all these algorithms and I hope this will help other people as well. The content of this post will also be necessary to develop extremely important image quality quantifiers including spot and fan diagrams, modulation transfer function, impulse response etc. It will also be the occasion to discuss about wavefronts, diffraction effects and 3^rd order aberrations theory. It will also serve as the basis for Monte-Carlo tolerance analysis much later. The next post will be about spot diagrams, which are the most straightforward way to me to introduce people to optical aberrations once they know the concepts of raytracing. I will then move to wavefront and diffraction theory before going to the concepts of Seidel aberrations. I m currently working on more advanced concepts for the series that I will reveal later. I have already some nice stuffs working for Seidel aberrations and paraxial tolerance analysis. This summer I also started working on a video tutorial on how to assemble the OpenRAMAN spectrometer but it s taking me more time than expected. I bought a BlackMagic Pocket 6K camera for the occasion but I took a lens that s a bit to small for the place I m recording the videos (I took a 35 mm lens but a 24 mm would have been more suited too bad!). [p]I would like to give a big thanks to James, Daniel, Naif, Lilith, Cam, Samuel, Themulticaster, Sivaraman and Arif who have supported this post through [∞] Patreon. I also take the occasion to invite you to donate through Patreon, even as little as $1. I cannot stress it more, you can really help me to post more content and make more experiments! [⇈] Top of Page Copyright The Pulsar (C) 2005-2024. All content of this website, including text, images, formula, and files are the property of The Pulsar unless otherwise explicitely noted. You are allowed to print, distribute page address and display the content. All other usages are prohibited. Some of the experiences and ideas presented here may be dangerous and might result in injuries even when operated accordingly to the author description. We cannot be held responsible for injuries, damage caused to third parties, uses and misuses resulting from applications of the content provided here.	{"url":"https://www.thepulsar.be/article/-devoptical-part-8--raytracing-101/","timestamp":"2024-11-13T18:54:35Z","content_type":"text/html","content_length":"39768","record_id":"<urn:uuid:2302dbae-8399-4626-b994-bcbf73fb284a>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00049.warc.gz"}
Density of rational points on del Pezzo surfaces of degree 1 Add to your list(s) Download to your calendar using vCal If you have a question about this talk, please contact Rong Zhou. Let X be an algebraic variety over an infinite field k. In arithmetic geometry we are interested in the set X(k) of k-rational points on X. For example, is X(k) empty or not? And if it is not empty, is X(k) dense in X with respect to the Zariski topology? Del Pezzo surfaces are surfaces classified by their degree d, which is an integer between 1 and 9 (for d ≥ 3, these are the smooth surfaces of degree d in P^d ). For del Pezzo surfaces of degree at least 2 over a field k, we know that the set of k-rational points is Zariski dense provided that the surface has one k-rational point to start with (that lies outside a specific subset of the surface for degree 2). However, for del Pezzo surfaces of degree 1 over a field k, even though we know that they always contain at least one k-rational point, we do not know if the set of k-rational points is Zariski dense in general. I will talk about density of rational points on del Pezzo surfaces, state what is known so far, and show a result that is joint work with Julie Desjardins, in which we give sufficient and necessary conditions for the set of k-rational points on a specific family of del Pezzo surfaces of degree 1 to be Zariski dense, where k is finitely generated over Q. This talk is part of the Number Theory Seminar series. This talk is included in these lists: Note that ex-directory lists are not shown.	{"url":"https://talks.cam.ac.uk/talk/index/180419","timestamp":"2024-11-08T19:18:20Z","content_type":"application/xhtml+xml","content_length":"14031","record_id":"<urn:uuid:38515a99-a157-4a71-983d-251dced598e7>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00816.warc.gz"}
Specifies parameters for the algebraic multigrid algorithm. ALGEBRAIC_MULTIGRID_PARAMETERS {parameters} This command has no qualifier. pressure_negative_coupling_tolerance (real) >=0 <=1 [=0.6] Tolerance within the algebraic multigrid (AMG) coarsening algorithm to determine when strongly negatively coupled variables for the pressure matrix are considered. Tolerance is the decimal fraction of the maximum value. The default value of 0.6 sets the tolerance to use values of 60 percent or greater of the most negative value. With a value of 1.0, negative coupling is ignored. Negative coupling tolerance is used when determining which entries from the fine matrix levels are retained at the coarser matrix levels. To select the coarse level grids for a matrix, only the values with a high contribution to the unknown equations are considered. Terms are considered to be highly negatively coupled if the negated value of the matrix entry exceeds the maximum negated value of the matrix multiplied by the negative coupling tolerance. The value of pressure_negative_coupling_tolerance can have significant influence on the structure of coarse level grids, influencing the convergence rate of the linear solution. pressure_positive_coupling_tolerance (real) >=0 <=1 [=1.0] Tolerance within the algebraic multigrid (AMG) coarsening algorithm to determine when strongly positively coupled variables for the pressure matrix are considered. Tolerance is the decimal fraction of the maximum value. With the default value of 1.0, positive coupling is ignored. Positive coupling tolerance is used when determining which entries from the fine matrix levels are retained at the coarser matrix levels. To select the coarse level grids for a matrix, only the values with a high contribution to the unknown equations are considered. Terms are considered to be highly positively coupled if the value of the matrix entry exceeds the maximum value of the matrix multiplied by the positive coupling tolerance. The value of pressure_positive_coupling_tolerance can have significant influence on the structure of coarse level grids, influencing the convergence rate of the linear solution. max_pressure_final_matrix (integer) >=1 [=100] Maximum number of entries present in the coarsest matrix of the AMG algorithm. With the default value of 100, coarsening will stop when the lowest level matrix is 10 by 10 or smaller. Increasing the size of this parameter may increase the time necessary to obtain a solution on the lowest level matrix. pressure_standard_interpolation (boolean) [=on] Interpolation method used to build the coarse multigrid matrices from fine matrices or to transfer the solution from a fine level to a coarse multigrid level, and vice versa. With this option set to off, direct interpolation will be used. Each equation in the fine level matrix that is considered to be important in the coarsening process gets interpolated to build the coarse level matrix. In direct interpolation, just neighboring points within the one fine level equation are used to calculate the interpolation weights. In standard interpolation, in addition to the neighboring points, other coupled equations are also considered in the weight calculations. More information is incorporated in the standard interpolation, leading to better convergence. In most cases, standard interpolation is recommended. pressure_positive_negative_separate (boolean) [=off] Determines if the positive and negative weights are separated during either standard or direct interpolation. In either direct or standard interpolation, weights are functions of off diagonal components of the fine level matrix and can be positive or negative. Interpolation can be conducted either separately for positive and negative weights, or these weights can be added together to have one interpolation. With this option set to on, summation of positive and negative weights will be calculated and used for interpolation. For pressure matrices, separation of weights is not pressure_truncated_interpolation (boolean) [=on] Determines if the small interpolation weights will be truncated when interpolating the fine level multigrid matrix to the coarse level matrix or when transferring the solution from the coarse matrix to the finer AMG matrix levels. This flag should be used together with the pressure_truncation_tolerance to set the tolerance to determine the small weights. Truncation can be used with both direct and standard interpolation. A minor increase in the computational time to reach convergence can be observed if this flag is set to off. pressure_truncation_tolerance (real) >=0 <=1 [=0.1] Sets the portion of the smallest interpolation weights to be truncated during interpolation from a fine level to a coarse level. Used when pressure_truncated_interpolation=on. With a value of zero, no interpolation weights will be truncated. With a value of one, all weights will be truncated. pressure_givens_scaling (boolean) [=on] Determines if two by two Givens matrix scaling is used for the multigrid smoothing process. With this option turned off, diagonal scaling is used. pressure_smoothing_type (enumerated) [=chebyshev] Smoothing method used at each grid level after interpolation when values are from a fine matrix to a coarse matrix, or from a coarse matrix to a fine matrix. Chebyshev polynomial smoothing. Chebychev smoothing provides better performance, accurate convergence with less computational time, and is recommended for most matrices. Jacobi smoothing. Jacobi smoothing is recommended for the fully coupled flow stagger. pressure_eigenvalue_tolerance (real) >0 [=1.e-2] Tolerance to stop the eigenvalue calculation iterations. Used with pressure_smoothing_type=chebyshev. Chebyshev polynomial smoothing requires the ratio of the largest to the smallest eigenvalues of the matrix. The Iterative Lanczos algorithm is used to calculate the maximum eigenvalue of the pressure matrix. max_pressure_eigenvalue_iterations (integer) >=0 [=20] Used with pressure_smoothing_type=chebyshev. Maximum number of iterations of the Lanczos algorithm for computing the largest eigenvalues at each multigrid level. The pressure_eigenvalue_tolerance and max_pressure_eigenvalue_iterations determine when iterations stop. Whichever constraint is met first stops the algorithm. In general, eigenvalues converge within 20 iterations. For cases where the eigenvalues do not converge, increasing the total number of iterations can lead to better estimates of the eigenvalues at the expense of increased total computation time. Note: Poor estimates of eigenvalues also increase solution time by slowing the convergence of the AMG algorithm. pressure_smoothing_order (integer) >=1 [=2] When pressure_smoothing_type=chebyshev, this parameter sets the polynomial order for the smoother. When pressure_smoothing_type=jacobi this parameter sets the number of smoothing passes during the downward and upward multigrid cycles. For both Chebyshev and Jacobi methods, increasing the smoothing order will reduce the total number of linear solver iterations required to converge the system. However, the time per iteration will also increase. There should be a balance between the number of iterations and the time per iteration. For pressure matrices, a value of two is found to give the best performance. pressure_chebyshev_max_min_ratio (real) >1 <=100 [=10] When pressure_smoothing_type=chebyshev, this parameter provides an estimate of the smallest eigenvalue by dividing the largest eigenvalue calculated by the Lanczos algorithm by this parameter. Chebyshev polynomial smoothing requires the ratio of the largest to the smallest eigenvalues of the matrix to calculate the polynomial coefficients. The smallest value may not be calculated. This parameter can be used to get an estimate. The default of 10 means that the smallest eigenvalue is one tenth of the largest. pressure_jacobi_relaxation_factor (real) >0 [=0.25] Used when pressure_smoothing_type=jacobi. Jacobi smoothing may not be stable and it should be relaxed. Conventional under relaxation is used to stabilize it. The default of 0.25 means updated values will be calculated by adding 25 percent of new values to 75 percent of the old values. pressure_setup_tolerance (real) >0 [=0] The nonlinear convergence tolerance used to control when the AMG setup process is performed. If the residual is higher than this value, the AMG setup process is performed. If the residual is lower than this tolerance, the AMG algorithm uses the existing setup information that is stored in RAM. With the