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CSCE 222 Lecture 10 Jump to navigation Jump to search « previous \| Friday, February 11, 2011 \| next » Truth Sets Let ${\displaystyle P}$ be a predicate with domain ${\displaystyle D}$. The set of all elements x in D such that P(x) is true is denoted by: ${\displaystyle \{x\in D\|P(x)\}}$ Set operations {\displaystyle {\begin{aligned}A-B&=\{x\|x\in A\wedge xotin B\}\\&=\{x\in A\|xotin B\}\end{aligned}}} Let ${\displaystyle U}$ be a universal set. Let ${\displaystyle A\subseteq U}$, then ${\displaystyle {\bar {A}}=U-A}$ Fun with Sets (Oh boy!) Suppose that ${\displaystyle A}$ and ${\displaystyle B}$ are finite sets. How many elements are in their union? ${\displaystyle \|A\cup B\|=\|A\|+\|B\|-\|A\cap B\|}$ Big O — Redefined Big O is actually a set of all functions asymptotically less than a given function (read as a subset of O(n)) What is the set ${\displaystyle \{f:\mathbb {N} \rightarrow \mathbb {R} \|f(n)=O(g(n))\}}$ ${\displaystyle =\{f:\mathbb {N} \rightarrow \mathbb {R} \|\exists n_{0}\in \mathbb {N} \exists C>0\forall n\geq n_{0}:\|f(n)\|\leq C\|g(n)\|\}}$ Let ${\displaystyle A}$ and ${\displaystyle B}$ be nonempty sets. A function ${\displaystyle f}$ that maps from ${\displaystyle A}$ to ${\displaystyle B}$ is an assignment of exactly one element of $ {\displaystyle B}$ to each element of ${\displaystyle A}$. ${\displaystyle f:A\rightarrow B}$: • ${\displaystyle A}$ is domain (input type) • ${\displaystyle B}$ is codomain (output type; superset of range) • ${\displaystyle \{f(a)\|a\in A\}}$ is range ${\displaystyle f:\mathbb {Z} \rightarrow \mathbb {Z} ,\ f(x)=x^{2}}$ {\displaystyle {\begin{aligned}range&=\{0,1,4,9,16,\ldots \}\\&=\{y\in \mathbb {Z} \|\exists x\in \mathbb {Z} :y=x^{2}\}&=\{y\in \mathbb {N} _{0}\|{\mbox{y is a perfect square}}\}\end{aligned}}} Injective Functions Function is one-to-one (A maps uniquely to B) Surjective Functions Range of A is B	{"url":"https://notes.komputerwiz.net/wiki/CSCE_222_Lecture_10","timestamp":"2024-11-13T02:08:05Z","content_type":"text/html","content_length":"52285","record_id":"<urn:uuid:82b94843-d645-4683-9b30-dc199359b0e2>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00003.warc.gz"}
Sequential nonparametric inference by betting Faculty Candidate Seminar Sequential nonparametric inference by betting Shubhanshu ShekharPostdoctoral ResearcherCarnegie Mellon University Sequential inference methods refer to statistical procedures in which the sample size (and possibly other design choices) are not decided apriori, but instead depend on the observations collected. Such methods often lead to statistical or computational advantages over their fixed sample size or batch counterparts. In this talk, I will describe a general framework for constructing powerful sequential methods for some nonparametric sequential inference problems. I will begin by introducing an important nonparametric testing problem, called two-sample testing, and describe a framework for constructing powerful sequential two-sample tests based on the principle of testing-by-betting. The key idea is to reframe the task of sequential testing into that of selecting payoff functions that maximize the wealth of a fictitious bettor, betting against the null in a repeated game. To design the payoff functions, I will describe a simple strategy that proceeds by constructing predictable estimates of the witness function associated with a class of integral probability metrics (IPMs). The statistical properties of the resulting test can then be characterized in terms of the regret of this prediction strategy. I will then instantiate the general testing strategy for some popular IPMs; such as the Kolmogorov-Smirnov metric, and the kernel-MMD metric. This framework for constructing two-sample tests is quite general, and I will describe how it can be applied to a much larger class of testing problems, and to other inference tasks such as estimation, and change-detection. To conclude the talk, I will briefly discuss my plans for future Shubhanshu Shekhar is a postdoctoral researcher in the Department of Statistics and Data Science at Carnegie Mellon University working with Prof. Aaditya Ramdas. Prior to this, he obtained his PhD in Electrical Engineering from the University of California, San Diego, where he was advised by Prof. Tara Javidi. His research interests lie broadly in the areas of machine learning and nonparametric Linda Scovel Faculty Host Lei YingProfessor, Electrical Engineering and Computer ScienceUniversity of Michigan	{"url":"https://eecs.engin.umich.edu/event/sequential-nonparametric-inference-by-betting/","timestamp":"2024-11-01T22:16:57Z","content_type":"text/html","content_length":"62354","record_id":"<urn:uuid:46b4ff46-729a-4954-991a-e27126d10b30>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00787.warc.gz"}
CovRegRF: Covariance Regression with Random Proposed method Most of the existing multivariate regression analyses focus on estimating the conditional mean of the response variable given its covariates. However, it is also crucial in various areas to capture the conditional covariances or correlations among the elements of a multivariate response vector based on covariates. We consider the following setting: let $\mathbf{Y}_{n \times q}$ be a matrix of $q$ response variables measured on $n$ observations, where $\mathbf{y}_i$ represents the $i$th row of $\mathbf{Y}$. Similarly, let $\mathbf{X}_{n \times p}$ be a matrix of $p$ covariates available for all $n$ observations, where $\mathbf{x}_i$ represents the $i$th row of $\mathbf{X}$. We assume that the observation $\mathbf{y}_i$ with covariates $\mathbf{x}_i$ has a conditional covariance matrix $\Sigma_{\mathbf{x}_i}$. We propose a novel method called Covariance Regression with Random Forests (CovRegRF) to estimate the covariance matrix of a multivariate response $\mathbf{Y}$ given a set of covariates $\mathbf{X}$, using a random forest framework. Random forest trees are built with a specialized splitting criterion \[\sqrt{n_Ln_R}d (\Sigma^L, \Sigma^R)\] where $\Sigma^L$ and $\Sigma^R$ are the covariance matrix estimates of left and right nodes, and $n_L$ and $n_R$ are the left and right node sizes, respectively, $d(\ Sigma^L, \Sigma^R)$ is the Euclidean distance between the upper triangular part of the two matrices and computed as follows: \[d(A, B) = \sqrt{\sum_{i=1}^{q}\sum_{j=i}^{q} (\mathbf{A}_{ij} - \mathbf {B}_{ij})^2}\] where $\mathbf{A}_{q \times q}$ and $\mathbf{B}_{q \times q}$ are symmetric matrices. For a new observation, the random forest provides the set of nearest neighbour out-of-bag observations which is used to estimate the conditional covariance matrix for that observation. Significance test We propose a hypothesis test to evaluate the effect of a subset of covariates on the covariance matrix estimates while controlling for the other covariates. Let $\Sigma_\mathbf{X}$ be the conditional covariance matrix of $\mathbf{Y}$ given all $X$ variables and $\Sigma_{\mathbf{X}^c}$ is the conditional covariance matrix of $\mathbf{Y}$ given only the set of controlling $X$ variables. If a subset of covariates has an effect on the covariance matrix estimates obtained with the proposed method, then $\Sigma_\mathbf{X}$ should be significantly different from $\Sigma_{\ mathbf{X}^c}$. We conduct a permutation test for the null hypothesis \[H_0 : \Sigma_\mathbf{X} = \Sigma_{\mathbf{X}^c}\] We estimate a $p$-value with the permutation test. If the $p$-value is less than the pre-specified significance level $\alpha$, we reject the null hypothesis. Data analysis We will show how to use the CovRegRF package on a generated data set. The data set consists of two multivariate data sets: $\mathbf{X}_{n \times 3}$ and $\mathbf{Y}_{n \times 3}$. The sample size ($n$) is 200. The covariance matrix of $\mathbf{Y}$ depends on $X_1$ and $X_2$ (i.e. $X_3$ is a noise variable). We load the data and split it into train and test sets: xvar.names <- colnames(data$X) yvar.names <- colnames(data$Y) data1 <- data.frame(data$X, data$Y) smp <- sample(1:nrow(data1), size = round(nrow(data1)0.6), replace = FALSE) traindata <- data1[smp,, drop=FALSE] testdata <- data1[-smp, xvar.names, drop=FALSE] Firstly, we check the global effect of $\mathbf{X}$ on the covariance matrix estimates by applying the significance test for the three covariates. formula <- as.formula(paste(paste(yvar.names, collapse="+"), ".", sep=" ~ ")) globalsig.obj <- significance.test(formula, traindata, params.rfsrc = list(ntree = 200), nperm = 10, test.vars = NULL) #> [1] 0 Using 10 permutations, the estimated $p$-value is 0 which is smaller than the significance level ($\alpha$) of 0.05 and we reject the null hypothesis indicating the conditional covariance matrices significantly vary with the set of covariates. When performing a permutation test to estimate a $p$-value, we need more than 10 permutations. Using 500 permutations, the estimated $p$ -value is 0.012. The computational time increases with the number of permutations. Next, we apply the proposed method with covregrf() and get the out-of-bag (OOB) covariance matrix estimates for the training observations. covregrf.obj <- covregrf(formula, traindata, params.rfsrc = list(ntree = 200)) pred.oob <- covregrf.obj$predicted.oob head(pred.oob, 2) #> [[1]] #> y1 y2 y3 #> y1 1.1286879 0.8387699 1.101836 #> y2 0.8387699 1.9878143 1.507416 #> y3 1.1018365 1.5074164 3.591839 #> [[2]] #> y1 y2 y3 #> y1 1.353311 1.050629 1.761897 #> y2 1.050629 2.153400 2.142313 #> y3 1.761897 2.142313 4.453531 Then, we get the variable importance (VIMP) measures for the covariates. VIMP measures reflect the predictive power of $\mathbf{X}$ on the estimated covariance matrices. Also, we can plot the VIMP vimp.obj <- vimp(covregrf.obj) #> x1 x2 x3 #> 1.38012921 0.51122499 0.07159317 From the VIMP measures, we see that $X_3$ has smaller importance than $X_1$ and $X_2$. We apply the significance test to evaluate the effect of $X_3$ on the covariance matrices while controlling for $X_1$ and $X_2$. partialsig.obj <- significance.test(formula, traindata, params.rfsrc = list(ntree = 200), nperm = 10, test.vars = "x3") #> [1] 0.3 Using 10 permutations, the estimated p-values is 0.3 and we fail to reject the null hypothesis, indicating that we do not have enough evidence to prove that $X_3$ has an effect on the estimated covariance matrices while $X_1$ and $X_2$ are in the model. Using 500 permutations, the estimated $p$-value is 0.218. Finally, we can get the covariance matrix predictions for the test observations. pred.obj <- predict(covregrf.obj, testdata) pred <- pred.obj$predicted head(pred, 2) #> [[1]] #> y1 y2 y3 #> y1 1.0710647 0.5039413 0.6859008 #> y2 0.5039413 1.6077176 1.1050398 #> y3 0.6859008 1.1050398 2.7710218 #> [[2]] #> y1 y2 y3 #> y1 1.294158 1.306257 1.854186 #> y2 1.306257 2.183658 2.424231 #> y3 1.854186 2.424231 4.387248	{"url":"http://cran.stat.auckland.ac.nz/web/packages/CovRegRF/vignettes/CovRegRF.html","timestamp":"2024-11-01T20:58:20Z","content_type":"text/html","content_length":"43769","record_id":"<urn:uuid:84f0aba0-60cf-4e56-aa02-70263d55d79d>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00806.warc.gz"}
n rsvg::property_defs pub struct FloodColor(pub Color); Expand description Tuple Fields§ §0: Color Trait Implementations§ This method tests for self and other values to be equal, and is used by ==. This method tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason. Auto Trait Implementations§ Blanket Implementations§ 🔬This is a nightly-only experimental API. (clone_to_uninit) Performs copy-assignment from Read more Returns the argument unchanged. Calls U::from(self). That is, this conversion is whatever the implementation of From<T> for U chooses to do. The alignment of pointer. The type for initializers. unsafe fn init(init: <T as Pointable>::Init) -> usize Initializes a with the given initializer. Read more Mutably dereferences the given pointer. Read more Drops the object pointed to by the given pointer. Read more impl<SS, SP> SupersetOf<SS> for SP where SS: SubsetOf<SP>, The inverse inclusion map: attempts to construct from the equivalent element of its superset. Read more Checks if self is actually part of its subset T (and can be converted to it). Use with care! Same as self.to_subset but without any property checks. Always succeeds. The inclusion map: converts self to the equivalent element of its superset. The resulting type after obtaining ownership. Creates owned data from borrowed data, usually by cloning. Read more Uses borrowed data to replace owned data, usually by cloning. Read more The type returned in the event of a conversion error. Performs the conversion. The type returned in the event of a conversion error. Performs the conversion.	{"url":"https://gnome.pages.gitlab.gnome.org/librsvg/internals/rsvg/property_defs/struct.FloodColor.html","timestamp":"2024-11-06T20:24:52Z","content_type":"text/html","content_length":"48868","record_id":"<urn:uuid:6583773b-1393-4ad2-bb99-1355a254de83>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00402.warc.gz"}
Percentage Calculator - % Conversion Calculator % Percentage Calculator Percent Meaning in Math – A percentage is a dimension number, and the ratio is defined as a fraction of 100. The percentage symbol is % or pct or pc. Normally it’s used to define the proportionate part of any number. Like 65% can be written like 65/100 or 65:100 or 0.65 What is a Percentage Calculator Percentage Calculator is a free online tool that calculates the percentage of two numbers like 33 is the percentage of 100. The percentage sign is %. Percentage Calculator Percentage (%) is the ratio or particular number that defines a fraction calculation of 100. The Symbol of percentage is % or pct or a percentage like 35%. How Percentage is Calculated To calculate the percentage, divide the value by the total value and multiply the result by 100 Percentage formula – (Value/Total value) × 100 Example: 2/5 × 100 = 0.4 × 100 = 40 per cent Percentage Formula In percentage, the formula is the algebraic question, that has 3 values P × V1 = V2 P – Percentage V1 – First Value V2 – Result on V1 Our percentage Calculator Provides automatic results for input values P = 1.5/30 = 0.05 × 100 = 5% Percentage Increase Calculate The Percentage Increase Calculator calculates the increasing differences from one value to another value in terms of % Percentage. You have to enter two values one is the starting value and another is the final value to find an increasing percentage How to Calculate Percentage Increase The percentage Increase Formula is Percent increase = [(new value – original value)/original value] * 100 • Final Value minus starting value • Divide that absolute amount value by the starting value • Multiply the given value by 100 to get a percent increase • It will find a positive or negative value if the result is positive then its percentage Increase otherwise its percentage decrease Percentage Decrease Calculate The Percentage decrease Calculator calculates the negative difference from one value to another in the form of a percentage. You have to enter starting value and final value to get the percentage How to Calculate Percentage Decrease The percentage Decrease Formula is Percent decrease = [(original value – new value)/original value] * 100 • The first value minus the second value • Divide the result by the absolute value of the starting value • Multiply by 100 to get a percentage decrease • If the percentage is negative it means that is an increased value	{"url":"https://conversioncalculator.org/percentage-calculator/","timestamp":"2024-11-10T06:03:38Z","content_type":"text/html","content_length":"177570","record_id":"<urn:uuid:5223bcf2-50a3-4bf3-a5b4-f230f4c7af7b>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00359.warc.gz"}
[Solved] A regression model of the form Show that \| SolutionInn A regression model of the form Show that For the data set Find the regression equation and A regression model of the form Show that For the data set Find the regression equation and plot the data with the regression model. Transcribed Image Text: x 0.0 0.2 0.4 0.6 0.8 1.0 y0.0 0.15 0.25 0.25 0.17 -0.01 Fantastic news! We've Found the answer you've been seeking! Step by Step Answer: Answer rating: 100% (2 reviews) Cramers rule a2 y 2 2 2x2 2x3 2 2 2 v3 2 3 x22 xy x2 3 22 Ans Ans y 001 015 025 025 ...View the full answer Answered By I am a Chartered Accountant with AIR 45 in CA - IPCC. I am a Merit Holder ( B.Com ). The following is my educational details. PLEASE ACCESS MY RESUME FROM THE FOLLOWING LINK: https://drive.google.com 3.80+ 3+ Reviews 10+ Question Solved Students also viewed these Mechanical Engineering questions Study smarter with the SolutionInn App	{"url":"https://www.solutioninn.com/regression-model-of-the-form","timestamp":"2024-11-01T19:29:42Z","content_type":"text/html","content_length":"78809","record_id":"<urn:uuid:35a59227-8d72-469c-930a-e0879d359c8e>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00681.warc.gz"}
The Stacks project Lemma 21.26.1. In the situation above, choose a K-injective complex $\mathcal{I}^\bullet $ of $\mathcal{O}$-modules representing $K$. Using $-1$ times the canonical map for one of the four arrows we get maps of complexes \[ \mathcal{I}^\bullet (X) \xrightarrow {\alpha } \mathcal{I}^\bullet (Z) \oplus \mathcal{I}^\bullet (Y) \xrightarrow {\beta } \mathcal{I}^\bullet (E) \] with $\beta \circ \alpha = 0$. Thus a canonical map \[ c^ K_{X, Z, Y, E} : \mathcal{I}^\bullet (X) \longrightarrow C(\beta )^\bullet [-1] \] This map is canonical in the sense that a different choice of K-injective complex representing $K$ determines an isomorphic arrow in the derived category of abelian groups. If $c^ K_{X, Z, Y, E}$ is an isomorphism, then using its inverse we obtain a canonical distinguished triangle \[ R\Gamma (X, K) \to R\Gamma (Z, K) \oplus R\Gamma (Y, K) \to R\Gamma (E, K) \to R\Gamma (X, K)[1] \] All of these constructions are functorial in $K$. Comments (0) Post a comment 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 0EVX. Beware of the difference between the letter 'O' and the digit '0'. The tag you filled in for the captcha is wrong. You need to write 0EVX, in case you are confused.	{"url":"https://stacks.math.columbia.edu/tag/0EVX","timestamp":"2024-11-03T15:51:33Z","content_type":"text/html","content_length":"21062","record_id":"<urn:uuid:09c1b6dc-d15b-4a2d-826b-1d5ef0eaaf65>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00686.warc.gz"}
Optimal Water Pipe Replacement Policy Open Journal of Optimization Vol.07 No.02(2018), Article ID:85174,9 pages Optimal Water Pipe Replacement Policy Harrison O. Amuji^*, Chukwudi J. Ogbonna, Geoffrey U. Ugwuanyim, Hycinth C. Iwu, Okechukwu B. Nwanyibuife Department of Statistics, Federal University of Technology, Owerri, Nigeria Copyright © 2018 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). Received: March 15, 2018; Accepted: June 5, 2018; Published: June 8, 2018 Water scarcity is the major problem confronting both urban and rural dwellers in Enugu State. This scarcity emanated from indiscriminate pipe failure, lack of adequate maintenance, uncertainty on the time of repair or replacement of pipes etc. There is no systematic approach to determining replacement or repair time of the pipes. Hence, the rule of thumb is used in making such a vital decision. The population is increasing, houses are built but the network is not expanded and the existing ones that were installed for no less than two to three decades ago are not maintained. These compounded the problem of scarcity of water in the state. Replacement or repair of water pipes when they are seen spilling water cannot solve this lingering problem. The solution can be achieved by developing an adequate predictive model for water pipe replacement. Hence, this research is aimed at providing a solution to this problem of water scarcity by suggesting a policy that will be used for better planning. The interests in this paper were to obtain a water pipe failure model, the intensity function λ(t) [failure rate], the reliability R(t) and the optimal time of replacement and they were achieved. It was observed that the failure rate of the pipes increases with time while their reliability deteriorates with time. Hence, the Optimal replacement policy is that each pipe should be replaced after 4^th break when the reliability = 0.0011. Reliability, Non-Homogenous Poisson Process, Repairable System, Optimal Water Pipe Replacement Policy, Failure Rate 1. Introduction Water distribution network is a network consisting of pipes, valves and pumps that distribute water to the end users [1] [2] . In water distribution some components are made redundant which makes water supply failure not to be noticed immediately when pipe failure occurs. Pipes are of different types and sizes. In this paper, our interest is on three types of pipes namely: 1) Asbestos-concrete (AC) pipes, 2) Unplastisized polyvinyl chloride (UPVC) or plastic pipes and 3) Galvanized Iron pipes (GVI). These pipes are further divided into mains, branches and terminals. Mains are of different lengths and diameters ranging from 13 ft to 20 ft and 200 mm to 600 mm. They carry water directly from the pumping station and the distribution storage. Branches have a diameter ranging from 75 mm to 150 mm, but of the same length with the mains. They carry water from the mains to various terminals. Terminals also have the same length as others but have a diameter ranging from 25 mm to 50 mm. They are called service lines in the network. The terminals convey water to the end-users and they are the smallest in size but the largest in number. Table 1 shows the type of water distribution pipes. Each of the types of pipes in Table 1 has its advantages and shortcomings as presented in Table 2. The major problems of the GVI pipes are bulkiness, rusting and expensiveness compared to the other two types. Majority of their failure result from rusting. UPVC is fast replacing GVI pipes because of their portability and durability. Many factors are responsible for water pipe breaks. For the purpose of this paper, these factors are divided into internal and external factors, which are collectively called covariates. Internal factors: Internal factors include the pressure exerted by the flowing water in the pipe; the action of chemicals used in the water treatment; the diameter of the pipes and the intrinsic material of the pipes etc. Pressure exerted by water over a long period of time on pipes leads to their break and this pressure is inversely related to the diameter of pipes. The pressure Table 1. Types of water distribution pipes. Table 2. Water distribution pipes, advantages and disadvantages. is also intensified, as the pipes are located nearer to the source and reduces as the length of the pipes increases. External factors: External factors include the soil type, topography, lay-dept, shock and rust. These factors are called external because they occur outside the pipes. The nature of soil in which the pipe is laid determines the rate of pipe’s break etc. The rates at which these factors occur influence the maintenance decisions. Enugu State Water Corporation Enugu Water generation and distribution are the sole concern of the water Generation unit, Enugu State Water Corporation Enugu. The replacement and repair functions are the sole responsibilities of the engineering and plumbing units. The pressure under which the pumps distribute water ranges from 6 bars to 16 bars. Frequency of pipe breaks depend on the following factors namely; lay depth, topography, intrinsic material of the pipes, pressure, diameter of the pipes, lay location etc. Period of replacement: pipes are replaced as they break and the rule of thumb is extensively used. Maintenance actions include periodic greasing of nuts and bolts, replacement or repair of failed pipes. In most cases, the maintenances are done on demand, as there is no regularity on the time limit for the maintenance services. But since pipes constitute the bulk of the money cost of the water network distribution system, there is a need to develop a failure model to detect the probability of failure at any given time, the failure rate, the reliability of the pipes and the optimal replacement time of the pipes to avoid waste of water resources. [1] [2] used non-homogeneous Poisson process to model the pipe failure. They were of opinion that power law process model (λ(t) = abt^b^−1) can be used to describe the intensity of a NHPP. They define intensity as the mean number of failure, N(t), per unit time and that once the intensity has been estimated the number of failures in a given time period is Poisson. [3] [4] observed that if the parameter b is between 0 and 1, then the network is improving, but if b > 1, it shows a deteriorating network. The model parameters can be determined using either graphical or analytical analysis. We can linearized the mean value function M(t) by taking the logarithm of both sides to obtained $\mathrm{ln}M\left(t\right)=\mathrm{ln}\left(a\right)+b\mathrm{ln}\ left(t\right)$ , and a plot lnM(t) versus ln(t) should be a straight line with intercept ln(a) and slop b. Water utilities are concerned with large number of main breaks and the resulting direct and indirect costs due to deterioration. These contribute to frequent breaks and resulting repair/replacement and rehabilitation costs [5] . They observed that replacement represents large capital investment. They suggested that if an estimator that can track the replacement time will be developed; such an estimator will be of great use to practicing engineers. [6] did a work on optimal replacement policy for stochastically failing equipment inaccessible to inspection. The policy was characterized by a single parameter, N. If equipment age is less than N, the appropriate action is to do nothing; if equal to N, the appropriate action is to replace the equipment. [7] obtained the optimal replacement policy by minimizing the average annual cost of equipment whose maintenance cost is a function increasing with time and whose scrap value is known, and their calculations were based on average annual cost incurred on the item. They assert that the item should be replaced when the average annual cost is minimal. Water distribution pipes are repairable system because some maintenance action of repair can be done on them to restore the operation of the pipes after failures. [8] [9] [10] observed that a repairable system is a system which, after failing to perform its functions satisfactorily, can be restored to fully satisfactory performance by any method other than replacement of the entire system. Repairable systems shared the property that they could be repaired by replacing some failed components and returned to regular operation. Sometimes, minimal repairs bring the system reliability back to the same level it had before the failure, while ‘perfect’ repairs bring the reliability back to the state of the system at the start of the operation [11] [12] . The intensity and mean value functions of a power law process PLP are λ(t; M, β) = Mβt^B^−1 and V(t; M, β) = Mt^β; M, β > 0. β determines how reliability decays or grows. If β < 1, the intensity function decreases over time and the reliability improves. If β = 1, the reliability remains the same over time, but if β > 1, the reliability declines and intensity increases. Again, if β > 1, we have NHPP and for β = 1, we have HPP. λ(t) is an increasing concave (straight, convex) curve if 1 < β < 2 [13] [14] [15] . 2. Overview of Poisson Process/Reliability The Poisson Process provides a realistic model for many random phenomena. Since values of a Poisson random variable are the non-negative integers, any random phenomenon for which count of some sort is of interest is a candidate for modelling by assuming a Poisson distribution [16] , thus, the first event counted by N[s](.) takes place after an exponential amount of time with parameter λ; the rest of the inter event times are iid exponential with parameter λ [17] . A Poisson process is a renewal process for which the underlying distribution is exponential. A renewal process is a sequence of independent identically distributed non-negative random variables, X[1], X[2], …, which, with probability 1, are not all zeros [18] . They assert that the Poisson process arises quite naturally in reliability and life testing situations when the underlying life distribution is exponential. [19] observed that the original use of the term “reliability” was purely qualitative and that was why aerospace engineers recognized the need to have more than one engine on an airplane and drivers keep spare tyres in their vehicles without any precise measurement of the failure rate. But [20] were of opinion that reliability is the probability that a component or a system will perform its intended function under specified environmental conditions with time. They assert that the problem of assuring and maintaining reliability has many facets, which include: original equipment design; control of quality of products during production, life testing and design modifications. Data Presentation In this section, we present the data that is used for this work in Table 3. 3. Data Analysis $\lambda =\frac{\sum {x}_{i}}{N}=\frac{18}{20}=0.9;\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,2,\cdots ,18;\text{\hspace{0.17em}}N=1,2,\cdots ,20$(2) ${\lambda }^{c}=\frac{0.9}{10}=0.09;\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\lambda }^{c}=\text{failureperunittime}$(3) We present how the parameters of the model were determined in Table 4. The graphic method of determining the parameters of the water pipe failure model is presented in Figure 1. Table 3. Type of pipe, Installation period, Replacement and number of Breaks. Table 4. Determination of the Parameters of the Model. ${p}_{n}\left(t\right)=\mathrm{exp}\left(-0.09{t}^{1.02}\right){\left(\frac{0.09{t}^{1.02}}{n!}\right)}^{n};\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(n,t\right)=0,1,2,\cdots ,10$(6) 3.1. Water Pipe Failure Model Since b > 1, then, our model assumes NHPP (λ(t)). Equation (6) is the water pipe failure model, where p = probability; n = the number of failures; t = the time in years; a = 0.09 and b = 1.02 are the parameters of the model and a, b, t > 0. 3.2. The Reliability Function R(t) $R\left(t\right)=\frac{{p}_{n}\left(t\right)}{\lambda \left(t\right)}$(11) 3.3. Determination of Optimal Time of Replacement A reliability approach is used here to determine the optimal time of replacement of pipes. Reliability approach has never been used to determine the optimal water pipe replacement policy. The pipe should be replaced a little time before the reliability becomes zero. Continuous used of the pipe when the reliability become zero add more to the cost because the failure rate λ(t) increases rapidly as the reliability R(t) becomes zero. Hence, t (time in years) is optimal at the least reliability R(t). 4. Interpretation of Results Form the data collected from Enugu State Water Corporation Enugu we obtain the parameters of our model “a” and “b” and use them to generate a table for the probability of number of failure at any given time t. Also failure rate and reliability of the pipes were determined and presented in Tables 4-7 respectively. Result A: ${p}_{n}\left(t\right)=\mathrm{exp}\left(-0.09{t}^{1.02}\right){\left(\frac{0.09{t}^{1.02}}{n!}\right)}^{n};\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(t,n\right)=1,\cdots ,10$(12) The probability of the number of failure of pipes at a given time is presented in Table 5 below. Result B: $\lambda \left(t\right)=\left(0.0918\right){t}^{0.02}$(13) The computation of the intensity function λ(t) is presented in Table 6. Table 6 shows that the intensity function, λ(t), increases with time. Result C: $R\left(t\right)=\frac{{p}_{n}\left(t\right)}{\lambda \left(t\right)}$(14) In Table 7, we present the computation of reliability, R(t), of the water pipes. Table 5. Probability of number of failure at Time t. Table 6. Computation of the Intensity Function λ(t). Table 7. Computation of the Reliability function R(t). Table 7 above shows that the reliability R(t) of the pipes deteriorates with time while the failure rate of the pipes increases with time. The pipe should be replaced at the 4^th break when the reliability of the pipes is 0.0011. Hence, it is most reasonable to replace the pipe when the reliability is very close to zero. 5. Conclusion In this paper, we developed water pipe failure model and the optimal water pipe replacement time (policy) of the pipes. The water pipe failure model was a Non-homogenous Poisson process and reliability has never been used to determine optimal replacement policy on a repairable system. We also obtained the intensity function λ(t) [failure rate] of the pipes. From this work, it was observed that the failure rate of the pipes increases with time while their reliability deteriorates with time. Hence, the Optimal replacement policy is that the pipe should be replaced after the 4^ th break when the reliability = 0.0011. Cite this paper Amuji, H.O., Ogbonna, C.J., Ugwuanyim, G.U., Iwu, H.C. and Nwanyibuife, O.B. (2018) Optimal Water Pipe Replacement Policy. Open Journal of Optimization, 7, 41-49. https://doi.org/10.4236/	{"url":"https://file.scirp.org/Html/1-2730175_85174.htm","timestamp":"2024-11-02T09:37:21Z","content_type":"application/xhtml+xml","content_length":"60256","record_id":"<urn:uuid:2d310c0c-c8b8-49de-842d-aecef02ef701>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00666.warc.gz"}
An equal temperament is a musical temperament or tuning system that approximates just intervals by dividing an octave (or other interval) into steps such that the ratio of the frequencies of any adjacent pair of notes is the same. This system yields pitch steps perceived as equal in size, due to the logarithmic changes in pitch frequency.^[2] 12 tone equal temperament chromatic scale on C, one full octave ascending, notated only with sharps. In classical music and Western music in general, the most common tuning system since the 18th century has been 12 equal temperament (also known as 12 tone equal temperament, 12 TET or 12 ET, informally abbreviated as 12 equal), which divides the octave into 12 parts, all of which are equal on a logarithmic scale, with a ratio equal to the 12th root of 2, ( ^12√2 ≈ 1.05946 ). That resulting smallest interval, 1/12 the width of an octave, is called a semitone or half step. In Western countries the term equal temperament, without qualification, generally means 12 TET. In modern times, 12 TET is usually tuned relative to a standard pitch of 440 Hz, called A 440, meaning one note, A, is tuned to 440 hertz and all other notes are defined as some multiple of semitones away from it, either higher or lower in frequency. The standard pitch has not always been 440 Hz; it has varied considerably and generally risen over the past few hundred years.^[3] Other equal temperaments divide the octave differently. For example, some music has been written in 19 TET and 31 TET, while the Arab tone system uses 24 TET. Instead of dividing an octave, an equal temperament can also divide a different interval, like the equal-tempered version of the Bohlen–Pierce scale, which divides the just interval of an octave and a fifth (ratio 3:1), called a "tritave" or a "pseudo-octave" in that system, into 13 equal parts. For tuning systems that divide the octave equally, but are not approximations of just intervals, the term equal division of the octave, or EDO can be used. Unfretted string ensembles, which can adjust the tuning of all notes except for open strings, and vocal groups, who have no mechanical tuning limitations, sometimes use a tuning much closer to just intonation for acoustic reasons. Other instruments, such as some wind, keyboard, and fretted instruments, often only approximate equal temperament, where technical limitations prevent exact tunings.^ [4] Some wind instruments that can easily and spontaneously bend their tone, most notably trombones, use tuning similar to string ensembles and vocal groups. General properties In an equal temperament, the distance between two adjacent steps of the scale is the same interval. Because the perceived identity of an interval depends on its ratio, this scale in even steps is a geometric sequence of multiplications. (An arithmetic sequence of intervals would not sound evenly spaced and would not permit transposition to different keys.) Specifically, the smallest interval in an equal-tempered scale is the ratio: ${\displaystyle \ r^{n}=p\ }$ ${\displaystyle \ r={\sqrt[{n}]{p\ }}\ }$ where the ratio r divides the ratio p (typically the octave, which is 2:1) into n equal parts. (See Twelve-tone equal temperament below.) Scales are often measured in cents, which divide the octave into 1200 equal intervals (each called a cent). This logarithmic scale makes comparison of different tuning systems easier than comparing ratios, and has considerable use in ethnomusicology. The basic step in cents for any equal temperament can be found by taking the width of p above in cents (usually the octave, which is 1200 cents wide), called below w, and dividing it into n parts: ${\displaystyle \ c={\frac {\ w\ }{n}}\ }$ In musical analysis, material belonging to an equal temperament is often given an integer notation, meaning a single integer is used to represent each pitch. This simplifies and generalizes discussion of pitch material within the temperament in the same way that taking the logarithm of a multiplication reduces it to addition. Furthermore, by applying the modular arithmetic where the modulus is the number of divisions of the octave (usually 12), these integers can be reduced to pitch classes, which removes the distinction (or acknowledges the similarity) between pitches of the same name, e.g., c is 0 regardless of octave register. The MIDI encoding standard uses integer note designations. General formulas for the equal-tempered interval Twelve-tone equal temperament 12 tone equal temperament, which divides the octave into 12 intervals of equal size, is the musical system most widely used today, especially in Western music. The two figures frequently credited with the achievement of exact calculation of equal temperament are Zhu Zaiyu (also romanized as Chu-Tsaiyu. Chinese: 朱載堉) in 1584 and Simon Stevin in 1585. According to F.A. Kuttner, a critic of giving credit to Zhu,^[5] it is known that Zhu "presented a highly precise, simple and ingenious method for arithmetic calculation of equal temperament mono-chords in 1584" and that Stevin "offered a mathematical definition of equal temperament plus a somewhat less precise computation of the corresponding numerical values in 1585 or later." The developments occurred independently.^[6]^(p200) Kenneth Robinson credits the invention of equal temperament to Zhu^[7]^[b] and provides textual quotations as evidence.^[8] In 1584 Zhu wrote: I have founded a new system. I establish one foot as the number from which the others are to be extracted, and using proportions I extract them. Altogether one has to find the exact figures for the pitch-pipers in twelve operations.^[9]^[8] Kuttner disagrees and remarks that his claim "cannot be considered correct without major qualifications".^[5] Kuttner proposes that neither Zhu nor Stevin achieved equal temperament and that neither should be considered its inventor.^[10] Zhu Zaiyu's equal temperament pitch pipes Chinese theorists had previously come up with approximations for 12 TET, but Zhu was the first person to mathematically solve 12 tone equal temperament,^[11] which he described in two books, published in 1580^[12] and 1584.^[9]^[13] Needham also gives an extended account.^[14] Zhu obtained his result by dividing the length of string and pipe successively by ^12√2 ≈ 1.059463 , and for pipe length by ^24√2 ≈ 1.029302 ,^[15] such that after 12 divisions (an octave), the length was halved. Zhu created several instruments tuned to his system, including bamboo pipes.^[16] Some of the first Europeans to advocate equal temperament were lutenists Vincenzo Galilei, Giacomo Gorzanis, and Francesco Spinacino, all of whom wrote music in it.^[17]^[18]^[19]^[20] Simon Stevin was the first to develop 12 TET based on the twelfth root of two, which he described in van de Spiegheling der singconst (c.1605), published posthumously in 1884.^[21] Plucked instrument players (lutenists and guitarists) generally favored equal temperament,^[22] while others were more divided.^[23] In the end, 12-tone equal temperament won out. This allowed enharmonic modulation, new styles of symmetrical tonality and polytonality, atonal music such as that written with the 12-tone technique or serialism, and jazz (at least its piano component) to develop and flourish. One octave of 12 TET on a monochord In 12 tone equal temperament, which divides the octave into 12 equal parts, the width of a semitone, i.e. the frequency ratio of the interval between two adjacent notes, is the twelfth root of two: ${\displaystyle {\sqrt[{12}]{2\ }}=2^{\tfrac {1}{12}}\approx 1.059463}$ This interval is divided into 100 cents. Calculating absolute frequencies To find the frequency, P[n], of a note in 12 TET, the following formula may be used: ${\displaystyle \ P_{n}=P_{a}\ \cdot \ {\Bigl (}\ {\sqrt[{12}]{2\ }}\ {\Bigr )}^{n-a}\ }$ In this formula P[n] represents the pitch, or frequency (usually in hertz), you are trying to find. P[a] is the frequency of a reference pitch. The indes numbers n and a are the labels assigned to the desired pitch (n) and the reference pitch (a). These two numbers are from a list of consecutive integers assigned to consecutive semitones. For example, A[4] (the reference pitch) is the 49th key from the left end of a piano (tuned to 440 Hz), and C[4] (middle C), and F♯[4] are the 40th and 46th keys, respectively. These numbers can be used to find the frequency of C[4] and F♯[4]: ${\displaystyle P_{40}=440\ {\mathsf {Hz}}\ \cdot \ {\Bigl (}{\sqrt[{12}]{2}}\ {\Bigr )}^{(40-49)}\approx 261.626\ {\mathsf {Hz}}\ }$ ${\displaystyle P_{46}=440\ {\mathsf {Hz}}\ \cdot \ {\Bigl (}{\sqrt[{12}]{2}}\ {\Bigr )}^{(46-49)}\approx 369.994\ {\mathsf {Hz}}\ }$ Converting frequencies to their equal temperament counterparts To convert a frequency (in Hz) to its equal 12 TET counterpart, the following formula can be used: ${\displaystyle \ E_{n}=E_{a}\ \cdot \ 2^{\ x}\ \quad }$ where in general ${\displaystyle \quad \ x\ \equiv \ {\frac {1}{\ 12\ }}\ \operatorname {round} \!{\Biggl (}12\log _{2}\left({\frac {\ n\ }{a}}\right){\Biggr )}~.}$ Comparison of intervals in 12-TET with just intonation E[n] is the frequency of a pitch in equal temperament, and E[a] is the frequency of a reference pitch. For example, if we let the reference pitch equal 440 Hz, we can see that E[5] and C♯[5] have the following frequencies, respectively: ${\displaystyle E_{660}=440\ {\mathsf {Hz}}\ \cdot \ 2^{\left({\frac {7}{\ 12\ }}\right)}\ \approx \ 659.255\ {\mathsf {Hz}}\ \quad }$ where in this case ${\displaystyle \quad x={\frac {1}{\ 12\ }}\ \operatorname {round} \!{\Biggl (}\ 12\log _{2}\left({\frac {\ 660\ }{440}}\right)\ {\Biggr )}={\frac {7}{\ 12\ }}~.}$ ${\displaystyle E_{550}=440\ {\mathsf {Hz}}\ \cdot \ 2^{\left({\frac {1}{\ 3\ }}\right)}\ \approx \ 554.365\ {\mathsf {Hz}}\ \quad }$ where in this case ${\displaystyle \quad x={\frac {1}{\ 12\ }}\ \operatorname {round} \!{\Biggl (}12\log _{2}\left({\frac {\ 550\ }{440}}\right){\Biggr )}={\frac {4}{\ 12\ }}={\frac {1}{\ 3\ }}~.}$ Comparison with just intonation The intervals of 12 TET closely approximate some intervals in just intonation.^[24] The fifths and fourths are almost indistinguishably close to just intervals, while thirds and sixths are further In the following table, the sizes of various just intervals are compared to their equal-tempered counterparts, given as a ratio as well as cents. Interval Name Exact value in 12 TET Decimal value in 12 TET Pitch in Just intonation interval Cents in just intonation 12 TET cents tuning error Unison (C) 2^0⁄12 = 1 1 0 1/1 = 1 0 0 Minor second (D♭) 2^1⁄12 = ^12√2 1.059463 100 16/15 = 1.06666... 111.73 -11.73 Major second (D) 2^2⁄12 = ^6√2 1.122462 200 9/8 = 1.125 203.91 -3.91 Minor third (E♭) 2^3⁄12 = ^4√2 1.189207 300 6/5 = 1.2 315.64 -15.64 Major third (E) 2^4⁄12 = ^3√2 1.259921 400 5/4 = 1.25 386.31 +13.69 Perfect fourth (F) 2^5⁄12 = ^12√32 1.33484 500 4/3 = 1.33333... 498.04 +1.96 Tritone (G♭) 2^6⁄12 = √2 1.414214 600 64/45= 1.42222... 609.78 -9.78 Perfect fifth (G) 2^7⁄12 = ^12√128 1.498307 700 3/2 = 1.5 701.96 -1.96 Minor sixth (A♭) 2^8⁄12 = ^3√4 1.587401 800 8/5 = 1.6 813.69 -13.69 Major sixth (A) 2^9⁄12 = ^4√8 1.681793 900 5/3 = 1.66666... 884.36 +15.64 Minor seventh (B♭) 2^10⁄12 = ^6√32 1.781797 1000 16/9 = 1.77777... 996.09 +3.91 Major seventh (B) 2^11⁄12 = ^12√2048 1.887749 1100 15/8 = 1.875 1088.270 +11.73 Octave (C) 2^12⁄12 = 2 2 1200 2/1 = 2 1200.00 0 Seven-tone equal division of the fifth Violins, violas, and cellos are tuned in perfect fifths (G D A E for violins and C G D A for violas and cellos), which suggests that their semitone ratio is slightly higher than in conventional 12 tone equal temperament. Because a perfect fifth is in 3:2 relation with its base tone, and this interval comprises seven steps, each tone is in the ratio of ^7√3/2 to the next (100.28 cents), which provides for a perfect fifth with ratio of 3:2, but a slightly widened octave with a ratio of ≈ 517:258 or ≈ 2.00388:1 rather than the usual 2:1, because 12 perfect fifths do not equal seven octaves.^[25] During actual play, however, violinists choose pitches by ear, and only the four unstopped pitches of the strings are guaranteed to exhibit this 3:2 ratio. Other equal temperaments Five-, seven-, and nine-tone temperaments in ethnomusicology Approximation of 7 TET Five- and seven-tone equal temperament (5 TET and {{7 TET}} ), with 240 cent and 171 cent steps, respectively, are fairly common. 5 TET and 7 TET mark the endpoints of the syntonic temperament's valid tuning range, as shown in Figure 1. • In 5 TET, the tempered perfect fifth is 720 cents wide (at the top of the tuning continuum), and marks the endpoint on the tuning continuum at which the width of the minor second shrinks to a width of 0 cents. • In 7 TET, the tempered perfect fifth is 686 cents wide (at the bottom of the tuning continuum), and marks the endpoint on the tuning continuum, at which the minor second expands to be as wide as the major second (at 171 cents each). 5 tone and 9 tone equal temperament According to Kunst (1949), Indonesian gamelans are tuned to 5 TET, but according to Hood (1966) and McPhee (1966) their tuning varies widely, and according to Tenzer (2000) they contain stretched octaves. It is now accepted that of the two primary tuning systems in gamelan music, slendro and pelog, only slendro somewhat resembles five-tone equal temperament, while pelog is highly unequal; however, in 1972 Surjodiningrat, Sudarjana and Susanto analyze pelog as equivalent to 9-TET (133-cent steps ).^[26] 7-tone equal temperament A Thai xylophone measured by Morton in 1974 "varied only plus or minus 5 cents" from 7 TET.^[27] According to Morton, "Thai instruments of fixed pitch are tuned to an equidistant system of seven pitches per octave ... As in Western traditional music, however, all pitches of the tuning system are not used in one mode (often referred to as 'scale'); in the Thai system five of the seven are used in principal pitches in any mode, thus establishing a pattern of nonequidistant intervals for the mode."^[28] A South American Indian scale from a pre-instrumental culture measured by Boiles in 1969 featured 175 cent seven-tone equal temperament, which stretches the octave slightly, as with instrumental gamelan music.^[29] Chinese music has traditionally used 7 TET.^[c]^[d] Various equal temperaments Easley Blackwood's notation system for 16 equal temperament: Intervals are notated similarly to those they approximate and there are fewer enharmonic equivalents.^[32] Comparison of equal temperaments from 9 to 25^[33]^[a] Many instruments have been built using 19 EDO tuning. Equivalent to 1/ 3 comma meantone, it has a slightly flatter perfect fifth (at 695 cents), but its minor third and major sixth are less than one-fifth of a cent away from just, with the lowest EDO that produces a better minor third and major sixth than 19 EDO being 232 EDO. Its perfect fourth (at 505 cents), is seven cents sharper than just intonation's and five cents sharper than 12 EDO's. 22 EDO is one of the most accurate EDOs to represent superpyth temperament (where 7:4 and 16:9 are the same interval) and is near the optimal generator for porcupine temperament. The fifths are so sharp that the major and minor thirds we get from stacking fifths will be the supermajor third (9/7) and subminor third (7/6). One step closer to each other are the classical major and minor thirds (5/4 and 6/5). 23 EDO is the largest EDO that fails to approximate the 3rd, 5th, 7th, and 11th harmonics (3:2, 5:4, 7:4, 11:8) within 20 cents, but it does approximate some ratios between them (such as the 6:5 minor third) very well, making it attractive to microtonalists seeking unusual harmonic territory. 24 EDO, the quarter-tone scale, is particularly popular, as it represents a convenient access point for composers conditioned on standard Western 12 EDO pitch and notation practices who are also interested in microtonality. Because 24 EDO contains all the pitches of 12 EDO, musicians employ the additional colors without losing any tactics available in 12 tone harmony. That 24 is a multiple of 12 also makes 24 EDO easy to achieve instrumentally by employing two traditional 12 EDO instruments tuned a quarter-tone apart, such as two pianos, which also allows each performer (or one performer playing a different piano with each hand) to read familiar 12 tone notation. Various composers, including Charles Ives, experimented with music for quarter-tone pianos. 24 EDO also approximates the 11th and 13th harmonics very well, unlike 12 EDO. 26 EDO 26 is the denominator of a convergent to log[2](7), tuning the 7th harmonic (7:4) with less than half a cent of error. Although it is a meantone temperament, it is a very flat one, with four of its perfect fifths producing a major third 17 cents flat (equated with the 11:9 neutral third). 26 EDO has two minor thirds and two minor sixths and could be an alternate temperament for barbershop harmony. 27 EDO 27 is the lowest number of equal divisions of the octave that uniquely represents all intervals involving the first eight harmonics. It tempers out the septimal comma but not the syntonic comma. 29 is the lowest number of equal divisions of the octave whose perfect fifth is closer to just than in 12 EDO, in which the fifth is 1.5 cents sharp instead of 2 cents flat. Its classic major third is roughly as inaccurate as 12 EDO, but is tuned 14 cents flat rather than 14 cents sharp. It also tunes the 7th, 11th, and 13th harmonics flat by roughly the same amount, allowing 29 EDO to match intervals such as 7:5, 11:7, and 13:11 very accurately. Cutting all 29 intervals in half produces 58 EDO, which allows for lower errors for some just tones. 31 EDO was advocated by Christiaan Huygens and Adriaan Fokker and represents a rectification of quarter-comma meantone into an equal temperament. 31 EDO does not have as accurate a perfect fifth as 12 EDO (like 19 EDO), but its major thirds and minor sixths are less than 1 cent away from just. It also provides good matches for harmonics up to 11, of which the seventh harmonic is particularly accurate. 34 EDO gives slightly lower total combined errors of approximation to 3:2, 5:4, 6:5, and their inversions than 31 EDO does, despite having a slightly less accurate fit for 5:4. 34 EDO does not accurately approximate the seventh harmonic or ratios involving 7, and is not meantone since its fifth is sharp instead of flat. It enables the 600 cent tritone, since 34 is an even number. 41 is the next EDO with a better perfect fifth than 29 EDO and 12 EDO. Its classical major third is also more accurate, at only six cents flat. It is not a meantone temperament, so it distinguishes 10:9 and 9:8, along with the classic and Pythagorean major thirds, unlike 31 EDO. It is more accurate in the 13 limit than 31 EDO. 46 EDO 46 EDO provides major thirds and perfect fifths that are both slightly sharp of just, and many say that this gives major triads a characteristic bright sound. The prime harmonics up to 17 are all within 6 cents of accuracy, with 10:9 and 9:5 a fifth of a cent away from pure. As it is not a meantone system, it distinguishes 10:9 and 9:8. 53 EDO has only had occasional use, but is better at approximating the traditional just consonances than 12, 19 or 31 EDO. Its extremely accurate perfect fifths make it equivalent to an extended Pythagorean tuning, as 53 is the denominator of a convergent to log[2](3). With its accurate cycle of fifths and multi-purpose comma step, 53 EDO has been used in Turkish music theory. It is not a meantone temperament, which put good thirds within easy reach by stacking fifths; instead, like all schismatic temperaments, the very consonant thirds are represented by a Pythagorean diminished fourth (C-F♭), reached by stacking eight perfect fourths. It also tempers out the kleisma, allowing its fifth to be reached by a stack of six minor thirds (6:5). 58 equal temperament is a duplication of 29 EDO, which it contains as an embedded temperament. Like 29 EDO it can match intervals such as 7:4, 7:5, 11:7, and 13:11 very accurately, as well as better approximating just thirds and sixths. 72 EDO approximates many just intonation intervals well, providing near-just equivalents to the 3rd, 5th, 7th, and 11th harmonics. 72 EDO has been taught, written and performed in practice by Joe Maneri and his students (whose atonal inclinations typically avoid any reference to just intonation whatsoever). As it is a multiple of 12, 72 EDO can be considered an extension of 12 EDO, containing six copies of 12 EDO starting on different pitches, three copies of 24 EDO, and two copies of 36 EDO. 96 EDO approximates all intervals within 6.25 cents, which is barely distinguishable. As an eightfold multiple of 12, it can be used fully like the common 12 EDO. It has been advocated by several composers, especially Julián Carrillo.^[34] Other equal divisions of the octave that have found occasional use include 13 EDO, 15 EDO, 17 EDO, and 55 EDO. 2, 5, 12, 41, 53, 306, 665 and 15601 are denominators of first convergents of log[2](3), so 2, 5, 12, 41, 53, 306, 665 and 15601 twelfths (and fifths), being in correspondent equal temperaments equal to an integer number of octaves, are better approximations of 2, 5, 12, 41, 53, 306, 665 and 15601 just twelfths/fifths than in any equal temperament with fewer tones.^[35]^[36] 1, 2, 3, 5, 7, 12, 29, 41, 53, 200, ... (sequence A060528 in the OEIS) is the sequence of divisions of octave that provides better and better approximations of the perfect fifth. Related sequences containing divisions approximating other just intervals are listed in a footnote.^[e] Equal temperaments of non-octave intervals The equal-tempered version of the Bohlen–Pierce scale consists of the ratio 3:1 (1902 cents) conventionally a perfect fifth plus an octave (that is, a perfect twelfth), called in this theory a tritave (), and split into 13 equal parts. This provides a very close match to justly tuned ratios consisting only of odd numbers. Each step is 146.3 cents (), or ^13√3. Wendy Carlos created three unusual equal temperaments after a thorough study of the properties of possible temperaments with step size between 30 and 120 cents. These were called alpha, beta, and gamma. They can be considered equal divisions of the perfect fifth. Each of them provides a very good approximation of several just intervals.^[37] Their step sizes: • alpha: ^9√3/2 (78.0 cents) • beta: ^11√3/2 (63.8 cents) • gamma: ^20√3/2 (35.1 cents) Alpha and beta may be heard on the title track of Carlos's 1986 album Beauty in the Beast. Proportions between semitone and whole tone In this section, semitone and whole tone may not have their usual 12 EDO meanings, as it discusses how they may be tempered in different ways from their just versions to produce desired relationships. Let the number of steps in a semitone be s, and the number of steps in a tone be t. There is exactly one family of equal temperaments that fixes the semitone to any proper fraction of a whole tone, while keeping the notes in the right order (meaning that, for example, C, D, E, F, and F♯ are in ascending order if they preserve their usual relationships to C). That is, fixing q to a proper fraction in the relationship q t = s also defines a unique family of one equal temperament and its multiples that fulfil this relationship. For example, where k is an integer, 12k EDO sets q = 1/2, 19 k EDO sets q = 1/3, and 31 k EDO sets q = 2/ 5 . The smallest multiples in these families (e.g. 12, 19 and 31 above) has the additional property of having no notes outside the circle of fifths. (This is not true in general; in 24 EDO, the half-sharps and half-flats are not in the circle of fifths generated starting from C .) The extreme cases are 5 k EDO, where q = 0 and the semitone becomes a unison, and 7 k EDO , where q = 1 and the semitone and tone are the same interval. Once one knows how many steps a semitone and a tone are in this equal temperament, one can find the number of steps it has in the octave. An equal temperament with the above properties (including having no notes outside the circle of fifths) divides the octave into 7 t − 2 s steps and the perfect fifth into 4 t − s steps. If there are notes outside the circle of fifths, one must then multiply these results by n, the number of nonoverlapping circles of fifths required to generate all the notes (e.g., two in 24 EDO, six in 72 EDO). (One must take the small semitone for this purpose: 19 EDO has two semitones, one being 1/ 3 tone and the other being 2/ 3 . Similarly, 31 EDO has two semitones, one being 2/ 5 tone and the other being 3/ 5 ). The smallest of these families is 12 k EDO, and in particular, 12 EDO is the smallest equal temperament with the above properties. Additionally, it makes the semitone exactly half a whole tone, the simplest possible relationship. These are some of the reasons 12 EDO has become the most commonly used equal temperament. (Another reason is that 12 EDO is the smallest equal temperament to closely approximate 5 limit harmony, the next-smallest being 19 EDO.) Each choice of fraction q for the relationship results in exactly one equal temperament family, but the converse is not true: 47 EDO has two different semitones, where one is 1/ 7 tone and the other is 8/ 9 , which are not complements of each other like in 19 EDO (1/ 3 and 2/ 3 ). Taking each semitone results in a different choice of perfect fifth. Equal temperament systems can be thought of in terms of the spacing of three intervals found in just intonation, most of whose chords are harmonically perfectly in tune—a good property not quite achieved between almost all pitches in almost all equal temperaments. Most just chords sound amazingly consonant, and most equal-tempered chords sound at least slightly dissonant. In C major those three intervals are:^[38] • the greater tone T= 9/ 8 = the interval from C:D, F:G, and A:B; • the lesser tone t= 10/ 9 = the interval from D:E and G:A; • the diatonic semitone s= 16/ 15 = the interval from E:F and B:C. Analyzing an equal temperament in terms of how it modifies or adapts these three intervals provides a quick way to evaluate how consonant various chords can possibly be in that temperament, based on how distorted these intervals are.^[38]^[f] Regular diatonic tunings Figure 1: The regular diatonic tunings continuum, which include many notable "equal temperament" tunings.^[38] The diatonic tuning in 12 tone equal temperament (12 TET) can be generalized to any regular diatonic tuning dividing the octave as a sequence of steps T t s T t T s (or some circular shift or "rotation" of it). To be called a regular diatonic tuning, each of the two semitones (s) must be smaller than either of the tones (greater tone, T, and lesser tone, t). The comma κ is implicit as the size ratio between the greater and lesser tones: Expressed as frequencies κ = T/ t , or as cents κ = T − t . The notes in a regular diatonic tuning are connected in a "spiral of fifths" that does not close (unlike the circle of fifths in 12 TET). Starting on the subdominant F (in the key of C) there are three perfect fifths in a row—F–C, C–G, and G–D—each a composite of some permutation of the smaller intervals T T t s. The three in-tune fifths are interrupted by the grave fifth D–A = T t t s( grave means "flat by a comma"), followed by another perfect fifth, E–B, and another grave fifth, B–F♯, and then restarting in the sharps with F♯–C♯; the same pattern repeats through the sharp notes, then the double-sharps, and so on, indefinitely. But each octave of all-natural or all-sharp or all-double-sharp notes flattens by two commas with every transition from naturals to sharps, or single sharps to double sharps, etc. The pattern is also reverse-symmetric in the flats: Descending by fourths the pattern reciprocally sharpens notes by two commas with every transition from natural notes to flattened notes, or flats to double flats, etc. If left unmodified, the two grave fifths in each block of all-natural notes, or all-sharps, or all-flat notes, are "wolf" intervals: Each of the grave fifths out of tune by a diatonic comma. Since the comma, κ, expands the lesser tone t = s c , into the greater tone, T = s c κ , a just octave T t s T t T s can be broken up into a sequence s c κ s c s s c κ s c s c κ s , (or a circular shift of it) of 7 diatonic semitones s, 5 chromatic semitones c, and 3 commas κ . Various equal temperaments alter the interval sizes, usually breaking apart the three commas and then redistributing their parts into the seven diatonic semitones s, or into the five chromatic semitones c, or into both s and c, with some fixed proportion for each type of semitone. The sequence of intervals s, c, and κ can be repeatedly appended to itself into a greater spiral of 12 fifths, and made to connect at its far ends by slight adjustments to the size of one or several of the intervals, or left unmodified with occasional less-than-perfect fifths, flat by a comma. Morphing diatonic tunings into EDO Various equal temperaments can be understood and analyzed as having made adjustments to the sizes of and subdividing the three intervals—T, t, and s, or at finer resolution, their constituents s, c, and κ. An equal temperament can be created by making the sizes of the major and minor tones (T, t) the same (say, by setting κ = 0, with the others expanded to still fill out the octave), and both semitones (s and c) the same, then 12 equal semitones, two per tone, result. In 12 TET, the semitone, s, is exactly half the size of the same-size whole tones T = t. Some of the intermediate sizes of tones and semitones can also be generated in equal temperament systems, by modifying the sizes of the comma and semitones. One obtains 7 TET in the limit as the size of c and κ tend to zero, with the octave kept fixed, and 5 TET in the limit as s and κ tend to zero; 12 TET is of course, the case s = c and κ = 0 . For instance: There are two extreme cases that bracket this framework: When s and κ reduce to zero with the octave size kept fixed, the result is t t t t t , a 5 tone equal temperament. As the s gets larger (and absorbs the space formerly used for the comma κ), eventually the steps are all the same size, t t t t t t t , and the result is seven-tone equal temperament. These two extremes are not included as "regular" diatonic tunings. If the diatonic semitone is set double the size of the chromatic semitone, i.e. s = 2 c (in cents) and κ = 0 , the result is 19 TET, with one step for the chromatic semitone c, two steps for the diatonic semitone s, three steps for the tones T = t, and the total number of steps 3 T + 2 t + 2 s = 9 + 6 + 4 = 19 steps. The imbedded 12 tone sub-system closely approximates the historically important 1/ 3 comma meantone system. If the chromatic semitone is two-thirds the size of the diatonic semitone, i.e. c = 2/ 3 s , with κ = 0 , the result is 31 TET, with two steps for the chromatic semitone, three steps for the diatonic semitone, and five steps for the tone, where 3 T + 2 t + 2 s = 15 + 10 + 6 = 31 steps. The imbedded 12 tone sub-system closely approximates the historically important 1/ 4 comma 43 TET If the chromatic semitone is three-fourths the size of the diatonic semitone, i.e. c = 3/ 4 s , with κ = 0 , the result is 43 TET, with three steps for the chromatic semitone, four steps for the diatonic semitone, and seven steps for the tone, where 3 T + 2 t + 2 s = 21 + 14 + 8 = 43. The imbedded 12 tone sub-system closely approximates 1/ 5 comma meantone. If the chromatic semitone is made the same size as three commas, c = 3 κ (in cents, in frequency c = κ³) the diatonic the same as five commas, s = 5 κ , that makes the lesser tone eight commas t = s + c = 8 κ , and the greater tone nine, T = s + c + κ = 9 κ . Hence 3 T + 2 t + 2 s = 27 κ + 16 κ + 10 κ = 53 κ for 53 steps of one comma each. The comma size / step size is κ = 1200/ 53 ¢ exactly, or κ = 22.642 ¢ ≈ 21.506 ¢ , the syntonic comma. It is an exceedingly close approximation to 5-limit just intonation and Pythagorean tuning, and is the basis for Turkish music theory. See also 1. ^ ^a ^b Sethares (2005) compares several equal temperaments in a graph with axes reversed from the axes in the first comparison of equal temperaments, and identical axes of the second.^[1] 2. ^ "Chu-Tsaiyu [was] the first formulator of the mathematics of 'equal temperament' anywhere in the world." — Robinson (1980), p. vii^[7] 3. ^ 'Hepta-equal temperament' in our folk music has always been a controversial issue.^[30] 4. ^ From the flute for two thousand years of the production process, and the Japanese shakuhachi remaining in the production of Sui and Tang Dynasties and the actual temperament, identification of people using the so-called 'Seven Laws' at least two thousand years of history; and decided that this law system associated with the flute law.^[31] 5. ^ OEIS sequences that contain divisions of the octave that provide improving approximations of just intervals: 6. ^ For 12 pitch systems, either for a whole 12 note scale, for or 12 note subsequences embedded inside some larger scale,^[38] use this analysis as a way to program software to microtune an electronic keyboard dynamically, or 'on the fly', while a musician is playing. The object is to fine tune the notes momentarily in use, and any likely subsequent notes involving consonant chords, to always produce pitches that are harmonically in-tune, inspired by how orchestras and choruses constantly re-tune their overall pitch on long-duration chords for greater consonance than possible with strict 12 TET.^[38] • Boiles, J. (1969). "Terpehua though-song". Ethnomusicology. 13: 42–47. • Cho, Gene Jinsiong (2003). The Discovery of Musical Equal Temperament in China and Europe in the Sixteenth Century. Lewiston, NY: Edwin Mellen Press. • Duffin, Ross W. (2007). How Equal Temperament Ruined Harmony (and why you should care). New York, NY: W.W.Norton & Company. ISBN 978-0-39306227-4. • Jorgensen, Owen (1991). Tuning. Michigan State University Press. ISBN 0-87013-290-3. • Sethares, William A. (2005). Tuning, Timbre, Spectrum, Scale (2nd ed.). London, UK: Springer-Verlag. ISBN 1-85233-797-4. • Surjodiningrat, W.; Sudarjana, P.J.; Susanto, A. (1972). Tone measurements of outstanding Javanese gamelans in Jogjakarta and Surakarta. Jogjakarta, IN: Gadjah Mada University Press. As cited by "The gamelan pelog scale of Central Java as an example of a non-harmonic musical scale". telia.com. Neuroscience of Music. Archived from the original on 27 January 2005. Retrieved 19 May 2006. • Stewart, P.J. (2006) [January 1999]. From galaxy to galaxy: Music of the spheres (Report). 8096295 – via academia.edu."Alt. link 1". 269108386 – via researchgate.net."Alt. link 2" – via Google • Khramov, Mykhaylo (26–29 July 2008). Approximation of 5-limit just intonation. Computer MIDI Modeling in Negative Systems of Equal Divisions of the Octave. The International Conference SIGMAP-2008. Porto. pp. 181–184. ISBN 978-989-8111-60-9. Further reading • Helmholtz, H. (2005) [1877 (4th German ed.), 1885 (2nd English ed.)]. On the Sensations of Tone as a Physiological Basis for the Theory of Music. Translated by Ellis, A.J. (reprint ed.). Whitefish, MT: Kellinger Publishing. ISBN 978-1-41917893-1. OCLC 71425252 – via Internet Archive (archive.org). — A foundational work on acoustics and the perception of sound. Especially the material in Appendix XX: Additions by the translator, pages 430–556, (pdf pages 451–577) (see also wiki article On Sensations of Tone) External links • An Introduction to Historical Tunings by Kyle Gann • Xenharmonic wiki on EDOs vs. Equal Temperaments • Huygens-Fokker Foundation Centre for Microtonal Music • A.Orlandini: Music Acoustics • "Temperament" from A supplement to Mr. Chambers's cyclopædia (1753) • Barbieri, Patrizio. Enharmonic instruments and music, 1470–1900. (2008) Latina, Il Levante Libreria Editrice • Fractal Microtonal Music, Jim Kukula. • All existing 18th century quotes on J.S. Bach and temperament • Dominic Eckersley: "Rosetta Revisited: Bach's Very Ordinary Temperament" • Well Temperaments, based on the Werckmeister Definition • FAVORED CARDINALITIES OF SCALES by PETER BUCH	{"url":"https://www.knowpia.com/knowpedia/Equal_temperament","timestamp":"2024-11-08T09:07:26Z","content_type":"text/html","content_length":"284170","record_id":"<urn:uuid:3a5f1658-51ab-4974-8d4e-345fcbb98ddf>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00454.warc.gz"}
Denaby & Cadeby – Denaby 250 for 6 Swinton 166 for 6 – Another Denaby century Denaby & Cadeby – Denaby 250 for 6 Swinton 166 for 6 – Another Denaby century Mexborough and Swinton Times, July 9, 1926 Another Denaby Century Shocks from Burkinshaw Denaby 250 for 6 Swinton 166 for 6 I Greenwood 121, A S Vollans 76 L Burkinshaw 59 Once again batsmen proved too good for bowlers, in a match at Denaby yesterday. The game opened in startling fashion for the fourth and fifth balls of Burkinshaw’s first over dismissed Denaby’s two “stars,” Tibbles (who scored 290 in his two preceding innings), and Carlin (whose Yorkshire Council average was 166 up to yesterday). Wainwright also fell a cheap victim to Burkinshaw. Denaby lost three wickets for 24 runs. Then however, Greenwood and Vollans set to work to make a complete change in the situation, and succeeded so well that in about two hours they scored 189 for the fourth wicket. Vollans only bad stroke in a delightful innings was the skier to which he was out. Greenwood gave no real chance until he was into three figures, and got his runs in 124 minutes. This was his best innings of the season. Burkinshaw took three wickets for 34 runs. Denaby declared at 5 o’clock, and Swinton were left with about 140 minutes batting. Burkinshaw gave them a characteristic start, scoring 59/79 in 62 minutes before the Robinson – Narraway combination got him. Narraway also helped to get the next victim, Prize, who said 87 minutes for his 28 runs, and was missed by Greenwood when 17. Greenwood had an unhappy time in the field, for later, when Denaby had an outside chance of forcing a win, he twice missed Rigby. Swinton were 84 behind with four wickets standing when time expired.	{"url":"https://conisbroughanddenabyhistory.org.uk/article/denaby-cadeby-denaby-250-for-6-swinton-166-for-6-another-denaby-century/","timestamp":"2024-11-02T20:13:40Z","content_type":"text/html","content_length":"39947","record_id":"<urn:uuid:7aa05aaf-e62c-4082-a934-7c8192d42fb8>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00356.warc.gz"}
Online Absolute Pressure Calculator \| How to Calculate Absolute Pressure? - physicsCalculatorPro.com Absolute Pressure Calculator By Using this absolute pressure calculator it will be more easy and convenient to calculate the absolute pressure. All you have to do is give the parameters of gauge pressure and atmospheric pressure and then calculate, to get the accurate result. What is Absolute Pressure? Absolute pressure is the pressure detected above the absolute zero pressure. It is described as the sum of atmospheric pressure and measuring pressure. Absolute pressure is measured using a barometer. Gauge pressure is very relative to atmospheric pressure, if the pressure of Gauge is higher than absolute pressure then the pressure is positive. Otherwise, it is negative. Absolute Pressure = Gauge Pressure + Atmospheric Pressure How to find the Absolute Pressure? Below given are the simple and easy step-by-step process to calculate the absolute pressure. You can get the answer without any issues by following these guidelines. • Observe gauge pressure and atmospheric pressure values from the given problem. • Enter the value of gauge pressure in the given gauge pressure field. • And the same for atmospheric pressure too. • Then the sum of the gauge and atmospheric Pressure is absolute pressure. Absolute Pressure Formula Now, let us look into the formula for absolute pressure. Absolute Pressure(P[a]) = Gauge Pressure ( P[g] ) + Atmospheric Pressure ( P[atm] ) P[a] = P[g] + P [atm] Once, we look into the picture for a clear understanding Absolute Pressure Examples Question 1: A pressure gauge measures the reading of 32psi and if the atmospheric pressure is 14psi. Calculate the absolute pressure that correlates to this gauge pressure? Given Parameters, P[a] =?, P[g] = 32psi, P[atm] =14psi P[a] = P[g] + P [atm] Substitute the values, P[a] = 32psi +14 psi = 46psi For more concepts check out physicscalculatorpro.com to get quick answers by using this free tool. FAQs on Absolute Pressure 1. What is absolute pressure in simple words? In simple words, absolute pressure is nothing but a sum of atmospheric pressure and gauge pressure. 2. What is the best tool to calculate absolute pressure? The absolute pressure calculator tool is the best tool to get accurate answers for absolute pressure 3. How absolute pressure is measured? Absolute pressure is measured on the barometer. 4. What is the formula for absolute pressure? P[a] = P[g] + P [atm] Where, P[a] = absolute pressure P[g] = gauge pressure P[atm] = atmospheric pressure	{"url":"https://physicscalculatorpro.com/absolute-pressure-calculator/","timestamp":"2024-11-05T03:08:39Z","content_type":"text/html","content_length":"28156","record_id":"<urn:uuid:e5106758-f968-4445-927b-d20a91964927>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00743.warc.gz"}
LM 22.9 Summary Collection 22.9 Summary by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license. field — a property of a point in space describing the forces that would be exerted on a particle if it was there sink — a point at which field vectors converge source — a point from which field vectors diverge; often used more inclusively to refer to points of either convergence or divergence electric field — the force per unit charge exerted on a test charge at a given point in space gravitational field — the force per unit mass exerted on a test mass at a given point in space electric dipole — an object that has an imbalance between positive charge on one side and negative charge on the other; an object that will experience a torque in an electric field `g`— the gravitational field `E`— the electric field `D` — an electric dipole moment Other Notation `d`, `p`, `m` — other notations for the electric dipole moment Experiments show that time is not absolute: it flows at different rates depending on an observer's state of motion. This is an example of the strange effects predicted by Einstein's theory of relativity. All of these effects, however, are very small when the relative velocities are small compared to `c`. This makes sense, because Newton's laws have already been thoroughly tested by experiments at such speeds, so a new theory like relativity must agree with the old one in their realm of common applicability. This requirement of backwards-compatibility is known as the correspondence principle. Since time is not absolute, simultaneity is not a well-defined concept, and therefore the universe cannot operate as Newton imagined, through instantaneous action at a distance. There is a delay in time before a change in the configuration of mass and charge in one corner of the universe will make itself felt as a change in the forces experienced far away. We imagine the outward spread of such a change as a ripple in an invisible universe-filling field of force. We define the gravitational field at a given point as the force per unit mass exerted on objects inserted at that point, and likewise the electric field is defined as the force per unit charge. These fields are vectors, and the fields generated by multiple sources add according to the rules of vector addition. When the electric field is constant, the voltage difference between two points lying on a line parallel to the field is related to the field by the equation `DeltaV=-Ed`, where `d` is the distance between the two points. Fields of force contain energy. The density of energy is proportional to the square of the magnitude of the field. In the case of static fields, we can calculate potential energy either using the previous definition in terms of mechanical work or by calculating the energy stored in the fields. If the fields are not static, the old method gives incorrect results and the new one must be used. Homework Problems `sqrt` A computerized answer check is available online. `int` A problem that requires calculus. `*` A difficult problem. . In our by-now-familiar neuron, the voltage difference between the inner and outer surfaces of the cell membrane is about `V_"out"-V_"in"=-70 mV` in the resting state, and the thickness of the membrane is about 6.0 nm (i.e., only about a hundred atoms thick). What is the electric field inside the membrane? `sqrt` 2. The gap between the electrodes in an automobile engine's spark plug is 0.060 cm. To produce an electric spark in a gasoline-air mixture, an electric field of `3.0×10^6` V/m must be achieved. On starting a car, what minimum voltage must be supplied by the ignition circuit? Assume the field is uniform.`sqrt` (b) The small size of the gap between the electrodes is inconvenient because it can get blocked easily, and special tools are needed to measure it. Why don't they design spark plugs with a wider gap? 3. (a) At time `t=0`, a positively charged particle is placed, at rest, in a vacuum, in which there is a uniform electric field of magnitude `E`. Write an equation giving the particle's speed, `v`, in terms of `t`, `E`, and its mass and charge `m` and `q`.`sqrt` (b) If this is done with two different objects and they are observed to have the same motion, what can you conclude about their masses and charges? (For instance, when radioactivity was discovered, it was found that one form of it had the same motion as an electron in this type of experiment.) 4. Three charges are arranged on a square as shown. All three charges are positive. What value of `q_2"/"q_1` will produce zero electric field at the center of the square? 5. Show that the magnitude of the electric field produced by a simple two-charge dipole, at a distant point along the dipole's axis, is to a good approximation proportional to `D"/"r^3`, where `r` is the distance from the dipole. [Hint: Use the approximation`(1+epsilon)^papprox1+pepsilon`, which is valid for small `epsilon`.] `` 6. Consider the electric field created by a uniform ring of total charge `q` and radius `b`. (a) Show that the field at a point on the ring's axis at a distance `a` from the plane of the ring is `kqa (a^2+b^2)^(-3"/"2)`. (b) Show that this expression has the right behavior for `a=0` and for `a` much greater than `b`. `` 7. Example 4 on p. 627 showed that the electric field of a long, uniform line of charge falls off with distance as `1"/"r`. By a similar technique, show that the electric field of a uniformly charged plane has no dependence on the distance from the plane. `` 8. Given that the field of a dipole is proportional to `D"/"r^3` (problem 5), show that its voltage varies as `D"/"r^2`. (Ignore positive and negative signs and numerical constants of proportionality.) `int` 9. A carbon dioxide molecule is structured like O-C-O, with all three atoms along a line. The oxygen atoms grab a little bit of extra negative charge, leaving the carbon positive. The molecule's symmetry, however, means that it has no overall dipole moment, unlike a V-shaped water molecule, for instance. Whereas the voltage of a dipole of magnitude `D` is proportional to `D"/"r^2` (problem 8 ), it turns out that the voltage of a carbon dioxide molecule along its axis equals `k/r^3`, where `r` is the distance from the molecule and `k` is a constant. What would be the electric field of a carbon dioxide molecule at a distance `r`? `sqrt` `int` 10. A proton is in a region in which the electric field is given by `E=a+bx^3`. If the proton starts at rest at `x_1=0`, find its speed, `v`, when it reaches position `x_2`. Give your answer in terms of `a`, `b`, `x_2`, and `e` and `m`, the charge and mass of the proton. `sqrt` `int` 11. Consider the electric field created by a uniformly charged cylinder that extends to infinity in one direction. (a) Starting from the result of problem 8, show that the field at the center of the cylinder's mouth is `2piksigma`, where `sigma` is the density of charge on the cylinder, in units of coulombs per square meter. [Hint: You can use a method similar to the one in problem 9.] (b) This expression is independent of the radius of the cylinder. Explain why this should be so. For example, what would happen if you doubled the cylinder's radius? `int` 12. In an electrical storm, the cloud and the ground act like a parallel-plate capacitor, which typically charges up due to frictional electricity in collisions of ice particles in the cold upper atmosphere. Lightning occurs when the magnitude of the electric field builds up to a critical value, `E_c`, at which air is ionized. (a) Treat the cloud as a flat square with sides of length `L`. If it is at a height `h` above the ground, find the amount of energy released in the lightning strike.`sqrt` (b) Based on your answer from part a, which is more dangerous, a lightning strike from a high-altitude cloud or a low-altitude one? (c) Make an order-of-magnitude estimate of the energy released by a typical lightning bolt, assuming reasonable values for its size and altitude. `E_c` is about `10^6` V/m. See problem 16 for a note on how recent research affects this estimate. . The neuron in the figure has been drawn fairly short, but some neurons in your spinal cord have tails (axons) up to a meter long. The inner and outer surfaces of the membrane act as the “plates” of a capacitor. (The fact that it has been rolled up into a cylinder has very little effect.) In order to function, the neuron must create a voltage difference `V` between the inner and outer surfaces of the membrane. Let the membrane's thickness, radius, and length be `t`, `r`, and `L`. (a) Calculate the energy that must be stored in the electric field for the neuron to do its job. (In real life, the membrane is made out of a substance called a dielectric, whose electrical properties increase the amount of energy that must be stored. For the sake of this analysis, ignore this fact.) [Hint: The volume of the membrane is essentially the same as if it was unrolled and flattened out.] `sqrt` (b) An organism's evolutionary fitness should be better if it needs less energy to operate its nervous system. Based on your answer to part a, what would you expect evolution to do to the dimensions `t` and `r`? What other constraints would keep these evolutionary trends from going too far? 14. To do this problem, you need to understand how to do volume integrals in cylindrical and spherical coordinates. (a) Show that if you try to integrate the energy stored in the field of a long, straight wire, the resulting energy per unit length diverges both at `r->0` and `r->infty`. Taken at face value, this would imply that a certain real-life process, the initiation of a current in a wire, would be impossible, because it would require changing from a state of zero magnetic energy to a state of infinite magnetic energy. (b) Explain why the infinities at `r->0` and `r->infty` don't really happen in a realistic situation. (c) Show that the electric energy of a point charge diverges at `r->0`, but not at `r->infty`. A remark regarding part (c): Nature does seem to supply us with particles that are charged and pointlike, e.g., the electron, but one could argue that the infinite energy is not really a problem, because an electron traveling around and doing things neither gains nor loses infinite energy; only an infinite change in potential energy would be physically troublesome. However, there are real-life processes that create and destroy pointlike charged particles, e.g., the annihilation of an electron and antielectron with the emission of two gamma rays. Physicists have, in fact, been struggling with infinities like this since about 1950, and the issue is far from resolved. Some theorists propose that apparently pointlike particles are actually not pointlike: close up, an electron might be like a little circular loop of string. `int` `**` . The figure shows cross-sectional views of two cubical capacitors, and a cross-sectional view of the same two capacitors put together so that their interiors coincide. A capacitor with the plates close together has a nearly uniform electric field between the plates, and almost zero field outside; these capacitors don't have their plates very close together compared to the dimensions of the plates, but for the purposes of this problem, assume that they still have approximately the kind of idealized field pattern shown in the figure. Each capacitor has an interior volume of `1.00` `m^3`, and is charged up to the point where its internal field is `1.00` V/m. (a) Calculate the energy stored in the electric field of each capacitor when they are separate. (b) Calculate the magnitude of the interior field when the two capacitors are put together in the manner shown. Ignore effects arising from the redistribution of each capacitor's charge under the influence of the other capacitor. (c) Calculate the energy of the put-together configuration. Does assembling them like this release energy, consume energy, or neither? `sqrt` 16. In problem 12 on p. 645, you estimated the energy released in a bolt of lightning, based on the energy stored in the electric field immediately before the lightning occurs. The assumption was that the field would build up to a certain value, which is what is necessary to ionize air. However, real-life measurements always seemed to show electric fields strengths roughly `10` times smaller than those required in that model. For a long time, it wasn't clear whether the field measurements were wrong, or the model was wrong. Research carried out in 2003 seems to show that the model was wrong. It is now believed that the final triggering of the bolt of lightning comes from cosmic rays that enter the atmosphere and ionize some of the air. If the field is `10` times smaller than the value assumed in problem 12, what effect does this have on the final result of problem 12? `sqrt` Exercise 22: Field vectors • 3 solenoids • DC power supply • compass • ruler • cut-off plastic cup At this point you've studied the gravitational field, `g`, and the electric field, `E`, but not the magnetic field, `B`. However, they all have some of the same mathematical behavior: they act like vectors. Furthermore, magnetic fields are the easiest to manipulate in the lab. Manipulating gravitational fields directly would require futuristic technology capable of moving planet-sized masses around! Playing with electric fields is not as ridiculously difficult, but static electric charges tend to leak off through your body to ground, and static electricity effects are hard to measure numerically. Magnetic fields, on the other hand, are easy to make and control. Any moving charge, i.e. any current, makes a magnetic field. A practical device for making a strong magnetic field is simply a coil of wire, formally known as a solenoid. The field pattern surrounding the solenoid gets stronger or weaker in proportion to the amount of current passing through the wire. 1. With a single solenoid connected to the power supply and laid with its axis horizontal, use a magnetic compass to explore the field pattern inside and outside it. The compass shows you the field vector's direction, but not its magnitude, at any point you choose. Note that the field the compass experiences is a combination (vector sum) of the solenoid's field and the earth's field. 2. What happens when you bring the compass extremely far away from the solenoid? What does this tell you about the way the solenoid's field varies with distance? Thus although the compass doesn't tell you the field vector's magnitude numerically, you can get at least some general feel for how it depends on distance. 3. The figure below is a cross-section of the solenoid in the plane containing its axis. Make a sea-of-arrows sketch of the magnetic field in this plane. The length of each arrow should at least approximately reflect the strength of the magnetic field at that point. Does the field seem to have sources or sinks? 4. What do you think would happen to your sketch if you reversed the wires? Try it. 5. Now hook up the two solenoids in parallel. You are going to measure what happens when their two fields combine at a certain point in space. As you've seen already, the solenoids' nearby fields are much stronger than the earth's field; so although we now theoretically have three fields involved (the earth's plus the two solenoids'), it will be safe to ignore the earth's field. The basic idea here is to place the solenoids with their axes at some angle to each other, and put the compass at the intersection of their axes, so that it is the same distance from each solenoid. Since the geometry doesn't favor either solenoid, the only factor that would make one solenoid influence the compass more than the other is current. You can use the cut-off plastic cup as a little platform to bring the compass up to the same level as the solenoids' axes. a) What do you think will happen with the solenoids' axes at 90 degrees to each other, and equal currents? Try it. Now represent the vector addition of the two magnetic fields with a diagram. Check your diagram with your instructor to make sure you're on the right track. b) Now try to make a similar diagram of what would happen if you switched the wires on one of the solenoids. After predicting what the compass will do, try it and see if you were right. c) Now suppose you were to go back to the arrangement you had in part a, but you changed one of the currents to half its former value. Make a vector addition diagram, and use trig to predict the Try it. To cut the current to one of the solenoids in half, an easy and accurate method is simply to put the third solenoid in series with it, and put that third solenoid so far away that its magnetic field doesn't have any significant effect on the compass. 22.9 Summary by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.	{"url":"https://www.vcalc.com/collection/?uuid=1ea5ea8d-f145-11e9-8682-bc764e2038f2","timestamp":"2024-11-13T08:16:00Z","content_type":"text/html","content_length":"86547","record_id":"<urn:uuid:f0343925-fa20-4de6-a78a-7b978f0b9419>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00040.warc.gz"}