default value of zero, the AMG setup process will occur every time the equations are solved, regardless of the residual. As the simulation converges, the change in solution between time steps decreases, minimizing the need to execute the AMG setup process. If AMG is used for more than one equation, such as pressure and turbulence, and this parameter is greater than zero, the memory requirements will be increased. Note: For each equation, a different setup tolerance can be used. This value should not be changed unless a convergence tolerance is set for the problem that is less than the default value. num_pressure_global_basis (integer) >=0 [=0] This parameter was retired in the 13.0 release. pressure_global_basis_tolerance (real) >0 [=1.e-6] This parameter was retired in the 13.0 release. max_pressure_global_basis_iterations (integer) >0 [=1000] This parameter was retired in the 13.0 release. num_pressure_initial_givens_rotations (integer) >=0 [=0] This parameter was retired in the 13.0 release. velocity_negative_coupling_tolerance (real) >=0 <=1 [=0.5] Tolerance within the algebraic multigrid (AMG) coarsening algorithm to determine when strongly negatively coupled variables for the velocity matrix are considered. Tolerance is the decimal fraction of the maximum value. The default value of 0.5 sets the tolerance to use values of 50 percent or greater of the most negative value. With a value of 1.0, negative coupling is ignored. Negative coupling tolerance is used when determining which entries from the fine matrix levels are retained at the coarser matrix levels. To select the coarse level grids for a matrix, only the values with a high contribution to the unknown equations are considered. Terms are considered to be highly negatively coupled if the negated value of the matrix entry exceeds the maximum negated value of the matrix multiplied by the negative coupling tolerance. The value of velocity_negative_coupling_tolerance can have significant influence on the structure of coarse level grids, influencing the convergence rate of the linear solution. velocity_positive_coupling_tolerance (real) >=0 <=1 [=1.0] Tolerance within the algebraic multigrid (AMG) coarsening algorithm to determine when strongly positively coupled variables for the velocity matrix are considered. Tolerance is the decimal fraction of the maximum value. With the default value of 1.0, positive coupling is ignored. Positive coupling tolerance is used when determining which entries from the fine matrix levels are retained at the coarser matrix levels. To select the coarse level grids for a matrix, only the values with a high contribution to the unknown equations are considered. Terms are considered to be highly positively coupled if the value of the matrix entry exceeds the maximum value of the matrix multiplied by the positive coupling tolerance. The value of velocity_positive_coupling_tolerance can have significant influence on the structure of coarse level grids, influencing the convergence rate of the linear solution. max_velocity_final_matrix (integer) >=1 [=100] Maximum number of entries present in the coarsest matrix of the AMG algorithm. With the default value of 100, coarsening will stop when the lowest level matrix is 10 by 10 or smaller. Increasing the size of this parameter may lead to longer duration of the solution calculation of the lowest level matrix. velocity_standard_interpolation (boolean) [=on] Interpolation method used to build the coarse multigrid matrices from fine matrices or to transfer the solution from a fine level to a coarse multigrid level, and vice versa. With this option set to off, direct interpolation will be used. Each equation in the fine level matrix that is considered to be important in the coarsening process gets interpolated to build the coarse level matrix. In direct interpolation, just neighboring points within the one fine level equation is used to calculate the interpolation weights. In standard interpolation, in addition to the neighboring points, other coupled equations are also considered in the weight calculations. More information is incorporated in the standard interpolation, leading to better convergence. In most cases, standard interpolation is recommended. velocity_positive_negative_separate (boolean) [=off] Determines if the positive and negative weights are separated during either standard or direct interpolation. In either direct or standard interpolation, weights are functions of off diagonal components of the fine level matrix and can be positive or negative. Interpolation can be conducted either separately for positive and negative weights, or these weights can be added together to have one interpolation. With this option set to on, summation of positive and negative weights will be calculated and used for interpolation. For velocity matrices, separation of weights is not velocity_truncated_interpolation (boolean) [=on] Determines if the small interpolation weights will be truncated when interpolating the fine level multigrid matrix to the coarse level matrix or when transferring the solution from the coarse matrix to the finer AMG matrix levels. This flag should be used together with the velocity_truncation_tolerance to set the tolerance to determine the small weights. Truncation can be used with both direct and standard interpolation. A minor increase in the computational time to reach convergence can be observed if this flag is set to off. velocity_truncation_tolerance (real) >=0 <=1 [=0.1] Sets the portion of the smallest interpolation weights to be truncated during interpolation from a fine level to a coarse level. Used when velocity_truncated_interpolation=on. With a value of zero, no interpolation weights will be truncated. With a value of one, all weights will be truncated. velocity_givens_scaling (boolean) [=off] Determines if two by two Givens matrix scaling is used for the multigrid smoothing process. With this option turned off, diagonal scaling is used. velocity_smoothing_type (enumerated) [=chebyshev] Smoothing method used at each grid level after interpolation when values are from a fine matrix to a coarse matrix, or from a coarse matrix to a fine matrix. Chebyshev polynomial smoothing. Chebychev smoothing provides better performance, accurate convergence with less computational time, and is recommended for most matrices. Jacobi smoothing. Jacobi smoothing is recommended for the fully coupled flow stagger. velocity_num_krylov_vectors (integer) >=0 [=30] Used with velocity_smoothing_type=chebyshev. Number of Krylov vectors in the Arnoldi algorithm for computing the largest eigenvalues at each multigrid level. In general, eigenvalues converge within 30 Krylov vectors. For cases where the eigenvalues do not converge, increasing the number of Krylov vectors can lead to better estimates of the eigenvalues at the expense of increased total computation time. Note: Poor estimates of eigenvalues also increase solution time by slowing the convergence of the AMG algorithm. velocity_smoothing_order (integer) >=1 [=2] When velocity_smoothing_type=chebyshev, this parameter sets the polynomial order for the smoother. When velocity_smoothing_type=jacobi this parameter sets the number of smoothing passes during the downward and upward multigrid cycles. For both Chebyshev and Jacobi methods, increasing the smoothing order will reduce the total number of linear solver iterations required to converge the system. However, the time per iteration will also increase. There should be a balance between the number of iterations and the time per iteration. For velocity matrices, a value of two is found to give the best performance. velocity_chebyshev_max_min_ratio (real) >1 <=100 [=10] When velocity_smoothing_type=chebyshev, this parameter provides an estimate of the smallest eigenvalue by dividing the largest eigenvalue calculated by the Lanczos algorithm by this parameter. Chebyshev polynomial smoothing requires the ratio of the largest to the smallest eigenvalues of the matrix to calculate the polynomial coefficients. The smallest value may not be calculated. This parameter can be used to get an estimate. The default of 10 means that the smallest eigenvalue is one tenth of the largest. velocity_jacobi_relaxation_factor (real) >0 [=0.25] Used when velocity_smoothing_type=jacobi. Jacobi smoothing may not be stable and it should be relaxed. Conventional under relaxation is used to stabilize it. The default of 0.25 means updated values will be calculated by adding 25 percent of new values to 75 percent of the old values. velocity_setup_tolerance (real) >0 [=0] The nonlinear convergence tolerance used to control when the AMG setup process is performed. If the residual is higher than this value, the AMG setup process is performed. If the residual is lower than this tolerance, the AMG algorithm uses the existing setup information that is stored in RAM. With the default value of zero, the AMG setup process will occur every time the equations are solved, regardless of the residual. As the simulation converges, the change in solution between time steps decreases, minimizing the need to execute the AMG setup process. If AMG is used for more than one equation, such as velocity and turbulence, and this parameter is greater than zero, the memory requirements will be increased. Note: For each equation, a different setup tolerance can be used. This value should not be changed unless a convergence tolerance is set for the problem that is less than the default value. flow_negative_coupling_tolerance (real) >=0 <=1 [=0.5] Tolerance within the algebraic multigrid (AMG) coarsening algorithm to determine when strongly negatively coupled variables for the flow matrix are considered. Tolerance is the decimal fraction of the maximum value. The default value of 0.5 sets the tolerance to use values of 50 percent or greater of the most negative value. With a value of 1.0, negative coupling is ignored. Negative coupling tolerance is used when determining which entries from the fine matrix levels are retained at the coarser matrix levels. To select the coarse level grids for a matrix, only the values with a high contribution to the unknown equations are considered. Terms are considered to be highly negatively coupled if the negated value of the matrix entry exceeds the maximum negated value of the matrix multiplied by the negative coupling tolerance. The value of flow_negative_coupling_tolerance can have significant influence on the structure of coarse level grids, influencing the convergence rate of the linear solution. flow_positive_coupling_tolerance (real) >=0 <=1 [=1.0] Tolerance within the algebraic multigrid (AMG) coarsening algorithm to determine when strongly positively coupled variables for the flow matrix are considered. Tolerance is the decimal fraction of the maximum value. With the default value of 1.0, positive coupling is ignored. Positive coupling tolerance is used when determining which entries from the fine matrix levels are retained at the coarser matrix levels. To select the coarse level grids for a matrix, only the values with a high contribution to the unknown equations are considered. Terms are considered to be highly positively coupled if the value of the matrix entry exceeds the maximum value of the matrix multiplied by the positive coupling tolerance. The value of flow_positive_coupling_tolerance can have significant influence on the structure of coarse level grids, influencing the convergence rate of the linear solution. max_flow_final_matrix (integer) >=1 [=100] Maximum number of entries present in the coarsest matrix of the AMG algorithm. With the default value of 100, coarsening will stop when the lowest level matrix is 10 by 10 or smaller. Increasing the size of this parameter may lead to longer duration of the solution calculation of the lowest level matrix. flow_standard_interpolation (boolean) [=off] Interpolation method used to build the coarse multigrid matrices from fine matrices or to transfer the solution from a fine level to a coarse multigrid level, and vice versa. With this option set to off, direct interpolation will be used. Each equation in the fine level matrix that is considered to be important in the coarsening process gets interpolated to build the coarse level matrix. In direct interpolation, just neighboring points within the one fine level equation are used to calculate the interpolation weights. In standard interpolation, in addition to the neighboring points, other coupled equations are also considered in the weight calculations. More information is incorporated in the standard interpolation, leading to better convergence. In most cases, standard interpolation is recommended. However, direct is recommended for the fully coupled flow stagger. flow_positive_negative_separate (boolean) [=off] Determines if the positive and negative weights are separated