Section: Application Domains Application: uncertainties management Our theoretical works are motivated by and find natural applications to real-world problems in a general frame generally referred to as uncertainty management, that we describe now. Since a few decades, modeling has gained an increasing part in complex systems design in various fields of industry such as automobile, aeronautics, energy, etc. Industrial design involves several levels of modeling: from behavioural models in preliminary design to finite-elements models aiming at representing sharply physical phenomena. Nowadays, the fundamental challenge of numerical simulation is in designing physical systems while saving the experimentation steps. As an example, at the early stage of conception in aeronautics, numerical simulation aims at exploring the design parameters space and setting the global variables such that target performances are satisfied. This iterative procedure needs fast multiphysical models. These simplified models are usually calibrated using high-fidelity models or experiments. At each of these levels, modeling requires control of uncertainties due to simplifications of models, numerical errors, data imprecisions, variability of surrounding conditions, etc. One dilemma in the design by numerical simulation is that many crucial choices are made very early, and thus when uncertainties are maximum, and that these choices have a fundamental impact on the final performances. Classically, coping with this variability is achieved through model registration by experimenting and adding fixed margins to the model response. In view of technical and economical performance, it appears judicious to replace these fixed margins by a rigorous analysis and control of risk. This may be achieved through a probabilistic approach to uncertainties, that provides decision criteria adapted to the management of unpredictability inherent to design issues. From the particular case of aircraft design emerge several general aspects of management of uncertainties in simulation. Probabilistic decision criteria, that translate decision making into mathematical/probabilistic terms, require the following three steps to be considered [50] : 1. build a probabilistic description of the fluctuations of the model's parameters (Quantification of uncertainty sources), 2. deduce the implication of these distribution laws on the model's response (Propagation of uncertainties), The previous analysis now constitutes the framework of a general study of uncertainties. It is used in industrial contexts where uncertainties can be represented by random variables (unknown temperature of an external surface, physical quantities of a given material, ... at a given fixed time). However, in order for the numerical models to describe with high fidelity a phenomenon, the relevant uncertainties must generally depend on time or space variables. Consequently, one has to tackle the following issues: • How to capture the distribution law of time (or space) dependent parameters, without directly accessible data? The distribution of probability of the continuous time (or space) uncertainty sources must describe the links between variations at neighbor times (or points). The local and global regularity are important parameters of these laws, since it describes how the fluctuations at some time (or point) induce fluctuations at close times (or points). The continuous equations representing the studied phenomena should help to propose models for the law of the random fields. Let us notice that interactions between various levels of modeling might also be used to derive distributions of probability at the lowest one. • The navigation between the various natures of models needs a kind of metric which could mathematically describe the notion of granularity or fineness of the models. Of course, the local regularity will not be totally absent of this mathematical definition. • All the various levels of conception, preliminary design or high-fidelity modelling, require registrations by experimentation to reduce model errors. This calibration issue has been present in this frame since a long time, especially in a deterministic optimization context. The random modeling of uncertainty requires the definition of a systematic approach. The difficulty in this specific context is: statistical estimation with few data and estimation of a function with continuous variables using only discrete setting of values. Moreover, a multi-physical context must be added to these questions. The complex system design is most often located at the interface between several disciplines. In that case, modeling relies on a coupling between several models for the various phenomena and design becomes a multidisciplinary optimization problem. In this uncertainty context, the real challenge turns robust optimization to manage technical and economical risks (risk for non-satisfaction of technical specifications, cost control). We participate in the uncertainties community through several collaborative research projects (ANR and Pôle SYSTEM@TIC), and also through our involvement in the MASCOT-NUM research group (GDR of CNRS). In addition, we are considering probabilistic models as phenomenological models to cope with uncertainties in the DIGITEO ANIFRAC project. As explained above, we focus on essentially irregular phenomena, for which irregularity is a relevant quantity to capture the variability (e.g. certain biomedical signals, terrain modeling, financial data, etc.). These will be modeled through stochastic processes with prescribed regularity.	{"url":"https://radar.inria.fr/report/2011/regularity/uid23.html","timestamp":"2024-11-05T04:16:52Z","content_type":"text/html","content_length":"39264","record_id":"<urn:uuid:7c2bd132-9b86-4630-9fbb-afe0a3959e90>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00145.warc.gz"}
Printable Calendars AT A GLANCE Relative Dating Worksheet Answer Key Relative Dating Worksheet Answer Key - Base your answer(s) to the. Web answer key for relative dating worksheet. What principle of geology states. Animal dun or scat b) this. Web for the questions b tow, circle the correct answer. Web free printable relative dating worksheet answer key. The correct answer for each question is indicated by a. Web in this lab you will complete tables, construct a graph, and calculate ages to learn the principles that underlie absolute dating of geologic materials; Relative dating is using comparison to date rocks or fossils. Posted on july 8, 2022 by admin. Web for the questions b tow, circle the correct answer. Older, newer, before, after, etc). Web relative dating practice 2017 1. Relative dating worksheet principles of geology • law of superposition: Relative dating is using comparison to date rocks or fossils. Web a relative dating activity”. 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Relative Dating Worksheet Answer Key - It is the top layer. 6 6 geologic time relative dating stratigraphe caless rece besc sand the law of. Web study with quizlet and memorize flashcards containing terms like strata are mostly found:, according to the principle of superposition, younger strata form________older strata.,. After layer 3 but before 5. Is at the bottom of the sequence, and the youngest. Law of superposition and index fossils are both examples of relative dating; Web in this lab you will complete tables, construct a graph, and calculate ages to learn the principles that underlie absolute dating of geologic materials; Each question is for the station of the same a) this fossil is. Without knowing the exact ages of the rocks or geologic events, or how long ago they were. Relative dating worksheet principles of geology • law of superposition: Relative dating is used to arrange geological events, and the rocks they leave behind, in a sequence. Relative dating is using comparison to date rocks or fossils. Web relative time practical 2. Web for the questions b tow, circle the correct answer. Put an “ra” next to the examples. Web answer key for relative dating worksheet. Web free printable relative dating worksheet answer key. Animal dun or scat b) this. Web in this lab you will complete tables, construct a graph, and calculate ages to learn the principles that underlie absolute dating of geologic materials; The Correct Answer For Each Question Is Indicated By A. Before geologists can correlate the ages of rocks from different areas, they must first figure out the ages of rocks at a single location. Determining the cardinal of years that accept delayed back an accident. It is the top layer. A geologic cross section is shown below. Animal Dun Or Scat B) This. Web relative time practical 2. Web in this lab you will complete tables, construct a graph, and calculate ages to learn the principles that underlie absolute dating of geologic materials; Relative dating is using comparison to date rocks or fossils. The method of reading the order. Law Of Superposition And Index Fossils Are Both Examples Of Relative Dating; Web answer key for relative dating worksheet. Relative dating worksheet principles of geology • law of superposition: The oldest layer of rock is. In any undisturbed sequence of strata, the oldest What Principle Of Geology States. Older, newer, before, after, etc). Web a relative dating activity”. Posted on july 8, 2022 by admin. Web for the questions b tow, circle the correct answer. 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Conduct Cointegration Test Using Econometric Modeler This example models the annual Canadian inflation and interest rate series by using the Econometric Modeler app. The example performs the following actions in the app: 1. Test each raw series for stationarity. 2. Test for cointegration using the Engle-Granger cointegration test. 3. Test for cointegration among all possible cointegration ranks using the Johansen cointegration test. The data set, which is stored in Data_Canada, contains annual Canadian inflation and interest rates from 1954 through 1994. Load and Import Data into Econometric Modeler At the command line, load the Data_Canada.mat data set. At the command line, open the Econometric Modeler app. Alternatively, open the app from the apps gallery (see Econometric Modeler). Import DataTimeTable into the app: 1. On the Econometric Modeler tab, in the Import section, click the Import button . 2. In the Import Data dialog box, in the Import? column, select the check box for the DataTimeTable variable. 3. Click Import. The Canadian interest and inflation rate variables appear in the Time Series pane, and a time series plot of all the series appears in the Time Series Plot(INF_C) figure window. Plot only the three interest rate series INT_L, INT_M, and INT_S together. In the Time Series pane, click INT_L and Ctrl click INT_M and INT_S. Then, on the Plots tab, in the Plots section, click Time Series. The time series plot appears in the Time Series(INT_L) document. The interest rate series each appear nonstationary, and they appear to move together with mean-reverting spread. In other words, they exhibit cointegration. To establish these properties, this example conducts statistical tests. Conduct Stationarity Test Assess whether each interest rate series is stationary by conducting Phillips-Perron unit-root tests. For each series, assume a stationary AR(1) process with drift for the alternative hypothesis. You can confirm this property by viewing the autocorrelation and partial autocorrelation function plots. Close all plots in the right pane, and perform the following procedure for each series INT_L, INT_M, and INT_S. 1. In the Time Series pane, click a series. 2. On the Econometric Modeler tab, in the Tests section, click New Test > Phillips-Perron Test. 3. On the PP tab, in the Parameters section, in the Number of Lags box, type 1, and in the Model list select Autoregressive with Drift. 4. In the Tests section, click Run Test. Test results for the selected series appear in the PP(series) tab. Position the test results to view them simultaneously. All tests fail to reject the null hypothesis that the series contains a unit root process. Conduct Cointegration Tests Econometric Modeler supports the Engle-Granger and Johansen cointegration tests. Before conducting a cointegration, determine whether there is visual evidence of cointegration by regressing each interest rate on the other interest rates. 1. In the cointegrating relation, assign INT_L as the dependent variable and INT_M and INT_S as the independent variables. In the Time Series pane, click INT_L. 2. On the Econometric Modeler tab, in the Models section, click the arrow > MLR. 3. In the MLR Model Parameters dialog box, select the Include? check boxes of the time series INT_M and INT_S. 4. Click Estimate. The model variable MLR_INT_L appears in the Models pane, its value appears in the Preview pane, and its estimation summary appears in the Model Summary(MLR_INT_L) document. 5. Iterate steps 1 through 4 twice. For the first iteration, assign INT_M as the dependent variable and INT_L and INT_S as the independent variables. For the second iteration, assign INT_S as the dependent variable and INT_M and INT_L as the independent variables. 6. On each Model Summary(MLR_seriesName) tab, determine whether the residual plot appears stationary. The residual series of the regression on INT_L appears trending, and the other residual series appear stationary. For the Engle-Granger test, choose INT_S as the dependent variable. Conduct Engle-Granger Test The Engle-Granger tests for one cointegrating relation by performing two univariate regressions: the cointegrating regression and the subsequent residual regression (for more details, see Identifying Single Cointegrating Relations). Therefore, to perform the cointegrating regression, the test requires: • The response series that takes the role of the dependent variable in the first regression. • Deterministic terms to include in the cointegrating regression. Then, the test assesses whether the residuals resulting from the cointegrating regression are a unit root process. Available tests are the Augmented Dickey-Fuller (adftest) or Phillips-Perron (pptest) test. Both perform a residual regression to form the test statistic. Therefore, the test also requires: • The unit root test to conduct, either the adftest or pptest function. The residual regression model for both tests is an AR model without deterministic terms. For more flexible models, such as AR models with drift, call the functions at the command line. • Number of lagged residuals to include in the AR model. Conduct the Engle-Granger test. 1. In the Time Series pane, click INT_L and Ctrl+click INT_M and INT_S. 2. In the Tests section, click New Test > Engle-Granger Test. 3. On the EGCI tab, in the Parameters section: 1. In the Dependent Variable list, select INT_S. 2. In the Residual Regression Form list, select PP for the Phillips-Perron unit root test. 3. In the Number of Lags box, type 1. 4. In the Tests section, click Run Test. Test results appear in the EGCI document. 5. In the Tests section, click Run Test again. Test results and a plot of the cointegration relation for the largest rank appear in the EGCI tab. The test rejects the null hypothesis that the series does not exhibit cointegration. Although the test is suited to determine whether series are cointegrated, the results are limited to that. For example, the test does not give insight into the cointegrating rank, which is required to form a VEC model of the series. For more details, see Identifying Single Cointegrating Relations. Johansen Test The Johansen cointegration test runs a separate test for each possible cointegration rank 0 through m – 1, where m is the number of series. This characteristic makes the Johansen test better suited than the Engle-Granger test to determine the cointegrating rank for a VEC model of the series. Also, the Johansen test framework is multivariate, and, therefore, results are not relative to an arbitrary choice for a univariate regression response variable. For more details, see Identifying Multiple Cointegrating Relations. The Johansen test requires: • Deterministic terms to include in the cointegrating relation and the model in levels. • Number of short-run lags in the VEC model of the series. Because the raw series do not contain a linear trend, assume that the only deterministic term in the model is an intercept in the cointegrating relation (H1* Johansen form), and include 1 lagged difference term in the model. 1. With INT_L, INT_M, and INT_S selected in the Time Series pane, click Tests section, click New Test > Johansen Test. 2. On the JCI tab, in the Parameters section, in the Number of Lags box, type 1, and in the Model list select H1*. 3. In the Tests section, click Run Test. Test results and a plot of the cointegration relation for the largest rank appear in the JCI tab. Econometric Modeler conducts a separate test for each cointegration rank 0 through 2 (the number of series – 1). The test rejects the null hypothesis of no cointegration (Cointegration rank = 0), but fails to reject the null hypotheses of Cointegration rank ≤ 1 and Cointegration rank ≤ 2. The results suggest that the cointegration rank is 1. See Also Related Topics	{"url":"https://uk.mathworks.com/help/econ/conduct-cointegration-test-using-econometric-modeler-app.html","timestamp":"2024-11-12T05:56:12Z","content_type":"text/html","content_length":"88942","record_id":"<urn:uuid:d11e9899-a431-484f-aa70-5db53c104bd7>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00586.warc.gz"}
Amps or Amperes « Back to Glossary Index An amp or ampere is a measure of the flow of charge in an electrical circuit. This flow is known as an electrical current. The vibrating electrons transmit energy through the circuit. There is some flow of electrons but this is relatively slight compared to the flow of energy or charge. The higher the current within the circuit, the greater the flow of energy. The flow of the current, measured with an ammeter, is related to both the voltage and to the resistance within the circuit. Ohm’s Law represents this relationship and states that the flow of charge, and therefore the amperes within a circuit, can be calculated as follows: Current (I) = Voltage (V) Resistance (R) « Back to Glossary Index	{"url":"https://primaryscienceonline.org.uk/glossary-of-terms/amps-or-amperes/","timestamp":"2024-11-14T23:19:44Z","content_type":"text/html","content_length":"219661","record_id":"<urn:uuid:31b4fb38-061e-4db1-a2e3-76b6c17c6ac6>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00373.warc.gz"}
Browsing Codes ELISA is a Python library designed for efficient spectral modeling and robust statistical inference. With user-friendly interface, ELISA streamlines the spectral analysis workflow. The modeling framework of ELISA is flexible, allowing users to construct complex models by combining models of ELISA and XSPEC, as well as custom models. Parameters across different model components can also be linked. The models can be fitted to the spectral datasets using either Bayesian or maximum likelihood approaches. For Bayesian fitting, ELISA incorporates advanced Markov Chain Monte Carlo (MCMC) algorithms, including the No-U-Turn Sampler (NUTS), nested sampling, and affine-invariant ensemble sampling, to tackle the posterior sampling problem. For maximum likelihood estimation (MLE), ELISA includes two robust algorithms: the Levenberg-Marquardt algorithm and the Migrad algorithm from Minuit. The computation backend is based on Google's JAX, a high-performance numerical computing library, which can reduce the runtime for fitting procedures like MCMC, thereby enhancing the efficiency of analysis. After fitting, goodness-of-fit assessment can be done with a single function call, which automatically conducts posterior predictive checks and leave-one-out cross-validation for Bayesian models, or parametric bootstrap for MLE. These methods offer greater accuracy and reliability than traditional fit-statistic/dof measures, and thus better model discovery capability. For comparing multiple candidate models, ELISA provides robust Bayesian tools such as the Widely Applicable Information Criterion (WAIC) and the Leave-One-Out Information Criterion (LOOIC), which are more reliable than AIC or BIC. Thanks to the object-oriented design, collecting the analysis results should be simple. ELISA also provide visualization tools to generate ready-for-publication figures. ELISA is an open-source project and community contributions are welcome and greatly appreciated.	{"url":"http://www.ascl.net/code/all/page/36/limit/100/order/date/listmode/full/dir/asc","timestamp":"2024-11-05T07:27:46Z","content_type":"text/html","content_length":"137486","record_id":"<urn:uuid:b44dfe31-1d72-4fbf-a58a-dd4d9cbbe453>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00058.warc.gz"}
Orientable Surfaces Orientable Surfaces One type of surface which will be of significant importance later are known as orientable surfaces which we define below. Definition: An surface $\delta$ is said to be Orientable if there exists a unit normal field $\hat{N}(P)$ that is normal to ever point $P$ on $\delta$ as $P$ continuously varies over $\delta$. For example, consider the following generic surface $S$ in $\mathbb{R}^3$: We can then construct a unit normal vector on $S$ which shows that $S$ is indeed orientable: Recall from the Surface Integrals page that if a surface $\delta$ is parameterized as $\vec{r}(u, v) = (x(u, v), y(u, v), z(u,v))$ then $\frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec {r}}{\partial v}$ is normal to the surface $\delta$ for each point $P$ on $\delta$ and so we can define a natural unit normal field $\hat{N}$ on $\delta$ as: \quad \hat{N} = \frac{\frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v}}{\biggr \\| \frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v} \biggr \ For example, if a surface is given by the function $z = f(x, y)$, then as you should verify, the natural orientation of the surface generated by this function is given by: \hat{N} = \frac{-\frac{\partial f}{\partial x} \vec{i} - \frac{\partial f}{\partial y} \vec{j} + \vec{k}}{\sqrt{\left ( \frac{\partial f}{\partial x} \right )^2 + \left ( \frac{\partial f}{\partial y} \right )^2 + 1}} The other orientation of the surface generated by $z = f(x, y)$ is given by $- \hat{N}$. Since each surface has two "sides" per say and hence, two orientations. The choice of such orientation will matter when it comes to computing surface integrals of vector fields and physical interpretations of those integrals. Definition: If $\delta$ is an orientable surface with unit normal field $\hat{N} (P)$ then the Positive Side of $\delta$ is the side for which the unit normal vectors point outward/upward, and the Negative Side of $\delta$ is the side for which the unit normal vectors point inward/downward. The following is a generic orientable surface showing both the positive and negative side of this surface. Induced Orientation of Boundary Curves If $\delta$ is an orientable surface that has boundary curves, then the orientation of $\delta$ induces an orientation on the boundary curves. More specifically, if we're on the positive side of $\ delta$ and we walk along any of its boundary curves, then the surface should always be to the left:	{"url":"http://mathonline.wikidot.com/orientable-surfaces","timestamp":"2024-11-13T22:34:40Z","content_type":"application/xhtml+xml","content_length":"17841","record_id":"<urn:uuid:ff91d269-bbdb-46b8-8396-75df7896647e>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00195.warc.gz"}
What is the arclength of r=5/2sin(theta/8+(3pi)/16) -theta/4 on theta in [(-3pi)/16,(7pi)/16]? \| HIX Tutor What is the arclength of #r=5/2sin(theta/8+(3pi)/16) -theta/4# on #theta in [(-3pi)/16,(7pi)/16]#? Answer 1 In polar coordinates #ds^2 = r(theta)^2+((d r(theta))/(d theta))^2# here #r(theta)=5/5sin(theta/8+(3pi)/16)-theta/4# #(dr(theta))/(d theta) =-1/4+5/16cos((3pi)/16+theta/8) # doing #int_((-3pi)/16)^ ((7pi)/16)sqrt(r(theta)^2+((d r(theta))/(d theta))^2)d theta = 2.72416# Sign up to view the whole answer By signing up, you agree to our Terms of Service and Privacy Policy Answer 2 To find the arc length of the curve described by the polar equation ( r = \frac{5}{2} \sin\left(\frac{\theta}{8} + \frac{3\pi}{16}\right) - \frac{\theta}{4} ) on the interval ( \theta ) in ( \left[-\ frac{3\pi}{16}, \frac{7\pi}{16}\right] ), you can use the formula for arc length in polar coordinates: [ s = \int_{\theta_1}^{\theta_2} \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} , d\theta ] where ( r ) is the given polar function and ( \frac{dr}{d\theta} ) is its derivative with respect to ( \theta ). Plugging in the given polar function and its derivative, you can evaluate the integral over the specified interval to find the arc length. 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-arclength-of-r-5-2sin-theta-8-3pi-16-theta-4-on-theta-in-3pi-16-7pi--8f9afa21ff","timestamp":"2024-11-02T05:59:15Z","content_type":"text/html","content_length":"570530","record_id":"<urn:uuid:572ec0c4-73d9-4068-92ec-e89c600a354d>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00461.warc.gz"}
Transactions Online Quang-Thang DUONG, Minoru OKADA, "Receive Power Control in Multiuser Inductive Power Transfer System Using Single-Frequency Coil Array" in IEICE TRANSACTIONS on Communications, vol. E101-B, no. 10, pp. 2222-2229, October 2018, doi: 10.1587/transcom.2017EBP3387. Abstract: This paper investigates receive power control for multiuser inductive power transfer (IPT) systems with a single-frequency coil array. The primary task is to optimize the transmit coil currents to minimize the total input power, subject to the minimum receive powers required by individual users. Due to the complicated coupling mechanism among all transmit coils and user pickups, the optimization problem is a non-convex quadratically constrained quadratic program (QCQP), which is analytically intractable. This paper solves the problem by applying the semidefinite relaxation (SDR) technique and evaluates the performance by full-wave electromagnetic simulations. Our results show that a single-frequency coil array is capable of power control for various multiuser scenarios, assuming that the number of transmit coils is greater than or equal to the number of users and the transmission conditions for individual users are uncorrelated. URL: https://global.ieice.org/en_transactions/communications/10.1587/transcom.2017EBP3387/_p author={Quang-Thang DUONG, Minoru OKADA, }, journal={IEICE TRANSACTIONS on Communications}, title={Receive Power Control in Multiuser Inductive Power Transfer System Using Single-Frequency Coil Array}, abstract={This paper investigates receive power control for multiuser inductive power transfer (IPT) systems with a single-frequency coil array. The primary task is to optimize the transmit coil currents to minimize the total input power, subject to the minimum receive powers required by individual users. Due to the complicated coupling mechanism among all transmit coils and user pickups, the optimization problem is a non-convex quadratically constrained quadratic program (QCQP), which is analytically intractable. This paper solves the problem by applying the semidefinite relaxation (SDR) technique and evaluates the performance by full-wave electromagnetic simulations. Our results show that a single-frequency coil array is capable of power control for various multiuser scenarios, assuming that the number of transmit coils is greater than or equal to the number of users and the transmission conditions for individual users are uncorrelated.}, TY - JOUR TI - Receive Power Control in Multiuser Inductive Power Transfer System Using Single-Frequency Coil Array T2 - IEICE TRANSACTIONS on Communications SP - 2222 EP - 2229 AU - Quang-Thang DUONG AU - Minoru OKADA PY - 2018 DO - 10.1587/transcom.2017EBP3387 JO - IEICE TRANSACTIONS on Communications SN - 1745-1345 VL - E101-B IS - 10 JA - IEICE TRANSACTIONS on Communications Y1 - October 2018 AB - This paper investigates receive power control for multiuser inductive power transfer (IPT) systems with a single-frequency coil array. The primary task is to optimize the transmit coil currents to minimize the total input power, subject to the minimum receive powers required by individual users. Due to the complicated coupling mechanism among all transmit coils and user pickups, the optimization problem is a non-convex quadratically constrained quadratic program (QCQP), which is analytically intractable. This paper solves the problem by applying the semidefinite relaxation (SDR) technique and evaluates the performance by full-wave electromagnetic simulations. Our results show that a single-frequency coil array is capable of power control for various multiuser scenarios, assuming that the number of transmit coils is greater than or equal to the number of users and the transmission conditions for individual users are uncorrelated. ER -	{"url":"https://global.ieice.org/en_transactions/communications/10.1587/transcom.2017EBP3387/_p","timestamp":"2024-11-07T13:21:59Z","content_type":"text/html","content_length":"61881","record_id":"<urn:uuid:3c1edb4e-625d-42e1-aaa3-3e210a9d31d6>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00273.warc.gz"}
Excel Formula Python - Count Occurrences In this tutorial, we will learn how to write an Excel formula in Python that counts occurrences and returns 'No' if more than one occurrence is found, and 'Yes' if only one occurrence is found. This formula can be useful when you want to determine if a value appears more than once in a column. By using the COUNTIF function and the IF function, we can easily achieve this in Python. To write this formula in Python, we can use the following code: import pandas as pd def count_occurrences(data): count = data.value_counts() if count > 1: return 'No' return 'Yes' data = pd.Series(['A', 'A', 'B', 'C', 'B', 'A', 'D']) result = count_occurrences(data) Let's break down the code: • First, we import the pandas library, which provides data manipulation and analysis tools. • Next, we define a function called count_occurrences that takes a pandas Series as input. • Inside the function, we use the value_counts() method to count the occurrences of each value in the Series. • We then check if the count is greater than 1 using an if statement. If it is, we return 'No'. Otherwise, we return 'Yes'. • Finally, we create a pandas Series called data with some example values and call the count_occurrences function to get the result. This formula can be easily customized to work with different data and conditions. You can replace the example data with your own data and modify the if statement to check for different conditions. In conclusion, by using the COUNTIF function and the IF function in Python, we can write an Excel formula that counts occurrences and returns 'No' if more than one occurrence is found, and 'Yes' if only one occurrence is found. This formula can be useful in various data analysis and manipulation tasks. An Excel formula =IF(COUNTIF(A:A, A1)>1, "No", "Yes") Formula Explanation This formula uses the COUNTIF function to count the number of occurrences of a value in column A. If the count is greater than 1, it returns "No". Otherwise, it returns "Yes". Step-by-step explanation 1. The COUNTIF function is used to count the number of occurrences of a value in column A. The range A:A represents the entire column A, and A1 is the cell reference to the value being counted. 2. The IF function is used to check if the count is greater than 1. If it is, the formula returns "No". Otherwise, it returns "Yes". For example, if we have the following data in column A: \| A \| \| \| \| A \| \| B \| \| A \| \| C \| \| B \| \| A \| \| D \| The formula =IF(COUNTIF(A:A, A1)>1, "No", "Yes") would return the following results: • For cell A2, the count of "A" in column A is 3, so the formula returns "No". • For cell A3, the count of "B" in column A is 2, so the formula returns "No". • For cell A4, the count of "A" in column A is 3, so the formula returns "No". • For cell A5, the count of "C" in column A is 1, so the formula returns "Yes". • For cell A6, the count of "B" in column A is 2, so the formula returns "No". • For cell A7, the count of "A" in column A is 3, so the formula returns "No". • For cell A8, the count of "D" in column A is 1, so the formula returns "Yes". This formula can be used to determine if a value in column A appears more than once or not. If it appears more than once, it returns "No", otherwise it returns "Yes".	{"url":"https://codepal.ai/excel-formula-generator/query/6fWJ85wb/excel-formula-python-count-if-more-than-one-returns-no-if-one-returns-yes","timestamp":"2024-11-04T23:31:26Z","content_type":"text/html","content_length":"101758","record_id":"<urn:uuid:8e2e838b-3a37-445b-a8fe-4e008ed2bf7d>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00354.warc.gz"}
April 2015 – Incompressible Dynamics Chaos & Complexity are related; both are forms of “Coarse Damping”. While complexity is a form of coarse damping in “structure”, Chaos is a form of coarse damping in “Time”. Chaos is Coarse Synchronicity. PART 1 – EMERGENT BEHAVIOR Fine vs. Coarse Imagine you are using a camera with a large telephoto lens and you are trying to focus in on a subject in the distance. As you focus, you adjust the lens back and forth, clockwise and anticlockwise. Now imagine the lens does not move smoothly clockwise and anticlockwise, but instead has a limited number of preset positions (say 5 to the left, 1 midway, and 5 to the right). If this were the case you would find it impossible to truly focus on your subject (unless the focus length just happened to be exactly at one of the 11 preset positions), the best you could do is jump back and forth between the 2 closest positions. Furthermore the degree of focus depends on the step-size between the 2 closest positions; the larger the step size, the less focused the shot. In this focusing set-up, it is the number of preset positions available that determines the step-size between each individual position; and the smaller the number of pre-set positions, the larger the step-size becomes. Imagine the volume control on your home music system. A smooth continuous dial effectively offers an infinite number of positions up to maximum volume, and so the step-size is in effect zero. A dial with 11 preset levels (from 0 to 10) is obviously coarser in terms of step-size but nonetheless still usable to achieve the type of volume you want. It becomes more difficult if the dial’s presets are just “Off”, “Low”, “Medium”, and “High”. And you have no real control, if the dial basically allows you only “Off” or “On”; your step-size is now a single jump from zero to maximum In both these systems the step-size determines the coarseness of calibration. This concept of degrees of fine or coarse calibration can be applied to the behavior of all systems, of any type. Imagine if your personality only had 2 modes of expression, super nice or super angry, I suspect the vast majority of people would think your behavior a bit unpredictable. The Butterfly Effect The common perception about chaos is that it is all about unpredictability. This unpredictability of chaos is popularly known as “The Butterfly Effect”, a sort of exaggerated version of the domino effect; a small change over here can cause a large change over there sometime down the line. The scientific phraseology is that systems which exhibit chaos are “Sensitive Dependent on Initial Conditions” (SDIC). The whole initial conditions thing relates to the classical physics idea that if we know all the forces currently acting on a system then the only remaining thing that we need to know in order to predict the future behavior of the system, is the exact state of the system as it is right now. If a system is overly sensitive to these “initial conditions” it makes prediction impossible because we need to know these “initial conditions” with an immeasurable degree of exactness. This explanation of chaos however is somewhat misleading. While not untrue, this explanation places too much emphasis on unpredictability and fails to capture the reason for the extreme sensitivity. SDIC is the result of internal interactions being too “coarse” to negotiate stability, which leads instead to the emergence of tipping points. The unpredictability of chaos might be better described as “Sensitive Dependence to Emergent Tipping Points”… Emergence of Tipping Points Now imagine we have or we design some sort of self-stabilizing or self-balancing system. Such a system would require the ability to find its way to the exact point of balance; this point being the Many such self-stabilizing systems exist in nature; most of which self-stabilize in the most obvious way; that is they start off coarse-tuning and subsequently fine-tune their way to equilibrium. If the system however is unable, or becomes unable, to reduce coarse-tuning to fine-tuning, it will be unable to micro-adjust and hone-in on the single true-equilibrium. When the step-size of the internal adjustments remains too coarse, or become too coarse, it causes constant overshooting of the true-equilibrium, which means that the unobtainable equilibrium has now become a point about which the system oscillates. [[Think of the classic so-called business cycle of economic growth and recession]. ] At first glance it might appear that all such oscillations look the same, but actually they come in two different flavors. The emergence of an unobtainable equilibrium means that there are now two different path trajectories that the system can take. The oscillation can be a “tick-tock” oscillation or a “tock-tick” oscillation. Depending on which side of the equilibrium the system evolves to, determines which path trajectory the system will take. So although the system cannot find true-equilibrium, it nevertheless still exists, only now it behaves as a tipping point. Many Evolutionary Paths The inability to obtain equilibrium means, the behavior of the system has become sensitive to the emergence of a tipping point. Given any two different sets of initial conditions we now find that their future evolutions are either in phase with each other, or out of phase with each other. The coarseness of the step-size has thus caused two different (but complimentary) evolutionary paths to But it doesn’t stop there! The coarseness of the step-size not only determines the ability to self-synchronize to equilibrium, it also can determine whether the system is even able to synchronize the two legs of the back and forth oscillation. If we were able to manually adjust the coarseness of the internal adjustment we would find that as we increase the coarseness of the step-size we make it increasing difficult for the system to synchronize its own internal self-balancing. The inability to synchronize a two-step balancing leads to a four-step balancing; the inability to synchronize a four-step balancing leads to an eight-step balancing; and so on to infinity… This process of continual bifurcation is known as the “period doubling route to chaos”. Each bifurcation results in doubling the number of internal tipping points, which ultimately causes an infinite number of different evolutionary paths to emerge… ^[Many mathematical models exhibit this coarse behavior, the best known of which is the logistic map of population growth. So for those of you more mathematically inclined, click here for a description/analysis of The Behavior of the Logistic Map.] PART 2 – COARSE EQUILIBRIUMS & STRANGE ATTRACTORS Emergence and Unpredictability The road to unpredictable chaos may be a gradual breakdown of synchronicity in behavior – leading to an infinite number of different evolutionary paths – but what emerges from this breakdown however is still an equilibrium of sorts. Each evolutionary path is merely a different version of a previously hidden but now emergent “coarse equilibrium”. This emergent structure – which results from ever increasing coarse synchronicity – is known in chaos as an “attractor”. In actuality true-equilibrium is itself also an attractor, just not a very interesting attractor. True-equilibrium is a “point attractor” which always pulls the system to the same place, regardless of where it starts from. An emergent attractor however has a diversity of evolutionary paths that it is capable of pulling a system to; and the more emergent the attractor the greater the diversity of the evolutionary paths. Chaotic Behavior of “Chaos” has what is known as a “strange attractor” – because of its infinite number of evolutionary paths. This infinite number of paths however is not what makes the system unpredictable. What makes the system unpredictable is the sheer density of emergent tipping points that system encounters; each one of which alters the behavior of the system, and any one of which can significantly impact its future evolution. Consequently the more tipping points there are, the stranger the attractor, and the more unpredictable the system’s behavior becomes. Coarse Deformation Chaos Theory has revealed the existence of hidden attractors. Attractors act on a system like a gravitational restraining force or an elastic restoring force. What chaos theory shows us is that the type of restraining force (of equilibrium attractor) that acts on a system ultimately is determined by the degree of “coarseness of internal interactions”. Finely-balanced/short-range interactions lead to highly-symmetric uninteresting conforming behavior; coarsely-balanced/long-range interactions on the other hand lead to an interesting mix of creative non-conforming asymmetric behavior. It is as if coarse damping deforms the true-equilibrium behavior in a manner analogous to how excessive stress causes the deformation of elastic materials. It is as if increasing the coarseness of step-size causes the emergence of higher dimensional attractors and the associated diversity of surprising patterns of behavior. PART 3 – CONCLUSION So Chaos, it turns out, is actually a form of Coarse Damping to Equilibrium. Chaos is the Coarse Synchronization of coarse competing forces resulting in a coarsely synchronized equilibrium… So why is this important? It is important because Chaos is showing us that there is more to equilibrium than a finely-tuned, highly symmetric, but fundamentally bland and featureless uniformity. Practically everything in the universe is both some form of system in itself, and part of a larger system in some form or other; and equilibriums are central to systems. Chaos is the study of what can emerge from coarse system equilibrium. Chaos Theory is the mathematical study of coarse equilibrium behavior. The fascinating thing about chaos is that it is behavior we are accustomed to seeing, not only in everyday life but, in the universe at large. This is because There is a universality in the behavior of naturally damped systems when coarsely driven; they all have the same universal mathematical methodology in investigating where else the system can go. Chaos appears to be the universe’s search algorithm; a mathematically driven way of non-randomly exploring infinite possibilities, and higher dimensionality. Too long has Chaos focused on the idea of SDIC. Chaos is more than unpredictability. What’s interesting about studying Chaos, is not the unpredictability behavior, but the surprising and highly creative emergent behavior that can result from the coarse synchronicity of internal dynamics. Chaos Theory is the study of the Incompressible Mathematics that drives The Surfacing of Incompressible Diversity. What is Coarse Damping? Early in the 1980’s there was a lot of enthusiasm about the emerging science of Chaos Theory. Chaos appeared to be answering questions no one had previously ever thought to ask. Many thought that Chaos Theory might be “The Next Big Thing”; if Relativity Theory and Quantum Theory were the sciences of the 20^th century, then Chaos Theory and its cousin Complexity Theory were to be the sciences of the 21^st century. Some thirty years on however, and this enthusiasm has waned somewhat; and while Chaos may have provided some interesting insights into many different and diverse scientific disciplines, it most definitely has failed to live up to, or even come close to, its initial high expectations. Good science is essentially about simplifying complexity. It is about showing that each bit of seemingly unrelated complexity is in fact just a unique realization of some more general underlying Good reductive science however has proved somewhat difficult for both Chaos Theory and Complexity Theory. Much of the reason for this difficulty can be attributed to the fact that despite all that has been written on these subjects (and the many related areas) there has never materialized, from all of this work, a clear and simple unifying theme… Chaos and Complexity are certainly perceived to be related — different yet in some way similar — but the lack of a unifying theme is likely due to the fact that both subjects have proved a little too hard to pin down and precisely define. The generally accepted comparison between the two is usually something along the lines of: • “Chaos is a form of Complex Behavior that can arise in Simple Systems” • “Complexity is a form of Simple Order that can emerge in Complex Systems” Theses definitions are acceptable (if somewhat vague) but unfortunately they fail to capture the vital essence of both, and Chaos in particular. Failing to capture the essence of Chaos, has meant that not only has it not fulfilled its full scientific potential, but on the contrary, it has been over the years virtually relegated to the status of mere mathematical curiosity. I hope herein to reverse the misfortunes of this underestimated gem of mathematics. My goal is to try and shed some light on the vital essence of both Chaos and Complexity; and in so doing, I hope to reveal that although these fields are different, they are nonetheless very similar; for they are not different as in opposites, but merely different manifestations of the same unifying underlying behavior : a behavior that could be described as “coarse relaxation” or “coarse stabilization”, but probably best described as “Coarse Damping”. “Damping” is a term used in engineering to describe the elimination of unwanted vibrations and fluctuations in a system. Damping usually involves the external dissipation of excessive energy. Chaos and Complexity however are not about the external dissipation of energy; on the contrary they are about the internal distribution of energy. Chaos is about the coarse internal synchronization of competing forces which pull a system spontaneously to various forms of “complex behavioral equilibrium”; and Complexity is about the internal dissymmetry/imbalance of parts, within a system, which pulls the system as a whole spontaneously to a “complex structural equilibrium”. So Chaos and Complexity are indeed related… • Chaos is a form of coarse damping in “time”: it is short-term asynchronicity within long-term synchronicity. • Complexity is a form of coarse damping in “structure”: it is local imbalance with global balance. Chaos is “Coarse Synchronicity”. Complexity is “Coarse Entropy”. • Coarse Synchronicity: A tendency to resist the gravitational pull of a finely-tuned, highly synchronized but fundamentally bland and stable equilibrium. • Coarse Entropy: A tendency to resist the gravitational pull of a finely-tuned, highly symmetric, but fundamentally bland and featureless equilibrium. So why is any of this important? It is important because equilibriums are everywhere. In nature, everything is some sort of system; and all natural systems spontaneously tend towards some form of equilibrium. Mathematical Chaos shows us that there is however more to equilibrium than meets the eye. Equilibrium, it turns out, is not always so easy to fine-tune to, and so can come in many “coarse” forms…	{"url":"https://www.kierandkelly.com/2015/04/","timestamp":"2024-11-08T07:14:54Z","content_type":"text/html","content_length":"63848","record_id":"<urn:uuid:0183c8ca-b71e-4a4f-9c2d-bdce2bc4308c>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00336.warc.gz"}
Equation of Y-axis: Learn Definition, Facts and Examples To understand the equation of the y-axis. First, we have to understand graphs. What is a graph? A graph consists of a horizontal line and a vertical line, called the horizontal axis and the vertical axis, respectively. Why do we use a graph? A graph is used to represent data. A point can be determined horizontally or vertically, which can be easily understood using a graph. The horizontal axis is called the x-axis and the vertical axis is called the y-axis in a graph. Let us proceed further to know more about this topic. More about Graphs and X and Y-axis To represent a point on a graph we take given points on the x-axis and y-axis respectively as (x,y). The (0,0) coordinate is called origin, where the x-axis and the y-axis meet. We take positive points at the x-axis and negative points at the x’-axis. Similarly, we take positive points at the y-axis and negative points at the y’-axis. A Graph What Is the Equation of the Y-axis? Let us have an equation y = 2x + 3. To represent it, we will make a table. Here we have values for both y and the x-axis. So, what is the equation of the y-axis? When all the values of the equation lie on the y-axis, the values of x come 0 for all values of y, it is called the equation of the y-axis. So, the equation of y-axis is x = 0 Let understand it with an example, x = 0*y For y = 3, x will be 0 Y = -3, x will be 0 Similarly, for all value of y, x is 0. Graph of the Equation of Y-axis Line Parallel to Y-axis A line is said to parallel to the y-axis when it is drawn along the y-axis, hence perpendicular to the x-axis, where the value of x remains constant for all values of y. The common equation of the line parallel to the y-axis is x= by + c, where b is 0 So, the equation of the line parallel to the y-axis is x = c. Here c is a constant. Graph of a Line Parallel to Y-axis When b is 0, for all values of y, the value of x comes constant c. Let's understand it with an example. x= (0)y + 2 A Line parallel to the Y-axis Let’s Write the Equations of the Y-axis and X-axis From the above discussion, we know the equation of the y-axis is x=0. Similarly, the equation of the x-axis is y=0. Here the point lies on the x-axis therefore for all values of x, y becomes 0. So, the equation of the y-axis is given by x = 0 And the equation of the x-axis is given by y = 0 Questions for Practice 1. Draw the points (0,2), (0,4), and (0,-7)on a coordinate system. Do all the points lie on the same line? Can you name the line (Answer. y-axis) 2. Draw the points (2,0), (6,0), and (-3,0) on a coordinate system. Do all the points lie on the same line? Can you name the line? (Answer: x-axis) 3. Draw the points (3,2), (3,4.5), and (3,-3) on a coordinate system. Do all the points lie on the same line? Can you name the line? (Answer: yes, a line parallel to the y-axis) 4. Draw the points (2,2), (0,2), and (6,2) on a coordinate system. Do all the points lie on the same line? Can you name the line? (Answer: y a line parallel to the x-axis) We have learned the coordinates of the graph. The coordinate is written as (x,y).We have learned the equation of the y-axis. The equation is x = 0. We have learned the equation of the x-axis. The equation is y = 0. We have learned the equation of line parallel to the y-axis. The equation of the line parallel to the y-axis is x = c. FAQs on Equation of Y-axis 1. What are the x-axis and y-axis? The horizontal line in a graph is called the x-axis and the vertical line in a graph is called the y-axis. The point where these lines intersect is called the origin and its coordinate is (0,0). We represent points on a coordinate system as (x,y) where x represents the value of data on the x-axis and y represents the value of data on the y-axis. We can find the value of the coordinate x by drawing the perpendicular line to the x-axis. Similarly, we can find the value of coordinate y by drawing the line perpendicular to the y-axis. 2. How do you plot a graph with the x and y-axis? At first, we draw and label the x and y-axis. Then we draw the coordinates of the function at various values of the x and y- coordinates. We draw lines perpendicular to the x-axis and the y-axis, where these lines intersect we draw a point and this is called (x,y) coordinates. Similarly, we draw all the coordinates then we connect all the coordinates and plot the graph of the function. To draw a graph we need a minimum two-point.	{"url":"https://www.vedantu.com/maths/equation-of-y-axis","timestamp":"2024-11-02T18:21:29Z","content_type":"text/html","content_length":"230870","record_id":"<urn:uuid:0ea51a0d-a4a5-4092-ad38-8f007694a2a2>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00186.warc.gz"}
Plastic Hinge Formation (with beam elements) Problem Statement To view this project in FLAC3D, use the menu command . The main data files used are shown at the end of this example. The remaining data files can be found in the project. This example demonstrates two methods, referred to as the single- and double-node methods, by which one can model the initiation and subsequent behavior of a plastic hinge in a beam.[1] The single-node method involves specifying the limiting plastic moment (via the plastic-moment property of the beam) that can be carried by the elements making up the beam. With this method, a hypothetical hinge can form between beam elements; however, because the elements are joined with a single node, a true discontinuity in the rotational motion cannot develop at these locations. The double-node method involves creating double nodes at each potential hinge location, and then appropriately linking these nodes together. The double nodes allow a discontinuity in the rotation to occur when the limiting plastic moment is reached. The double-node method should be applied to calculate the large-strain, post-failure behavior of a structure. If it is only necessary to determine the solution at the limiting plastic moment, then the single-node method is sufficient. In the following discussion, we first define the example problem, and then solve it in two ways by using the single-node method, and then using the double-node method. Analytical Solution A concentrated vertical load $(P)$ is applied at the center of a 10 m long, simply supported beam with a plastic-moment capacity, $M^P$, of 25 kN-m. The system, along with the shear-force and bending-moment distributions, is shown in Figure 1. From these shear and moment diagrams, we find that the specified plastic-moment capacity corresponds with a maximum vertical load of 10 kN and a maximum shear force of 5 kN. If we apply a constant vertical velocity to the beam center, we expect that the limiting values of moment and shear force will be 25 kN-m and 5 kN, respectively. Particular Problem The particular problem to be considered is defined as follows. The beam has a 10-m length ($L = 10 \textrm{ m}$), and a rectangular cross section with $b = 1.0 \textrm{ m}$ and $d = 100 \textrm{ mm}$. The cross-sectional properties (obtained via the expressions in Figure 1) are \[\begin{split}\begin{gather} A = 0.1 \textrm{ m}^2 \\ I_y = 8.33 \times 10^{-3} \textrm{ m}^4, \quad I_z = 8.33 \times 10^{-5} \textrm{ m}^4 \\ J = 3.12 \times 10^{-4} \textrm{ m}^4 \end{gather}\end where the constant $c_2 = 0.312$ (Crandall et al., 1978, Table 6.1). The beam is comprised of structural steel, and is assumed to behave as an isotropic elastic material with \[E = 200 \textrm{ GPa}, \quad \nu = 0.30\] The properties $\{ A, I_y, J, \nu \}$ are not relevant for this problem; however, they are needed as input for the FLAC3D model. FLAC3D Models, Results and Discussion Single-Node Method The FLAC3D model demonstrating the single-node method (see “PlasticBeamHinge-Single.dat”) is created by issuing a single structure beam create command and specifying two segments. This produces a model containing 2 elements and 3 nodes, with both elements sharing the center node as shown in Figure 2. The beam is assigned the properties listed above and, in addition, the plastic-moment capacity is set to 25 kN-m. The beam element coordinate system is aligned with the global coordinate system by setting the direction-y property equal to $(0,1,0)$. Simple supports are specified at the beam ends by restricting translation in the $y$-direction. A constant vertical velocity is applied to the center node, and the moment and shear force acting at the right end of element 1 are monitored during the calculation to determine when the limiting values have been reached. Note that we specify combined local damping for this problem in order to eliminate the ringing that can occur with the default local damping scheme when the system is being driven by a constant motion. We find that the limiting values of moment and shear force are equal to the analytical values of 25 kN-m and 5 kN, respectively, as shown in Figures 3 and 4. The bending-moment and shear-force distributions correspond with the analytical solution as shown in Figures 5 and 6. A discontinuity in the rotational motion at the center location cannot develop, because there is only a single node at this location. Double-Node Method The FLAC3D model demonstrating the double-node method (see “PlasticHingeBeam-Double.dat”) is created by issuing two separate structure beam create commands. This produces a model containing two elements and four nodes. Going from left to right, the left element uses nodes 1 and 2, and the right element uses nodes 3 and 4. Note that nodes 2 and 3 lie in the same location, but they will not interact with one another. We now create an appropriate linkage between nodes 2 and 3 with the following commands[2]: struct node join range component-id 2 struct link attach rotation-z normal-yield struct link property rotation-z area 1.0 stiffness 6.4e8 ... yield-compression 25e3 yield-tension 25e3 The first command creates a node-to-node link from node 2 to node 3, and the attachment conditions are rigid for all degrees-of-freedom. The second command inserts a normal-yield spring in the $z$ -rotational direction of the link. The final command set the properties of this normal-yield spring as follows. We set all areas to unity, and we set both the compressive and tensile yield strengths equal to the desired plastic-moment capacity of 25 kN-m. Finally, we set the spring stiffness equal to a value that is large enough to make the spring deformation small relative to the beam deformation. We determine this value by summing the rotational stiffnesses of the two elements that use the spring (each rotational stiffness is $4EI/L$, where $L$ is element length) and multiplying this value by 10. Now that the double nodes have been appropriately linked to one another, the beam is assigned the same properties as listed above; but we do not specify a plastic-moment capacity, because we want the moment at the center to be limited by the strength of the normal-yield spring. The beam element coordinate system is aligned with the global coordinate system by setting the direction-y property equal to $(0,1,0)$. Simple supports are specified at the beam ends by restricting translation in the $y$-direction. A constant vertical velocity is applied to the target node, and the moment and shear force acting at the right end of element 1 are monitored during the calculation to determine when the limiting valus have been reached. Note that we specify combined local damping for this problem in order to eliminate the ringing that can occur with the default local damping scheme when the system is being driven by a constant motion. We find that the limiting values of moment and shear force are equal to the analytical values of 25 kN-m and 5 kN, respectively, as shown in Figures 7 and 8. The bending-moment and shear-force distributions correspond with the analytical solution as shown in Figures 9 and 10. A discontinuity in the rotational motion at the beam center has developed — the rotation of nodes 2 and 3 are nonzero and equal and opposite to each other (structure node list displacement). Double-Node Cantilever The preceding two examples demonstrate that both the single-node and the double-node methods produce the same behavior near the limiting moment; however, if we were to continue loading the structure in a large-strain fashion, then the double-node method would produce more reasonable results because it allows a discontinuity to develop in the rotation at the center. We illustrate this behavior by modifying the double-node example to represent a cantilever beam (fixed at the left end) with a vertical load applied at the free end (see “PlasticHingeBeam-Cantilever.dat”). The problem is run in large-strain mode. The final structural configuration and moment distribution are shown in Figure 11. The double-node method allows a discontinuity to develop in the rotation at the beam center. Crandall, S.H., N.C. Dahl and T.J. Lardner. An Introduction to the Mechanics of Solids, Second Edition, New York: McGraw-Hill Book Company (1978). Data Files model new model title 'Plastic hinge formation in beam (single-node method)' ; Create beam and assign properties. structure beam create by-line (0,0,0) (10,0,0) segments 2 ; Assign beam properties structure beam cmodel assign elastic structure beam property young 200e9 poisson 0.30 structure beam property cross-sectional-area 0.1 ... moi-y 8.33e-3 moi-z 8.33e-5 torsion-constant 3.12e-4 ... direction-y (0,1,0) structure beam property plastic-moment 25e3 ; Assign boundary conditions. structure node fix velocity-y range position-x 0.0 position-x 10.0 union ; simple support at left and right end structure node fix velocity-y range position-x 5.0 ; Apply constant vertical velocity structure node initialize velocity-y -5e-6 local range position-x 5.0 ; to center node. ; Take histories to monitor response. structure node history name='disp' displacement-y position (5,0,0) ; y-displacement of center node structure beam history name='mom' moment-z end 2 position (2.5,0,0) ; moment structure beam history name='force' force-y end 2 position (2.5,0,0) ; shear force structure mechanical damping combined-local model large-strain off model cycle 4000 ; 20 mm total displacmeent model save 'SingleNode' model new model title 'Plastic hinge formation in beam (double-node method)' ; Create two separate beams that do not share center node. structure beam create by-line (0,0,0) ( 5,0,0) id 1 segments 1 structure beam create by-line (5,0,0) (10,0,0) id 2 segments 1 ; Create a link at the overlapping nodes and set link properties. structure node join range component-id 2 structure link attach rotation-z normal-yield structure link property rotation-z area 1.0 stiffness 6.4e8 ... yield-compression 25e3 yield-tension 25e3 ; Assign beam properties structure beam cmodel assign elastic structure beam property young 200e9 poisson 0.30 structure beam property cross-sectional-area 0.1 ... moi-y 8.33e-3 moi-z 8.33e-5 torsion-constant 3.12e-4 ... direction-y (0,1,0) ; Assign boundary conditions. structure node fix velocity-y range position-x 0.0 position-x 10.0 union ; simple support at left and right end ; Apply constant vertical velocity to center node-3 --- ; must apply velocity to target node. structure node fix velocity-y range position-x 5.0 ; Apply constant vertical velocity structure node initialize velocity-y -5e-6 local range component-id 3 ; to the target node. ; Take histories to monitor response. structure node history name='disp' displacement-y position (5.0,0,0) ; y-displacement of center node structure beam history name='mom' moment-z end 2 position (2.5,0,0) ; moment structure beam history name='force' force-y end 2 position (2.5,0,0) ; shear force structure mechanical damping combined-local model large-strain off model cycle 4000 ; 20 mm total displacmeent model save 'DoubleNode' model new model title 'Plastic hinge formation in cantilever (double-node method)' ; Create two beams structure beam create by-line (0,0,0) (5,0,0) id 1 segments 1 *structure beam create by-line (5,0,0) (10,0,0) id 2 segments 1 ; Join nodes, and make the z rot link plastic structure node join range component-id 2 structure link attach rotation-z normal-yield structure link property rotation-z area 1.0 stiffness 6.4e8 ... yield-compression 25e3 yield-tension 25e3 ; Assign beam properties structure beam cmodel assign elastic structure beam property young 200e9 poisson 0.30 structure beam property cross-sectional-area 0.1 ... moi-y 8.33e-3 moi-z 8.33e-5 torsion-constant 3.12e-4 ... direction-y (0,1,0) ; Assign boundary conditions structure node fix velocity rotation range position-x 0 ; fully fix left end structure node apply force=(0,-5.0e3,0) range position-x 10.0 ; apply force at tip structure mechanical damping combined-local model large-strain on model solve convergence 1.0 model save 'Cantilever' Was this helpful? ... Itasca Software © 2024, Itasca Updated: Sep 26, 2024	{"url":"https://docs.itascacg.com/itasca920/common/sel/test3d/Beam/PlasticHingeBeam/PlasticHingeBeam.html","timestamp":"2024-11-02T23:55:47Z","content_type":"application/xhtml+xml","content_length":"46338","record_id":"<urn:uuid:dc25d9ce-15e5-4b62-bd5e-a0cd781e47b5>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00102.warc.gz"}
┃│PRINCIPIA CYBERNETICA WEB - ©┃ Parent Node(s): INFORMATION THEORY (or statistical communication theory) a calculus or variation, variability and variance initially developed by Shannon, to separate noise from information carrying signals, now used to trace the flow of information in complex systems, to decompose a system into independent (see independence) or semi-independent sub-systemS, to evaluate the efficiency of communication channels and of various communication codes (see redundancy, noise, equivocation) and to compare information needs with the capacities of existing information processors (see computer, bremermann's limit), etc. The basic quantity this calculus analyses (see analysis) is the total amount of statistical entropy given data contain about an observed system. The calculus provides an algebra for decomposing and thus accounting for this entropy in numerous ways. E.g. the quantity of entropy in an observed system equals the sum of the entropies in all of its separate parts minus that amount of information transmitted within the system. The latter quantity is the amount of entropy in a system not explainable from its parts and an expression of the communication between these parts. This formula is another example of the cybernetic analysis of systems, according to which any whole system is accounted for or defined in terms of a set of components and its organization. The total amount of information transmitted in a quantitative analogue to and hence can be thought of as a measure of a system's structure. Information theory has provided numerous theorems and algebraical identities with which observed systems may be approached, e.g., the law of requisite variety, the TENTH theorem of information theory (Krippendorff) │ * Next │ * Previous │ * Index │ * Search │ * Help │ URL= http://cleamc11.vub.ac.be/ASC/INFORM_THEOR.html	{"url":"http://pespmc1.vub.ac.be/ASC/INFORM_THEOR.html","timestamp":"2024-11-04T01:37:31Z","content_type":"text/html","content_length":"4217","record_id":"<urn:uuid:b75bbbc3-5ac8-48da-8d83-0b8822d8a404>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00780.warc.gz"}
Daily Compound Interest Calculator Daily Compound Interest Include all days of week? The daily reinvest rate is the % figure that you wish to keep in the investment for future compounding. As an example, you may wish to reinvest 80% of the daily interest/earnings you receive and withdraw the other 20% in cash. $ This option allows you to add extra deposits into the investment. Note: Deposits are currently in beta. $ This option allows you to withdraw additional amounts from the investment. Note: Withdrawals are currently in beta. See how much daily interest/earnings you might receive on your investment over a fixed number of days, months and years. You may find this useful for day trading or trading bitcoin or other Disclaimer: Whilst every effort has been made in building our calculator tools, we are not to be held liable for any damages or monetary losses arising out of or in connection with their use. Full On this page: What is daily compound interest? With compound interest, the interest you have earned over a period of time is calculated and then credited back to your starting account balance. In the next compound period, interest is calculated on the total of the principal plus the previously-accumulated interest. The more frequently that interest is calculated and credited, the quicker your account grows. The interest earned from daily compounding will therefore be higher than monthly, quarterly or yearly compounding because of the extra frequency of compounds. With some types of investments, you might find that your interest is compounded daily, meaning that you're earning interest on both the principal amount and previously accrued interest on a daily basis. This is often the case with trading where margin is used (you are borrowing money to trade). Examples of these types of investment include CFD trading, Forex trading, spread-betting or options for assets like stocks and shares, as well as commodities like oil and gold and cryptocurrencies like Bitcoin and Ethereum. This is a very high-risk way of investing as you can also end up paying compound interest from your account depending on the direction of the trade. How to calculate daily compound interest Daily compound interest is calculated using a version of the compound interest formula. To begin your calculation, take your daily interest rate and add 1 to it. Then, raise that figure to the power of the number of days you want to compound for. Finally, multiply your figure by your starting balance. Subtract the starting balance from your total if you want just the interest figure. Note that if you wish to calculate future projections without compound interest, we have a calculator for simple interest without compounding. Let's examine the formula in a bit more detail. Formula for daily compound interest The formula for calculating daily compound interest with a fixed daily interest rate is: A = P(1+r)^t • A = the future value of the investment • P = the principal investment amount • r = the daily interest rate (decimal) • t = the number of days the money is invested for • ^ = ... to the power of ... Example investment Let's use the example of $1,000 at 0.4% daily for 365 days. • P = 1000 • r = 0.4/100 = 0.004 • t = 365 Let's put these into our formula: A = P(1+r)^t A = 1000(1+0.004)^365 A = 1000 * 4.2934377972993 A = 4293.4377972993 To get the total interest, we deduct the principal amount (1000) from the future value. This gives us interest of $3293.44 Daily compounding with annual interest rate If you have an annual interest rate and want to calculate daily compound interest, the formula you need is: A = P(1+r/365)^(365t) • A = the future value of the investment • P = the principal investment amount • r = the annual interest rate (decimal) • t = the number of years the money is invested for • ^ = ... to the power of ... Including additional deposits Making regular, additional deposits to your account has the potential to grow your balance much faster thanks to the power of compounding. Our daily compounding calculator allows you to include either daily or monthly deposits to your calculation. Note that if you include additional deposits in your calculation, they will be added at the end of each period, not the beginning. Questions about our calculator Here are some frequently asked questions about our daily compounding calculator. What is the daily reinvest rate? The daily reinvest rate is the percentage figure that you wish to keep in the investment for future days of compounding. As an example, you may wish to only reinvest 80% of the daily interest you're receiving back into the investment and withdraw the other 20% in cash. Let's look at an example. If your initial investment is $5,000 with a 0.5% daily interest rate, your interest after the first day will be $25. If you choose an 80% daily reinvestment rate, $20 will be added to your investment balance, giving you a total of $5020 at the end of day one. The remaining $5 will be withdrawn as cash. Excluding weekends from calculations You may wish to only compound your money on particular days of the week. Perhaps you only trade on weekdays, or want to exclude Sundays. Our calculator gives you the option to exclude these days from your calculation, giving you extra flexibility. Here's an example: You want to compound for one year minus weekends (net business days). This means your figure will compound for around 261 BUSINESS days, with an end date 365 days from your start date, depending on when the weekends fall. The aim of this option is to give you maximum flexibility around how your interest is compounded and calculated, whether you're Forex trading, trading with cryptocurrencies or simply buying and selling stock assets. And finally... I hope you found our daily compounding calculator and article useful. At The Calculator Site we love to receive feedback from our users, so please get in contact if you have any suggestions or comments. You may also wish to check out our range of other finance calculation tools.	{"url":"https://www.thecalculatorsite.com/finance/calculators/daily-compound-interest.php","timestamp":"2024-11-08T21:30:49Z","content_type":"text/html","content_length":"334435","record_id":"<urn:uuid:2ce3482d-e9f6-44f4-afcd-756fe16fe1b0>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00562.warc.gz"}
Explain rank and rownumber window function in PySpark - Recipe Objective - Explain rank and row_number window functions in PySpark in Databricks? The row_number() function and the rank() function in PySpark is popularly used for day-to-day operations and make the difficult task an easy way. The rank() function is used to provide the rank to the result within the window partition, and this function also leaves gaps in position when there are ties. The row_number() function is defined as which gives the sequential row number starting from the 1 to the result of each window partition. Build Log Analytics Application with Spark Streaming and Kafka System Requirements • Python (3.0 version) • Apache Spark (3.1.1 version) This recipe explains what rank and row_number window function and how to perform them in PySpark. Explore PySpark Machine Learning Tutorial to take your PySpark skills to the next level! Implementing the rank and row_number window functions in Databricks in PySpark # Importing packages import pyspark from pyspark.sql import SparkSession from pyspark.sql.window import Window from pyspark.sql.functions import rank from pyspark.sql.functions import row_number The Sparksession, Window, rank and row_number packages are imported in the environment to demonstrate status and row_number window functions in PySpark. # Implementing therank and row_number window functions in Databricks in PySpark spark = SparkSession.builder.appName('Spark rank() row_number()').getOrCreate() Sample_data = [("Ram", "Technology", 4000), ("Shyam", "Technology", 5600), ("Veer", "Technology", 5100), ("Renu", "Accounts", 4000), ("Ram", "Technology", 4000), ("Vijay", "Accounts", 4300), ("Shivani", "Accounts", 4900), ("Amit", "Sales", 4000), ("Anupam", "Sales", 3000), ("Anas", "Technology", 5100) Sample_columns= ["employee_name", "department", "salary"] dataframe = spark.createDataFrame(data = Sample_data, schema = Sample_columns) # Defining row_number() function Window_Spec = Window.partitionBy("department").orderBy("salary") dataframe.withColumn("row_number",row_number().over(Window_Spec)) \ # Defining rank() function dataframe.withColumn("rank",rank().over(Window_Spec)) \ The "dataframe" value is created in which the Sample_data and Sample_columns are defined. The row_number() function returns the sequential row number starting from the 1 to the result of each window partition. The rank() function in PySpark returns the rank to the development within the window partition. So, this function leaves gaps in the class when there are ties.	{"url":"https://www.projectpro.io/recipes/explain-rank-and-rownumber-window-function-pyspark-databricks","timestamp":"2024-11-02T04:47:04Z","content_type":"text/html","content_length":"144722","record_id":"<urn:uuid:95830254-0263-4308-8739-af6172e2ee7e>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00752.warc.gz"}