during either standard or direct interpolation. In either direct or standard interpolation, weights are functions of off diagonal components of the fine level matrix and can be positive or negative. Interpolation can be conducted either separately for positive and negative weights, or these weights can be added together to have one interpolation. With this option set to on, summation of positive and negative weights will be calculated and used for interpolation. For flow matrices, separation of weights is not flow_truncated_interpolation (boolean) [=on] Determines if the small interpolation weights will be truncated when interpolating the fine level multigrid matrix to the coarse level matrix or when transferring the solution from the coarse matrix to the finer AMG matrix levels. This flag should be used together with the flow_truncation_tolerance to set the tolerance to determine the small weights. Truncation can be used with both direct and standard interpolation. A minor increase in the computational time to reach convergence can be observed if this flag is set to off. flow_truncation_tolerance (real) >=0 <=1 [=0.1] Sets the portion of the smallest interpolation weights to be truncated during interpolation from a fine level to a coarse level. Used when flow_truncated_interpolation=on. With a value of zero, no interpolation weights will be truncated. With a value of one, all weights will be truncated. flow_givens_scaling (boolean) [=off] Determines if two by two Givens matrix scaling is used for the multigrid smoothing process. With this option turned off, diagonal scaling is used. flow_smoothing_type (enumerated) [=jacobi] Smoothing method used at each grid level after interpolation when values are from a fine matrix to a coarse matrix, or from a coarse matrix to a fine matrix. Chebyshev polynomial smoothing. Chebychev smoothing provides better performance, accurate convergence with less computational time, and is recommended for most matrices. Jacobi smoothing. Jacobi smoothing is recommended for the fully coupled flow stagger. flow_num_krylov_vectors (integer) >=0 [=30] Used with flow_smoothing_type=chebyshev. Number of Krylov vectors in the Arnoldi algorithm for computing the largest eigenvalues at each multigrid level. In general, eigenvalues converge within 30 Krylov vectors. For cases where the eigenvalues do not converge, increasing the number of Krylov vectors can lead to better estimates of the eigenvalues at the expense of increased total computation time. Note: Poor estimates of eigenvalues also increase solution time by slowing the convergence of the AMG algorithm. flow_smoothing_order (integer) >=1 [=2] When flow_smoothing_type=chebyshev, this parameter sets the polynomial order for the smoother. When flow_smoothing_type=jacobi this parameter sets the number of smoothing passes during the downward and upward multigrid cycles. For both Chebyshev and Jacobi methods, increasing the smoothing order will reduce the total number of linear solver iterations required to converge the system. However, the time per iteration will also increase. There should be a balance between the number of iterations and the time per iteration. For flow matrices, a value of two is found to give the best performance. flow_chebyshev_max_min_ratio (real) >1 <=100 [=10] When flow_smoothing_type=chebyshev, this parameter provides an estimate of the smallest eigenvalue by dividing the largest eigenvalue calculated by the Lanczos algorithm by this parameter. Chebyshev polynomial smoothing requires the ratio of the largest to the smallest eigenvalues of the matrix to calculate the polynomial coefficients. The smallest value may not be calculated. This parameter can be used to get an estimate. The default of 10 means that the smallest eigenvalue is one tenth of the largest. flow_jacobi_relaxation_factor (real) >0 [=0.2] Used when flow_smoothing_type=jacobi. Jacobi smoothing may not be stable and it should be relaxed. Conventional under relaxation is used to stabilize it. The default of 0.2 means updated values will be calculated by adding 20 percent of new values to 80 percent of the old values. flow_setup_tolerance (real) >0 [=0] The nonlinear convergence tolerance used to control when the AMG setup process is performed. If the residual is higher than this value, the AMG setup process is performed. If the residual is lower than this tolerance, the AMG algorithm uses the existing setup information that is stored in RAM. With the default value of zero, the AMG setup process will occur every time the equations are solved, regardless of the residual. As the simulation converges, the change in solution between time steps decreases, minimizing the need to execute the AMG setup process. If AMG is used for more than one equation, such as flow and turbulence, and this parameter is greater than zero, the memory requirements will be increased. Note: For each equation a different setup tolerance can be used. This value should not be changed unless a convergence tolerance is set for the problem that is less than the default value. temperature_negative_coupling_tolerance (real) >=0 <=1 [=0.5] Tolerance within the algebraic multigrid (AMG) coarsening algorithm to determine when strongly negatively coupled variables for the temperature matrix are considered. Tolerance is the decimal fraction of the maximum value. The default value of 0.5 sets the tolerance to use values of 50 percent or greater of the most negative value. With a value of 1.0, negative coupling is ignored. Negative coupling tolerance is used when determining which entries from the fine matrix levels are retained at the coarser matrix levels. To select the coarse level grids for a matrix, only the values with a high contribution to the unknown equations are considered. Terms are considered to be highly negatively coupled if the negated value of the matrix entry exceeds the maximum negated value of the matrix multiplied by the negative coupling tolerance. The value of temperature_negative_coupling_tolerance can have significant influence on the structure of coarse level grids, influencing the convergence rate of the linear solution. temperature_positive_coupling_tolerance (real) >=0 <=1 [=1.0] Tolerance within the algebraic multigrid (AMG) coarsening algorithm to determine when strongly positively coupled variables for the temperature matrix are considered. Tolerance is the decimal fraction of the maximum value. With the default value of 1.0, positive coupling is ignored. Positive coupling tolerance is used when determining which entries from the fine matrix levels are retained at the coarser matrix levels. To select the coarse level grids for a matrix, only the values with a high contribution to the unknown equations are considered. Terms are considered to be highly positively coupled if the value of the matrix entry exceeds the maximum value of the matrix multiplied by the positive coupling tolerance. The value of temperature_positive_coupling_tolerance can have significant influence on the structure of coarse level grids, influencing the convergence rate of the linear solution. max_temperature_final_matrix (integer) >=1 [=100] Maximum number of entries present in the coarsest matrix of the AMG algorithm. With the default value of 100, coarsening will stop when the lowest level matrix is 10 by 10 or smaller. Increasing the size of this parameter may lead to longer duration of the solution calculation of the lowest level matrix. temperature_standard_interpolation (boolean) [=on] Interpolation method used to build the coarse multigrid matrices from fine matrices or to transfer the solution from a fine level to a coarse multigrid level, and vice versa. With this option set to off, direct interpolation will be used. Each equation in the fine level matrix that is considered to be important in the coarsening process gets interpolated to build the coarse level matrix. In direct interpolation, just neighboring points within the one fine level equation are used to calculate the interpolation weights. In standard interpolation, in addition to the neighboring points, other coupled equations are also considered in the weight calculations. More information is incorporated in the standard interpolation, leading to better convergence. In most cases, standard interpolation is recommended. temperature_positive_negative_separate (boolean) [=off] Determines if the positive and negative weights are separated during either standard or direct interpolation. In either direct or standard interpolation, weights are functions of off diagonal components of the fine level matrix and can be positive or negative. Interpolation can be conducted either separately for positive and negative weights, or these weights can be added together to have one interpolation. With this option set to on, summation of positive and negative weights will be calculated and used for interpolation. For temperature matrices, separation of weights is not recommended. temperature_truncated_interpolation (boolean) [=on] Determines if the small interpolation weights will be truncated when interpolating the fine level multigrid matrix to the coarse level matrix or when transferring the solution from the coarse matrix to the finer AMG matrix levels. This flag should be used together with the temperature_truncation_tolerance to set the tolerance to determine the small weights. Truncation can be used with both direct and standard interpolation. A minor increase in the computational time to reach convergence can be observed if this flag is set to off. temperature_truncation_tolerance (real) >=0 <=1 [=0.1] Sets the portion of the smallest interpolation weights to be truncated during interpolation from a fine level to a coarse level. Used when temperature_truncated_interpolation=on. With a value of zero, no interpolation weights will be truncated. With a value of one, all weights will be truncated. temperature_givens_scaling (boolean) [=off] Determines if two by two Givens matrix scaling is used for the multigrid smoothing process. With this option turned off, diagonal scaling is used. temperature_smoothing_type (enumerated) [=chebyshev] Smoothing method used at each grid level after interpolation when values are from a fine matrix to a coarse matrix, or from a coarse matrix to a fine matrix. Chebyshev polynomial smoothing. Chebychev smoothing provides better performance, accurate convergence with less computational time, and is recommended for most matrices. Jacobi smoothing. Jacobi smoothing is recommended for the fully coupled flow stagger. temperature_num_krylov_vectors (integer) >=0 [=30] Used with temperature_smoothing_type=chebyshev. Number of Krylov vectors in the Arnoldi algorithm for computing the largest eigenvalues at each multigrid level. In general, eigenvalues converge within 30 Krylov vectors. For cases where the eigenvalues do not converge, increasing the number of Krylov vectors can lead to better estimates of the eigenvalues at the expense of increased total computation time. Note: Poor estimates of eigenvalues also increase solution time by slowing the convergence of the AMG algorithm. temperature_smoothing_order (integer) >=1 [=2] When temperature_smoothing_type=chebyshev, this parameter sets the polynomial order for the smoother. When temperature_smoothing_type=jacobi this parameter sets the number of smoothing passes during the downward and upward multigrid cycles. For both Chebyshev and Jacobi methods, increasing the smoothing order will reduce the total number of linear solver iterations required to converge the system. However, the time per iteration will also increase. There should be a balance between the number of iterations and the time per iteration. For temperature matrices, a value of two is found to give the best performance. temperature_chebyshev_max_min_ratio (real) >1 <=100 [=10] When temperature_smoothing_type=chebyshev, this parameter provides an estimate of the smallest eigenvalue by dividing the largest eigenvalue calculated by the Lanczos algorithm by this parameter. Chebyshev polynomial smoothing requires the ratio of the largest to the smallest eigenvalues of the matrix to calculate the polynomial coefficients. The smallest value may not be calculated. This parameter can be used to get an estimate. The default of 10 means that the smallest eigenvalue is one tenth of the largest. temperature_jacobi_relaxation_factor (real) >0 [=0.25] Used when temperature_smoothing_type=jacobi. Jacobi smoothing may not be stable and it should be relaxed. Conventional under relaxation is used to stabilize it. The default of 0.25 means updated values will be calculated by adding 25 percent of new values to 75 percent of the old values. temperature_setup_tolerance (real) >0 [=0] The nonlinear convergence tolerance used to control when the AMG setup process is performed. If the residual is higher than