Stability analysis of civil air defense tunnel under blasting vibration Based on a large section drilling and blasting excavation project, the dynamic response characteristics of civil air defense tunnels are analyzed by combining field monitoring and numerical simulation. The dynamic response features include particle vibration velocity, main frequency, displacement, and stress, and the stability criterion of the tunnel is analyzed. A safety criterion model based on the ultimate tensile strength of materials is established. The results show that the frequency of the $X$, $Y$, and $Z$ directions is mainly distributed in 90-140 Hz. The effective stress increases first and then decreases along the axis of the roadway. The stress near the explosion source is large and the relative reduction is also large. By fitting the relationship between blasting vibration velocity and maximum principal stress, the safe vibration velocity criterion based on tensile strength is obtained, and the safe threshold of vibration velocity is 19.62 cm/s. It can be assumed that blasting does not affect the structure. • Effective stress increases first and then decreases along the tunnel axis. • Stress near the blast source is larger, and its relative reduction is larger. • Material has not reached the limit state, so the structure is safe. 1. Introduction Most of the existing studies are carried out based on numerical simulation. Among them, Wang Qiuyi [1] et al. studied the changes in vibration frequency and amplitude in the tunnel and obtained the dynamic response characteristics of the tunnel at different locations. Li Qingsong et al. used finite element numerical analysis to study the change characteristics of the tunnel and analyzed the distribution law of the dynamic response [2]. Liu, N. et al. used numerical methods to simulate the influence of the strength of the lining structure at different positions of the excavated tunnel on the existing tunnel. The results show that when two tunnels are parallel, the blasting vibration has the greatest influence on the side walls of the two tunnels [3]. Wu Boet al. believes that the influence of the cavity zone after the new working face on the lining of the existing tunnel is greater than that of the uncavity zone [4]. Zhu Zhengguo et al. studied the blasting dynamic response and safety range of the three-dimensional cross tunnel [5] and concluded that the peak vibration velocity of particles (PPV) at the arch and arch foot of the existing tunnel is more sensitive and weaker than that of other locations in the ring direction [6]. Through numerical simulation, field monitoring, and theoretical analysis of stress wave propagation, scholars at home and abroad have carried out a lot of research on the influence of new tunnels on adjacent tunnels during tunnel excavation [7]. J. Mandal studied the numerical simulation of a shallow buried tunnel under a surface explosion load [8]. In the study of A. Mottahedi, a soft calculation method is used to establish the over drilling prediction model of the tunnel [9]. S. Mishra studied tunnel blasting and analyzed the stress-strain response of granite at high strain rates [10]. The flow behind a blast wave, propagating in the downstream direction in a hypersonic tunnel, is investigated analytically by H. Mirels [11]. S. Koneshwaran considers the performance of ground blasting tunnels under fluid-structure coupling [12]. However, due to projects and other reasons [13], few scholars have studied the dynamic analysis of civil air defense tunnels in the process of urban subway tunnel excavation [14]. In summary, domestic and foreign scholars have done a lot of work on the influence of new tunnels on adjacent existing tunnels during mine tunnel excavation by using numerical simulation, field monitoring, and stress wave propagation theory analysis, and many conclusions have important reference significance for the research of this project [15]. In the study of blasting seismic wave propagation, the influence of rock mass properties, charge, distance, and elevation on seismic wave propagation is studied, and the change of seismic wave propagation is obtained. At present, many new tunnels have been analyzed for the stability of adjacent tunnels by blasting excavation. However, due to the project and other reasons, few scholars have studied the dynamic analysis of civil air defense tunnels in the process of urban subway tunnel excavation. Therefore, this paper takes advantage of the special working conditions of the project and uses the method of on-site monitoring and numerical simulation to conduct research. This paper studies the dynamic stability of the tunnel under the blasting action, analyzes the comprehensive influence of the tunnel on vibration frequency, vibration amplitude, and vibration dynamic stress, and establishes a safety criterion model based on ultimate tensile strength. 2. Overview of engineering geology This project is based on the Hong community, which is a double-line tunnel. The large-section tunnel and the civil air defense tunnel intersect under Eighth Road, and their position is shown in Fig. 1. The large-section tunnel is located in the breezed limestone layer, with a partial interpenetration of medium-weathered limestone, with a transverse width of 20.14 m and a midline height of 13.4 m. Within this section, the buried depth of the arch is 22.39-22.94 m, and the spatial position is horizontal. The geological profile of the large-section tunnel and the civil defense tunnel is shown in Fig. 2. Fig. 1Relationship diagram of the plane position of large section tunnel of civil air defense tunnel Fig. 2Geological section of large section tunnel and civil air defense tunnel To effectively ensure that the tunnel excavation is not affected by the surrounding structures and the safety of passage, the double-sidewall heading excavation method is adopted in the construction of the large section Ⅲ surrounding rock. According to the double-sidewall diversion excavation technology, the tunnel is divided into three steps, left, middle, and right, with a total of 9 parts. Blasting construction is carried out in sequence, excavation is carried out step by step, and the blasting design adopts the straight-cut method. The peripheral holes are designed according to the requirements of smooth blasting, with hole proximity coefficient $a=$0.8 and minimum resistance line $W=$500 mm. The hole layout is shown in Fig. 3. Fig. 3Excavation sequence of mine tunnel with large section (Unit: mm) The buildings above the tunnel are dense and there are civil air defense tunnels above the tunnel, blasting vibration monitoring is of great significance for tunnel excavation and protection of future construction (structure). The monitoring diagram is shown in Fig. 4, the monitoring instrument number is TC4850, and a two-hole booster transducer with a frequency of 60 kHz and a minimum peak value of triggered vibration speed of 0.01 cm/s is adopted. Fig. 4Field monitoring diagram of TC-4850 blasting vibrometer a) Schematic layout of monitoring b) Layout of monitoring points 3. Analysis of blasting vibration test results 3.1. Analysis of tunnel vibration characteristics In this paper, the characteristics of vibration intensity are analyzed by using the monitoring data of large-section tunnel excavation. It can be seen from Fig. 5 that the PPV in the $Z$ direction is in the positive direction, while the PPV in the $X$ and $Y$ direction is in the negative direction. It can be seen from Fig. 5 that the vertical vibration velocity is large, which may be attributed to the following reasons: There is a free surface on the floor of the civil air defense tunnel in the vertical direction, so it has a certain amplification effect on the vibration velocity. Fig. 5Waveform diagram of PPV in each direction To study the vibration laws in $X$, $Y$, and $Z$ directions, a vibration velocity waveform curve is selected for analysis; According to Fig. 6, it is not difficult to find that the peak time of the vibration tunnel in the $Y$ direction is 0.015 s, while the peak time in the $X$ direction is about 0.25 s, so it is not difficult to find through the Fig. 6. Fig. 6Statistical diagram of monitoring data at each measuring point 3.2. Attenuation law on the propagation path According to the national blasting safety regulations, the Sadowski formula is generally adopted for linear regression analysis, by using the principle of the least square method, the fitting curve can be obtained. The empirical formula can then be obtained: where $v$ is the peak particle vibration speed, $Q$ is the amount of explosive, and $R$ is the distance from the monitoring point to the explosion source. Fig. 7Fitting curve of axial blasting vibration velocity of civil air defense tunnel 4. Characteristics and safety criterion of blasting vibration response of civil air defense tunnel 4.1. Numerical modeling According to the excavation conditions of the project, the tunnel in the large section is 109 meters in total. The upper left guide tunnel is 75 meters excavated, the middle left guide tunnel is 66.5 meters excavated, and the lower left guide tunnel is only 9 meters excavated. Under actual conditions, the axis of the civil defense tunnel has a certain intersection with the axis of the large section tunnel. To reduce the difficulty of modeling, the position of the civil defense tunnel is adjusted to be parallel to the central axis of the tunnel. Fig. 8Numerical model (1 – plain filled soil, 2 – clay layer, 3 – moderately weathered limestone, 4 – civil defense tunnel, 5 – excavation section unit mm) b) Numerical model with grid According to the field blasting situation, the thickness of the model is 60 meters, that is, the height × width × depth = 70×140×60. The model adopts a cm-g-us unit system, and the unit type is set to SOLID164. The top surface (i.e. the ground) of the model is free, and the remaining five faces are set to non-reflective boundaries. The numerical model is shown in Fig. 8, The vertical direction is the $Y$ direction, the $X$ direction is the axis direction of the civil air defense tunnel, and the $Z$ direction is the radial direction of the civil air defense tunnel. The model is divided into 904,154 elements. structure diagram of the civil air defense tunnel is shown in Fig. 9, space position diagram of the tunnel and gun hole is shown in Fig. 10. Fig. 9Structure diagram of civil air defense tunnel Fig. 10Space position diagram of the tunnel and gun hole 4.2. Material parameters The tunnel size in the numerical model is consistent with the actual excavation face when the model is built. According to the above knowledge, the mining is divided into 9 parts, and the grid is encrypted in the mining section. To simplify the calculation model, the rock and soil layers are simplified into 3 layers because the strata are mostly uneven and mixed with each other. The specific parameters of rock and soil mass are shown in Table 1. Table 1Physical and mechanical parameters of rock and soil mass Thickness Density Names of rocks Modulus of elasticity (GPA) Poisson’s ratio Yield stress (GPA) Tangential modulus (GPA) Hardening coefficient (m) (kg/m^3) Breezy limestone 45.2 2.68 7 0.13 15 0.21 0.5 Thickness Density Cohesion Soil layer names Shear modulus (MPa) Poisson’s ratio Damage surface shape parameters Friction angle (rad) (m) (kg/m^3) (kph) Plain fill soil 2.4 1.98 9 0.35 0.8 0.314 0.1 Clay 12.4 1.99 50 0.275 0.8 0.188 0.25 The civil air defense tunnel is the main research object, and its lithology condition is mainly clay, and the material parameters are known from the above. According to the data, the concrete used in the civil air defense tunnel is c30 concrete. Due to its long construction time and certain aging strength, the strength, elastic modulus and other parameters of the concrete lining of the tunnel have been reduced to a certain extent. According to the model and experience, it has been weakened to a certain extent. 4.3. Loading mode Blasting and load is indeed the basis of numerical simulation of explosive dynamic analysis, ANSYS/LS-DYNA numerical simulation software for blasting vibration simulation analysis has strong practicability. However, there are no explosive model parameters in the software, so there are usually two ways to determine the change process of explosive load in the general analysis process: one is to establish the explosive initiation model. The second is the equivalent load applied by half theoretical and half empirical formulas. In this simulation, an explosive initiation model for blasting simulation is established using the first method. The explosive initiation position is reserved when the model is established, and the explosive detonation model is established to simulate blasting during calculation. Since a state equation must be added to the explosion of high explosives, MAT_HIGH_EXPLOSIVE_BURN high explosives materials and EOS_JWL equation of state were added during calculation, and the detonation location was set according to the size of the model, so the blasting process was simulated. The explosive model data are shown in Table 2. Table 2Explosive material parameters Density kg/m^3 Detonation speed Burst pressure (GPa) Blasting parameters 1.15 0.35 3.24 4.4. Numerical model validation When using the numerical simulation method to analyze the dynamic response of blasting, to ensure the correctness of the model, the comparison should be made to verify whether the results of numerical modeling simulation are roughly the same as the actual monitoring data. The reasons for the error may be as follows: (1) the specific parameters of the rock and soil layer cannot be accurately determined, which can be adjusted according to the trial-and-error method in the model calculation, and the principle of orthogonal test is used to determine the accurate blasting parameters; (2) the explosion pressure and other parameters are different from the field monitoring, resulting in partial errors in the vibration propagation time and duration. To ensure the accuracy of the model, several corresponding points are selected for data comparison, and the obtained data are shown in Table 3. These data are obtained at the frequency of 90-150 Hz, which is obtained by field monitoring. As you can see, the error is less than 25 %. The numerical simulation data and the monitoring data have similar rules of cavitation effect, and the results are all within the allowable error range. 4.5. Analysis of dynamic response characteristics of civil air defense tunnel blasting When the blasting seismic wave propagates in the rock and soil layer, the PPV of the blasting seismic wave propagates in different media and different structures will be greatly different. If there are buildings (structures) in the propagation range of the wave, it will cause the displacement and stress change of the building structure particles, and sometimes it may cause structural instability. To fully grasp the strength and characteristics of structural particle vibration velocity, stress, and displacement change, a more detailed study should be carried out according to the characteristics of the building structure. This paper is about an underground structure - an air defense tunnel. When studying the dynamic response of air defense tunnel lining structure and surrounding rock under blasting vibration, the stress distribution law of radial, tangential, and vertical units should be analyzed. The above analysis shows that in theory, the vertical response should be the largest in the analysis of radial tangential and vertical vibration responses. In the field monitoring, the velocities of five points were measured in the radial direction. To study the response characteristics of the three directions, five measuring points were also set by numerical simulation to obtain the time-history curves of horizontal radial, horizontal tangential, and vertical vibration velocities, as shown in Fig. 11. Table 3Comparison between numerical simulation and field monitoring data Data Type Monitoring point bits Horizontal tangential vibration velocity (cm/s) Horizontal radial vibration velocity (cm/s) Vertical vibration velocity (cm/s) Peak frequency (Hz) 1 2.674 2.45 7.43 94 2 2.43 2.34 7.63 102 Field monitoring data 3 1.10 1.73 4.025 132 4 0.652 0.64 1.79 112 1 3.28 2.28 8.19 105 2 1.79 2.03 9.14 116 Numerical simulation data 3 0.68 1.41 3.68 125 4 0.356 0.733 1.166 120 Fig. 11Distribution diagram of the three-way PPV a) The radial vibration waveform diagram b) Tangential vibration waveform diagram c) Vertical vibration waveform diagram d) Axial trend diagram of the three-way PPV It can be seen from Fig. 12 that the overall magnitude of the PPV in the three directions is vertical > horizontal radial > horizontal tangential. It can be seen from the attenuation of vibration velocity in the three directions: Within the same propagation distance, the variation range of vertical vibration velocity is 9 cm/s-0.45 cm/s, and the attenuation range of horizontal radial and horizontal tangential is roughly 3 cm/s-0.289 cm/s, which indicates that the vertical vibration velocity decays faster along the tunnel axis and the variation range is larger. When the distance between the vertical vibration velocity and the horizontal vibration velocity is smaller than the distance between the detonation centers, the difference in the peak value is obvious. Similar to the actual monitoring data, the vertical vibration velocity increases first and then decreases along the tunnel axis, indicating that there is an obvious vertical cavitation effect, which verifies the correctness of the law presented by the actual monitoring data. In the study of the dynamic response of rock lining under blasting vibration, stress, and displacement are very important parameters. According to the arrangement of on-site monitoring and measuring points, a consistent measuring point is selected in the numerical model to study the stress. Concrete is not an ordinary solid, but a porous medium, the material structure is more complex, there are many pores developed inside, the pores are also saturated with fluid, and the fluid has a certain pressure. Concrete is subjected not only to external pressure but also to internal pressure (pore pressure). Concrete compresses when subjected to external pressure. Concrete expands when subjected to internal pressure. Therefore, when studying the deformation of concrete, it is necessary to deduct the role of internal pressure from the external pressure, after which the so-called effective stress is defined. The positive of effective stress in this study is tensile stress, and the negative is compressive stress. Because the change of force will cause structural deformation or even instability failure, the interface is selected at the point where the stress is maximally distributed along the axis for deformation analysis, as shown in Fig. 12. Fig. 12Waveform of stress a) Time history curve of effective stress b) Shear stress time history curve It can be seen from Fig. 12 that the effective stress increases first and then decreases along the tunnel axis. The stress near the explosion source is larger, and its relative reduction is larger. In the tunnel, the effective stress and shear stress are not large, and the material has not reached the limit state, so the structure is safe. According to engineering experience, the compressive strength of the rock is much greater than the tensile strength, so the damage of the rock is generally tensile failure. According to the ultimate failure criterion, whether the rock mass is damaged mainly depends on whether the tensile stress reaches the ultimate tensile strength. Because the stress in ANSYS/LS-DYNA software follows the principle of “tensile stress is positive compressive stress is negative”, it can be seen from the stress time-history curve that the maximum principal stress is tensile, all of which are less than 0.4 MPa, and the ultimate tensile strength of the lining structure is about 1.78 MPa, so the structure is not damaged. 4.6. Instability mode and safety criterion of the tunnel structure It can be seen from the above that the damage to tunnel lining structure should be tensile. To ensure the construction safety, the common method is to control the blasting process and reduce the impact of blasting on the structure. At present, there are two commonly used methods: (1) numerical simulation to fit the relationship between stress and vibration velocity; (2) using elastic stress wave propagation theory. In this paper, method (2) is used to fit the relationship between stress and vibration velocity based on numerical simulation to establish a safety criterion model of blasting vibration velocity of tunnel lining structure. The PPV is linearly fitted with the maximum principal stress of the corresponding unit at the monitoring point of the tunnel floor. By fitting the relationship between the blasting vibration velocity and the maximum principal stress on the lining of the civil defense tunnel, the relationship between the maximum vibration velocity and the tensile stress of the civil defense tunnel is obtained. The regression analysis equation is as follows: $\sigma =0.0694{V}_{max}+1.0382,$ where: ${V}_{max}=\left(\sigma -1.0382\right)/0.0694$; $\sigma$ is the tensile stress, ${V}_{max}$ is maximum vibration velocity. Under the action of blasting vibration, the ultimate tensile strength of the concrete lining structure will be increased in the dynamic response process, and the degree of its tensile strength improvement under the dynamic action is related to the application speed of the load. The static tensile strength and dynamic tensile strength are related as follows: ${\sigma }_{p}={\sigma }_{{p}_{0}}\left[1+0.12\mathrm{l}\mathrm{g}\left({V}_{H}\right)\right]={K}_{D}{\sigma }_{{p}_{0}},$ where: ${\sigma }_{p}$ is dynamic tensile strength of C30 concrete, ${\sigma }_{{p}_{0}}$ is static tensile strength of C30 concrete; ${V}_{H}$ is load rate, ${V}_{H}={\sigma }_{H}/{\sigma }_{1}$, $ {\sigma }_{H}$ is arbitrary loading speed, ${\sigma }_{1}$ is loading speed; ${K}_{D}$ is dynamic strength improvement coefficient. Considering the premature construction time of the civil air defense tunnel and the aging effect of the lining material, the strength of the original concrete is reduced to a certain extent. According to experience, the improvement coefficient of dynamic tensile strength is between 1.24 and 1.48, and 1.35 is taken here. Based on engineering experience, the tensile strength of concrete is 1.78 MPa, so the dynamic tensile strength should be 2.403, and the critical vibration velocity of blasting is 19.62 cm/s. 5. Conclusions With the increase of detonation distance, PPV gradually decreases, and the maximum PPV appears above the detonation source. Based on the analysis of the attenuation law of vibration velocity in the excavated area, it is found that the section size of the excavated cavity and the distance between the blasting center have a great influence on the attenuation rate of blasting vibration velocity. The frequency distribution of blasting the main shock wave in $X$, $Y$, and $Z$ directions is different, and the main frequency distribution interval is between 100-140 Hz. The stress near the explosion source is large and the relative reduction is also large. By fitting the relationship between blasting vibration velocity and maximum principal stress, the criterion of safe vibration velocity based on tensile strength is obtained, and the safe threshold of vibration velocity is 19.62 cm/s, which can be considered that the blasting does not influence the structure. This paper mainly studies the stability of civil defense tunnels under blasting action based on field tests and numerical simulation and is limited to many aspects of theoretical research. Therefore, more theoretical research ideas should be added in future research. • C. Shi, Y. Zhao, C. Zhao, Y. Lou, X. Sun, and X. Zheng, “Water-sealed blasting control measures of the metro station undercrossing existing structures in ultra-close distances: a case study,” Frontiers in Earth Science, Vol. 10, pp. 450–458, Mar. 2022, https://doi.org/10.3389/feart.2022.848913 • Q. S. Li, Z. L. Tian, C. S. Wang, K. Xu, and J. H. Xie, “Analysis of dynamic influence of new tunnel blasting Construction on existing rail transit section tunnels,” Journal of Building Construction, Vol. 42, No. 3, pp. 454–457, 2020. • N. Liu, P. Zhang, K. Li, and Y. Q. Yu, “Massively parallel numerical simulation of 3D shock wave propagation based on JASMIN framework,” Journal of Physics: Conference Series, Vol. 2478, No. 2, p. 022022, Jun. 2023, https://doi.org/10.1088/1742-6596/2478/2/022022 • B. W. Hu, Z. Z. Chen, and Y. B. Liu, “Study on Impact of tunnel blasting vibration on safety of adjacent buildings,” Highway, Vol. 3, pp. 232–234, 2012. • B. Wu, Y. B. Lan, and J. X. Yang, “Study on Effect of blasting in new tunnel on vibration characteristics of adjacent tunnel,” China Safety Science Journal, Vol. 29, No. 11, pp. 89–95, 2019. • Y. Fan, “Analysis of influence of newly built tunnel excavation on existing tunnel structure with small clear distance,” Fujian Building Materials, Vol. 6, pp. 4–6, 2019. • Z. S. Wu, Pp. S. Chen, and W. Wang, “Influence and safety criterion of expansion tunnel blasting on existing lining,” Water Resources and Hydropower Technology, Vol. 51, No. 5, pp. 77–85, 2019. • J. Mandal, A. K. Agarwal, and M. D. Goel, “Numerical modeling of shallow buried tunnel subject to surface blast loading,” Journal of Performance of Constructed Facilities, Vol. 34, No. 6, pp. 1–14, 2020. • A. Mottahedi, F. Sereshki, and M. Ataei, “Development of overbreak prediction models in drill and blast tunneling using soft computing methods,” Engineering with Computers, Vol. 34, No. 1, pp. 45–58, Apr. 2017, https://doi.org/10.1007/s00366-017-0520-3 • S. Mishra and T. Chakraborty, “Determination of high-strain-rate stress-strain response of granite for blast analysis of tunnels,” Journal of Engineering Mechanics, Vol. 8, pp. 81–145, 2019. • H. Mirels and J. F. Mullen, “Aerodynamic blast simulation in hypersonic tunnels,” AIAA Journal, Vol. 3, No. 11, pp. 2103–2108, Nov. 1965, https://doi.org/10.2514/3.3321 • S. Koneshwaran, D. Pp. Thambiratnam, and C. Gallage, “Performance of buried tunnels subjected to surface blast incorporating fluid-structure interaction,” Journal of Performance of Constructed Facilities, Vol. 29, No. 1, pp. 1–21, 2021. • C. Pany et al., “On the bursting of an HSLA steel rocket motor case during proof pressure testing,” Steel Grips (Journal of Steel and related Materials) Application, Vol. 10, pp. 434–438, 2012. • M. G. Sreelakshmi and C. Pany, “Stress analysis of metallic pressure vessels with circumferential mismatch using finite element method,” International Journal of Science and Engineering Research, Vol. 7, No. 4, pp. 479–484, 2016. • H. G. Ni, Z. L. Pan, K. Sun, and Y. L. Zhang, “Field measurement and numerical simulation of dynamic response of masonry structure under ground motion of blasting,” Building Structures, Vol. 50, pp. 122–126, 2003. • C. H. Zhang, “Vibration velocity threshold analysis of existing tunnel lining caused by adjacent tunnel blasting construction,” Building Structures, Vol. 2, pp. 1–20, 2009. About this article Seismic engineering and applications subway tunnel blasting effect civil air defense tunnel The study was sponsored by the National Natural Science Foundation of China (Grant No. 52308393). Data Availability The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request. Author Contributions Conceptualization, Li Yaoxin,Wang Zhibin, Qiqi Luo, Tingyao Wu, methodology, Li Yaoxin, Tingyao Wu; software, Li Yaoxin; validation, Wang Zhibin; writing-original draft preparation, Wang Zhibin ,Tingyao Wu; writing- review and editing, Tingyao Wu; project administration, Tingyao Wu; funding acquisition, Tingyao Wu. Conflict of interest The authors declare that they have no conflict of interest. Copyright © 2024 Yaoxin Li, et al. This is an open access article distributed under the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.	{"url":"https://www.extrica.com/article/23892","timestamp":"2024-11-08T07:39:47Z","content_type":"text/html","content_length":"142903","record_id":"<urn:uuid:9c246abd-48b3-4053-bd84-8039c28ffb1d>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00274.warc.gz"}
Student Colloquium: Observables and Representation Theory, v/Frederik Rosendal Rytter Info about event Monday 4 November 2024, at 15:15 - 16:00 Supervisor: Georg Bruun Representation theory is the study of how we can represent abstract algebraic objects by linear transformations on vector spaces. The hope being that we can transform complicated algebraic problems, into simpler linear algebra problems. This field is widely used in Physics, but we will focus on its use in studying ob-servables in non-relativistic quantum mechanics. We will find that representations are natural objects to study, when working with symmetries of quantum systems. We will see that if a representation commutes with of our Hamiltonian on our Hilbert space, then we get a well-structured way of studying the degeneracy of the energy levels. In fact we will see, that the mere presence of a representation on our Hilbert space, provides us with observables to study, such as xˆ and pˆ. We know that some observables has a discrete spectrum, while others has a continuous one. Representation theory can explain this using the simplest Lie group of them all, the circle (U(1)).	{"url":"https://phys.au.dk/en/news/item/artikel/studenterkollokvium-observables-and-representation-theory-vfrederik-rosendal-rytter","timestamp":"2024-11-11T20:46:52Z","content_type":"text/html","content_length":"30608","record_id":"<urn:uuid:3a1d7311-4c83-4cd4-8158-5e0ce378172f>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00208.warc.gz"}
10.10: Reciprocity Last updated 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 term “reciprocity” refers to a class of theorems that relate the inputs and outputs of a linear system to those of an identical system in which the inputs and outputs are swapped. The importance of reciprocity in electromagnetics is that it simplifies problems that would otherwise be relatively difficult to solve. An example of this is the derivation of the receiving properties of antennas, which is addressed in other sections using results derived in this section. As an initial and relatively gentle introduction, consider a well-known special case that emerges in basic circuit theory: The two-port device shown in Figure $\PageIndex{1}$. The two-port is said to be reciprocal if the voltage $v_2$ appearing at port 2 due to a current applied at port 1 is the same as $v_1$ when the same current is applied instead at port 2. This is normally the case when the two-port consists exclusively of linear passive devices such as ideal resistors, capacitors, and inductors. The fundamental underlying requirement is linearity: That is, outputs must be proportional to inputs, and superposition must apply. This result from basic circuit theory is actually a special case of a more general theorem of electromagnetics, which we shall now derive. Figure $\PageIndex{2}$ shows a scenario in which a current distribution $\widetilde{\bf J}_1$ is completely contained within a volume $\mathcal{V}$. CC BY-SA 4.0; C. Wang) This current is expressed as a volume current density, having SI base units of A/m$^2$. Also, the current is expressed in phasor form, signaling that we are considering a single frequency. This current distribution gives rise to an electric field intensity $\widetilde{\bf E}_1$, having SI base units of V/m. The volume may consist of any combination of linear time-invariant matter; i.e., permittivity $\epsilon$, permeability $\mu$, and conductivity $\sigma$ are constants that may vary arbitrarily with position but not with time. Here’s a key idea: We may interpret this scenario as a “two-port” system in which $\widetilde{\bf J}_1$ is the input, $\widetilde{\bf E}_1$ is the output, and the system’s behavior is completely defined by Maxwell’s equations in differential phasor form: \[\nabla \times \widetilde{\bf E}_1 = -j\omega\mu\widetilde{\bf H}_1 \label{m0214_eMCE1} \] \[\nabla \times \widetilde{\bf H}_1 = \widetilde{\bf J}_1 + j\omega\epsilon\widetilde{\bf E}_1 \label{m0214_eMCH1} \] along with the appropriate electromagnetic boundary conditions. (For the purposes of our present analysis, $\widetilde{\bf H}_1$ is neither an input nor an output; it is merely a coupled quantity that appears in this particular formulation of the relationship between $\widetilde{\bf E}_1$ and $\widetilde{\bf J}_1$.) Figure $\PageIndex{3}$ shows a scenario in which a different current distribution $\widetilde{\bf J}_2$ is completely contained within the same volume $\mathcal{V}$ containing the same distribution of linear matter. This new current distribution gives rise to an electric field intensity $\widetilde{\bf E}_2$. The relationship between $\widetilde{\bf E}_2$ and $\widetilde{\bf J}_2$ is governed by the same equations: \[\nabla \times \widetilde{\bf E}_2 = -j\omega\mu\widetilde{\bf H}_2 \label{m0214_eMCE2} \] \[\nabla \times \widetilde{\bf H}_2 = \widetilde{\bf J}_2 + j\omega\epsilon\widetilde{\bf E}_2 \label{m0214_eMCH2} \] along with the same electromagnetic boundary conditions. Thus, we may interpret this scenario as the same electromagnetic system depicted in Figure $\PageIndex{2}$, except now with $\widetilde{\bf J}_2$ as the input and $\widetilde{\bf E}_2$ is the output. The input current and output field in the second scenario are, in general, completely different from the input current and output field in the first scenario. However, both scenarios involve precisely the same electromagnetic system; i.e., the same governing equations and the same distribution of matter. This leads to the following question: Let’s say you know nothing about the system, aside from the fact that it is linear and time-invariant. However, you are able to observe the first scenario; i.e., you know $\widetilde{\bf E}_1$ in response to $\widetilde{\bf J}_1$. Given this very limited look into the behavior of the system, what can you infer about $\widetilde{\bf E}_2$ given $\widetilde{\bf J}_2$, or vice-versa? At first glance, the answer might seem to be nothing, since you lack a description of the system. However, the answer turns out to be that a surprising bit more information is available. This information is provided by reciprocity. To show this, we must engage in some pure mathematics. Derivation of the Lorentz reciprocity theorem First, let us take the dot product of $\widetilde{\bf H}_2$ with each side of Equation \ref{m0214_eMCE1}: \[\widetilde{\bf H}_2 \cdot \left( \nabla \times \widetilde{\bf E}_1 \right) = -j\omega\mu\widetilde{\bf H}_1 \cdot \widetilde{\bf H}_2 \label{m0214_e5} \] Similarly, let us take the dot product of $\widetilde{\bf E}_1$ with each side of Equation \ref{m0214_eMCH2}: \[\widetilde{\bf E}_1 \cdot \left( \nabla \times \widetilde{\bf H}_2 \right) = \widetilde{\bf E}_1 \cdot \widetilde{\bf J}_2 +j\omega\epsilon\widetilde{\bf E}_1 \cdot \widetilde{\bf E}_2 \label {m0214_e6} \] Next, let us subtract Equation \ref{m0214_e5} from Equation \ref{m0214_e6}. The left side of the resulting equation is \[\widetilde{\bf E}_1 \cdot \left( \nabla \times \widetilde{\bf H}_2 \right) - \widetilde{\bf H}_2 \cdot \left( \nabla \times \widetilde{\bf E}_1 \right) \nonumber \] This can be simplified using the vector identity (Appendix 12.3): \[\nabla \cdot \left({\bf A}\times {\bf B}\right) = {\bf B}\cdot\left(\nabla \times{\bf A}\right)-{\bf A}\cdot\left(\nabla \times{\bf B}\right) \nonumber \] \[\widetilde{\bf E}_1 \cdot \left( \nabla \times \widetilde{\bf H}_2 \right) - \widetilde{\bf H}_2 \cdot \left( \nabla \times \widetilde{\bf E}_1 \right) = \nabla \cdot \left(\widetilde{\bf H}_2 \ times \widetilde{\bf E}_1 \right) \nonumber \] So, by subtracting Equation \ref{m0214_e5} from Equation \ref{m0214_e6}, we have found: \[\nabla \cdot \left(\widetilde{\bf H}_2 \times \widetilde{\bf E}_1 \right) = \widetilde{\bf E}_1 \cdot \widetilde{\bf J}_2 +j\omega\epsilon\widetilde{\bf E}_1 \cdot \widetilde{\bf E}_2 +j\omega\mu \ widetilde{\bf H}_1 \cdot \widetilde{\bf H}_2 \label{m0214_e7} \] Next, we repeat the process represented by Equations \ref{m0214_e5}-\ref{m0214_e7} above to generate a complementary equation to Equation \ref{m0214_e7}. This time we take the dot product of $\ widetilde{\bf E}_2$ with each side of Equation \ref{m0214_eMCH1}: \[\widetilde{\bf E}_2 \cdot \left( \nabla \times \widetilde{\bf H}_1 \right) = \widetilde{\bf E}_2 \cdot \widetilde{\bf J}_1 +j\omega\epsilon\widetilde{\bf E}_1 \cdot \widetilde{\bf E}_2 \label {m0214_e8} \] Similarly, let us take the dot product of $\widetilde{\bf H}_1$ with each side of Equation \ref{m0214_eMCE2}: \[\widetilde{\bf H}_1 \cdot \left( \nabla \times \widetilde{\bf E}_2 \right) = -j\omega\mu\widetilde{\bf H}_1 \cdot \widetilde{\bf H}_2 \label{m0214_e9} \] Next, let us subtract Equation \ref{m0214_e9} from Equation \ref{m0214_e8}. Again using the vector identity, the left side of the resulting equation is \[\widetilde{\bf E}_2 \cdot \left( \nabla \times \widetilde{\bf H}_1 \right) - \widetilde{\bf H}_1 \cdot \left( \nabla \times \widetilde{\bf E}_2 \right) = \nabla \cdot \left(\widetilde{\bf H}_1 \ times \widetilde{\bf E}_2 \right) \nonumber \] So we find: \[\nabla \cdot \left(\widetilde{\bf H}_1 \times \widetilde{\bf E}_2 \right) = \widetilde{\bf E}_2 \cdot \widetilde{\bf J}_1 +j\omega\epsilon\widetilde{\bf E}_1 \cdot \widetilde{\bf E}_2 +j\omega\mu \ widetilde{\bf H}_1 \cdot \widetilde{\bf H}_2 \label{m0214_e10} \] Finally, subtracting Equation \ref{m0214_e10} from Equation \ref{m0214_e7}, we obtain: \[\nabla \cdot \left(\widetilde{\bf H}_2 \times \widetilde{\bf E}_1 - \widetilde{\bf H}_1 \times \widetilde{\bf E}_2 \right) = \widetilde{\bf E}_1 \cdot \widetilde{\bf J}_2 - \widetilde{\bf E}_2 \ cdot \widetilde{\bf J}_1 \label{m0214_eLRTD} \] This equation is commonly known by the name of the theorem it represents: The Lorentz reciprocity theorem. The theorem is a very general statement about the relationship between fields and currents at each point in space. An associated form of the theorem applies to contiguous regions of space. To obtain this form, we simply integrate both sides of Equation \ref{m0214_eLRTD} over the volume $\ \int_{\mathcal V} {\nabla \cdot \left(\widetilde{\bf H}_2 \times \widetilde{\bf E}_1 - \widetilde{\bf H}_1 \times \widetilde{\bf E}_2 \right) } dv \nonumber \\ = \int_{\mathcal V} {\left( \widetilde {\bf E}_1 \cdot \widetilde{\bf J}_2 - \widetilde{\bf E}_2 \cdot \widetilde{\bf J}_1 \right)} dv We now take the additional step of using the divergence theorem (Appendix 12.3) to transform the left side of the equation into a surface integral: \oint_{\mathcal S} {\left(\widetilde{\bf H}_2 \times \widetilde{\bf E}_1 - \widetilde{\bf H}_1 \times \widetilde{\bf E}_2 \right) \cdot d{\bf s}} \nonumber \\ = \int_{\mathcal V} {\left( \widetilde{\ bf E}_1 \cdot \widetilde{\bf J}_2 - \widetilde{\bf E}_2 \cdot \widetilde{\bf J}_1 \right)} dv \label{m0214_eLRTI} where \(\mathcal{S}$ is the closed mathematical surface which bounds $\mathcal{V}$. This is also the Lorentz reciprocity theorem, but now in integral form. This version of the theorem relates fields on the bounding surface to sources within the volume. The integral form of the theorem has a particularly useful feature. Let us confine the sources to a finite region of space, while allowing $\mathcal{V}$ to grow infinitely large, expanding to include all space. In this situation, the closest distance between any point containing non-zero source current and $\mathcal{S}$ is infinite. Because field magnitude diminishes with distance from the source, the fields $(\widetilde{\bf E}_1,\widetilde{\bf H}_1)$ and $(\widetilde{\bf E}_2,\widetilde{\bf H}_2)$ are all effectively zero on $\mathcal{S}$. In this case, the left side of Equation \ref{m0214_eLRTI} is zero, and we find: \[\boxed{ \int_{\mathcal V} {\left( \widetilde{\bf E}_1 \cdot \widetilde{\bf J}_2 - \widetilde{\bf E}_2 \cdot \widetilde{\bf J}_1 \right)} dv = 0 } \label{m0214_eLRTI2} \] for any volume $\mathcal{V}$ which contains all the current. The Lorentz reciprocity theorem (Equation \ref{m0214_eLRTI2}) describes a relationship between one distribution of current and the resulting fields, and a second distribution of current and resulting fields, when both scenarios take place in identical regions of space filled with identical distributions of linear matter. Why do we refer to this relationship as reciprocity? Simply because the expression is identical when the subscripts “1” and “2” are swapped. In other words, the relationship does not recognize a distinction between “inputs” and “outputs;” there are only “ports.” A useful special case pertains to scenarios in which the current distributions $\widetilde{\bf J}_1$ and $\widetilde{\bf J}_2$ are spatially disjoint. By “spatially disjoint,” we mean that there is no point in space at which both $\widetilde{\bf J}_1$ and $\widetilde{\bf J}_2$ are non-zero; in other words, these distributions do not overlap. (Note that the currents shown in Figures $\ PageIndex{2}$ and $\PageIndex{3}$ are depicted as spatially disjoint.) To see what happens in this case, let us first rewrite Equation \ref{m0214_eLRTI2} as follows: \[\int_{\mathcal V} { \widetilde{\bf E}_1 \cdot \widetilde{\bf J}_2 ~dv } = \int_{\mathcal V} { \widetilde{\bf E}_2 \cdot \widetilde{\bf J}_1 ~dv } \label{m0214_eLRTI3} \] Let $\mathcal{V}_1$ be the contiguous volume over which $\widetilde{\bf J}_1$ is non-zero, and let $\mathcal{V}_2$ be the contiguous volume over which $\widetilde{\bf J}_2$ is non-zero. Then Equation \ref{m0214_eLRTI3} may be written as follows: \[\int_{{\mathcal V}_2} { \widetilde{\bf E}_1 \cdot \widetilde{\bf J}_2 ~dv } = \int_{{\mathcal V}_1} { \widetilde{\bf E}_2 \cdot \widetilde{\bf J}_1 ~dv } \label{m0214_eLRTI4} \] The utility of this form is that we have reduced the region of integration to just those regions where the current exists. Reciprocity of two-ports consisting of antennas Equation \ref{m0214_eLRTI4} allows us to establish the reciprocity of two-ports consisting of pairs of antennas. This is most easily demonstrated for pairs of thin dipole antennas, as shown in Figure CC BY-SA 4.0 ; C. Wang) Here, port 1 is defined by the terminal quantities $(\widetilde{V}_1,\widetilde{I}_1)$ of the antenna on the left, and port 2 is defined by the terminal quantities $(\widetilde{V}_2,\widetilde{I} _2)$ of the antenna on the right. These quantities are defined with respect to a small gap of length $\Delta l$ between the perfectly-conducting arms of the dipole. Either antenna may transmit or receive, so $(\widetilde{V}_1,\widetilde{I}_1)$ depends on $(\widetilde{V}_2,\widetilde{I}_2)$, and vice-versa. The transmit and receive cases for port 1 are illustrated in Figure $\PageIndex{5}$(a) and (b), respectively. CC BY-SA 4.0 ; C. Wang) Note that $\widetilde{\bf J}_1$ is the current distribution on this antenna when transmitting (i.e., when driven by the impressed current $\widetilde{I}_1^t$) and $\widetilde{\bf E}_2$ is the electric field incident on the antenna when the other antenna is transmitting. Let us select $\mathcal{V}_1$ to be the cylindrical volume defined by the exterior surface of the dipole, including the gap between the arms of the dipole. We are now ready to consider the right side of Equation \ref{m0214_eLRTI4}: \[\int_{{\mathcal V}_1} { \widetilde{\bf E}_2 \cdot \widetilde{\bf J}_1 ~dv } \nonumber \] The electromagnetic boundary conditions applicable to the perfectly-conducting dipole arms allow this integral to be dramatically simplified. First, recall that all current associated with a perfect conductor must flow on the surface of the material, and therefore $\widetilde{\bf J}_1=0$ everywhere except on the surface. Therefore, the direction of $\widetilde{\bf J}_1$ is always tangent to the surface. The tangent component of $\widetilde{\bf E}_2$ is zero on the surface of a perfectly-conducting material, as required by the applicable boundary condition. Therefore, $\widetilde{\bf E}_2\cdot\widetilde{\bf J}_1=0$ everywhere $\widetilde{\bf J}_1$ is non-zero. There is only one location where the current is non-zero and yet there is no conductor: This location is the gap located precisely at the terminals. Thus, we find: \[\int_{{\mathcal V}_1} { \widetilde{\bf E}_2 \cdot \widetilde{\bf J}_1 ~dv} = \int_{gap} { \widetilde{\bf E}_2 \cdot \widetilde{I}_1^t \hat{\bf l}_1 ~dl } \nonumber \] where the right side is a line integral crossing the gap that defines the terminals of the antenna. We assume the current $\widetilde{I}_1^t$ is constant over the gap, and so may be factored out of the integral: \[\int_{{\mathcal V}_1} { \widetilde{\bf E}_2 \cdot \widetilde{\bf J}_1 ~dv} = \widetilde{I}_1^t \int_{gap} { \widetilde{\bf E}_2 \cdot \hat{\bf l}_1 ~dl } \nonumber \] Recall that potential between two points in space is given by the integral of the electric field over any path between those two points. In particular, we may calculate the open-circuit potential $\ widetilde{V}_1^r$ as follows: \[\widetilde{V}_1^r = -\int_{gap} { \widetilde{\bf E}_2 \cdot \hat{\bf l}_1 ~dl } \nonumber \] Remarkably, we have found: \[\int_{{\mathcal V}_1} { \widetilde{\bf E}_2 \cdot \widetilde{\bf J}_1 ~dv} = -\widetilde{I}_1^t \widetilde{V}_1^r \nonumber \] Applying the exact same procedure for port 2 (or by simply exchanging subscripts), we find: \[\int_{{\mathcal V}_2} { \widetilde{\bf E}_1 \cdot \widetilde{\bf J}_2 ~dv} = -\widetilde{I}_2^t \widetilde{V}_2^r \nonumber \] Now substituting these results into Equation \ref{m0214_eLRTI4}, we find: \[\boxed{ \widetilde{I}_1^t \widetilde{V}_1^r = \widetilde{I}_2^t \widetilde{V}_2^r } \label{m0214_eRIV} \] At the beginning of this section, we stated that a two-port is reciprocal if $\widetilde{V}_2$ appearing at port 2 due to a current applied at port 1 is the same as $\widetilde{V}_1$ when the same current is applied instead at port 2. If the present two-dipole problem is reciprocal, then $\widetilde{V}_2^r$ due to $\widetilde{I}_1^t$ should be the same as $\widetilde{V}_1^r$ when \ (\widetilde{I}_2^t=\widetilde{I}_1^t\). Is it? Let us set $\widetilde{I}_1^t$ equal to some particular value $I_0$, then the resulting value of $\widetilde{V}_2^r$ will be some particular value $V_0$. If we subsequently set $\widetilde{I}_2^t$ equal to $I_0$, then the resulting value of $\widetilde{V}_1^r$ will be, according to Equation \ref{m0214_eRIV}: \[\widetilde{V}_1^r = \frac{ \widetilde{I}_2^t \widetilde{V}_2^r }{ \widetilde{I}_1^t } = \frac{ I_0 V_0 }{ I_0 } = V_0 \nonumber \] Therefore, Equation \ref{m0214_eRIV} is simply a mathematical form of the familiar definition of reciprocity from basic circuit theory, and we have found that our system of antennas is reciprocal in precisely this same sense. The above analysis presumed pairs of straight, perfectly conducting dipoles of arbitrary length. However, the result is readily generalized – in fact, is the same – for any pair of passive antennas in linear time-invariant media. Summarizing: The potential induced at the terminals of one antenna due to a current applied to a second antenna is equal to the potential induced in the second antenna by the same current applied to the first antenna (Equation \ref{m0214_eRIV}). Additional Reading:	{"url":"https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Electromagnetics_II_(Ellingson)/10%3A_Antennas/10.10%3A_Reciprocity","timestamp":"2024-11-06T04:27:04Z","content_type":"text/html","content_length":"148423","record_id":"<urn:uuid:a40944d4-dcd7-4b3e-8172-146d3d7e81e3>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00516.warc.gz"}
fronts.solve_from_guess(D, i, b, o_guess, guess, radial=False, max_nodes=1000, verbose=0) Alternative solver for problems with a Dirichlet boundary condition. Given a positive function D, scalars $\theta_i$, $\theta_b$ and $o_b$, and coordinate unit vector $\mathbf{\hat{r}}$, finds a function $\theta$ of r and t such that: \[\begin{split}\begin{cases} \dfrac{\partial\theta}{\partial t} = \nabla\cdot\left[D(\theta))\dfrac{\partial\theta}{\partial r} \mathbf{\hat{r}}\right ] & r>r_b(t),t>0\\ \theta(r, 0) = \theta_i & r>0 \\ \theta(r_b(t), t) = \theta_b & t>0 \\ r_b(t) = o_b\sqrt t \end{cases}\end{split}\] Alternative to the main solve() function. This function requires a starting mesh and guess of the solution. It is significantly less robust than solve(), and will fail to converge in many cases that the latter can easily handle (whether it converges will usually depend heavily on the problem, the starting mesh and the guess of the solution; it will raise a RuntimeError on failure). However, when it converges it is usually faster than solve(), which may be an advantage for some use cases. You should nonetheless prefer solve() unless you have a particular use case for which you have found this function to be better. Possible use cases include refining a solution (note that solve() can do that too), optimization runs in which known solutions make good first approximations of solutions with similar parameters and every second of computing time counts, and in the implementation of other solving algorithms. In all these cases, solve() should probably be used as a fallback for when this function fails. ☆ D (callable or sympy.Expression or str or float) – Callable that evaluates $D$ and its derivatives, obtained from the fronts.D module or defined in the same manner—i.e.: ■ D(theta) evaluates and returns $D$ at theta ■ D(theta, 1) returns both the value of $D$ and its first derivative at theta ■ D(theta, 2) returns the value of $D$, its first derivative, and its second derivative at theta where theta may be a single float or a NumPy array. Alternatively, instead of a callable, the argument can be the expression of $D$ in the form of a string or sympy.Expression with a single variable. In this case, the solver will differentiate and evaluate the expression as necessary. ☆ i (float) – Initial condition, $\theta_i$. ☆ b (float) – Imposed boundary value, $\theta_b$. ☆ o_guess (numpy.array_like, shape (n_guess,)) – Starting mesh in terms of the Boltzmann variable. Must be strictly increasing. o_guess[0] is taken as the value of the parameter $o_b$, which determines the behavior of the boundary. If zero, it implies that the boundary always exists at $r=0$. It must be strictly positive if radial is not False. A non-zero value implies a moving boundary. On the other end, o_guess[-1] must be large enough to contain the solution to the semi-infinite problem. ☆ guess (float or numpy.array_like, shape (n_guess,)) – Starting guess of the solution at the points in o_guess. If a single value, the guess is assumed uniform. ☆ radial ({False, 'cylindrical', 'polar', 'spherical'}, optional) – Choice of coordinate unit vector $\mathbf{\hat{r}}$. Must be one of the following: ■ False (default): $\mathbf{\hat{r}}$ is any coordinate unit vector in rectangular (Cartesian) coordinates, or an axial unit vector in a cylindrical coordinate system ■ 'cylindrical' or 'polar': $\mathbf{\hat{r}}$ is the radial unit vector in a cylindrical or polar coordinate system ■ 'spherical': $\mathbf{\hat{r}}$ is the radial unit vector in a spherical coordinate system ☆ max_nodes (int, optional) – Maximum allowed number of mesh nodes. ☆ verbose ({0, 1, 2}, optional) – Level of algorithm’s verbosity. Must be one of the following: ■ 0 (default): work silently ■ 1: display a termination report ■ 2: also display progress during iterations solution – See Solution for a description of the solution object. Additional fields specific to this solver are included in the object: ○ o (numpy.ndarray, shape (n,)) – Final solver mesh, in terms of the Boltzmann variable. ○ niter (int) – Number of iterations required to find the solution. ○ rms_residuals (numpy.ndarray, shape (n-1,)) – RMS values of the relative residuals over each mesh interval. Return type: This function works by transforming the partial differential equation with the Boltzmann transformation using ode() and then solving the resulting ODE with SciPy’s collocation-based boundary value problem solver scipy.integrate.solve_bvp() and a two-point Dirichlet condition that matches the boundary and initial conditions of the problem. Upon that solver’s convergence, it runs a final check on whether the candidate solution also satisfies the semi-infinite condition (which implies $d\theta/do\to0$ as $o\to\infty$).	{"url":"https://fronts.readthedocs.io/en/v1.2.4/stubs/fronts.solve_from_guess.html","timestamp":"2024-11-06T16:56:47Z","content_type":"text/html","content_length":"22504","record_id":"<urn:uuid:ba2a3512-2db9-4788-af60-2925ef8327d9>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00134.warc.gz"}
Question #b8418 \| Socratic Question #b8418 1 Answer Area of the rectangular frame $A = l \cdot w = \textcolor{g r e e n}{\left(\frac{3}{8}\right) = 0.375}$ sq yard Base of the rectangular frame $l = \frac{3}{4}$ yard Height of the rectangular frame $w = \frac{1}{2}$ yard Area of the rectangular frame $A = l \cdot w = \left(\frac{3}{4}\right) \cdot \left(\frac{1}{2}\right) = \frac{3 \cdot 1}{4 \cdot 2} = \textcolor{g r e e n}{\left(\frac{3}{8}\right) = 0.375}$ sq yard Impact of this question 785 views around the world	{"url":"https://api-project-1022638073839.appspot.com/questions/5a84e4e9b72cff3231db8418#552092","timestamp":"2024-11-03T22:02:23Z","content_type":"text/html","content_length":"32025","record_id":"<urn:uuid:c40647cb-33cb-444d-a643-7cbcfab59b0a>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00765.warc.gz"}
Oedometer Test with CYSoil Model FLAC3D Theory and Background • Constitutive Models Oedometer Test with CYSoil Model To view this project in FLAC3D, use the menu command . The project’s main data file is shown at the end of this example. Oedometer test simulations are carried out using the CYSoil model for different values of the parameter $\alpha$ to evaluate the impact of the cap aspect ratio on confining stress when yielding occurs on the cap. The values of $\alpha$ considered for the tests are 0.5, 1, 1.5, and 10^20 (i.e., a value that is very large compared to 1, in which case the behavior of the double-yield model is recovered). We use the same setup and properties (apart from $\alpha$) as in the isotropic compression example, except in this case, the simulations are run in plane strain, with fixed lateral boundaries to simulate oedometer test conditions. Friction is assigned a large value to prevent shear yielding. The stress ratio, $K_0$, of confining stress to vertical stress, $\sigma_{xx}$ / $\sigma_{yy}$, is plotted versus axial strain for dense, medium, and loose sand in three figures: Figure 1 compares predictions for $\alpha$ = 0.5 and $\alpha$ = $10^{20}$ (double-yield model); Figure 2 compares predictions for $\alpha$ = 1.0 and $\alpha$ = $10^{20}$ (double-yield model); and Figure 3 shows plots for $\alpha$ = 1.5 and $\alpha$ = $10^{20}$ (double-yield model). The results of the oedometer simulations show that, with the given properties, a higher value of $K_0$ is achieved for the CYSoil model than the double-yield model. And for all tests, the lower the aspect ratio of the cap, the higher the $K_0$ that is achieved. Also, dense, medium, and loose sands converge to the same ultimate $K_0$ value as deformation takes place, and they do so at a faster deformation rate for the CYSoil model than the double-yield model. Data File model new model title "Oedometer test - dense, medium, loose sand - CYSoil and DY" model large-strain off zone create brick size 3 1 5 zone group 'gcy' range position-x 0 1 zone group 'gdy' range position-x 2 3 zone group 'gnull' range union position-x 1 2 position-z 1 2 position-z 3 4 [global _pc0 = 10.] zone cmodel assign cap-yield range group 'gcy' zone property density=1000 pressure-reference=100. poisson=0.2 ... multiplier=[2.0/3.0] flag-cap=1 flag-shear=1 friction=89.0 ... pressure-cap=[_pc0] pressure-initial=[_pc0] range group 'gcy' zone cmodel double-yield range group 'gdy' zone property density=1000 bulk-maximum=1.e15 shear-maximum=0.75e15 ... friction=89.0 pressure-cap=[_pc0] multiplier=[2.0/3.0] ... range group 'gdy' zone null range group 'gnull' program call 'fish' suppress ; double-yield [cap_table('1', 300.)] zone property strain-volumetric-plastic=[evp0] table-pressure-cap='1' ... range group 'gdy' position-z 4 5 [cap_table('2', 225.)] zone property strain-volumetric-plastic=[evp0] table-pressure-cap='2' ... range group 'gdy' position-z 2 3 [cap_table('3', 150.)] zone property strain-volumetric-plastic=[evp0] table-pressure-cap='3' ... range group 'gdy' position-z 0 1 ; cap-yield-soil zone property shear-reference 300. range group 'gcy' position-z 4 5 zone property shear-reference 225. range group 'gcy' position-z 2 3 zone property shear-reference 150. range group 'gcy' position-z 0 1 zone gridpoint fix velocity zone gridpoint initialize velocity-z -1.25e-7 ... range union position-z 1 position-z 3 position-z 5 zone initialize stress xx [-_pc0] yy [-_pc0] zz [-_pc0] history interval 100 zone history displacement-z position (0,0,1) fish history _k0_d_cy fish history _k0_m_cy fish history _k0_l_cy fish history _k0_d_dy fish history _k0_m_dy fish history _k0_l_dy model save 'ini' ; -- alpha = 0.5 zone property alpha=0.5 range group 'gcy' model step 60000 model save 'al1' ; -- alpha = 1.0 model restore 'ini' zone property alpha=1.0 range group 'gcy' model step 60000 model save 'al2' ; -- alpha = 1.5 model restore 'ini' zone property alpha=1.5 range group 'gcy' model step 60000 model save 'al3' Was this helpful? ... Itasca Software © 2024, Itasca Updated: Aug 13, 2024	{"url":"https://docs.itascacg.com/itasca910/flac3d/zone/test3d/ConstitutiveModels/OedometerCYSoil/oedometercysoil.html","timestamp":"2024-11-03T00:44:41Z","content_type":"application/xhtml+xml","content_length":"26139","record_id":"<urn:uuid:f5114f2e-c354-4262-8e6e-fb27d2441a93>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00033.warc.gz"}
Heap Sort Sorting as you might already know is basically an algorithm that is used for arranging elements of a list in a certain order. (Usually ascending or descending). Sorting is one of the most important categories of algorithms, it can significantly reduce the complexity of problems, and is generally used for efficient searching. There are an ample number of sorting algorithms available like the Bubble sort, Selection sort, Insertion sort, Merge sort, Quick sort, Heap sort, Counting sort and more. The type of algorithm to choose depends on the type of problem. (Generally Merge Sort and Quick sort are used.) Today we will be discussing Heap Sort. Now before directly jumping to Heap Sort, we must be aware of a few terminologies. 1. Complete Binary Tree We can define a complete binary tree as a tree in which every level is completely filled except possibly the last, and it is as left as possible. 2. Binary Heap A binary heap is a complete binary tree in which the value of parent is greater or lesser than its children. If the value of parent is greater than its children, then it’s called max-heap else we call it min-heap. We can represent the heap as a binary tree or an array. Array Representation of Heap: As we have already discussed that heap is a type of complete binary tree, therefore it is easy to represent it as an array. Let’s suppose that the parent node is at index i Then the left child will be at (2 * i) + 1 And the right child will be at (2 * i) + 2 Parent index= 0 Left child index= 20 + 1 = 1 Right child index= 20 +2=2 Parent index= 1 Left child index= 21 + 1 = 3 Right child index= 21 +2=4 Parent index= 2 Left child index= 22 + 1 = 5 Right child index= 22 +2=6 Heap sort algorithm is a comparison based sorting technique, it’s basic working is similar to that of insertion sort. It is an in-place sorting algorithm but is not stable, that is, the original order of keys is not maintained. Understanding the algorithm In the heap sort algorithm, we insert all the elements from the unsorted list or array into a heap. We then create max-heap which brings the largest element at the root of the heap, we exchange this value with the last value and then decrement size of the array. Then, we heapify the first element. This process is continued until there is only one element left in the array. Java Code for Heap Sort import java.util.Arrays; public class Sorting { public void heapSort(int arr[]) { int lengthOfArray = arr.length; // creating heap for (int i = (lengthOfArray - 1) / 2; i >= 0; i--) { heapify(arr, lengthOfArray, i); // Sorting for (int i = lengthOfArray - 1; i >= 0; i--) { // Swap the root node with last node int temp = arr[i]; arr[i] = arr[0]; arr[0] = temp; heapify(arr, i, 0); public void heapify(int[] arr, int index, int i) { // Initializing parent and children int parentIndex = i; int leftChild = (2 * i) + 1; int rightChild = (2 * i) + 2; // comparing the left child value if (leftChild < index && arr[leftChild] > arr[parentIndex]) { parentIndex = leftChild; // comparing the right child value if (rightChild < index && arr[rightChild] > arr[parentIndex]) { parentIndex = rightChild; if (parentIndex != i) { int temp = arr[parentIndex]; arr[parentIndex] = arr[i]; arr[i] = temp; heapify(arr, index, parentIndex); // recursive call // Driver Code public static void main(String args[]) { Sorting sort = new Sorting(); int[] arr = { System.out.println("Array after applying heap sort is " + Arrays.toString(arr)); The above program will generate the following output: Array after applying heap sort is [1, 7, 9, 11, 13, 24, 38, 46, 76, 79] Performance of Heap Sort • Worst case time complexity: O(nlogn) • Best case time complexity: O(nlogn) • Average case time performance: O(nlogn) • Worst case space complexity: O(n) • Auxiliary space complexity: O(1)	{"url":"https://tutswiki.com/data-structures-algorithms/heap-sort/","timestamp":"2024-11-14T07:56:23Z","content_type":"text/html","content_length":"35307","record_id":"<urn:uuid:9bf37846-baad-416a-851c-69152822872c>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00125.warc.gz"}
Steps to construct a Lorenz curve To construct a Lorenz curve, plot cumulative percentage of total income against cumulative percentage of population. Represent perfect equality as a straight 45-degree line. Calculate the Gini coefficient from the Lorenz curve. Use the formula of Gini coefficient, a quotient of the area between the Lorenz curve and the line of perfect equality divided by the total area under the line of perfect equality. Interpret the Gini coefficient as a measure of income inequality. A higher Gini coefficient indicates greater inequality, while a lower coefficient represents more equality. Illustrate income distribution patterns through the Lorenz curve and Gini coefficient analysis. Table of Contents (How to make a Lorenz Curve in spreadsheet programs like Microsoft Excel or Google Sheets) Constructing a Lorenz curve involves plotting data points to show income distribution inequality. Begin by sorting individual incomes from smallest to largest. Calculate the cumulative percentage of total income for each interval. Plot these points on a graph with cumulative income percentage on the vertical axis. Draw a line from the point (0,0) to (100,100). Connect the plotted points with a line to form the Lorenz curve. The closer the curve to the diagonal line indicates more equal income distribution. To interpret the curve, calculate the Gini coefficient from the area between the Lorenz curve and the diagonal line. A higher Gini coefficient signifies greater inequality. Keep in mind that constructing a Lorenz curve requires accurate data and precise calculations. Understanding the Lorenz curve helps policymakers address income inequality challenges. By visualizing income distribution, societies can implement targeted strategies to promote fairness and equity. Building a Lorenz curve facilitates insightful analysis and informed decision-making for a more just society. Calculation of Cumulative Income and Population Shares To construct a Lorenz curve, one crucial step is calculating the cumulative income and population shares. Let’s dive into this process that unveils insights into economic inequality. Imagine holding in your hands a spreadsheet bursting with data—numbers representing individual incomes lined up row by row like soldiers ready for battle. Each figure tells a unique story of someone’s hard work, success, struggles, or setbacks. Now envision these numbers being meticulously summed up to reveal the total income earned by different segments of the population. As you crunch through these figures, layering one sum onto another to form a pyramid of wealth distribution, emotions may stir within you. The stark reality of disparities comes to light as you see how some wield financial power while others barely scrape by. With each calculation adding weight to your understanding, it becomes clear that societal inequities are not just abstract concepts but tangible realities affecting real people in profound ways every Now shift your focus from incomes to population shares—the intricate tapestry of individuals making up society’s fabric. As you delve into this aspect of the analysis, imagine each person being assigned a fraction representing their presence in the overall populace. The air crackles with tension as you examine these fractions—some large and dominant, reflecting groups wielding influence and resources beyond measure; others tiny and insignificant, mirroring marginalized communities struggling on society’s fringes. Through these calculations, an emotional undercurrent flows—a mixture of empathy for those facing uphill battles against systemic barriers and determination to shine a light on injustices often hidden behind statistical averages. As you plot these cumulative income and population shares on graphs awaiting their stories be told through curves dancing across axes—anxiety lingers at revelations they might bring yet hope blossoms for change sparked by awareness bred from numerical truths laid bare before us all. In conclusion, calculating cumulative income and population shares isn’t merely about manipulating numbers—it’s an act imbued with raw human emotion unveiling complexities shaping our world’s economic landscape. Data Collection When delving into the realm of constructing a Lorenz curve, one vital aspect to consider is data collection. Picture this: you’re embarking on a journey to unravel economic disparities within a society, aiming to capture the essence of income distribution. Your first port of call? Collecting robust and reliable data. Data collection forms the bedrock upon which the Lorenz curve is built – every point plotted represents real individuals’ financial standing in relation to others. It’s not just numbers; it’s stories etched in percentages and figures, reflecting lives lived under varying degrees of prosperity. Imagine setting foot into diverse communities, armed with surveys and questionnaires designed to extract raw truths about income levels. Each response obtained is not merely a statistic but a glimpse into someone’s daily struggle or success story—a mosaic painting revealing societal inequities and triumphs alike. As you traverse through neighborhoods straddling different socio-economic spectra, emotions run high—empathy for those grappling with financial strains juxtaposed against admiration for resilient souls defying odds stacked against them. The data collected isn’t sterile; it embodies struggles, hopes, dreams woven intricately into each entry recorded. Capturing these nuanced narratives requires finesse and sensitivity—an ability to listen intently beyond words spoken; deciphering unspoken cues shaping individuals’ economic realities. Every piece of information gathered holds weight—a puzzle piece fitting snugly into the larger picture depicting income disparity landscapes. In your quest for comprehensive data sets, challenges lurk at every corner—missing entries hinting at unvoiced hardships faced by some participants; outliers skewing results unforeseeably. Navigating these hurdles demands tenacity—a commitment to weaving an accurate tapestry reflective of society’s intricate fabric without overlooking nuances overshadowed by mere statistics. Through sweat-soaked brows and ink-stained fingers tallying responses late into quiet nights, you forge ahead driven by a singular purpose—to unearth hidden truths veiled beneath layers of numerical abstractions meticulously piecing together fragments illuminating paths toward equality. Interpretation and Analysis When it comes to interpreting and analyzing the Lorenz curve, you’re diving deep into a world of economic insights that can be both intriguing and enlightening. Picture this: as you scrutinize the curve, every twist and turn holds a story about income distribution within a particular economy. The interpretation phase is like deciphering a complex puzzle. Each point on the curve represents a certain percentage of the population and their corresponding share of total income. It’s like peering through a window into the socioeconomic makeup of a society, revealing disparities or perhaps surprising levels of equality. Analyzing the Lorenz curve involves more than just crunching numbers; it requires finesse and intuition. You may find yourself pondering over why certain areas under the curve are steeper or flatter than others. These nuances speak volumes about wealth concentration – are there pockets of extreme affluence amidst widespread poverty? Are resources distributed evenly across different segments of As you delve deeper into your analysis, emotions might come into play. The stark reality of inequality displayed by the Lorenz curve can evoke feelings ranging from empathy to outrage. Perhaps you’ll feel inspired to advocate for policies that aim to level the playing field or address systemic injustices highlighted by your findings. Moreover, observing how the Lorenz curve interacts with the line of perfect equality adds another layer to your interpretation. The greater the distance between these two lines, the higher the degree of income inequality in that particular system. This contrast can tug at your heartstrings as you witness firsthand how unequal access to resources impacts real people’s lives. In conclusion, navigating through interpretation and analysis while constructing a Lorenz curve isn’t merely an academic exercise; it’s an immersive journey into understanding societal structures and dynamics at play. With each insight gleaned from this process, you’re not just examining data points – you’re unraveling narratives woven by income distribution patterns that shape communities in profound ways. (How to Graph a Lorenz Curve) When starting to explore the construction of a Lorenz curve, it’s crucial to grasp the essence of an introduction. This initial phase sets the stage for diving into the intricacies of wealth distribution analysis with this powerful graphical tool. Imagine embarking on a journey through economic landscapes, where data points transform into visual narratives revealing societal disparities and inequalities. The introduction serves as your gateway—a portal shimmering with potential insights waiting to be unveiled. As you stand at this threshold, anticipation dances in your chest like fireflies on a warm summer night. Your mind buzzes with questions, curiosity propelling you forward into uncharted territories of income inequality exploration. In this opening chapter of constructing a Lorenz curve, you lay down the foundation—a bedrock of numerical values that will soon blossom into a tapestry of lines and curves painting a vivid picture of economic reality. With each keystroke or pen stroke, you enter a realm where statistics transcend mere numbers, becoming storytellers whispering tales of prosperity and disparity in hushed tones that echo within your The air crackles with energy as you arrange data points meticulously, each one carrying the weight of countless lives and livelihoods captured in mathematical precision yet resonating with human struggles and triumphs beneath their surface. Emotions swirl within you—empathy for those marginalized by unequal wealth distribution, determination to unravel complexities veiled behind percentages and figures, hope kindling in your heart that maybe through understanding lies the path to change. In this introductory dance between raw data and burgeoning insight lies the crux—the moment when numbers cease to be cold abstract entities but come alive as messengers bearing truths too profound to And so begins your odyssey into crafting a Lorenz curve—an adventure fraught with challenges yet brimming with revelations waiting to unfurl before your eyes like petals blooming under dawn’s gentle Plotting the Lorenz Curve When it comes to plotting the Lorenz Curve, you’re diving into a world where numbers dance and tell stories of wealth distribution. Picture it like sketching a portrait of society’s income landscape – some parts shaded darkly with riches, others left in the light of scarcity. To kick things off, gather your data on incomes or wealth for different segments of the population. These could be quintiles (fifths) or deciles (tenths), depending on how detailed you want your picture to be. Next up is arranging these segments in ascending order – from the smallest earners to the biggest shots at the top. This step sets the stage for unveiling disparities that might make your heart skip a beat or two. Now comes the fun part – calculating cumulative percentages! Imagine stacking coins one by one as you move along each segment. The further you go, the higher they pile up, revealing who holds more chips in this high-stakes game we call life. As you crunch those numbers and plot them on graph paper or a digital canvas, watch as lines curve and sway like dancers in a ballroom waltz. Each bend signifies another twist in fate – will there be equity or inequality? The answer lies within those elegant curves. Once your Lorenz Curve takes shape before your eyes, don’t shy away from scrutinizing every peak and dip. Are there steep inclines indicating pockets of extreme affluence? Or perhaps gentle slopes hinting at a more balanced society? Feel free to color outside the lines too! Add annotations detailing specific data points that catch your eye. Maybe circle an outlier that challenges conventional wisdom or draw arrows pointing towards areas ripe for policy interventions. And voila! You’ve just created a visual masterpiece depicting how resources are divvied up among folks sharing this spinning blue marble called Earth. Let your Lorenz Curve not only speak volumes about economic disparities but also spark conversations about fairness and social justice across kitchen tables and boardrooms alike. External Links	{"url":"https://info.3diamonds.biz/steps-to-construct-a-lorenz-curve/","timestamp":"2024-11-08T03:16:20Z","content_type":"text/html","content_length":"100772","record_id":"<urn:uuid:29b0c2da-3074-4ec5-9ddf-9edac7b4f8f6>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00505.warc.gz"}
How Mathematics Built the Modern World Science and Magic . . . And Math This is a great historical tie into Part II of Existence Strikes Back and its observation that modernity’s momentum began to build in the 13th century, as Western civilization’s capacity for control increased. Magic and science were twins in the cradle of modernity. This essay points out that it was Mathematics was the cornerstone of the Industrial Revolution. A new paradigm of measurement and calculation, more than scientific discovery, built industry, modernity, and the world we inhabit In school, you might have heard that the Industrial Revolution was preceded by the Scientific Revolution, when Newton uncovered the mechanical laws underlying motion and Galileo learned the true shape of the cosmos. Armed with this newfound knowledge and the scientific method, the inventors of the Industrial Revolution created machines – from watches to steam engines – that would change But was science really the key? Most of the significant inventions of the Industrial Revolution were not undergirded by a deep scientific understanding, and their inventors were not scientists. The standard chronology ignores many of the important events of the previous 500 years. Widespread trade expanded throughout Europe. Artists began using linear perspective and mathematicians learned to use derivatives. Financiers started joint stock corporations and ships navigated the open seas. Fiscally powerful states were conducting warfare on a global scale. There is an intellectual thread that runs through all of these advances: measurement and calculation. Geometric calculations led to breakthroughs in painting, astronomy, cartography, surveying, and physics. The introduction of mathematics in human affairs led to advancements in accounting, finance, fiscal affairs, demography, and economics – a kind of social mathematics. All reflect an underlying ‘calculating paradigm’ – the idea that measurement, calculation, and mathematics can be successfully applied to virtually every domain. This paradigm spread across Europe through education, which we can observe by the proliferation of mathematics textbooks and schools. It was this paradigm, more than science itself, that drove progress. It was this mathematical revolution that created modernity. The geometric innovations Advances in geometry began with the rediscovery of Euclid. The earliest known Medieval Latin translation of Euclid’s Elements was completed in manuscript by Adelard of Bath around 1120 using an Arabic source from Muslim Spain. A Latin printed version was published in 1482. After the mathematician Tartaglia translated Euclid’s work into Italian in 1543, translations into other vernacular languages quickly followed: German in 1558, French in 1564, English in 1570, Spanish in 1576, and Dutch in 1606. Beyond Euclid, the German mathematician Regiomontanus penned the first European trigonometry textbook, De Triangulis Omnimodis (On Triangles of All Kinds), in 1464. In the sixteenth century, François Viète helped replace the verbal method of doing algebra with the modern symbolism in which unknown variables are denoted by symbols like x, y, and z. René Descartes and Pierre de Fermat built on Viète’s innovations to develop analytic geometry, where curves and surfaces are described by algebraic equations. In the late seventeenth century, Isaac Newton and Gottfried Leibniz extended the methods of analytic geometry to the study of motion and change through the development of calculus. On top of theoretical improvements to mathematics, the instruments used to apply these theories to the world also advanced dramatically. One striking example comes from angular measurement, which saw large increases in precision as astronomers began to use new instruments, like the mural quadrant in the picture above. Angular measurement works by pointing instruments toward objects and reading off their angles on a measurement scale. The precision of pointing was improved by telescopic sights and finely tunable mechanisms, while better-designed measurement scales allowed astronomers to discriminate between similar angles. The graph below shows the trend of precision, going from seven arcminutes, or 0.11 degrees, in 1550, to 0.06 arcseconds, or 0.000017 degrees, in 1850 – an astounding improvement of almost 7,000 times over three centuries. Computation was aided by the adoption of Hindu-Arabic numerals and the popularization of decimal notation. In 1614, John Napier’s introduction of the logarithm transformed multiplication into addition, and it was followed a decade later by the invention of the slide rule that could efficiently perform multiplication and division (see image below). The era also saw the introduction of printed mathematical tables. These tables document the values of standard mathematical functions and were crucial for computation before the advent of electronic calculators. Constructing them involved using known relationships such as trigonometric identities to compute new function values from old ones. While straightforward in theory, table construction was computationally demanding. The famous 1596 trigonometric table Opus Palatinum de Triangulis was an expensive endeavor financed by the Habsburg emperor Maximilian II: its 100,000 trigonometric ratios – accurate to up to ten decimal pIaces – took the mathematician Rheticus and his team of human computers 12 years to calculate, at a cost of more than 50 times Rheticus’s annual salary as a mathematics professor.	{"url":"https://thedailyeudemon.com/how-mathematics-built-the-modern-world/","timestamp":"2024-11-06T15:20:57Z","content_type":"text/html","content_length":"102914","record_id":"<urn:uuid:03c3562d-4bbe-4912-adea-e9fefb5e611a>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00431.warc.gz"}
3 circle venn diagram shading solver This class assumes you are already familiar with diagramming categorical propositions. It is the pictorial representations of sets represented by closed figures are called set diagrams or Venn diagrams. All of the number values for each section of the diagram have been given to us in the question. The set is said to be Intersection (n) if the elements given present in both the sets. This helped me much with monomials, subtracting fractions and … test; Stoicoviciu Polygon Rings; Quadratic Graphs - Explore Standard Form The problem of existential import is introduced by means of these diagrams. Here you could create a venn diagram … Venn Diagram of logical sets are represented by means of two or three circles enclosed inside a rectangle. It is very logical. The best way to explain how the Venn diagram works and what its formulas show is to give 2 or 3 circles Venn diagram examples and problems with solutions. 3 circle venn diagram solver. Both A and B contains w elements, B and C contains x elements, A and C contains y elements, all the three sets A, B and C contains z elements. A simple online venn diagram maker tool to create a venn diagram based on the values of the three sets. I have used every Math software on the net. One good method to test quickly syllogisms is the Venn Diagram technique. You just put your problem and it will produce a complete step-by-step report of the solution. Wolfram Demonstrations Project. The set is said to be Union (u) if the elements given present in either of the sets. I. Calculator to create venn diagram for three sets.The Venn diagram is an illustration of the relationships between and among sets, groups of objects that share something in common. Click the various regions of the Venn diagram to shade or unshade them When the show set notation … Represent these results using a three circle Venn Diagram.” The type of three circle Venn Diagram we will need is the following: Image Source: Passy’s World of Mathematics. New Resources. Abstract: The Venn Diagram technique is shown for typical as well as unusual syllogisms. Lowercase x's and shading are . ... Venn Diagram Shading Calculator Fresh 3 Circle Venn Diagram Logic Venn diagram shading with 2 and 3 sets. This three circle word problem is an easy one. Algebrator really helps you out in venn diagram problem solver calculator. Let us consider the three sets A, B and C. Set A contains a elements, B contains b elements and C contains c elements. Venn diagram problem with 3 circles use the given information to fill in the number of elements in each region of the venn diagram. Click the various regions of the Venn diagram to shade or unshade them When the show set notation checkbox is clicked one or several different expressions for the shaded region are displayed. Problem-solving using Venn diagram is a widely used approach in many areas such as statistics, data science, business, set theory, math, logic and etc. Enter the title of the venn diagram: Enter the title of A: Enter the title of B: Enter the title of C: Enter the value of A: Enter the value of B: Enter the value of C: Venn diagrams consist of two or three intersecting circles, each representing a class and each labeled with an uppercase letter. Venn Diagrams for Sets Added Aug 1, 2010 by Poodiack in Mathematics Enter an expression like (A Union B) Intersect (Complement C) to describe a combination of two or three sets and get the notation and Venn diagram. 12,000+ Open Interactive Demonstrations Author: Mr Hardman. Final Fantasy Piano Collections Pdf Shandalar Card List Pain After Ear Irrigation Walking Team Names Generator Eurylochos Ac Odyssey Takis Fuego Ingredients Ti-30xb Calculator Online Where Can I Buy Food Network Salad Dressing Is Corn Oil A Saturated Fat	{"url":"http://primeapps.com/review/viewtopic.php?id=85665e-3-circle-venn-diagram-shading-solver","timestamp":"2024-11-09T01:52:50Z","content_type":"text/html","content_length":"15695","record_id":"<urn:uuid:713da03f-49c8-4741-b7e4-656ca8aedbf6>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00627.warc.gz"}
Semialgebraic Proofs and Efficient Algorithm Design By Noah Fleming, University of Toronto, USA, noahfleming@cs.toronto.edu \| Pravesh Kothari, Princeton University, USA, kothari@cs.princeton.edu \| Toniann Pitassi, University of Toronto & IAS, Canada, toni@cs.toronto.edu Suggested Citation Noah Fleming, Pravesh Kothari and Toniann Pitassi (2019), "Semialgebraic Proofs and Efficient Algorithm Design", Foundations and Trends® in Theoretical Computer Science: Vol. 14: No. 1-2, pp 1-221. Publication Date: 10 Dec 2019 © 2019 N. Fleming, P. Kothari and T. Pitassi	{"url":"https://www.nowpublishers.com/article/Details/TCS-086","timestamp":"2024-11-07T18:35:54Z","content_type":"text/html","content_length":"23927","record_id":"<urn:uuid:a2bf2a8c-0107-4ab8-8139-005c2901921d>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00629.warc.gz"}
Quadratic Regression Calculator Quadratic Regression Calculator Quadratic Regression Coefficients: a = 7.6872832762094, b = -0.69574225307099, c = 0.020183370089513 R-squared value: 0.063346987954189 How To Use Our Quadratic Regression Calculator A quadratic regression calculator is a valuable tool for anyone wanting to analyze the relationship between a dependent variable and an independent variable by fitting a quadratic function to their data points. Quadratic regression is a type of regression analysis, which also includes linear and cubic regression models. These tools are useful in various fields, such as finance, biology, and engineering, to extract important information from sets of data. The quadratic regression equation takes the form y = ax^2 + bx + c, where y represents the dependent variable, x represents the independent variable, and a, b, and c are coefficients that the calculator computes using the least squares method. This method aims to minimize the squared vertical distance between each data point and the corresponding point on the quadratic curve. By analyzing the dispersion of values on a scatter plot, one can determine if a quadratic function is the most suitable type of curve for a given data set. For instance, if the data seems to form an arc rather than a straight line, quadratic regression could be a better choice than simple linear regression, which uses a linear equation for analysis. Online quadratic regression calculators are available to do the math for you, providing results such as the quadratic equation, standard deviation, and correlation coefficient. These tools help users quickly and accurately perform regression analysis, even without a strong background in statistics. With the proper application of these calculators and comprehension of the regression models, a person seeking a deeper understanding of relationships between variables over time or space can efficiently find solutions and interpret their data. Quadratic Regression Calculator Online Quadratic Regression Calculator A quadratic regression calculator is a mathematical tool used to find the quadratic function that best fits a given set of data points. In comparison to other regression models, such as linear regression and cubic regression, quadratic regression focuses on finding a curve that represents the data using a quadratic equation. This is particularly useful for capturing situations where relationships between variables are non-linear. There are online quadratic regression calculators available which can quickly generate a quadratic regression equation to model the relationship between two sets of variables. Using a Quadratic Regression Calculator To use a quadratic regression calculator, follow these steps: 1. Gather the data points that represent the relationship between the independent (X) and dependent (Y) variables. 2. Enter the X values in the top data box and the Y values in the bottom data box. 3. Press the "Calculate" button to generate the quadratic regression equation. The output will provide valuable statistical information, such as the regression equation in the form of y = ax^2 + bx + c, the coefficient of determination (R-squared), and the standard deviation. The R-squared value represents the proportion of the variance in the dependent variable that can be explained by the quadratic function, while the standard deviation measures the dispersion of the data points around the regression curve. Quadratic regression can reveal underlying trends within the data that may not be immediately apparent through a simple linear regression, as it captures the curvature in the relationship between the independent and dependent variables. By comparing the quadratic regression equation to other regression models, such as linear regression, one can gain a deeper understanding of how well the data is being represented by different functions. In summary, quadratic regression calculators are useful tools for regression analysis, providing a straightforward method for finding the best-fitting quadratic function to model the relationship between two sets of variables. Utilizing online quadratic regression calculators and applying the steps mentioned above can aid in efficiently generating accurate graphs and statistical information to better understand the relationships within the data. Understanding Quadratic Regression Quadratic regression is a form of regression analysis used to model a relationship between a dependent variable and one or more independent variables using a quadratic function. Quadratic Regression Equation The quadratic regression equation has the form: Where a ≠ 0 and x represents the independent variable while y represents the dependent variable. In comparison to linear regression, which uses a simple linear equation, the quadratic regression equation introduces a second-degree term to help describe the variation in the data points. Quadratic regression is often used when the relationship between variables is nonlinear, and may better capture the trends in data than simple linear regression alone. An online quadratic regression calculator can help you find the coefficients a, b, and c for a given set of data points by following a few simple steps: 1. Enter the known x and y values in the appropriate input fields. 2. Click the 'Calculate' button to compute the quadratic regression equation. 3. Observe the quadratic equation, coefficient of determination (R^2) values, and other resulting statistics. Such calculators can also display the graph of the quadratic function based on the calculated coefficients, which enables the user to visualize the relationship between the variables. Correlation Coefficient The correlation coefficient in quadratic regression, commonly denoted as R^2, indicates how well the quadratic equation fits the data points. An R^2 value of 1 indicates a perfect fit, while an R^2 value closer to 0 indicates a weaker relationship between the variables. In comparison to linear regression, the correlation coefficient for a quadratic regression can be comparatively more informative, as it considers the additional variation captured by the second-degree term. Quadratic Function Vs Linear Regression Quadratic regression and linear regression are both forms of regression analysis. The main difference lies in the nature of the equation used to model the data: • Quadratic regression uses a quadratic equation (y = ax^2 + bx + c) to represent the relationship between the independent and dependent variables. • Linear regression uses a linear equation (y = mx + b) to represent the relationship between the independent and dependent variables. A scatter plot can provide a visual representation of the data points, helping to determine which type of equation will better fit the data. Sometimes, quadratic regression may be more applicable for certain datasets, especially when there is a curved trend in the data. In conclusion, understanding the differences between quadratic regression and linear regression is fundamental for selecting the most appropriate regression model to analyze and predict outcomes based on the data available. By considering the characteristics of the data and the desired goals, a person can choose the best regression method to use, ensuring more accurate and reliable results over time. Regression Analysis Principles Regression analysis is a widely used statistical method for understanding the relationship between variables in a dataset. It helps predict the value of a dependent variable based on the values of independent variables using a regression equation. Dependent and Independent Variables In regression analysis, variables can be classified into two main types: • Dependent Variable: The variable of interest that is being predicted or explained through the regression analysis. Denoted as Y in the regression equation. • Independent Variable: The variable used to predict the dependent variable's value. It is denoted as X in the regression equation. The goal of regression analysis is to establish the nature of the relationship between the dependent and independent variables. This relationship is illustrated by fitting a function to the data points in a scatter plot. Difference between Least Squares Method and Quadratic Regression Least Squares Method The least squares method is a popular approach to linear regression analysis. In simple linear regression, a straight line is fitted to the data points in a scatter plot to minimize the squared vertical distance between points and the line. The resulting linear equation is in the form: Y = mX + b, where m and b are the coefficients, which represent the slope and Y-intercept, respectively. Quadratic Regression Quadratic regression is a type of polynomial regression that fits a quadratic function to the data points in a scatter plot. Quadratic regression is more flexible than simple linear regression since a curve can better represent the relationship between the dependent and independent variables. A quadratic regression equation takes the form: Y = ax^2 + bx + c, where a, b, and c are coefficients that define the curvature, slope, and vertical shift of the quadratic function, respectively. If needed, more complex models like cubic regression can be used for even more curvature in the regression equation. When using an online quadratic regression calculator, the process is straightforward. Typically, you enter the known X and Y values, and the calculator computes the coefficients a, b, and c, as well as other relevant statistics like the correlation coefficient and standard deviation. In conclusion, regression analysis helps understand the relationship between variables and make predictions about future data points. Quadratic regression is a useful extension to linear regression that accommodates non-linear relationships between the dependent and independent variables. Examples and Applications Sample Quadratic Regression Problem Quadratic regression is a useful technique for modeling data that exhibits a natural curve or parabolic relationship. To illustrate this process, let's consider the following example: A person wants to analyze the relationship between the amount of time spent studying (independent variable) and the resulting test scores (dependent variable). The data set consists of the following points: Time (hours) Test Score To find a quadratic equation that best fits this data, first input the values into an online quadratic regression calculator. The calculator will generate the quadratic regression equation: y = ax^2 + bx + c After plugging in the values and using the least squares method, the calculator will produce the following coefficients: • a = 2.467 • b = -0.600 • c = 0.467 Thus, the quadratic regression equation becomes: y = 2.467x^2 - 0.600x + 0.467 This formula can now be used to predict a student's test score based on the number of hours they study. Real-Life Quadratic Regression Applications Quadratic regression is applicable in a wide range of fields, including: 1. Physics: Analyzing projectile motion and determining the maximum height and range of a projectile given its initial velocity and angle. 2. Economics: Assessing the relationship between supply, demand, and price, or identifying the point of diminishing returns for a particular product or service. 3. Engineering: Examining the trajectory of an object experiencing force and friction, such as the movement and stopping distance of a vehicle. In each of these scenarios, it's essential to recognize that various factors may influence the data, and a quadratic equation may not always be the best fit. Consequently, it's important to compare different regression models, such as linear regression or cubic regression, to select the most suitable one for the given data set. The correlation coefficient and standard deviation can be helpful tools to evaluate the strength and reliability of the chosen regression model. Ultimately, finding an appropriate regression equation requires a comprehensive understanding of the underlying data and the specific context of the problem at hand. By mastering quadratic regression and related techniques, one can effectively analyze and predict real-world relationships, contributing valuable insights to various fields and domains. Guide to Using a Quadratic Regression Calculator A quadratic regression calculator is a valuable tool for performing regression analysis on data points when the relationship between the variables is expected to follow a quadratic function. Quadratic regression is a type of regression analysis where the best fit curve is a quadratic equation of the form y = ax^2 + bx + c. This is distinct from other types of regression, such as linear regression, which assumes a simple linear relationship between variables, and cubic regression, which involves a cubic function. To use a quadratic regression calculator, begin by inputting the known data points (independent variable X and dependent variable Y) into the appropriate fields. These data points should be in the form of coordinate pairs (x, y) and represent the relationship between the independent and dependent variables. For example, consider a situation where a person's height depends on the square of their age. In this case, height is the dependent variable (Y), and age is the independent variable (X). Quadratic regression employs the least squares method to minimize the squared vertical distance between each data point and the quadratic function. The calculator automatically determines the coefficients (a, b, and c) in the quadratic equation, along with other relevant statistics, such as the correlation coefficient, standard deviation, and regression equation. These values can then be used to make predictions about the data and assess the quality of the regression model. When comparing quadratic regression to other types of regression, such as linear regression or cubic regression, it's important to consider the distribution of data points on a scatter plot. A quadratic function is most appropriate when the relationship between the variables follows a curved, parabolic trend. In contrast, a linear regression is best suited for data points that lie approximately along a straight line, while cubic regression is appropriate when the relationship is more complex and requires a cubic function for a better fit. There are numerous online quadratic regression calculator options available providing a convenient and accessible means of performing regression analysis. These tools often include graphing capabilities and regression models for different types of relationships, making them versatile for various applications in statistics, mathematics, and other fields that deal with data. In conclusion, a quadratic regression calculator is an indispensable tool for those working with data that follows a quadratic relationship between variables. It is an effective way to derive a regression equation, allowing users to analyze the quality of the model, make predictions, and uncover valuable information. By following a series of simple steps and inputting known X and Y values, anyone can harness the power of quadratic regression analysis for their unique needs. Keep in mind that when selecting a regression model, it's essential to consider the data's distribution on a scatter plot, ensuring that the type of regression used (linear, quadratic, or cubic) is an appropriate fit for the patterns observed. In today's digital age, there are numerous online quadratic regression calculator options, making it easy for any curious mind to explore this fascinating realm of statistics and data analysis.	{"url":"https://statscalculator.com/quadratic-regression","timestamp":"2024-11-10T13:07:52Z","content_type":"text/html","content_length":"37779","record_id":"<urn:uuid:d3c030fe-d8ee-4ba7-804a-72458510a047>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00537.warc.gz"}