this value, the AMG setup process is performed. If the residual is lower than this tolerance, the AMG algorithm uses the existing setup information that is stored in RAM. With the default value of zero, the AMG setup process will occur every time the equations are solved, regardless of the residual. As the simulation converges, the change in solution between time steps decreases, minimizing the need to execute the AMG setup process. If AMG is used for more than one equation, such as temperature and turbulence, and this parameter greater than zero, the memory requirements will be increased. Note: For each equation, a different setup tolerance can be used. This value should not be changed unless a convergence tolerance is set for the problem that is less than the default value. species_negative_coupling_tolerance (real) >=0 <=1 [=0.5] Tolerance within the algebraic multigrid (AMG) coarsening algorithm to determine when strongly negatively coupled variables for the species matrix are considered. Tolerance is the decimal fraction of the maximum value. The default value of 0.5 sets the tolerance to use values of 50 percent or greater of the most negative value. With a value of 1.0, negative coupling is ignored. Negative coupling tolerance is used when determining which entries from the fine matrix levels are retained at the coarser matrix levels. To select the coarse level grids for a matrix, only the values with a high contribution to the unknown equations are considered. Terms are considered to be highly negatively coupled if the negated value of the matrix entry exceeds the maximum negated value of the matrix multiplied by the negative coupling tolerance. The value of species_negative_coupling_tolerance can have significant influence on the structure of coarse level grids, influencing the convergence rate of the linear solution. species_positive_coupling_tolerance (real) >=0 <=1 [=1.0] Tolerance within the algebraic multigrid (AMG) coarsening algorithm to determine when strongly positively coupled variables for the species matrix are considered. Tolerance is the decimal fraction of the maximum value. With the default value of 1.0, positive coupling is ignored. Positive coupling tolerance is used when determining which entries from the fine matrix levels are retained at the coarser matrix levels. To select the coarse level grids for a matrix, only the values with a high contribution to the unknown equations are considered. Terms are considered to be highly positively coupled if the value of the matrix entry exceeds the maximum value of the matrix multiplied by the positive coupling tolerance. The value of species_positive_coupling_tolerance can have significant influence on the structure of coarse level grids, influencing the convergence rate of the linear solution. max_species_final_matrix (integer) >=1 [=100] Maximum number of entries present in the coarsest matrix of the AMG algorithm. With the default value of 100, coarsening will stop when the lowest level matrix is 10 by 10 or smaller. Increasing the size of this parameter may lead to longer duration of the solution calculation of the lowest level matrix. species_standard_interpolation (boolean) [=on] Interpolation method used to build the coarse multigrid matrices from fine matrices or to transfer the solution from a fine level to a coarse multigrid level, and vice versa. With this option set to off, direct interpolation will be used. Each equation in the fine level matrix that is considered to be important in the coarsening process gets interpolated to build the coarse level matrix. In direct interpolation, just neighboring points within the one fine level equation are used to calculate the interpolation weights. In standard interpolation, in addition to the neighboring points, other coupled equations are also considered in the weight calculations. More information is incorporated in the standard interpolation, leading to better convergence. In most cases, standard interpolation is recommended. species_positive_negative_separate (boolean) [=off] Determines if the positive and negative weights are separated during either standard or direct interpolation. In either direct or standard interpolation, weights are functions of off diagonal components of the fine level matrix and can be positive or negative. Interpolation can be conducted either separately for positive and negative weights, or these weights can be added together to have one interpolation. With this option set to on, summation of positive and negative weights will be calculated and used for interpolation. For species matrices, separation of weights is not species_truncated_interpolation (boolean) [=on] Determines if the small interpolation weights will be truncated when interpolating the fine level multigrid matrix to the coarse level matrix or when transferring the solution from the coarse matrix to the finer AMG matrix levels. This flag should be used together with the species_truncation_tolerance to set the tolerance to determine the small weights. Truncation can be used with both direct and standard interpolation. A minor increase in the computational time to reach convergence can be observed if this flag is set to off. species_truncation_tolerance (real) >=0 <=1 [=0.1] Sets the portion of the smallest interpolation weights to be truncated during interpolation from a fine level to a coarse level. Used when species_truncated_interpolation=on. With a value of zero, no interpolation weights will be truncated. With a value of one, all weights will be truncated. species_givens_scaling (boolean) [=off] Determines if two by two Givens matrix scaling is used for the multigrid smoothing process. With this option turned off, diagonal scaling is used. species_smoothing_type (enumerated) [=chebyshev] Smoothing method used at each grid level after interpolation when values are from a fine matrix to a coarse matrix, or from a coarse matrix to a fine matrix. Chebyshev polynomial smoothing. Chebychev smoothing provides better performance, accurate convergence with less computational time, and is recommended for most matrices. Jacobi smoothing. Jacobi smoothing is recommended for the fully coupled flow stagger. species_num_krylov_vectors (integer) >=0 [=30] Used with species_smoothing_type=chebyshev. Number of Krylov vectors in the Arnoldi algorithm for computing the largest eigenvalues at each multigrid level. In general, eigenvalues converge within 30 Krylov vectors. For cases where the eigenvalues do not converge, increasing the number of Krylov vectors can lead to better estimates of the eigenvalues at the expense of increased total computation time. Note: Poor estimates of eigenvalues also increase solution time by slowing the convergence of the AMG algorithm. species_smoothing_order (integer) >=1 [=2] When species_smoothing_type=chebyshev, this parameter sets the polynomial order for the smoother. When species_smoothing_type=jacobi this parameter sets the number of smoothing passes during the downward and upward multigrid cycles. For both Chebyshev and Jacobi methods, increasing the smoothing order will reduce the total number of linear solver iterations required to converge the system. However, the time per iteration will also increase. There should be a balance between the number of iterations and the time per iteration. For species matrices, a value of two is found to give the best performance. species_chebyshev_max_min_ratio (real) >1 <=100 [=10] When species_smoothing_type=chebyshev, this parameter provides an estimate of the smallest eigenvalue by dividing the largest eigenvalue calculated by the Lanczos algorithm by this parameter. Chebyshev polynomial smoothing requires the ratio of the largest to the smallest eigenvalues of the matrix to calculate the polynomial coefficients. The smallest value may not be calculated. This parameter can be used to get an estimate. The default of 10 means that the smallest eigenvalue is one tenth of the largest. species_jacobi_relaxation_factor (real) >0 [=0.25] Used when species_smoothing_type=jacobi. Jacobi smoothing may not be stable and it should be relaxed. Conventional under relaxation is used to stabilize it. The default of 0.25 means updated values will be calculated by adding 25 percent of new values to 75 percent of the old values. species_setup_tolerance (real) >0 [=0] The nonlinear convergence tolerance used to control when the AMG setup process is performed. If the residual is higher than this value, the AMG setup process is performed. If the residual is lower than this tolerance, the AMG algorithm uses the existing setup information that is stored in RAM. With the default value of zero, the AMG setup process will occur every time the equations are solved, regardless of the residual. As the simulation converges, the change in solution between time steps decreases, minimizing the need to execute the AMG setup process. If AMG is used for more than one equation, such as species and turbulence, and this parameter is greater than zero, the memory requirements will be increased. Note: For each equation, a different setup tolerance can be used. This value should not be changed unless a convergence tolerance is set for the problem that is less than the default value. turbulence_negative_coupling_tolerance (real) >=0 <=1 [=0.5] Tolerance within the algebraic multigrid (AMG) coarsening algorithm to determine when strongly negatively coupled variables for the turbulence matrix are considered. Tolerance is the decimal fraction of the maximum value. The default value of 0.5 sets the tolerance to use values of 50 percent or greater of the most negative value. With a value of 1.0, negative coupling is ignored. Negative coupling tolerance is used when determining which entries from the fine matrix levels are retained at the coarser matrix levels. To select the coarse level grids for a matrix, only the values with a high contribution to the unknown equations are considered. Terms are considered to be highly negatively coupled if the negated value of the matrix entry exceeds the maximum negated value of the matrix multiplied by the negative coupling tolerance. The value of turbulence_negative_coupling_tolerance can have significant influence on the structure of coarse level grids, influencing the convergence rate of the linear solution. turbulence_positive_coupling_tolerance (real) >=0 <=1 [=1.0] Tolerance within the algebraic multigrid (AMG) coarsening algorithm to determine when strongly positively coupled variables for the turbulence matrix are considered. Tolerance is the decimal fraction of the maximum value. With the default value of 1.0, positive coupling is ignored. Positive coupling tolerance is used when determining which entries from the fine matrix levels are retained at the coarser matrix levels. To select the coarse level grids for a matrix, only the values with a high contribution to the unknown equations are considered. Terms are considered to be highly positively coupled if the value of the matrix entry exceeds the maximum value of the matrix multiplied by the positive coupling tolerance. The value of turbulence_positive_coupling_tolerance can have significant influence on the structure of coarse level grids, influencing the convergence rate of the linear solution. max_turbulence_final_matrix (integer) >=1 [=100] Maximum number of entries present in the coarsest matrix of the AMG algorithm. With the default value of 100, coarsening will stop when the lowest level matrix is 10 by 10 or smaller. Increasing the size of this parameter may lead to longer duration of the solution calculation of the lowest level matrix. turbulence_standard_interpolation (boolean) [=on] Interpolation method used to build the coarse multigrid matrices from fine matrices or to transfer the solution from a fine level to a coarse multigrid level, and vice versa. With this option set to off, direct interpolation will be used. Each equation in the fine level matrix that is considered to be important in the coarsening process gets interpolated to build the coarse level matrix. In direct interpolation, just neighboring points within the one fine level equation are used to calculate the interpolation weights. In standard interpolation, in addition to the neighboring points, other coupled equations are also considered in the weight calculations. More information is incorporated in the standard interpolation, leading to better convergence. In most cases, standard interpolation is recommended. turbulence_positive_negative_separate (boolean) [=off] Determines if the positive and negative weights are separated during either standard or direct interpolation. In either direct or standard interpolation, weights are functions of off diagonal components of the fine level matrix and can be positive or negative. Interpolation can be conducted either separately for positive and