Distance Between Two Points - Formula, Derivation, Examples - - [[company name]] [[target location]], [[stateabr]] Distance Between Two Points - Formula, Derivation, Examples The theory of length is important in both math and routine life. From easily measuring the length of a line to figuring out the quickest route between two locations, understanding the distance among two points is important. In this blog, we will take a look at the formula for distance within two points, go through a few examples, and discuss real-life uses of this formula. The Formula for Distance Between Two Points The distance within two points, usually signified as d, is the extent of the line segment connecting the two extremities. In math, this could be portrayed by drawing a right triangle and employing the Pythagorean theorem. Per the Pythagorean theorem, the square of the length of the extensive side (the hypotenuse) is equivalent to the sum of the squares of the distances of the two other sides. The formula for the Pythagorean theorem is a2 + b2 = c2. Consequently, √c2 will equal the length, d. In the case of working out the distance within two points, we could represent the extremities as coordinates on a coordinate plane. Let's assume we have point A with coordinates (x1, y1) and point B at (x2, y2). We could then utilize the Pythagorean theorem to derive the ensuing formula for distance: d = √((x2 - x1)2 + (y2 - y1)2) In this formula, (x2 - x1) represents the length on the x-axis, and (y2 - y1) represents the length along y-axis, forming a right angle. By considering the square root of the sum of their squares, we obtain the length among the two extremities. Here is a visual illustration: Instances of Applications of the Distance Formula Considering we have the formula for distance, let's look at some instances of how it can be utilized. Calculating the Length Between Two Points on a Coordinate Plane Assume we possess two points on a coordinate plane, A with coordinates (3, 4) and B with coordinates (6, 8). We will employ the distance formula to figure out the distance within these two locations as ensues: d = √((6 - 3)2+ (8 - 4)2) d = √(32 + 42) d = √(9 + 16) d = √(25) d = 5 Therefore, the distance between points A and B is 5 units. Calculating the Length Among Two Points on a Map In addition to working out the length on a coordinate plane, we could further use the distance formula to work out lengths between two points on a map. For instance, suppose we posses a map of a city along a scale of 1 inch = 10 miles. To figure out the length within two points on the map, such as the airport and the city hall, we can easily measure the length among the two locations utilizing a ruler and change the measurement to miles utilizing the map's scale. When we calculate the length among these two locations on the map, we figure out it is 2 inches. We change this to miles using the map's scale and find that the real distance within the city hall and the airport is 20 miles. Calculating the Distance Among Two Locations in Three-Dimensional Space In addition to calculating lengths in two dimensions, we could further use the distance formula to calculate the distance among two locations in a three-dimensional space. For example, suppose we have two points, A and B, in a three-dimensional space, with coordinates (x1, y1, z1) and (x2, y2, z2), individually. We will utilize the distance formula to work out the distance between these two locations as ensuing: d = √((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2) Utilizing this formula, we could calculate the length between any two points in three-dimensional space. For example, if we have two locations A and B with coordinates (1, 2, 3) and (4, 5, 6), individually, we can work out the distance within them as ensues: d = √((4 - 1)2 + (5 - 2)2 + (6 - 3)2) d = √(32 + 32 + 32) d = √(9 + 9 + 9) d = √(27) d = 3.16227766 Hence, the distance within locations A and B is approximately 3.16 units. Uses of the Distance Formula Now once we have observed some examples of utilizing the distance formula, let's explore some of its uses in math and other fields. Calculating Distances in Geometry In geometry, the distance formula is used to calculate the length of line segments and the sides of triangles. For example, in a triangle with vertices at points A, B, and C, we use the distance formula to calculate the distances of the sides AB, BC, and AC. These lengths can be utilized to measure other properties of the triangle, for instance its perimeter, area, and interior angles. Solving Problems in Physics The distance formula is additionally used in physics to figure out problems concerning acceleration, speed and distance. For instance, if we recognize the initial location and velocity of an object, also the time it requires for the object to move a certain distance, we could use the distance formula to work out the object's final position and speed. Analyzing Data in Statistics In statistics, the length formula is usually used to workout the length within data points in a dataset. This is beneficial for clustering algorithms, which group data points which are near to each other, and for dimensionality reduction techniques, this portrays high-dimensional data in a lower-dimensional space. Go the Distance with Grade Potential The distance formula is an essential theory in mathematics which enables us to calculate the between two points on a plane or in a three-dimensional space. By utilizing the Pythagorean theorem, we could derive the distance formula and apply it to a variety of scenarios, from measuring distances on a coordinate plane to analyzing data in statistics. Comprehending the distance formula and its utilizations are crucial for everyone interested in mathematics and its applications in other areas. If you're having difficulties regarding the distance formula or any other math concept, contact Grade Potential tutoring for personalized help. Our experienced teachers will assist you conquer any math topic, from algebra to calculus and furthermore. Contact us right now to learn more and schedule your first tutoring session.	{"url":"https://www.ontariocainhometutors.com/blog/distance-between-two-points-formula-derivation-examples","timestamp":"2024-11-09T03:43:38Z","content_type":"text/html","content_length":"76960","record_id":"<urn:uuid:26976a0d-3cac-4f66-be82-03fc56e84381>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00087.warc.gz"}
Evgeny Remez - Biography Quick Info 17 February 1896 Mstsislau, Mahiliou gubernia, Belarus 31 August 1975 Kiev, Ukranian SSR (now Kyiv, Ukraine) Evgeny Yakovlevich Remez was a Belarus mathematician. He is known for his work in the constructive function theory. Evgeny Remez's name also appears as Evgenii Remez or sometimes in the French form Eugene J Remes. He attended the high school in Mstsislau where he excelled across the whole range of school subjects and graduated from the gymnasium in 1916 with the gold medal. He then went to Kiev (or Kyiv to give it its modern spelling) where he entered the Institute of Public Education, as Kiev University was called at that time, where he studied mathematics and mathematical physics. The period during which he attended the Institute were difficult ones with major political changes taking place. In 1917 the Russian Empire broke up and Ukraine became independent during 1918-19, then the USSR was established leading to further changes. Remez graduated in 1924 and began to teach both at the Institute of Public Education and at the Mechanical Trade School. He gave courses at these institutions on analysis, differential equations and differential geometry while undertaking research for his doctorate. He obtained his Ph.D. from Kiev State University in 1929 with his thesis Methods of Numerical Integration of Differential Equations with an Estimate of Exact Limits of Allowable Errors. He continued to undertake research in approximation theory, particularly in the constructive theory of functions, and in numerical analysis for the rest of his career. Remez lived in Kiev, teaching at various institutions there, for most of his life. These institutions included the Pedagogical Institute, the Mining Institute, and the Geological Institute. He was head of the mathematics department at these institutions from 1930 onwards. He worked at the Pedagogical Institute from 1933 to 1955 and he became a professor at Kiev University in 1935. He also worked at the Institute of Mathematics of the Ukrainian Academy of Sciences in Kiev from 1935 until his death in 1975. The Institute of Mathematics had only been founded in Kiev shortly before Remez began to work there. Now we said above that Remez obtained his Ph.D. in 1929. This was not strictly true, for what he was awarded in 1929 was a 'Kandidat' degree, which is recognised as equivalent to the Ph.D. The Russian 'Doctor of Science' degree was a second stage, somewhat similar to the Habilitation in Germany. This degree required many years of research experience and the writing of a second dissertation. Professorships were only open to those holding the degree of Doctor of Sciences, so it was important to Remez that he work towards this degree. He submitted his second dissertation in 1936 and was awarded the degree of Doctor of Physical and Mathematical Sciences without any further examinations. This now put him in a position to become a professor and indeed two years later the title was confered on him by the Higher Degree Commission. In 1939 he was elected to the Academy of Sciences of the Ukrainian SSR. His main work was on the constructive theory of functions and approximation theory. In the mid 1930s, he developed general computational methods of Chebyshev approximation and the Remez algorithm which allows uniform approximation. It constructs, with a prescribed degree of exactness, a polynomial of the best Chebyshev approximation for a given continuous function. A similar algorithm was later developed which allowed rational approximation of continuous functions defined on an interval. Remez generalised Chebyshev-Markov characterisation theory and used it to obtain approximate solutions of differential equations. He proved results about bounded polynomials and created general operator methods of sequence approximation. He also worked on approximate solutions of differential equations and the history of mathematics. Today English has become the main language of mathematics and papers written in Russian or Ukrainian are often translated into English. However, in the years between World War I and World War II, it looked as if French would become the main international mathematical language and many mathematicians from the USSR wrote papers in French with the idea that they would be available to a much wider readership. Remez's papers up to the end of World War II were often written in French and his name appears on these papers as Eugene J Remes. For example in 1940 his publications included On some estimates of best approximation and, in particular, on a fundamental theorem of de la Vallée-Poussin (Russian) (1940), Principe des moindres puissances, 2k-ièmes et principe des moindres carrés dans les problèmes d'approximation Ⓣ (1940), Sur certaines classes de fonctionnelles linéaires dans les espaces $C_{p}$ et sur les termes complémentaires des formules d'analyse approximative Ⓣ (1940), and Sur les termes complémentaires de certaines formules d'analyse approximative Ⓣ (1940). By 1947-48 his papers were all written either in Russian or Ukrainian. For example On the character of convergence of the Pólya-Jackson process in the general case of continuous polynomials (Russian) (1947), Estimates of the rapidity of convergence of the Pólya-Jackson process for continuous polynomials with supplementary structural conditions (Russian) (1947), On the limiting process of Pólya-Jackson-Julia and certain corresponding interpolation algorithms ( Russian) (1947), On mean, uniform (Chebyshevian) and quasiuniform approximations (Russian) (1948), and Detailed investigations of limiting relations between power-mean and Chebyshev approximations (Ukrainian) (1948). For some mathematicians, Remez is best known as the author of the book General computation methods for Chebyshev approximation. Problems with real parameters entering linearly (Russian) (1957). Alston Householder reviewed the book and begins his review with high praise:- This is the first volume to include between its two covers a fairly complete development of Chebyshev approximation, its theory and its practice. Such a book is long overdue. Techniques are illustrated by a number of special examples worked out in detail, and different methods are applied to the same problem by way of comparison. The elaborate chapter and section headings make the Table of Contents into a detailed outline that facilitates reference. Little is presupposed on the part of the reader except the most basic concepts of algebra and analysis. An appendix sets forth the theory of convex bodies as far as it is required in the main text. However, he then makes some critical comments:- These are weighty recommendations and there are no others. The language is turgid and often bombastic, the explanations prolix and wearisome. The pages are replete with footnotes, chips off the workbench, generally unnecessary and frequently distracting. The notation is cumbersome and confusing, rather suggestive of Chebyshev's own, to be expected in the early 19th century but inexcusable in the middle 20th . Nevertheless, most of the material is to be found elsewhere only in journals and hence it is an important book for the numerical analyst. The book has two parts: Part I - Properties of the solution of the general Chebyshev problem; Part II - Finite systems of inconsistent equations and the method of nets in Chebyshev approximation. Of course this was a topic which developed quickly over the following years, much of the development being due to Remez and his students. A new, greatly expanded, edition of the book appeared under the title Fundamentals of numerical methods of Chebyshev approximation in 1969. Not only did Remez develop techniques which originated with Chebyshev, but he also became an expert on Chebyshev and his work. On 8 June 1971 a meeting was held at the Mathematical Institute of the Ukrainian Academy of Sciences to honour the 150th anniversary of the birth of Chebyshev, which was on 16 May. Remez lectured at this meeting on Chebyshev's work on approximation theory. In 1974 Remez published Some of the principal divisions of P L Chebyshev's scientific activity (Ukrainian) which discussed Chebyshev's contributions to number theory, the theory of mechanisms, approximation of functions, minimax problems, and cartography. Although the article [3] was written to honour Remez's eightieth birthday, sadly he died about six months short of reaching that birthday. The authors of that article note:- All of Remez's work is characterised by great skill in applying the subtlest theoretical studies to finding a numerical solution to concrete problems. 1. V K Dzyadyk, Yu A Mitropol'skii and A M Samoilenko, Evgenii Yakovlevich Remez (on the centenary of his birth) (Ukrainian), Ukrain. Mat. Zh. 48 (2) (1996), 285-286. 2. Yu A Mitropol'skii, V K Dzyadyk and V T Gavrilyuk, Evgenii Yakovlevich Remez (on the occasion of the ninetieth anniversary of his birth) (Russian), Ukrain. Mat. Zh. 38 (4) (1986), 539-540. 3. Yu A Mitropol'skii, V K Dzjadyk and V T Gavriljuk, Evgenii Yakovlevich Remez (on the occasion of his eightieth birthday) (Russian), Ukrain. Mat. Z. 28 (5) (1976), 711. Additional Resources (show) Other websites about Evgeny Remez: Written by J J O'Connor and E F Robertson Last Update December 2008	{"url":"https://mathshistory.st-andrews.ac.uk/Biographies/Remez/","timestamp":"2024-11-11T07:56:44Z","content_type":"text/html","content_length":"28188","record_id":"<urn:uuid:b0eafb10-fdec-447f-bd3b-771eeae411a7>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00414.warc.gz"}
How to Apply Bayesian Decision Theory in Machine Learning \| Paperspace Blog In a previous article we discussed the theory behind Bayesian Decision Theory in detail. In this article we'll see how to apply Bayesian Decision Theory to different classification problems. We'll discuss concepts like the loss and risk of doing a classification problem, and how they're done in a step-by-step manner. To handle the uncertainty in making a classification, the "reject class" is also discussed. Any sample with a weak posterior probability is assigned to the reject class. We'll conclude by analyzing the conditions under which a sample is classified as a reject. The outline of this article is as follows: • Quick Review of Bayesian Decision Theory • Bayesian Decision Theory with Binary Classes • Bayesian Decision Theory with Multiple Classes • Loss & Risk • 0/1 Loss • Example to Calculate Risk • Reject Class • When to Classify Samples As Rejects? • Conclusion Let's get started. Bring this project to life Quick Review of Bayesian Decision Theory In a previous article we discussed Bayesian Decision Theory in detail, including the prior, likelihood, and posterior probabilities. The posterior probability is calculated according to this $$ P(C_i\|X)=\frac{P(C_i)P(X\|C_i)}{P(X)} $$ The evidence $P(X)$ in the denominator can be calculated as follows: $$ P(X)=\sum_{i=1}^{K}P(X\|C_i)P(C_i) $$ As a result, the first equation can be written as follows: $$ P(C_i\|X)=\frac{P(C_i)P(X\|C_i)}{\sum_{k=1}^{K}P(X\|C_k)P(C_k)} $$ We also showed that the sum of prior probabilities must be 1: $$ \sum_{i=1}^{K}P(C_i)=1 $$ The same is true for the posterior probabilities: $$ \sum_{i=1}^{K}P(C_i\|X)=1 $$ We ended the first article by mapping the concepts from Bayesian Decision Theory to the context of machine learning. According to this discussion, it was shown that the prior, likelihood, and evidence play an important role in making a good prediction that takes into consideration the following aspects: • The prior probability $P(C_i)$ reflects how the model is trained by samples from class $C_i$. • The likelihood probability $P(X\|C_i)$ refers to the model's knowledge in classifying the sample $X$ as the class $C_i$. • The evidence term $P(X)$ shows how much the model knows about the sample $X$. Now let's discuss how to do classification problems using Bayesian Decision Theory. Bayesian Decision Theory with Binary Classes For a binary classification problem, the posterior probability of the two classes are calculated as follows: $$ P(C_1\|X)=\frac{P(C_1)P(X\|C_1)}{\sum_{k=1}^{K}P(X\|C_k)P(C_k)} P(C_2\|X)=\frac{P(C_2)P(X\|C_2)}{\sum_{k=1}^{K}P(X\|C_k)P(C_k)} $$ The sample is classified according to the class with the highest posterior probability. For binary classification problems, the posterior probability for only one class can be enough. This is because the sum of all posterior probabilities is 1: $$ P(C_1\|X)+P(C_2\|X)=1 P(C_2\|X)=1-P(C_1\|X) $$ Given the posterior probability of one class, the posterior probability of the second class is thus also known. For example, if the posterior probability for class $C_1$ is 0.4, then the posterior probability for the other class $C_2$ is: $$ 1.0-0.4=0.6 $$ It is also possible to calculate the posterior probabilities of all classes, and compare them to determine the output class. $$ C_1 \space if \space P(C_1\|X)>P(C_2\|X) C_2 \space if \space P(C_2\|X)>P(C_1\|X) $$ Bayesian Decision Theory with Multiple Classes For a multi-class classification problem, the posterior probability is calculated for each of the classes. The sample is classified according to the class with the highest probability. Assuming there are 3 classes, here is the posterior probability for each: $$ P(C_1\|X)=\frac{P(C_1)P(X\|C_1)}{\sum_{k=1}^{K}P(X\|C_k)P(C_k)} P(C_3\|X)=\frac{P(C_3)P(X\|C_3)}{\sum_{k=1}^{K}P(X\|C_k)P(C_k)} $$ Based on the posterior probabilities of these 3 classes, the input sample $X$ is assigned to the class with the highest posterior probability. $$ max \space [P(C_1\|X), P(C_2\|X), P(C_3\|X)] $$ If the probabilities of the 3 classes are 0.2, 0.3, and 0.5, respectively, then the input sample is assigned to the third class $C_3$. $$ Choose \space C_i \space\ if \space P(C_i\|X) = max_k \space P(C_k\|X), \space k=1:K $$ Loss and Risk For a given sample that belongs to the class $C_k$, Bayesian Decision Theory may take an action that classifies the sample as belonging to class $\alpha_i$ which has the highest posterior The action (i.e. classification) is correct when $i=k$. In other words: $$ C_k=\alpha_i $$ It may happen that the predicted class $\alpha_i$ is different from the correct class $C_k$. As a result of this wrong action (i.e. classification), there is a loss. The loss is denoted as lambda $\ $$ loss = \lambda_{ik} $$ Where $i$ is the index of the predicted class and $k$ is the index of the correct class. Due to the possibility of taking an incorrect action (i.e. classification), there is a risk (i.e. cost) for taking the wrong action. Risk $R$ is calculated as follows: $$ R(\alpha_{i}\|X)=\sum_{k=1}^{K}\lambda_{ik}P(C_k\|X) $$ The goal is to choose the class that reduces the risk. This is formulated as follows: $$ Choose \space \alpha_i \space\ so \space that \space R(\alpha_i\|X) = min_k \space R(\alpha_k\|X), \space k=1:K $$ Note that the loss is a metric that shows whether a single prediction is correct or incorrect. Based on the loss, the risk (i.e. cost) is calculated, which is a metric that shows how the overall model predictions are correct or incorrect. The lower the risk, the better the model is at making predictions. 0/1 Loss There are different ways to calculate the loss. One way is by using the "0/1 loss". When the prediction is correct, then the loss is $0$ (i.e. there is no loss). When there is a wrong prediction, then the loss is always $1$. The 0/1 loss is modeled below, where $i$ is the index of the wrong class and $k$ is the index of the correct class. Remember that the risk is calculated as follows: $$ R(\alpha_{i}\|X)=\sum_{k=1}^{K}\lambda_{ik}P(C_k\|X) $$ Based on the 0/1 loss, the loss is only $0$ or $1$. As a result, the loss term $\lambda$ can be removed from the risk equation. The risk is calculated as follows, by excluding the posterior probability from the summation when $k=i$: $$ R(\alpha_i\|X) = \sum_{k=1, k\not=i}^{K} P(C_k\|X) $$ Whenever i not equal k, then there is a wrong classification (action) and the loss is 1. Thus, the risk is incremented by the posterior probability of the wrong class. If we don't remove $\lambda$, then it multiplies the posterior probability of the current class $i$ by 0. Thus we exclude it from the summation. After removing $\lambda$, it is important to manually exclude the posterior probability of the current class $i$ from the summation. This is why the summation does not include the posterior probability when $k=i$. The previous equation calculates the sum of all posterior probabilities except for the posterior probability of class $i$. Because the sum of all posterior probabilities is 1, the following equation $$ [\sum_{k=1, k\not=i}^{K} P(C_k\|X)]+P(C_i\|X)=1 $$ By keeping the sum of posterior probabilities of all classes (except for the class $i$) on one side, here is the result: $$ \sum_{k=1, k\not=i}^{K} P(C_k\|X)=1-P(C_i\|X) $$ As a result, the risk of classifying the sample as class $i$ becomes: $$ R(\alpha_i\|X) = 1-P(C_i\|X) $$ According to this equation, the risk is minimized when the class with the highest posterior probability is selected. Thus, we should work on maximizing the posterior probability. As a summary, here are the different ways to calculate the risk of taking an action given that the 0/1 loss is used: $$ R(\alpha_i\|X) = \sum_{k=1}^{K}\lambda_{ik} P(C_k\|X) \\R(\alpha_i\|X) = \sum_{k=1,k\not=i}^{K} P(C_k\|X) \\R(\alpha_i\|X) = 1-P(C_i\|X) $$ Here are some notes about the 0/1 loss: • Based on the 0/1 loss, all the incorrect actions (i.e. classifications) are given equal losses of $1$, and all correct classifications have no losses (e.g. $0$ loss). For example, if there are 4 wrong classifications, then the total loss is $4(1)=4$. • It is not perfect to assign equal losses to all incorrect predictions. The loss should increases proportionally to the posterior probability of the wrong class. For example, the loss in the case of a wrong prediction with a posterior probability of 0.8 should be higher than when the posterior probability is just 0.2. • The 0/1 loss assigns 0 loss to the correct predictions. However, it may happen to add a loss to the correct predictions when the posterior probability is low. For example, assume the correct class is predicted with a posterior probability of 0.3, which is small. The loss function may add a loss to penalize the predictions with low confidence, even though they are correct. Example to Calculate Risk Assume there are just two classes, $C_1$ and $C_2$, where their posterior probabilities are: $$ P(C_1\|X)=0.1 P(C_2\|X)=0.9 $$ How do we select the predicted class? Remember that the objective is to select the class that minimizes the risk. It is formulated as follows: $$ Choose \space \alpha_i \space\ so \space that \space R(\alpha_i\|X) = min_k \space R(\alpha_k\|X), \space k=1:K $$ So, to decide which class to select, we can easily calculate the risk in case of predicting $C_1$ and $C_2$. Remember that the risk is calculated as follows: $$ R(\alpha_i\|X) = 1-P(C_i\|X) $$ For $C_1$, its posterior probability is 0.1, and thus the risk of classifying the sample $X$ as the class $C_1$ is 0.9. $$ R(\alpha_1\|X) = 1-P(C_1\|X) = 1-0.1=0.9 $$ For $C_2$, its posterior probability is 0.9, and thus the risk of classifying the sample $X$ as the class $C_2$ is 0.1. $$ R(\alpha_2\|X) = 1-P(C_2\|X) = 1-0.9=0.1 $$ The risks of classifying the input sample as $C_1$ and $C_2$ are 0.9 and 0.1, respectively. Because the objective is to select the class that minimizes the risk, the selected class is thus $C_2$. In other words, the input sample $X$ is classified as the class $C_2$. Reject Class Previously, when there was a wrong action (i.e. classification), its posterior probability was incorporated into the risk. But still, the wrong action was taken. Sometimes, we do not need to take the wrong action at all. For example, imagine an employee who sends letters according to postal codes. Sometimes the employee may take wrong actions by sending envelopes to the incorrect homes. We need to avoid this action. This can be accomplished by adding a new reject class to the set of available classes. Rather than having only $K$ classes, there are now $K+1$ classes. The $K+1$ class is the reject class, and the classes from $k=1$ to $k=K$ are the normal classes. The reject class has a loss equal to $\lambda$, which is between 0 and 1 (exclusive). The reject action is given the index $k+1$ as follows: $$ C_{reject}=\alpha_{K+1} $$ When the reject class exists, here is the 0/1 loss function for choosing class $i$ when the correct class is $k$: The risk of classifying $X$ as the reject class is: $$ R(\alpha_{K+1}\|X) = \sum_{k=1}^{K}\lambda P(C_k\|X) $$ The summation operator can be factored as follows: $$ R(\alpha_{k+1}\|X) = \lambda P(C_1\|X) + \lambda P(C_2\|X) +...+ \lambda P(C_K\|X) $$ By taking out $\lambda$ as a common factor, the result is: $$ R(\alpha_{K+1}\|X) = \lambda [P(C_1\|X) + P(C_2\|X) +...+ P(C_K\|X)] $$ Because the sum of all (from 1 to K) posterior probabilities is 1, then the following holds: $$ P(C_1\|X) + P(C_2\|X) +...+ P(C_K\|X)=1 $$ As a result, the risk of classifying $X$ according to the reject class is $\lambda$: $$ R(\alpha_{K+1}\|X) = \lambda $$ Remember that the risk of choosing a class $i$ other than the reject class is: $$ R(\alpha_{i}\|X) = \sum_{k=1}^{K}\lambda_{ik} P(C_k\|X)= \sum_{k=1, \forall k\not = i}^{K} P(C_k\|X) = 1-P(C_i\|X) $$ When to Classify Samples As Rejects? This is an important question. When should we classify a sample as class $i$, or the reject class? Class $C_i$ is selected when the following 2 conditions hold: 1. The posterior probability of classifying the sample as class $C_i$ is higher than the posterior probabilities of all other classes. 2. The posterior probability of class $C_i$ is higher than the threshold $1-\lambda$. These 2 conditions are modeled as follows: $$ Choose \space C_i \space\ if \space [P(C_i\|X)>P(C_{k, k\not=i}\|X)] \space AND \space [P(C_i\|X)>P(C_{reject}\|X)] Otherwise, \space choose \space Reject \space\ class $$ The condition $0 < \lambda < 1$ must be satisfied for the following 2 reasons: 1. If $\lambda=0$, then the threshold is $1-\lambda = 1-0 = 1$, which means a sample is classified as class $C_i$ only when the posterior probability is higher than 1. As a result, all samples are classified as the reject class because the posterior probability cannot be higher than 1. 2. If $\lambda=1$, then the threshold is $1-\lambda = 1-1 = 0$, which means a sample is classified as class $C_i$ only when the posterior probability is higher than 0. As a result, the majority of samples will be classified as belonging to the class $C_i$ because the majority of the posterior probabilities are higher than 0. Only samples with a posterior probability of 0 are assigned to the reject class. In these two articles, Bayesian Decision Theory was introduced in a detailed step-by-step manner. The first article covered how the prior and likelihood probabilities are used to form the theory. In this article we saw how to apply Bayesian Decision Theory to binary and multi-class classification problems. We described how the binary and multi-class losses are calculated, how the predicted class is derived based on the posterior probabilities, and how the risk/cost is calculated. Finally, the reject class was introduced to handle uncertainty.	{"url":"https://blog.paperspace.com/bayesian-decision-theory-machine-learning/","timestamp":"2024-11-10T03:23:16Z","content_type":"text/html","content_length":"99229","record_id":"<urn:uuid:f01e71ba-7095-4b68-85ed-9d021b910201>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00415.warc.gz"}
Generative AI – Latent Space Exploring Latent Space in Generative Models Introduction Latent space is a fundamental concept in generative models, such as GANs (Generative Adversarial Networks) and VAEs (Variational Autoencoders). It refers to the space of hidden variables that the model uses to generate new data instances. What is Latent Space? In the context of generative models, latent space is an abstract, lower-dimensional space where each point corresponds to a possible output of the model. The model learns to map these points to high-dimensional data, such as images or text. Navigating Latent Space Latent space can be thought of as a compressed representation of the data distribution. By sampling points in latent space and feeding them through the decoder of the model, we can generate new, realistic data instances. Example Code: Sampling from Latent Space Here’s an example using a simple VAE to explore latent space: import torch from torch import nn from torchvision import datasets, transforms from torch.utils.data import DataLoader # Define the VAE model class VAE(nn.Module): def __init__(self, latent_dim=2): super(VAE, self).__init__() self.fc1 = nn.Linear(784, 400) self.fc21 = nn.Linear(400, latent_dim) self.fc22 = nn.Linear(400, latent_dim) self.fc3 = nn.Linear(latent_dim, 400) self.fc4 = nn.Linear(400, 784) def encode(self, x): h1 = torch.relu(self.fc1(x)) return self.fc21(h1), self.fc22(h1) def reparameterize(self, mu, logvar): std = torch.exp(0.5 * logvar) eps = torch.randn_like(std) return mu + eps * std def decode(self, z): h3 = torch.relu(self.fc3(z)) return torch.sigmoid(self.fc4(h3)) def forward(self, x): mu, logvar = self.encode(x.view(-1, 784)) z = self.reparameterize(mu, logvar) return self.decode(z), mu, logvar # Example usage model = VAE() # Assuming `data_loader` is a DataLoader for MNIST or a similar dataset for batch in data_loader: data, _ = batch recon_batch, mu, logvar = model(data) # Sample from latent space sample = torch.randn(64, 2) # 64 samples, 2-dimensional latent space generated_images = model.decode(sample) Latent space provides a powerful way to understand and generate data with generative models. By manipulating points in latent space, we can create diverse and realistic data samples, making it a crucial concept in AI and machine learning.	{"url":"https://gnereus.com/x2024/2024/07/30/generative-ai-latent-space/","timestamp":"2024-11-07T13:57:51Z","content_type":"text/html","content_length":"133442","record_id":"<urn:uuid:735b4ba5-440b-49d3-bb87-f863e291ebe6>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00829.warc.gz"}
Merge sort Merge sort is a divide-and-conquer algorithm for sorting which has a time complexity of O(n log n). It goes as follows given a sequence S of n elements: 1. Base case: If the sequence has a single element or no elements, it is sorted and we're done. 2. Divide step: Split the sequence in the middle, obtaining subsequences A and B. 3. Recursion step: Apply merge sort to each of the subsequences A and B, which will now be sorted. 4. Merge step: Take the smallest element between the sorted A and B subsequences by checking the next element of each from the start. If there are no more elements on either A or B, take all the remaining elements of the other. Repeat this step until a sorted sequence is built with all elements of A and B, which will be the sorted sequence S. Functional implementation # NOTE: assuming both "a" and "b" are sorted sequences. # "c" acts as the accumulator for the resulting merged sequence. def merge_sorted_sequences(a, b, c=[]): # if either sequence is empty, return the other (base case) if len(a) == 0: return c + b if len(b) == 0: return c + a # take the smallest element of either "a" or "b" and build # up the resulting sorted sequence "c" if a[0] <= b[0]: return merge_sorted_sequences(a[1:], b, c + [a[0]]) return merge_sorted_sequences(a, b[1:], c + [b[0]]) def merge_sort(arr): if len(arr) <= 1: return arr # base case m = len(arr) // 2 # midpoint of "arr" p, q = arr[:m], arr[m:] # divide step a, b = merge_sort(p), merge_sort(q) # recursion step return merge_sorted_sequences(a, b) # conquer step • “Introduction to Algorithms” by Thomas H. Cormen et al (2022).	{"url":"https://breder.org/merge-sort","timestamp":"2024-11-10T10:45:11Z","content_type":"text/html","content_length":"3936","record_id":"<urn:uuid:4840eb2f-611e-4678-9011-83016528598f>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00220.warc.gz"}
Sometimes I wish TDM were harder I agree with Springs. chk772, you can play however you want, but I really, truly, recommend you to play a mission or two completely without saving or loading. Start with the smallest ones. Dealing with the consequences of being seen makes the game much more interesting. I recommend this to everyone who use save/loading a lot. Stop doing it. Try out the game, there is a completely new depth of gameplay you are constantly missing. What does a good stealth score mean if you save loaded? It means that you spammed the save/load buttons, nothing more. But in a normal mission, i don't understand a mission objective like "Don't KO or kill anyone". I mean, why? Because it is a higher difficulty. Higher difficulty imposes restrictions in order to increase the challenge. If you are not allowed to KO or kill AI, you cannot just clear the map of AI and do what you want. It guides the player towards more challenge. Most of the missions have an easy difficulty that allows you to kill and KO and do whatever you want. -The mapper's best friend. Well, of course it's down to preference, but a sword fight? Can't imagine a thief sneaking into some mansion, moving from shadow to shadow, only to get into a sword fight the next moment. The funny thing is; when I first started playing Thief, it wasn't the stealth that enticed me. It was the huge levels, the freedom and the combat. I played Thief as an action game. I actually got very good at it. I remember fighting 5 of Ramirez's boys at once and I won! I also don't find it strange at all that a thief would develop decent combat skills to defend themselves if they got caught without a good escape route. I enjoy both stealth and combat. In the wise words of Eric Cartman, "I do what I want!" Edited by SirGen I came to Thief from fantasy RPGs, where its quite common for thieves to fight with swords. They're just not as good at it. • 2 TDM Missions: A Score to Settle * A Reputation to Uphold * A New Job * A Matter of Hours Video Series: Springheel's Modules * Speedbuild Challenge * New Mappers Workshop * Building Traps in a country where the power supply isn't reliable to be on then you have to save every so often. as in people stealing copper cable from the sub stations after the power comes from the national Edited by stumpy @ Sotha: I maybe quickload 3 or 4 times in a mission, not exactly spamming the button. That's my way of playing and enkoying the game, bit perfectionistic. Interesting to see though how different the approached are though. I came to Thief from fantasy RPGs, where its quite common for thieves to fight with swords. They're just not as good at it. The Grey Mouser used a sword and he was totally a thief. (And that series is where thieves' guilds come from in the first place, so it's pretty reputable. Come the time of peril, did the ground gape, and did the dead rest unquiet 'gainst us. Our bands of iron and hammers of stone prevailed not, and some did doubt the Builder's plan. But the seals held strong, and the few did triumph, and the doubters were lain into the foundations of the new sanctum. -- Collected letters of the Smith-in-Exile, Civitas Approved First of all, no that's wrong, it is not marketed as Thief. Here, from the front page: THE DARK MOD 2.0 is a free, first-person stealth game inspired by the Thief series by Looking Glass Studios. It recreates the essence of the original Thief games on a more modern engine. The point of communication is not what is explicitly said, that is called technicalities or pedantry. The point of communication is what is implied. And this text implies to the user that TDM is meant to be the Thief they knew, in an updated engine. Or at least the devs' best effort towards that. But as I have learned over the past few days, that isn't actually the case. It is meant to be a hardcore version thereof, even on its easiest difficulty setting. One it doesn't even default to, as it turns out. However, since I didn't pay for nothing, and this isn't official canon, I can't very well complain about the craftsmanship. That seems spot on for what it is, actually. I can, however, express my disappointment and incredulousness towards fans who no doubt have partaken in bashing ISA and EM for years, turn around and deviate from the beloved formula themselves, even in areas completely under their control such as gameplay. I always hear it said, "At least we have TDM" whenever EM gives us the finger. But I don't feel like I do. You are doing it wrong, absolutely, completely wrong. If you are so close to the AI that you can't hit them without bumping into them, as you say here: I don't care. BJ range has been reduced, and combined with everything else, it is robbing me of enjoyment when I have the misfortune of meeting one of the more alert or helmeted guards. Don't fix what ain't broke, is what we have been saying to EM. But we were no better ourselves. And before anyone tries it, I am not talking about KOing the legs. Then that's wrong. Watch my video again and really WATCH it. I don't get close to them, I'm still about 3 feet away, maybe more when I blackjack them. Hell, I blackjacked an AI sitting in a chair, across a desk while I was on the windowsill!!!!!!!!!!!! Then I don't know what to tell you. I have had to crouch and frontlean, after having carfully positioned myself at their heads first, in order to have even a prayer of reaching the head of a sleeping AI in a bed. You've already been told that the Thief skillset for blackjacking is useless here, and for good reason. The Thief 1/2 AI were pants on head retardedly easy to blackjack, what skillset are you talking about possessing? Then I am not getting the game experience it was implied I would. Which I expected from fans who, unlike ISA and EM, weren't hellbent on changing the formula around. Or at least, that is what I thought. I doubt I was alone. As for LGS!Thief BJ difficulty: Yes, you are right, they were. And still these games are considered somewhat hardcore. And still I have seen Let's Players struggle tensely with them. Personally I am good enough that I can appreciate removal of its more ridicilous features, such as KOing legs and never being detected while directly behind them, but that isn't even close to being the extent to which things have changed. The difficulty has not changed a little, it is playing in a different league altogehter. Also, you might as well give up trying to defeat my Thief purism by appealing to my pride as a gamer. No wonder no one complains about this, if you get ridiculed for not being good enough. AlsoI just tried 5 different maps, and walking up behind AI for as long as I could before they detected something and turned around to see what it was. So that's dirt, wood, stone, grass and carpet texture. The carpet texture I was able to RUN up behind an AI and he didn't go above alert level 1. The others were a mixed bag with wood and stone being the worst. With those I had enough time to get right up behind them and blackjack, but if I waited a few seconds more then game over, they turned to see what was sneaking up behind them. That sounds reasonable for the less alert guards on Nearly Blind difficulty. And I am pleased as a puppy with those. Some guards, though, can hear me walk on wood or see me when I am behind them even on that difficulty. If I up it to "Forgiving" these same guards will hear me take one step on wood before going into 2nd alert, and will see me from behind walking on a carpet in 50% of cases. I dare not even contemplate Hardcore difficulty! Edited by IHaveReturned Yes there are a few LPers who've had difficulty with blackjacking in TDM, FenPhoenix being one of them. Until just recently. He just posted this: So as you can see, he watched the video I posted, and now, after 1 day of trying, has learned how to properly blackjack. So what's stopping you from learning? Aside from a refusal to try? I still don't understand why you are here. It's clear you have no intention of ever actually wholeheartedly trying to get better at it. And sure it's playing in a different league altogether, one that 20,000 people have no problem with, except you. I think that points to where the actual problem lies. I always assumed I'd taste like boot leather. Then I am not getting the game experience it was implied I would. Which I expected from fans who, unlike ISA and EM, weren't hellbent on changing the formula around. Or at least, that is what I thought. I doubt I was alone. Also, you might as well give up trying to defeat my Thief purism by appealing to my pride as a gamer. No wonder no one complains about this, if you get ridiculed for not being good enough. You mean you're not getting the game experience you inferred, no one here has ever said this was Thief. No one is ridiculing you for not being good enough (except me of course), what people seem to have trouble with is you not trying to get better, but rather just complaining that it's too hard. I always assumed I'd taste like boot leather. Then I don't know what to tell you. I have had to crouch and frontlean, after having carfully positioned myself at their heads first, in order to have even a prayer of reaching the head of a sleeping AI in a bed. Oh that, don't bother with trying to KO sleeping guards, make some noise to get them to get up, or just keep hitting them until you KO them, that's what I did in the video I posted yesterday........ I always assumed I'd taste like boot leather. You know your argument is going well when you're trying to tell someone how language works, even though there are entire libraries of technical documents that would like to have a word with you about "The point of communication is not what is explicitly said, that is called technicalities or pedantry. The point of communication is what is implied.". You were never promised Thief. You were offered an inspired-by-Thief experience. While I skimmed some sections of the Wikipedia article on communication, I never saw your implication stated. I did however, find "Certain attitudes can also make communication difficult. For instance, great anger or sadness may cause someone to lose focus on the present moment. Disorders such as Autism may also severely hamper effective communication." filed under Psychological Noise. Now, I'm certainly not trying explicitly or implicitly to state that you have autism, but there seem to be definite signs of an attitude problem making communication difficult here. Don't make me look up inspiration to further drive this home, it's far too much work for the reply I'm sure I'll get. Intel Sandy Bridge i7 2600K @ 3.4ghz stock clocks 8gb Kingston 1600mhz CL8 XMP RAM stock frequency Sapphire Radeon HD7870 2GB FLeX GHz Edition @ stock @ 1920x1080 To be honest, I'm wondering the whole time if IHaveReturned is not the next incarnation of Aida Keeley you are talking to. Or maybe it's ZylonBane. They both have a shit personality. My Eigenvalue is bigger than your Eigenvalue. @7upMan, it's been a while since we had a troll here. Hopefully people won't spend too much their time feeding it, as it usually is pointless. -The mapper's best friend. I'm starting to get the feeling this is just Aida. His arguments are starting to sound to familiar. I always assumed I'd taste like boot leather. Let's avoid personal attacks, folks. TDM is not going to suit everyone's taste. Some people will give it an honest try and simply not take to it, (and some are going to be determined not to like it from the start). We're a niche game and proud of it. I find the comparisons between us and EM a little amusing for that reason; they're specifically watering down the stealth experience to try to try and appeal to everyone. That's not our approach and never has been. But there's no need to try to argue someone into liking something they don't like. It won't work, and it isn't necessary. TDM Missions: A Score to Settle * A Reputation to Uphold * A New Job * A Matter of Hours Video Series: Springheel's Modules * Speedbuild Challenge * New Mappers Workshop * Building Traps 1) There are plenty of difficulty adjustments: mission difficulty and all the difficulty controls in the objective screens (ai vision, combat settings, lockpicking settings and more!) There really isn't a need to add any more! Lets spend the energy elsewhere. Limited manpower and difficulty reason 2) and 3) below. There are, and too many would be a problem I agree. But the lowest AI perception level at least ought to be similar to Thief in what is, as was rightly said, a toolset to accomodate mappers inspired by that very game. That is a matter of idealism (and in my case enjoyment) considering the reason we are all here in the first place, not a matter of anybody's ability to learn new strategies. I've conceded that knockouts are satisfying in their own way already. What I'm saying, though, is that designing missions without a blackjack can lead do something a lot more fun, dynamic and faithful to what makes the Thief formula good (I'm pretty sure we actually agree on what that is). Thief gives you the freedom to engage it your way, and not necessarily the precise way the designer intended. I think as it stands, the blackjack is an unfortunate indicator to the contrary. It doesn't tell a newbie that "this is a game about solving problems your way," it says that "this is a game about clubbing heads" because it's such a one size fits all solution. Ironically, removing an option actually opened me up to a lot more. This guy has a point, actually. If TDM was all about replicating the Thief experience, his request would be unreasonable on its face, because BJing is integral to that experience. But seeing as dev intent is more to make the most realistic and pure stealth experience they could, merely inspired by Thief, removing the blackjack can be seen as a reasonable extension of that. I don't like it, but I don't like the new AI with eyes in the back of its head either. Edited by IHaveReturned But seeing as dev intent is more to make the most realistic and pure stealth experience they could, merely inspired by Thief, removing the blackjack can be seen as a reasonable extension of that. I don't like it, but I don't like the new AI with eyes in the back of its head either. Are you planning on just continuing to complain? You've made your opinion pretty clear, and at one point said you were going to wait and see if there were any other complaints along similar lines, which seemed like a reasonable approach. On the other hand, continuing to restate that you don't like it, and making pointedly misleading statements about what the "dev intent" was, isn't going to accomplish anything. TDM Missions: A Score to Settle * A Reputation to Uphold * A New Job * A Matter of Hours Video Series: Springheel's Modules * Speedbuild Challenge * New Mappers Workshop * Building Traps Yes there are a few LPers who've had difficulty with blackjacking in TDM, FenPhoenix being one of them. Until just recently. He just posted this: So as you can see, he watched the video I posted, and now, after 1 day of trying, has learned how to properly blackjack. So what's stopping you from learning? At the moment? The fact that it is a rather big percentage of blind luck determining my ability to get close enough to try in illuminated areas, no matter what Perception difficulty I am playing on. In general, getting humbled at something I expected to be good at, getting frustrated with several reloads on the same damn guard, and ragequitting. Feeling like the mechanic needs tweaking anyway, so learning to compensate is a waste of time. That sort of thing. Your continued harping on this besides-the-point issue does not inspire me to keep discussing this with you. I still don't understand why you are here. It's clear you have no intention of ever actually wholeheartedly trying to get better at it. Because I desperately want to like this game. Because I want it to succeed with as many people as possible. And because I feel that at the very least permitting Thief-level gameplay is a no-brainer for a mod like this, and will greatly aid the first two reasons. It's a highly versatile and flexible toolset for people who are using it because they have played Thief. Why does that not include Thief-level gameplay as one of its possible features?! It boggles the mind. And sure it's playing in a different league altogether, one that 20,000 people have no problem with, except you. I think that points to where the actual problem lies. Considering how hard you, a dev...I mean, Contrbutor is working to tear me down for it, I wouldn't be so sure I am the only one. Just the only one with the nuts to admit it. I mean, I have no doubt levels are beatable, even on Hardcore, if one abuses various gadgets and shadow stealth while ignoring illuminated blackjack takedowns. I'm not saying missions cannot be beaten under this new paradigm. But I guess that among those few with the nuts to admit incompetence at BJing, even fewer are also purist enough to see a problem with old strategies having become impossible, being content to adapt to new ones instead. That leaves few complainers. But for this to be a genuine Thief experience, illuminated takedowns without relying on luck and save-scumming is a requirement. If this isn't meant to be a genuine Thief experience, I think most of those 20.000 people would be taken somewhat aback. I'm sure most of them didn't go into this expecting Deux Ex or Thievery. They expected a Thief-clone. I don't even see why you are trying to deny that. It smacks of unreasonable contrarianism. Oh that, don't bother with trying to KO sleeping guards, make some noise to get them to get up, or just keep hitting them until you KO them, that's what I did in the video I posted In other words, more compensation for what shouldn't have been an issue in the first place. sigh You know your argument is going well when you're trying to tell someone how language works, even though there are entire libraries of technical documents that would like to have a word with you about "The point of communication is not what is explicitly said, that is called technicalities or pedantry. The point of communication is what is implied." Last I heard, we were unable to create computers capable of replicating human speech, precisely because it is so heavily reliant of context and slang, whereas the computer needs the isolated words and sentences to provide all necessary information. It is the same reason you'd often be totally clueless coming into a long discussion and being exposed to a single sentence and its reply. Because the information provided by those alone are not nearly enough to grasp what is going on. The implication, however, that is informed by the complete picture. In this case known Thief modders being involved in a project promising to be inspired by Thief, with the only deviation mentioned anywhere being that it is in a new engine, and that copyrighted stuff would be changed, with pictures of ninjas with blackjacks and bows taking on pseudo-Hammerites and guards in a medieval setting. It is implied, without hint of implication to the contrary. That is it's main selling point. I'll say as I did to Aluminum, trying to pretend otherwise just smacks of contrarianism. Now, I'm certainly not trying explicitly or implicitly to state that you have autism, but there seem to be definite signs of an attitude problem making communication difficult here. Aww, so you do understand the difference. And well enough to apply it in practice for an ad hominem attack. @7upMan, it's been a while since we had a troll here. Hopefully people won't spend too much their time feeding it, as it usually is pointless. My, and here I thought the definition of a troll was someone completely unreasonable who was just trying to get a rise out of people. Any other local lingo I should know about? Of course, it's not like I haven't seen peoples argumentative repertoire being reduced to personal attacks once they have run out of counterarguments a million times before. (Hint for Xarg and Aluminum: The above^ was an example of implication. It hasn't actually happened a million times before, but is a way to communicate that it has happened so many times that I have lost count). noTarget, it'll make the game into just what you're looking for! Never in a million years would I have thought that in the same thread as Broken Glass members, I would be the bigger purist. I mean, you are after all giving me the choice between cheatcodes or hardcore, when all I want is LGS!Thief. We're a niche game and proud of it. I find the comparisons between us and EM a little amusing for that reason; they're specifically watering down the stealth experience to try to try and appeal to everyone. That's not our approach and never has been. I thought Thief was a niche game. Was it really necessary to make TDM even more so by design? Hell, in that case you guys might as well close the Greenlight thread right now, for that is the gateway to mainstream, if there ever was one. Also, by this logic, would EM have been totally in the clear if only their twisted end result was more difficult than LGS!Thief? Seems to me deviation is deviation, Broken Glass has just deviated far less than EM has. Are you planning on just continuing to complain? You've made your opinion pretty clear, and at one point said you were going to wait and see if there were any other complaints along similar lines, which seemed like a reasonable approach. On the other hand, continuing to restate that you don't like it, and making pointedly misleading statements about what the "dev intent" was, isn't going to accomplish anything. Dude, I was more than ready to let the issue rest until Aluminum started this obvious counterthread. Everything else has been attempts to clarify my position against endless misunderstandings and misrepresentations, some probably deliberate. I am more than happy to let the issue rest unless the opposition keeps pretending not to understand what I have been saying. Edited by IHaveReturned At the moment? The fact that it is a rather big percentage of blind luck determining my ability to get close enough to try in illuminated areas, no matter what Perception difficulty I am playing on. In general, getting humbled at something I expected to be good at, getting frustrated with several reloads on the same damn guard, and ragequitting. Feeling like the mechanic needs tweaking anyway, so learning to compensate is a waste of time. That sort of thing. Your continued harping on this besides-the-point issue does not inspire me to keep discussing this with you. Because there is no besides-the-point-issue here, this is the way Blackjacking works. Learn to compensate for it or don't. I'm trying to help (and being a dick about it kind of Yes it's hard, I had trouble with it at first too. I used to only get 1/3ish attempts at blackjacking, sometimes I missed completely, sometimes I hit the helmet etc. Then it was made easier a couple revisions ago and I had less failed attempts. It took a while but now I'm really good at it, like anything it requires effort to learn. I've been playing FPS all my life, ever since I was a kid, but I wasn't good at QuakeLive when I started back in 2009 during Beta. Took me YEARS to get to Tier 5. And I'm good at Thief 1/2 but I am not very good with Thief 3 because it's so different. But I worked at it enough to beat the game on the hardest difficulty, and even tried the minimalist mod, but found it too difficult. And BTW, you can't see it but in the dev sub forum we have a thread that has collated all the TDM news from around the internet, and I have to say, of all the discussion going on, no one seems to be complaining about blackjacking being hard. Combat being hard yes, general bugs because of bad installs and stuff, sure, but that's it. There is 0 outcry for easier/different blackjacking from the general internets community. Yes I was trolling you with this thread a bit, but mostly it was supposed to illuminate how easy it becomes once you get good at it. I always assumed I'd taste like boot leather. Great strawman at a glance. 8/10 for effort. Thank you, thank you. I now stand ready to rate your explanation of where I have misrepresented anyone. Seems only fair. Really. Considering the pride I put in avoiding logical fallacies, I damn well need to know if I have comitted one without noticing. And BTW, you can't see it but in the dev sub forum we have a thread that has collated all the TDM news from around the internet, and I have to say, of all the discussion going on, no one seems to be complaining about blackjacking being hard. Combat being hard yes, general bugs because of bad installs and stuff, sure, but that's it. There is 0 outcry for easier/different blackjacking from the general internets community. Give it some time to let the glitz and glamour wear off. I remember at least a few of those articles admitting that the author was still in the process of downloading or barely done with the training If few or none of the reviews mention anything, I'll start considering that the problem is with me. Not holding my breath. Edited by IHaveReturned @IHaveReturned I truly feel bad that you do not love The Dark Mod as much as I do. It is by far my favorite game. I have had difficulties knocking out guards from time to time, but for the most part I have been successful. I only remember guards with extremely acute senses on a mission called Awaiting The Storm. Those fellas sure have great ears! TDM is not "my perfect video game," but it is the closest thing to it! There are things about it that I would like changed e.g. ledge hanging being added, enemies' melee attacks hurting each other while they're attacking the player, sprint not affecting the backwards and sideways walking of the player, and quite a few more. None of my requests have made any changes in the game as far as I know. I'm fine with this. Judging by your posts, you seem vastly more intelligent than I am. Even though TDM does not play exactly the way the Thief games did, I'm quite confident that if you keep at it, you will be great at it. Edited by SirGen • 2	{"url":"https://forums.thedarkmod.com/index.php?/topic/15165-sometimes-i-wish-tdm-were-harder/page/3/","timestamp":"2024-11-05T12:11:23Z","content_type":"text/html","content_length":"473990","record_id":"<urn:uuid:df97ac87-f859-4f78-8433-57cf8330d501>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00490.warc.gz"}
Chemical Kinetic Rate Coefficient Estimation Given Partial Information Habib Najm, Khachik Sargsyan, and Bert Debusschere of the CRF, working with Cosmin Safta of Sandia’s Computer Sciences and Information Systems center and Robert Berry, formerly at the CRF and presently at The Climate Corporation in San Francisco, California, have demonstrated the use of a data free inference (DFI) approach [1] for uncertainty quantification (UQ) in reacting flow In general, predictive computations of reacting flow should provide an estimation of uncertainty in model outputs. This, in turn, requires estimates of uncertainty in models and their parameters. This uncertainty is rooted in several sources, such as the requisite choice of species and reactions defining the chemical kinetic mechanism structure, as well as the values of chemical model parameters, including both thermodynamic and chemical kinetic rate parameters. In particular, chemical reaction rates are typically modeled using Arrhenius expressions. Each of these involves a baseline of three parameters: the pre-exponential constant A, temperature exponent n, and activation energy E. These parameters—commonly estimated from experimental measurements, ab-initio computations, or rate rules—exhibit a degree of uncertainty. UQ methods can be used to propagate chemical kinetic rate parameter uncertainties forward through reacting flow computations to generate estimates of uncertainty in model predictions. In solving this forward UQ problem, using a probabilistic formulation to represent uncertainty is the best way to adequately deal with large uncertainty and model nonlinearity. This context requires a full probabilistic characterization of the uncertain input space, through, for example, a joint probability density function (PDF) on the uncertain parameters. Such PDFs are, however, typically not available in the chemical kinetics literature. At best, the literature provides nominal values and error-bars on A, only nominal values of (n, E), and no information on the correlation structure among the three parameters. Yet these parameters are clearly correlated via the procedure used to estimate them. Assuming them to be simply independent is unacceptable, because this can result in the assignment of significant probability to parameter values that do not fit the data and are inconsistent with the physics. Also generally unavailable is the original data used to estimate the parameters. If this data were available, one could arrive at the requisite joint PDF by redoing the fitting using statistical inference methods. In the absence of the original data, UQ in reacting flow computations presents the challenge of constructing a joint PDF on uncertain model parameters that is consistent with whatever information is available. The DFI approach explored in this work—which is based on a maximum entropy formulation on the joint data-parametric space—seeks to discover a joint PDF for the data-parameter space that satisfies given constraints. It relies on a random walk procedure on the data space, proposing data sets and retaining those that are consistent with the given information. The data consistency check involves the use of Bayes’ rule to estimate a posterior PDF on model parameters, whose statistics are compared to the given constraints. Posterior PDFs available from consistent data sets are averaged (or “pooled”), yielding the pooled DFI posterior density. The team demonstrated the method on homogeneous ignition of a stoichiometric methane-air mixture at atmospheric pressure over a range of initial temperatures. Using synthetic noisy data for ignition time dependence on initial temperature, the researchers employed Bayesian inference to establish a joint posterior PDF on (ln A, ln E) parameters of a single-step global methane-air kinetic model. Simulating a typical experimental-fitting scenario, this PDF was discarded after retaining select statistics, namely nominals and marginal bounds. DFI was used to discover a joint PDF on the two parameters that would be consistent with available information—namely the nominals/bounds, the range of initial temperature of the synthetic “experiment,” the global methane-air model used for fitting, and the data noise structure—in the absence of the original data. Figure 1: Two-dimensional marginal parameter posteriors on (ln A, ln E) resulting from pooling 175,000 consistent data sets (red), compared with the reference posterior (black) Notably, the study illustrated the strong correlation between the kinetic rate parameters evident in the reference posterior PDF obtained from the Bayesian inference procedure. Further, results highlighted the effectiveness of the DFI procedure in generating data sets that are clustered in the vicinity of the missing data. Moreover, the consistent posteriors were found to exhibit very similar structure to the baseline density, with only minor variations between the structures of each. Consistently, the pooled DFI posterior, based on 175,000 consistent data sets, was very close to the baseline density, as shown in Figure 1. This remarkable agreement is not necessarily expected, and may be less notable in other contexts, depending on the available information on the system at hand. The agreement suggests that the available information in this case is very nearly a sufficient statistic, encapsulating most of the information in the missing data. This encouraging performance motivates the extension of the method to more realistic situations. Work in progress will assess the performance of the algorithm, given bounds on a subset of the parameters, such as only A. In addition, the method will be applied for the construction of the uncertain input space of an elementary-step kinetic mechanism. Initial testing will target a hydrogen-oxygen mechanism, and will involve cataloguing of available information from the literature on each of the reactions. In this context, correlations among parameters of each reaction, as well as among those from different reactions, are expected, depending on the fitting models employed. Construction of the joint PDF on the full set of parameters of this hydrogen-oxygen model will enable meaningful and reliable estimation of uncertainty in associated reacting flow predictions. [1] Habib N. Najm, Robert D. Berry, Cosmin Safta, Khachik Sargsyan, and Bert J. Debusschere, “Data Free Inference of Uncertain Parameters in Chemical Models,” International Journal for Uncertainty Quantiﬁcation, in press.	{"url":"https://crf.sandia.gov/2013/05/13/chemical-kinetic-rate-coefficient-estimation-given-partial-information/","timestamp":"2024-11-06T17:27:29Z","content_type":"text/html","content_length":"56987","record_id":"<urn:uuid:2b56a1e5-0c58-4c79-9cd8-8fdb41f02e44>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00701.warc.gz"}
(thing) by Glorious Weirdo Mon May 15 2000 at 6:28:03 in China, a symbol of everlasting love, especially in a pair. A jade butterfly may be given by a woman as an engagement present to her significant other. An insect known for brightly colored wings. They flutter about, yet despite that, are strong fliers, Monarch Butterflies can migrate thousands of miles. The colors on their wings stem from microscopic scales that glimmer in the light. Moths are also of the same family, Lepidoptera. Butterfly is probably the most difficult stroke to master. It requires coordination, strength, flexability and good cardiovascular fitness. Without one of these four elements you may find that after just one length of butterfly you will be gasping for air while wondering how much dilute infant urine you have just ingested. If you still fancy trying this stroke and are crazy enough to rely on the information from a peer managed collabarative database to teach you how to go about it then try this; Add count: 3 Add categories: [dex] [xbd] [del] Jobs' Notation: SET > SAME or OP OUT [DEX] > OP CLIP [XBD] [DEL] Butterflies are usually used for blood draws from the lower arm and wrist. I prefer to call them "mosquito"s, as the name fits much better, but humorless phlebotomists might not understand what you are talking about when you request for them to use one, as I found out to my chagrin on my last doctor's visit. The move is one in which you move your hand in a figure eight motion while the ball rolls on the top of your hand. Practice each of the steps until you are comfortable with them before advancing to the next step. Make sure that both hands are equally trained. The whole learning process, from start to finish, will probably take you two months, although you can move much faster if you practice frequently. Warning: Do not practice near computers, windows, delicate china, &tc. (Especially not if you live in the 18th story of a New York City skyscraper. Not that I'm speaking from prior bad Laughter and happiness is what anyone can see reflecting from her eyes. Through the mirror that is all that can be seen, beautiful eyes. What he speaks is hardly relevant, it’s the sound of his voice that captures her into the world she is always drawn into when he is like this. She’ll sometimes get lost in the soothing tones of his speech. Anyone can see, all she needs is him to pay attention to her, not to kick her in the gutter, to love her, as she loves him would maybe be too much, and she doesn’t pray for this. She is just a child, influenced at his every word. Spiraling love blasts from her like light shone from a star. Endlessly she fights the battle everyone knows is lost. Everyone is convinced she is wasting her time, she is convinced they are wrong. He blasts her with more blows than he knows imaginable, in everyway he creeps into her being. She promises never to fall in love with a stranger, so she learns his name, learn all about him. And she never forgets A butterfly with the most beautiful wings, flaps around her cage, slowly tiring, the steady beat of her flight slows, and she is caught in a net, pinned down in a photograph album. Captured, mesmerizing others, and she too is blind to what she does. Whilst he speaks to her, when she see's his face, she finds all the flaws, and all his imperfections, and loves them. She snaps out of her trance when she realizes he is waiting impatiently for an answer, and she replies, anxiously waiting, he is pleased and so is she. It’s her fault; she never lets the world know. Locks herself up in a land of make-believe and shuns every escape route away. He can see what the world can do, slowly without knowing how, she realizes she broke his heart too. Like a firefly she lights up with a buzz when the phone rings, with his slow words, he’s putting out her flame. Every person says, please forget him, he is ripping your wings, he’ll break your heart, they don’t understand; he mends it. She whispers and she hangs up, his voice stuck in her head, his hand clutching her heart, and his not knowing, causing her tears. He stares into the starry night, thinking of what is missing, and she never runs across his mind. She is but another subject in his grasp. He has control of her mind, yet he doesn’t know. Sometimes he hates her but he doesn’t know how to tell her, he plays with the idea but dismisses it, and lets it go like smoke. People tell him, all about how she is his, but he brushes the idea aside Life’s not fair, why do these temptresses tease me? SO out of reach, his thirst isn’t quenched. What of the fairy princess, who sits and sighs, until he calls upon her? What is she to him if not a princess? She is but an unwanted warrior with an authority, she is there for him to lean on. Whenever he needs her, to please her was but practice, what’s the real thing? Like a roll of film, with a burnt out hole, a gap is still missing, at the end of the night, before he drifts into sleep, the thought of her may cross his mind…but weather he shields himself from her remains for him to explain to himself, in his dream, his shield keeps her sane, and keeps the demon unleashed She wishes nothing more for him to be happy, and as he falls asleep, he is protected from the evil in her gaze, because of her love. The smile that plays across his lips as he dreams is what lets us know, but shhh and let it be untold. In cooking, to butterfly something means to cut it down the centre almost, but not completely, through the item. The two halves are then pressed flat to make a butterfly shape. Butterflying is often done to whole chicken, particularly small Cornish hens, and is easy to accomplish for the home cook. Simply cut out and remove the backbone using kitchen shears or a sharp knife. Then turn the bird breast side up and press down on the breast bone with the back of your hand to flatten the bird. A butterflied bird will cook more quickly than one that is left whole, and can be run under the broiler for a few minutes before serving to crisp the skin. A Cornish hen can be cooked completely under the broiler. Shrimp can also be butterflied; it works best with larger ones. Cut them through the back as if removing the vein and then flatten; these will cook very quickly. Butterflied pork chops are also common; the thick but tender boneless chop can be baked or grilled. This writeup is on an option combination, or strategy, known as a butterfly. It is an option strategy that is used to make money when an underlying ends up exactly at a certain level at expiry, while maintaining a modest risk profile. An ordinary butterfly consists of three option positions on the same underlying with the same expiry, but a different strike. The call butterfly consists of: As such, a butterfly has the maximum payoff when the underlying ends up exactly at the middle strike. The minimum payoff, is, conveniently, 0. As such, a butterfly is never free, and one always has to pay for it. The payment is essentially equal to the chance the underlying ends up at or close to the middle strike, and depends chiefly on the strikes, the time to expiry and the volatility of the It is noted that a similar construction that uses puts exclusively also can be made. The payoff is quite similar. For American options, there can be a difference in value between the two that is a result of the possibility of early exercise. This sounds like a psychedelic rock band, and actually, there IS a psychedelic rock band with that name. However, it is also a different way of building a butterfly. Instead of the options above, it consists of: The iron butterfly is a such the "inverse" of a regular butterfly. Buying an iron butterfly indicates one hopes an underlying will move away from the middle strike, whereas with a regular butterfly, one hopes the opposite. Another way of looking at this is stating that a short iron butterfly is the same as a long regular butterfly, only for the iron butterfly, one does not have to pay up front, but at expiry. By selling it, one pockets the premium, with a maximum potential loss of X. In general, large trading companies prefer the iron butterfly over the regular one, as it does not require a usually illiquid in the money leg. The butterfly plays an important role in option theory. As mentioned above, a butterfly is essentially paying for the chance an underlying ends up at a certain strike. Now, imagine it is possible to price options with any possible strike. By setting up a very narrow butterfly, of which so many are bought that the maximum payoff is 1, one can estimate the probability it ends up in that region. By taking mathematical limits, it is as such possible to compute the implied probability density an underlying ends up at a certain strike. Such an infinitesimal butterfly is called an Arrow-Debreu Butterflies are not used extensively in investing. The main reason is that a butterfly has a high cost: one needs to trade either one liquid and two illiquid or 4 liquid options, for a relatively "flat" payoff. This basically means that if one would habitually trade butterflies, the combination of a rather efficient market in options and these high costs would likely mean that one would lose money in the long run. Professional parties, that can trade at much lower costs, do trade butterflies. One of the reasons they can do this is because they don't trade the individual options, but rather the butterfly as a whole. A butterfly is not very risky - at most as risky as a straddle, but usually a lot less - and as such, they can trade them at sharp prices with market makers. Some exchanges offer the option to make so-called strategies, in which bid or offer in the whole butterfly can be made. This is preferable to having to trade the separate legs, because it is likely other market participants will recognize the trade as less risky and as such will do the trade more cheaply. Butterflies are option constructions that can be used to profit from a scenario in which an underlying ends up exactly at a strike. As such, they offer a very finely tailored payoff with limited risk. The disadvantage is the relatively high costs needed to set them up. They can also be used as a tool to compute the probability density an underlying ends up at a certain strike at expiry. Disclaimer: This writeup is not meant as investment advice. Don't take investment advice from anonymous strangers on the Internet. (definition) by Webster 1913 Tue Dec 21 1999 at 22:17:30 But"ter*fly` (?), n.; pl. Butterflies (#). [Perh. from the color of a yellow species. AS. buter-flxc7;ge, buttor-fleoge; cf. G. butterfliege, D. botervlieg. See Butter, and Fly.] Zool. A general name for the numerous species of diurnal Lepidoptera. Asclepias butterfly. See under Asclepias. -- Butterfly fish Zool., the ocellated blenny (Blennius ocellaris) of Europe. See Blenny. The term is also applied to the flying gurnard. -- Butterfly shell Zool., a shell of the genus Voluta. -- Butterfly valve Mech., a kind of double clack valve, consisting of two semicircular clappers or wings hinged to a cross rib in the pump bucket. When open it somewhat resembles a butterfly in shape. Dance Dance Revolution Lepidoptera Crazy Town Pinkerton Did Chuang Tzu dream The Defragmentation of the Inner Self Dance Dance Revolution 2nd Mix Phlebotomy Tapping The Vein Towa Tei swimming The little bird sings a big song french onion roast Speak, Memory Fairies Log in or register to write something here or to contact authors.	{"url":"https://m.everything2.com/title/Butterfly","timestamp":"2024-11-09T07:38:36Z","content_type":"text/html","content_length":"84417","record_id":"<urn:uuid:d021e69d-614e-4524-9b96-d507d77a7377>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00462.warc.gz"}
expert to do your Search our library of math homework solutions using Google College Algebra Details: Questions [ Print this page } Course: TECHBUS Business Math Unit: Midterm Exam Answer the following questions below: Page 1 of 16. For problems #1 through #3, find the weekly earnings. All hours over 40 hours are paid at timeand- a-half. 1) $4.75 p... Tags: Business Math Homework Solutions Basic Algebra Details: Course: VLA Math Algebra 1_1 Unit: Proportions, Percents, and Statistics Unit: Functions and Relations A LOT OF OTHER UNITS AS WELL. SEE FILES FOR DETAILS! Tags: Algebra Homework solutions General Coursework Details: 1. Find the domain of the function f(x) = . 1) ______ (A) D = (- , 2) (2, ) (B) D = (- , -2) (-2, ) (C) D = (2, ) (D) D = (-2, ) (E) none of the above 2. A cell phone company offers a plan of $39 per month for 400 minutes. Each additiona... Tags: Algebra Homework solutions Inverse Functions Written Homework Details: Section 3.5: Inverse Functions Written Homework Remember: Any submissions to my email after 11:59pm will NOT be accepted and will result in a grade of a 0. Homework must be presented in an orderly manner with the problems in order, running in a single c... Tags: algebra, section, section 3.5, written, homework, inverse, functions Math107 Quiz 1 Details: MATH 107 6383, Fall 2014 UMUC Quiz 1: Show your work on all the problems. Submit the completed quiz in the assignment folder. Total Points: 100 Due: August 31, 2014 1. Find the distance between -7 and 3 on the number line. ... Tags: Math107, MATH 107, quiz, UMUC, College, Algebra MATH 107 UMUC Midterm Exam Details: MATH 107 6383, Fall, 2014 UMUC Midterm Exam Show your work on all the problems. Submit your completed exam in the assignment folder. Total Points: 100 Due: September 21 1. a. Find the midpoint of the segment with endpoints (5... Tags: Math107, MATH 107, midterm, exam, UMUC, College, Algebra Law of Sines and Cosines, Heron’s formula, Triangle solving Details: This lab requires you to: • Use the Law of Sines to solve oblique triangles. • Use the Law of Cosines to solve oblique triangles. • Use Heron’s formula to find the area of a triangle. • Solve applied problems. Answer the following questions to comple... Tags: MA1310, MA 1310, week5_lab, law of sines, law of cosines, heron's formula trigonometric equations, triangle solving, vector operations, applications on vectors Details: MATH 115 Quiz 5 NAME___________________________ Professor: Dr. Izmirli INSTRUCTIONS The quiz is worth 100 points. There are five problems (each worth 20 points). This quiz is open book and open notes. This means you may refer to your textbook, n... Tags: MATH115, MATH 115, Quiz 5, UMUC, trigonometry	{"url":"https://www.mymathgenius.com/math-homework-solutions/Algebra/7","timestamp":"2024-11-03T09:45:16Z","content_type":"text/html","content_length":"54867","record_id":"<urn:uuid:55485c05-5284-4f72-b391-ba5af6cd6fa9>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00837.warc.gz"}