negative weights, or these weights can be added together to have one interpolation. With this option set to on, summation of positive and negative weights will be calculated and used for interpolation. For turbulence matrices, separation of weights is not turbulence_truncated_interpolation (boolean) [=on] Determines if the small interpolation weights will be truncated when interpolating the fine level multigrid matrix to the coarse level matrix or when transferring the solution from the coarse matrix to the finer AMG matrix levels. This flag should be used together with the turbulence_truncation_tolerance to set the tolerance to determine the small weights. Truncation can be used with both direct and standard interpolation. A minor increase in the computational time to reach convergence can be observed if this flag is set to off. turbulence_truncation_tolerance (real) >=0 <=1 [=0.1] Sets the portion of the smallest interpolation weights to be truncated during interpolation from a fine level to a coarse level. Used when turbulence_truncated_interpolation=on. With a value of zero, no interpolation weights will be truncated. With a value of one, all weights will be truncated. turbulence_givens_scaling (boolean) [=off] Determines if two by two Givens matrix scaling is used for the multigrid smoothing process. With this option turned off, diagonal scaling is used. turbulence_smoothing_type (enumerated) [=chebyshev] Smoothing method used at each grid level after interpolation when values are from a fine matrix to a coarse matrix, or from a coarse matrix to a fine matrix. Chebyshev polynomial smoothing. Chebychev smoothing provides better performance, accurate convergence with less computational time, and is recommended for most matrices. Jacobi smoothing. Jacobi smoothing is recommended for the fully coupled flow stagger. turbulence_num_krylov_vectors (integer) >=0 [=30] Used with turbulence_smoothing_type=chebyshev. Number of Krylov vectors in the Arnoldi algorithm for computing the largest eigenvalues at each multigrid level. In general, eigenvalues converge within 30 Krylov vectors. For cases where the eigenvalues do not converge, increasing the number of Krylov vectors can lead to better estimates of the eigenvalues at the expense of increased total computation time. Note: Poor estimates of eigenvalues also increase solution time by slowing the convergence of the AMG algorithm. turbulence_smoothing_order (integer) >=1 [=3] When turbulence_smoothing_type=chebyshev, this parameter sets the polynomial order for the smoother. When turbulence_smoothing_type=jacobi this parameter sets the number of smoothing passes during the downward and upward multigrid cycles. For both Chebyshev and Jacobi methods, increasing the smoothing order will reduce the total number of linear solver iterations required to converge the system. However, the time per iteration will also increase. There should be a balance between the number of iterations and the time per iteration. For turbulence matrices, a value of two is found to give the best performance. turbulence_chebyshev_max_min_ratio (real) >1 <=100 [=10] When turbulence_smoothing_type=chebyshev, this parameter provides an estimate of the smallest eigenvalue by dividing the largest eigenvalue calculated by the Lanczos algorithm by this parameter. Chebyshev polynomial smoothing requires the ratio of the largest to the smallest eigenvalues of the matrix to calculate the polynomial coefficients. The smallest value may not be calculated. This parameter can be used to get an estimate. The default of 10 means that the smallest eigenvalue is one tenth of the largest. turbulence_jacobi_relaxation_factor (real) >0 [=0.25] Used when turbulence_smoothing_type=jacobi. Jacobi smoothing may not be stable and it should be relaxed. Conventional under relaxation is used to stabilize it. The default of 0.25 means updated values will be calculated by adding 25 percent of new values to 75 percent of the old values. turbulence_setup_tolerance (real) >0 [=0] The nonlinear convergence tolerance used to control when the AMG setup process is performed. If the residual is higher than this value, the AMG setup process is performed. If the residual is lower than this tolerance, the AMG algorithm uses the existing setup information that is stored in RAM. With the default value of zero, the AMG setup process will occur every time the equations are solved, regardless of the residual. As the simulation converges, the change in solution between time steps decreases, minimizing the need to execute the AMG setup process. If AMG is used for more than one equation, such as temperature and turbulence, and this parameter is greater than zero, the memory requirements will be increased. Note: For each equation, a different setup tolerance can be used. This value should not be changed unless a convergence tolerance is set for the problem that is less than the default value. mesh_negative_coupling_tolerance (real) >=0 <=1 [=0.5] Tolerance within the algebraic multigrid (AMG) coarsening algorithm to determine when strongly negatively coupled variables for the mesh matrix are considered. Tolerance is the decimal fraction of the maximum value. The default value of 0.5 sets the tolerance to use values of 50 percent or greater of the most negative value. With a value of 1.0, negative coupling is ignored. Negative coupling tolerance is used when determining which entries from the fine matrix levels are retained at the coarser matrix levels. To select the coarse level grids for a matrix, only the values with a high contribution to the unknown equations are considered. Terms are considered to be highly negatively coupled if the negated value of the matrix entry exceeds the maximum negated value of the matrix multiplied by the negative coupling tolerance. The value of mesh_negative_coupling_tolerance can have significant influence on the structure of coarse level grids, influencing the convergence rate of the linear solution. mesh_positive_coupling_tolerance (real) >=0 <=1 [=1.0] Tolerance within the algebraic multigrid (AMG) coarsening algorithm to determine when strongly positively coupled variables for the mesh matrix are considered. Tolerance is the decimal fraction of the maximum value. With the default value of 1.0, positive coupling is ignored. Positive coupling tolerance is used when determining which entries from the fine matrix levels are retained at the coarser matrix levels. To select the coarse level grids for a matrix, only the values with a high contribution to the unknown equations are considered. Terms are considered to be highly positively coupled if the value of the matrix entry exceeds the maximum value of the matrix multiplied by the positive coupling tolerance. The value of mesh_positive_coupling_tolerance can have significant influence on the structure of coarse level grids, influencing the convergence rate of the linear solution. max_mesh_final_matrix (integer) >=1 [=100] Maximum number of entries present in the coarsest matrix of the AMG algorithm. With the default value of 100, coarsening will stop when the lowest level matrix is 10 by 10 or smaller. Increasing the size of this parameter may lead to longer duration of the solution calculation of the lowest level matrix. mesh_standard_interpolation (boolean) [=on] Interpolation method used to build the coarse multigrid matrices from fine matrices or to transfer the solution from a fine level to a coarse multigrid level, and vice versa. With this option set to off, direct interpolation will be used. Each equation in the fine level matrix that is considered to be important in the coarsening process gets interpolated to build the coarse level matrix. In direct interpolation, just neighboring points within the one fine level equation are used to calculate the interpolation weights. In standard interpolation, in addition to the neighboring points, other coupled equations are also considered in the weight calculations. More information is incorporated in the standard interpolation, leading to better convergence. In most cases, standard interpolation is recommended. mesh_positive_negative_separate (boolean) [=on] Determines if the positive and negative weights are separated during either standard or direct interpolation. In either direct or standard interpolation, weights are functions of off diagonal components of the fine level matrix and can be positive or negative. Interpolation can be conducted either separately for positive and negative weights, or these weights can be added together to have one interpolation. With this option set to on, summation of positive and negative weights will be calculated and used for interpolation. For mesh matrices, separation of weights is mesh_truncated_interpolation (boolean) [=on] Determines if the small interpolation weights will be truncated when interpolating the fine level multigrid matrix to the coarse level matrix or when transferring the solution from the coarse matrix to the finer AMG matrix levels. This flag should be used together with the mesh_truncation_tolerance to set the tolerance to determine the small weights. Truncation can be used with both direct and standard interpolation. A minor increase in the computational time to reach convergence can be observed if this flag is set to off. mesh_truncation_tolerance (real) >=0 <=1 [=0.1] Sets the portion of the smallest interpolation weights to be truncated during interpolation from a fine level to a coarse level. Used when mesh_truncated_interpolation=on. With a value of zero, no interpolation weights will be truncated. With a value of one, all weights will be truncated. mesh_givens_scaling (boolean) [=off] Determines if two by two Givens matrix scaling is used for the multigrid smoothing process. With this option turned off, diagonal scaling is used. mesh_smoothing_type (enumerated) [=chebyshev] Smoothing method used at each grid level after interpolation when values are from a fine matrix to a coarse matrix, or from a coarse matrix to a fine matrix. Chebyshev polynomial smoothing. Chebychev smoothing provides better performance, accurate convergence with less computational time, and is recommended for most matrices. Jacobi smoothing. Jacobi smoothing is recommended for the fully coupled flow stagger. mesh_eigenvalue_tolerance (real) >0 [=1.e-2] Tolerance to stop the eigenvalue calculation iterations. Used with mesh_smoothing_type=chebyshev. Chebyshev polynomial smoothing requires the ratio of the largest to the smallest eigenvalues of the matrix. The Iterative Lanczos algorithm is used to calculate the maximum eigenvalue of the mesh matrix. max_mesh_eigenvalue_iterations (integer) >=0 [=20] Used with mesh_smoothing_type=chebyshev. Maximum number of iterations of the Lanczos algorithm for computing the largest eigenvalues at each multigrid level. The mesh_eigenvalue_tolerance and max_mesh_eigenvalue_iterations determine when iterations stop. Whichever constraint is met first stops the algorithm. In general, eigenvalues converge within 20 iterations. For cases where the eigenvalues do not converge, increasing the total number of iterations can lead to better estimates of the eigenvalues at the expense of increased total computation time. Note: Poor estimates of eigenvalues also increase solution time by slowing the convergence of the AMG algorithm. mesh_smoothing_order (integer) >=1 [=2] When mesh_smoothing_type=chebyshev, this parameter sets the polynomial order for the smoother. When mesh_smoothing_type=jacobi this parameter sets the number of smoothing passes during the downward and upward multigrid cycles. For both Chebyshev and Jacobi methods, increasing the smoothing order will reduce the total number of linear solver iterations required to converge the system. However, the time per iteration will also increase. There should be a balance between the number of iterations and the time per iteration. For mesh matrices, a value of two is found to give the best performance. mesh_chebyshev_max_min_ratio (real) >1 <=100 [=10] When mesh_smoothing_type=chebyshev, this parameter provides an estimate of the smallest eigenvalue by dividing the largest eigenvalue calculated by the Lanczos algorithm by this parameter. Chebyshev polynomial smoothing requires the ratio of the largest to the smallest eigenvalues of the matrix to calculate the polynomial coefficients. The smallest value may not be calculated. This parameter can be used to get an estimate. The default of 10 means that the smallest eigenvalue is one tenth of the largest. mesh_jacobi_relaxation_factor (real) >0 [=0.25] Used when mesh_smoothing_type=jacobi. Jacobi smoothing may not be stable and it should be relaxed. Conventional under relaxation is used to stabilize it. The default of 0.25 means updated values will be calculated by adding 25 percent of new values to 75 percent of the old