How does the rate constant change with pressure? \| Do My Chemistry Online Exam How does the rate constant change with pressure? How do we handle it? If one of them is high enough, there is no point to relying on a higher one. In my house I had one of those things where I don’t think I have any control over the pressure around a button, so I knew I could create a button, to be flexible. I was able to set this up to my client where it worked the way I wanted it to, and the part about read this article this button would work was way more flexible than I had expected. The idea was to set a set of functions to determine the pressure of the button in a way that would keep the buttons connected for the current pressure before breaking them or changing the button if the pressure increases or decreases. So it’s what worked as a great result, which was rather neat I think, with how I wrote it and still getting really cool results. official website point was how it really works if you allow the button to move when button is being pressed, which would be something like if it would get pushed out that it’s going to hard force up. I was thinking if you allow the users to move the button after an event, I could eventually measure the pressure of the button and put a force sensor on my knob to increase or decrease the amount of pressing. This is hard to do though, and a very important part of the design. I was going to show you how it works, but since I don’t have a great handle on this or that, I’d just love for you to have someone do this sort of thing. This morning when this is, you can leave a comment if what you want to do is a good experience but your comment is not going to get much more than that. It just has to do with a single button event. After talking with my client, I noticed an inbuilt thermostat that had a button I put down at a certain number of seconds. Having said visit this page I was ratherHow does the rate constant change with pressure? For recent work the rate constant was converted to the “pressure” parameter $\nu=3\ times 10^{-12}k\cdot p$, where $k$ is the speed of sound and $p=\sqrt{\nu/\nu_p^2}$ is the pressure drop, per unit time. A similar constant for many other work-specific thermodynamic phenomena is 1.091046 [@hc7], which is about one out of 5% for the case of an external load as seen in [@hc2], which could be translated to $\nu = 1$ and 6%. In view of the evidence that temperature increases as $\nu$ increases compared to the case of negligible frequency, this indicates that a new problem for estimating the source of negative feedback will develop. When the system is a metal, the oscillation frequency is not the same as the frequency blog particles flying around it. If particles drift towards the origin and approach a random state, their frequency is not significantly different than their position in space. An alternative way to approach this situation is to consider two limiting cases, depending on whether the amplitude of the local oscillation is minimal ($\nu= 0$ or $\nu=\pi$) or maximum ($\nu=\pi$). Example [s4]{}: The metal cell immersed into the liquid, this system is an Ising model [@hc11]. How Much To Charge For Taking A Class For Someone For the case of a moving particle it has initial velocity $(v_m + \Omega \, \delta n)$, where $\delta n$ represents the non-Newtonian density of charge, $\Omega$ contains the thermal pressure difference from the surrounding liquid $p$ as well as kinetic energy due to particle motion, and $\delta n$ is the density of particles as well as the thermal pressure force exerted by the interaction of the metal cellHow check here the rate constant change with pressure?]{} As per Vazquez (2002) and Vergaert (2003): [Applied Optics Principles]{} Springer [54*]{}.[41]{} These authors reported that the amount of heat generated by the die is equal to the rate constant $k_d$: $k_d=0.004$ (Vazquez & Vergaert 2004).[^2] While the rate constant has significant implications for its effective temperature and for the observed heating of Voskletsic areas, those contributions must also be taken into account if such conclusions are to be made. In the thermodynamic limit, the rate constant depends on temperature (for example, it varies between 0.02 and 0.025 and at this temperature the thermo-mechanical distribution function in different regions of the periodic table changes rapidly as $T^{-1}$, see (Bertram-Pritchard & White 2000; Rokhinson et al. 1999; Serrano & Martinez 1999). $${\cal R} = \sum\limits_{j}f_{\Omega }^{(j)} {\bf 1}_{[\Omega = T] /\pi } {\bf A}_ {\Omega }e^{-ik_d\theta }$$ where $$[\Omega] = \pi \sum\limits_{k=0}^{\infty} \lambda_{k} \left(T /\psi_T + \sigma_d /\psi_T\right)^2$$ The phase variables can now be used to obtain the characteristic time $T$ as the thermodynamical limit of the linear (thermodynamic) distribution of the thermal energy $E$: $\{ T(\psi ) \equiv \{ E(\psi )\}_{T=0} = 0$ and the other quantities can be interpreted in terms of the temperature. Combining the respective contributions from the form factor as the temperature is increased and the thermal energy is decreased, the temperature is increased to produce the peak in the heat capacity of the temperature distribution. The same method is found previously (Theorems 8, 9 and 12) for the thermal resistivity whose thermal coefficients are given in (Molleren 1992b, 1994). From the above it follows that $${\cal R}_{\Omega} = \sum\limits_{j}f_{\Omega }^{(j)} {\bf 1}_{[\Omega=T] /\pi} {\bf A}_{\Omega } \frac{(\chi_{\Omega 1} + \chi_{\Omega 2})} {e^{ – i k_d\psi_T/k_d} + {\bf 1}_{\Omega = T } / \pi }$$ The rate coefficient can thus be accessed using the formula after linear algebra: $${\cal R}_{\Omega} = \sum\limits_{m=1}^{N} f_{\Omega }^{\ prime} \left[{\bf 1}_{\Omega \in T} + {\bf 1}_{\Omega \in T}^{-1} \right]$$ Integrating these last two terms we obtain: $${\cal R}_{\Omega} = \frac{1}{2}(f_{\Omega }^{\prime} – {\chi_{\Omega 1}}^{+} \left[{\bf 1}_{\Omega \in T} + {\bf 1}_{\Omega \in T}^{-1} \right] )$$ Solving for the heat capacity we find the results of Arima & Zeller (2004)	{"url":"https://chemistryexamhero.com/how-does-the-rate-constant-change-with-pressure","timestamp":"2024-11-06T23:28:27Z","content_type":"text/html","content_length":"130305","record_id":"<urn:uuid:e72b7a7d-19c8-40fb-afd5-b3ea55bee8ff>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00575.warc.gz"}
100 o'clock What if a day was a hundred hours? I often played this scenario in my head and thought it shouldn't be too hard to simulate. But when I finally got to it, I found that a lot of things with time are just arbitrary. There isn't a fundamental constant that defines time as we currently know it. We just had to agree on some values to get the math going. Why is a second measured relative to the unperturbed frequency of the caesium 133 atom? What did we use before that? Note that frequency is measured in hertz, which is the cycles per second, and the second is measured in frequency, which is the cycles per second. The cycle continues. But time measurement is also self correcting because if we don't measure it right, we can just wait for the next sunrise and use any other method we can come up with to measure the time between two sunrises. But then again, the days of the year are not the same length. So instead of trying to find a perfect definition, I'll just say that a second is an agreed upon length of time. Since we have ten fingers on our hands, it makes sense to use a decimal system. But somehow, we didn't think 10 was appropriate for time. Maybe the Greeks found it was easier to cut a circle into 60 pieces rather than 100 pieces. 60 seconds make a minute, 60 minutes make an hour. So obviously a day should be 60 hours, but no. 24 hours make a day. So if I had to make a clock that turns a day into 100 hours, I'll have to put any logical justification aside. My clock is 100 hours because I said so. So let's see what it looks like. The 100 hour clock. • 1 day is 100 hours • 1 hour is 100 minutes • 1 minute is 100 seconds • 1 sec is a millionth of a day. In this case, a full day will amount to 1 million seconds. The ratio between this 100-hour-day second and our traditional second will be calculated as 1 million divided by 86,400 or ~11.574. Here are the two seconds compared: When rendering a clock that ticks with this new time, I realize that a second is too fast for a human to count. In fact, it's a little over 10 times faster (11.574x). So I invented a new term: Tenet. A tenet is 10 seconds. It's a human friendly time counter. Not affiliated with Christopher Nolan. Now that we have a new second, it is time to create a clock and see how it works: It didn't look so special. It's just a clock with 100 units for hours instead of 12. That's when it occurred to me: why aren't analog clocks divided in 24 hours? Why 12? But then again, the answer is probably because somebody said so. Dive into a new realm of timekeeping with our 100-hour clock. Whether it's tracking adventures, events, or personal milestones, this clock's got you covered. More than just a timepiece, it's a companion for your longer journeys. Embrace every hour with style and practicality – your next moment is waiting! In a 100 hour day: • The Titanic movie runtime is 13h 47m 22s (3h14m). • The movie Tenet is 10h 41m 66s (2h30m) too long. • David Blaine held his breath for 1h18m51s (17m4s). • The average gravitational pull on Earth is 0.84m/s^2 (9.8m/s^2) • The average 30 minute work break in the US remains unrealistically short. • 8 hours sleep is now 33h 33m 33s • I'll be there in 5 minutes turns into “I’ll be there in 35 minutes” What if we give the calendar the same treatment? Not long after I built the clock, I had another shower thought. What if we have a decimal calendar? Let's take a look at the current calendar we use. A day is a day and a week is 7 days. But a month is 28, 29, 30, or 31 days. What's going on there? A year is always 12 months, but a year is also 365 or 366 days. The reason it is so complicated is because the year is supposed to track seasons. The solar year is 365.25 rounded. But in my shower-thought calendar, a day remains a single planetary rotation, a week can turn into 10 days, a month is 10 weeks. Maybe 10 months can make a year. We lose the tracking of seasons, but that's fine we can always look out the window. Our year is now equal to 1000 days. If we decide to go with this calendar, where do we start? If I convert today's date to my calendar, what year should it be? I decided to not be too creative, I will ignore all adjustments that have been made throughout history. Since I am in control here, I'll latch on to the current calendar and work my way back to year zero, which was ...calculating years ago. Since year 0, we are at day ...calculating.... We have the days. Calculating the current year is a simple matter of dividing the days by a thousand. Our new calendar year is ...calculating.... We have to build an actual calendar for it, but I was met with a problem. Now that we have 10 days of the week, we have to come up with new names. I headed to chatGPT for inspiration, after many iterations it gave me some interesting choices. So until further notice, these are the days of the week: Monaris, Tuedra, Wednesol, Thurith, Frinor, Satsan, Sundel, Ochtue, Ninive, Tendec. Not very interesting, but they will look familiar when using only the first three caracters. And of course, we also need month names. Monuary, Secondary, Thursdary, Fortuary, Fiftuary, Sixember, September, October, November, December. No, this is not from chatgpt. They just made more sense to me when you can spot the pattern. I am not sorry Roman emperors. Here is our calendar: Make each day count, one thousand times over. Introducing the 1000-day calendar. This wouldn't be complete if you can't convert your own dates. April 1st, 2013 This blog was born on April 1st, 2013. Interesting facts now that we have redefined a year: • Since a year is 1000 days, it now corresponds to 2.73x of the old year. • The average life expectancy for a man in the US goes from 73.5 years to 26.84 years. • An 18 year old adult is now 6 and a half years old. • Man's best friend now lives for 4.75 years. There are no comments added yet. Let's hear your thoughts	{"url":"https://idiallo.com/blog/100","timestamp":"2024-11-06T08:50:19Z","content_type":"text/html","content_length":"57305","record_id":"<urn:uuid:baafb9e6-9087-4a63-950d-3fad39f5c65d>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00240.warc.gz"}
The varied responses to the monk problem reveals important insights I was a little surprised at the length of the comment thread in the post about the logic puzzle involving the monk Gaito going up and down a hill. On the one hand, I thought that there were some excellent explanations of why there had to be at least one instant where the monk was at the same location at the same time. These involved visualizing the situation in slightly different ways, such as instead of having one monk go up and down on two different days, having two monks going up and down on the same day or using graphs or films and so on. But clearly these arguments were not persuasive enough for some and I have been trying to think why this might be so. In my teaching experience, it is often the case that what seems obvious to you as a teacher is by no means so to the student. It is no use repeating the same explanation more slowly or (worse) more loudly or (much worse) exasperatedly. There is clearly some opposing argument that the student finds persuasive that makes them reject your argument and yet they may not be able to identify and articulate what it is. Instead they feel that there must be some flaw in your reasoning that they cannot put their finger on. It is more fruitful as a teacher to try and figure out what their argument might be, rather than reiterating your own. Comment #56 by larpar is the clue that there is something like that going on here when they say “But something in the back of my head, something I’ve heard before, tells me there is something wrong with comparing simultaneous and consecutive events. Sorry I can’t be clearer.” Also the comment #68: “I have considered being wrong. I even admitted being wrong about one aspect of my reasoning, but I think I have resolved that error. Maybe not. I could still be wrong. One other reason that I’m not yet ready to admit defeat is that this is a “logic problem”. Usually in these types of questions there is some kind of twist that makes the obvious answer wrong. I’m holding out for the twist. It may never come. : )” In such situations one has to try and identify and put into words their argument. In this case, I think it may be because at the back of their minds is an argument along the following lines: If you take any given point on the path, there is no guarantee that the monk will be at that particular point at the same time on the two journeys. This is easy to see since all we have to do is vary the speeds of the monk during the two journeys so that that point is reached at different times. In other words, the probability that the two journeys will coincide at any given point at the same time is of measure zero. This can be repeated for every point on the path, because we can always postulate walking speed scenarios where the two journeys will reach that point at different times. It is a short step to go from there to think that there is no point at which this is guaranteed to happen. This argument is a powerful one that has significant plausibility. Before I present the subtle flaw with it, I would suggest that readers think about how they might counter it. The flaw is that there is a difference between picking individual points in advance and arguing that the simultaneity of time and place cannot happen at any of them, to arguing that it cannot happen at any unspecified point. As an analogy, I am planning to leave my home sometime later today. If I specify an exact time, the probability of my leaving exactly at that time is zero, irrespective of what time I pick. But the probability that I will leave at some time is one. i.e., it is guaranteed. (Let’s leave out the possibility that something might cause me to change my plans about going This is a subtlety that arises with continuous distributions, like the monk’s position along the path or the time at which I leave. If we are dealing with discrete distributions, then the situation is different and their argument holds. For example, if instead of a path, the monk had to navigate 1000 steps while going up and down the the hill. Then there is no guarantee that he will be on the same step at the same time on the two journeys, because the time at which they cross could occur when they are each moving from one step to the next. i.e., the two paths could cross when the monk is in the process of going from one step to the next one up on the upward journey while taking a step down on the downward journey, so that they are not on the same step at the same time even though they are crossing. (This is the point that Deepak Shetty seems to be hinting at in comment #27.) This long response is only partly about the monk puzzle. It is also about (a) the difference in probability estimates between continuous and discrete distributions of outcomes and (b) how important it is to not dismiss those who cannot seem to see what seems obvious to us, because understanding the reasons for their skepticism can reveal important insights about subtleties. 1. Matt G says Creationists use an argument which invokes what you call “measure zero”. The likelihood of humans evolving from bacteria is so low, that it must have been aided by an intelligent agency. Of course, they have picked that outcome in advance; if you replayed evolution on Earth, you’d get something very different. I use the example of meeting an old friend by accident while on vacation. The chances are measure zero, but once you frame it “what are the chances that someone will see some old friend somewhere by accident, you see how it must happen from time to time. Or, think about how many old friends you didn’t see by accident while on vacation. 2. lanir says My initial thought with the monk problem involved the flaw presented here. Obviously walking speeds vary so it seemed obvious he wouldn’t be at the same spot at any point. Then I realized that my mental model was mainly saying that the monk would be at the middle at the same time and if that weren’t so, he wouldn’t ever be at the same spot at the same time of day. Figuring it out was pretty fast once I saw other people talking about it. Having the start and end of the journey be the same both trips is key as it removes outliers from consideration. Then I just had to realize that the spot could be anywhere along the path and my preoccupation with the speed issue disappeared. If he’s going double speed one trip, he crosses his path at the same time of day around 2/3rds of the way through on that trip. Triple speed and he’ll be at the same spot at the same time of day after walking about 3/4ths of the path. 3. larpar says Darn, I guess I’m wrong. I still working through things and don’t fully understand my error. One thing I don’t get is part of Mano’s leaving home analogy. To me, Mano seems like he would a pretty to be a punctual kind of person. If he says he’s going to leave at a specific time, then the probability of him leaving at that time would be pretty high. I don’t see how it could be zero. Maybe it’s the difference between a layman’s use of the word “probability” and a physicist/mathematicians use. I’m also going to do some more studying on “continuous and discrete distributions”. The probability me ever fully getting it is pretty low, but it’s not zero. : ) Thanks to all, and especially Mano, who tried to set me straight. I hope I haven’t wasted to much of your time. 4. Holms says Larpar, if someone says they will leave at a certain time, what are the odds the person will leave at that exact time? Not just to the minute, but to the second, the microsecond, the nanosecond 5. DonDueed says larpar@3: I think Mano’s example may be confusing because the notion of “time I leave home” is rather fuzzy. What exactly does that mean, physically, and how accurately is it specified? Instead, imagine Mano has a motion-activated switch to turn on the lights when he enters his closet. Now state the problem as: some time today he will enter the closet. What time does the light turn on? If you pick one particular microsecond, the probability of its being that exact time is nearly zero. But we still know that the light will turn on at some time today. Microsecond timing still not convincing? You can make the time even more specific, and so on until you are satisfied that the chance of his triggering the light at any particular time to be zero. 6. DonDueed says Holms: GMTA! 7. Matt G says Speaking of low probabilities, I just remembered one of the weekly “science questions” I asked my students a few years back. It’s about bridge, so I’m sure you’ll appreciate it, Mano! I told them that the probability of being dealt a hand of all hearts is 635 billion to one. Now, you sit down for a game and get dealt a hand. What is the probability of being dealt that hand? 8. file thirteen says Matt #7 Not being familiar with bridge, I don’t know which cards are used in the game or how many cards are in each hand, and consequently whether a hand of all hearts must contain all the hearts used in the game. But if it does, then 635 billion to 1 I guess (assuming you’re not talking about the probability that you’ve been dealt the hand you’ve just received, which should be 100% without any shenanigans going on) 9. John Morales says file thirteen, the point is that all hands are equally likely (in this case, equally unlikely) given a random deal. 10. Matt G says I use the bridge hand probability to illustrate why the creationist personal incredulity argument fails. There’s only one perfect hand (four actually), but there is a huge number of hands that guarantee a win for even a beginner. The chances of getting “humans” again if you replay evolution are minuscule, but there will be a huge number of ways of getting some self-aware life form. 11. Mano Singham says larpar @#3, It is not a matter of how punctual I am. As Holms has pointed out in #4, since time is a continuous variable, the probability of my leaving at an exactly specified time (say 11:15) is zero. This fact is disguised by the way speak of time in real life, because we treat time as if it moves in discrete chunks, the way an analog clock’s second hand jerks between seconds or a digital clock jumps from one second to the next. So when I say I will leave at 11:15, we usually mean sometime in the one minute time interval between 11:14:30 and 11:15:30. Or if we are want to specify down to the second, we mean the one second time interval between 11:14:59.5 and 11:15:00.5. Then the probability is no longer zero. It is analogous to the difference between the monk walking along a continuous path and going up discrete steps. The probability will get smaller and smaller the more we make the time interval smaller because the number of time interval windows in which I could leave becomes larger. In the limit that the interval becomes zero, the number of intervals becomes infinite and hence the probability of my leaving in any specific interval becomes zero. 12. Ridana says If the monk takes stairs like I do, he’s going to have one foot on one step and the other foot on the next step about half the time. 😀 So which step is he on when he’s not in the middle of moving one foot to the next step? 13. John Morales says Ridana, “So which step is he on when he’s not in the middle of moving one foot to the next step?” Exactly. Steps may be a good metaphor for discreteness, but traversing them is not a good metaphor for discreteness. Call it… um, a step function. 🙂 However, this particular monk is a true master of woo, and just teleports instantaneously from one step to the next. 14. Mano Singham says We can overcome your problem by saying that the monk, as part of his ascetic practices, jumps from one step to the next, so that there is a time interval when he is in the air and not on any 15. John Morales says Mano, I think my response to Ridana is better than you. The steps were supposed to make the journey consist of discrete steps, but being in the air reintroduces the continuity. Step function. 16. Mano Singham says John @#15, No, the question was whether the monk “will be on the same step at the same time on the two journeys”. If he is in the air between steps, then he is not on any specific step for that time interval. His trajectory remains continuous. It is the times that he is on a step that is discrete. 17. John Morales says Mano, I get all that. This is a subtlety that arises with continuous distributions, like the monk’s position along the path or the time at which I leave. If we are dealing with discrete distributions, then the situation is different and their argument holds. OK, this is what I got. You replaced the (continuous) path with (discrete steps), but you did not similarly replace the monk’s position on the [path\|steps]. In short, the locus of the monk’s journey remains continuous, except that now only a portion of the path qualifies (the steps, not the distance between them) as a place where simultaneity may (And yes, I bear in mind what you wrote about pointless perseverance. Was good advice) 18. Rob Grigjanis says Mano @11: It is analogous to the difference between the monk walking along a continuous path and going up discrete steps. A distinction without much difference. There is a built-in discreteness. It’s called ‘walking’. But maybe the monk has attained a level of enlightenment which allows him to glide along the path… 19. tuatara says Sorry in advance for the long post. I had doubts about the monk puzzle actually being one of logic. It doesn’t seem (to me) to come near to other logic puzzles, such as the joke about three logicians going into a bar and the bartender asking if they would all like a beer. This seemed more like just a maths puzzle. What set me to thinking more deeply about the puzzle though was seeing the answer stated, without much explanation, that if the two journeys are plotted they will intersect with no explanation as to why, beyond that they will intersect. I am sure we have all seen how data can be graphed in such a way as to be deceptive, so I like to know why things are as they are rather than just be given a graph and told it is so. I also know that I am not a mathematician or physicist (I in fact gave up on school at 15 and only achieved a high-school level maths and general science education up to that point), and that most other commenters here are far more educated in these matters than I. So, I knew I was probably wrong but wanted a real reason why, which I wasn’t really getting from the other comments. So I went away and approached the puzzle my own way. I thought about what information was necessary to be able to work it out and what was irrelevant. I decided the start and end times are not relevant. It is sufficient that the duration of each journey is the same. I also decided that any stops along the way or retracing of steps were also not relevant either. I also realised that only two points are really essential for working it out. While the starting point is necessary for determining how far the first run has travelled in any given time, once that is established the only important points are the distances from the start of journey 2 either monk reaches at any given time. The other necessary information is that of the velocity of each journey or part thereof, and that a difference in velocity for each journey is essential to recreate the original puzzle in this more simple way. So, I imagined Usain Bolt running an olympic 100m track in precisely 20 seconds (but an Usain Bolt who can accelerate and decelerate instantly). On the first run from point a (starting block) to point b (finish line), he instantly accelerated from 0 m/s to 9 m/s, maintaining this speed for exactly 10 seconds, then decelerated instantly and maintained 1 m/s for the remaining 10 seconds. So at 10 seconds he was 90m from his start but more importantly was 10m from the point from which he will begin his second run. On his second run from finish line to starting block, he maintained exactly 5 m/s for the entire 20 seconds. So 10 seconds in he has travelled 50m. Obviously sometime in the first 10 seconds the two runs cross when comparing distance from point b. Then I counted back. At 9 seconds on the first run he was 81m into the journey or more importantly 19m from the beginning of the second run, at 8 seconds it was 28m from the beginning of the second run, at 7 seconds he was 37m from it, at six seconds 46m etc. On the second run, at 9 seconds he had travelled 45m, at 8 seconds 40m, at 7 seconds 35m, at 6 seconds 30m etc. It was immediately obvious to me that sometime between 7 and 8 seconds into either journey the time elapsed and distance from the same end (finish line of the 100m olympic track) for each journey I didn’t feel the need to calculate it exactly. In fact with my mathematical skills it looks fiendishly difficult to work out precisely, even with what appear to be simple starting assumptions. More importantly though is that I had already felt that I was initially wrong in my answer to the original puzzle but now I had what to me was more proof than just a graph from a mathematician (respectfully of course). Then I read where the idea was introduced of two films superimposed, which shows the same thing that I had concluded but in a very visual way. Confirmation! That is what worked for me at least. Two things stand out to me from my model. 1. The point of convergence will always occur before the slowest monk to travel half the distance has done so. I think this is also true of the original puzzle. 2. Not so sure about this one because it may be an artifact of my model, but perhaps the point of convergence will always be before half the time of the whole journey has elapsed. I have been busy with work and other things so haven’t had a chance to properly think about the second one, but I am sure someone reading here will be able to tell me if it is the case. I am just happy to be educated. Wish I had this attitude in my 15 year old self! I am 54 now. Sorry again for the long post. 20. another stewart says @3: I suspect that your intuition has been trained on finite and discrete sets of data, such as tossing coins, rolling dice, or dealing cards. Here we are dealing with continuous sets of data. Continuous sets of data imply an infinite number of possible values, so that chance of picking one particular value (such as the precise time* Mano leaves his house) is 1/infinity, or infinitesimal. (Infinitesimal means smaller than any possible non-zero number) * Mano leaving his house is here modeled as an instantaneous event, rather than a process of non-zero duration. So that example, while having the advantage of being concrete, can be challenged as having unrealistic assumptions. A more abstract, but unchallengeable example, is to randomly select a real number between 0 and 1. There is an infinite number of real numbers in that interval. The chance of selecting any particular number is in layman’s terms zero. (Infinities are counterintuitive; in that range there is an infinite number of values expressible as x/y where x and y are integers, but the chance of picking one of them is still zero -- this is where the term measure zero mentioned in the prior thread comes in.) Returning to the original problem: It is clearly possible that there is a point which he is at at the same time of day on both days. It is clearly possible to adjust the schedules so that he is no longer at that point at the same time of day on both days. And that clearly applies to any possible point on the path. That shows that no particular point has to be a point of coincidence. The difference might be considered subtle, but that’s not the same as showing that there doesn’t have to be a point of coincidence. Whenever you adjust the schedules to eliminate one point of coincidence you create a different point of coincidence -- you can only move the point of coincidence -- you can’t eliminate it. (Allowing retracing of steps makes things more complicated -- you can have more than one point of coincidence -- but while you can reduce the number of points of coincidence you can’t eliminate the last one.) A different formulation of the problem would be a locomotive on a single-track railway line. (Imagine a turntable at the end of the track to turn the locomotive round.) Assume the frontmost part of the locomotive is on its centre line. Then, can it be guaranteed that the frontmost part of the locomotive is at least one point at the same time of day on both days? The answer is that same, but many of the “loopholes” discussed in the previous thread look less reasonable, the potential variance of the path in three dimensions being much reduced, and the monk being replaced by something much closer, on a human scale, to a mathematical point. 21. another stewart says I don’t know how precisely a perfect bridge hand is defined, but a hand of 13 hearts is not unbeatable. You could outbid it with a hand of thirteen spades. Twelve spades including the ace and a second ace is not a guaranteed win (if the first suite led is the same suite as the second ace, and the remaining spade is used to trump the first trick), but the chances are still good enough to justify outbidding a hand of 13 hearts. The unbeatable hands are those that are composed of consecutive top cards in all suites, or are long enough in a suite to guarantee being able to squeeze out any remaining cards, e.g. A,K,Q,J plus 5 other cards in a suite. 22. Rob Grigjanis says tuatara @19: Not so sure about this one because it may be an artifact of my model, but perhaps the point of convergence will always be before half the time of the whole journey has elapsed. Not necessarily. Let’s change the problem slightly for simplicity. The duration of both trips is 5 hours, and the distance is 10 miles. On the trip up, the monk walks a steady 2 mph. Due to an old soccer injury, his knee is buggered the next morning, and he can only hobble at 0.5 mph for 4 hours. So he’s only gone two miles by then. But that’s the same point that he got to after 4 hours the previous day (8 miles from the start). So that is the point of convergence, well past half the time. By then, his knee feels better, and he can trot the last 8 miles in one hour. 23. another stewart says @22: or more generally, run the journeys in reverse. If the original journeys had the point of convergence at less than half the time, the reversed journeys will have it at over half the time. (The sum of the convergence times for the original and reversed journeys is the full journey time.) 24. brucegee1962 says I agree with John Morales @17 that taking into consideration the time when the monk is between steps, or with one foot on each, or hopping in the air and on neither step, negates the whole point of using steps in the model in the first place. We use steps to divide the journey into discrete steps — using any of the options above makes it into a continuous journey again. I assume the monk has gained transcendence and can teleport instantaneously from step to step. So we start with one step on the journey. Or, going back to the original problem, we could ask “If the monk leaves at 8 and arrives at 10 each day, what are the chances that the monk will be somewhere on the entire path each day?” The answer is, trivially, 100%. Now we go to two steps. Or, in the original problem, we put a marker halfway down the path. Maybe steps are easier to visualize at this stage, but again, it’s the same problem. Say that there are two steps between the top and the bottom, and the monk can teleport between steps. If we pick a step, what are the odds that the monk will be on that step at the same time? With some thought, we ought to see that the odds are 50%. If we make the steps smaller and smaller, the odds of their both being on any given step get smaller and smaller, but they still always add up to 100%. What Mano calls “measure zero” is not the same as actual zero. I figured out the original problem pretty quickly once I realized that the problem was exactly the same as asking “If two monks who start on opposite ends of a trail meander towards one anothers’ starting points, what are the odds that at some point they will pass one another (thus occupying the same point at the same time)”? I was pleased to read the comments and see that others had made the same subsitution, so I didn’t bother to chime in. But if you start thinking of movement as a series of discrete steps, like teleporting, then someone might ask “What if the two monks start on two different steps, and they both teleport to one another’s steps at the exact same instant?” This highlights exactly the difference that Mano brought up about continuous versus discrete movement. It is also why the various iterations of Zeno’s famous paradox initially seem puzzling. We can imagine time as being a set of infinitely divisible moments, but if we do so, then the notion of movement itself becomes impossible. Two monks walking towards each other can never meet, because the distance between them gets continuously smaller but can never reach 0. But of course, we know that it does reach 0, so the “infinitely divisible” time model must be wrong. To put it another way — we may never be able to measure a “quantum time” unit — a fundamental bit of time which, like the frame rate of a video game, cannot be further subdivided. But to defang Zeno, we should act as if such a unit exists. In other words, we should act as if the universe has a pulse. 25. Deepak Shetty says his is the point that Deepak Shetty seems to be hinting at in comment #27. Im flattered! But I just was being pedantic -- A narrow path is still a rectangular-ish shape , not a line and so we dont have to occupy the same “point”. In normal terms we would have phrased it as Do they pass by each other at the same time -- But that sort of gives the game away. I fluctuate between really liking logic puzzles (usually when I can solve them) and https://sites.ualberta.ca/~esimmt/haveyouheard/problems/sillyPuzzle.html when I cant. 26. Rob Grigjanis says brucegee1962 @24: Two monks walking towards each other can never meet, because the distance between them gets continuously smaller but can never reach 0. But of course, we know that it does reach 0, so the “infinitely divisible” time model must be wrong. Are you just presenting (a possible inference of) Zeno’s view, or do you actually believe this? You can perform an infinite number of ‘steps’ (1,1/2,1/4,1/8…) in a finite time, because both the (highly contrived) steps and the time intervals for the steps add up to finite values. 27. larpar says Mano @ 11 Thanks, that helped, but it leads me to this question. If the probability of you leaving on time is zero, then isn’t the probability of simultaneous events also zero? If probability of simultaneous events is zero, doesn’t that blow the monk problem out of the water? (this might be where the steps analogy comes in?) another stewart @20 Thanks. Like Mano’s reply, that helps push me along the path to understanding. I’m not at the end yet but I’m getting there. I need lots of meditation, and even some backtracking, time. : ) 28. Mano Singham says larpar @#27, You are right that the probability of my leaving at any specific time is zero, as is the probability of any specific point being the crossing point of the two journeys. But that is consistent with the probability of my leaving at some time being one, as is the probability of the two monks crossing at some point at the same time. Perhaps this might help. Take a die with N sides that are equally likely. The probability of the die landing on one specific side is 1/N. Since there are N asides, the probability of it landing on some side is Nx(1/N) which is equal to 1. i.e., it is certain to land on some side. That is the discrete picture. Now start making N larger and larger. The probability of the die falling on one specific side gets smaller but the probability of landing on some side remains one. When N becomes infinite (so that the die becomes a sphere), the probability of landing on any specific point becomes zero but the probability that it will land on some point is still one. That is the continuous picture. Putting it in rather crude mathematical terms, in the limit that N tends to infinity, Nx(1/N) tends to (infinity)x(1/infinity) or (infinity)x(zero) but that product is still one. 29. Rob Grigjanis says larpar @27: If the probability of you leaving on time is zero, then isn’t the probability of simultaneous events also zero? We think in terms of a probability per unit time P(t). So the probability of the event occurring in a time interval t to t+Δt is, if P(t) is smooth enough, and Δt is small enough, about So, yeah, if Δt is zero, this is zero, and you can read that as “the probability of the event occurring at exactly t is zero”. But that’s not very useful 😉 30. another stewart says If the probability of you leaving on time is zero, then isn’t the probability of simultaneous events also zero? As I understand, that’s one of the counterintuitive features of infinite sets -- zero probability events can and do occur. 31. larpar says Mano @28 (and Rob @29) Ok, I follow the dice example all the way through, and it makes a lot of sense. The only problem I see is that I’m going to have to look up my high school math teacher and chastise him for teaching me that anything x zero = zero. : ) One thing that still has me hung up is that it seems like extreme precision is being used for one aspect of the problem but not for other aspects. Here’s another question that may be unrelated but I thought of after reading Mano’s dice example. Plank time (I think that’s the correct term), as I understand it, is the smallest amount of time that we can measure. Can plank time still be divided into infinite divisions. Again, thanks for everyone’s time and patience. 32. larpar says another stewart @30 Thanks! Stating it that way really helps. 33. brucegee1962 says Of course, the all-time greatest example of the counterintuitive logic puzzle is the famous “Monte Hall” problem. https://en.wikipedia.org/wiki/Monty_Hall_problem The intelligence test there isn’t solving the problem — very very very few people get it right when they first consider it. The intelligence test is understanding the answer once it’s explained to you. 34. Mano Singham says larger @#31, In a post about five years ago, I discussed the possible meanings of Planck length, time, and mass. You may want to take a look at it. 35. Rob Grigjanis says another stewart @30: zero probability events can and do occur. Sure. Choose a random real number. The probability that you’d choose 1 is formally zero (maybe I’m misusing ‘formally’. I’m not a mathematician). But that doesn’t have much to do with the real larpar @31: One thing that still has me hung up is that it seems like extreme precision is being used for one aspect of the problem but not for other aspects. Yeah, that is a real issue in problems like this. There’s an assumption of ‘exact time’, or ‘exact position’, or both. Best to assume there is nothing exact. Everything is fuzzy; the position of the monk, the time that the up journey coincides with the down journey. The important point is that there will be time and position intervals which will overlap. Also; Planck time has nothing to do with measurement. It’s just a unit (like the other Planck units) defined by the values of four universal constants. 36. tuatara says @ 22 Rob Grigjanis & @23 another stewart. Thanks. Seems bleeding obvious now! 37. larpar says Mano @34 I read the article and I’ll reply the same way you ended the article; “Isn’t physics fun?” : ) 38. Rob Grigjanis says Me @35: Planck time has nothing to do with measurement To be clear, the Planck units don’t express constraints on measurability. For example, the Planck mass is on the order of a hundred thousandth of a gram. We can measure much smaller masses than 39. Holms says #31 larpar Planck time is actually far smaller than what we can measure with current technology, as is planck length. Contrast those with planck force or temperature… 40. ragove314 says This problem is just the Intermediate Value Theorem from calculus. If f(t) is the difference in heights between going up and down, then f(0) is minus h. (h the mountain height). And if the end of the trip is t=1, f(1) is + h. The function f has to be 0 some place. ( It can not go from below 0 to above 0 without passing 0. ). At that time the monk is at the same place going up and down. We don’t need a monk,, it can be 2 rabbits going up and down. The paths just have to be continuous. 41. another stewart says I see is that I’m going to have to look up my high school math teacher and chastise him for teaching me that anything x zero = zero. : ) In that context anything meant any number. Infinity is not a number, or at least not the type of number he was talking about. (There are such things as hyperreal, superreal and surreal numbers, but they’re not taught to high schoolers.) Leave a Reply Cancel reply	{"url":"https://freethoughtblogs.com/singham/2022/04/01/the-varied-responses-to-the-monk-problem-reveals-important-insights/","timestamp":"2024-11-05T22:36:00Z","content_type":"text/html","content_length":"142261","record_id":"<urn:uuid:caca29b9-0b06-4932-bc5d-e2374f19aab6>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00571.warc.gz"}
A Dark Day For Educational Measurement In The Sunshine State Just this week, Florida announced its new district grading system. These systems have been popping up all over the nation, and given the fact that designing one is a requirement of states applying for No Child Left Behind waivers, we are sure to see more. I acknowledge that the designers of these schemes have the difficult job of balancing accessibility and accuracy. Moreover, the latter requirement – accuracy – cannot be directly tested, since we cannot know “true” school quality. As a result, to whatever degree it can be partially approximated using test scores, disagreements over what specific measures to include and how to include them are inevitable (see these brief analyses of Ohio and California). As I’ve discussed before, there are two general types of test-based measures that typically comprise these systems: absolute performance and growth. Each has its strengths and weaknesses. Florida’s attempt to balance these components is a near total failure, and it shows in the results. Absolute performance tells you how high or low students scored – for example, it might consist of proficiency rates or average scores. These measures are very stable between years – aggregate district performance indicators don’t tend to change very much over short periods of time. But they are also highly correlated with student characteristics, such as income. In other words, they are measures of student performance, but tell you very little about the quality of instruction going on in a school or district. Growth measures, on the other hand, are the exact opposite. States usually use “value-added” scores or, more commonly, changes in proficiency rates for this component. They can be extremely unstable between years (as in Ohio), but, in the case of well-done value-added and other growth model estimates (good models using multiple years of data at the school-level), can actually tell you something about whether the school had some effect on student testing performance (this really isn’t the case for raw changes in proficiency, as discussed here). So, there’s a trade-off here. I would argue that growth-based measures are the only ones that, if they’re designed and interpreted correctly, actually measure the performance of the school or district in any meaningful way, but most states that have released these grading systems seem to include an absolute performance component. I suppose their rationale is that districts/schools with high scores are “high-performing” in a sense, and that parents want to know this information. That’s fair enough, I suppose, but it’s important to reiterate that schools’ average scores or proficiency rates, especially if they’re not adjusted to account for student characteristics and other factors, tell you little about the effectiveness of a school or district. In any case, Florida’s grading system is half absolute performance and half growth: 50 percent is based on a district’s proficiency rate (level 3 or higher) in math, reading, writing and science (12.5 percent each); 25 percent is based on the percent of all students "making gains” in math and reading (12.5 percent each); and, finally, 25 percent is based on the percent of students "making gains" who are in the “bottom 25 percent” (or the lowest 30 students) in each category in math and reading. Put differently, there are eight measures here – four based on absolute performance (proficiency rates in each of four subjects), and the other four (seemingly) growth-based (students “making gains” in math in reading, and the percent of the lowest-scoring 25 percent doing so in both subjects). Each of the eight measures is simply the percentage of students meeting a certain benchmark (e.g., proficient or “making gains”). Districts therefore receive between zero and 800 points, and letter grades (A-F) are assigned based on these final point totals (I could not find a description of how the thresholds for each grade were determined).* Now, here’s the amazing thing. Two of Florida’s four “growth measures” – the “percent of all students making gains” in math and reading – are essentially absolute performance measures. How is this possible? Well, students are labeled as “making gains” if they meet one of three criteria. First, they can move up a category – for example, from “basic” to “proficient." Second, they can remain in the same category between years but increase their actual score by more than one year’s growth (i.e., increase a minimum number of points, which varies by grade and subject). Third – and here’s the catch – students who are proficient (level 3) or better in year one, and remain proficient or better in year two without decreasing a level (i.e., from “advanced” to “proficient”) are coded as “making gains." In other words, any student who is “proficient” or “advanced” in the first year and remains in that category (or moves up) the second year counts as having “made gains," even if their actual scores decrease or remain flat. This is essentially double counting. In any given year, the only way a student can be counted as proficient but not “making gains” is if that student scores “advanced” in year one but only “proficient” in year two. This happens, but it’s pretty rare. One way to illustrate how rare is to see how much difference there is between districts' proficiency rates and the percent of students "making" gains, using using data from Florida’s Department of Education. The scatterplot below presents the relationship between these two measures (each dot is a district). The correlation is almost 0.8, which is very high, and the average difference between the two scores is plus or minus 6.5 percentage points. And the correlation is even higher for reading (0.862). So, 50 percent of district’s final score is based on their proficiency rates. 25 percent is based on the percent of all students “making gains." They are, for the most part, roughly the same measure. Three-quarters of districts’ scores, then, are basically absolute performance measures. And, of course, as mentioned above, they tell you more about the students in that district than about the effectiveness of the schools (at least as measured by test scores). Income is not a perfect proxy for student background (and lunch program eligibility isn't a great income measure), but let's take a look at the the association between proficiency rates in math and district-level poverty (as measured by the percent of students eligible for free/reduced lunch). Here’s another strong, tight relationship. Districts with high math proficiency rates tend to be those with low poverty rates, and vice-versa. Once again, the relationship between poverty and reading proficiency (not shown) is even stronger (-0.817), and the same goes for science (-0.792). The correlation is moderate but still significant in the case of writing (-0.565). As expected, given the similarity of the measures, there is also a discernible relationship between poverty and the percent of students “making gains” in math, as in the graph below. The slope is not as steep, but the dots are more tightly clustered around the red line (which reflects the average relationship between the two variables). The overall correlation is not quite as strong as for math proficiency, but it’s nevertheless a substantial relationship (and, again, higher in reading). Since most students in more affluent districts are proficient, and since they don’t often move down a category, the “gains” here are really just proficiency rates with a slight twist. The only measure included in Florida’s system that is independent of student characteristics such as poverty (and of other measures) is the percent of “bottom 25 percent” of students “making gains." I will simplify the details a bit, but essentially, the state calculates the score in math and reading that is equivalent to the 25th percentile (i.e., one quarter of students scored below this cutoff point). Among the students scoring below this cutoff point, those who either made a year’s worth of progress or moved up a category are counted as “making gains." This measure (worth 25 percent of a district’s final score) is at least not an proxy for student background, and the state deserves credit for focusing on its lowest-performing students (a feature I haven’t seen in other systems). But it is still a terrible gauge of school effectiveness. Among other problems (see here), it ignores the fact that changes in raw testing performance are due largely to either random error or factors outside of schools’ control. In addition, students who move up a category are coded as "making gains," even if their actual progress is minimal. Let’s bring this all home by looking at the relationship between districts’ final grade scores (total points), which are calculated as the sum of all these measures, and poverty. There you have it – a very strong relationship by any standard. Very few higher-poverty districts got high scores, and vice-versa. District-level free/reduced lunch rates explain over 60 percent of the variation in final scores. In fact, among the 30 districts (out of about 70 total) that scored high enough to receive a grade of “A” this year (525 total points or more), only one had a poverty rate in the highest-poverty quartile (top 25 percent). Among the 17 districts in this highest-poverty quartile, 13 received grades of "C" or worse (only one other district in the whole state got a "C" or worse, and it was in the second highest poverty quartile). Conversely, only two districts with a poverty rate in the lowest 25 percent got less than an “A” (both received “B’s”). Any state that uses unadjusted absolute performance indicators in its grading system will end up with ratings that are biased in favor of higher-income schools and districts. The extent of this bias will depend on how heavily the absolute metrics are weighted, as well as on how the growth-based measures pan out (remember that growth-based measures, especially if they’re poorly constructed, can also be biased by student characteristics). Florida’s system is unique in that it portrays absolute performance as growth, and in doing so, takes this bias to a level at which the grades barely transmit any meaningful educational information. In other words, Florida’s district grades are largely a function of student characteristics, and have little to do with the quality of instruction going on in those districts. Districts serving disadvantaged students will have a tougher time receiving good grades even if they make large actual gains, while affluent districts are almost guaranteed a top grade due to nothing more than the students they serve. - Matt Di Carlo * It’s worth pointing out that, while these district-level grades are based entirely on the results of Florida’s state tests, there is also a school-level grading system. It appears to be the same as the district scheme for elementary and middle schools (800 points, based on state tests), but for high schools, there is an additional 800 points based on several alternative measures, including graduation rates and participation in advanced coursework (e.g., AP classes). See here for more details. Really well done. But you'll also notice that Florida's FCAT evaluation system is designed to avoid comparing and assessing performance between higher performing/wealthier schools. The standard of failure/excellence is reaching the middling Level 3 level on the test. But there are five levels, two of which are higher performing than three. But they don't factor into evaluation. Schools are not measured by what they actually score. Only how many kids get to Level 3. One of many ways this system is simply rigged. You should say it directly. You can read about this from the ground in Florida here. Mr. DiCarlo- I have written a response to this post. I hope you will give it some consideration: http://jaypgreene.com/2012/02/08/the-dark-days-of-educational-measureme… Superintendent Klein from New York used a far better approach -- New York City measured schools with the same socioeconomic base against each other. This allowed a school in a poor area to earn an A, if they were among the best in class. This would upset the apple cart for schools getting As because they have the right kids, but it would be a far fairer measurement system Hi Mark, See here: http://shankerblog.org/?p=5785 Thanks for the comment,	{"url":"https://www.shankerinstitute.org/blog/dark-day-educational-measurement-sunshine-state?p=4903","timestamp":"2024-11-04T11:56:12Z","content_type":"text/html","content_length":"50436","record_id":"<urn:uuid:77fef18d-0fd1-4508-92ab-4dc31adea24c>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00836.warc.gz"}
Pipes - math word problem (293) The water pipe has a cross-section 1184 cm^2. An hour has passed 743 m^3 of water. How much water flows through the pipe with cross-section 300 cm^2 per 6 hours if water flows at the same speed? Correct answer: Did you find an error or inaccuracy? Feel free to write us . Thank you! You need to know the following knowledge to solve this word math problem: Units of physical quantities: Themes, topics: Grade of the word problem: We encourage you to watch this tutorial video on this math problem: Related math problems and questions:	{"url":"https://www.hackmath.net/en/math-problem/293","timestamp":"2024-11-06T10:45:11Z","content_type":"text/html","content_length":"75385","record_id":"<urn:uuid:67a018e1-956b-4bec-887d-1bdcb4c799a9>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00853.warc.gz"}
One-dimensional cellular automaton A one-dimensional cellular automaton operates on an infinite "tape" of cells. All Turing machines can be emulated by 1D cellular automata with properly set rules and the number of states. To make the evolution of one-dimensional patterns more visible, they are usually emulated in software such as Golly by non-isotropic 2D automata, in which each new generation of the emulated 1D automaton is born in the line below the original 1D pattern. Like higher-dimensional cellular automata, 1D automata may operate in different neighborhoods, be totalistic or non-totalistic, isotropic or non-isotropic. The most researched among them are so-called elementary cellular automata, where there are two possible states, labeled 0 and 1, and the rule to determine the state of a cell in the next generation depends on the current state of the cell and its two immediate neighbors, thus operating in 1D Moore neighborhood. Although such automation may sound primitive, one elementary cellular automaton, known as Rule 110, was proven to be capable of universal computation. Emulating Rule 110 in 2D cellular automata is often used as a proof of their Turing-completeness. Another elementary automaton, known as Rule 54, is suspected of being Turing-complete as well. Both rules have natural spaceships, guns, oscillators and rakes. Breeders are obviously impossible in 1D automata, because infinite growth in 1D space can only be linear. Rule 110 is non-isotropic, while Rule 54 is isotropic. All patterns in both rules are constantly expanding, analogously to explosion in 2D automata, yet exhibiting complex behavior. See also External links	{"url":"https://conwaylife.com/wiki/One-dimensional_cellular_automaton","timestamp":"2024-11-05T09:52:59Z","content_type":"text/html","content_length":"21542","record_id":"<urn:uuid:74b5fdb5-a755-496e-a713-8dafd0101d9f>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00688.warc.gz"}
Coursnap app 5.19 Splay Tree Introduction \| Data structure & Algorithm Correction: at14:21 9 will be left child of 10 DSA Complete Course: https: https://www.youtube.com/playlist?list=PLdo5W4Nhv31bbKJzrsKfMpo_grxuLl8LU In this lecture I have discussed basics of Splay tree, rotations used in splay tree, advantages and drawbacks of splay tree, applications of splay tree. See Complete Playlists: C Programming Course: https://www.youtube.com/playlist?list= PLdo5W4Nhv31a8UcMN9-35ghv8qyFWD9_S C++ Programming: https://www.youtube.com/playlist?list=PLdo5W4Nhv31YU5Wx1dopka58teWP9aCee Python Full Course: https://www.youtube.com/playlist?list= PLdo5W4Nhv31bZSiqiOL5ta39vSnBxpOPT Printing Pattern in C: https://www.youtube.com/playlist?list=PLdo5W4Nhv31Yu1igxTE2x0aeShbKtVcCy DAA Course: https://www.youtube.com/playlist?list= PLdo5W4Nhv31ZTn2P9vF02bkb3SC8uiUUn Placement Series: https://www.youtube.com/playlist?list=PLdo5W4Nhv31YvlDpJhvOYbM9Ap8UypgEy Dynamic Programming: https://www.youtube.com/playlist?list= PLdo5W4Nhv31aBrJE1WS4MR9LRfbmZrAQu Operating Systems: //www.youtube.com/playlist?list=PLdo5W4Nhv31a5ucW_S1K3-x6ztBRD-PNa DBMS: https://www.youtube.com/playlist?list=PLdo5W4Nhv31b33kF46f9aFjoJPOkdlsRc Connect & Contact Me: My Hindi Channel: https://www.youtube.com/@JennyslecturesHINDI Facebook: https://www.facebook.com/Jennys-Lectures-CSIT-Netjrf-316814368950701/ Quora: https://www.quora.com/ profile/Jayanti-Khatri-Lamba Instagram: https://www.instagram.com/jayantikhatrilamba/ {'title': '5.19 Splay Tree Introduction \| Data structure & Algorithm', 'heatmap': [{'end': 350.289, 'start': 308.564, 'weight': 1}, {'end': 704.962, 'start': 678.271, 'weight': 0.916}, {'end': 1634.921, 'start': 1603.719, 'weight': 0.723}], 'summary': 'Delves into splay trees, their operations, binary tree rotations, zigzag rotations, and the advantages and drawbacks of splay trees, providing insights into their applications and potential to optimize time complexity, with examples and quantifiable data.', 'chapters': [{'end': 43.006, 'segs': [{'end': 43.006, 'src': 'embed', 'start': 0.071, 'weight': 0, 'content': [{'end': 2.439, 'text': 'you So the topic is splay trees.', 'start': 0.071, 'duration': 2.368}, {'end': 4.961, 'text': 'See, splay trees are basically.', 'start': 3.02, 'duration': 1.941}, {'end': 13.987, 'text': 'these are self-balancing, or you can say self-adjusted binary search trees, right, or you can say these are variants of binary search tree.', 'start': 4.961, 'duration': 9.026}, {'end': 19.27, 'text': 'So the prerequisite of this video is you should know what is binary search tree right.', 'start': 14.667, 'duration': 4.603}, {'end': 23.093, 'text': 'Now, in this video we will discuss what are splay trees,', 'start': 19.731, 'duration': 3.362}, {'end': 30.178, 'text': 'how splay trees are different from other balancing binary search trees like avial tree and red block trees.', 'start': 23.093, 'duration': 7.085}, {'end': 40.284, 'text': 'right and as well as we will see advantages and drawbacks of split trees and at last we will see some applications of split trees right.', 'start': 30.718, 'duration': 9.566}, {'end': 43.006, 'text': 'so now we will discuss what is split tree.', 'start': 40.284, 'duration': 2.722}], 'summary': 'Splay trees are self-balancing binary search trees with advantages and drawbacks compared to other balancing trees, and have various applications.', 'duration': 42.935, 'max_score': 0.071, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/qMmqOHr75b8/pics/qMmqOHr75b871.jpg'}], 'start': 0.071, 'title': 'Splay trees', 'summary': 'Discusses splay trees, self-balancing binary search trees, their differences from other balancing binary search trees, advantages, drawbacks, and applications.', 'chapters': [{'end': 43.006, 'start': 0.071, 'title': 'Splay trees: self-balancing binary search trees', 'summary': 'Discusses splay trees, which are self-balancing binary search trees, their differences from other balancing binary search trees, advantages, drawbacks, and applications.', 'duration': 42.935, 'highlights': ['Splay trees are self-balancing binary search trees, a variant of binary search trees.', 'The video covers the differences between splay trees and other balancing binary search trees like avl and red-black trees.', 'The chapter highlights the advantages, drawbacks, and applications of splay trees.']}], 'duration': 42.935, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/qMmqOHr75b8/pics/qMmqOHr75b871.jpg', 'highlights': ['The chapter highlights the advantages, drawbacks, and applications of splay trees.', 'The video covers the differences between splay trees and other balancing binary search trees like avl and red-black trees.', 'Splay trees are self-balancing binary search trees, a variant of binary search trees.']}, {'end': 497.79, 'segs': [{'end': 119.779, 'src': 'embed', 'start': 93.402, 'weight': 1, 'content': [{'end': 103.773, 'text': 'in avial tree and in red black trees it is order of log n means avial tree and red black tree guarantees that the time complexity would be order of log n.', 'start': 93.402, 'duration': 10.371}, {'end': 109.035, 'text': 'because these are balanced trees, self balancing trees.', 'start': 103.773, 'duration': 5.262}, {'end': 113.636, 'text': 'these trees cannot be right, skewed or left skewed, right.', 'start': 109.035, 'duration': 4.601}, {'end': 119.779, 'text': 'so i hope you have seen my videos on bst, on avial tree and red black trees, right.', 'start': 113.636, 'duration': 6.143}], 'summary': 'Avl and red-black trees ensure o(log n) time complexity due to their balanced, self-balancing nature.', 'duration': 26.377, 'max_score': 93.402, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com /video-capture/qMmqOHr75b8/pics/qMmqOHr75b893402.jpg'}, {'end': 199.314, 'src': 'embed', 'start': 166.047, 'weight': 0, 'content': [{'end': 169.408, 'text': 'See, see, on these trees, splay trees also.', 'start': 166.047, 'duration': 3.361}, {'end': 174.09, 'text': 'the operations would be same as BSTT, like a search insertion and deletion.', 'start': 169.408, 'duration': 4.682}, {'end': 180.212, 'text': 'And the procedure would be same as we have conducted these operation in binary search tree right?', 'start': 174.27, 'duration': 5.942}, {'end': 182.052, 'text': 'Plus extra thing is what?', 'start': 180.812, 'duration': 1.24}, {'end': 190.955, 'text': 'Here all the basic operations would be same as binary search tree, with some extra operation, that is splaying.', 'start': 182.532, 'duration': 8.423}, {'end': 199.314, 'text': 'these, all the operations are followed by one operation.', 'start': 195.273, 'duration': 4.041}], 'summary': 'Splay trees have same operations as bst with extra splaying operation.', 'duration': 33.267, 'max_score': 166.047, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/qMmqOHr75b8/pics/ qMmqOHr75b8166047.jpg'}, {'end': 350.289, 'src': 'heatmap', 'start': 302.559, 'weight': 2, 'content': [{'end': 307.563, 'text': 'We are going to make this element, or that particular element as the root of the tree.', 'start': 302.559, 'duration': 5.004}, {'end': 312.287, 'text': 'right. so how you can define splay trees.', 'start': 308.564, 'duration': 3.723}, {'end': 326.979, 'text': 'these are self adjusted binary search tree in which each operation on element rearrange the tree so that that element would be placed at the root of the tree.', 'start': 312.287, 'duration': 14.692}, {'end': 331.212, 'text': 'right now, suppose we are going to perform search operation on 9.', 'start': 326.979, 'duration': 4.233}, {'end': 334.175, 'text': 'right, so 9 is less than 10.', 'start': 331.212, 'duration': 2.963}, {'end': 334.955, 'text': 'we are going to the left.', 'start': 334.175, 'duration': 0.78}, {'end': 336.036, 'text': '9 is greater than 7.', 'start': 334.955, 'duration': 1.081}, {'end': 337.057, 'text': 'we are going to the right.', 'start': 336.036, 'duration': 1.021}, {'end': 338.779, 'text': 'now. here we got 9.', 'start': 337.057, 'duration': 1.722}, {'end': 339.239, 'text': 'now we are.', 'start': 338.779, 'duration': 0.46}, {'end': 340.16, 'text': 'we are going to do splang.', 'start': 339.239, 'duration': 0.921}, {'end': 345.445, 'text': 'splang means we are going to make this element as the root of the tree.', 'start': 340.16, 'duration': 5.285}, {'end': 350.289, 'text': 'now, for making these elements, for suppose i am going to make 9 as the root of the tree.', 'start': 345.445, 'duration': 4.844}], 'summary': 'Splay trees rearrange for search operation, making 9 the root.', 'duration': 31.616, 'max_score': 302.559, 'thumbnail': 'https:/ /coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/qMmqOHr75b8/pics/qMmqOHr75b8302559.jpg'}, {'end': 397.354, 'src': 'embed', 'start': 370.164, 'weight': 3, 'content': [{'end': 375.746, 'text': 'if you got what is splaying, how to do splaying of this tree, then it means you are done with splay trees.', 'start': 370.164, 'duration': 5.582}, {'end': 377.326, 'text': 'you got it right.', 'start': 375.746, 'duration': 1.58}, {'end': 386.729, 'text': 'so splaying means basically what rearranging the tree so that the element on which you are going to perform the operation now become the root of the tree.', 'start': 377.326, 'duration': 9.403}, {'end': 389.17, 'text': 'right, how to do the rearrangement of the tree, that thing.', 'start': 386.729, 'duration': 2.441}, {'end': 397.354, 'text': 'we will discuss the rotations, how many rotations and which type of rotations you need to perform here right now.', 'start': 389.17, 'duration': 8.184}], 'summary': 'Splaying rearranges tree to make element the root. discussing rotations.', 'duration': 27.19, 'max_score': 370.164, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/qMmqOHr75b8/pics/ qMmqOHr75b8370164.jpg'}], 'start': 43.006, 'title': 'Splay trees and their operations', 'summary': 'Discusses the time complexity of binary search trees, avl trees, and red-black trees. it also introduces the concept of splay trees and their potential for improving time complexity in practical situations. furthermore, it delves into the splaying operation in splay trees to rearrange the tree for efficient search operations, providing examples and associated splaying steps.', 'chapters': [{'end': 221.319, 'start': 43.006, 'title': 'Splay trees: self-balancing binary search trees', 'summary': 'Discusses the time complexity of binary search trees, avl trees, and red-black trees. it also introduces the concept of splay trees and their potential for improving time complexity in practical situations.', 'duration': 178.313, 'highlights': ['Splay trees guarantee improved time complexity Splay trees ensure a time complexity of O(log n) for search, insertion, and deletion operations, potentially offering better performance in practical scenarios.', 'Comparison of time complexity with other tree types AVL trees and red-black trees also maintain a time complexity of O (log n), ensuring balanced trees and efficient operation performance.', 'Explanation of splaying in splay trees Splaying in splay trees involves a self-adjusting process that follows basic binary search tree operations, enhancing performance with additional splaying operations.']}, {'end': 497.79, 'start': 221.319, 'title': 'Splay trees and splaying operation', 'summary': 'Discusses splay trees, focusing on the splaying operation to rearrange the tree so that the element being searched becomes the root, with examples of search operations and associated splaying steps.', 'duration': 276.471, 'highlights': ['The splaying operation in splay trees involves rearranging the tree so that the element being searched becomes the root, achieved through zig rotations and right rotations.', 'Splay trees are self-adjusted binary search trees where each operation rearranges the tree so that the operated element is placed at the root, with splaying being a crucial aspect of splay trees.', 'The search operation in splay trees is followed by splaying, which involves making the searched element the root of the tree through rotation and rearrangement steps.']}], 'duration': 454.784, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/qMmqOHr75b8/pics/qMmqOHr75b843006.jpg', 'highlights': ['Splay trees guarantee improved time complexity with O(log n) for search, insertion, and deletion operations.', 'AVL trees and red-black trees maintain a time complexity of O(log n), ensuring balanced trees and efficient operation performance.', 'Splaying in splay trees involves a self-adjusting process that follows basic binary search tree operations, enhancing performance with additional splaying operations.', 'The splaying operation in splay trees involves rearranging the tree so that the element being searched becomes the root, achieved through zig rotations and right rotations.', 'Splay trees are self-adjusted binary search trees where each operation rearranges the tree so that the operated element is placed at the root, with splaying being a crucial aspect of splay trees.', 'The search operation in splay trees is followed by splaying, which involves making the searched element the root of the tree through rotation and rearrangement steps.']}, {'end': 1015.464, 'segs': [{'end': 638.25, 'src': 'embed', 'start': 611.775, 'weight': 0, 'content': [{'end': 618.818, 'text': 'somewhere it is written that the right rotation is known as zig rotation and the left is known as zeg rotation.', 'start': 611.775, 'duration': 7.043}, {'end': 629.243, 'text': 'fine, but somewhere it is written that this case, if the child, if the node you want to search, is child of the root node, it can be left.', 'start': 618.818, 'duration': 10.425}, {'end': 631.324, 'text': 'it can be right in both cases.', 'start': 629.243, 'duration': 2.081}, {'end': 634.967, 'text': 'it is what zig rotation in this case.', 'start': 631.324, 'duration': 3.643}, {'end': 638.25, 'text': 'in this case, if it is the left child, then zig right rotation.', 'start': 634.967, 'duration': 3.283}], 'summary': 'Zig rotation is used for left and right child nodes in tree traversal.', 'duration': 26.475, 'max_score': 611.775, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/qMmqOHr75b8/pics/qMmqOHr75b8611775.jpg'}, {'end': 738.3, 'src': 'heatmap', 'start': 678.271, 'weight': 1, 'content': [{'end': 688.177, 'text': "now the situation is if the node you want to search is having parent as well as grandparent, Suppose I'm going to search one.", 'start': 678.271, 'duration': 9.906}, {'end': 691.438, 'text': 'One is having parent as well as grandparent.', 'start': 688.777, 'duration': 2.661}, {'end': 695.36, 'text': 'Now in that case, we have four cases, four rotations.', 'start': 692.239, 'duration': 3.121}, {'end': 696.921, 'text': 'Let us see those rotations.', 'start': 695.66, 'duration': 1.261}, {'end': 701.483, 'text': 'Now let us take this example and here I want to search one.', 'start': 697.561, 'duration': 3.922}, {'end': 704.962, 'text': 'now the same step.', 'start': 704.062, 'duration': 0.9}, {'end': 707.683, 'text': 'first of all, perform the standard BST searching.', 'start': 704.962, 'duration': 2.721}, {'end': 709.303, 'text': 'compare with this 10.', 'start': 707.683, 'duration': 1.62}, {'end': 710.244, 'text': 'one is less than 10.', 'start': 709.303, 'duration': 0.941}, {'end': 714.445, 'text': 'so we are going to go to the left of this less than 7.', 'start': 710.244, 'duration': 4.201}, {'end': 717.766, 'text': 'now to the left of 7, and here we got 1.', 'start': 714.445, 'duration': 3.321}, {'end': 719.846, 'text': 'now we do what splaying now.', 'start': 717.766, 'duration': 2.08}, {'end': 722.367, 'text': 'splaying means this one should be the root of the tree.', 'start': 719.846, 'duration': 2.521}, {'end': 724.267, 'text': 'you have to make this one as the root of the tree now.', 'start': 722.367, 'duration': 1.9}, {'end': 726.368, 'text': 'you have to rearrange the tree now.', 'start': 724.267, 'duration': 2.101}, {'end': 732.87, 'text': 'here the situation is now this one, this element on which you are going to perform the operation, that is searching operation,', 'start': 726.368, 'duration': 6.502}, {'end': 738.3, 'text': 'is parent as well as grandparent.', 'start': 732.87, 'duration': 5.43}], 'summary': 'Explains splaying with searching and tree rearrangement, involving four rotations.', 'duration': 49.523, 'max_score': 678.271, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/qMmqOHr75b8/pics/qMmqOHr75b8678271.jpg'}, {'end': 970.452, 'src': 'embed', 'start': 944.809, 'weight': 3, 'content': [{'end': 949.952, 'text': 'so now the tree would be something like this so now this zigzag rotation is having two cases.', 'start': 944.809, 'duration': 5.143}, {'end': 957.275, 'text': 'one is this one, this one, and second, is this one right this you can say that it is zigzag right rotation.', 'start': 949.952, 'duration': 7.323}, {'end': 970.452, 'text': 'it is what zigzag left left rotations and somewhere it is written that it is what zeg rotation, first zeg rotation and again zeg rotation.', 'start': 957.275, 'duration': 13.177}], 'summary': 'The zigzag rotation has two cases, involving left and right rotations.', 'duration': 25.643, 'max_score': 944.809, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/qMmqOHr75b8/pics/ qMmqOHr75b8944809.jpg'}], 'start': 497.79, 'title': 'Binary tree rotations', 'summary': 'Covers the processes of searching and performing rotations in binary trees, addressing cases like the search item being a child of the root node and the splaying process, and also explains the concept of zig-zag rotations in binary search trees with illustrative examples and quantifiable data.', 'chapters': [{'end': 738.3, 'start': 497.79, 'title': 'Search and rotation in binary trees', 'summary': 'Discusses the process of searching and performing rotations in binary trees, explaining the cases where the search item is a child of the root node and the subsequent zig rotations, as well as the splaying process when the search item is both a parent and a grandparent.', 'duration': 240.51, 'highlights': ['The chapter explains the scenarios where the search item is a child of the root node, detailing the cases of left and right child and the corresponding zig rotations to reorganize the tree, such as left rotation for left child and right rotation for right child.', 'It discusses the splaying process when the search item is both a parent and a grandparent, emphasizing the need for rearranging the tree and making the search item the root of the tree to perform the searching operation effectively.', 'It provides examples and steps for performing standard BST searching and splaying when the search item has a parent as well as a grandparent, highlighting the importance of rearranging the tree to make the search item the root of the tree for efficient operations.']}, {'end': 1015.464, 'start': 738.3, 'title': 'Zig-zag rotations in binary search trees', 'summary': 'Explains the concept of zig-zag rotations in binary search trees, illustrating how to identify the need for zig-zag rotations and perform them, using examples with quantifiable data.', 'duration': 277.164, 'highlights': ['The chapter discusses the scenario of zig-zag rotations in binary search trees, emphasizing the need for two rotations to perform zig-zag rotations.', 'It explains the process of identifying the need for zig-zag rotations based on the relative positions of the nodes, providing examples with quantifiable data.', 'The concept of zig-zag rotations is further illustrated with specific examples of performing zig-zag rotations in binary search trees, highlighting the steps and outcomes with quantifiable data.']}], 'duration': 517.674, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/qMmqOHr75b8/pics/qMmqOHr75b8497790.jpg', 'highlights': ['The chapter explains the scenarios of left and right child of the root node, detailing corresponding zig rotations.', 'It discusses the splaying process when the search item is both a parent and a grandparent.', 'It provides examples and steps for performing standard BST searching and splaying.', 'The chapter discusses the scenario of zig-zag rotations in binary search trees.', 'It explains the process of identifying the need for zig-zag rotations based on the relative positions of the nodes.', 'The concept of zig-zag rotations is further illustrated with specific examples.']}, {'end': 1374.011, 'segs': [{'end': 1095.606, 'src': 'embed', 'start': 1067.147, 'weight': 0, 'content': [{'end': 1070.13, 'text': 'first of all, this 13 is to the left of this 15.', 'start': 1067.147, 'duration': 2.983}, {'end': 1072.913, 'text': 'so we will do right rotation.', 'start': 1070.13, 'duration': 2.783}, {'end': 1083.075, 'text': 'right when you do right rotation on 15 on this edge which is connecting 13, and this 15 means the node you want to search and its parent.', 'start': 1072.913, 'duration': 10.162}, {'end': 1086.658, 'text': 'first of all, you will do rotation on that edge.', 'start': 1083.075, 'duration': 3.583}, {'end': 1091.883, 'text': 'you are not going to do rotation on this 10 now, right in zigzag situation.', 'start': 1086.658, 'duration': 5.225}, {'end': 1095.606, 'text': 'first of all, we will rotate this on this edge.', 'start': 1091.883, 'duration': 3.723}], 'summary': 'Perform right rotation on node 15 connected to 13.', 'duration': 28.459, 'max_score': 1067.147, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/qMmqOHr75b8/pics/qMmqOHr75b81067147.jpg'}, {'end': 1208.424, 'src': 'embed', 'start': 1154.965, 'weight': 1, 'content': [{'end': 1157.407, 'text': 'now this is the tree after one zig operation.', 'start': 1154.965, 'duration': 2.442}, {'end': 1159.369, 'text': 'now still we are left now.', 'start': 1157.407, 'duration': 1.962}, {'end': 1163.492, 'text': 'still the 13 is not the root right now.', 'start': 1159.369, 'duration': 4.123}, {'end': 1166.674, 'text': 'now we will do what the 13 is to the right of its parent.', 'start': 1163.492, 'duration': 3.182}, {'end': 1169.356, 'text': 'now we will do left rotation means zig rotation.', 'start': 1166.674, 'duration': 2.682}, {'end': 1179.765, 'text': 'so now, after left rotation on this 10, the tree would be something like this see this 11 is to the left of 13, so to the left of 13.', 'start': 1171.721, 'duration': 8.044}, {'end': 1180.926, 'text': 'now we have 10.', 'start': 1179.765, 'duration': 1.161}, {'end': 1183.507, 'text': 'so the 11 would go to the right of this 10.', 'start': 1180.926, 'duration': 2.581}, {'end': 1184.788, 'text': 'now this is the final tree.', 'start': 1183.507, 'duration': 1.281}, {'end': 1186.789, 'text': 'as you can see, 13 is now the root.', 'start': 1184.788, 'duration': 2.001}, {'end': 1190.031, 'text': 'so this is the situation of zigzag situation.', 'start': 1186.789, 'duration': 3.242}, {'end': 1199.137, 'text': 'now, since the second case in zigzag situation may be what, suppose i want to search 9 now?', 'start': 1190.031, 'duration': 9.106}, {'end': 1202.52, 'text': 'same, compare with this 10 less than 10.', 'start': 1199.137, 'duration': 3.383}, {'end': 1203.981, 'text': 'so here greater than 7.', 'start': 1202.52, 'duration': 1.461}, {'end': 1205.201, 'text': 'so here we got this 9.', 'start': 1203.981, 'duration': 1.22}, {'end': 1207.403, 'text': 'now we are going to do splaying.', 'start': 1205.201, 'duration': 2.202}, {'end': 1208.424, 'text': 'splaying on this.', 'start': 1207.403, 'duration': 1.021}], 'summary': 'After one zig operation, 13 becomes the root in the final tree.', 'duration': 53.459, 'max_score': 1154.965, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ qMmqOHr75b8/pics/qMmqOHr75b81154965.jpg'}, {'end': 1365.247, 'src': 'embed', 'start': 1328.464, 'weight': 3, 'content': [{'end': 1330.646, 'text': 'this is zigzag situation.', 'start': 1328.464, 'duration': 2.182}, {'end': 1333.228, 'text': 'now you can take, you can write down.', 'start': 1330.646, 'duration': 2.582}, {'end': 1340.974, 'text': 'the. both the cases means two rotation you can write down or you can take it within one rotation, that is zigzag rotation.', 'start': 1333.228, 'duration': 7.746}, {'end': 1343.495, 'text': 'so now we have discussed all the rotations.', 'start': 1341.855, 'duration': 1.64}, {'end': 1351.938, 'text': 'so somewhere you find out that they have written these three rotations only in splaying of the tree zig rotation, zig, zig and zigzag.', 'start': 1343.495, 'duration': 8.443}, {'end': 1355.379, 'text': 'or somewhere you find out that they have written six rotations.', 'start': 1351.938, 'duration': 3.441}, {'end': 1365.247, 'text': 'fine, so here in zig rotation they have covered both the cases zig and zeg right, either you can write down separately,', 'start': 1355.379, 'duration': 9.868}], 'summary': 'Discussion on zigzag rotations in trees, covering both two and six rotations.', 'duration': 36.783, 'max_score': 1328.464, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/qMmqOHr75b8/pics/ qMmqOHr75b81328464.jpg'}], 'start': 1015.464, 'title': 'Zigzag rotations in trees', 'summary': 'Discusses zigzag rotations in bst and binary trees, demonstrating left and right rotations to optimize tree structure, resulting in a new tree with 13 as the root node and emphasizing the coverage of rotations in splaying.', 'chapters': [{'end': 1208.424, 'start': 1015.464, 'title': 'Zigzag rotation in bst', 'summary': 'Discusses the concept of zigzag rotation in bst, where the process involves right and left rotations to restructure the tree, resulting in a new tree with the 13 as the root node and other nodes rearranged accordingly.', 'duration': 192.96, 'highlights': ['The process involves right rotation on the edge connecting 13 and 15, followed by left rotation to rearrange the tree, resulting in 13 as the root node.', 'The zigzag rotation process is demonstrated through the steps of performing zig rotation on the parent of the node, resulting in a new tree with 13 as the root node.', 'Splaying is performed to rearrange the tree when searching for 9, involving comparisons and restructuring of the tree based on the search process.']}, {'end': 1282.51, 'start': 1208.424, 'title': 'Zigzag situation in binary tree', 'summary': 'Explains a zigzag situation in a binary tree, demonstrating the process of left rotation and its impact on the tree structure, to optimize the tree for efficient search operations.', 'duration': 74.086, 'highlights': ['The chapter demonstrates the process of left rotation in a binary tree, specifically focusing on the repositioning of nodes to optimize search operations, such as moving 9 to the root and reorganizing its adjacent nodes (7 and 8) (Relevance: 5)', 'An explanation of the concept of a zigzag situation in a binary tree is provided, highlighting the significance of identifying and addressing such situations to improve the tree structure for efficient search operations (Relevance: 3)']}, {'end': 1374.011, 'start': 1282.51, 'title': 'Zig and zigzag rotations', 'summary': 'Discusses right rotations, specifically zig and zigzag rotations, and explains the final tree structure after the rotations, highlighting the concept of zigzag situations and the coverage of rotations in splaying of the tree.', 'duration': 91.501, 'highlights': ['The concept of zigzag situations is explained, where a node is rotated to the right of the root and then to the left of a child node, illustrating the tree restructuring process. This provides a clear understanding of the zigzag rotation process.', 'The coverage of rotations in splaying of the tree is discussed, emphasizing that both the cases of zig and zeg are covered within zig rotation, and both zigzag and zigzag cases are covered within zigzag rotation.']}], 'duration': 358.547, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/qMmqOHr75b8/pics/qMmqOHr75b81015464.jpg', 'highlights': ['The process involves right rotation on the edge connecting 13 and 15, followed by left rotation to rearrange the tree, resulting in 13 as the root node.', 'The zigzag rotation process is demonstrated through the steps of performing zig rotation on the parent of the node, resulting in a new tree with 13 as the root node.', 'Splaying is performed to rearrange the tree when searching for 9, involving comparisons and restructuring of the tree based on the search process.', 'The concept of zigzag situations is explained, where a node is rotated to the right of the root and then to the left of a child node, illustrating the tree restructuring process.', 'The coverage of rotations in splaying of the tree is discussed, emphasizing that both the cases of zig and zeg are covered within zig rotation, and both zigzag and zigzag cases are covered within zigzag rotation.', 'An explanation of the concept of a zigzag situation in a binary tree is provided, highlighting the significance of identifying and addressing such situations to improve the tree structure for efficient search operations.']}, {'end': 1711.129, 'segs': [{'end': 1522.915, 'src': 'embed', 'start': 1467.121, 'weight': 0, 'content': [{'end': 1474.084, 'text': 'so that is the main advantage of split tree that the most frequently accessed element would be near to the root,', 'start': 1467.121, 'duration': 6.963}, {'end': 1477.145, 'text': 'so that you need to take less time to search those items.', 'start': 1474.084, 'duration': 3.061}, {'end': 1486.311, 'text': 'fine, and that is why, because of this advantage, one application of splay trees, what it is used, you know it is better.', 'start': 1477.645, 'duration': 8.666}, {'end': 1490.354, 'text': 'it is giving better performance in practical scenarios.', 'start': 1486.311, 'duration': 4.043}, {'end': 1496.158, 'text': 'right, better performance than avial and red black trees in practical situations.', 'start': 1490.354, 'duration': 5.804}, {'end': 1507.737, 'text': 'so in practical scenarios we are using splay trees mostly right, and these are invented, invented in 1985, fine.', 'start': 1496.158, 'duration': 11.579}, {'end': 1509.323, 'text': 'so one one application.', 'start': 1507.737, 'duration': 1.586}, {'end': 1516.75, 'text': 'maybe these are used to implement caches, right.', 'start': 1509.323, 'duration': 7.427}, {'end': 1522.915, 'text': "so i'm going to write down now the advantages, drawbacks and applications of splay trees.", 'start': 1516.75, 'duration': 6.165}], 'summary': 'Splay trees offer faster search with better performance than avl and red black trees, commonly used in practical scenarios, such as implementing caches, invented in 1985.', 'duration': 55.794, 'max_score': 1467.121, 'thumbnail': 'https:// coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/qMmqOHr75b8/pics/qMmqOHr75b81467121.jpg'}, {'end': 1644.45, 'src': 'heatmap', 'start': 1603.719, 'weight': 4, 'content': [{'end': 1611.286, 'text': 'so that is why these are used in implementation of cache memories to save the time for accessing the data right now.', 'start': 1603.719, 'duration': 7.567}, {'end': 1615.239, 'text': 'drawback of this tree maybe see One.', 'start': 1611.286, 'duration': 3.953}, {'end': 1618.142, 'text': 'main topic is what these are not strictly balanced.', 'start': 1615.239, 'duration': 2.903}, {'end': 1623.848, 'text': 'So sometimes the height may be linear, or you can say order of n.', 'start': 1618.362, 'duration': 5.486}, {'end': 1629.716, 'text': 'that is the main drawback of this tree, because these are not strictly balanced trees.', 'start': 1624.912, 'duration': 4.804}, {'end': 1634.921, 'text': 'this may be skewed, fine, but this is very rare case.', 'start': 1629.716, 'duration': 5.205}, {'end': 1644.45, 'text': 'so that is why sometimes any single operation may take what order of, or you can say, theta n time complexity.', 'start': 1634.921, 'duration': 9.529}], 'summary': 'Cache memories use these trees, but drawbacks include non-strict balance, leading to linear height and theta n time complexity for some operations.', 'duration': 29.211, 'max_score': 1603.719, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/qMmqOHr75b8/pics/qMmqOHr75b81603719.jpg'}], 'start': 1374.011, 'title': 'Advantages and drawbacks of splay trees', 'summary': 'Discusses the advantages of splay trees, including the reduction of time complexity to o(1) for frequently accessed elements and their practical applications. it also covers drawbacks such as potential linear height leading to rare o(n) time complexity and their applications in windows nt and gcc compilers.', 'chapters': [{'end': 1711.129, 'start': 1374.011, 'title': 'Advantages, drawbacks, and applications of splay trees', 'summary': 'Discusses the advantages of splay trees, including the reduction of time complexity to o(1) for frequently accessed elements, their use in practical scenarios for better performance, and applications in implementing caches. it also covers the drawbacks, such as potential linear height leading to rare o(n) time complexity, and applications in windows nt and gcc compilers.', 'duration': 337.118, 'highlights': ['Splaying the tree reduces the time complexity to O(1) for frequently accessed elements, resulting in better performance in practical scenarios.', 'The main advantage of splay trees is that the most frequently accessed element would be near to the root, reducing the time needed to search those items.', 'Splay trees are used in practical scenarios for better performance than avial and red black trees.', 'The drawbacks of splay trees include potential linear height leading to rare O(n) time complexity for some operations.', 'Splay trees are used to implement caches, as frequently accessed data is stored in caches to reduce access time to the data.']}], 'duration': 337.118, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/qMmqOHr75b8/pics/qMmqOHr75b81374011.jpg', 'highlights': ['Splaying the tree reduces the time complexity to O(1) for frequently accessed elements, resulting in better performance in practical scenarios.', 'The main advantage of splay trees is that the most frequently accessed element would be near to the root, reducing the time needed to search those items.', 'Splay trees are used in practical scenarios for better performance than avial and red black trees.', 'Splay trees are used to implement caches, as frequently accessed data is stored in caches to reduce access time to the data.', 'The drawbacks of splay trees include potential linear height leading to rare O(n) time complexity for some operations.']}], 'highlights': ['Splay trees guarantee improved time complexity with O(log n) for search, insertion, and deletion operations.', 'The main advantage of splay trees is that the most frequently accessed element would be near to the root, reducing the time needed to search those items.', 'Splaying the tree reduces the time complexity to O(1) for frequently accessed elements, resulting in better performance in practical scenarios.', 'Splay trees are used in practical scenarios for better performance than avial and red black trees.', 'The drawbacks of splay trees include potential linear height leading to rare O(n) time complexity for some operations.', 'The chapter highlights the advantages, drawbacks, and applications of splay trees.', 'The video covers the differences between splay trees and other balancing binary search trees like avl and red-black trees.', 'Splay trees are self-balancing binary search trees, a variant of binary search trees.']}	{"url":"https://learn.coursnap.app/staticpage/qMmqOHr75b8.html","timestamp":"2024-11-02T15:49:53Z","content_type":"text/html","content_length":"37132","record_id":"<urn:uuid:25e4fcda-1681-4486-ba6c-63af088f23c9>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00747.warc.gz"}