values. mesh_setup_tolerance (real) >0 [=0] The nonlinear convergence tolerance used to control when the AMG setup process is performed. If the residual is higher than this value, the AMG setup process is performed. If the residual is lower than this tolerance, the AMG algorithm uses the existing setup information that is stored in RAM. With the default value of zero, the AMG setup process will occur every time the equations are solved, regardless of the residual. As the simulation converges, the change in solution between time steps decreases, minimizing the need to execute the AMG setup process. If AMG is used for more than one equation, such as mesh and turbulence, and this parameter is greater than zero, the memory requirements will be increased. Note: For each equation, a different setup tolerance can be used. This value should not be changed unless a convergence tolerance is set for the problem that is less than the default value. viscoelastic_negative_coupling_tolerance (real) >=0 <=1 [=0.5] Tolerance within the algebraic multigrid (AMG) coarsening algorithm to determine when strongly negatively coupled variables for the viscoelastic matrix are considered. Tolerance is the decimal fraction of the maximum value. The default value of 0.5 sets the tolerance to use values of 50 percent or greater of the most negative value. With a value of 1.0, negative coupling is ignored. Negative coupling tolerance is used when determining which entries from the fine matrix levels are retained at the coarser matrix levels. To select the coarse level grids for a matrix, only the values with a high contribution to the unknown equations are considered. Terms are considered to be highly negatively coupled if the negated value of the matrix entry exceeds the maximum negated value of the matrix multiplied by the negative coupling tolerance. The value of viscoelastic_negative_coupling_tolerance can have significant influence on the structure of coarse level grids, influencing the convergence rate of the linear solution. viscoelastic_positive_coupling_tolerance (real) >=0 <=1 [=0.3] Tolerance within the algebraic multigrid (AMG) coarsening algorithm to determine when strongly positively coupled variables for the viscoelastic matrix are considered. Tolerance is the decimal fraction of the maximum value. The default value of 0.3 sets the tolerance to use values of 30 percent or greater of the most positive value. With a value of 1.0, positive coupling is ignored. Positive coupling tolerance is used when determining which entries from the fine matrix levels are retained at the coarser matrix levels. To select the coarse level grids for a matrix, only the values with a high contribution to the unknown equations are considered. Terms are considered to be highly positively coupled if the value of the matrix entry exceeds the maximum value of the matrix multiplied by the positive coupling tolerance. The value of viscoelastic_positive_coupling_tolerance can have significant influence on the structure of coarse level grids, influencing the convergence rate of the linear solution. max_viscoelastic_final_matrix (integer) >=1 [=100] Maximum number of entries present in the coarsest matrix of the AMG algorithm. With the default value of 100, coarsening will stop when the lowest level matrix is 10 by 10 or smaller. Increasing the size of this parameter may lead to longer duration of the solution calculation of the lowest level matrix. viscoelastic_standard_interpolation (boolean) [=on] Interpolation method used to build the coarse multigrid matrices from fine matrices or to transfer the solution from a fine level to a coarse multigrid level, and vice versa. With this option set to off, direct interpolation will be used. Each equation in the fine level matrix that is considered to be important in the coarsening process gets interpolated to build the coarse level matrix. In direct interpolation, just neighboring points within the one fine level equation are used to calculate the interpolation weights. In standard interpolation, in addition to the neighboring points, other coupled equations are also considered in the weight calculations. More information is incorporated in the standard interpolation, leading to better convergence. In most cases, standard interpolation is recommended. viscoelastic_positive_negative_separate (boolean) [=off] Determines if the positive and negative weights are separated during either standard or direct interpolation. In either direct or standard interpolation, weights are functions of off diagonal components of the fine level matrix and can be positive or negative. Interpolation can be conducted either separately for positive and negative weights, or these weights can be added together to have one interpolation. With this option set to on, summation of positive and negative weights will be calculated and used for interpolation. For viscoelastic matrices, separation of weights is not recommended. viscoelastic_truncated_interpolation (boolean) [=on] Determines if the small interpolation weights will be truncated when interpolating the fine level multigrid matrix to the coarse level matrix or when transferring the solution from the coarse matrix to the finer AMG matrix levels. This flag should be used together with the viscoelastic_truncation_tolerance to set the tolerance to determine the small weights. Truncation can be used with both direct and standard interpolation. A minor increase in the computational time to reach convergence can be observed if this flag is set to off. viscoelastic_truncation_tolerance (real) >=0 <=1 [=0.1] Sets the portion of the smallest interpolation weights to be truncated during interpolation from a fine level to a coarse level. Used when viscoelastic_truncated_interpolation=on. With a value of zero, no interpolation weights will be truncated. With a value of one, all weights will be truncated. viscoelastic_givens_scaling (boolean) [=off] Determines if two by two Givens matrix scaling is used for the multigrid smoothing process. With this option turned off, diagonal scaling is used. viscoelastic_smoothing_type (enumerated) [=chebyshev] Smoothing method used at each grid level after interpolation when values are from a fine matrix to a coarse matrix, or from a coarse matrix to a fine matrix. Chebyshev polynomial smoothing. Chebychev smoothing provides better performance, accurate convergence with less computational time, and is recommended for most matrices. Jacobi smoothing. Jacobi smoothing is recommended for the fully coupled flow stagger. viscoelastic_num_krylov_vectors (integer) >=0 [=30] Used with viscoelastic_smoothing_type=chebyshev. Number of Krylov vectors in the Arnoldi algorithm for computing the largest eigenvalues at each multigrid level. In general, eigenvalues converge within 30 Krylov vectors. For cases where the eigenvalues do not converge, increasing the number of Krylov vectors can lead to better estimates of the eigenvalues at the expense of increased total computation time. Note: Poor estimates of eigenvalues also increase solution time by slowing the convergence of the AMG algorithm. viscoelastic_smoothing_order (integer) >=1 [=2] When viscoelastic_smoothing_type=chebyshev, this parameter sets the polynomial order for the smoother. When viscoelastic_smoothing_type=jacobi this parameter sets the number of smoothing passes during the downward and upward multigrid cycles. For both Chebyshev and Jacobi methods, increasing the smoothing order will reduce the total number of linear solver iterations required to converge the system. However, the time per iteration will also increase. There should be a balance between the number of iterations and the time per iteration. For viscoelastic matrices, a value of two is found to give the best performance. viscoelastic_chebyshev_max_min_ratio (real) >1 <=100 [=10] When viscoelastic_smoothing_type=chebyshev, this parameter provides an estimate of the smallest eigenvalue by dividing the largest eigenvalue calculated by the Lanczos algorithm by this parameter. Chebyshev polynomial smoothing requires the ratio of the largest to the smallest eigenvalues of the matrix to calculate the polynomial coefficients. The smallest value may not be calculated. This parameter can be used to get an estimate. The default of 10 means that the smallest eigenvalue is one tenth of the largest. viscoelastic_jacobi_relaxation_factor (real) >0 [=0.25] Used when viscoelastic_smoothing_type=jacobi. Jacobi smoothing may not be stable and it should be relaxed. Conventional under relaxation is used to stabilize it. The default of 0.25 means updated values will be calculated by adding 25 percent of new values to 75 percent of the old values. viscoelastic_setup_tolerance (real) >0 [=0] The nonlinear convergence tolerance used to control when the AMG setup process is performed. If the residual is higher than this value, the AMG setup process is performed. If the residual is lower than this tolerance, the AMG algorithm uses the existing setup information that is stored in RAM. With the default value of zero, the AMG setup process will occur every time the equations are solved, regardless of the residual. As the simulation converges, the change in solution between time steps decreases, minimizing the need to execute the AMG setup process. If AMG is used for more than one equation, such as viscoelastic and turbulence, and this parameter is greater than zero, the memory requirements will be increased. Note: For each equation, a different setup tolerance can be used. This value should not be changed unless a convergence tolerance is set for the problem that is less than the default value. This command specifies the parameters of the algebraic multigrid algorithm. This option can be used to speed up the linear solution on problems in which the solver takes a large number of iterations at each step. The AMG algorithm works by interpolating the full matrix equation down to smaller sized matrices that can be solved directly with very little effort. The solution on the coarse matrix is then propagated back up through the successively finer matrices until it reaches the full matrix. At this point, the AMG solution is used to precondition the standard solver used for each stagger. This operation carries a significant amount of additional compute expense to perform, but can provide significant acceleration of the solution in some cases. This method also carries additional memory overhead. For most applications, turning this feature on will lead to approximately 50 percent greater memory consumption than using the standard preconditioning. For example, to specify the use of AMG on the pressure projection equation when performing the fully coupled flow solve, the following commands can be used: pressure_projection = on pressure_algebraic_multigrid = on pressure_standard_interpolation = on pressure_truncated_interpolation = on pressure_negative_coupling_tolerance = 0.6 pressure_positive_coupling_tolerance = 1 pressure_truncation_tolerance = 0.1 max_pressure_final_matrix = 100 pressure_eigenvalue_tolerance = 0.01 max_pressure_eigenvalue_iterations = 20 pressure_smoothing_order = 2 pressure_chebyshev_max_min_ratio = 10 pressure_jacobi_relaxation_factor = 0.25 pressure_smoothing_type = chebyshev pressure_positive_negative_separate = off pressure_givens_scaling = on pressure_setup_tolerance = 0	{"url":"https://2022.help.altair.com/2022.2/hwcfdsolvers/acusolve/topics/acusolve/algebraic_multigrid_parameters_acusolve_com_ref.htm","timestamp":"2024-11-01T19:29:24Z","content_type":"application/xhtml+xml","content_length":"160036","record_id":"<urn:uuid:0a24e7b3-88a7-4d32-8f22-6d9621f6f31f>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00456.warc.gz"}
perplexus.info :: Word Problems : 3 lowest score words Let's substitute each letter of a given word by the value of its ordinal count in the ABC: A=>1, B==>2 .. Z==>26. Evaluate the total for that word and call it f(k), k being the number of letters in the word. Let MW(K) be a word for which f(k) is minimal. Example: assuming the word CAB generates the lowest f(3) then MW(3) is CAB and its f(3)=6. Question: What triplet of common English words will generate the lowest f(7)+f(9)+ f(11)?	{"url":"http://perplexus.info/show.php?pid=10990&cid=59054","timestamp":"2024-11-13T14:57:01Z","content_type":"text/html","content_length":"12819","record_id":"<urn:uuid:8474647b-e351-40df-ad17-1cf9e46b0292>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00681.warc.gz"}