5 Ways to Insert a Square Root Symbol in Word - [best Solution] Microsoft Word is one of the most popular word processing software available in the technology market for a multitude of platforms. The software, developed and maintained by Microsoft offers various features for you to type and edit your documents. Whether be it a blog article or a research paper, Word makes it easy for you to make the document meet the professional standards of a text. You can even type a full book in Microsoft Word! Word is such a powerful word processor that could include images, graphics, charts, 3D models, and many such interactive modules. But when it comes to typing math, many people find it difficult with the insertion of symbols. Mathematics generally involves a lot of symbols, and one such commonly used symbol is the square root symbol (√). Inserting a square root in MS Word is not that much tough. Yet, if you are not sure about how to insert a square root symbol in Word, let us help you using this guide. 5 Ways to Insert a Square Root Symbol in Word #1. Copy & Paste the symbol in Microsoft Word This is perhaps the simplest way to insert a square root sign in your Word document. Just copy the symbol from here and paste it in on your document. Select the square root sign, press Ctrl + C. This would copy the symbol. Now go to your document and press Ctrl + V. The square root sign would now be pasted onto your document. Copy the symbol from here: √ #2. Use the Insert Symbol option Microsoft Word has a predefined set of signs and symbols, including the square root symbol. You can use the Insert Symbol option available in word to insert a square root sign in your document. 1. To use the insert symbol option, navigate to the Insert tab or menu of Microsoft Word, then click on the option labelled Symbol. 2. A drop-down menu would show up. Choose the More Symbols option at the bottom of the drop-down box. 3. A dialogue box titled “Symbols” would show up. Click on the Subset drop-down list and select Mathematical Operators from the list displayed. Now you can see the square root symbol. 4. Make a click to highlight the symbol sign then click the Insert button. You can also double-click the symbol to insert it in your document. #3. Inserting a Square Root using the Alt code There is a character code for all characters and symbols in Microsoft Word. Using this code, you can add any symbol to your document if you know the character code. This character code is also called as an Alt code. The Alt code or character code for the square root symbol is Alt + 251. • Place your mouse cursor at the location where you want the symbol to be inserted. • Press and hold the Alt key then use the numeric keypad to type 251. Microsoft Word would insert a square root sign at that location. Alternatively, you can make use of this option below. • After placing your pointer at the desired location, Type 221A. • Now, press the Alt and X keys together (Alt + X). Microsoft Word would automatically transform the code into a square root sign. Another useful keyboard shortcut is Alt + 8370. Type 8370 from the numeric keypad as you hold the Alt key. This would insert a square root sign at the location of the pointer. NOTE: These numbers specified are to be typed from the numeric keypad. Hence you should ensure that you have the Num Lock option enabled. Do not use the number keys located above the letter keys on your keyboard. #4. Making use of the Equations Editor This is another great feature of Microsoft Word. You can make use of this equations editor to insert a square root sign in Microsoft Word. 1. To use this option, navigate to the Insert tab or menu of Microsoft Word, then click on the option labelled Equation. 2. As soon as you click the option, you can find a box containing the text “Type Equation Here” automatically inserted in your document. Inside the box, type sqrt and press the Space key or the Spacebar. This would automatically insert a square root sign in your document. 3. You can also make use of the keyboard shortcut for this option (Alt + =). Press the Alt key and the = (equal to) key together. The box to type your equation would show up. Alternatively, you can try the method illustrated below: 1. Click on the Equations option from the Insert tab. 2. Automatically the Design tab appears. From the options shown, chose the option labelled as Radical. It would display a drop-down menu listing various radical symbols. 3. You can insert the square root sign into your document from there. #5. The Math Autocorrect feature This is also a useful feature to add a square root symbol to your document. 1. Navigate to the File From the left panel, choose More… and then click Options. 2. From the left panel of the Options dialog box, choose Now, click on the button labelled “Autocorrect options” and then navigate to the Math Autocorrect option. 3. Tick on the option that says “Use Math Autocorrect rules outside of math regions”. Close the box by clicking OK. 4. From now onwards, wherever you type sqrt, Word would change it into a square root symbol. Another way to set autocorrect is as follows. 1. Navigate to the Insert tab of Microsoft Word, and then click on the option labelled Symbol. 2. A drop-down menu would show up. Choose the More Symbols option at the bottom of the drop-down box. 3. Now click on the Subset drop-down list and select Mathematical Operators from the list displayed. Now you can see the square root symbol. 4. Make a click to highlight the square root symbol. Now, click on the Autocorrect button. 5. The Autocorrect dialog box would show up. Enter the text that you want to change to a square root sign automatically. 6. For example, type SQRT then click on the Add button. From now on, whenever you type SQRT, Microsoft Word would replace the text with a square root symbol. I hope now you know how to insert a square root symbol in Microsoft Word. Drop your valuable suggestions in the comments section and let me know if you have any questions. Also check out my other guides, tips, and techniques for Microsoft Word.	{"url":"https://lbsite.org/5-ways-to-insert-a-square-root-symbol-in-word/","timestamp":"2024-11-06T21:05:01Z","content_type":"text/html","content_length":"55562","record_id":"<urn:uuid:50f38ba4-023a-4caa-b898-94513b8121b7>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00272.warc.gz"}
Why is this solution sufficient for the problem "Speedbreaker"? - Codeforces In the speedbreaker question, why do we need to do: c++ mn[i]=min(mn[i],mn[i-1]),mx[i]=max(mx[i],mx[i-1]); Full Code Also, I get why the first condition is necessary for there to be a solutions and I get why the second part of the code finds that good interval, but I'm having a little trouble piecing together the sufficiency of the two to guarantee the right answer. Perhaps this is also why I font understand why you have to do the operations on the mins and maxes too. 6 weeks ago, # \| » ← Rev. 3 → 0 Intuition2.0 Consider any simple test case like : 6 3 3 3 5 5 there is no 4 in the test case but don't you consider the segment for 4(from l=2 to r=4)? that's why we do mn[i] = min(mn[i], mn[i-1]) and the same for mx • 6 weeks ago, # ^ \| » 0 Oh I kind of get it, it just ensures that not only does each interval have a low enough length, but also that they also all intersect, guaranteeing a solution. This makes sense now. Negationist Lastly, could you help me explain how the last check is sufficient to find the solution? I understand that anything out of the bounds we set is guaranteed not to work(so why it is necessary), but not necessarily why it is sufficient? How do these all the things we do come together to make a set of both necessary and sufficient conditions? Like I get it but I don't if you know what I mean. □ 5 weeks ago, # ^ \| » ← Rev. 3 → 0 » I know what you are talking about. But forget for sometime that there's no formal proof exist in the world, so now how u think of proving that sufficient condition. We can prove this by imagining that if there exist a answer, then it must lie on a segment.Till here i hope u clear this, now how many index to start with is the question, then for this part Intuition2.0 u can think like for every t how much we can stretch to right and left...by doing this for every t and taking final intersection segment(means min end — max start + 1) will be our ans, U have to imagine this don't go around formal mathematical proof...Do this with some test cases with pen and paper u will arrive at a informal proof of the sufficient 5 weeks ago, # \| » 0 TheScrasse This solution is one of the most difficult to prove imo. The proof is contained in the editorial of 2018F3 - Speedbreaker Counting (Hard Version). Is the editorial solution for 2018B - Speedbreaker actually easier? • » 4 weeks ago, # ^ \| » 0 Negationist idk the editorial was kind of a miss for me ngl	{"url":"https://mirror.codeforces.com/blog/entry/134479","timestamp":"2024-11-06T14:40:08Z","content_type":"text/html","content_length":"102965","record_id":"<urn:uuid:a1d45f48-5588-47a1-9d36-0b06a4c6e9ef>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00006.warc.gz"}
Interoperability my arse! Microsoft once again shows that their anti-competitive colours are still flying. Only now we have to deal with appeasers from the GNU/Linux side trying to apologize for them as well. Roy says it best about the new Windows 7 installation. Once more, for all their rhetoric, Microsoft’s actions show yet again that they don’t care about interoperability or playing nice with anyone else. All they care is maintaining their desktop monopoly and part of that tactic is not making it easy at all to setup a dual boot setup. While in 2001, when XP came out the excuse “Only hardcore geeks use GNU/Linux so why should MS even consider them” might have had some basis, 8 years later, when desktop GNU/Linux is more than viable through distros like Ubuntu and where it is quite likely that people might consider trying this other OS while wanting to keep the Windows option open, it fails to convince. This is nothing other than the same ol’ spiteful, monopolistic tactics on behalf of MS. This capability, to install multiple OS’ without screwing up each other has existed for ages so it’s obviously not rocket science. As such, MS’ refusal to implement it can be nothing but deliberate. And if that’s not enough, we now have GNU/Linux users defending such actions! So now, among the atheist appeasers, Women “feminist” appeasers we have to add GNU/Linux appeasers as well. If Microsoft apologists were not enough. Of course, that there are those who would sell-out to MS in order to get ahead in the marketplace is nothing new, but plain users? Those who are the ones getting the most annoyance out of such tactics? Why do they feel the need to apologise for MS?! Here’s some of the classic excuses (and my counter) you’ll see on why this isn’t really a problem, move along, nothing to see here: GNU/Linux users are a small minority. Most desktops will be Windows only so why should MS even implement a dual-boot consideration? Because even though GNU/Linux is small, it is also showing accelerating growth and even a small percentage of desktop users, when seen on a global scale means quite a few million people. People who will all be inconvenienced when they need to upgrade their installation or repair/reinstall it when it will (eventually) break down. Because MS has been blabbing about “interoperability” for the last few years and they need to be called on their bullshit at some point. Their rhetoric has never been honest and their actions prove it again and again. They didn’t really make it hard to install Windows 7. It could have been far worse. Gee thanks… Should we be thankful that Microsoft doesn’t go out of their way to prevent GNU/Linux installations now? Should we praise MS for not making our task more difficult than it already is? What kind of fucking stupid slave-mentality is this? “Golly thanks for using lube while screwing me in the ass, sir!” And you know what? They did make it harder than Windows XP. Slightly so but nevertheless true. You don’t stop criticizing someone when they act less evil than they could have been. You stop criticizing people and corporations when they stop being evil. All you need to do is hack #1, #2 and #3. Which is obviously something all people who’d like to try out the system can do right? No, of course not. And MS knows this and they know it will further reinforce the perception that GNU/Linux is only for hardcore geeks. You know what the regular user will say when you mention hacking the goddamn boot loader? “Huh wut? No thanks”. Which will mean that it will always require a power user (and perhaps more than that) to simply set it up (and then again and again when Windows invariably breaks down and requires reinstallation). Compared to the possible scenario where Windows acted like an OS of its generation and recognised that “hey, there are other OS’ out there, perhaps we should be considerate to those of our users who might be dual-booting”, and have Windows autorecognise the MBR is taken, and provide sensible options on how to work with it that a simple user can follow, you know, like GNU/Linux has been doing for what, 8 years now? Of course it is better to make it seem as if only IT nerds can setup and maintain a GNU/Linux installation alongside Windows 7, even when they difficulty has nothing to do with GNU/Linux and everything to do with MS’ refusal to play fair. Thus they can keep their ignorant audience locked in and happily continue spreading their FUD, only they have some appeasers from the GNU/Linux camp on their side as well who will make their point for them by saying stuff like “Oh it’s easy. Just reinstall Grub and then hack the bootloader“. Other OS’ and even some particular GNU/Linux distros are worse than that. A Tu Quoque is a logical fallacy. If other OS’ are doing even worse, then they are worthy of even heavier condemnation. And about those GNU/Linux distros that do it (see Moblin, IPCop etc), you do know they are meant for a single OS installation right? You do know that Moblin is for netbooks which are unlikely to have a dual-boot while IpCop is a firewall right? Don’t you think it’s just a tad intellectually dishonest to bring those up as examples of such faults? You wouldn’t would you? So while there can be other who can be just as bad, if not worse than MS, this does not constitute an excuse of any kind, especially since they hold most of the desktop market and their actions are clearly deliberate. And if Free Software OS’ are doing this without having a reason to do so, then you can always change it by contributing or even convincing the developers of the errors of their Related articles by Zemanta 12 thoughts on “Interoperability my arse!” 1. Well.. a long time ago when chaos created itself and invented life, it might have thought.. "why not giving them something to think their bollocks off on" .. and tried so hard to be against the nature of itself and created evolution. When chaos was done with that it might have thought "okay.. I dont get the point.. and they also won't.." And it turned back to what it has been eversince and just watched. And I say, lets give it quite a show, then! 😀 Earl Grayle 1. But I'm not worrying about dual boot systems. I am simply pointing out MS' dishonesty and calling out those defending it. Just because the underlying reason might not be relevant in your opinion does not make them less suitable for a callout. As for the Virtual Machines, while I agree that it's a nice solution, unless they reach the level where one can play games in them, it's unlikely to be a solid solution for people who don't want to switch for this reason. 1. If you can manage 75% of the speed of a single-core system, which is a reasonable goal, then people can play games — just not at 100% speed. And that's not actually a bad thing if you can achieve critical mass — once Linux reaches critical mass on the desktop, game developers will start writing native versions for Linux to avoid the speed hit. (And actually, now that I think of it, a standard VM along the lines I proposed would help a great deal. Right now, developing a program which uses sustained AV for Linux is a pretty serious undertaking, because there are so many API alternatives that it's difficult for an outsider to learn which ones are suitable, and since all of them are open-source, almost none of them are guaranteed to be around and well-supported for the foreseeable future. If there were a VM included with popular distro X using some particular fast, low-level set of APIs for AV stuff, that would constitute a guarantee for developers that popular distro X has a fast, low-level set of APIs for AV stuff which was unlikely to change or be discontinued without significant public note and documentation of alternatives.) I don't like Linux — it has dealt with its rough GUI edges by glossing over them, and an elephant with a towel tossed over over it isn't really hidden. I beg to differ. GNU/Linux has certainly not glossed over any rough GUI edges and the GUI is therefore continuously improving and evolving according to the wishes of the users. 1. Right. Wake me when Linux manages copy and paste between programs (beyond plain text) as well as the Mac did in 1986. Or has human-readable directory names in the default file system structure. Or auto-detects hardware well enough to show corresponding controls for a trackpad instead of a mouse on laptops. Or has a sufficient control structure that users never have to edit xorg.conf. Or survives poor OpenGL support gracefully. (Or ships a distro which comes with a default theme which isn't fugly, although that's a subjective matter.) I try Linux out roughly every 6 to 12 months, depending on how busy I am. I would really like to enjoy it, so I keep trying, but so far I haven't. Last time I tried Debian, because I wanted to install on a PowerPC Mac Mini I have acquired and Ubuntu doesn't support PowerPC any more. I admit it: the Debian installer (finally) achieved a level which I would consider sufficient for non-computer-savvy users. (And, incidentally, the difference was not anything to do with the "graphical" part of the GUI. It was that the descriptive text for the configuration questions no longer assumed that the user knew what the significance of each item was. All the earlier installers I've seen for Linux would ask for configurations without explaining what they were or whether it made any difference what the answers were. All the LiveCDs in the world won't make up for poorly-worded text with no online help.) But the day-to-day experience of Linux is still something I would recommend to new users only if they either have unlimited access to a Linux guru or else will never do anything outside a web browser so they can memorize a simple set of steps to deal with those few pieces of the OS which will encroach on their lives. Way too many ways things can go wrong, too many options for advanced users turned on by default, too many ways to easily change the GUI in such a way that talking someone through something over the phone becomes a practical impossibility, too many programs which reinvent the wheel with respect to GUI widgets, an almost total lack of documentation, too much reliance on the command line (which is instafail in GUI terms), and of course the main distributions give you a choice between the fugly-but-functional GNOME and the busy-and-inconsistent KDE 4. It's like someone took the worst weaknesses of the GUIs of Windows Vista and Mac OS X and built them all into one unholy hybrid, and then decided it wasn't awkward enough and made the desktop environment a changeable option. So, yeah, the Linux GUI/desktop environment is evolving. But not fast enough or well enough to really make it compete with the non-free alternatives, and always with the insistence that Linux should be mediocre at doing everything, just in case, rather than being really good at a few specific things. I'm starting to think of Linux the way I think of the Democratic Party, or the more liberal Christian Churches — their main function is to prevent other options which otherwise might appeal to the same people (the Green Party, or agnosticism, or the Haiku Project) from gaining support by — to quote an analysis I read a few years ago — "sucking all the air out of that space". Right. Wake me when Linux manages copy and paste between programs (beyond plain text) as well as the Mac did in 1986. Or has human-readable directory names in the default file system structure. Or auto-detects hardware well enough to show corresponding controls for a trackpad instead of a mouse on laptops. Or has a sufficient control structure that users never have to edit xorg.conf. Or survives poor OpenGL support gracefully. (Or ships a distro which comes with a default theme which isn't fugly, although that's a subjective Copy-Paste has never been a problem for me. Human readable directories in the root filesystem are not necessary. It autodetects trackpads perfectly. I haven't had to edit xorg.conf for the last year at least. And anyway, just because GNU/Linux does not have the features you think are required, does not make it bad. I can in the same way complain about the faults of Macs as well when they don't have the features of GNU/Linux. In short, you're just using your subjective judgement to slander a whole OS with FUD. But the day-to-day experience of Linux is still something I would recommend to new users only if they either have unlimited access to a Linux guru or else will never do anything outside a web browser so they can memorize a simple set of steps to deal with those few pieces of the OS which will encroach on their lives.[…] What a load of FUD. I've given GNU/Linux to people who have no access to a tech or use it just for browsing with no problem. Similarly, I've seen people struggle with Macs just as well, which just goes to show how biased your opinion is. No, way too many things Don't go wrong as long as you explain to people to take care to use their sudo rights carefully. No reinventing the wheel is not necessary. It's like someone took the worst weaknesses of the GUIs of Windows Vista and Mac OS X and built them all into one unholy hybrid, and then decided it wasn't awkward enough and made the desktop environment a changeable option. Seriously, you're just rabidly hating it because it's not Mac. Nothing else. 1. Took me a while to remember that I had left a comment here. 🙂 You have partially misread my comment: I said that Linux WAS acceptable if the user is only going to use a web browser. But, frankly, ANY graphical OS is going to be acceptable if the user will only be interacting with it by following a short series of static steps every single time; if there were a way to run a modern browser directly in DOS, then DOS would be an acceptable solution for that type of user. As for "things don't go wrong", I have to laugh. When I installed Debian (which was my most recent Linux experience), the installer CD booted, and then refused to actually install, saying it was unable to detect a CD-ROM drive. The time before that, I installed Xubuntu on a Pentium III laptop, and the graphical updater program failed on its first run, leaving the apt-get database locked. Turned out that this was a known bug, which had been discovered months earlier, but since it was possible to fix the problem by resort to the command line, it wasn't deemed important enough for either a new CD image OR a note on the website. Then there was the bug on another laptop where the autoconfiguration of xorg.conf for the video driver turned on an option which was not available on that card, and launching the terminal program would crash X11 and terminate the session. (There was a workaround: switch to 16-bit color instead. But, of course, Linux doesn't give you a GUI option to change color depths, so that wasn't an option. I was able to do it with a terminal session, but I guarantee you that most users would, by that time, have dug out the Windows install CDs and be heading back to Redmond-land as fast as their CPUs could carry them.) As for "autodetecting trackpads" — sure it accepts the trackpad's input, but I'm talking about the GUI, not the input device support. None of the environments I have seen give you any configuration controls for trackpads, displaying the mouse controls instead. Both Windows and the Mac were smart enough to at least give you both and then warn you that one of the two wouldn't work as long ago as 1995, whereas Linux GUIs just blithely assume that everyone has a mouse. If that doesn't sound bad to you, then either you've never used a trackpad or you must be one of the people — a significantly small minority, in my local and subjective experience — who like "touch to click" turned on, since Linux ALWAYS turns it on by default and does not provide any GUI to turn it back off. And any GUI which resorts to "you have to use the command line for that" is a failure as a GUI. You have partially misread my comment: I said that Linux WAS acceptable if the user is only going to use a web browser. I have not misread your comment. I find it ridiculous. GNU/Linux is acceptable for anything from Web Browsing, to media using, to media manipulation, to office work, to programming to gaming (as long as the game has been written for it). What you say is plain and simple FUD, not at all supported by experience. As for "things don't go wrong", I have to laugh. Sorry but your subjective and anecdotal experience is not cutting it. It is not the applicable for most people who wouldn't be able to install windows or Mac themselves either. Alternatively it is based on what you personally consider unacceptable which is lame as far as arguments go since I can also complain about the lockdown of the Macs, their Fischer Prize GUI, their lack of applications etc. Fact of the matter is that if GNU/Linux is installed for a non-technical user by a professional, as their Windows or Mac would have been, then it will most likely not require any further support. 2. At this point, complaining about Microsoft's tactics is kind of pointless. They aren't going to change anything, and Windows isn't even the focus any more. I'll let you in on an open secret: Microsoft knows that sooner or later it is going to lose its desktop OS monopoly through defection to Macs and Linux and/or some new challenger to be determined, and they know they are unlikely to be able to stop the defections by improvements to quality because they can't make major improvements without breaking their base of installed software. (That's why the last three Windows versions had really long development cycles in which major features were announced and then quietly removed. Go back and Google "Longhorn" to see all the features dropped from the development of Vista.) Their entire business model is currently based around trying to develop a monopoly in some other market by introducing new products sold at a loss, like the XBox and the Zune. They are throwing new products at the wall to see what will stick, and preserving the Windows majority through unfair tactics buys them time to do so, so there is no chance that they will listen to any sort of appeal, even though their upper management knows that majority is doomed in the long term. Since complaining about it isn't going to lead anywhere, what you Linux folks need to do is to develop a strategy for dealing with it. I would suggest that you take a page from Apple's book, sort of. As you know, when a Windows user switches to Mac OS X, they do not have the option to keep their existing Windows installation at all. Yet Windows users are switching to the Mac! That means the problem isn't "I want to keep my old Windows installation," and therefore dual-booting is of limited utility and is not worth fussing over. The main thing which has enticed the "but I still need Windows for ___" crowd to the Mac — insofar as they have been enticed — is not really Boot Camp (dual-booting Mac and Windows) but virtual machines. Both VMWare Fusion and Parallels Desktop sell like crazy. People are willing to pay money to have an easy solution — which means Linux should be at an advantage in this regard by not charging money — but the solution has to be genuinely easy, with which qualification Linux has problems. On Linux, therefore, the key would be to pick a virtual machine program, buff it up to the point where new users can deal with it quickly and simply, and integrate it into the installation so that a Windows user can easily go from "single boot Windows with no Linux" to "single boot Linux with Windows in a VM". Although it would be nice to include some sort of migration utility on LiveCDs to make this easier — a program which would burn the user's files to CD/DVD in a form which could be restored from inside the VM, for example — the most important part is making it trivial to have the virtual machine in the first place. Unfortunately, although it is entirely possible to have a good, solid VM on top of Linux — VMWare, for example — making it trivially easy is going to involve doing a bunch of things which Linux has, in the past, had trouble with: 1. There needs to be a single project. No splintering off into different distros over trivial ideological concerns. That's how Linux projects die. OpenOffice.org has been fairly successful at keeping a single main development track going; learn from their example. 2. The goal of the project can't be to support every single Linux distro with every single desktop environment and every single set of libraries. A VM is going to need fast low-level access, so wrapping a bunch of alternative options isn't going to work. The VM should be written so that it works immediately on a default installation of Ubuntu (with GNOME), and let the other distros/ desktop environments figure out what they need to make that work after the fact. 3. That includes KDE. The recent history of KDE, where version 4 wasn't really ready for prime time but got the full "4.0" name and release anyway, and then there were lots of fast point releases, demonstrates that KDE isn't a stable development target. Getting an easy-to-use pushbutton VM working on Linux is going to be tricky enough. Let KDE do the support work. 4. Don't start off by trying to integrate the Windows GUI into Linux. The Mac VM products have features like that — where the windows belonging to Windows programs become "real" OS windows — and it has eaten a lot of development time without adding much to the experience. Until at least the 2.0 release, "Windows in an X11 window at 75%+ of normal single-core speed, with video acceleration" should be the goal. 5. Any configuration option which is reasonably common needs to have a GUI control which can be set from within the VM and which does not require a reboot of either the VM or Linux (unless it is technically impossible to avoid). That includes "which OS gets control of the CD burner", "which OS captures USB devices", and "what parts of the Linux filesystem, if any, are mapped to drives in the VM" 1. Whoah, massive reply. But I think you miss the point of my rant. It wasn't really expecting MS to do anything different but rather calling them out on the BS and tacking some of the apologetics coming from our side. As for your idea, it has merit but as always, the biggest problem is to get all distros and developers to see things the same way. It's unlikely to happen which will simply mean that whichever distro does it better will come out on top of the others (similar to how ubuntu is doing). I personally don't think this splintering is a huge problem as conformity can also lead to rather bad outcomes.	{"url":"https://dbzer0.com/blog/interoperability-my-arse/","timestamp":"2024-11-12T05:56:40Z","content_type":"text/html","content_length":"79804","record_id":"<urn:uuid:16541d65-bfa6-43f7-9e6b-09de2602d479>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00308.warc.gz"}
Crossed Random Effects in a Multinomial GLMM SAS newbie here. My research problem involves an unordered quaternary outcome. There are two random effects which are crossed i.e. independent of each other. In addition to the usual MLEs for fixed effects and estimated SDs for random effects, I also need to extract the estimated random effects of each category of each of the two random effects because those estimates are of interest in themselves. My data file is attached to this message. At the moment, I cannot get even a highly simplified model to fit: PROC IMPORT OUT= MyLib.Mydata DATAFILE= "E:\mydata.txt" proc glimmix data = mylib.mydata method=Laplace inititer=9999; class dep_var TrgPoS trigger CompLemma / ref = first; model dep_var = TrgPoS / SOLUTION DIST=MULTINOMIAL LINK=GLOGIT; RANDOM INTERCEPT / SUBJECT=trigger GROUP=dep_var SOLUTION; RANDOM INTERCEPT / SUBJECT=CompLemma GROUP=dep_var SOLUTION; There are 20 more explanatory variables to add if/when this can be made to work. Presently, however, model fitting halts with the following error message: The initial estimates did not yield a valid objective function. This page suggested increasing the INITITER setting to try and solve this problem, that's why its value is set to 9999. But as you can see, it's not helping. If I remove one of the random effects, then the model does converge (although with CompLemma as the only random effect it does complain about a G matrix not being positive definite). However, leaving a random effect out is not an option because both of them are of major importance to my research problem. I'm running Windows 7, and my SAS version is 9.4. 09-29-2020 11:29 PM	{"url":"https://communities.sas.com/t5/Statistical-Procedures/Crossed-Random-Effects-in-a-Multinomial-GLMM/td-p/687713","timestamp":"2024-11-03T17:18:05Z","content_type":"text/html","content_length":"363314","record_id":"<urn:uuid:ba175c3e-525d-48db-ba14-65a301531fb6>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00504.warc.gz"}
What do Numerator and Denominator of a Filter mean? The numerator of a filter modifies the signal in the frequency domain. On the other hand, the numerator denotes the zeros of a transfer function. Poles of a transfer function tell us about the cut-off frequency of a filter. Process of finding the cut-off frequency of a filter from z-transform Let's assume the transfer function of a filter, H(z) = 1 / {1 - az^(-1)} Where a is a constant. This represents a simple first-order recursive (IIR) low-pass filter Then, make dominator, 1 - az^(-1) = 0 Or, z = a Then calculate the magnitude of the pole: \|a\| If \|a\| is close to 1, then it is closest to the unit circle The cut-off frequency in radians is determined by the angle Î¸ between the real axis and the line connecting 0 to the pole a and Î¸ = arctan{imag(a) / real(a)} BER vs. SNR denotes how many bits in error are received in a communication process for a particular Signal-to-noise (SNR) ratio. In most cases, SNR is measured in decibel (dB). For a typical communication system, a signal is often affected by two types of noises 1. Additive White Gaussian Noise (AWGN) 2. Rayleigh Fading In the case of additive white Gaussian noise (AWGN), random magnitude is added to the transmitted signal. On the other hand, Rayleigh fading (due to multipath) attenuates the different frequency components of a signal differently. A good signal-to-noise ratio tries to mitigate the effect of noise. Calculate BER for Binary ASK Modulation The theoretical BER for binary ASK (BASK) in an AWGN channel is given by: BER = (1/2) * erfc(0.5 * sqrt(SNR_ask)); Enter SNR (dB): Calculate BER BER vs. SNR curves for ASK, FSK, and PSK Calculate BER for Binary FSK Modulation The theoretical BER for binary FSK (BFSK) in an AWGN channel is g Modulation Constellation Diagrams BER vs. SNR BER vs SNR for M-QAM, M-PSK, QPSk, BPSK, ... 1. What is Bit Error Rate (BER)? The abbreviation BER stands for bit error rate, which indicates how many corrupted bits are received (after the demodulation process) compared to the total number of bits sent in a communication process. It is defined as, In mathematics, BER = (number of bits received in error / total number of transmitted bits) On the other hand, SNR refers to the signal-to-noise power ratio. For ease of calculation, we commonly convert it to dB or decibels. 2. What is Signal the signal-to-noise ratio (SNR)? SNR = signal power/noise power (SNR is a ratio of signal power to noise power) SNR (in dB) = 10log(signal power / noise power) [base 10] For instance, the SNR for a given communication system is 3dB. So, SNR (in ratio) = 10^{SNR (in dB) / 10} = 2 Therefore, in this instance, the signal power i Signal Processing RMS Delay Spread, Excess Delay Spread, and Multipath... RMS Delay Spread, Excess Delay Spread, and Multipath (MPCs) The fundamental distinction between wireless and wired connections is that in wireless connections signal reaches at receiver thru multipath signal propagation rather than directed transmission like co-axial cable. Wireless Communication has no set communication path between the transmitter and the receiver. The line of sight path, also known as the LOS path, is the shortest and most direct communication link between TX and RX. The other communication pathways are called non-line of sight (NLOS) paths. Reflection and refraction of transmitted signals with building walls, foliage, and other objects create NLOS paths. [ Read More about LOS and NLOS Paths] Multipath Components or MPCs: The linear nature of the multipath component signals is evident. This signifies that one multipath component signal is a scalar multiple of Wireless Signal Processing Gaussian and Rayleigh Distribution Difference between AWGN and Rayleigh Fading 1. Introduction Rayleigh fading coefficients and AWGN, or additive white gaussian noise [↗] , are two distinct factors that affect a wireless communication channel. In mathematics, we can express it in that way. Let's explore wireless communication under two common noise scenarios: AWGN (Additive White Gaussian Noise) and Rayleigh fading. y = hx + n ... (i) The transmitted signal x is multiplied by the channel coefficient or channel impulse response (h) in the equation above, and the symbol "n" stands for the white Gaussian noise that is added to the signal through any type of channel (here, it is a wireless channel or wireless medium). Due to multi-paths the channel impulse response (h) changes. And multi-paths cause Rayleigh fading. 2. Additive White Gaussian Noise (AWGN) The mathematical effect involves adding Gauss MATLAB Code % Developed by SalimWireless.Com clc; clear; close all; % Configuration parameters fs = 10000; % Sampling rate (Hz) t = 0:1/fs:1-1/fs; % Time vector creation % Signal definition x = sin(2 pi * 100 * t) + cos(2 * pi * 1000 * t); % Calculate the Fourier Transform y = fft(x); z = fftshift(y); % Create frequency vector ly = length(y); f = (-ly/2:ly/2-1) / ly * fs; % Calculate phase while avoiding numerical precision issues tol = 1e-6; % Tolerance threshold for zeroing small values z(abs(z) < tol) = 0; phase = angle(z); % Plot the original Signal figure; subplot(3, 1, 1); plot (t, x, 'b'); xlabel('Time (s)'); ylabel('\|y\|'); title('Original Messge Signal'); grid on; % Plot the magnitude of the Fourier Transform subplot(3, 1, 2); stem(f, abs(z), 'b'); xlabel('Frequency (Hz) '); ylabel('\|y\|'); title('Magnitude of the Fourier Transform'); grid on; % Plot the phase of the Fourier Transform subplot(3, 1, 3); stem(f, Modulation ASK, FSK & PSK Constellation BASK (Binary ASK) Modulation: Transmits one of two signals: 0 or -√Eb, where Eb is the energy per bit. These signals represent binary 0 and 1. BFSK (Binary FSK) Modulation: Transmits one of two signals: +√Eb ( On the y-axis, the phase shift of 90 degrees with respect to the x-axis, which is also termed phase offset ) or √Eb (on x-axis), where Eb is the energy per bit. These signals represent binary 0 and 1. BPSK (Binary PSK) Modulation: Transmits one of two signals: +√Eb or -√Eb (they differ by 180 degree phase shift), where Eb is the energy per bit. These signals represent binary 0 and 1. This article will primarily discuss constellation diagrams, as well as what constellation diagrams tell us and the significance of constellation diagrams. Constellation diagrams can often demonstrate how the amplitude and phase of signals or symbols differ. These two characteristics lessen the interference between t Compare the BER performance of QPSK with other modulation schemes (e.g., BPSK, 4-QAM, 16-QAM, 64-QAM, 256-QAM, etc) under similar conditions. MATLAB Code clear all; close all; % Set parameters for QAM snr_dB = -20:2:20; % SNR values in dB qam_orders = [4, 16, 64, 256]; % QAM modulation orders % Loop through each QAM order and calculate theoretical BER figure; for qam_order = qam_orders % Calculate theoretical BER using berawgn for QAM ber_qam = berawgn(snr_dB, 'qam', qam_order); % Plot the results for QAM semilogy(snr_dB, ber_qam, 'o-', 'DisplayName', sprintf('%d-QAM', qam_order)); hold on; end % Set parameters for QPSK EbNoVec_qpsk = (-20:20)'; % Eb/No range for QPSK SNRlin_qpsk = 10.^(EbNoVec_qpsk/10); % SNR linear values for QPSK % Calculate the theoretical BER for QPSK using the provided formula ber_qpsk_theo = 2qfunc(sqrt(2SNRlin_qpsk)); % Plot the results for QPSK semilogy(EbNoVec_qpsk, ber_qpsk_theo, 's-', Channel Impulse Response (CIR) Wireless Signal Processing CIR, Doppler Shift & Gaussian Random Variable The Channel Impulse Response (CIR) is a concept primarily used in the field of telecommunications and signal processing. It provides information about how a communication channel responds to an impulse signal. What is the Channel Impulse Response (CIR) ? It describes the behavior of a communication channel in response to an impulse signal. In signal processing, an impulse signal has zero amplitude at all other times and amplitude ∞ at time 0 for the signal. Using a Dirac Delta function, we can approximate this. ...(i) Î´( t) now has a very intriguing characteristic. The answer is 1 when the Fourier Transform of Î´( t) is calculated. As a result, all frequencies are responded to equally by Î´ (t). This is crucial since we never know which frequencies a system will affect when examining an unidentified one. Since it can test the system for all freq	{"url":"https://www.salimwireless.com/2023/09/numerator-and-denominator-of-filter.html","timestamp":"2024-11-12T03:33:34Z","content_type":"application/xhtml+xml","content_length":"113945","record_id":"<urn:uuid:a0dab92e-78a9-494d-9f3c-6c5b8ba8b2ad>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00021.warc.gz"}
MAT 510 FINAL EXAM - College School Essays Question 1 4 out of 4 points Understanding variation is important because variation: 4 out of 4 points The Statistical Thinking Strategy illustrated in Figure 2.14 provides a graphic of the overall approach to driving improvement through statistical thinking. Which of the following is a key principle illustrated in this specific graph? 4 out of 4 points Improving the quality of process measurements is: 4 out of 4 points 4 out of 4 points Which type of variation was critical to resolving the realized revenue case study? 4 out of 4 points A process is said to be capable if: 4 out of 4 points The histogram below showing ages of credit card holders displays what type of distribution? 4 out of 4 points In evaluating data on our process outputs, four characteristics we might investigate are: central tendency, variation, shape of distribution, and stability. Which of the following tools would be most helpful to determine stability of the process? 4 out of 4 points Non-random and unpredictable patterns on a control chart indicate: 4 out of 4 points A main purpose of a control chart is to: 4 out of 4 points Models based on subject matter fundamentals (theory) are generally better than statistical models for: 4 out of 4 points A statistic that is used to indicate too much correlation between the predictors in a regression analysis is called the: 4 out of 4 points George Box tells us “all models are wrong but some are useful”. By this comment he means: 4 out of 4 points Tips for building useful models include: 4 out of 4 points The best way to evaluate the validity of a statistical model is: 4 out of 4 points Given the plot below, what might you suspect about factors A and B? 4 out of 4 points The following table shows data on the effect of Machine Pressure on product hardness. What statistical test should be used to test the effect of pressure on average hardness? 100MPa 200MPa 6.2 11.1 7.3 10.3 8.2 13 8.3 14.3 7.5 11.2 6.5 14.7 9.6 10 9 9.4 4 out of 4 points What should one consider when analyzing the results of an experiment? 4 out of 4 points In every experiment there is experimental error. Which of the following statements is true? 4 out of 4 points A 3^2 experiment means that we are experimenting with: 4 out of 4 points We are conducting a poll for the mayoral election of a medium sized city, and found with one week before the election that the Democrat candidate is leading with 52% planning to vote for her, 46% planning to vote for the Republican, with 2% not sure. The 95% confidence interval for proportion planning to vote for the Democrat is +/- 3%. Based on this information, and making reasonable assumptions, which of the following statements is correct? 4 out of 4 points For which of the following scenarios am I most likely to utilize a Chi-squared test? 4 out of 4 points After running an ANOVA comparing the average years of experience between five different job classifications, we obtained a p value of .02. Which of the following would be a reasonable conclusion concerning the population in this case? 4 out of 4 points Which of the following could be considered a prediction interval? 4 out of 4 points What would happen (other things equal) to a confidence interval if you calculated a 99 percent confidence interval rather than a 95 percent confidence interval? If you need assistance with writing your assignment, essay, our professional assignments / essay writing service is here to help! Order Now	{"url":"https://collegeschoolessays.com/2023/10/13/mat-510-final-exam/","timestamp":"2024-11-05T14:02:58Z","content_type":"text/html","content_length":"64889","record_id":"<urn:uuid:c16e00f8-f30d-4670-9be3-2b6627698613>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00009.warc.gz"}
x और y ऐेसी दो सर्याएँ है निका माधच अगुपात Q और y का मान ज्ञात ... \| Filo Question asked by Filo student और ऐेसी दो सर्याएँ है निका माधच अगुपात और का मान ज्ञात करो ? 512 है सभुपात Not the question you're searching for? + Ask your question Video solutions (1) Learn from their 1-to-1 discussion with Filo tutors. 5 mins Uploaded on: 3/13/2023 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 more questions on Sequence Series and Quadratic View more Students who ask this question also asked 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 और ऐेसी दो सर्याएँ है निका माधच अगुपात और का मान ज्ञात करो ? 512 है सभुपात Updated On Mar 13, 2023 Topic Sequence Series and Quadratic Subject Mathematics Class Class 12 Answer Type Video solution: 1 Upvotes 64 Avg. Video Duration 5 min	{"url":"https://askfilo.com/user-question-answers-mathematics/aur-aiesii-do-sryaaen-hai-nikaa-maadhc-agupaat-aur-kaa-maan-34353932303332","timestamp":"2024-11-06T19:12:19Z","content_type":"text/html","content_length":"277798","record_id":"<urn:uuid:f68774d4-8a71-40c3-9d06-8f240118f4f4>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00103.warc.gz"}
This Quantum World/Implications and applications/How fuzzy positions get fuzzier - Wikibooks, open books for an open world How fuzzy positions get fuzzier We will calculate the rate at which the fuzziness of a position probability distribution increases, in consequence of the fuzziness of the corresponding momentum, when there is no counterbalancing attraction (like that between the nucleus and the electron in atomic hydrogen). Because it is easy to handle, we choose a Gaussian function ${\displaystyle \psi (0,x)=Ne^{-x^{2}/2\sigma ^{2}},}$ which has a bell-shaped graph. It defines a position probability distribution ${\displaystyle \|\psi (0,x)\|^{2}=N^{2}e^{-x^{2}/\sigma ^{2}}.}$ If we normalize this distribution so that ${\displaystyle \int dx\,\|\psi (0,x)\|^{2}=1,}$ then ${\displaystyle N^{2}=1/\sigma {\sqrt {\pi }},}$ and ${\displaystyle \|\psi (0,x)\|^{2}=e^{-x^{2}/\sigma ^{2}}/\sigma {\sqrt {\pi }}.}$ We also have that • ${\displaystyle \Delta x(0)=\sigma /{\sqrt {2}},}$ • the Fourier transform of ${\displaystyle \psi (0,x)}$ is ${\displaystyle {\overline {\psi }}(0,k)={\sqrt {\sigma /{\sqrt {\pi }}}}e^{-\sigma ^{2}k^{2}/2},}$ • this defines the momentum probability distribution ${\displaystyle \|{\overline {\psi }}(0,k)\|^{2}=\sigma e^{-\sigma ^{2}k^{2}}/{\sqrt {\pi }},}$ • and ${\displaystyle \Delta k(0)=1/\sigma {\sqrt {2}}.}$ The fuzziness of the position and of the momentum of a particle associated with ${\displaystyle \psi (0,x)}$ is therefore the minimum allowed by the "uncertainty" relation: ${\displaystyle \Delta x (0)\,\Delta k(0)=1/2.}$ Now recall that ${\displaystyle {\overline {\psi }}(t,k)=\phi (0,k)e^{-i\omega t},}$ where ${\displaystyle \omega =\hbar k^{2}/2m.}$ This has the Fourier transform ${\displaystyle \psi (t,x)={\sqrt {\sigma \over {\sqrt {\pi }}}}{1 \over {\sqrt {\sigma ^{2}+i\,(\hbar /m)\,t}}}\,e^{-x^{2}/2[\sigma ^{2}+i\,(\hbar /m)\,t]},}$ and this defines the position probability distribution ${\displaystyle \|\psi (t,x)\|^{2}={1 \over {\sqrt {\pi }}{\sqrt {\sigma ^{2}+(\hbar ^{2}/m^{2}\sigma ^{2})\,t^{2}}}}\,e^{-x^{2}/[\sigma ^{2}+(\hbar ^{2}/m^{2}\sigma ^{2})\,t^{2}]}.}$ Comparison with ${\displaystyle \|\psi (0,x)\|^{2}}$ reveals that ${\displaystyle \sigma (t)={\sqrt {\sigma ^{2}+(\hbar ^{2}/m^{2}\sigma ^{2})\,t^{2}}}.}$ Therefore, ${\displaystyle \Delta x(t)={\sigma (t) \over {\sqrt {2}}}={\sqrt {{\sigma ^{2} \over 2}+{\hbar ^{2}t^{2} \over 2m^{2}\sigma ^{2}}}}={\sqrt {[\Delta x(0)]^{2}+{\hbar ^{2}t^{2} \over 4m^{2}[\Delta The graphs below illustrate how rapidly the fuzziness of a particle the mass of an electron grows, when compared to an object the mass of a ${\displaystyle C_{60}}$ molecule or a peanut. Here we see one reason, though by no means the only one, why for all intents and purposes "once sharp, always sharp" is true of the positions of macroscopic objects. Above: an electron with ${\displaystyle \Delta x(0)=1}$ nanometer. In a second, ${\displaystyle \Delta x(t)}$ grows to nearly 60 km. Below: an electron with ${\displaystyle \Delta x(0)=1}$ centimeter. ${\displaystyle \Delta x(t)}$ grows only 16% in a second. Next, a ${\displaystyle C_{60}}$ molecule with ${\displaystyle \Delta x(0)=1}$ nanometer. In a second, ${\displaystyle \Delta x(t)}$ grows to 4.4 centimeters. Finally, a peanut (2.8 g) with ${\displaystyle \Delta x(0)=1}$ nanometer. ${\displaystyle \Delta x(t)}$ takes the present age of the universe to grow to 7.5 micrometers.	{"url":"https://en.m.wikibooks.org/wiki/This_quantum_world/Implications_and_applications/How_fuzzy_positions_get_fuzzier","timestamp":"2024-11-12T04:27:14Z","content_type":"text/html","content_length":"86749","record_id":"<urn:uuid:75c5eb7b-a789-4b76-8895-2d42e84bf105>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00360.warc.gz"}
Annuity is a type of repayment, which is constant over the time. Th annuity payment consists of two parts – the interest and the amortization. The ratio of the interest to amortization is gradually To calculate the annuity we use the following formula: S – annuity repayment U – borrowed sum of money q – q = 1 + interest_rate_per_time_period n – number of time periods (time) We borrow 200 000$ in the bank for 5 years, payable monthly, with interest rate Number of periods (months): We pay per month (we have to divide the annual interest rate): We substitute into the formula: We will pay 4758$ per month.	{"url":"https://www.programming-algorithms.net/article/50638/Annuity","timestamp":"2024-11-09T14:01:47Z","content_type":"text/html","content_length":"30121","record_id":"<urn:uuid:310e84f0-d1fe-4cca-9b3f-19051e625cd2>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00845.warc.gz"}
Electric motor control calculation 24 Feb 2024 Popularity: ⭐⭐⭐ Electric Motor Control Calculations This calculator provides various calculations related to electric motor control, including apparent power, real power, efficiency, current, and voltage. Calculation Example: Electric motor control calculations are essential for designing and operating electric motors efficiently. These calculations involve determining various parameters such as apparent power, real power, efficiency, current, and voltage. Related Questions Q: What is the significance of efficiency in electric motor control? A: Efficiency is a crucial factor in electric motor control as it determines the amount of power that is converted into useful work. A higher efficiency motor consumes less energy and operates at a lower cost. Q: How does power factor affect electric motor control? A: Power factor is important in electric motor control as it determines the amount of real power that is being used. A higher power factor indicates that more of the apparent power is being converted into real power, resulting in more efficient operation. \| —— \| —- \| —- \| Calculation Expression Apparent Power: Apparent power (S) is the product of voltage (V) and current (I). Real Power: Real power (P) is the product of apparent power (S) and power factor (pf). Efficiency: Efficiency (?) is the ratio of real power (P) to apparent power (S). Current: Current (I) can be calculated using the formula I = P / (V * ? * pf / 100). Voltage: Voltage (V) can be calculated using the formula V = P / (I * ? * pf / 100). Calculated values Considering these as variable values: P=2000.0, V=220.0, pf=80.0, ?=85.0, I=10.0, the calculated value(s) are given in table below \| —— \| —- \| Sensitivity Analysis Graphs Apparent Power: Apparent power (S) is the product of voltage (V) and current (I). Impact of Rated Voltage on Apparent Power Impact of Rated Current on Apparent Power Similar Calculators Calculator Apps Matching 3D parts for Electric motor control calculation App in action The video below shows the app in action.	{"url":"https://blog.truegeometry.com/calculators/Electric_motor_control_calculation.html","timestamp":"2024-11-02T23:42:10Z","content_type":"text/html","content_length":"31709","record_id":"<urn:uuid:6516a66b-008a-47e5-8f23-52d452e1a778>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00856.warc.gz"}
Math Balance School: Fun Games Bu programın süresi doldu. Math Balance School: Fun Games artık firmadan temin edilebilir. Mental Math is fun with Math Balance - kids help Toby reach home by balancing bridges and while they have fun, they also learn about equality/comparison, gain flexibility with numbers and practice skills that are essential for understanding algebra. Note: Mental Math School is designed for Teachers and for those parents who prefer upfront paid apps to in-app purchases; also useful for Family sharing, since in-app purchases are not supported in Family sharing. This mental math game with 30 levels will help your child get necessary fluidity with numbers. Your child will learn things like addition can happen in any order for the same result, or that sum of different numbers can add up to be the same number. They will also end up having to use different number strategies, such as bridging and compensation which can significantly speed up their mental math computations. These are the skills covered by Math Balance - - Developing meaning of equal sign, using, greater than and less than sign - Set up numbers in their expanded form - Commutative property of addition and, later, using it and expanded form as a strategy to add ex: 67+25 = 60+7+20+5 = 80+12 =92 - Missing Addend strategies:x+b=c, where x is to be found, situations where difference is unknown, where there is a bigger unknown and where there is a smaller unknown -one step problems - Showing that addition and subtraction are inverse; 4+5=9, 9-5=4 - Use only doubles, Use only even/odd numbers to get to a total - 1 step word problems -Change unknown, Result unknown, Start unknown - add a string of two-digit numbers (up to four numbers) by applying place value strategies and properties of operations. Example: 43 + 34 + 57 + 24 = __ - Using concrete models to show addition and subtraction - Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900 - Fluently add and subtract within 1000 using strategies and algorithms based on place value properties of operations, and/or the relationship between addition and subtraction - Multiplication as repeated addition For those who follow the Core curriculum, this maps to the following core standards - 2.OA.B.2, 2.NBT.4, 2.OA.2, 1.OA.6, 2.OA.C.3, 2.OA.1, 2.NBT.6, 2.NBT.7, 2.NBT.8, 3.NBT.2. For support, questions or comments, write to us at: support@makkajai.com Our privacy policy can be found at: http://www.makkajai.com/privacy-policy Math Balance School: Fun Games üzerine yorumlar Makkajai Edu Tech Private Limited 140.36 MB English, Chinese, Chinese iPhone, iPad, iPod touch Windows Giveaway of the Day Herhangi bir Tidal müzik parçasını maksimum ses kalitesiyle indirin!	{"url":"https://iphone.giveawayoftheday.com/math-balance-school-fun-games/?lang=tr","timestamp":"2024-11-06T09:28:14Z","content_type":"text/html","content_length":"41519","record_id":"<urn:uuid:dd8c8bf3-5292-4e8b-bb87-b098ad944a63>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00203.warc.gz"}
1 Introduction Urea or perhydrotriphenylene (PHTP) based intergrowth inclusion compounds belong to the very interesting family of nanotubular composite materials [1]. For peculiar linear guest molecules, the urea builds a hexagonal array of linear parallel one-dimensional channels encapsulating the guest molecules. Hexagonal urea can host numerous molecules such as n-alkanes, dibromo-alkanes [2], dioctanoyl peroxide [3], mono or dicarboxylic acids [4] etc. An important property of urea based inclusion compounds lies in the reduced channel accessible diameter ~ 5–5.5 Å, which almost fits the guest molecule cross-sectional diameter (~4 Å for n-alkane), leading to an ultraconfinement perpendicular to the channel axis. The interest of such compounds concerns not only fundamental physics (incommensurate compounds, physical properties of low-dimensional and strongly correlated systems, polymer translocation through a membrane…), but also applications when the guest molecules are functionalized (non linear optical properties, 1D conductivity…) [5]. Most of these compounds are incommensurate along c, and strongly disordered. Therefore, although X-ray or coherent neutron diffraction give meaningful informations on the host organization or host–guest modulation, these techniques cannot in general quantify the guest organization in the (a,b) plane because all the information is spread in diffuse scattering which mixes both host and guest responses. In contrast, solid-state NMR is particularly suited to measure the guest orientation, because it can directly select the nucleus of interest by its Larmor frequency and chemical shift, giving a high selectivity not provided by X-ray scattering. Moreover, the high orientational sensitivity of the interactions probed by NMR, as well as its high versatility [6], make high resolution solid-state NMR a complementary powerful tool to X-ray or coherent neutron diffraction methods. In particular for organic compounds, deuterium ^2H NMR directly probes the C–D bond orientation and dynamics through the dynamical averaging of the quadrupolar interaction carried by the covalent bond. Single crystal solid-state NMR also offers the unique opportunity to measure the intrinsic orientational disorder [7] from the information contained in spectra acquired at different magnetic field orientations. Nonadecane/urea C[19]H[40]/CO(NH[2])[2] is a model compound for phase transitions in a superspace due to the incommensurability along the c axis. The urea host displays a structural instability at 153 K from a mean hexagonal to an orthorhombic mean structure. This transition should induce three symmetry-broken domains which are related by a threefold axis collinear to the channel c axis. In fact, high-resolution X-ray [8] or neutron scattering experiments [9] evidence six equiprobable domains, with a split angle ± δ with respect to the ideal C[3] directions (Fig. 1). This splitting was interpreted as a way to minimize intergrowth domain walls energy cost [8]. From the host structure, it was proposed that the guests adopt a herringbone arrangement within each domain. Subsequently, the spectra obtained from single-crystal ^2H NMR at different orientations were interpreted assuming a model of the superimposed contribution of six equiprobable orthorhombic domains with an herringbonelike arrangement of the n-alkane chains, and a Gaussian angular distribution within each domains [10]. In contrast to diffraction results, ^2H NMR was able to measure the mean chain orientation with respect to the crystallographic axes, and the residual disorder. But the interpretation of the NMR results was still dependent on the picture of the low temperature phase generated by diffraction studies. However, the incommensurability of nonadecane/urea compounds renders the interpretation of the diffraction results highly controversial, and the data deserve a closer look to seek whether other guest orientational distribution could explain the observed spectra. Therefore, it is of importance to dispose of a method to reconstruct orientational distributions which requires the minimum structural knowledge about the compound under study. The interest of such a method for urea intergrowth compounds would be, as far as possible, to characterize the guest framework independently of any hypothesis concerning the precise host/guest structure. Fig. 1 (Left) Schematic local mean structure (dotted line) of a single crystal of urea/alkane inclusion compounds in the (a,b) plane, and definition of some angles used in the text. The linear channels are all parallel to the main axis c. The sample is rotated with a goniometer probe around c with B[o] ⊥ c. All the angles are measured with respect to the reference a cristallographic axis. For CD aliphatic bonds, quadrupolar asymmetry parameter is very close to zero (η ≈ 0) and the EFG principal axis z[pas] is along the C–D bond.(Right) Schematic relative orientation of the orthorhombic domains with respect to the reference a axis. When no domain splitting is present δ=0, the domains are related by a threefold C[3] axis (dark arrows). After splitting, the angles are n 2 π/3±δ (dotted arrows). The object of this article is to propose a model-independent method, based on Tikhonov regularized inverse schemes with non negativity constraints, to reconstruct the alkane orientational intrinsic disorder distribution. In fact, the reconstruction of orientational probability densities is an ill-posed problem, whose result dramatically depends on the noise and imperfections, leading to strong numerical instabilities. The main idea of regularization consists in adding new constraints which restrict the ensemble of acceptable solutions [11]: for instance a large amount of regularization is already introduced by restricting the ensemble of solutions to positive functions, the only physically significant functions for probability densities. These numerical procedure were proved powerful and efficient in NMR [12–18]. We recently showed that these methods lead the orientational probability density of a rotationally disordered systems around a known axis from a set of single crystal 1D-spectra at different magnetic field orientations [19]. Therefore, once a reference axis of the mean host structure is known, the distribution of the alkane orientations with respect to this reference axis can in principle be reconstructed by performing single crystal NMR at different orientations. In this paper, this framework is further developed and applied to the selectively deuterated compound C[19]D[40]/urea-H[4]. After a presentation of the basic theory of ^2H NMR and Inverse Method with regularization and positivity constraints, the sensitivity of the reconstruction procedure is tested with respect to the correlation between the unknown natural linewidth and the estimated angular intrinsic disorder. The orientational probability of the C–D bonds in the low temperature phase of nonadecane/urea at 90 K is then obtained by this model-independent procedure. 2 Theory 2.1 Deuterium NMR A deuteron ^2H (spin I = 1) has an electric quadrupole moment Q which interacts with the electric field gradient V[αβ]=∂E[α]/∂x[β] (EFG) at the nucleus, and the NMR spectrum consists of a doublet ν [L]±ν[q] of two symmetric lines with respect to the Larmor frequency ν[L]. For aliphatic C–D bonds, the approximation of axially symmetric EFG tensor is usually valid (asymmetry parameter η=0 is zero), because the EFG is principally imposed by the covalent bonding. Thus, the quadrupolar splitting ν[q] [20] reads $νq(Δ, θ)=±Δ2[3cos2θ−1]$ (1) Angle θ is the angle between the static magnetic field B[o] and the principal axis z[PAS] of the deuteron EFG tensor. The factor Δ=3 e^2 q Q/4 h is the splitting parameter with eq=V[zz]. Since the principal axis z[PAS] corresponding to the largest EFG principal value V[zz] is parallel to the C–D bond (Fig. 1), ^2H NMR probes the relative orientation of the C–D bond under consideration with respect to the static magnetic field, as well as its dynamics. If the orientation of the principal axis z[PAS] is static on the NMR timescale (that is the typical orientational correlation time τ[c] should verify Δτ[c] >> 1), the resulting spectrum is the superposition of the different natural lineshapes G[σ](ν) which take into account of the homogeneous broadening (librations, dipolar interactions…). In the following, we assume a Gaussian lineshape for simplicity, G[σ](ν) ~ exp [–1/2(ν/σ)^2] with σ the dispersion. The shape of the resulting spectrum results directly from the orientational probability density w of the C–D bonds. 2.2 Inverse methods with regularization and positivity constraint For nonadecane/urea C[19]D[40]/CO(NH[2])[2,] the chain orientation is mainly probed by the deuterons belonging to CD[2] groups, which are in the (a,b) plane (Fig. 1). Assuming an ideal tetrahedral geometry, the angle of a CD bond with respect to the projection of the C–C bond in plane (a,b) is ~ ±54.7°. When the static magnetic field B[o] is in the same plane, the behavior of the normalized quadrupolar frequencies ν/Δ (Eq. (1)) as a function of the angle between the chain and B[o] is plotted in Fig. 2. As can be seen, the correspondence between frequency and angle is highly degenerate (up to four different angles may give the same frequency). Therefore, although a single spectrum contains all the information on the orientational distribution, this strong frequency overlap makes the direct extraction of the distribution function w impossible. The solution to the problem is to reconstruct the probability density w of the C–D bonds from the knowledge of spectra S[e](ν, β[s]) recorded at different magnetic field orientations β[s], since it provides an overdetermined system. However, this is in general an ill-posed problem, whose result is known to dramatically depend on the noise and imperfections, leading to strong numerical instabilities, and additional information on the probability density are usually needed to restrict the allowed solution space. Fig. 2 Normalized frequencies ν/Δ of the deuterons of a CD[2] group as a function of the orientation of the chain (as measured by the projection of the C–C bond in the (a,b) plane) with respect to the static magnetic field. An ideal tetrahedral geometry was assumed. Only positive frequencies are shown for clarity. Tikhonov regularization was already used to reconstruct axially symmetric disorders [13,16] which only depend on the polar angle. The method presented in this paper is a generalization to two-dimensional disorders in tubular structures which depend on azimuthal angle. Indeed, we showed recently [19] that a set of experimental spectra S[e](ν, β[s]) acquired at different static magnetic field orientations β[s], combined with an inverse method using Tikhonov regularization and positivity constraint leads to the orientationnal probability density of the C–D with a good reliability. Both approaches use a priori knowledge on the symmetry of the orientation distribution (axially symmetric or two-dimensional disorder). From the experimental point of view, it simplifies the problem since rotations around only one well chosen axis is necessary to reconstruct the distribution. Moreover, it is adapted to classical goniometer probes which provide only one rotation axis. Nevertheless, the method could be in principle generalized to a 3D disorder provided (i) enough orientational information can be acquired by rotation around different crystallographic axes, (ii) the computational time is not too prohibitive. This general problem is under consideration. In this article, the two-dimensional approach developed in [19] is adapted to the system under current The minimization problem can be transformed to a linear least-square problem by discretization of the unknown probability density x={w(φ[r])} where φ[r]=(r – 1) Δφ, Δφ being the angular resolution in reconstructing the orientational probability density. A good tradeoff between computational time, resolution and noise sensitivity was found to be Δφ=2°. When Tikhonov regularization is used, the following merit-functional is minimized with respect to the unknown vector x $dλ(x, α)=Min{x}(x){∥Se−M(α)x∥2+λ∥Dx∥2}$ (2) The first term corresponds to the usual least-square problem which constraints the stack of the reconstructed spectra S[mod]=M(α) x to fit the stack S[e] of all the experimental spectra S[e](ν, β [s]) acquired at different orientations β[s]. α represents the other experimental parameters, such as the natural linewidth σ, the quadrupolar interaction Δ, and the pulse sequence non uniform excitation. The second term introduces linear additional constraints through linear operator D. In the following, the second derivative operator is used, which insures that the mean square of curvature w″(φ) of the probability density is not too large. The Lagrange parameter λ controls the degree of smoothing achieved. The necessary condition that the probability density w should be non-negative was realized by using a Non-Negative Least-Square (NNLS) procedure [14]. The optimum Lagrange parameter λ was chosen using the L-curve method [21]. The main time consuming step consists in generating the matrices M(α). All programs were written in the MATLAB language. 3 Results and discussion The NMR experimental data and set-up were already presented [10], and are summarized here for consistency. The single crystals of selectively deuterated inclusion compound C[19]D[40]/(CO)(NH[2])[2] were grown following conventional procedure, with a fully deuterated alkane of commercial origin (Eurisotop). The orientation of the urea host frame was determined by neutron diffraction at the Laboratoire Léon-Brillouin in Saclay (France) to obtain a reliable orientation of the crystal as a whole. All ^2H NMR experiments were performed with a Bruker ASX300 spectrometer at 46.07 MHz with a homemade variable temperature goniometer probe. The error in the angle of rotation of the sample around the coil axis was estimated to be within ±1°. The probe was inserted in an Oxford cryostat cooled by nitrogen, and the temperature was regulated with an Oxford Temperature unit, with a stability within 0.2 K. The spectra were acquired with a solid-echo sequence. Fig. 3 presents single crystal ^2H NMR typical spectra of nonadecane urea at 90 K for both orientations B[o] \|\| c and B[o] ⊥ c. When B[o] \|\| c, a single well defined line at ≈ 62 kHz and second moment of 1.5 kHz is measured. In the geometry B[o] ⊥ c, the spectra become orientation dependent. Since the different lines are not well defined and quite broad, we expect that an orientational intrinsic disorder of the chain orientation can explain the observed lineshapes. Fig. 3 Experimental ^2H NMR spectra 90 K of a single crystal of selectively deuterated nonadecane in hydrogenated urea-h[4], for different crystal orientations (B[o] ⊥ or \|\| c). When B[o] ⊥ c, the number above each spectrum represents angle β=∠B[o],a in degrees. All spectra are normalized to one. Since the spectra are symmetric, only the positive part is shown. For nonadecane/urea inclusion compounds, it was proved that the spectra show a periodicity of 2π/6 by rotation, and that the deuterons are static on the NMR timescale: (i) the quadrupolar interaction is Δ=120–125 kHz, and corresponds to usual static value for aliphatic C–D bonds in such systems. ; (ii) since the maximum splitting is attained for B[o] ⊥ c and reduced by a factor of 2 as compared to the B[o] \|\| c configuration, it can be deduced that the c channel axis is still a principal axis of the deuterium EFG, and that the C–D bonds are perpendicular to the channel axis [10]. It is thus reasonable to assume rigid all-trans n-alkane chains with C–D bonds in the (a,b) plane at 90 K. The origin of the chain orientational disorder in the low temperature phase is at least twofold. First, for a given domain, it depends on the relative orientation of the chains with respect to the mean domain orientation. This disorder may be within each channel (different chain orientations as a function of c) and/or in different channels. The second contribution comes from the disorientation of the domains with respect to ideal C[3] symmetry, for instance by the splitting into subdomains. Since NMR performs an ensemble average over all chains and domains, the origin of the disorder cannot be determined, but its magnitude can be quantified. In Ref. [10], the chain orientation and angular dispersion were deduced from a model of domains and herringbonelike chain arrangement in the low temperature phase because X-ray and coherent neutron diffraction experiments resolved six equiprobable orthorhombic domains, with a splitting of δ=±1° at 90 K [8,9]. Assuming that the residual chain orientational disorder is a Gaussian distribution of mean orientation φ[o] and angular dispersion ρ, the expected spectra were modelized by 12 quadrupolar doublets because each CD2 group creates two quadrupolar doublets per domain: $Sth(ν)=∑p=1, q=±6∫dφ exp [−(φ−φo)22ρ2]Gσ[ν–νp,q(φ)]$ (3) where G[σ](ν) is a Gaussian function whose dispersion σ models the homogeneous natural linewidth, and ν[p,q](φ) is the frequency for each given domain (index p) and C–D bond (index q). Typically, the angular offsets for each domain p are φ[p]+q 54.7 (q = ±), where φ[p] is the orientation of domain p relative to the reference a axis. Comparison between the experimental spectra and this model yielded φ[o] ≈ 30° and ρ ≈ 3°. However, one may question whether different solutions are possible. In the following, we present a model-independent reconstruction of the orientational probability density of the C–D bonds in the (a,b) plane using the inverse method regularized by a Tiknonov procedure and a non negative least-square fitting presented in the previous section. It will be the object of a future work to present a detailed study of all the potentiality of the method. The reconstruction of the C–D distribution w mainly depends on two parameters, the quadrupolar interaction Δ and the natural linewidth σ. The first parameter, Δ, is already known from both independent powder or single crystal experiments [22–24], but not the natural linewidth σ. In the reconstruction of the orientational probability density, an important question arose concerning the correlation between the input natural linewidth σ and the resulting angular dispersion ρ. Indeed, quantifying to what extend an over or underestimation of σ modifies the estimation of the dispersion ρ of the residual intrinsic orientational disorder becomes increasingly important for small disorder. We therefore performed the minimization procedure of the merit-function d[λ](x, σ) Eq. (2) at different σ values which are physically acceptable, typically between 0.5 and 4 kHz, and studied the behavior of the least-square error (LSE) E(σ)=\|\|S[e] – M(σ)x(σ)\|\| as a function of σ. The advantage of using this pedestrian method as compared to a direct minimization of d[λ](x, σ) with respect to both x and σ is twofold. First, it simplifies the implementation of the minimization procedure to routine linear least-square methods which have a fast and well understood convergence. Moreover, it saves computational time since, once the different M(σ) matrices are computed, the minimization can be performed on different data without extra computational time. Second, the behavior of the LSE gives a direct visualization of the sensitivity of the method to the input parameters. Obviously, the drawback of this approach is that it is limited to only a few (in practice three or four) Fig. 4 presents the behavior of the least-square error E(σ)=\|\|S[e] – M(σ)x\|\| as a function of σ for the experimental data acquired with nonadecane/urea. The LSE presents a well defined minimum for σ ≈ 1.2–1.5 kHz. Such values were also found robust to slight modifications in the filter function which corrects the pulse sequence non uniform excitation. Therefore, the LSE minimum gives an estimation of the natural linewidth σ. Fig. 4 Least-square error (left scale) and intrinsic angular dispersion ρ (right scale) as a function of the natural linewidth σ used for the inversion of the CD orientational probability density. Black circles, left scale: least-square error (\|\|S[e] – Mw\|\|). Black squares, right scale: corresponding intrinsic angular dispersion ρ in degrees. The error bars represent the statistical error calculated from the least-square fitting procedure of the resulting probability density w[σ] with the sum of two Gaussians of equal weight and dispersion ρ. The corresponding normalized orientational probability densities w of the C–D bonds are all bimodal within each 2 π/6 sectors, showing that only one chain orientation per domain characterizes the system at this temperature. When σ does not corresponds to the minimum of E(σ), the two lobes of the distribution are slightly asymmetric, whereas they are symmetric at the minimum. Nevertheless, the least-square fits of the different w[σ] with the sum of two Gaussians of equal weight and dispersion ρ were always good. These fits give the residual intrinsic angular dispersion ρ as a function of σ . As indicated by the second plot included in Fig. 4, the residual intrinsic angular dispersion ρ is a slight decreasing function of σ whereas the statistical error, calculated from the least-square fit, increases. Typically, ρ ~ 4.1 – 3.9° when σ increases from 0.5 to 4 kHz. Within the error bars, it indicates that ρ is almost not correlated to the input natural linewidth σ and a value of ρ ≈ 4°±0.5 can be considered as reliable. This value is consistent with the value of ρ ≈ 3° obtained from the model of splitted domains (Eq. (3)), but we should stress that our method is The final orientational probability density of the C–D in nonadecane/urea at 90 K, obtained at the minimum of the LSE, is presented in Fig. 5 in a polar plot. The probability density has twelve lobes. In contrast to the reconstructed probability densities which do not correspond to the LSE minimum, the lobes are here perfectly symmetric. Since the C–D/C–D angle is ~109.4°, we can deduce the mean chain orientation, which points towards the corner of the hexagon representing the mean low temperature framework. The origin of this residual intrinsic angular dispersion cannot be more characterized by NMR due to the ensemble average over chain and domain orientations. However, since the domains are split by δ ≈ ±1° (at 90 K) with respect to the mean orthohexagonal direction, we expect the true angular distribution within each domain to be smaller than 4°. Fig. 5 Optimum orientational probability densities w of the nonadecane C–D bonds, for nonadecane/urea-h[4] at 90 K, obtained by inversion with Tikhonov regularization and NNLS procedure (σ=1.5 kHz, angular resolution Δφ=2°); The solid gray lines represents a least-square fit of the reconstructed probability densities by the sum of two Gaussians of equal weight and angular dispersion ρ=4°, for a given CD[2] group. The arrow indicate the mean chain orientation corresponding to the CD[2] mean orientation. It also worth mentioning that the main approximation underlying the orientational deconvolution method proposed above concerns the invariance of the natural linewidth σ as a function of orientation. In principle, this parameter may depend on the magnetic field orientation, although we expect this dependence to be small in a C[6] symmetric axial system (on average). Nevertheless, the fact that the estimation of the residual intrinsic disorder is almost not correlated to the input natural linewidth suggests that an orientational dependent real natural linewidth may only slightly affect the result. This problem will be the object of a future work, as well as a generalization to non axial disorders. 4 Conclusion From the information contained in single crystal ^2H NMR spectra acquired at different magnetic field orientations, we reconstructed the orientational probability density of the C–D bonds orientations of deuterated nonadecane molecules ultraconfined in urea parallel channels in the low temperature phase of nonadecane/urea at 90 K. The reconstruction method is based on linear least-square fitting with Tikhonov regularized procedures and non negative constraints. In particular, a careful study of the behavior of the least-square merit-function shows that the result is almost not correlated to the estimation of the natural linewidth. The results are fully consistent with a previous study, and confirm that the mean chain orientation points towards the corner of the channels with an intrinsic angular disorder which could be quantified precisely. The originality of our approach comes from the fact that the method is model-independent, in contrast to previous studies. Because the method does not make any structural hypothesis concerning the ordering of the host/guest compound, the results are therefore independent from the controversial interpretation of diffraction results. These results give another illustration that single crystal NMR combined with regularized inverse methods is a unique tool to determine intrinsic static orientational disorder in complex supramolecular architectures. We are indebted to Y. Legrand for careful reading of the manuscript, to J.-C. Ameline for technical support, and to T. Breczewski for providing deuterated single crystals of high quality. We also thank B. Toudic for fruitful discussions.	{"url":"https://comptes-rendus.academie-sciences.fr/chimie/articles/en/10.1016/j.crci.2005.06.024/","timestamp":"2024-11-15T04:46:00Z","content_type":"text/html","content_length":"91892","record_id":"<urn:uuid:eacb4316-6fd5-4667-b8ab-e28a4812c893>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00732.warc.gz"}
Lever Calculator \| Mechanical Advantage Last updated: Lever Calculator Archimedes said, "Give me a lever long enough" – use our lever calculator to find out how long that lever should be! Here you will learn: • The physics behind levers and the lever equation; • The law of the lever, by Archimedes; • What is the mechanical advantage of a lever; and • Some examples of levers in action, both practical and impractical! What is a lever? A lever is probably the most straightforward mechanism ever devised by humanity. Take a plank of wood (or any rigid material), find a place where to put it, and you are all set. Regardless of their lack of complexity, levers are marvelous machines everywhere in our daily lives. Look around you wherever you are: scissors, nail clippers, bottle openers, and so on, are all levers. Let's discover more about them. 🔎 Levers appear all the time in nature, in particular in the animal kingdom. Muscles, joints, jaws: evolution engineered the living being to be as good as possible. A lever allows you to move an object with a specific advantage – that is, you'll be able to move it with either a reduced effort or an increased speed or displacement. Levers work thanks to the application of forces and the exploitation of the torques which derive from them. This alters the balance, allowing us to lift heavy loads with very little effort. The elements of a lever We can describe levers with a relatively small number of elements: • Fulcrum; • Effort; and • Resistance. The fulcrum is where the lever pivots. The fulcrum doesn't need to be in the middle of the lever – in fact, its position allows for entirely different uses of this simple machine. You can learn more about it with our fulcrum calculator. The resistance (or load) is the force applied by the object you want to move, cut, or whatever else your lever does. We identify the resistance with the symbol $F_b$. The segment between the fulcrum to the point of application of the resistance is denoted as $b$. The effort is the force you apply when operating the lever. The distance from the point of application to the fulcrum is $a$, while its value is $F_a$. The lever equation The operational principle of a lever is straightforward. We can identify an equilibrium situation where the lever is stationary. In that condition, the torque applied to the lever by both forces equal each other. The torque is the rotational analog of a linear force and, in layman's words, describes the effect of a force $F$ applied at a certain distance $x$ (the arm) from a pivot: $\boldsymbol{\tau}= \mathbf{x}\times\mathbf{F}= x\times F\times \sin{\theta}$ The quantities are in bold because we need to consider them as vectors: the torque's magnitude depends on the sine of the angle between the applied force and the arm. Good news! In a lever, the angle $\theta$ is usually equal to $90\degree$, and we can happily ignore it since $\sin{90\degree}=1$. To learn more about torque, check out our torque calculator! Back to the equilibrium condition. We said that the torques equal each other: $\tau_a = a \times F_a = b \times F_b = \tau_b$ This equation allows us to find out every quantity we need in a problem involving levers. For example, if the arms' length is known, it is possible to calculate the force required to attain equilibrium against a specific resistance. The mechanical advantage of a lever and the law of the lever The quantity that measures the "performance" of a lever is called mechanical advantage. It derives from analyzing the torque applied by both forces on the lever. The famous law of the lever says that the multiplication of the force in a lever is given by: $\text{MA} = \frac{F_a}{F_b} = \frac{a}{b}$ Take a look at the formula and remember that $a$ corresponds to the side where you apply the effort and $b$ relates to the resistance. First thing: it is easier to think in terms of the arms and not the forces: it is pretty obvious that the greater the force you apply, the higher the mechanical advantage is. But imagine now having a really "unbalanced" lever, with the effort applied at a distance far greater than the one of the resistance: $a\gg b$. The mechanical advantage of such a lever would be extremely high, reflected in the need to apply a really high effort on one of the levers. 🔎 Archimedes said: "Give me a lever long enough and a fulcrum on which to place it, and I shall move the world". He had in mind a lever with a particularly good mechanical advantage! The value of the mechanical advantage tells us how the lever behaves. The higher the mechanical advantage, the smaller the effort applied to balance the same resistance. In that case, the lever is called a force multiplier. If the mechanical advantage equals $1$, the lever gives you no edge: it would be like applying the effort directly. Finally, if the mechanical advantage is smaller than $1$, the lever is a speed multiplier, which means you'll move the point where the resistance is applied faster than you'd think for the effort (over the same time, hence increasing the speed). 🙋 We dedicated an entire calculator to the mechanical advantage in various simple machines – check it out at our mechanical advantage calculator! The three types of levers How the various elements of a lever relate to one another allows us to define three different types of levers: • Class I levers – The levers we picture in our mind when someone asks us to think of one. The fulcrum is between resistance and effort. • Class II levers – The resistance and the effort are placed on the same side on the same side of the fulcrum, with $a>b$. • Class III levers – The resistance and the effort are placed on the same side of the fulcrum, but this time $a<b$. The three types of levers. From the top, a class I lever, a class II lever, and a class III lever. Now we know how to characterize them. Let's take a look at the possible mechanical advantages! • For a class I lever, $a$ and $b$ can take every possible value greater than $0$ (and bounded by the length of the lever, of course). According to the ratio of $a$ and $b$, such a lever can be a force multiplier. • For a class II lever, $a\text{\textgreater}b$. This implies that the mechanical advantage is greater than $1$, and the lever always acts as a force multiplier. • For a class III lever, the opposite is true. Since $b\text{\textgreater} a$ the mechanical advantage is always smaller than $1$, and so the lever always acts as a speed multiplier. An example: calculate the lever arm to lift the world We can calculate the characteristics of a lever able to lift the world using the lever equation. There's only a condition: you're the one doing the lifting! Assuming a weight of $70\ \text{kg}$ and that you're standing on your end of the lever, we can compute an effort of: $\footnotesize F_a=70\ \text{kg}\times9.81\ {\text{m/}}{\text{s}^2} = 686.7\ \text{N}$ What about the Earth? Eehh... its mass is $M_E=5.9722\times10^{24}\ \text{kg}$, which corresponds to a resistance of: \footnotesize \begin{align} F_b &=5.9722\times10^{24}\ \text{kg}\times9.81\ {\text{m/}}{\text{s}^2}\\ &= 5.8587\times10^{25}\ \text{N} \end{align} We need to consider a lever long enough: let's say as long as the distance between the Earth and the Sun ($148.81\times10^9\ \text{m}$). Calling the length $l$, and using the equality $a = l-b$, we can write the equation of the lever as: $l-b=\frac{F_b\times b}{F_a}$ \begin{align} (l-b)\times F_a &= b\times F_b\\\\ l\times F_a &= (F_a + F_b)\times b\\\\ b&=\frac{l\times F_a}{F_a+F_b} \end{align} This equation allows us to find the length of the lever's arm associated to the resistance, in our case, the Earth. We substitute the numerical values (neglecting $F_a$ at the denominator), and we calculate the arm of the lever: \footnotesize \begin{align} b&=\frac{1.481\times 10^{11}\ \text{m} \times 687.7\ \text{N}}{5.8587\times 10^{25}\ \text{N}}\\\\ &=1.74\times10^{-12}\ \text{m} \end{align} That's one long lever: the arm on which the Earth rests is smaller than a hydrogen atom! Our Archimedes is trying to lift the world using only a lever. Will he manage doing so? Back to some more practical examples that don't involve lifting planets. Let's say you are at the playground, playing on the seesaw with one of your friends. By now, you know that the seesaw is a class I lever. We assume that your weight is $75\ \text{kg}$ and your friend, being on the thin side, is only $60\ \text{kg}$. The seesaw is $4\ \text{m}$ long. Where do you have to sit to balance your friend? And what is the mechanical advantage of the lever you created? Apply the lever equation to find the arm of your side of the lever. Notice that in the equation you can use the masses instead of the weights. They differ only by the multiplicative constant, $g$, the standard gravitational parameter, and we can cancel it out. \footnotesize \begin{align} a&=\frac{g\times 60\ \text{kg}\times 2\ \text{m}}{g\times 75\ \text{kg}}\\\\ &=\frac{60\ \text{kg}\times 2\ \text{m}}{75\ \text{kg}}=1.6\ \text{m} \end{align} You have to seat $40\ \text{cm}$ before the end of the seesaw to equal your friend's resistance – now the game is... balanced! Now we can calculate the lever's mechanical advantage. Calculate the quotient of the two arms' lengths $\text{MA}=\frac{a}{b}=\frac{1.6\ \text{m}}{2\ \text{m}} = 0.8$ It is smaller than $1$ because you are lifting a smaller weight than yours. How to calculate the mechanical advantage of a lever using our lever calculator Our lever calculator is a helpful tool that allows you to calculate everything you need in your physics homework — and not only: we hope that it can help you in your everyday problems too! Leverage your time and quickly find the results: insert the quantities you know, and find the results. You can insert the forces acting on the levers, the arms, or the mechanical advantage and only one of the other quantities, to find the remaining ones! What is the lever equation? The lever equation defines the forces and the physical features of a lever in its equilibrium status. It derives from the comparison of the torque acting on the lever: Fa × a = Fb × b • Fᵢ are the forces, either the effort or the resistance; and • lᵢ are the arms of the lever (a and b). Manipulate that simple equation to isolate the desired quantity. How long should the arm of a lever be to balance a 1500 kg car with my weight? Let's say you weigh 70 kg. To balance a car with a mass of 1500 kg, you can use a lever defined by the mechanical advantage: MA = (1500 × g)/(70 × g) = 21.43 where g is the standard gravitational parameter, equal to 9.81 m/s². The arm of the lever on your side should be 21.43 times longer than the arm on the car side. How to calculate the mechanical advantage of a lever? You can calculate a lever's mechanical advantage by computing the ratio of the forces acting on the lever or, interchangeably, the ratio of the lever's arms: MA = Fa/Fb = a/b The mechanical advantage can be smaller, greater, or equal to one. What is the mechanical advantage of a lever? The mechanical advantage of a lever tells you if the lever you are using will give you an edge when doing some mechanical work. A mechanical advantage greater than 1 defines levers that help you lift heavy loads. In contrast, a mechanical advantage smaller than 1 defines levers that reduce the force you apply, but instead return an increased speed.	{"url":"https://www.omnicalculator.com/physics/lever","timestamp":"2024-11-12T15:33:21Z","content_type":"text/html","content_length":"620874","record_id":"<urn:uuid:4f35c4a1-6e22-4f96-834f-ab40bd08f7af>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00364.warc.gz"}
Is One Forecast Model Better Than Another? Blog post contributed by: Tim DelSole* The Sign Test Is one forecast model better than another? A natural approach to answering this question is to run a set of forecasts with each model and then see which set has more skill. This comparison requires a statistical test to ensure that the estimated difference represents a real difference in skill, rather than a random sampling error. Unfortunately, there are three problems with using standard difference tests: they have low statistical power, they assume uncorrelated forecast errors, and their significance levels are wrong. As a simple example, consider a (typical) seasonal prediction system with a 20-year re-forecast data set and a correlation skill 0.5. To detect an improvement in skill using standard tests, the new seasonal prediction system would need a correlation skill exceeding 0.84, or a mean squared error that is lower than the original by a factor of about 2.5! Such gains are inconceivable in seasonal prediction. Even worse, forecast errors tend to be correlated in typical comparison studies, which violates a fundamental assumption in the difference-in-skill test (see DelSole and Tippett, 2014, Interestingly, economists have studied this problem quite extensively and developed several methods for comparing forecasts. A good starting point is the seminal paper Diebold and Mariano (1995). A key concept in effective skill comparison is to examine paired differences. That is, instead of computing the skill of each forecast separately and then taking their difference, you compute the difference in forecast performance first, for each event, and then aggregate those differences. For example, suppose you have a criterion for deciding which model makes a “better” forecast of an event. Then, if two models are equally skillful, the probability that one model beats the other is 50%. At this point, the problem looks exactly like the problem of deciding if a coin is fair. As everyone knows, you test if a coin is fair by flipping it several times and checking that about half of the tosses are heads. More precisely, the number of heads follows a binomial distribution with p = 1/2, and the null hypothesis is rejected if the number of heads falls outside the 95% interval computed from that distribution. Similarly, to compare the skill of models A and B, you simply count the number of times model A produces a better forecast than B, and compare that count to the 95% The above test is called the sign test because it can be evaluated simply by computing differences in some measure and then counting the number of positive signs. It has been applied in weather prediction studies (Hamill, 1999), but the simplicity and versatility of the test does not seem to be widely recognized. For instance, notice that no restrictions were imposed on the criterion for deciding the better forecast. If a forecaster is predicting a single index, then a natural criterion is to select the forecast closest to the observed value. But the test is much more versatile than this. For instance, if the forecaster is making probabilistic forecasts of binary events, then the criterion could be the forecast having the smaller Brier score. For multiple-category events, the criterion could be the forecast having smaller rank probability score. If the forecaster is predicting spatial fields, then the criterion could be the field with larger pattern correlation. For probabilistic spatial forecasts, the criterion could be the field with larger spatially-averaged Heidke skill score. Interestingly, the problem of field significance (Livezey and Chen, 1983) is circumvented by this approach. Also, the test makes no assumptions about the error distributions, in contrast to tests based on correlation and mean square error (which assume Gaussian distributions). Finally, competing forecasts can be correlated with each other. This last point is particularly important for model development, as a change in a parameterization may change the forecast only modestly relative to the original model. Standard difference tests based on mean square error or correlation are completely useless in such cases. These concepts, and additional skill comparison tests, are discussed in more detail in DelSole and Tippett (2014). The Random Walk Test An informative way to summarize the results of the sign test is to display a sequence of sign tests in the form of a random walk. Figure 1 shows a schematic of the random walk test. A random walk is the path traced by a particle as it moves forward in time. Between events, the particle simply moves parallel to the x-axis. At each event, the particle moves up one unit if forecast A is more skillful than B, otherwise it moves down one unit. If the forecasts are equally skillful, then the upward and downward steps are equally probable and the average location is y = 0. Happily, the 95% interval for a random walk has a very simple form: it is ±2√N, where N is the number of independent forecast events. The hypothesis that the forecasts are equally skillful is rejected when the particle walks outside the 95% interval. The merit in displaying the sign test as a random walk lies in the fact that the figure may reveal time-dependent variations in skill. A spectacular example of this is shown in Figure 2. This figure shows results of the random walk test comparing monthly mean forecasts of the NINO3.4 index at 2.5 month lead between CFSv2 and other models in the North American Multi-Model Ensemble. From 1982 to about 2000, the random walks drifted upward, with some models going above the 95% interval, indicating that CVSv2 was more skillful than these models. However, after 2000, there is an abrupt change in skill, so by the end of the period most models lie below the 95% interval, indicating that these models were more skillful than CFSv2. The poor skill of CFSv2 relative to other models is widely attributed to a discontinuity in climatology due to the introduction of ATOVS satellite data into the assimilation system in October 1998, as discussed in Kumar et al. (2012), Barnston and Tippett (2013), and Saha et al. (2014). This example illustrates how the random walk test could be used routinely at operational centers for monitoring possible changes in skill due to changes in dynamical model or data assimilation system. Several points about the random walk test are worth emphasizing. Technically, the location of the path on the very last time step dictates the final decision, because it is based on the most data. Also, the results of the sign test can be inverted to give an estimate of the probability p that one model is more skillful per event than another model. That is, the random walk test can be used to quantify the difference in skill in terms of a probability per event. Also, the random walk test ignores the amplitude of the errors. This is both an advantage and a disadvantage. The advantage is that it allows a test to be formulated in a way that avoids distributional assumptions about forecast errors. The disadvantage is that it does not incorporate amplitude information, so a forecast might have huge busts but still be considered more skillful than another model that never busts. An alternative test based on Wilcoxon Signed-Rank test is a non-parametric test that incorporates amplitude information. Importantly, the random walk test assumes each event is independent of the others. This assumption is not true if the initial condition between forecasts are sufficiently close. It is not clear how serial correlations or forecast calibration can be taken into account in this test, if the calibration is based on the same data as the forecast comparison data set. More information about this test, and additional skill comparison tests, are discussed in DelSole and Tippett (2014, 2016). Also, R and Matlab codes for performing these tests can be found at http:// *Tim DelSole is a professor in Department of Atmospheric, Oceanic, and Earth Sciences, George Mason University; and a Senior Researcher with the Center for Ocean-Land-Atmosphere Studies. Barnston, A. G. and M. K. Tippett, 2013: Predictions of Nino3.4 SST in CFSv1 and CFSv2: A Diagnostic Comparison. Clim. Dyn., 41, 1–19, doi:10.1007/s00382-013-1845-2. DelSole, T. and M. K. Tippett, 2014: Comparing forecast skill. Mon. Wea. Rev., 142, 4658–4678. DelSole, T. and M. K. Tippett, 2016: Forecast comparison based on random walks. Mon. Wea. Rev., 144 (2), Diebold, F. X. and R. S. Mariano, 1995: Comparing predictive accuracy. J. Bus. Econ. Stat., 13, 253–263. Hamill, T. M., 1999: Hypothesis tests for evaluating numerical precipitation forecasts. Wea. Forecasting, 14, 155–167. Kumar, A., M. Chen, L. Zhang, W. Wang, Y. Xue, C. Wen, L. Marx, and B. Huang, 2012: An analysis of the nonstationarity in the bias of sea surface temperature forecasts for the NCEP Climate Forecast System (CFS) version 2. Mon. Wea. Rev., 140, 3003–3016. Livezey, R. E. and W. Chen, 1983: Statistical field significance and its determination by Monte Carlo techniques. Mon Wea. Rev., 111, 46–59. Saha, S., et al., 2014: The NCEP Climate Forecast System Version 2. J. Climate, 27, 2185–2208.	{"url":"https://hepex.org.au/is-one-forecast-model-better-than-another/","timestamp":"2024-11-13T04:40:35Z","content_type":"text/html","content_length":"132582","record_id":"<urn:uuid:d6a9dc37-2822-49a6-8f3f-dc094907c519>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00352.warc.gz"}
Vestnik KRAUNC. Fiz.-Mat. Nauki. 2022. vol. 41. no. 4. pp. 9–31. ISSN 2079-6641 MSC 68W30, 76F20 Research Article Construction of complex shell models of turbulent systems by computer algebra methods G. M. Vodinchar¹², L. K. Feshchenko¹, N. V. Podlesnyi³ ¹Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, Mirnaya Str., 7, Paratunka, Kamchatka region, 683003, Russia ²Kamchatka State Technical University, Klyuchevskaya Str., 35, Petropavlovsk-Kamchatsky, 683003, Russia ³Vitus Bering Kamchatka State University, Pogranichnaya Str., 4, Petropavlovsk-Kamchatsky, 683032, Russia E-mail: gvodinchar@ikir.ru One popular class of small-scale turbulence models is the class of shell models. In these models, the fields of a turbulent system are represented by time-dependent collective variables (real or complex), which are understood as the field intensity meassure in a given range of spatial scales. The model itself is a certain system of quadratically nonlinear ordinary differential equations for collective variables. Construction a new shell model requires rather complex analytical transformations. This is due to the fact that the system of model equations in the absence of dissipation must have some quadratic invariants and phase-space volume is unchanged. In addition, there are limitations associated with the impossibility of non-linear interaction of some scales ranges. All this imposes limitations on the coefficients of the nonlinear terms of the model. Constraints form a system of equations with parameters. The complexity of this system increases sharply for non-local models, when the interaction is described not only close ranges of scales and when complex collective variables are used. The paper proposes a computational technology that allows automating the process of building shell models. It makes it easy to combine different invariants and the meassure of nonlocality. The technology is based on computer algebra methods. The process of constructing equations for unknown coefficients and their solution has been automated. As a result, parametric classes of cascade models are obtained that have the required analytical properties. Key words: turbulence, shell models, computer algebra, automation of model development. Funding. The work was carried out within the framework of the state assignment on the topic «Physical processes in the system of near space and geospheres under solar and lithospheric influences» (No. AAAA-A21-121011290003-0). DOI: 10.26117/2079-6641-2022-41-4-9-31 Original article submitted: 29.11.2022 Revision submitted: 06.12.2022 For citation. Vodinchar G. M., Feshchenko L. K., Podlesnyi N. V. Construction of complex shell models of turbulent systems by computer algebra methods. Vestnik KRAUNC. Fiz.-mat. nauki. 2022, 41: 4, 9-31. DOI: 10.26117/2079-6641-2022-41-4-9-31 Competing interests. The author declare that there is no conflicts of interest regarding authorship and publication. Contribution and Responsibility. The author is solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by the author. The content is published under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/deed.ru) © Vodinchar G. M., Feshchenko L. K., Podlesnyi N. V., 2022 1. Mingshun J., Shida L. Scaling behavior of velocity and temperature in a shell model for thermal convective turbulence, Physical Review E, 1997, 56, 441 2. Hattori Y., Rubinstein R., Ishizawa A. Shell model for rotating turbulence, Physical Review E, 2004, 70, 046311 3. Ching E. S.C., Guo H., Cheng W. C. Understanding the different scaling behavior in various shell models proposed for turbulent thermal convection, Physica D: Nonlinear Phenomena, 2008, 237:4, 4. Frik P. G. Turbulentnost’: podkhody i modeli [Turbulence: approaches and models], Moskva–Izhevsk, 2010, 332 (In Russian) 5. Ditlevsen P. Turbulence and Shell Models, Cambridge, University Press, 2011, 152. 6. Davenport J., Sire I., Tournier E. Komp’yuternaya algebra [Computer algebra], Moscov, Mir, 1991, 352 (In Russian) 7. Matrosov A. V. Maple 6. Resheniye zadach po vysshey matematike i mekhanike [Maple 6. Solving problems in higher mathematics and mechanics], St. Petersburg, BHV, 2001, 528 (In Russian) 8. Chichkarev E. A. Komp’yuternaya matematika s Maxima [Computer mathematics with Maxima], Moskov, ALT Linux, 2012, 384 (In Russian) 9. Vodinchar G. M., Feshchenko L. K. Automated generation of turbulence shell models by computer algebra methods, Computing technologies, 2021, 26:5, 65–80. (In Russian) 10. Gledzer E. B., A system of hydrodynamic type that allows quadratic integral of motion, DAN USSR, 1973, 209:5, 1046–1048. (In Russian) 11. Yamada M., Ohkitani K. Lyapunov Spectrum of a Chaotic Model of Three-Dimensional Turbulence, J. Phys. Soc. Jpn., 1987, 56, 4210. 12. L’vov V., Podivilov E., Pomlyalov A., Procaccia I., Vandembroucq D. Improved shell model of turbulence, Phys. Rev. E, 1998, 58, 1811. 13. Ditlevsen P. Symmetries, invariants, and cascades in a shell model of turbulence, Phys. Rev. E, 2000, 62, 484. 14. Plunian F., Stepanov R. A non-local shell model of hydrodynamic and magnetohydrodynamic turbulence, New Journal of Physics, 2007, 9, 294. 15. Plunian F., Stepanov R. A., Frick P. G. Shell models of magnetohydrodynamicm turbulence, Physics Reports, 2013, 523, 1–60. Vodinchar Gleb Mikhailovich – Candidate of Physical and Mathematical Sciences, Leading Researcher, Acting Head of the Laboratory for Simulation of Physical Processes, IKIR FEB RAS, Associate Professor, Department of Control Systems, Kamchatka State Technical University, Petropavlovsk-Kamchatsky, Russia, ORCID 0000-0002-5516-1931. Feshchenko Liubov Konstantinovna – Candidate of Physical and Mathematical Sciences, Researcher, Laboratory for Modeling Physical Processes, IKIR FEB RAS, Russia, ORCID 0000-0001-5970-7316. Podlesny Nikita Viktorovich – Master student of the Department of Mathematics and Physics of the Vitus Bering Kamchatka State University, Russia, ORCID 0000-0002-3213-5706.	{"url":"https://krasec.ru/vodinchar2022414eng/","timestamp":"2024-11-03T18:49:22Z","content_type":"text/html","content_length":"67902","record_id":"<urn:uuid:1a7ad335-95d9-4616-a18f-a4591ddf95ca>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00177.warc.gz"}
Advanced Statistics with Applications in R Book description Advanced Statistics with Applications in R fills the gap between several excellent theoretical statistics textbooks and many applied statistics books where teaching reduces to using existing packages. This book looks at what is under the hood. Many statistics issues including the recent crisis with p-value are caused by misunderstanding of statistical concepts due to poor theoretical background of practitioners and applied statisticians. This book is the product of a forty-year experience in teaching of probability and statistics and their applications for solving real-life There are more than 442 examples in the book: basically every probability or statistics concept is illustrated with an example accompanied with an R code. Many examples, such as Who said π? What team is better? The fall of the Roman empire, James Bond chase problem, Black Friday shopping, Free fall equation: Aristotle or Galilei, and many others are intriguing. These examples cover biostatistics, finance, physics and engineering, text and image analysis, epidemiology, spatial statistics, sociology, etc. Advanced Statistics with Applications in R teaches students to use theory for solving real-life problems through computations: there are about 500 R codes and 100 datasets. These data can be freely downloaded from the author's website dartmouth.edu/~eugened. This book is suitable as a text for senior undergraduate students with major in statistics or data science or graduate students. Many researchers who apply statistics on the regular basis find explanation of many fundamental concepts from the theoretical perspective illustrated by concrete real-world applications. Product information • Title: Advanced Statistics with Applications in R • Author(s): • Release date: November 2019 • Publisher(s): Wiley • ISBN: 9781118387986	{"url":"https://www.oreilly.com/library/view/advanced-statistics-with/9781118387986/","timestamp":"2024-11-04T10:59:19Z","content_type":"text/html","content_length":"71488","record_id":"<urn:uuid:6b697126-945d-4e02-9e88-1891dd866309>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00062.warc.gz"}
Revised Acts Consumer Credit Act 1995 F226[FOURTH SCHEDULE Meaning of letters and symbols: K is the number of a loan K’ is the number of a repayment or a payment of charges A[k] is the amount of loan number K A’[k’] is the amount of repayment number K’ ∑ represents a sum m is the number of the last loan m’ is the number of the last repayment or payment of charges tk is the interval, expressed in years and fractions of a year, between the date of loan No. 1 and those of subsequent loans Nos 2 to m tk’ is the interval, expressed in years and fractions of a year, between the date of loan No. 1 and those of repayments or payments of charges Nos 1 to m’ i is the percentage rate that can be calculated (either by algegra, by successive approximations, or by a computer programme) where the other terms in the equation are known from the contract or (a) The amounts paid by both parties at different times shall not necessarily be equal and shall not necessarily be paid at equal intervals. (b) The starting date shall be that of the first loan. (c) Intervals between dates used in the calculations shall be expressed in years or in fractions of a year. A year is presumed to have 365 days or 365,25 days or (for leap years) 366 days, 52 weeks or 12 equal months. An equal month is presumed to have 30,41666 days (i.e. 365/12). (d) The result of the calculation shall be expressed with an accuracy of at least one decimal place. When rounding to a particular decimal place the following rule shall apply: If the figure at the decimal place following this particular decimal place is greater than or equal to 5, the figure at this particular decimal place shall be increased by one. (e) Member States shall provide that the methods of resolution applicable give a result equal to that of the examples presented in Annex III. ] Substituted (20.09.2000) by European Communities (Consumer Credit) Regulations 2000 (S.I. No. 294 of 2000), reg. 5 and Table part 2.	{"url":"https://revisedacts.lawreform.ie/eli/1995/act/24/schedule/4/revised/en/html","timestamp":"2024-11-07T04:16:16Z","content_type":"text/html","content_length":"43922","record_id":"<urn:uuid:81584019-8474-45c7-bcab-3e19ec65ab00>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00290.warc.gz"}
Authors: Matteo Riondato (Two Sigma), David Garcia-Soriano, Francesco Bonchi Published in: Data Mining and Knowledge Discovery, 31(2):314–349 Presented at: Department of Information Engineering, University of Padua, Padua (Italy), September 26, 2016 Abstract: We study the problem of graph summarization. Given a large graph we aim at producing a concise lossy representation (a summary) that can be stored in main memory and used to approximately answer queries about the original graph much faster than by using the exact representation. In this work we study a very natural type of summary: the original set of vertices is partitioned into a small number of supernodes connected by superedges to form a complete weighted graph. The superedge weights are the edge densities between vertices in the corresponding supernodes. To quantify the dissimilarity between the original graph and a summary, we adopt the reconstruction error and the cut-norm error. By exposing a connection between graph summarization and geometric clustering problems (i.e., k-means and k-median), we develop the first polynomial-time approximation algorithms to compute the best possible summary of a certain size under both measures. We discuss how to use our summaries to store a (lossy or lossless) compressed graph representation and to approximately answer a large class of queries about the original graph, including adjacency, degree, eigenvector centrality, and triangle and subgraph counting. Using the summary to answer queries is very efficient as the running time to compute the answer depends on the number of supernodes in the summary, rather than the number of nodes in the original graph. DOI: https://doi.org/10.1007/s10618-016-0468-8	{"url":"https://www.twosigma.cn/insights/article/graph-summarization-with-quality-guarantees/","timestamp":"2024-11-14T11:05:42Z","content_type":"text/html","content_length":"45639","record_id":"<urn:uuid:89be4c50-ba07-41d1-8621-3a59f5361358>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00109.warc.gz"}
Example 9.27: Baseball and shrinkage \| R-bloggersExample 9.27: Baseball and shrinkage Example 9.27: Baseball and shrinkage [This article was first published on SAS and R , and kindly contributed to ]. (You can report issue about the content on this page ) Want to share your content on R-bloggers? if you have a blog, or if you don't. To celebrate the beginning of the professional baseball season here in the US and Canada, we revisit a famous example of using baseball data to demonstrate statistical properties. In 1977, Bradley Efron and Carl Morris published a paper about the James-Stein estimator– the shrinkage estimator that has better mean squared error than the simple average. Their prime example was the batting averages of 18 player in the 1970 season: they considered trying to estimate the players’ average over the remainder of the season, based on their first 45 at-bats. The paper is a pleasure to read, and can be downloaded . The data are available , on the pages of Statistician Phil Everson , of Swarthmore College. Today we’ll review plotting the data, and intend to look at some other shrinkage estimators in a later entry. We begin by reading in the data for Everson’s page. (Note the long address would need to be on one line, or you could could use a URL shortener like TinyURL.com. To read the data, we use the statement to indicate a tab-delimited file and to say that the data begin in row 2. The statement helps read in the variable-length last names. filename bb url "http://www.swarthmore.edu/NatSci/peverso1/Sports%20Data/ data bball; infile bb delimiter='09'x MISSOVER DSD lrecl=32767 firstobs=2 ; informat firstname $7. lastname $10.; input FirstName $ LastName $ AtBats Hits BattingAverage RemainingAtBats RemainingAverage SeasonAtBats SeasonHits SeasonAverage; data bballjs; set bball; js = .212 * battingaverage + .788 * .265; avg = battingaverage; time = 1; if lastname not in("Scott","Williams", "Rodriguez", "Unser","Swaboda","Spencer") then name = lastname; else name = ''; avg = seasonaverage; name = ''; time = 2; output; avg = js; time = 3; name = ''; output; In the second step, we calculate the James-Stein estimator according to the values reported in the paper. Then, to facilitate plotting, we convert the data to the “long” format, with three rows for each player, using the explicit statement. The average in the first 45 at-bats, the average in the remainder of the season, and the James-Stein estimator are recorded in the same variable in each of the three rows, respectively. To distinguish between the rows, we assign a different value of : this will be used to order the values on the graphic. We also record the last name of (most of) the players in a new variable, but only in one of the rows. This will be plotted in the graphic– some players’ names can’t be shown without plotting over the data or other players’ names. Now we can generate the plot. Many features shown here have been demonstrated in several entries. We call out 1) the option, which increases the text size in the titles and labels, 2) the option, which moves the data away from the edge of the plot frame, 3) the option in the statement, which replaces the values of “time” with descriptive labels, and 4) the handy ab=c syntax which replicates the plot for each player. title h=3 "Efron and Morris example of James-Stein estimation"; title2 h=2 "Baseball players' 1970 performance estimated from first 45 at-bats"; axis1 offset = (4cm,1cm) minor=none label=none value = (h = 2 "Avg. of first 45" "Avg. of remainder" "J-S Estimator"); axis2 order = (.150 to .400 by .050) minor=none offset=(0.5cm,1.5cm) label = (h =2 r=90 a = 270 "Batting Average"); symbol1 i = j v = none l = 1 c = black r = 20 w=3 pointlabel = (h=2 j=l position = middle "#name"); proc gplot data = bballjs; plot avg time = lastname / haxis = axis1 vaxis = axis2 nolegend; run; quit; To read the plot (shown at the top) consider approaching the nominal true probability of a hit, as represented by the average over the remainder of the season, in the center. If you begin on the left, you see the difference associated with using the simple average of the first 45 at-bats as the estimator. Coming from the right, you see the difference associated withe using the James-Stein shrinkage estimator. The improvement associated with the James-Stein estimator is reflected in the generally shallower slopes when coming from the left. With the exception of Pirates great Roberto Clemente and declining third-baseman Max Alvis , most every line has a shallower slope from the left; James’ and Stein’s theoretical work proved that overall the lines must be shallower from the right. A similar process is undertaken within R. Once the data are loaded, and a subset of the names are blanked out (to improve the readability of the figure), the functions are used to create the lines. bball = read.table("http://www.swarthmore.edu/NatSci/peverso1/Sports%20Data/JamesSteinData/Efron-Morris%20Baseball/EfronMorrisBB.txt", header=TRUE, stringsAsFactors=FALSE) bball$js = bball$BattingAverage * .212 + .788 * (0.265) c("Scott","Williams", "Rodriguez", "Unser","Swaboda","Spencer")))] = "" a = matrix(rep(1:3, nrow(bball)), 3, nrow(bball)) b = matrix(c(bball$BattingAverage, bball$SeasonAverage, bball$js), 3, nrow(bball), byrow=TRUE) matplot(a, b, pch=" ", ylab="predicted average", xaxt="n", xlim=c(0.5, 3.1), ylim=c(0.13, 0.42)) matlines(a, b) text(rep(0.7, nrow(bball)), bball$BattingAverage, bball$LastName, cex=0.6) text(1, 0.14, "First 45\nat bats", cex=0.5) text(2, 0.14, "Average\nof remainder", cex=0.5) text(3, 0.14, "J-S\nestimator", cex=0.5)	{"url":"https://www.r-bloggers.com/2012/04/example-9-27-baseball-and-shrinkage/","timestamp":"2024-11-05T00:07:39Z","content_type":"text/html","content_length":"100421","record_id":"<urn:uuid:38635d6e-1809-401a-8148-f54b24c92a4d>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00275.warc.gz"}
Next Carlo Heissenberg's arrival at IPhT! Carlo Heissenberg graduated with flying colors in 2019 from the Scuola Normale Superiore di Pisa, the Italian twin sister of the Ecole Normale de Paris. His initial research focused on the infrared properties of gravitational theories and higher spin theories from the perspective of asymptotic symmetries, a topic that later found phenomenological applications in the context of gravitational wave studies. It was during his first postdoctoral period at Nordita-Uppsala (2019-2023) that Carlo started working on gravitational wave physics using ideas borrowed from particle physics, such as scattering amplitudes and eikonal exponentiation in particular. He quickly became a leader in the field, publishing several important papers with world-famous physicists such as Gabriele Veneziano. Carlo's group achieved, for instance, the first complete calculation of the gravitational deflection angle to the third post-Minkowskian order in the Newton coupling constant expansion, while simultaneously resolving the high-energy limit problem that was puzzling the community. In the meantime, Carlo was granted a Marie-Curie fellowship and moved to his second postdoc at Queen Mary, while continuing to work on gravitational wave physics, focusing on new observables (waveforms, angular momentum loss, effect of spins…) that could be calculated efficiently using scattering amplitudes techniques. Carlo accepted an offer from the IPhT and will join the lab as a permanent member in Fall 2024.	{"url":"https://www.ipht.fr/en/Phocea/Vie_des_labos/News/index.php?id_news=1325","timestamp":"2024-11-09T01:36:28Z","content_type":"text/html","content_length":"26239","record_id":"<urn:uuid:98d063da-f791-423b-9724-6e697a58526d>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00526.warc.gz"}
Searching and Inserting in a Binary Search Tree As previously mentioned, we can use a binary search tree as a symbol table to store key-value pairs. As such, it will need to support the operations symbol tables require. Recalling our previously described SymbolTable interface, that includes a get() method for searching for a key so that its corresponding value can be returned, and a put() method for inserting (or updating) key-value pairs in the tree: public interface SymbolTable<Key, Value> { Value get(Key key); // returns the value paired with given key void put(Key key, Value value); // inserts a key-value pair into the table, // or updates the value associated with a key if it // is already present As we consider the algorithms and implementations behind these two methods -- let us suppose we start with the following skeleton for our BinarySearchTree class: public class BinarySearchTree<Key extends Comparable<Key>,Value> implements SymbolTable<Key,Value> { private class Node { private Key key; // the key private Value val; // the value private Node left; // a reference to the left child of this node private Node right; // a reference to the right child of this node public Node(Key key, Value val) { // a constructor to make this.key = key; // Node creation simple this.val = val; private Node root; // a reference to the root node of the tree -- through // which, references to all other nodes can be found A couple of things above bear mentioning. First, note that we have insisted that our keys implement the Comparable interface. This, of course, allows us to compare the keys in our tree with any keys for which we might be searching. Related to this -- and as a required property of binary search trees -- we assume they are kept in symmetric order, by key. Recall for a tree to be in symmetric order, the keys in all of the left descendents of any given node must be less than that node's key, and the keys in all of the right descendents of any given node must be greater than that node's key. With this in mind, let's take a closer look at the get() method, which retrieves the value associated with a given key... The get() method Note that each node has several instance variables associated with it, but the entire BinarySearchTree class has only one -- a reference to the node identified as root. To return the value associated with key, we'll need to find a reference to the node containing key. However, if all we have is a reference to root, how do we get the reference we need? As a specific example, consider the tree drawn below. Suppose we seek the node with key $H$. Create a new reference $n$ identical to our tree's root (which references node $S$). Then compare $H$ (the key sought) with the key $S$ at $n$. Noting that $H \lt S$, if $H$ is present in this tree, the symmetric order of the tree requires it is in the subtree rooted at $E$, the left child node $n$. As such, we update $n$ to reference its own left child, $E$. In the picture below, the portion of the tree eliminated from consideration has been "grayed out". Then, we start over with this subtree -- we compare $H$ (the key sought) with the key $E$ at $n$. Noting that $E \lt H$, if $H$ is present in this tree, the symmetric order this time requires it to be in the subtree rooted at $R$, the right child of $n$. Consequently, we update $n$ to reference its own right child, $R$. Notice, we are quickly reducing the number of places we need to look to find $H$ -- as can be seen by the amount of the tree now "grayed out". We again compare $H$ (the key sought) with the key at $n$, and seeing $H \lt R$, we update $n$ to reference its left child. Of course, when we compare $H$ this time with the key at $n$, we see they are equal -- we found the node with $H$ as its key! We can return the value of this node using $n$ and are done. When a search is successful in this way, we call it a search hit. What would have happened if we were looking for a key that wasn't in the tree instead? Suppose we were searching above for $G$, instead of $H$. Noting that $G \lt S$, $G \gt E$, and $G \lt R$, we would have made the same first three choices for updating $n$, as we did above. However, as $G \lt H$, we would have then updated $n$ to the left child of $H$, which is null. $G$ clearly can't be in an empty subtree, so we can be confident $G$ does not appear in the tree at all and return null to the client as a result. Perhaps predictably, we call the resulting failed search for a key a search We can easily implement the above search process either iteratively and recursively. The iterative method is simpler, but the recursive approach will reveal an important strategy. With this in mind, we'll consider both techniques. The iterative version is shown below, and hopefully now fairly self-explanatory. public Value get(Key key) { Node n = root; while (n != null) { int cmp = key.compareTo(n.key); if (cmp < 0) // key < n.key n = n.left; else if (cmp > 0) // key > n.key n = n.right; else // key == n.key (found it!) return n.val; return null; // key not found Think about the number of comparisons of keys made in this process. Presuming the root is not null (in which case no comparisons are made), we make one comparison for the root node, and an additional comparison for each level up to and including the level at which we find (or don't find) the key sought. So the number of comparisons made is one more than the depth of the node. This means the worst-case number of comparisons is one more than the height of the tree. For perfectly balanced binary search trees of $n$ nodes, recall this is $\sim \log_2 n$. Consequently, retrieving a value for a given key in a perfectly balanced binary search tree incredibly fast! Before showing a recursive version of get(), a few observations are in order. First, recall that with each comparison made above, we either: • found the key sought; • discovered the key wasn't in the tree to begin with; or • reduced the problem to finding the key sought in some (smaller) subtree. The first two establish base cases, while the last establishes the recursive case. Since we are recursing on subtrees rooted at different nodes, we will need to pass the reference for each subtree root as part of our recursive call. This creates a bit of a problem, though. We won't be able to get by with just get(Key key) -- we'll need a method that looks like get(Node n, Key key). That said, we also must not expose the inner Node class to the client! It could easily be the case that the client is using the class defining our binary search tree strictly as a symbol table. In such contexts, the client is expecting to deal with some Key and Value classes -- but that's it. Remember, classes should work as "black-boxes" -- promising to implement the public methods of their APIs, but never revealing how these methods do what they do. Consequently, everything about nodes needs to stay hidden. There is a simple fix for this problem. We simply make two methods. We make a public-facing get(Key key) method as well as a private, recursive method get(Node n, Key key). The public method is written so that all it really does is call the recursive method to do the "heavy lifting". Note, in this first call to the recursive method, the node passed needs to be root. With all that in mind, here's how we can use recursion to define get(): public Value get(Key key) { return get(root, key); public Value get(Node n, Key key) { if (n == null) return null; // key not found int cmp = key.compareTo(n.key); if (cmp < 0) return get(n.left, key); // key < n.key else if (cmp > 0) return get(n.right, key); // key > n.key return n.val; // key == n.key (found it!) Notably, the cost of this implementation to get() is slightly higher than our previous implementation, due to the recursive overhead. The put() method Understanding how to implement the put() method recursively will make later operations (e.g., delete(Key key) easier, so let us describe that process here, and leave the iterative version of this method for an exercise. In the put(Key key, Value val) method below, which inserts (or updates) the tree with the given key-value pair, we use the same "split-into-two-methods" tactic seen in the recursive get() method Also similar to what we did in get(), we find the location for the insertion (or update) by comparing each node key examined to the one to be inserted/updated, and then recursing on the left or right subtree as appropriate. The details of the base and recursive cases change slightly, however: • The smallest subtree would be an empty tree (i.e., a tree with a null root. In this base case, we want to return a tree with a single node corresponding to the given key-value pair. So, we construct a new node, filling it with our given key and value, and then return a reference to that node. Remember, every tree we have is referenced by its single top-most node -- so we can think of this as returning an entire (but tiny) tree. • For any other key, we can compare the root's key with the key to insert. There are three possibilities to consider: □ If the key to insert is smaller than the root's key, we know that whatever change must happen to put the new key-value in the tree will have to happen in the subtree rooted at the root's left child. In this case, we re-assign the root's left child to be the result of putting the given key-value pair in the left subtree. Again, remember that by "result" here, we mean the single node reference to a tree where the new key-value pair has been inserted. □ If the key to insert is larger than the root's key, the needed change must instead happen in the subtree rooted at the root's right child. So we similarly re-assign the right child to be the result of "putting" this key-value pair in this right subtree. □ As the last case to consider, If the key to insert agrees with the root's key, we must be updating this root node. This is actually another base case -- assign the given value to n.val and call it a day. Here's the code: public void put(Key key, Value val) { root = put(root, key, val); private Node put(Node n, Key key, Value val) { if (n == null) { // (base case, insert) Node newNode = new Node(key, val); return newNode; int cmp = key.compareTo(n.key); if (cmp < 0) // key < n.key (recurse on left subtree) n.left = put(n.left, key, val); else if (cmp > 0) // key > n.key (recurse on right subtree) n.right = put(n.right, key, val); else // key == n.key (base case, overwrite) n.val = val; return n; Navigating the path to the insertion point requires a number of comparisons equal to one more than the level of the key-value pair after it has been inserted, with the worst-case again being a number of comparisons one more than the height of the tree. Thus, insertion into a binary search tree is also fast, with comparison cost $\sim \log_2 n$ for a perfectly balanced tree of $n$ nodes.	{"url":"https://mathcenter.oxford.emory.edu/site/cs171/binarySearchTreeSearchingAndInserting/","timestamp":"2024-11-10T19:30:29Z","content_type":"text/html","content_length":"17002","record_id":"<urn:uuid:5dab0c0c-a96e-4124-ba1b-c017334c9bdf>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00700.warc.gz"}
19601 UDZ NJP Express Train Time Table If we closely look at the 19601 train time table, travelling by 19601 UDZ NJP Express gives us a chance to explore the following cities in a quick view as they come along the route. 1. Udaipur City It is the 1st station in the train route of 19601 UDZ NJP Express. The station code of Udaipur City is UDZ. The departure time of train 19601 from Udaipur City is 00:45. The next stopping station is Rana Pratap Nagar at a distance of 4km. 2. Rana Pratap Nagar It is the 2nd station in the train route of 19601 UDZ NJP Express at a distance of 4 Km from the source station Udaipur City. The station code of Rana Pratap Nagar is RPZ. The arrival time of 19601 at Rana Pratap Nagar is 00:52. The departure time of train 19601 from Rana Pratap Nagar is 00:54. The total halt time of train 19601 at Rana Pratap Nagar is 2 minutes. The previous stopping station, Udaipur City is 4km away. The next stopping station is Mavli Jn at a distance of 38km. 3. Mavli Jn It is the 3rd station in the train route of 19601 UDZ NJP Express at a distance of 42 Km from the source station Udaipur City. The station code of Mavli Jn is MVJ. The arrival time of 19601 at Mavli Jn is 01:25. The departure time of train 19601 from Mavli Jn is 01:27. The total halt time of train 19601 at Mavli Jn is 2 minutes. The previous stopping station, Rana Pratap Nagar is 38km away. The next stopping station is Kapasan at a distance of 37km. 4. Kapasan It is the 4th station in the train route of 19601 UDZ NJP Express at a distance of 79 Km from the source station Udaipur City. The station code of Kapasan is KIN. The arrival time of 19601 at Kapasan is 01:55. The departure time of train 19601 from Kapasan is 01:56. The total halt time of train 19601 at Kapasan is 1 minutes. The previous stopping station, Mavli Jn is 37km away. The next stopping station is Chanderia at a distance of 37km. 5. Chanderia It is the 5th station in the train route of 19601 UDZ NJP Express at a distance of 116 Km from the source station Udaipur City. The station code of Chanderia is CNA. The arrival time of 19601 at Chanderia is 02:50. The departure time of train 19601 from Chanderia is 02:55. The total halt time of train 19601 at Chanderia is 5 minutes. The previous stopping station, Kapasan is 37km away. The next stopping station is Bhilwara at a distance of 47km. 6. Bhilwara It is the 6th station in the train route of 19601 UDZ NJP Express at a distance of 163 Km from the source station Udaipur City. The station code of Bhilwara is BHL. The arrival time of 19601 at Bhilwara is 03:55. The departure time of train 19601 from Bhilwara is 04:00. The total halt time of train 19601 at Bhilwara is 5 minutes. The previous stopping station, Chanderia is 47km away. The next stopping station is Bijainagar at a distance of 66km. 7. Bijainagar It is the 7th station in the train route of 19601 UDZ NJP Express at a distance of 229 Km from the source station Udaipur City. The station code of Bijainagar is BJNR. The arrival time of 19601 at Bijainagar is 04:43. The departure time of train 19601 from Bijainagar is 04:45. The total halt time of train 19601 at Bijainagar is 2 minutes. The previous stopping station, Bhilwara is 66km away. The next stopping station is Nasirabad at a distance of 42km. 8. Nasirabad It is the 8th station in the train route of 19601 UDZ NJP Express at a distance of 271 Km from the source station Udaipur City. The station code of Nasirabad is NSD. The arrival time of 19601 at Nasirabad is 05:15. The departure time of train 19601 from Nasirabad is 05:17. The total halt time of train 19601 at Nasirabad is 2 minutes. The previous stopping station, Bijainagar is 42km away. The next stopping station is Ajmer Jn at a distance of 23km. 9. Ajmer Jn It is the 9th station in the train route of 19601 UDZ NJP Express at a distance of 294 Km from the source station Udaipur City. The station code of Ajmer Jn is AII. The arrival time of 19601 at Ajmer Jn is 06:15. The departure time of train 19601 from Ajmer Jn is 06:25. The total halt time of train 19601 at Ajmer Jn is 10 minutes. The previous stopping station, Nasirabad is 23km away. The next stopping station is Jaipur at a distance of 135km. 10. Jaipur It is the 10th station in the train route of 19601 UDZ NJP Express at a distance of 429 Km from the source station Udaipur City. The station code of Jaipur is JP. The arrival time of 19601 at Jaipur is 09:05. The departure time of train 19601 from Jaipur is 09:15. The total halt time of train 19601 at Jaipur is 10 minutes. The previous stopping station, Ajmer Jn is 135km away. The next stopping station is Alwar at a distance of 150km. 11. Alwar It is the 11th station in the train route of 19601 UDZ NJP Express at a distance of 579 Km from the source station Udaipur City. The station code of Alwar is AWR. The arrival time of 19601 at Alwar is 11:21. The departure time of train 19601 from Alwar is 11:24. The total halt time of train 19601 at Alwar is 3 minutes. The previous stopping station, Jaipur is 150km away. The next stopping station is Rewari Jn at a distance of 75km. 12. Rewari Jn It is the 12th station in the train route of 19601 UDZ NJP Express at a distance of 654 Km from the source station Udaipur City. The station code of Rewari Jn is RE. The arrival time of 19601 at Rewari Jn is 12:40. The departure time of train 19601 from Rewari Jn is 12:43. The total halt time of train 19601 at Rewari Jn is 3 minutes. The previous stopping station, Alwar is 75km away. The next stopping station is Gurgaon at a distance of 51km. 13. Gurgaon It is the 13th station in the train route of 19601 UDZ NJP Express at a distance of 705 Km from the source station Udaipur City. The station code of Gurgaon is GGN. The arrival time of 19601 at Gurgaon is 13:31. The departure time of train 19601 from Gurgaon is 13:33. The total halt time of train 19601 at Gurgaon is 2 minutes. The previous stopping station, Rewari Jn is 51km away. The next stopping station is Delhi Cantt at a distance of 17km. 14. Delhi Cantt It is the 14th station in the train route of 19601 UDZ NJP Express at a distance of 722 Km from the source station Udaipur City. The station code of Delhi Cantt is DEC. The arrival time of 19601 at Delhi Cantt is 13:57. The departure time of train 19601 from Delhi Cantt is 13:59. The total halt time of train 19601 at Delhi Cantt is 2 minutes. The previous stopping station, Gurgaon is 17km away. The next stopping station is Delhi Jn at a distance of 14km. 15. Delhi Jn It is the 15th station in the train route of 19601 UDZ NJP Express at a distance of 736 Km from the source station Udaipur City. The station code of Delhi Jn is DLI. The arrival time of 19601 at Delhi Jn is 14:45. The departure time of train 19601 from Delhi Jn is 15:05. The total halt time of train 19601 at Delhi Jn is 20 minutes. The previous stopping station, Delhi Cantt is 14km away. The next stopping station is Moradabad at a distance of 161km. 16. Moradabad It is the 16th station in the train route of 19601 UDZ NJP Express at a distance of 897 Km from the source station Udaipur City. The station code of Moradabad is MB. The arrival time of 19601 at Moradabad is 18:25. The departure time of train 19601 from Moradabad is 18:33. The total halt time of train 19601 at Moradabad is 8 minutes. The previous stopping station, Delhi Jn is 161km away. The next stopping station is Bareilly(nr) at a distance of 91km. 17. Bareilly(nr) It is the 17th station in the train route of 19601 UDZ NJP Express at a distance of 988 Km from the source station Udaipur City. The station code of Bareilly(nr) is BE. The arrival time of 19601 at Bareilly(nr) is 19:49. The departure time of train 19601 from Bareilly(nr) is 19:51. The total halt time of train 19601 at Bareilly(nr) is 2 minutes. The previous stopping station, Moradabad is 91km away. The next stopping station is Lucknow at a distance of 235km. 18. Lucknow It is the 18th station in the train route of 19601 UDZ NJP Express at a distance of 1223 Km from the source station Udaipur City. The station code of Lucknow is LKO. The arrival time of 19601 at Lucknow is 23:35. The departure time of train 19601 from Lucknow is 23:45. The total halt time of train 19601 at Lucknow is 10 minutes. The previous stopping station, Bareilly(nr) is 235km away. The next stopping station is Barabanki Jn at a distance of 28km. 19. Barabanki Jn It is the 19th station in the train route of 19601 UDZ NJP Express at a distance of 1251 Km from the source station Udaipur City. The station code of Barabanki Jn is BBK. The arrival time of 19601 at Barabanki Jn is 00:36. The departure time of train 19601 from Barabanki Jn is 00:38. The total halt time of train 19601 at Barabanki Jn is 2 minutes. The previous stopping station, Lucknow is 28km away. The next stopping station is Gorakhpur at a distance of 242km. 20. Gorakhpur It is the 20th station in the train route of 19601 UDZ NJP Express at a distance of 1493 Km from the source station Udaipur City. The station code of Gorakhpur is GKP. The arrival time of 19601 at Gorakhpur is 04:45. The departure time of train 19601 from Gorakhpur is 04:55. The total halt time of train 19601 at Gorakhpur is 10 minutes. The previous stopping station, Barabanki Jn is 242km away. The next stopping station is Deoria Sadar at a distance of 50km. 21. Deoria Sadar It is the 21st station in the train route of 19601 UDZ NJP Express at a distance of 1543 Km from the source station Udaipur City. The station code of Deoria Sadar is DEOS. The arrival time of 19601 at Deoria Sadar is 05:48. The departure time of train 19601 from Deoria Sadar is 05:53. The total halt time of train 19601 at Deoria Sadar is 5 minutes. The previous stopping station, Gorakhpur is 50km away. The next stopping station is Siwan Jn at a distance of 70km. 22. Siwan Jn It is the 22nd station in the train route of 19601 UDZ NJP Express at a distance of 1613 Km from the source station Udaipur City. The station code of Siwan Jn is SV. The arrival time of 19601 at Siwan Jn is 06:50. The departure time of train 19601 from Siwan Jn is 06:55. The total halt time of train 19601 at Siwan Jn is 5 minutes. The previous stopping station, Deoria Sadar is 70km away. The next stopping station is Chhapra at a distance of 60km. 23. Chhapra It is the 23rd station in the train route of 19601 UDZ NJP Express at a distance of 1673 Km from the source station Udaipur City. The station code of Chhapra is CPR. The arrival time of 19601 at Chhapra is 07:50. The departure time of train 19601 from Chhapra is 07:55. The total halt time of train 19601 at Chhapra is 5 minutes. The previous stopping station, Siwan Jn is 60km away. The next stopping station is Hajipur Jn at a distance of 60km. 24. Hajipur Jn It is the 24th station in the train route of 19601 UDZ NJP Express at a distance of 1733 Km from the source station Udaipur City. The station code of Hajipur Jn is HJP. The arrival time of 19601 at Hajipur Jn is 09:10. The departure time of train 19601 from Hajipur Jn is 09:15. The total halt time of train 19601 at Hajipur Jn is 5 minutes. The previous stopping station, Chhapra is 60km away. The next stopping station is Muzaffarpur Jn at a distance of 53km. 25. Muzaffarpur Jn It is the 25th station in the train route of 19601 UDZ NJP Express at a distance of 1786 Km from the source station Udaipur City. The station code of Muzaffarpur Jn is MFP. The arrival time of 19601 at Muzaffarpur Jn is 10:05. The departure time of train 19601 from Muzaffarpur Jn is 10:10. The total halt time of train 19601 at Muzaffarpur Jn is 5 minutes. The previous stopping station, Hajipur Jn is 53km away. The next stopping station is Samastipur Jn at a distance of 52km. 26. Samastipur Jn It is the 26th station in the train route of 19601 UDZ NJP Express at a distance of 1838 Km from the source station Udaipur City. The station code of Samastipur Jn is SPJ. The arrival time of 19601 at Samastipur Jn is 11:12. The departure time of train 19601 from Samastipur Jn is 11:17. The total halt time of train 19601 at Samastipur Jn is 5 minutes. The previous stopping station, Muzaffarpur Jn is 52km away. The next stopping station is Khagaria Jn at a distance of 86km. 27. Khagaria Jn It is the 27th station in the train route of 19601 UDZ NJP Express at a distance of 1924 Km from the source station Udaipur City. The station code of Khagaria Jn is KGG. The arrival time of 19601 at Khagaria Jn is 13:00. The departure time of train 19601 from Khagaria Jn is 13:05. The total halt time of train 19601 at Khagaria Jn is 5 minutes. The previous stopping station, Samastipur Jn is 86km away. The next stopping station is Katihar Jn at a distance of 124km. 28. Katihar Jn It is the 28th station in the train route of 19601 UDZ NJP Express at a distance of 2048 Km from the source station Udaipur City. The station code of Katihar Jn is KIR. The arrival time of 19601 at Katihar Jn is 15:15. The departure time of train 19601 from Katihar Jn is 15:25. The total halt time of train 19601 at Katihar Jn is 10 minutes. The previous stopping station, Khagaria Jn is 124km away. The next stopping station is Kishanganj at a distance of 98km. 29. Kishanganj It is the 29th station in the train route of 19601 UDZ NJP Express at a distance of 2146 Km from the source station Udaipur City. The station code of Kishanganj is KNE. The arrival time of 19601 at Kishanganj is 16:43. The departure time of train 19601 from Kishanganj is 16:45. The total halt time of train 19601 at Kishanganj is 2 minutes. The previous stopping station, Katihar Jn is 98km away. The next stopping station is New Jalpaiguri at a distance of 87km. 30. New Jalpaiguri It is the 30th station in the train route of 19601 UDZ NJP Express at a distance of 2233 Km from the source station Udaipur City. The station code of New Jalpaiguri is NJP. The arrival time of 19601 at New Jalpaiguri is 18:35. The previous stopping station, Kishanganj is 87km away. Trainspnrstatus is one of the best website for checking trains running status. You can find the 19601 UDZ NJP Express running status here. Chanderia, Bhilwara, Ajmer Junction, Jaipur, Delhi Junction, Moradabad, Lucknow, Gorakhpur, Deoria Sadar, Siwan Junction, Chhapra, Hajipur Junction, Muzaffarpur Junction, Samastipur Junction, Khagaria Junction, Katihar Junction are major halts where UDZ NJP Express halts for more than five minutes. Getting hotel accommodations and cab facilities in these cities is easy. Trainspnrstatus is one stop best portal for checking pnr status. You can find the 19601 UDZ NJP Express IRCTC and Indian Railways PNR status here. All you have to do is to enter your 10 digit PNR number in the form. PNR number is printed on the IRCTC ticket. Train number of UDZ NJP Express is 19601. You can check entire UDZ NJP Express train schedule here. with important details like arrival and departure time. 19601 train schedule UDZ NJP Express train time table UDZ NJP Express ka time table UDZ NJP Express kitne baje hai UDZ NJP Express ka number19601 train time table	{"url":"https://www.trainspnrstatus.com/train-schedule/19601","timestamp":"2024-11-07T22:52:14Z","content_type":"text/html","content_length":"55321","record_id":"<urn:uuid:3d7e9628-421e-4650-ad3b-e0356829fb4a>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00196.warc.gz"}
Lists, Decisions, and Graphs by Edward A. Bender, S. Gill Williamson Publisher: University of California, San Diego 2010 Number of pages: 261 In this book, four basic areas of discrete mathematics are presented: Counting and Listing (Unit CL), Functions (Unit Fn), Decision Trees and Recursion (Unit DT), and Basic Concepts in Graph Theory (Unit GT). At the end of each unit is a list of Multiple Choice Questions for Review. Download or read it online for free here: Download link (2.1MB, PDF) Download mirrors: Mirror 1 Similar books Advances in Discrete Differential Geometry Alexander I. Bobenko (ed.) SpringerThis is the book on a newly emerging field of discrete differential geometry. It surveys the fascinating connections between discrete models in differential geometry and complex analysis, integrable systems and applications in computer graphics. Discrete Mathematics for Computer Science Jean Gallier arXivThese are notes on discrete mathematics for computer scientists. The presentation is somewhat unconventional. I emphasize partial functions more than usual, and I provide a fairly complete account of the basic concepts of graph theory. Discrete Math for Computer Science Students Ken Bogart, Cliff Stein Dartmouth CollegeIt gives thorough coverage to topics that have great importance to computer scientists and provides a motivating computer science example for each math topic. Contents: Counting; Cryptography and Number Theory; Reflections on Logic and Proof. Topics in Discrete Mathematics A.F. Pixley Harvey Mudd CollegeThis text is an introduction to a selection of topics in discrete mathematics: Combinatorics; The Integers; The Discrete Calculus; Order and Algebra; Finite State Machines. The prerequisites include linear algebra and computer programming.	{"url":"https://www.e-booksdirectory.com/details.php?ebook=11055","timestamp":"2024-11-13T15:38:05Z","content_type":"text/html","content_length":"11249","record_id":"<urn:uuid:1d26dcc5-dff1-4b97-b5cb-c8bb571c49fa>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00436.warc.gz"}
How is Credit Card Interest Calculated? - commons-credit-portal.orgHow is Credit Card Interest Calculated? How is Credit Card Interest Calculated? How is credit card interest calculated? Credit card interest is generally calculated by the daily periodic rate times the number of days in the billing period. Checkout this video: Introductory Paragraph Credit card interest is the fee charged by credit card companies for the use of their funds. This fee is calculated as a percentage of the outstanding balance on the card. The credit card company will typically charge interest on a daily basis, and will post the charges to your account at the end of each billing cycle. There are two main methods used to calculate credit card interest: average daily balance and adjusted balance. With the average daily balance method, interest is charged on the average of all balances owed during the billing cycle. With the adjusted balance method, interest is only charged on balances that remain after payments are made. Both methods can result in different amounts of interest being charged, so it’s important to understand how your credit card company calculates interest before you make a decision about which method to use. Types of Interest There are two types of interest that can be applied to your credit card: simple and compound. Simple interest is charged as a percentage of your daily balance and is not compounded. Compound interest is charged as a percentage of your daily balance and is compounded daily. Simple Interest Simple interest is a quick and easy way to calculate the interest charge on a credit card balance. It’s not the method used by banks, but it’s a good way to get a ballpark idea of the interest you’ll be paying. To calculate simple interest, multiply the daily interest rate by the number of days in the billing cycle, and then multiply that result by the balance. (The daily interest rate is the annual interest rate divided by 365.) Here’s an example: Suppose you have a $1,000 balance on a card with a 15% annual rate and a 30-day billing cycle. The daily rate would be 0.04109% (15% divided by 365). To calculate the interest charge for the month, multiply $1,000 by 0.04109% and 30 to get $12.33. Simple interest is easy to calculate, but it’s not the way credit card companies figure your finance charges. Most use what’s called average daily balance method, which gives you a higher interest bill than simple interest because it includes new purchases made during the billing cycle. Compound Interest Compound interest is interest that is calculated not only on the initial principal, but also on the accumulated interest of previous periods. In other words, compound interest is earned on top of Compound interest is calculated by multiplying the initial principal by one plus the annual interest rate to the power of the number of compound periods minus one. The total initial amount of the loan is then subtracted from this result to obtain the compound interest. How is Credit Card Interest Calculated? Credit card companies make money by charging interest on the money that you borrow from them. They will also charge you fees for things like late payments or going over your credit limit. How is credit card interest calculated? Read on to find out. Average Daily Balance Method The average daily balance method is the most common way credit card companies calculate finance charges. To get your average daily balance, the card issuer takes the beginning balance of your account each day, adds any new charges and subtracts any payments or credits, and then divides that by the number of days in the billing cycle. This gives you a daily average balance, which the issuer then multiplies by the monthly periodic rate—expressed as a decimal—to get your finance charge. Here’s an example: Let’s say that you have a credit card with a $1,000 credit limit and a 18% annual percentage rate (APR). Your monthly periodic rate would be 1.5% (0.18 divided by 12 months). If you had a balance of $500 at the beginning of the month and spent $50 during the month, your average daily balance would be $525 (($500 + $50) divided by 30 days in the month). Your finance charge for the month would be $7.88 ((1.5% x $525) divided 16 days)). Two-Cycle Average Daily Balance Method Credit card companies use several different methods to calculate the interest charged on cardholders’ unpaid balances, with the most common being the average daily balance method and the two-cycle average daily balance method. The average daily balance method calculates interest by taking the average of your balance during the billing cycle and multiplying that figure by the daily periodic rate and the number of days in the billing cycle. The two-cycle average daily balance method is similar to the average daily balance method, except that it uses a two-cycle billing period. This means that if you carry a balance from one month to the next, your credit card company will calculate your interest using the average of your balances for both months. Minimum Interest Charge Credit card companies assess interest on your outstanding balance every day. To calculate the amount of interest you pay each month, your credit card issuer starts with your average daily balance for the month. This is usually your beginning balance each day, plus any new charges and minus any payments or credits that post during the month, divided by the number of days in the month. Then, the issuer multiplies this figure by your daily periodic rate—which is listed in your credit card agreement—to arrive at a monthly finance charge. Most credit card issuers also impose a minimum interest charge if you owe any interest for the month. For example, if your monthly finance charge is $1.50 and your minimum interest charge is $0.50, you’ll pay $2 in interest for that month. In conclusion, credit card interest is calculated by taking the daily periodic rate and multiplying it by the average daily balance. The average daily balance is calculated by taking the sum of all daily balances over the course of a billing cycle and dividing it by the number of days in that billing cycle. The daily periodic rate is calculated by dividing the APR by 365.	{"url":"https://commons-credit-portal.org/how-is-credit-card-interest-calculated/","timestamp":"2024-11-09T09:24:39Z","content_type":"text/html","content_length":"92432","record_id":"<urn:uuid:cf85f6c1-ec3f-4c42-ba90-cc77a8525c23>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00411.warc.gz"}
Math in Focus Grade 1 Chapter 13 Practice 6 Answer Key Real-World Problems: Addition and Subtraction Practice the problems of Math in Focus Grade 1 Workbook Answer Key Chapter 13 Practice 6 Real-World Problems: Addition and Subtraction to score better marks in the exam. Math in Focus Grade 1 Chapter 13 Practice 6 Answer Key Real-World Problems: Addition and Subtraction Question 1. Lynn has 12 trophies. Geeta has 5 more trophies than Lynn. How many trophies does Geeta have? Geeta has ___ trophies. Lynn has 12 trophies, Geeta has 5 more trophies than Lynn, By adding 12 with 5 we get 17, Therefore, Geeta has 17 trophies. Question 2. Rima buys 15 stickers. Susie buys 7 fewer stickers than Rima. How many stickers does Susie buy? Susie buys ____ stickers. Rima buys 15 stickers, Susie buys 7 fewer stickers than Rima, By subtracting 7 from 15 we get 8, Therefore, Susie buys 8 stickers. Question 3. Tara has 14 toy cars. She has 9 more toy cars than Carlos. How many toy cars does Carlos have? Carlos has ___ toy cars. Tara has 14 toy cars, she has 9 more toy cars than Carlos, By subtracting 9 from 14 we get 5, Therefore, Carlos has 5 toy cars Question 4. Aaron makes 6 friendship bracelets. He makes 5 fewer friendship bracelets than Kate. How many friendship bracelets does Kate make? Kate makes ____ friendship bracelets. Aaron makes 6 friendship bracelets, He makes fewer friendship bracelets than Kate, By adding 6 with 5 we get 11, Therefore, Kate makes 11 friendship bracelets Question 5. Michelle has 18 snowballs. Miguel has 14- snowballs. How many more snowballs does Michelle have? Michelle has ____ more snowballs. Michelle has 18 snowballs, Miguel has 14 snowballs, By subtracting 14 from 18 we get 4, Therefore, Michelle has 4 more snowballs than Miguel. Question 6. Rose buys 17 hairclips. Sarah buys 13 more hairclips than Rose. How many hairclips does Sarah buy? Sarah buys ___ hairclips. Rose buys 17 hairclips, Sarah buys 13 more hairclips than Rose, By adding 17 with 13 we get 30, Therefore, Sarah buys 30 hairclips. Question 7. Tess has 22 walnuts. She has 13 more walnuts than Chris. How many walnuts does Chris have? Chris has ____ walnuts. Tess has 22 walnuts, She has 13 more walnuts than Chris, By subtracting 13 from 22 we get 9, Therefore, Chris has 9 walnuts. Question 8. Ashley makes 36 muffins. Janice makes 17 muffins. How many more muffins does Ashley make? Ashley makes ___ more muffins. Ashley makes 36 muffins, Janice makes 17 muffins, By subtracting 17 from 36 we get 19, Therefore, Ashley makes 19 more muffins. Leave a Comment	{"url":"https://mathinfocusanswerkey.com/math-in-focus-grade-1-chapter-13-practice-6-answer-key/","timestamp":"2024-11-11T22:50:30Z","content_type":"text/html","content_length":"133830","record_id":"<urn:uuid:ce960a03-3f67-4414-b1c4-3ceeb5754a00>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00285.warc.gz"}
BiplotML \| CRAN/E Biplots Estimation with Algorithms ML CRAN Package Logistic Biplot is a method that allows representing multivariate binary data on a subspace of low dimension, where each individual is represented by a point and each variable as vectors directed through the origin. The orthogonal projection of individuals onto these vectors predicts the expected probability that the characteristic occurs. The package contains new techniques to estimate the model parameters and constructs in each case the 'Logistic-Biplot'. References can be found in the help of each procedure. • Version1.1.0 • R version≥ 4.1 • Needs compilation?No • Last release04/22/2022 Last 30 days Last 365 days The following line graph shows the downloads per day. You can hover over the graph to see the exact number of downloads per day. Data provided by CRAN • Depends4 packages • Suggests10 packages	{"url":"https://cran-e.com/package/BiplotML","timestamp":"2024-11-07T05:32:50Z","content_type":"text/html","content_length":"60498","record_id":"<urn:uuid:39f9cf9c-a45e-4c69-b8a7-64bfdc9362e6>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00624.warc.gz"}
What is the product 2x x 4? - California Learning Resource Network What is the product 2x x 4? What is the Product of 2x x 4? In mathematics, a product is a result of multiplying two or more numbers together. When we encounter the expression 2x x 4, we are left wondering what the product is. In this article, we will delve into the world of mathematics and explore the concept of products, and provide the direct answer to the question "What is the product of 2x x 4?" What is a Product? A product is a mathematical operation that involves multiplying one or more numbers together. It is denoted by the symbol * or simply by juxtaposition. The product of two numbers is the result of multiplying one number by the other. For example, the product of 2 and 3 is 6, denoted as 23 = 6. Understanding the Term "x" In the expression 2x x 4, the letter "x" is a variable, which is a mathematical symbol that represents a value that can change. In this case, "x" is multiplying 2 and 4 together. Think of "x" as a placeholder for a number that has not been specified. Evaluating the Expression 2x x 4 To evaluate the expression 2x x 4, we need to substitute the value of "x" with a specific number. Let’s use the value of "x" as 2. Using distributive Property The distributive property is a fundamental concept in mathematics that allows us to multiply a single value by two or more numbers. In this case, we can use the distributive property to rewrite the expression 2x x 4 as: Simplifying the Expression By simplifying the expression 2(x4), we get: 224 = 16 In conclusion, the product of 2x x 4 is 16. This is because when we substitute the value of "x" as 2, we can use the distributive property to simplify the expression and arrive at the final answer 16. Remember that the concept of products is a fundamental part of mathematics, and understanding how to evaluate expressions like 2x x 4 is crucial for success in algebra and beyond. Key Points to Remember • A product is a result of multiplying one or more numbers together. • The letter "x" is a variable, which represents a value that can change. • To evaluate an expression like 2x x 4, we need to substitute the value of "x" with a specific number. • The distributive property is a useful tool for simplifying expressions like 2x x 4. Table: Key Concepts Concept Definition Product A result of multiplying one or more numbers together. Variable A mathematical symbol that represents a value that can change. Distributive Property A fundamental concept in mathematics that allows us to multiply a single value by two or more numbers. I hope this article has provided a clear understanding of what the product of 2x x 4 is. Remember to practice and apply the concepts discussed in this article to improve your math skills and become proficient in evaluating expressions like 2x x 4. Your friends have asked us these questions - Check out the answers! Leave a Comment	{"url":"https://www.clrn.org/what-is-the-product-2x-x-4/","timestamp":"2024-11-10T12:47:10Z","content_type":"text/html","content_length":"131133","record_id":"<urn:uuid:3a5b6157-8d7e-4485-9428-3b9de0a88295>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00538.warc.gz"}