EViews Help: @pinverse Moore-Penrose pseudo-inverse of matrix. Syntax: @pinverse(m) m: matrix, sym Return: matrix, sym Returns the Moore-Penrose pseudo-inverse of a matrix object or sym. The pseudo-inverse has the property that both pre-multiplying and post-multiplying the pseudo-inverse by the source matrix returns the source matrix, and that both pre-multiplying and post-multiplying the source matrix by the pseudo-inverse will return the pseudo-inverse. Calling the function to produce the pseudo-inverse of a matrix returns a matrix, while the pseudo-inverse of a sym returns a sym. Note that pseudo-inverting a sym is much faster than inverting a matrix m1 = @mnrnd(10, 10) matrix m1p = @pinverse(m1) computes the pseudo-inverse of a randomly generated matrix M1. matrix m2p = @pinverse(@inner(m1)) computes the pseudo-inverse of the PSD inner product of M1. The following validate the properties of the pseudo-inverse: matrix diff1 = m1 * m1p * m1 - m1 matrix diff2 = m1p * m1 * m1p - m1p as both DIFF1 and DIFF2 equal zero.	{"url":"https://help.eviews.com/content/functionref_p-@pinverse.html","timestamp":"2024-11-12T07:02:41Z","content_type":"application/xhtml+xml","content_length":"10063","record_id":"<urn:uuid:fce15c9f-c562-419c-99ac-3ae99be36068>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00806.warc.gz"}
What are the standard deviation and the inter quartile range of the d0 condition? - Uni Essay Help What are the standard deviation and the inter quartile range of the d0 condition? Textbook #1: Lane et al. Introduction to Statistics, David M. Lane et al., 2013. ( http://onlinestatbook.com/Online_Statistics_Education.pdf ) Textbook #2:Illowsky et al. Introductory Statistics, Barbara Illowsky et al., 2013. ( http://openstaxcollege.org/files/textbook_version/hi_res_pdf/15/col11562-op.pdf ) Lane – Chapter 2: 7,9 and Chapter 3: 6,8,30,31 7. For the data from the 1977 Stat. and Biom. 200 class for eye color, construct: a. pie graph b. horizontal bar graph c. vertical bar graph d. a frequency table with the relative frequency of each eye colour 9. Which of the box plots on the graph has a large positive skew? Which has a large negative skew? 6. You recorded the time in seconds it took for 8 participants to solve a puzzle. These times appear in the table on the right. However, when the data was entered into the statistical program, the score that was supposed to be 22.1 was entered as 21.2. You had calculated the following measures of central tendency: the mean, the median, and the mean trimmed 25%. Which of these measures of central tendency will change when you correct the recording error? 8. You know the minimum, the maximum, and the 25th, 50th, and 75th percentiles of a distribution. Which of the following measures of central tendency or variability can you determine? Mean, Median, Mode, Trimean, Geometric Mean, Range, Interquartile Range, Variance, Standard Deviation For #30 and #31 see the ADHD Treatment Case Study (Page 624, http://onlinestatbook.com/2/case_studies/adhd.html) 30. What is the mean number of correct responses of the participants after taking the placebo (0 mg/kg)? 31. What are the standard deviation and the interquartile range of the d0 condition? 78. Twenty-five randomly selected students were asked the number of movies they watched the previous week. The results are as follows. a. Construct a histogram of the data. b. Complete the columns of the chart. 80. Use the following information: Suppose one hundred eleven people who shopped in a special t-shirt store were asked the number of t-shirts they own costing more than $19 each. If the data were collected by asking the first 111 people who entered the store, then the type of sampling is a. cluster b. simple random c. stratified d. convenience 84. Given the following box plot: a. which quarter has the smallest spread of data? What is that spread? b. which quarter has the largest spread of data? What is that spread? c. find the interquartile range (IQR) d. are there more data in the interval 5–10 or in the interval 10–13? How do you know this e. which interval has the fewest data in it? How do you know this? v. need more information 88. Given the following box plots, answer the questions. a. In complete sentences, explain why each statement is false. i. Data 1 has more data values above two than Data 2 has above two. ii. The data sets cannot have the same mode. iii. For Data 1, there are more data values below four than there are above four. b. For which group, Data 1 or Data 2, is the value of “7” more likely to be an outlier? Explain why in complete sentences	{"url":"https://www.uniessayhelp.com/2015/08/25/what-are-the-standard-deviation-and-the-inter-quartile-range-of-the-d0-condition/","timestamp":"2024-11-01T20:52:10Z","content_type":"text/html","content_length":"104787","record_id":"<urn:uuid:1fe61575-dbd8-4d24-88a3-0d1487568d4d>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00253.warc.gz"}
Computing mixed strategies equilibria in presence of switching costs by the solution of nonconvex QP problems In this paper we address a game theory problem arising in the context of network security. In traditional game theory problems, given a defender and an attacker, one searches for mixed strategies which minimize a linear payoff functional. In the problem addressed in this paper an additional quadratic term is added to the minimization problem. Such term represents switching costs, i.e., the costs for the defender of switching from a given strategy to another one at successive rounds of the game. The resulting problem is a nonconvex QP problem with linear constraints. We prove that, though simple when one of the two terms in the objective function is missing, the problem is, in general, a NP-hard one. The proof is based on a reduction from the decision version of the clique problem. Next, we discuss different reformulations and bounds for such problem. Finally, we show, through an extensive set of computational experiments, that the most promising approach to tackle this problem appears to be a branch-and-bound approach where a predominant role is played by an optimality-based domain reduction, based on the multiple solutions of LP problems at each node of the branch-and-bound tree. The computational cost per node is, thus, high but this is largely compensated by the rather small size of the corresponding tree. arXiv:2002.12599 [math.OC] View Computing mixed strategies equilibria in presence of switching costs by the solution of nonconvex QP problems	{"url":"https://optimization-online.org/2020/03/7656/","timestamp":"2024-11-10T18:18:48Z","content_type":"text/html","content_length":"84808","record_id":"<urn:uuid:111b5661-21ec-4b01-b53b-e6754dff36bc>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00720.warc.gz"}
Exceed Integer Limits ◀ Assignment Operator Between Two Structure Instances▶ Short-Circuit EvaluationAmazon A s most of you know, there are limits to how big and small an integer can go. If you go beyond those limits, funny things happen. In this section we will discuss this topic. There are basically three C++ integer types: , and . Standard C++ guarantees a minimum size for each of them: • A short is at least 16 bits. • An int is at least as big as short. • A long is at least 32 bits and at least as big as int. There is an easy way to find out the exact size of your system’s integers. You can simply print symbolic constants provided by <climits> (or <limits.h> for older implementations) in a program. You can also use sizeof on a type to see how big it is in bytes. For example, sizeof(int) gives you 4 and sizeof(short) gives you 2 on most systems. Now that we understand that, let’s turn our attention to the potential traps we could fall into if we are not careful enough. I have the following declarations: int john=0; unsigned int cheryl=0; I deduct one from and print their values: cout<<"John: " <<john<<" Cheryl: "<<cheryl; I got John: -1 Cheryl: 4294967295 . Now I reinitialize their values: cheryl= INT_MAX; Add one to and print their values: cout<<"John: " <<john<<" Cheryl: "<<cheryl; I got John: -2147483648 Cheryl: 2147483648 Your intuition probably tells you that –2147483648 is the minimum value of and 4294967295 is the maximum value of unsigned int , and you hit the nail right on the head! is an int, which ranges from -2147483648 to 2147483647. is an unsigned int, which ranges from 0 to 4294967295. As we can see, when you subtract one from the minimum value of unsigned int, you get the maximum value of unsigned int; when you add one to the maximum value of int, you get the minimum value of int. The overflow and underflow behavior can be depicted as a circle: if you move past the limit of a type, you get the value at the other extreme of the range. You can also try the following code in your program: cout<<INT_MAX2<<endl; / print out –2 */ Try to figure out why the output is –2. You can use a narrower analogous situation such as use 5 as the maximum value and –6 as the minimum value. Next we’ll look at short-circuit evaluation and how we can use it effectively! ◀ Assignment Operator Between Two Structure Instances▶ Short-Circuit Evaluation	{"url":"https://cppprogramming.chtoen.com/exceed-integer-limits.html","timestamp":"2024-11-08T21:20:34Z","content_type":"text/html","content_length":"33808","record_id":"<urn:uuid:ff74147b-45d3-43bb-9a10-2dd0e82bf538>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00684.warc.gz"}
Time And Work Aptitude Time And WorkPage 1 1. If A can do a piece of work in n days, then A's 1 day's work = 1/n 2. If A's 1 day's work = 1/n, then A can finish the work in n days 3. If A is thrice as good as in work B, then: Ratio of work done by A and B = 3 : 1. Ratio of time taken by A and B to finish a work = 1 : 3. 4. Short Tricks If M1 can do a W1 work in D1 days working H1 hour per day and M2 can do a W2 work in D2 days working H2 hour per day, Then M1D1W2 = M2D2W1 M1D1T1W2 = M2D2T2W1 (When time is given) M1D1T1E1W2 = M2D2T2E2W1 (When efficiency is added) M = Person D = Days W = Work T = Time E = Efficiency A and B can do a project in 20 days and 30 respectively. They were told to complete the project in 8 days and were paid in 1170 rupees. They took the help of C and completed the work in time. Find C's share in the total money earned by them? 1. (a)200 Answer is: CA and B can do (1/20)th and (1/30)th part of total work in a day respectively. in 8 days they can do 8(1/20 + 1/30) = 8/12 = (2/3)rd of the total work. So, C has to do (1/3)rd of total work. his share is (1/3)rd of total money. i.e. (1/3) X 1170 = 390 Ajay can do a certain work in 16 days. He start the work, works for 4 days and then quits. Bharat takes up the job and does the remaining work. If Bharat alone takes 24 days to do the entire work, in total how many days will the work be completed? 2. (a)22 Answer is: AAjay can do the work in 16 days. so , ajay's one day's work = (1/16)th of the work. in 4 days he can do (4 X 1)/16 = (1/4)th of the work. Bharat has to do 1 - (1/4) = 3/4 of the work. so, To do 3/4th of the work he takes (3/4) X 24 = 18 days. Total days = 4 +18 = 22 days P and Q can do certain work in 28 days and 56 days respectively. P work for 7 days, and then Q joins P. In how many more days can they complete the work? (a)7 days 3. (b)14 days (c)21 days (d)28 days Answer is: BP can do (1/28)th of the work in a day i.e. in 7 days he completes (1/28) X 7 = (1/4)th of the work. the remaining part of work (i.e. 3/4 of the work) is completed by P and Q working together at rate of (1/28 + 1/56) = 3/56 of the par day. so , they take (3/4) X (56/3) = 14 days more to complete remaining work together. P can do a certain work in 4 days and Q can do the same work in 12 days. They work together for a few days after P leaves and Q alone completes the remaining work. If it takes 6 days to complete the entire work, after how many days does P quit? 4. (a)3 days (b)4 days (c)2 days (d)5 days Answer is: CP can do (1/4)th the work in 1 day and Q can do (1/12)th of the work in 1 day. if they work for x days together,they complete x{1/4 + 1/12}th of work i.e. (x/3)rd of work. So, {1 - x/3} work is done by Q alone. He completes this work in {(3 - x)/3 X 12} = (12 - 4x) days. Total no. of days taken x + (12 - 4x) = 6 days. So, x = 2 days, P quits after 2 days. A can do a work in 6 days, which B can do in 9 days and C can do in 12 days. If a similar work is done in 24 days by all three of them working together, how many days will B alone take to complete that work? 5. (a)68 days (b)48 days (c)54 days (d)78 days Answer is: DInitially let the amount of work be x units which A, B, C do in 6, 9,12 days respectively. A does x/6 units of work in a day. B does x/9 units of work in a day. C does x/12 units of work in a day. So, they can do {x/6 + x/9 + x/12} units of work by working together in 1 day. In 24 days the amount of work done = 24 X (13x/36) = 26x/3 units. B can do 26x/3 units of work done by in 26x X 9/3x = 78 days. B can complete in 78 days. A group of 5 people can do a certain work in a certain number of days. If 4 more people join the group, they take 12 days less to do the same work. in how many days a group of 3 people can do the 6. (a)30 days (b)45 days (c)15 days Answer is: BLet 5 people do the work in x days. Then 9 people will do the work in (x - 12) days. But, work done by both sets of people is the same So, 5 X x = 9(x - 12) 5x = 9x - 108 x = 27 days If 3 people take y days, then 5 X 27 = 3 X y y = 45 days. B takes 18 days more than A to do a work. If A is thrice as efficient as B, and if they work together, in how many days do they complete the work? (a)6 days 7. (b)7 days (c)31/4 days (d)27/4 days Answer is: DLet, A takes x days for complete work then B takes (x + 18) days for complete work. A is thrice more efficient than B. So, M1D1 = M2D2 3 X x = 1 X (x + 18) 2x = 18 x = 9 A takes 9 days and B takes (9 + 18) days. They work together = 1/9 + 1/18 = 4/27 They finish work 27/4 days. P and Q can do a certain piece of work in 10 and 15 days respectively. P and Q work for 3 days each alternately till the work is completed. If P start the work, in how many days will the work be 8. (a)6 Answer is: DP and Q can do (1/10)th and (1/15)th of the total work in la day respectively. In 6 days, they do (3/10 + 3/15)th of the total work i.e. (1/2) of the total work. So, they complete the work in 12 days Terrific article This is the type of information that should be shared across the net. Disgrace on the search engines for now not positioning this post higher Come on over and talk over with my website . Thank you = Concepts are not fully clear.... Please Explain more better then this....	{"url":"https://zcos.in/aptitude/time-and-work","timestamp":"2024-11-04T16:49:45Z","content_type":"text/html","content_length":"32341","record_id":"<urn:uuid:46b6b1de-472b-46f3-a57c-5249ff3bc65e>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00179.warc.gz"}