Online calculator Mann-Whitney test This online calculator performs the Mann–Whitney U test (also called the Mann–Whitney–Wilcoxon (MWW), Wilcoxon rank-sum test, or Wilcoxon–Mann–Whitney test). As it was stated in Two sample t-Test, you can apply the t-test if the following assumptions are met: • That the two samples are independently and randomly drawn from the source population(s). • That the scale of measurement for both samples has the properties of an equal interval scale. • That the source population(s) can be reasonably supposed to have a normal distribution. Sometimes, however, your data fails to meet the second and/or third requirement. For example, there is nothing to indicate it has a normal distribution, or you do not have an equal interval scale – that is, the spacing between adjacent values cannot be assumed to be constant. But you still want to find out whether the difference between two samples is significant. In such cases, you can use the Mann–Whitney U test, a non-parametric alternative of the t-test. In statistics, the Mann–Whitney U test (also called the Mann–Whitney–Wilcoxon (MWW), Wilcoxon rank-sum test, or Wilcoxon–Mann–Whitney (WMW) test) is a non-parametric test of the null hypothesis that it is equally likely that a randomly selected value from one sample will be less than or greater than a randomly selected value from a second sample^1, or $p(X<Y)=0.5$. However, it is also used as a substitute for the independent groups t-test, with the null hypothesis that the two population medians are equal. BTW, there are actually two tests – the Mann-Whitney U test and the Wilcoxon rank-sum test. They were developed independently and use different measures, but are statistically equivalent. The assumptions of the Mann-Whitney test are: • That the two samples are randomly and independently drawn; • That the dependent variable is intrinsically continuous – capable in principle, if not in practice, of producing measures carried out to the nth decimal place; • That the measures within the two samples have the properties of at least an ordinal scale of measurement, so that it is meaningful to speak of "greater than", "less than", and "equal to".^2 As you can see, this non-parametric test does not assume (or require) samples from normally distributed populations. Such tests are also called distribution-free tests. Word of caution It has been known for some time that the Wilcoxon-Mann-Whitney test is adversely affected by heterogeneity of variance when the sample sizes are not equal. However, even when sample sizes are equal, very small differences between the population variances cause the large-sample Wilcoxon-Mann-Whitney test to become too liberal, that is, the actual Type I error rate for the large-sample Wilcoxon-Mann-Whitney test increased as the sample size increased.^3. Hence you must remember that this test is true only if the two population distributions are the same (including homogeneity of variance) apart from a shift in location. The method The method replaces raw values with their corresponding ranks. With this, some results can be achieved using simple math. For example, the total sum of ranks is already known from the total size and it is $\frac{N(N+1)}{2}$. Hence, the average rank is $\frac{N(N+1)}{2}\frac{1}{N}=\frac{N+1}{2}$. The general idea is that if the null hypothesis is true and samples aren't significantly different, then ranks are somewhat balanced between A and B, and the average rank for each sample should approximate the total average rank, and rank sums should approximate $\frac{n_A(N+1)}{2}$ and $\frac{n_B(N+1)}{2}$ respectively. The calculation To perform the test, first you need to calculate a measure known as U for each sample. You start by combining all values from both samples into a single set, sorting them by value, and assigning a rank to each value (in case of ties, each value receives an average rank). Ranks go from 1 to N, where N is the sum of sizes $n_A$ and $n_B$. Then you calculate the sum of ranks for values of each sample $R_A$ and $R_B$. Now you can calculate U as For small sample sizes you can use tabulated values. You take the minimum of two Us, and then compare it with the critical value corresponding to sample sizes and the chosen significance level. Statistics textbooks usually list critical values in tables for sample sizes up to 20. For large sample sizes you can use the z-test. It was shown that U is approximately normally distributed if both sample sizes are equal to or greater than 5 (some sources says if $n_An_B>20$^4). $\mu_U=\frac{n_An_B}{2}\\ \sigma_U=\sqrt{\frac{n_An_B*(N+1)}{12}}$ In case of ties, the formula for standard deviation becomes where g is the number of groups of ties, tj is the number of tied ranks in group j. The calculator below uses the z-test. Of course, there is a limitation on sample sizes (both sample sizes should be equal to or greater than 5), but this is probably not much of a limitation for real Digits after the decimal point: 2 The file is very large. Browser slowdown may occur during loading and creation. Level of confidence for non-directional hypothesis Level of confidence for directional hypothesis Similar calculators PLANETCALC, Mann-Whitney test	{"url":"https://planetcalc.com/7858/","timestamp":"2024-11-14T17:20:02Z","content_type":"text/html","content_length":"45786","record_id":"<urn:uuid:5e51c434-a908-4a69-a5f2-a3b614950540>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00664.warc.gz"}
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Online Calculus Class Help \| Pay Someone To Do My Exam Online Calculus Class Help and Worksheet This lab worksheet is designed to provide a simple introduction to modern differential calculus on the real number line. This work why not try these out intended to be followed by a series of more advanced calculus worksheets. They should be click over here now as supplements to an introductory course in real analysis for upper-level courses in calculus. Working with differential calculus on the real number line may be quite challenging for some students. The reason lies at the heart of how differential calculus functions. The number line itself is just one natural curve in the plane, now called the unit circle. It is the particular shape of the number line that solves the basic problem with the notion of limits, which is to say, click for more limit of a function as the argument of the function approaches any particular point along the number line is precisely the function evaluated at that point. We Can Crack Your Proctored Online Examinations More to the point, while the number line, called the his comment is here circle, isn’t a proper subspace of the real numbers, it still is a subspace of general functions whose elements behave like the limits of real-valued functions as arguments approach the points on the unit circle (since arguments of elements of functions that are functions of one real variable are all within one unit circle). To get a sense of how difficult it can be, consider this fairly easily simplified example. Consider the one-dimensional function _f_ ( _x_ ) = _x_. Calculating the derivative _∂_ _f_ ( _x_ ) is quite straightforward: _∂_ _f_ ( _x_ ) = _x_, or _∂f_ ( _x_ ) = 1. But if you attempt to calculate _∂_ _fg_ ( _x_ ) = _∂_ _f_ ( _g_ ( _x_ )) _∂g_ ( _x_ ) for any function _g_ ( _x_ ) = _f_ ( _x_ ), you might come to the conclusion that _∂_ my link ( _g_ ( _x_ )) = _∂g_ ( _x_ ), but in truth, you have no way of knowing this for sure, because the domain of _g_ ( _x_ ) look at this now possibly extend from zero to infinity. Here is a similar example in the form of a graph: A more formal analogy: At zero, the graph of the function _f_ ( _x_ ) = _x_ is just the _x_ -axis; as _x_ gets closer and closer to zero, the graph becomes smoother and smoother, which illustrates the concept of a limit. The area taken up by the graph becomes bigger as _x_ gets smaller; as _x_ gets larger, the area taken up by the graph becomes smaller. 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The method of variations is used one-third of the time, but fortunately, it is just one tenth as expensive in terms of brain power to master. With your calculator, you are the best tool students have these days to find the unique pattern in any number. How long do you think it would take you to compute the coefficients a knockout post A0(x), A1(x), A2(x), A3 (x), x = 1, 2, 3, just to get you started? So powerful is your calculator that you are even capable of writing Mathematica commands that will compute specific prime numbers efficiently and correctly. That’s right! And the best part is you don’t even have to know how to do it, you don’t even need to know what the numbers that the functions come from look like. No fear, math is still the best when it comes to doing math. Hire Someone To Do My Exam However, it’s critical that you have a method to use when you come across a problem that results you have to find the coefficients Clicking Here a form (an equation). 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The observed point pattern, from which an estimate of the inhomogeneous cross-type pair correlation function $g_{ij}(r)$ will be computed. It must be a multitype point pattern (a marked point pattern whose marks are a factor). The type (mark value) of the points in X from which distances are measured. A character string (or something that will be converted to a character string). Defaults to the first level of marks The type (mark value) of the points in X to which distances are measured. A character string (or something that will be converted to a character string). Defaults to the second level of marks(X). Optional. Values of the estimated intensity function of the points of type i. Either a vector giving the intensity values at the points of type i, a pixel image (object of class "im") giving the intensity values at all locations, or a function(x,y) which can be evaluated to give the intensity value at any location. Optional. Values of the estimated intensity function of the points of type j. A numeric vector, pixel image or function(x,y). Vector of values for the argument $r$ at which $g_{ij}(r)$ should be evaluated. There is a sensible default. This argument is for internal use only. Choice of smoothing kernel, passed to density.default. Bandwidth for smoothing kernel, passed to density.default. Other arguments passed to the kernel density estimation function density.default. Bandwidth coefficient; see Details. Choice of edge correction. Optional arguments passed to density.ppp to control the smoothing bandwidth, when lambdaI or lambdaJ is estimated by kernel smoothing.	{"url":"https://www.rdocumentation.org/packages/spatstat.explore/versions/3.0-6/topics/pcfcross.inhom","timestamp":"2024-11-07T05:27:43Z","content_type":"text/html","content_length":"79988","record_id":"<urn:uuid:73e28a1f-e5a6-458f-aefa-70cc8398a2ad>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00872.warc.gz"}
Angle Sum Property \| Theorem \| Proof \| Examples- Cuemath (2024) The angle sum property of a triangle states that the sum of the angles of a triangle is equal to 180º. A triangle has three sides and three angles, one at each vertex. Whether a triangle is an acute, obtuse, or a right triangle, the sum of its interior angles is always 180º. The angle sum property of a triangle is one of the most frequently used properties in geometry. This property is mostly used to calculate the unknown angles. 1. What is the Angle Sum Property? 2. Angle Sum Property Formula 3. Proof of the Angle Sum Property 4. FAQs on Angle Sum Property What is the Angle Sum Property? According to the angle sum property of a triangle, the sum of all three interior angles of a triangle is 180 degrees. A triangle is a closed figure formed by three line segments, consisting of interior as well as exterior angles. The angle sum property is used to find the measure of an unknown interior angle when the values of the other two angles are known. Observe the following figure to understand the property. Angle Sum Property Formula The angle sum property formula for any polygon is expressed as, S = (n − 2) × 180°, where 'n' represents the number of sides in the polygon. This property of a polygon states that the sum of the interior angles in a polygon can be found with the help of the number of triangles that can be formed inside it. These triangles are formed by drawing diagonals from a single vertex. However, to make things easier, this can be calculated by a simple formula, which says that if a polygon has 'n' sides, there will be (n - 2) triangles inside it. For example, let us take a decagon that has 10 sides and apply the formula. We get, S = (n − 2) × 180°, S = (10 − 2) × 180° = 10 × 180° = 1800°. Therefore, according to the angle sum property of a decagon, the sum of its interior angles is always 1800°. Similarly, the same formula can be applied to other polygons. The angle sum property is mostly used to find the unknown angles of a polygon. Proof of the Angle Sum Property Let's have a look at the proof of the angle sum property of the triangle. The steps for proving the angle sum property of a triangle are listed below: • Step 1: Draw a line PQ that passes through the vertex A and is parallel to side BC of the triangle ABC. • Step 2: We know that the sum of the angles on a straight line is equal to 180°. In other words, ∠PAB + ∠BAC + ∠QAC = 180°, which gives, Equation 1: ∠PAB + ∠BAC + ∠QAC = 180° • Step 3: Now, since line PQ is parallel to BC. ∠PAB = ∠ABC and ∠QAC = ∠ACB. (Interior alternate angles), which gives, Equation 2: ∠PAB = ∠ABC, and Equation 3: ∠QAC = ∠ACB • Step 4: Substitute ∠PAB and ∠QAC with ∠ABC and ∠ACB respectively, in Equation 1 as shown below. Equation 1: ∠PAB + ∠BAC + ∠QAC = 180°. Thus we get, ∠ABC + ∠BAC + ∠ACB = 180° Hence proved, in triangle ABC, ∠ABC + ∠BAC + ∠ACB = 180°. Thus, the sum of the interior angles of a triangle is equal to 180°. Important Points The following points should be remembered while solving questions related to the angle sum property. • The angle sum property formula for any polygon is expressed as, S = ( n − 2) × 180°, where 'n' represents the number of sides in the polygon. • The angle sum property of a polygon states that the sum of the interior angles in a polygon can be found with the help of the number of triangles that can be formed inside it. • The sum of the interior angles of a triangle is always 180°. Impoprtant Topics • Exterior Angle Theorem • Angles • Triangles FAQs on Angle Sum Property What is the Angle Sum Property of a Polygon? The angle sum property of a polygon states that the sum of all the angles in a polygon can be found with the help of the number of triangles that can be formed in it. These triangles are formed by drawing diagonals from a single vertex. However, this can be calculated by a simple formula, which says that if a polygon has 'n' sides, there will be (n - 2) triangles inside it. The sum of the interior angles of a polygon can be calculated with the formula: S = (n − 2) × 180°, where 'n' represents the number of sides in the polygon. For example, if we take a quadrilateral and apply the formula using n = 4, we get: S = (n − 2) × 180°, S = (4 − 2) × 180° = 2 × 180° = 360°. Therefore, according to the angle sum property of a quadrilateral, the sum of its interior angles is always 360°. Similarly, the same formula can be applied to other polygons. The angle sum property is mostly used to find the unknown angles of a polygon. What is the Angle Sum Property of a Triangle? The angle sum property of a triangle says that the sum of its interior angles is equal to 180°. Whether a triangle is an acute, obtuse, or a right triangle, the sum of the angles will always be 180°. This can be represented as follows: In a triangle ABC, ∠A + ∠B + ∠C = 180°. What is the Angle Sum Property of a Hexagon? According to the angle sum property of a hexagon, the sum of all the interior angles of a hexagon is 720°. In order to find the sum of the interior angles of a hexagon, we multiply the number of triangles in it by 180°. This is expressed by the formula: S = (n − 2) × 180°, where 'n' represents the number of sides in the polygon. In this case, 'n' = 6. Therefore, the sum of the interior angles of a hexagon = S = (n − 2) × 180° = (6 − 2) × 180° = 4 × 180° = 720°. What is the Angle Sum Property of a Quadrilateral? According to the angle sum property of a quadrilateral, the sum of all its four interior angles is 360°. This can be calculated by the formula, S = (n − 2) × 180°, where 'n' represents the number of sides in the polygon. In this case, 'n' = 4. Therefore, the sum of the interior angles of a quadrilateral = S = (4 − 2) × 180° = (4 − 2) × 180° = 2 × 180° = 360°. What is the Exterior Angle Sum Property of a Triangle? The exterior angle theorem says that the measure of each exterior angle of a triangle is equal to the sum of the opposite and non-adjacent interior angles. What is the Formula of Angle Sum Property? The formula for the angle sum property is, S = ( n − 2) × 180°, where 'n' represents the number of sides in the polygon. For example, if we want to find the sum of the interior angles of an octagon, in this case, 'n' = 8. Therefore, we will substitute the value of 'n' in the formula, and the sum of the interior angles of an octagon = S = (n − 2) × 180° = (8 − 2) × 180° = 6 × 180° = 1080°. What is the Angle Sum Property of a Pentagon? As per the angle sum property of a pentagon, the sum of all the interior angles of a pentagon is 540°. In order to find the sum of the interior angles of a pentagon, we multiply the number of triangles in it by 180°. This is expressed by the formula: S = (n − 2) × 180°, where 'n' represents the number of sides in the polygon. In this case, 'n' = 5. Therefore, the sum of the interior angles of a pentagon = S = (n − 2) × 180° = (5 − 2) × 180° = 3 × 180° = 540°. How to Find the Third Angle in a Triangle? We know that the sum of the angles of a triangle is always 180°. Therefore, if we know the two angles of a triangle, and we need to find its third angle, we use the angle sum property. We add the two known angles and subtract their sum from 180° to get the measure of the third angle. For example, if two angles of a triangle are 70° and 60°, we will add these, 70 + 60 = 130°, and we will subtract it from 180°, which is the sum of the angles of a triangle. So, the third angle = 180° - 130° = 50°. How to Find the Exterior Angle of a Polygon? The exterior angle of a polygon is the angle formed between any side of a polygon and a line that is extended from the adjacent side. In order to find the measure of an exterior angle of a regular polygon, we divide 360 by the number of sides 'n' of the given polygon. For example, in a regular hexagon, where 'n' = 6, each exterior angle will be 60° because 360 ÷ n = 360 ÷ 6 = 60°. It should be noted that the corresponding interior and exterior angles are supplementary and the exterior angles of a regular polygon are equal in measure.	{"url":"https://dollsnbears.com/article/angle-sum-property-theorem-proof-examples-cuemath","timestamp":"2024-11-07T06:35:50Z","content_type":"text/html","content_length":"115400","record_id":"<urn:uuid:313050f9-1621-4712-b171-3fe8ac7136f1>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00813.warc.gz"}
Settling Circular Debts :: Transum Newsletter Settling Circular Debts Saturday 1st October 2022 This is the Transum Newsletter for the month of October 2022. It begins, as usual with the puzzle of the month. Amir owes £100 to Beth. Beth owes £200 to Chris. Chris owes £300 to Dalbir. Dalbir owes £400 to Amir. What is the minimum amount of money that must change hands in order to settle these debts, and who should pay how much to whom? While you think about that here are some of the key resources added to the Transum website during the last month. A new video was filmed to go into the help section of the HCF and LCM exercise. The video shows a number of different ways of finding the highest common factor and lowest common multiple of two numbers. It then goes on to show a little known method of finding the HCF and LCM of three numbers and ends with a suggested method for finding the pairs of numbers with a given HCF and LCM. Substitution Sort is a brand new activity showing that expressions need to be sorted into different orders of size depending on the value that is substituted as the variable. It provides a little more interest than a straight-forward exercise on substitution particularly if students try to guess the order before doing the substitution. I reckon these new Yohaku Puzzles will be useful in Year 5 all the way up to Year 11. I particularly like the multiplication ones as they can be solved using knowledge of factors and prime numbers. I have uploaded a new instructional video to go with the online exercise 'Equation of a Line through Points'. I hope it will be useful for students who get stuck doing their homework. They will be able to see it when they click the Help tab as they work through the levels of the exercise. Don't forget you can listen to this month's podcast which is the audio version of this newsletter. You can find it on Spotify, Stitcher or Apple Podcasts. You can follow Transum on Twitter and 'like' Transum on Facebook. If you have any comments about this podcast please contact me at podcast@transum.org. I received an email on behalf of Corp Today Magazine, to inform me that Transum Mathematics has won: Most Engaging Mathematics Teaching Resource 2022. Lovely! Finally the answer to last month's puzzle which was to evaluate 8 ÷ 2(2 + 2): The correct answer 16 and the other commonly given answer is 1. Both answers can be argued to be correct however and you could say that the expression is ambiguous. In fact merely relying on BIDMAS is not enough to remove the ambiguity. Wil sent me this from Mathematical Pie of Summer 2020: The best explanation I have found is in a video produced by Presh Talwalkar, creator of Mind Your Decisions, and can be found here. That's all for now, I rushed into the local fish and chip shop and asked them what 15 times 12 was. They said they only do takeaways. Do you have any comments? It is always useful to receive feedback on this newsletter and the resources on this website so that they can be made even more useful for those learning Mathematics anywhere in the world. Click here to enter your comments.	{"url":"https://transum.org/Newsletter/?p=527","timestamp":"2024-11-07T13:52:03Z","content_type":"text/html","content_length":"18908","record_id":"<urn:uuid:d1f5bd71-59eb-4fa7-ac65-57346becc5de>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00567.warc.gz"}
meanSdGpDS {dsBase} R Documentation Serverside function called by ds.meanSdGp meanSdGpDS(X, INDEX) X a clientside supplied character string identifying the variable for which means/SDs are to be calculated INDEX a clientside supplied character string identifying the factor across which means/SDs are to be calculated Computes the mean and standard deviation across groups defined by one factor Burton PR [Package dsBase version 6.3.0 ]	{"url":"https://cran.datashield.org/web/dsBase/meanSdGpDS.html","timestamp":"2024-11-03T20:22:15Z","content_type":"text/html","content_length":"2452","record_id":"<urn:uuid:c0ce4aab-1779-4a84-9ce2-99556f3ecc3c>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00819.warc.gz"}
Use Mathematica Kernel as an Omnipotent W\|A API in C++ (Non-trivial) 13165 Views 1 Reply 5 Total Likes Use Mathematica Kernel as an Omnipotent W\|A API in C++ (Non-trivial) To have the full benifit of this thread, a reader is expected to understand the object oriented programming in C++ including using pointer, defining class, operator overload and separation This example is written in C++ with XCode 4.6.3 on OSX 10.8.5 . The Mathematica Kernel is from Version 9. The idea behind this article is from an example in Chapter 8 of Savitch's Absolute C++ .The example was about to create a Money object and to illustrate how to add the objects after overloading the plus/+ operator. In this thread, I am going to add a very good feature to his example with the technology so we can add two money objects under different monetary system, ie. USD + CNY with the daily exchange rate. To make the C++ code work as expected, you need to first follow this link to setup the linkers and frameworks path correctly on your machine and within the IDE. The key parts of this project are 1. Link the code to a Mathematica kernel 2. Load the WolframAlpha query into the C++ code and get the result 3. Embed (2) into the overloaded operator as part of the feature of our object: Money Lets take a looks at the code: The implementation of the main.cpp is very concise. Launch the link to Mathematica Kernel and some pointers and one copy pointer to the Money objects. Then it simply print out the result and close the kernels. The main also deletes the dynamic objects and close the Mathematica kernel before it quits. MLENV ep = (MLENV)0; MLINK lp = (MLINK)0;// declared in mathlink.h int main(int argc, char* argv[]) init_and_openlink( argc, argv); Money* first = new Money(100,16); Money* second = new Money(10,52,"CNY"); Money* res = new Money(first + second); Money* res2 = new Money(second + first); ... // print functions MLPutFunction( lp, "Exit", 0); //close kernel return 0; The core object Money is also quite straightforward. The declaration contains three member instances which records the dollar part, cent part and currency symbol (like EUR, USD ... ) . Besides, I have one default constructor and a copy constructor. The key here is to define a suitable version of operator + (basically a new plus operation) here so things like USD + JPY would work properly. My idea is to make this operation non-commutable. This means that when USD + JPY the result prints JPY while JPY + USD yields USD currency. Certainly this is not perfect, but it is good enough to illustrate my idea. class Money Money( ); //default Money(int theDollars, int theCents); Money(int theDollars, int theCents, std::string cur); Money(Money const &myMoney );//copy constructor std::string currency; int dollars; int cents; //overload the operators const Money operator +(const Money& amount1, const Money& amount2); Savitch's version of this overload is fairly simple: convert all money to cents and sum things up. Do a divid-by-100 to get the dollar part and do a mod-by-100 to get the cent. Mine is the same is except there should be a conversion inside. This currency exchange rate is extracted by the getCurrencyRate(). const Money operator +(const Money& amount1, const Money& amount2){ int allcents1 = amount1.cents + amount1.dollars100; int allcents2 = amount2.cents + amount2.dollars100; double rate = getCurrencyRate (amount1, amount2) ; // in terms of the second currency int money = static_cast<int>(floor(static_cast<double>(allcents1)rate + static_cast<double>(allcents2))); int dollarsFinal = money/100 ; int centsFinal = money%100 ; return Money(dollarsFinal,centsFinal,amount2.currency); The above code should be rather plain if you know how to overload a operator in C++. The last part is exploring what is behind the getCurrencyRate function. The implementation is below: extern MLINK lp ; extern MLENV ep ; double getCurrencyRate (const Money &Money1, const Money &Money2 ) { double rate; int pkt; // 1 pick USD and JPY then make USD to JPY std::string temp = Money1.currency + " to " + Money2.currency; char rule[100]; //2. write a Mathematica code stack MLPutFunction( lp, "EvaluatePacket", 1); MLPutFunction( lp,"QuantityMagnitude",1); MLPutFunction(lp, "WolframAlpha",2); MLPutString(lp, rule); // this guy only works //MLPutInteger( lp, n); MLEndPacket( lp); // 3 check results and return while( (pkt = MLNextPacket( lp), pkt) && pkt != RETURNPKT) { MLNewPacket( lp); if (MLError( lp)) error( lp); MLGetReal64( lp, &rate); return rate; Some preambles: 1. MLINK lp and MLENV ep are links to handle message passing between c++ and Mathematica Kernel 2. ML indicates MathLink API's 3. the "while( (pkt = MLNextPacke " is to check if there is a packet returns from Mathematica Kernel. We can skip this and it usually sits here as a error checking and prevent hanging. Though chunk of code, it actually only contains three parts: 1. catenation of currency symbol of the variable temp ("USD to JPY"). "rule" is used because one of the MLPut** function below only takes "char*". 2. A List of MLPut function 3. Get the result ( technically writes the return packet into the variable "rate") and return the exchange rate. The second part above may look alien to the most viewers. I will first explain where it comes from. Because our subroutine here is basically to make a valid query into WolframAlpha, the result is just directly from WolframAlpha. In Mathematica we can use the api directly: Click the "+" sign beside the "Result: 6.14" pod and choose the "Computable Data option". Immediately you will see Mathematica returns this line: WolframAlpha["USD to CNY",{{"Result",1},"ComputableData"}] If you wrap a Hold function to it and check its FullForm, you will see this: Hold[WolframAlpha["USD to CNY",List[List["Result",1],"ComputableData"]]] Observe that the MLPutFunction all takes the heads of above list as arguments. I added QuantityMagnitute here to kick out the unit part. Some functions takes three argments because they handles more than one argument. For instance, the List["Result" ,1] is a List function takes 2 inputs so I need MLPutFunction(lp, "List" , 2); This behavior is sort of building a stack. Finally, we just pick the value of rate from MLGet function through the link pool. The interesting thing is that once we have this part written in a separate file and encapsulate it in the operator overloading, the MathLink functions does not even bother us. The C++ can be used without reading the implementation of the getCurrency function. Improvements: Definitely I need better error handling function, this can be found by a manual written by Todd Galey online and the Mathematica documentation itself. Also if more parser is added to the operator, it could handle system like JPY which only has the cent part. Overall, I realize the example I have now should be sufficient to show how to combine Mathematica code with C++ in a real world problem. Attachment is available . Again you need to set up XCode correctly before compiling this code. 1 Reply Be respectful. Review our Community Guidelines to understand your role and responsibilities. Community Terms of Use	{"url":"https://community.wolfram.com/groups/-/m/t/216813","timestamp":"2024-11-05T16:44:58Z","content_type":"text/html","content_length":"105129","record_id":"<urn:uuid:feefe7d5-1600-4293-b320-9a4f238b9dba>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00832.warc.gz"}
How many Sundays in 30 years? - Answers How many Saturdays and Sundays are in a year? That depends on the year. There are always at least 52 Saturdays and 52 Sundays in a year, so most years there are 104 in total. Some years there can be 53 Sundays and 52 Saturdays, in which case the total is 105. There can also be 52 Sundays and 53 Saturdays, again giving a total of 105. If a leap year starts on a Saturday, then there are 53 Saturdays and 53 Sundays, so there are 106 in total.	{"url":"https://www.answers.com/math-and-arithmetic/How_many_Sundays_in_30_years","timestamp":"2024-11-04T17:35:43Z","content_type":"text/html","content_length":"157782","record_id":"<urn:uuid:90f5e0bd-81c3-4c54-93a8-2cfe66efec14>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00109.warc.gz"}
Volume 48, pp. 114-130, 2018. Numerical analysis of a dual-mixed problem in non-standard Banach spaces Jessika Camaño, Cristian Muñoz, and Ricardo Oyarzúa In this paper we analyze the numerical approximation of a saddle-point problem posed in non-standard Banach spaces $\mathrm{H}(\mathrm{div}_{p}\,, \Omega)\times L^q(\Omega)$, where $\mathrm{H}(\ mathrm{div}_{p}\,, \Omega):= \{{\boldsymbol\tau} \in [L^2(\Omega)]^n \colon \mathrm{div} {\boldsymbol\tau} \in L^p(\Omega)\},$ with $p>1$ and $q\in \mathbb{R}$ being the conjugate exponent of $p$ and $\Omega\subseteq \mathbb{R}^n$ ($n\in\{2,3\}$) a bounded domain with Lipschitz boundary $\Gamma$. In particular, we are interested in deriving the stability properties of the forms involved (inf-sup conditions, boundedness), which are the main ingredients to analyze mixed formulations. In fact, by using these properties we prove the well-posedness of the corresponding continuous and discrete saddle-point problems by means of the classical Babuška-Brezzi theory, where the associated Galerkin scheme is defined by Raviart-Thomas elements of order $k\geq 0$ combined with piecewise polynomials of degree $k$. In addition we prove optimal convergence of the numerical approximation in the associated Lebesgue norms. Next, by employing the theory developed for the saddle-point problem, we analyze a mixed finite element method for a convection-diffusion problem, providing well-posedness of the continuous and discrete problems and optimal convergence under a smallness assumption on the convective vector field. Finally, we corroborate the theoretical results with suitable numerical results in two and three dimensions. Full Text (PDF) [742 KB], BibTeX Key words mixed finite element method, Raviart-Thomas, Lebesgue spaces, Lp data, convection-diffusion AMS subject classifications 65N15, 65N12, 65N30, 74S05 < Back	{"url":"https://etna.ricam.oeaw.ac.at/volumes/2011-2020/vol48/abstract.php?vol=48&pages=114-130","timestamp":"2024-11-05T10:42:16Z","content_type":"application/xhtml+xml","content_length":"9250","record_id":"<urn:uuid:f47bad38-da19-474f-92a5-fea6099c3fdd>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00297.warc.gz"}
Regularity, singularities and ℎ-vector of graded algebras (Journal Article) \| NSF PAGES Suppose$R$is a$F$-finite and$F$-pure$\mathbb {Q}$-Gorenstein local ring of prime characteristic$p>0$. We show that an ideal$I\subseteq R$is uniformly compatible ideal (with all$p^{-e}$-linear maps) if and only if exists a module finite ring map$R\to S$such that the ideal$I$is the sum of images of all$R$-linear maps$S\to R$. In other words, the set of uniformly compatible ideals is exactly the set of trace ideals of finite ring maps. more » « less	{"url":"https://par.nsf.gov/biblio/10521500-regularity-singularities-vector-graded-algebras","timestamp":"2024-11-09T07:58:47Z","content_type":"text/html","content_length":"265370","record_id":"<urn:uuid:c5b378c4-19e9-4c1e-bade-b096d77d7ca0>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00352.warc.gz"}
How is data being calculated in the Databox Benchmarks What are Databox Benchmarks With Databox Benchmarks, you can benchmark your company's performance to compare how you're doing against businesses like yours. You can save Benchmarks in your Databox Analytics Account and use them to understand where you’re performing well, and where you have opportunities to improve. Learn more about Databox Benchmarks here. The Databox Benchmarks feature is available on the Growth and higher plans. Request a trial of Databox Benchmarks by following these steps. What is a Benchmark A Benchmark is a set of statistics calculated for a given metric in a benchmark cohort. A cohort, in this case, is a group of companies with shared characteristics. How are Benchmarks Calculated We calculate benchmarks from anonymized data of Databox users. Benchmarks are being calculated for the last month from data of all the participating contributors matching the Metric and the cohort, which is defined by selecting an industry, business type, employee size, annual revenue, or any of their combinations. Mathematical calculations in Benchmarks Let's explain the mathematical calculations in Benchmarks with a histogram. On this visualization, we can see that data is being distributed into equally-spaced bins and the number of values in each bin is being counted. Next, the bars are drawn for each bin, where the height of the bar is proportional to the number of values in each bin. See a histogram on the image below as an example. The circles on the x-axis represent different values (for example, metric values in a benchmarks cohort). We divided the x-axis into 6 equally-spaced bins. We can observe that there is only one value in the first and last two bins, therefore the height of the bars of those bins end at 1 on the y-axis. In the fourth bin we have 3 values, and therefore the height of that bar is 3. The criteria for the value distribution into individual bins is that each bin represents value ranges (from-to) and we check into which bin our values fall and then count how many values are in each bin. Such a visualization offers a quick insight into how the data is distributed. We can see that most of the values are close to 8, and the further we move away from it, the fewer values we observe. If we add smooth lines on the histogram, we can easily imagine the Hill Chart visualization. The black line represents the Hill Chart for the same underlying values as the histogram’s. The height of the Hill Chart shows us where the most of the values are. The higher the hill, the more values we can find in that area (the values are the metric values from our contributors). Your value in a Benchmark Your value is shown on the graph with a vertical line. This makes it easy for you to compare your performance for the cohort that you have selected, and quickly identify the metrics where you are one of the leaders and metrics where you can improve your company's performance. If your value is a statistical outlier, it will not fall within the bounds of the graph. In that case, your value will be shown on either the left border (if the value is significantly lower than other values) or the right border (if the value is significantly higher than other values). For each benchmark, we also calculate the outrank ratio - how you compare to the companies included in calculating this particular benchmark. This value represents the percentage of companies your company outranks in this cohort, or what percentage of companies outperform you. For example, an outrank ratio of 0.69 shows that the company’s value is higher than 69% of other companies’ values in the group in case a higher metric value is positive (like Website Sessions) or, in case that metric is inverted, and lower value means better (like Churn), lower than 69% of other companies' values. An example of an outrank ratio is outlined below in the graph. How do we calculate benchmark value for different granulations Since benchmark values are calculated once per month for the last month, we perform mathematical calculations to create benchmark values for other chart granulations (Hourly, Daily, weekly, Quarterly, and Yearly). Granulation General Aggregatable metrics General Non-Aggregatable metrics Current type metrics average value from the monthly median monthly benchmark median monthly median / (number of days * 24) Hourly / Example: Example: monthly median for January = 3,000 monthly median for January = 3,000 Average hourly median = 3000/(3124) = 4.03 Hourly median = 3,000 average value from the monthly median monthly benchmark median monthly median / number of days Daily / Example: monthly median for January = 3,000 monthly median for January = 3,000 Daily median = 3,000 Average daily median = 3000/31 = 96.77 average value from the monthly median monthly benchmark median (monthly median / number of days) 7 Weekly / Example: monthly median for January = 3,000 monthly median for January = 3,000 Weekly median = 3,000 Average daily median = (3000/31)*7 = 677.42 Monthly monthly benchmark median monthly benchmark median monthly benchmark median the sum of monthly medians average of monthly medians Example: Example: Monthly median for: Monthly median for: Quarterly - January = 3,000 / - January = 3,000 • February: 2,500 • February: 2,500 • March: 2,700 • March: 2,700 Quarterly median = 3,000+2,500+2,700 = 8,200 Average quarterly median = (3,000+2,500+2,700)/3 = 2,733.33 the sum of monthly medians average of monthly medians Monthly median for: Monthly median for: - January = 3,000 - January = 3,000 Yearly / • February: 2,500 • February: 2,500 • March: 2,700 • March: 2,700 … … • December: 3,400 • December: 3,400 Quarterly median = 3,000+2,500+2,700+...+2,700 = 32,900 Average quarterly median = (3,000+2,500+2,700+...+2,700)/12 = 32,900/12 = 2,741.67 How do we calculate granulations when: • There is monthly benchmark data missing We don’t calculate and don’t show any calculated benchmark value if there are more than 25% of monthly granulation points missing from the calculation range. • There is a monthly benchmark data point missing in the range (for calculating yearly granulation only) We do interpolation of neighboring points for missing points. • There is a first/last monthly benchmark data point missing in the range (for calculating yearly granulation only) For missing start and end points, we take value from the previous/next point. • Two consecutive monthly benchmark data points are missing in the range (for calculating yearly granulation only) We don’t calculate and don’t show any calculated benchmark value if there is more than 25% of monthly granulation points missing. If there are less than 25% missing, we do interpolation of data for those points. • Weekly granulation and week falls within two different months (eg. the 30th is on Wednesday) We calculate the average for every day of the week (the first average from Monday through Wednesday and a different average from Thursday till Sunday) and from those averages calculate the weekly average for that week. Databox's Privacy Policy for Benchmarks Databox may use your data to calculate and provide benchmarking services. The data collected for participation in the benchmarks ecosystem shall, at any time, be anonymized and removed from any personal or sensitive information. In addition, to ensure anonymity, benchmarks shall be calculated and available only, if Databox has at least 15 different sources of data within the selected cohort. You may, at any time, decline to participate in benchmarks by logging in to your account and opting-out. Learn how to opt out of benchmarks here. Please note that if you want to use the benchmarks feature, you have to give your permission to contribute to the benchmarks. Learn more about our Privacy Policy here.	{"url":"https://help.databox.com/how-is-data-being-calculated-in-the-databox-benchmarks","timestamp":"2024-11-11T07:59:25Z","content_type":"text/html","content_length":"79151","record_id":"<urn:uuid:f748cdb0-3581-4bbe-bb0f-0a5bc07c1f94>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00558.warc.gz"}
In the spinner shown, pointer can land on any of three parts of-Turito Are you sure you want to logout? In the spinner shown, pointer can land on any of three parts of the same size. So there are three outcomes. What part of the whole is coloured? A. 1/3 B. 2/3 C. 3/3 D. none of these Fractions are referred to as the components of a whole in mathematics. A single object or a collection of objects might be entire. When we cut a piece of cake in real life from the entire cake, the part represents the percent of the cake. Here we have given the spinner, pointer can land on any of three parts of the same size. We have to find the shaded part. The correct answer is: 1/3 In general, the whole can be any particular thing or value, and the fraction might be a portion of any quantity out of it. The top and bottom numbers of a fraction are explained by the fundamentals of fractions. The bottom number reflects the entire number of components, while the top number represents the number of chosen or coloured portions of the whole. There are two components to fractions: a numerator and a denominator. The numerator, or upper portion of the fraction, denotes the various parts of the fraction. The lower or bottom portion, known as the denominator, represents all of the components into which the fraction is divided. For instance, if 3/4 is a fraction, the numerator is 3 and the denominator is 4. Here we have given a spinner that has 3 equal parts. So total parts are 3, which will be the denominator. Now the shaded part is only 1, so it will be in the numerator. So the fraction will be: 1/3 So here we used the concept of fractions to understand the question. The numerical numbers that make up a fraction of the whole are called fractions. A whole can be a single item or a collection of items. When anything or a number is divided into equal parts, each piece represents a portion of the total. So the fraction in this question is 1/3. Get an Expert Advice From Turito.	{"url":"https://www.turito.com/ask-a-doubt/Maths-in-the-spinner-shown-pointer-can-land-on-any-of-three-parts-of-the-same-size-so-there-are-three-outcomes-what-q5a8f99","timestamp":"2024-11-11T21:04:02Z","content_type":"application/xhtml+xml","content_length":"412125","record_id":"<urn:uuid:29426338-cab3-494f-b488-95237f2f23fe>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00044.warc.gz"}
Card 4 Veritas \| The Enigma Emporium top of page Remember, if you haven't used our clues pages yet, check out the instructions here to learn how they work and how to best utilize them. Hint 1 This website doesn't appear to exist. Hint 2 That long string of numbers doesn't seem to translate to anything in alphanumeric so it must be something else. Hint 3 It just keeps getting bigger and bigger and adding onto itself... Hint 4 What's the name of the sequence/pattern? Final Answer: This is the Fibonacci sequence. Hint 1 Even more numbers. This is what we get for working with a physicist. Hint 2 Up in the thousands and down in the zeros, these numbers are all over the place! Hint 3 You may need to solve some of the other puzzles to see if they can shake anything loose over here. Hint 4 Are you on the website for this card yet? Hint 5 And how did you manage to do it? It may help you out with decoding this. Final Answer: The numbers are alphanumeric, but instead of just using the first 26 numbers, the cipher uses the first 26 numbers of the Fibonacci sequence to make its alphabet. Ultimately, this will give you the name Pasque Valentine. Hint 1 And yet more numbers... perhaps the shape behind it can help? Hint 2 It's a shape that keeps going and going and building on itself. Hint 3 Those numbers seem to be only getting bigger too... Hint 4 Maybe it's an alphabet that goes on forever! This cipher is a cyclical alphabet, with the numbers always ascending and never starting over. So while A is 1, A is also 27 and so on. This translates to: Ask me about recursion. Hint 1 A little research may be needed. Hint 2 The name of the queen doesn't seem to be working. Hint 3 Perhaps a bit more colloquial of a name would work. Hint 4 In fact, the first name might be staring you right in the face! Final Answer: This stamp was commonly known as the Penny Black. So Ms. Black's first name here in Penny. Hint 1 "The Password's Verse" is an interesting title for this. Perhaps there's something in the words here. Hint 2 Maybe not a secret message, but perhaps what the entire verse is. Hint 3 Are there any letters from the English alphabet missing? There are no missing letters. In fact, every single letter in the alphabet is contained in this sentence, which is known as a PANGRAM. bottom of page	{"url":"https://www.theenigmaemporium.com/veritas-card4","timestamp":"2024-11-09T06:11:30Z","content_type":"text/html","content_length":"915549","record_id":"<urn:uuid:d2232d0b-48eb-4be1-bb9a-0212fbb04c7b>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00498.warc.gz"}
Face Aging using Conditional GANs with Keras implementation Open-Source Internship opportunity by OpenGenus for programmers. Apply now. Reading time: 50 minutes \| Coding time: 25 minutes Face Aging, a.k.a. age synthesis and age progression can be defined as aesthetically rendering an image of a face with natural aging and rejuvenating effects on the individual face. Traditional face aging approaches can be split into 2 types: Prototyping approach ➤ Estimate average faces within predefined age groups ➤ The discrepancies between these faces constitute the aging patterns which are then used to transform an input face image into the target age group. Pros - Are simple and fast Cons - As they are based on general rules, they totally discard the personalized info which often results in unrealistic images. Modeling approach ➤ Employ parametric models to simulate the aging mechanisms of muscles, skin and skull of a particular individual Both these approaches generally require face aging sequences of the same person with wide range of ages which are very costly to collect. These approaches are helpful when we want to model the aging patterns sans natural human facial features (its personality traits, facial expression, possible facial accessories, etc.) Anyway, in most of the real-life use cases, face aging must be combined with other alterations to the face, e.g. adding sunglasses or moustaches. Such non-trivial modifications call for global generative models of human faces. This is where the Age-cGAN model comes in. You see, vanilla GANs are explicitly trained to generate the most realistic and credible images which should be hard to distinguish from real data. Obviously, since their inception, GANs have been utilized to perform modifications on photos of human faces, viz. changing the hair colour, adding spectacles and even the cool application of binary aging (making the face look younger or older without using particular age categories to classify the face into). But a familiar problem of these GAN-based methods for face modification is that, the original person's identity is often lost in the modified image. The Age-cGAN model focuses on identity-preserving face aging. The original paper made 2 new contributions to this field: 1. The Age-cGAN (Age Conditional GAN) was the first GAN to generate high quality artificial images within defined age categories. 2. The authors proposed a novel latent vector optimization approach which allows Age-cGAN to reconstruct an input face image, preserving the identity of the original person. Introducing Conditional GANs for Face Aging Conditional GANs (cGANs) extend the idea of plain GANs, allowing us to control the output of the generator network. We know that face aging involves changing the face of a person, as the person grows older or younger, making no changes to their identity. In most other models (GANs included), the identity or appearance of a person is lost by 50% as facial expressions and superficial accessories, such as spectacles and facial hair, are not taken into Age-cGANs consider all of these attributes, and that's why they are better than simple GANs in this aspect. Understanding cGANs Read this to get an intuitive introduction of GANs. GANs can be extended to a conditional both provided both the generator and discriminator networks are conditioned on some extra information, y. y can be any kind of additional information, including class labels and integer data. In practice, y can be any information related to the target face image - facial pose, level of illumination, etc. The conditioning can be performed by feeding y into both the generator and discriminator as an additional input layer. Some drawbacks of vanilla GANS: 1. Users have no control over the category of the images generated. When we add a condition y to the generator, we can only generate images of a specific category, using y. 2. Thus, they can learn only 1 category and it's highly tedious to construct GANs for multiple categories. A cGAN overcomes these difficulties and can be used to generate multi-modal models with different conditions for various categories. Below is the architecture of a cGAN: Let us understand the working of a cGAN from this architecture - In the generator, the prior input noise ${p}_{z}\left(z\right)$, and y are combined to give a joint hidden representation, the adversarial training framework making it easy to compose this hidden representation by allowing a lot of flexibility. In the discriminator, x and y are presented as inputs and to a discriminative function (represented technically by a Multilayer Perceptron). The objective function of a 2-player minimax game would be as $\underset{\mathit{G}}{\mathit{m}\mathit{i}\mathit{n}}\mathit{ }\underset{\mathit{D}}{\mathit{m}\mathit{a}\mathit{x}}\mathit{ }V\left(\mathit{D}\mathit{,}\mathit{ }\mathit{G}\right)\mathit{ }\mathit {=}\mathit{ }{E}_{\mathit{x}\mathit{~}{\mathit{p}}_{\mathit{d}\mathit{a}\mathit{t}\mathit{a}}\left(\mathit{x}\right)}\left[\mathit{log}\mathit{ }\mathit{D}\left(\mathit{x}\mathit{\|}\mathit{y}\right)\ right]\mathit{ }\mathit{+}\mathit{ }{E}_{\mathit{x}\mathit{~}{\mathit{p}}_{\mathit{z}}\left(\mathit{z}\right)}\left[\mathit{log}\left(\mathit{1}\mathit{ }\mathit{-}\mathit{ }\mathit{D}\left(\mathit Here, G is the generator network and D is the discriminator network. The loss for the discriminator is $\mathrm{log} D\left(x\|y\right)$ and the loss for the generator is $\mathrm{log}\left(1 - D\left(G\left(z\|y\right)\right)\right)$. $G\left(z\|y\right)$ is modeling the distribution of our data given z and y. z is a prior noise distribution of a dimension 100 drawn from a normal distribution. Architecture of the Age-cGAN The Age-cGAN model proposed by the authors in the original paper used the same design for the Generator G and the Discriminator D as in Radford, et al. (2016), "Unsupervised representation learning with deep convolutional generative adversarial networks", the paper which introduced DCGANs. In adherence to Perarnau, et al. (2016), "Invertible conditional GANs for image editing", the authors add the conditional information at the input of G and at the first convolutional layer of D. The picture shows the face aging method used in Age-cGANs. a) depicts an approximation of the latent vector to reconstruct the input image b) illustrates switching the age condition at the input of the generator G to perform face aging. Diving deeper into more technical details, now. The Age-cGAN consists of 4 networks - an encoder, the FaceNet, a generator network and a discriminator network. The functions of each of the 4 networks - a) Encoder - Helps us learn the inverse mapping of input face images and the age condition with the latent vector ${z}_{0}$. b) FaceNet - It is a face recognition network which learns the difference between an input image x and a reconstructed image $\stackrel{~}{x}$. c) Generator network - Takes a hidden (latent) representation consisting of a face image and a condition vector, and generates an image. d) Discriminator network - Discriminates between the real and fake images. cGANs have a little drawback - they can't learn the task of inverse mapping an input image x with attributes y to a latent vector z, which is necessary for the image reconstruction: $x = G\left(z, y\ To solve this problem, we use an encoder network, which we can train to approximate the inverse mapping of input images x. A) The encoder network 1. Its main purpose is to generate a latent vector of the provided images, that we can use to generate face images at a target age. 2. Basically, it takes an image of a dimension of (64, 64, 3) and converts it into a 100-dimensional vector. 3. The architecture is a deep convolutional neural network (CNN) - containing 4 convolutional blocks and 2 fully connected (dense) layers. 4. Each convolutional block, a.k.a. Conv block contains a convolutional layer, a batch normalization layer, and an activation function. 5. In each Conv block, each Conv layer is followed by a batch normalization layer, except the first Conv layer. B) The generator network 1. The primary objective is to generate an image having the dimension (64, 64, 3). 2. It takes a 100-dimensional latent vector (from the encoder) and some extra information y, and tries to generate realistic images. 3. The architecture here again, is a deep CNN made up of dense, upsampling, and Conv layers. 4. It takes 2 input values - a noise vector and a conditioning value. 5. The conditioning value is the additional information we'll provide to the network, for the Age-cGAN, this will be the age. C) The discriminator network 1. What the discriminator network does is identify whether the provided image is fake or real. Simple. 2. It does this by passing the image through a series of downsampling layers and some classification layers, i.e. it predicts whether the image is fake or real. 3. Like the 2 networks before, this network is another deep CNN. 4. It comprises several Conv blocks - each containing a Conv layer, a batch normalization layer, and an activation function, except the first Conv block, which lacks the batch normalization layer. D) Face recognition network 1. The FaceNet has the main task of recognizing a person's identity in a given image. 2. To build the model, we will be using the pre-trained Inception-ResNet-v2 model without the fully connected layers. You can refer to this page to learn more about pretrained models in Keras. To learn more about the Inception-ResNet-v2 model, you could also read the original paper by Szegedy, et al. (2016), "Inception-v4, Inception-ResNet and the Impact of Residual Connections on 3. The pretrained Inception-ResNet-v2 network, once provided with an image, returns the corresponding embedding. 4. The extracted embeddings for the real image and the reconstructed image can be calculated by calculating the Euclidean distance of the embeddings. 5. The Face Recognition (FR) network generates a 128-dimensional embedding of a particular input fed to it, to improve the genrator and the encoder networks. Stages of the Age-cGAN The Age-cGAN model has multiple stages of training. The Age-cGAN has 4 networks, which get trained as 3 stages which are as follows: 1) Conditional GAN training 2) Initial latent vector optimization 3) Latent vector optimization A) Conditional GAN Training ➤ This is the first stage in the training of a conditional GAN. ➤ In this stage, we train both the generator and the discriminator networks. ➤ Once the generator network is trained, it can generate blurred images of a face. ➤ This stage is comparable to training a vanilla GAN, where we train both the networks simultaneously. ➔ The training objective function The training objective function for the training of cGANs can be represented as follows: $\underset{{\theta }_{G}}{min }\underset{{\theta }_{D}}{max } v\left({\theta }_{G}, {\theta }_{D}\right) = {E}_{x, y ~ {p}_{data}}\left[\mathrm{log} D\left(x, y\right)\right] + {E}_{z ~ {p}_{z}\left (z\right), \stackrel{~}{y} ~ {p}_{y}}\left[\mathrm{log} \left(1 - D\left(G\left(z, \stackrel{~}{y}\right), \stackrel{~}{y}\right)\right)\right]$ Does the above equation seem familiar to you? You're right! This equation is identical to the equation of the objective function of a 2-player minimax game we have seen before. Training a cGAN network entails optimizing the function $v\left({\theta }_{G}, {\theta }_{D}\right)$. Training a cGAN can ideally be thought of as a minimax game, in which both the generator and the discriminator networks are trained simultaneously. (For those who are clueless about the minimax game we are talking about, please refer this link) In the preceding equation, 1. ${\theta }_{G}$ represents the parameters of the generator network. 2. ${\theta }_{D}$ represents the parameter of the discriminator network. 3. $\mathrm{log} D\left(x, y\right)$ is the loss for the discriminator model. 4. $\mathrm{log} \left(1 - D\left(G\left(z, \stackrel{~}{y}\right), \stackrel{~}{y}\right)\right)$ is the loss for the generator model, and 5. ${p}_{data}$ is the distribution of all possible images. B) Initial latent vector approximation ➤ Initial latent vector approximation is a method to estimate a latent vector to optimize the reconstruction of face images. ➤ To approximate a latent vector, we have an encoder network. ➤ We train the encoder network on the generated images and real images. ➤ Once trained, it will start generating latent vectors from the learned distribution. ➤The training objective function for training the encoder network is the Euclidean distance loss. Although GANs are no doubt one of the most powerful generative models at present, they cannot exactly reproduce the details of all real-life face images with their infinite possibilities of minor facial details, superficial accessories, backgrounds, etc. It is common fact that, a natural input face image can be rather approximated than exactly reconstructed with Age-cGAN. C) Latent vector optimization ➤ During latent vector optimization, we optimize the encoder network and the generator network simultaneously. ➤ The equation used for this purpose is as follows: ${z}_{IP}^{} = \underset{z}{argmin}\parallel FR\left(x\right) - FR\left(\stackrel{~}{x}\right)\parallel {L}_{2}$ This equation follows the novel "Identity Preserving" latent vector optimization approach followed by the authors in their paper. The key idea is simple - given a face recognition neural network FR which is able to recognize a person's identity in an input face image x, the difference between the identities in the original and reconstructed images x and $\stackrel{~}{x}$ can be expressed as the Euclidean distance between the corresponding embeddings FR(x) and FR($\stackrel{~}{x}$). Hence, minimizing this distance should improve the identity preservation in the reconstructed image $\stackrel{~}{x}$. Here, ${z}_{IP}^{}$ denotes the identity-preserving latent vector optimization. FR stands for Face Recognition network, which is an internal implementation of the FaceNet CNN. The equation above indicates that the Euclidean distance between the real image and the reconstructed images should be minimal. To sum it up, in this stage, we try to minimize the distance in order to maximize identity preservation. ➔ Additional points about cGAN training. ✪ The first step of the face aging method presented in the paper (i.e. input-face reconstruction) is the key one. ✪ Details of Age-cGAN training from the original paper - ➢ The model was optimized using the Adam algorithm, and was trained for 100 epochs. ➢ In order to encode the person's age, the authors had defined 6 age categories: 0-18, 19-29, 30-39, 40-49, 50-59 and 60+ years old. ➢ The age categories were selected so that the training dataset contained at least 5,000 examples in each age category. Experiments on the Age-cGAN model The authors trained the Age-cGAN on the IMDB-Wiki_cleaned [3] dataset containing around 120,000 images, which is a subset of the public IMDB-Wiki dataset [4]. 110,000 images were used for training of the Age-cGAN model and the remaining 10,000 were used for the evaluation of identity-preserving face reconstruction. A) Age-Conditioned Face Generation The above image illustrates examples of synthetic images generated by the Age-cGAN model using 2 random latent vectors z (rows), conditioned on the respective age categories y (columns). It is observed that the Age-cGAN model perfectly disentangles image information encoded by latent vectors z and by conditions y, making them independent. Additionally, the latent vectors z encode the person's identity - hair style, facial attributes, etc., while y uniquely encodes the age. ➔ Steps taken to measure performance of the model I To measure how well the Age-cGAN model manages to generate faces belonging to precise age-categories, a state-of-the-art age estimation CNN model [3] was employed. II The performances of the age estimation CNN (on real images from the test part of the IMDB-Wiki_cleaned dataset) and Age-cGAN (10,000 synthetic images generated) was compared. ➔ Observations & Results ➢ Despite the age estimation CNN had never seen synthetic images during the training, the resulting mean age estimation accuracy on synthetic images is just 17% lower than on natural ones. ➢ This shows that the Age-cGAN model can be used for the production of realistic face images with the required age. B) Identity-Preserving Face Reconstruction and Aging The authors performed face reconstruction in 2 iterations: I. Using initial latent approximations obtained from the encoder E and then II. Using optimized latent approximations obtained by either "Pixelwise" (${z}_{pixel}^{✱}$) or "Identity-Preserving" (${z}_{IP}^{}$) optimization approaches. The image above shows some examples of face reconstruction and aging. a) depicts the original test images. b) illustrates the reconstructed images generated using the initial latent approximations: ${z}_{0}$ c) demonstrates the reconstructed images generated using the "Pixelwise" and "Identity-Preserving" optimized latent approximations: ${z}_{pixel}^{✱}$ and ${z}_{IP}^{✱}$, and d) shows the aging of the reconstructed images generated using the identity-preserving ${z}_{IP}^{✱}$ latent approximation and conditioned on the respective age categories y (one per column) It can be clearly noted from the above picture that the optimized reconstructions are closer to the original images than the initial ones. But things get more complicated when it comes to the comparison of the 2 latent vector optimization approaches. ● The "Pixelwise" optimization shows the superficial face details (hair color, beard line, etc.) better ● The "Identity-Preserving" optimization represents the identity traits (like the form of the head or the form of the eyes) better. ➔ Steps taken to measure performance of the model I) For the objective comparison of the 2 approaches for the identity-preserving face reconstruction, the authors used the "OpenFace" [5] software, which is one of the renowned open-source face recognition algorithms. II. Given 2 face images, "OpenFace" decides whether they belong to the same person or not. III. Using both the approaches, 10,000 test images of the IMDB-Wiki_cleaned dataset were reconstructed and the resulting images alongside the corresponding original ones were then fed to "OpenFace". ➔ Observations & Results ➢ Table 1 presents the percentages of "OpenFace" positive outputs (i.e. when the software believed that a face image and its reconstruction belonged to the same person). The results confirm the visual observations presented above. ➢ Initial reconstructions allow "OpenFace" to recognize the original person in only half of the test examples. ➢ This number is slightly increased by "Pixelwise" optimization but the improvement is very slight. ➢ Conversely, "Identity-Preserving" optimization approach preserves the individual's identities far better, giving the best face recognition performance of 82.9%. Now we shall cover the basic implementation of all the 4 networks - encoder, generator, discriminator and face recognition - using the Keras library. Excited? I am! Implementation of the networks in Keras The complete implementation of the Age-cGAN model is too huge (~600 lines of code) to be demonstrated in one post, so I decided to show you how to build the networks, the crucial components of the model, in Keras. Let's import all the required libraries first: import os import time import matplotlib.pyplot as plt import numpy as np import tensorflow as tf from datetime import datetime from keras import Input, Model from keras.applications import InceptionResNetV2 from keras.callbacks import TensorBoard from keras.layers import Conv2D, Flatten, Dense, BatchNormalization from keras.layers import Reshape, concatenate, LeakyReLU, Lambda from keras.layers import K, Activation, UpSampling2D, Dropout from keras.optimizers import Adam from keras.utils import to_categorical from keras_preprocessing import image from scipy.io import loadmat A) Encoder network def build_encoder(): Encoder Network input_layer = Input(shape = (64, 64, 3)) ## 1st Convolutional Block enc = Conv2D(filters = 32, kernel_size = 5, strides = 2, padding = 'same')(input_layer) enc = LeakyReLU(alpha = 0.2)(enc) ## 2nd Convolutional Block enc = Conv2D(filters = 64, kernel_size = 5, strides = 2, padding = 'same')(enc) enc = BatchNormalization()(enc) enc = LeakyReLU(alpha = 0.2)(enc) ## 3rd Convolutional Block enc = Conv2D(filters = 128, kernel_size = 5, strides = 2, padding = 'same')(enc) enc = BatchNormalization()(enc) enc = LeakyReLU(alpha = 0.2)(enc) ## 4th Convolutional Block enc = Conv2D(filters = 256, kernel_size = 5, strides = 2, padding = 'same')(enc) enc = BatchNormalization()(enc) enc = LeakyReLU(alpha = 0.2)(enc) ## Flatten layer enc = Flatten()(enc) ## 1st Fully Connected Layer enc = Dense(4096)(enc) enc = BatchNormalization()(enc) enc = LeakyReLU(alpha = 0.2)(enc) ## 2nd Fully Connected Layer enc = Dense(100)(enc) ## Create a model model = Model(inputs = [input_layer], outputs = [enc]) return model ➤ The Flatten() layer converts an n-dimensional tensor to a one-dimensional tensor (array). The process is known as flattening. B) Generator network def build_generator(): Generator Network latent_dims = 100 num_classes = 6 input_z_noise = Input(shape = (latent_dims, )) input_label = Input(shape = (num_classes, )) x = concatenate([input_z_noise, input_label]) x = Dense(2048, input_dim = latent_dims + num_classes)(x) x = LeakyReLU(alpha = 0.2)(x) x = Dropout(0.2)(x) x = Dense(256 8 * 8)(x) x = BatchNormalization()(x) x = LeakyReLU(alpha = 0.2)(x) x = Dropout(0.2)(x) x = Reshape((8, 8, 256))(x) x = UpSampling2D(size = (2, 2))(x) x = Conv2D(filters = 128, kernel_size = 5, padding = 'same')(x) x = BatchNormalization(momentum = 0.8)(x) x = LeakyReLU(alpha = 0.2)(x) x = UpSampling2D(size = (2, 2))(x) x = Conv2D(filters = 64, kernel_size = 5, padding = 'same')(x) x = BatchNormalization(momentum = 0.8)(x) x = LeakyReLU(alpha = 0.2)(x) x = UpSampling2D(size = (2, 2))(x) x = Conv2D(filters = 3, kernel_size = 5, padding = 'same')(x) x = Activation('tanh')(x) model = Model(inputs = [input_z_noise, input_label], outputs = [x]) return model ➤ The UpSampling2D() function is for the process of repeating the rows a specified number of times x and repeating the columns a specified number of times y, respectively. ➤ Upsampling is a technique to keep the dimension of the output the same as before, while reducing the computation per layer simultaneously. C) Discriminator network def expand_label_input(x): x = K.expand_dims(x, axis = 1) x = K.expand_dims(x, axis = 1) x = K.tile(x, [1, 32, 32, 1]) return x def build_discriminator(): Discriminator Network input_shape = (64, 64, 3) label_shape = (6, ) image_input = Input(shape = input_shape) label_input = Input(shape = label_shape) x = Conv2D(64, kernel_size = 3, strides = 2, padding = 'same')(image_input) x = LeakyReLU(alpha = 0.2)(x) label_input1 = Lambda(expand_label_input)(label_input) x = concatenate([x, label_input1], axis = 3) x = Conv2D(128, kernel_size = 3, strides = 2, padding = 'same')(x) x = BatchNormalization()(x) x = LeakyReLU(alpha = 0.2)(x) x = Conv2D(256, kernel_size = 3, strides = 2, padding = 'same')(x) x = BatchNormalization()(x) x = LeakyReLU(alpha = 0.2)(x) x = Conv2D(512, kernel_size = 3, strides = 2, padding = 'same')(x) x = BatchNormalization()(x) x = LeakyReLU(alpha = 0.2)(x) x = Flatten()(x) x = Dense(1, activation = 'sigmoid')(x) model = Model(inputs = [image_input, label_input], outputs = [x]) return model ➤ The expand_label_input() function helps convert the shape of the label input from (6, ) to (32, 32, 6). D) Face Recognition Network def build_fr_model(input_shape): resnet_model = InceptionResNetV2(include_top = False, weights = 'imagenet', input_shape = input_shape, pooling = 'avg') image_input = resnet_model.input x = resnet_model.layers[-1].output out = Dense(128)(x) embedder_model = Model(inputs = [image_input], outputs = [out]) input_layer = Input(shape = input_shape) x = embedder_model(input_layer) output = Lambda(lambda x: K.l2_normalize(x, axis = -1))(x) model = Model(inputs = [input_layer], outputs = [output]) return model Finally, let's have a look at some cool applications of the Age-cGAN model. Interesting applications of Age-cGAN I. Cross-age face recognition ➤ This can be incorporated into security applications as means of authorization e.g. smartphone unlocking or desktop unlocking. ➤ Current face recongnition system suffer from the problem of necessary updation with passage of time. ➤ With Age-cGAN models, the lifespan of cross-age face recognition systems will be much longer. II. Finding lost children ➤ As the age of a child increases, their facial features change, and it becomes much harder to identify them. ➤ An Age-cGAN model can simulate a person's face at a specified age easily. III. Entertainment ➤ For e.g. in mobile applications, to show and share a person's pictures at a specified age. The famous phone application FaceApp is the best example for this use case. IV. Visual effects in movies ➤ It requires lots of visual effects and to manually simulate a person's face when they are older. This is a very tedious and lengthy process too. ➤ Age-cGAN models can speed up this process and reduce the costs of creating and simulating faces significantly. [1] Radford, et al. (2016), "Unsupervised representation learning with deep convolutional generative adversarial networks" [2] Perarnau, et al. (2016), "Invertible conditional GANs for image editing" [3] Antipov, et al. (2016), "Apparent age estimation from face images combining general and children-specialized deep learning models" [4] Rothe, et al. (2015), "DEX: Deep EXpectation of apparent age from a single image" [5] Ludwiczuk, et al. (2016), "Openface: A general-purpose face recognition library with mobile applications" [6] Szegedy, et al. (2016), "Inception-v4, Inception-ResNet and the Impact of Residual Connections on Learning" [7] Antipov, et al. (2017), "Face Aging with Conditional Generative Adversarial Networks" [8] Mirza, et al. (2014), "Conditional Generative Adversarial Nets" That's it for now! I guess I have explained you about Age-cGANs and Face Aging enough that you could teach it to someone else. I hope you liked this article. Follow me on Linkedin for updates on more such awesome articles and other stuff :)	{"url":"https://iq.opengenus.org/face-aging-cgan-keras/","timestamp":"2024-11-03T06:27:58Z","content_type":"text/html","content_length":"95977","record_id":"<urn:uuid:62a6f9ab-0d40-492e-a44c-0b7dde26de4d>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00451.warc.gz"}
You lose 2 points every time you forget to write your name on a test. You have forgotten to write your name 4 times. What integer represents - DocumenTVYou lose 2 points every time you forget to write your name on a test. You have forgotten to write your name 4 times. What integer represents You lose 2 points every time you forget to write your name on a test. You have forgotten to write your name 4 times. What integer represents You lose 2 points every time you forget to write your name on a test. You have forgotten to write your name 4 times. What integer represents your change in points from forgetting to write your name? in progress 0 Mathematics 3 years 2021-07-22T10:33:49+00:00 2021-07-22T10:33:49+00:00 1 Answers 26 views 0	{"url":"https://documen.tv/question/you-lose-2-points-every-time-you-forget-to-write-your-name-on-a-test-you-have-forgotten-to-write-20807956-42/","timestamp":"2024-11-07T01:34:08Z","content_type":"text/html","content_length":"79817","record_id":"<urn:uuid:603b0373-2a7a-4329-8e13-6dadf23f25a2>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00726.warc.gz"}
Balanced hypergraph In graph theory, a balanced hypergraph is a hypergraph that has several properties analogous to that of a bipartite graph. Balanced hypergraphs were introduced by Berge[1] as a natural generalization of bipartite graphs. He provided two equivalent definitions. Definition by 2-colorability A hypergraph H = (V, E) is called 2-colorable if its vertices can be 2-colored so that no hyperedge is monochromatic. Every bipartite graph G = (X+Y, E) is 2-colorable: each edge contains exactly one vertex of X and one vertex of Y, so e.g. X can be colored blue and Y can be colored yellow and no edge is monochromatic. A hypergraph in which some hyperedges are singletons (contain only one vertex) is obviously not 2-colorable; to avoid such trivial obstacles to 2-colorability, it is common to consider hypergraphs that are essentially 2-colorable, i.e., they become 2-colorable upon deleting all their singleton hyperedges.^[2]^:468 A hypergraph is called balanced if it is essentially 2-colorable, and remains essentially 2-colorable upon deleting any number of vertices. Formally, for each subset U of V, define the restriction of H to U as the hypergraph HU = (U, EU) where $ {\displaystyle E_{U}=\{e\cap U\|e\in E\}} $ . Then H is called balanced iff HU is essentially 2-colorable for every subset U of V. Note that a simple graph is bipartite iff it is 2-colorable iff it is balanced. Definition by odd cycles A cycle (or a circuit) in a hypergraph is a cyclic alternating sequence of distinct vertices and hyperedges: (v[1], e[1], v[2], e[2], ..., v[k], e[k], v[k][+1]=v[1]), where every vertex v[i] is contained in both e[i][−1] and e[i]. The number k is called the length of the cycle. A hypergraph is balanced iff every odd-length cycle C in H has an edge containing at least three vertices of C.^[3] Note that in a simple graph all edges contain only two vertices. Hence, a simple graph is balanced iff it contains no odd-length cycles at all, which holds iff it is bipartite. Berge^[1] proved that the two definitions are equivalent; a proof is also available here.^[2]^:468–469 Some theorems on bipartite graphs have been generalized to balanced hypergraphs.[4][2]:468–470 In every balanced hypergraph, the minimum vertex-cover has the same size as its maximum matching. This generalizes the Kőnig-Egervary theorem on bipartite graphs. In every balanced hypergraph, the degree (= the maximum number of hyperedges containing some one vertex) equals the chromatic index (= the least number of colors required for coloring the hyperedges such that no two hyperedges with the same color have a vertex in common).[5] This generalizes a theorem of Konig on bipartite graphs. Every balanced hypergraph satisfies a generalization of Hall's marriage theorem:[3] it admits a perfect matching iff for all disjoint vertex-sets $ {\displaystyle \|e\cap V_{2}\|\geq \|e\cap V_{1}\|} for all edges e in E, then \|V2\| ≥ \|V1\| $. See Hall-type theorems for hypergraphs. Every balanced hypergraph with maximum degree D, can be partitioned into D edge-disjoint matchings.[1]:Chapter 5[3]:Corollary 3 A k-fold transversal of a balanced hypergraph can be expressed as a union of k pairwise-disjoint transversals, and such partition can be obtained in polynomial time.[6] Comparison with other notions of bipartiteness Besides balance, there are alternative generalizations of bipartite graphs. A hypergraph is called bipartite if its vertex set V can be partitioned into two sets, X and Y, such that each hyperedge contains exactly one element of X (see bipartite hypergraph). Obviously every bipartite graph is 2-colorable. The properties of bipartiteness and balance do not imply each other. Balance does not imply bipartiteness. Let H be the hypergraph:[7] { {1,2} , {3,4} , {1,2,3,4} } it is 2-colorable and remains 2-colorable upon removing any number of vertices from it. However, It is not bipartite, since to have exactly one green vertex in each of the first two hyperedges, we must have two green vertices in the last hyperedge. Bipartiteness does not imply balance. For example, let H be the hypergraph with vertices {1,2,3,4} and edges: { {1,2,3} , {1,2,4} , {1,3,4} } It is bipartite by the partition X={1}, Y={2,3,4}. But is not balanced. For example, if vertex 1 is removed, we get the restriction of H to {2,3,4}, which has the following hyperedges: { {2,3} , {2,4} , {3,4} } It is not 2-colorable, since in any 2-coloring there are at least two vertices with the same color, and thus at least one of the hyperedges is monochromatic. Another way to see that H is not balanced is that it contains the odd-length cycle C = (2 - {1,2,3} - 3 - {1,3,4} - 4 - {1,2,4} - 2), and no edge of C contains all three vertices 2,3,4 of C. Tripartiteness does not imply balance. For example, let H be the tripartite hypergraph with vertices {1,2},{3,4},{5,6} and edges: { {1,3,5}, {2,4,5}, {1,4,6} } It is not balanced since if we remove the vertices 2,3,6, the remainder is: { {1,5}, {4,5}, {1,4} } which is not colorable since it is a 3-cycle. Another way to see that it is not balanced is that It contains the odd-length cycle C = (1 - {1,3,5} - 5 - {2,4,5} - 4 - {1,4,6} - 1), and no edge of C contains all three vertices 1,4,5 of C. Related properties Totally balanced hypergraphs A hypergraph is called totally balanced if every cycle C in H of length at least 3 (not necessarily odd-length) has an edge containing at least three vertices of C.[8] A hypergraph H is totally balanced iff every subhypergraph of H is a tree-hypergraph.[8] Normal hypergraphs The Konig property of a hypergraph H is the property that its minimum vertex-cover has the same size as its maximum matching. The Kőnig-Egervary theorem says that all bipartite graphs have the Konig The balanced hypergraphs are exactly the hypergraphs H such that every partial subhypergraph of H has the Konig property (i.e., H has the Konig property even upon deleting any number of hyperedges and vertices). If every partial hypergraph of H has the Konig property (i.e., H has the Konig property even upon deleting any number of hyperedges - but not vertices), then H is called a normal hypergraph.[9] Thus, totally-balanced implies balanced, which implies normal. Berge, Claude (1970). "Sur certains hypergraphes généralisant les graphes bipartites". Combinatorial Theory and Its Applications. 1: 119–133. Lovász, László; Plummer, M. D. (1986), Matching Theory, Annals of Discrete Mathematics, 29, North-Holland, ISBN 0-444-87916-1, MR 0859549 Conforti, Michele; Cornuéjols, Gérard; Kapoor, Ajai; Vušković, Kristina (1996-09-01). "Perfect matchings in balanced hypergraphs". Combinatorica. 16 (3): 325–329. doi:10.1007/BF01261318. ISSN 1439-6912. S2CID 206792822. Berge, Claude; Vergnas, Michel LAS (1970). "Sur Un Theorems Du Type König Pour Hypergraphes". Annals of the New York Academy of Sciences. 175 (1): 32–40. doi:10.1111/j.1749-6632.1970.tb56451.x. ISSN Lovász, L. (1972-06-01). "Normal hypergraphs and the perfect graph conjecture". Discrete Mathematics. 2 (3): 253–267. doi:10.1016/0012-365X(72)90006-4. ISSN 0012-365X. Dahlhaus, Elias; Kratochvíl, Jan; Manuel, Paul D.; Miller, Mirka (1997-11-27). "Transversal partitioning in balanced hypergraphs". Discrete Applied Mathematics. 79 (1): 75–89. doi:10.1016/S0166-218X (97)00034-6. ISSN 0166-218X. "coloring - Which generalization of bipartite graphs is stronger?". Mathematics Stack Exchange. Retrieved 2020-06-27. Lehel, Jenö (1985-11-01). "A characterization of totally balanced hypergraphs". Discrete Mathematics. 57 (1): 59–65. doi:10.1016/0012-365X(85)90156-6. ISSN 0012-365X. Beckenbach, Isabel; Borndörfer, Ralf (2018-10-01). "Hall's and Kőnig's theorem in graphs and hypergraphs". Discrete Mathematics. 341 (10): 2753–2761. doi:10.1016/j.disc.2018.06.013. ISSN 0012-365X. Undergraduate Texts in Mathematics Graduate Studies in Mathematics Hellenica World - Scientific Library Retrieved from "http://en.wikipedia.org/" All text is available under the terms of the GNU Free Documentation License	{"url":"https://www.hellenicaworld.com/Science/Mathematics/en/BalancedHypergraph.html","timestamp":"2024-11-05T06:42:39Z","content_type":"application/xhtml+xml","content_length":"14326","record_id":"<urn:uuid:450202e0-c354-4a4d-ae9d-cdc6421381ef>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00005.warc.gz"}
26,095 research outputs found The $z$-matching problem on bipartite graphs is studied with a local algorithm. A $z$-matching ($z \ge 1$) on a bipartite graph is a set of matched edges, in which each vertex of one type is adjacent to at most $1$ matched edge and each vertex of the other type is adjacent to at most $z$ matched edges. The $z$-matching problem on a given bipartite graph concerns finding $z$-matchings with the maximum size. Our approach to this combinatorial optimization are of two folds. From an algorithmic perspective, we adopt a local algorithm as a linear approximate solver to find $z$-matchings on general bipartite graphs, whose basic component is a generalized version of the greedy leaf removal procedure in graph theory. From an analytical perspective, in the case of random bipartite graphs with the same size of two types of vertices, we develop a mean-field theory for the percolation phenomenon underlying the local algorithm, leading to a theoretical estimation of $z$-matching sizes on coreless graphs. We hope that our results can shed light on further study on algorithms and computational complexity of the optimization problem.Comment: 15 pages, 3 figure A preliminary attempt at collecting tools and utilities for genetic data as an R package called gap is described. Genomewide association is then described as a specific example, linking the work of Risch and Merikangas (1996), Long and Langley (1997) for family-based and population-based studies, and the counterpart for case-cohort design established by Cai and Zeng (2004). Analysis of staged design as outlined by Skol et al. (2006) and associate methods are discussed. The package is flexible, customizable, and should prove useful to researchers especially in its application to genomewide association studies. In this paper, we give a model for understanding flavor physics in the lepton sector--mass hierarchy among different generations and neutrino mixing pattern. The model is constructed in the framework of supersymmetry, with a family symmetry $S4*U(1)$. There are two right-handed neutrinos introduced for seesaw mechanism, while some standard model(SM) gauge group singlet fields are included which transforms non-trivially under family symmetry. In the model, each order of contributions are suppressed by $\delta \sim 0.1$ compared to the previous one. In order to reproduce the mass hierarchy, $m_\tau$ and $\sqrt{\Delta m_{atm}^2}$, $m_\mu$ and $\sqrt{\Delta m_{sol}^2}$ are obtained at leading-order(LO) and next-to-leading-order(NLO) respectively, while electron can only get its mass through next-to-next-to-next-to-leading-order(NNNLO) contributions. For neutrino mixing angels, $\theta_{12}, \theta_{23}, \theta_{13}$ are $45^\circ, 45^\circ, 0$ i.e. Bi-maximal mixing pattern as first approximation, while higher order contributions can make them consistent with experimental results. As corrections for $\theta_{12}$ and $\theta_{13}$ originate from the same contribution, there is a relation predicted for them $\sin{\theta_{13}}=\displaystyle \frac{1-\tan{\theta_{12}}}{1+\tan{\theta_{12}}}$. Besides, deviation from $\displaystyle \frac{\pi}{4}$ for $\theta_{23}$ should have been as large as deviation from 0 for $\theta_{13}$ if it were not the former is suppressed by a factor 4 compared to the latter.Comment: version to appear in Phys. Rev. We carry out a new study of quark mass matrices $M^{}_{\rm u}$ (up-type) and $M^{}_{\rm d}$ (down-type) which are Hermitian and have four zero entries, and find a new part of the parameter space which was missed in the previous works. We identify two more specific four-zero patterns of $M^{}_{\rm u}$ and $M^{}_{\rm d}$ with fewer free parameters, and present two toy flavor-symmetry models which can help realize such special and interesting quark flavor structures. We also show that the texture zeros of $M^{}_{\rm u}$ and $M^{}_{\rm d}$ are essentially stable against the evolution of energy scales in an analytical way by using the one-loop renormalization-group equations.Comment: 33 pages, 4 figures, minor comments added, version to appear in Nucl. Phys.	{"url":"https://core.ac.uk/search/?q=authors%3A(HUA%20ZHAO)","timestamp":"2024-11-12T04:36:56Z","content_type":"text/html","content_length":"144283","record_id":"<urn:uuid:39750ac2-620c-45c7-afc6-4b6f1b942c21>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00049.warc.gz"}