Developing a CFD Digital Twin: 4 Steps for Successful Outcomes When other experimental approaches are too limited to effectively predict a fluid production process across a variety of operating conditions, engineers can create a CFD digital twin. Here are four steps for the industrial practitioner. A digital twin is a digital replica of a physical system. Designed to run real-time numerical experiments based on relevant mathematical models, the predictions from a digital twin should be identical to real experimental data—hence, twin. Thanks to modern algorithms and advancements in graphics processing unit (GPU) hardware, digital twins are becoming an optimal method for engineers who are trying to optimize processes with complex fluid mechanics and design limitations that make other approaches inapplicable or too limited. For a successful digital twin, four things need to be true: 1. Literature correlations and numerical modeling are not applicable and/or physical experiments are too limited. 2. The digital twin produces repeatable results that are indistinguishable to experimental data. 3. The digital twin takes less time and resources to develop compared to experimental measurement, at a much lower cost. 4. The digital twin generates process design correlations. For certain applications, developing a digital twin makes sense. Let’s take a look at an example to see why. Use Case for a CFD Digital Twin Model In the biopharmaceutical manufacturing process, it’s common to mix two miscible liquids with differences between fluid density and viscosity. But these differences make predicting mixing processes within agitated tanks complicated, causing significant processing and scale-up challenges. To optimize pharmaceutical blending and mixing, manufacturers can hybridize the three traditional approaches—literature correlations, numerical modeling, and experiments—with a digital twin. This is because too few literature correlations are applicable to multi-fluid systems with large variations in density and viscosity, experiments are limited by costs and access to equipment, and the transport physics involved limit the applicability of numerical modeling approaches. In this example, a digital twin that pairs Lattice Boltzmann–based transport algorithms with GPU resources allowed the pharmaceutical manufacturer to simulate minutes/hours of fluid mechanics within hours/days of computer wall time. The transient processing insights that the twins generated rivaled experimental data—but at a cost orders-of-magnitude lower. For similar results, follow these four steps. Developing a CFD Digital Twin Model: Four Steps 1. Identify Modeling & Operating Requirements To develop a digital twin, you must first identify the requirements that your model must satisfy. In the example above, there are three primary requirements based on the fluid properties of the mixing system: 1. The model must approximate the three-dimensional mixing systems (as opposed to two-dimensional). 2. The framework must integrate spatiotemporal variations in fluid properties directly into the solution of time-dependent fluid flow equations. 3. The twin must solve quickly to produce predictions within industrially relevant analysis time scales. For successful outcomes, practitioners need to understand influential operating parameters and the transport physics within the system you’re trying to model. 2. Determine Numerical Approach After you identify modeling and operating requirements, you can determine the numerical approach that satisfies those requirements. In our miscible blending example, we identified three equations to solve: 1. The three-dimensional, time-dependent, incompressible Navier-Stokes equation, which models the conservation of momentum of a fluid particle. 2. The advection-diffusion equation, which models the conservation of species. 3. The Boltzmann transport equation, which models the conservation of transport carrier probability density. These three equations together can be solved via the Lattice Boltzmann method. Interested in reading more details about using a CFD digital twin to understand miscible blending? Check out our academic paper with AbbVie. 3. Build the Digital Twin Model To build the digital twin model—and solve the above equations—you need a tool that supports the Lattice Boltzmann method. The Lattice Boltzmann method is high-resolution, which means direct numerical simulation and/or large-eddy simulation can be used to build a digital twin that can handle laminar, transitional, and turbulent flow regimes with no reconfiguration. Compared to traditional finite element and finite difference approaches, the Lattice Boltzmann approach can model multiphase and multiphysics transport processes in fluid mechanical systems at much faster computational speeds. This speed is only further amplified when run in a GPU-based computing environment. This is where computational fluid dynamics software (CFD) comes in. Modern CFD software solves Lattice Boltzmann algorithms on GPUs, which can be used to build digital twins and quickly produce detailed, accurate process simulations. 4. Validate the Model But before you can apply the model, you have to validate it. In our pharmaceutical blending example, the manufacturer validated the twin by first comparing the single fluid blend times and power numbers predicted from the twin to experimental data across a wide range of Reynolds numbers. From there, they used the twin to explore blending in two-fluid, density stratified systems. Then, they verified output against experimental data taken at multiple impeller speeds. The takeaway: Before you can use the twin for process optimization and design on unmeasured systems, you must first measure and supply relevant fluid properties to the twin. Appropriate experimental data is crucial for validating output. The initial development of a digital twin does not eliminate the need for experiment. However, once developed, it can be used to generate processing insights with a fidelity that rivals experimental measurement at a much lower cost. When you’re dealing with complex fluid mechanics, the initial set up of a digital twin is worth it—especially when backed by modern CFD software. Build advanced fluid models in minutes, predict real-time dynamics with precision, and solve more complex fluid flow problems faster with M-Star CFD—CFD software for the real world.	{"url":"https://mstarcfd.com/blog/cfd-digital-twin-4-steps/","timestamp":"2024-11-08T08:22:25Z","content_type":"text/html","content_length":"99843","record_id":"<urn:uuid:bf7c6041-7f46-48e5-9687-18b7d9cdbb2d>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00242.warc.gz"}
How to solve mathematics? - California Learning Resource Network How to solve mathematics? How to Solve Mathematics? Mathematics can be a daunting task for many students, but with the right approach, it can be mastered. In this article, we will explore the most effective ways to solve mathematics problems, covering the fundamental concepts, strategies, and methods to help you overcome your math anxiety and become a math whiz. Understanding the Basics: The Foundation of Solving Mathematics Before diving into the solutions, it is essential to have a solid understanding of the basics. Here are the fundamental concepts that will help you build a strong foundation: • Number Systems: Be familiar with the concepts of numbers, such as decimals, fractions, and percentages. • Algebraic Expressions: Understand the rules of exponents, roots, and order of operations. • Equations and Inequalities: Be able to identify and solve linear and quadratic equations, as well as linear and nonlinear inequalities. Breaking Down the Problem: Deconstructing the Complexity When faced with a math problem, it’s easy to feel overwhelmed. To combat this, break down the problem into smaller, manageable parts. Here’s how: • Identify the Goal: Understand what you are trying to solve. • Break Down the Problem: Identify specific components of the problem. • Simplify the Problem: Use mathematical operations to simplify the problem. • Eliminate Variables: Identify and eliminate any unnecessary variables. Strategy 1: Visual Aids for Better Understanding Visual aids can be powerful tools in helping you understand and solve math problems. Here are some strategies to consider: • Graphs and Charts: Use visual representations to visualize the problem, such as graphs, charts, and diagrams. • Number Lines: Utilize number lines to help track the solution and identify patterns. • Block Models: Use blocks or manipulatives to create a physical representation of the problem. Strategy 2: Word Problems for Real-World Context Word problems can be a great way to understand the relevance of math in real-life situations. Here are some ways to make the most of word problems: • Read Carefully: Pay attention to the context and identify the key information. • Identify the Question: Determine what is being asked. • Break Down the Problem: Break down the problem into smaller, manageable parts. • Solve the Problem: Use math operations to find the solution. Strategy 3: The Power of formulas and Theorems Formulas and theorems can be powerful tools in solving math problems. Here are some ways to make the most of them: • Understand the Concept: Understand the underlying concept behind the formula or theorem. • Apply Correctly: Apply the formula or theorem correctly to the problem. • Check Your Work: Verify the solution by plugging in the values. Common Math Mistakes to Avoid Here are some common math mistakes to avoid: • Ignoring the Order of Operations: Failing to follow the correct order of operations (PEMDAS/BODMAS). • Rounding Off: Rounding off intermediate values, which can affect the accuracy of the solution. • Inconsistent Units: Using inconsistent units, which can lead to incorrect solutions. Solving mathematics is not a one-size-fits-all solution. By understanding the basics, breaking down problems, using visual aids, and applying formulas and theorems, you can overcome math anxiety and become proficient in solving math problems. Remember to avoid common mistakes and practice regularly to build your math skills and confidence. Additional Tips and Resources Here are some additional tips and resources to help you master math: • Practice Regularly: Practice consistently to build your math skills and confidence. • Seek Help When Needed: Don’t hesitate to ask for help when you’re struggling. • Use Online Resources: Utilize online resources, such as Khan Academy, MIT OpenCourseWare, and other online resources. • Join a Study Group: Join a study group or online community to collaborate and learn from others. By following the strategies outlined in this article and utilizing the additional tips and resources, you can overcome math anxiety and become proficient in solving math problems. Your friends have asked us these questions - Check out the answers! Leave a Comment	{"url":"https://www.clrn.org/how-to-solve-mathematics/","timestamp":"2024-11-13T06:26:31Z","content_type":"text/html","content_length":"132636","record_id":"<urn:uuid:849f754f-1cd8-4a7c-a070-8bb75199b2a6>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00623.warc.gz"}